HuangUMAT\u5355\u6676\u5851\u6027\u5b50\u7a0b\u5e8f\u8be6\u89e3<\/title>\n<title data-lang=\"en\" style=\"display:none\">Huang’s UMAT Crystal Plasticity Subroutine<\/title>\n<style>\n:root {\n --bg: #f5f7fa;\n --bg2: #eef1f6;\n --ink: #1a2332;\n --muted: #5a6a7e;\n --rule: #d0d7e2;\n --accent: #2563eb;\n --accent2: #0ea5e9;\n --code-bg: #1a2332;\n --code-fg: #e2e8f0;\n}\n* { box-sizing: border-box; margin: 0; padding: 0; }\nhtml { scroll-behavior: smooth; }\nbody {\n font-family: \"Microsoft YaHei\", \"PingFang SC\", system-ui, -apple-system, sans-serif;\n background: var(--bg);\n color: var(--ink);\n line-height: 1.75;\n font-size: 16px;\n}\nbody.en-mode {\n font-family: system-ui, -apple-system, \"Segoe UI\", sans-serif;\n}\n.lang-bar {\n position: sticky;\n top: 0;\n z-index: 1000;\n background: linear-gradient(90deg, #1a2332, #2c3e5a);\n padding: 10px 20px;\n display: flex;\n justify-content: center;\n gap: 12px;\n box-shadow: 0 2px 8px rgba(0,0,0,0.15);\n}\n.lang-bar button {\n 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document.querySelectorAll('[data-lang]').forEach(function(el) {\n if (el.tagName.toLowerCase() === 'title') {\n if (el.getAttribute('data-lang') === lang) el.style.display = '';\n else el.style.display = 'none';\n return;\n }\n if (el.getAttribute('data-lang') === lang) {\n el.style.display = '';\n } else {\n el.style.display = 'none';\n }\n });\n document.querySelectorAll('.lang-bar button').forEach(function(btn) {\n btn.classList.toggle('active', btn.getAttribute('data-set-lang') === lang);\n });\n if (lang === 'en') document.body.classList.add('en-mode');\n else document.body.classList.remove('en-mode');\n if (window.mermaid) {\n mermaid.initialize({ startOnLoad: true, theme: 'neutral', securityLevel: 'loose' });\n }\n}\ndocument.addEventListener('DOMContentLoaded', function() {\n switchLanguage('zh');\n});\n<\/script>\n<\/head>\n<body>\n\n<div class=\"lang-bar\">\n <button data-set-lang=\"zh\" onclick=\"switchLanguage('zh')\" class=\"active\">\n \u4e2d\u6587<\/span>\n Chinese<\/span>\n <\/button>\n <button data-set-lang=\"en\" onclick=\"switchLanguage('en')\">\n English<\/span>\n English<\/span>\n <\/button>\n<\/div>\n\n<div class=\"container\">\n\n\n<div class=\"cover\">\n <h1>\n Huang UMAT \u5355\u6676\u5851\u6027\u5b50\u7a0b\u5e8f\u8be6\u89e3<\/span>\n Detailed Explanation of Huang’s UMAT Crystal Plasticity Subroutine<\/span>\n <\/h1>\n <h2>\n Crystal Plasticity Finite Element Method (CPFEM) \/ \u6676\u4f53\u5851\u6027\u6709\u9650\u5143\u65b9\u6cd5<\/span>\n Crystal Plasticity Finite Element Method (CPFEM)<\/span>\n <\/h2>\n <div class=\"subtitle\">\n \u57fa\u4e8e Abaqus\/Standard \u9690\u5f0f\u6c42\u89e3\u5668<\/span>\n For Abaqus\/Standard Implicit Solver<\/span>\n <\/div>\n <div class=\"meta\">\n \u5b8c\u6574\u672c\u6784\u7406\u8bba\u3001UMAT\u63a5\u53e3\u3001\u4ee3\u7801\u7ed3\u6784\u4e0e\u5b9e\u73b0\u7ec6\u8282<\/span>\n Complete Constitutive Theory, UMAT Interface, Code Structure and Implementation Details<\/span>\n <\/div>\n<\/div>\n\n\n\n\n<div class=\"toc\" id=\"toc\">\n <h2>\n \u76ee\u5f55 \/ Table of Contents<\/span>\n Table of Contents<\/span>\n <\/h2>\n <ul>\n <li><a href=\"#ch1\">\n Chapter 1 \u6982\u8ff0 \/ Overview<\/span>\n Chapter 1 Overview<\/span>\n <\/a><\/li>\n <li><a href=\"#ch2\">\n Chapter 2 \u672c\u6784\u7406\u8bba \/ Constitutive Theory<\/span>\n Chapter 2 Constitutive Theory<\/span>\n <\/a><\/li>\n <li><a href=\"#ch3\">\n Chapter 3 Abaqus UMAT\u63a5\u53e3 \/ UMAT Interface<\/span>\n Chapter 3 Abaqus UMAT Interface<\/span>\n <\/a><\/li>\n <li><a href=\"#ch4\">\n Chapter 4 \u6750\u6599\u53c2\u6570 \/ Material Parameters<\/span>\n Chapter 4 Material Parameters<\/span>\n <\/a><\/li>\n <li><a href=\"#ch5\">\n Chapter 5 \u4ee3\u7801\u7ed3\u6784 \/ Code Structure<\/span>\n Chapter 5 Code Structure<\/span>\n <\/a><\/li>\n <li><a href=\"#ch6\">\n Chapter 6 Jacobian\u77e9\u9635 \/ Jacobian Matrix<\/span>\n Chapter 6 Jacobian Matrix<\/span>\n <\/a><\/li>\n <li><a href=\"#ch7\">\n Chapter 7 \u5178\u578b\u53c2\u6570 \/ Typical Parameters<\/span>\n Chapter 7 Typical Parameters<\/span>\n <\/a><\/li>\n <li><a href=\"#ch8\">\n Chapter 8 \u4f7f\u7528\u6ce8\u610f\u4e8b\u9879 \/ Usage Notes<\/span>\n Chapter 8 Usage Notes<\/span>\n <\/a><\/li>\n <li><a href=\"#refs\">\n \u53c2\u8003\u6587\u732e \/ References<\/span>\n References<\/span>\n <\/a><\/li>\n <\/ul>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch1\">\n <h2>\n 01<\/span>\n \u6982\u8ff0 \/ Overview<\/span>\n Overview<\/span>\n <\/h2>\n\n <h3 id=\"ch1-1\">\n 1.1 \u672c\u6784\u6a21\u578b\u7b80\u4ecb \/ Constitutive Model Introduction<\/span>\n 1.1 Constitutive Model Introduction<\/span>\n <\/h3>\n \n Huang\uff08Yonggang Huang\uff09\u6559\u6388\u5f00\u53d1\u7684UMAT\u5355\u6676\u5851\u6027\u5b50\u7a0b\u5e8f\u662f\u6676\u4f53\u5851\u6027\u6709\u9650\u5143\u65b9\u6cd5\uff08CPFEM\uff09\u9886\u57df\u6700\u5e7f\u6cdb\u4f7f\u7528\u7684\u5f00\u6e90\u5b9e\u73b0\u4e4b\u4e00\u3002\n \u8be5\u5b50\u7a0b\u5e8f\u57fa\u4e8eAbaqus\/Standard\u7528\u6237\u6750\u6599\u5b50\u7a0b\u5e8f\uff08UMAT\uff09\u63a5\u53e3\uff0c\u91c7\u7528\u9690\u5f0f\u65f6\u95f4\u79ef\u5206\u65b9\u6848\uff0c\n \u80fd\u591f\u51c6\u786e\u6a21\u62df\u5355\u6676\u53ca\u591a\u6676\u6750\u6599\u5728\u590d\u6742\u52a0\u8f7d\u6761\u4ef6\u4e0b\u7684\u5851\u6027\u53d8\u5f62\u884c\u4e3a\u3002\n \u8be5\u6a21\u578b\u8003\u8651\u4e86\u6676\u4f53\u7684\u5404\u5411\u5f02\u6027\u5f39\u6027\u3001\u6676\u4f53\u5b66\u6ed1\u79fb\u7cfb\u4e0a\u7684\u526a\u5207\u53d8\u5f62\uff0c\u4ee5\u53ca\u4f4d\u9519\u4ea4\u4e92\u4f5c\u7528\u5bfc\u81f4\u7684\u786c\u5316\u6548\u5e94\u3002\n <\/p>\n \n The UMAT single-crystal plasticity subroutine developed by Professor Yonggang Huang\n is one of the most widely used open-source implementations in the field of Crystal Plasticity Finite Element Method (CPFEM).\n Based on the Abaqus\/Standard User Material (UMAT) interface with an implicit time integration scheme,\n it accurately simulates the plastic deformation behavior of single-crystal and polycrystalline materials under complex loading conditions.\n The model accounts for anisotropic elasticity, crystallographic slip system shear deformation,\n and hardening effects due to dislocation interactions.\n <\/p>\n \n \u672c\u7a0b\u5e8f\u7684\u6838\u5fc3\u7279\u5f81\u5305\u62ec\uff1a\u57fa\u4e8e\u4e2d\u95f4\u6784\u578b\u7684\u4e58\u6cd5\u5206\u89e3\u8fd0\u52a8\u5b66\u6846\u67b6\u3001Schmid\u5b9a\u5f8b\u9a71\u52a8\u7684\u6ed1\u79fb\u7cfb\u6fc0\u6d3b\u5224\u636e\u3001\n \u7387\u76f8\u5173\u5e42\u5f8b\u6d41\u52a8\u6cd5\u5219\u3001\u53ef\u9009\u62e9\u7684\u786c\u5316\u6a21\u578b\uff08\u53cc\u66f2\u6b63\u5272\u786c\u5316\u6216Bassani-Wu\u786c\u5316\uff09\uff0c\n \u4ee5\u53ca\u5b8c\u6574\u7684\u9690\u5f0f\u5e94\u529b\u66f4\u65b0\u7b97\u6cd5\u548c\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635\uff08Jacobian\uff09\u8ba1\u7b97\u3002\n <\/p>\n \n Key features of this subroutine include: multiplicative decomposition kinematics based on the intermediate configuration,\n Schmid’s law driven slip system activation criterion, rate-dependent power-law flow rule,\n selectable hardening models (hyperbolic secant hardening or Bassani-Wu hardening),\n and a complete implicit stress update algorithm with consistent tangent stiffness matrix (Jacobian) computation.\n <\/p>\n\n <h3 id=\"ch1-2\">\n 1.2 \u7406\u8bba\u57fa\u7840 \/ Theoretical Foundation<\/span>\n 1.2 Theoretical Foundation<\/span>\n <\/h3>\n \n \u9ec4\u6c38\u521aUMAT\u7684\u7406\u8bba\u57fa\u7840\u4e3b\u8981\u6765\u6e90\u4e8ePeirce\u3001Asaro\u548cNeedleman\uff081983\uff09\u4ee5\u53caPeirce\u3001Shih\u548cNeedleman\uff081984\uff09\u7684\u7ecf\u5178\u5de5\u4f5c\u3002\n \u8fd9\u4e9b\u6587\u732e\u5efa\u7acb\u4e86\u6676\u4f53\u5851\u6027\u7406\u8bba\u5728\u73b0\u4ee3\u6709\u9650\u5143\u6846\u67b6\u4e0b\u7684\u6570\u5b66\u8868\u8ff0\uff0c\u5305\u62ec\uff1a\n <\/p>\n \n The theoretical foundation of Huang’s UMAT is primarily derived from the seminal works of\n Peirce, Asaro and Needleman (1983) and Peirce, Shih and Needleman (1984).\n These papers established the mathematical formulation of crystal plasticity theory within the modern finite element framework, including:\n <\/p>\n <ul>\n <li>\n \u603b\u53d8\u5f62\u68af\u5ea6F\u7684\u4e58\u6cd5\u5206\u89e3\u4e3a\u5f39\u6027\u90e8\u5206Fe\u548c\u5851\u6027\u90e8\u5206Fp<\/span>\n Multiplicative decomposition of the total deformation gradient F into elastic Fe and plastic Fp parts<\/span>\n <\/li>\n <li>\n \u57fa\u4e8eSchmid\u5206\u89e3\u7684\u5206\u89e3\u5e94\u529b\uff08resolved shear stress\uff09\u6982\u5ff5<\/span>\n The concept of resolved shear stress based on Schmid decomposition<\/span>\n <\/li>\n <li>\n \u7387\u76f8\u5173\u6d41\u52a8\u6cd5\u5219\uff0c\u907f\u514d\u7387\u65e0\u5173\u6a21\u578b\u7684\u975e\u552f\u4e00\u6027\u548c\u6570\u503c\u4e0d\u7a33\u5b9a\u6027<\/span>\n Rate-dependent flow rule to avoid non-uniqueness and numerical instability of rate-independent models<\/span>\n <\/li>\n <li>\n \u81ea\u786c\u5316\uff08self-hardening\uff09\u548c\u6f5c\u786c\u5316\uff08latent hardening\uff09\u7684\u7edf\u4e00\u63cf\u8ff0<\/span>\n Unified description of self-hardening and latent hardening<\/span>\n <\/li>\n <\/ul>\n \n \u540e\u7eedAsaro\u548cNeedleman\uff081985\uff09\u4ee5\u53caHarren\u3001Deve\u548cAsaro\uff081988\uff09\u7684\u5de5\u4f5c\u8fdb\u4e00\u6b65\u5b8c\u5584\u4e86\u6676\u4f53\u5851\u6027\u7406\u8bba\u5728\u6709\u9650\u53d8\u5f62\u6761\u4ef6\u4e0b\u7684\u6570\u503c\u5b9e\u73b0\u3002\n \u9ec4\u6c38\u521a\u5728\u5176\u5b50\u7a0b\u5e8f\u4e2d\u6574\u5408\u4e86\u8fd9\u4e9b\u7406\u8bba\u6210\u679c\uff0c\u5e76\u9488\u5bf9Abaqus\/Standard\u7684\u9690\u5f0f\u6c42\u89e3\u5668\u8fdb\u884c\u4e86\u7b97\u6cd5\u4f18\u5316\u3002\n <\/p>\n \n Subsequent works by Asaro and Needleman (1985) and Harren, Deve and Asaro (1988) further refined\n the numerical implementation of crystal plasticity theory under finite deformation conditions.\n Huang integrated these theoretical achievements into his subroutine and optimized the algorithm\n specifically for the Abaqus\/Standard implicit solver.\n <\/p>\n\n <h3 id=\"ch1-3\">\n 1.3 \u9002\u7528\u8303\u56f4 \/ Scope<\/span>\n 1.3 Scope<\/span>\n <\/h3>\n \n \u672c\u5b50\u7a0b\u5e8f\u9002\u7528\u4e8e\u4ee5\u4e0b\u7814\u7a76\u573a\u666f\uff1a\n <\/p>\n \n This subroutine is applicable to the following research scenarios:\n <\/p>\n <ul>\n <li>\n FCC\u3001BCC\u548cHCP\u6676\u4f53\u7ed3\u6784\u7684\u5355\u6676\u548c\u591a\u6676\u91d1\u5c5e\u6750\u6599\u7684\u529b\u5b66\u884c\u4e3a\u6a21\u62df<\/span>\n Mechanical behavior simulation of single-crystal and polycrystalline metals with FCC, BCC, and HCP crystal structures<\/span>\n <\/li>\n <li>\n \u5355\u8c03\u52a0\u8f7d\u3001\u5faa\u73af\u52a0\u8f7d\u53ca\u5e94\u53d8\u8def\u5f84\u53d8\u5316\u6761\u4ef6\u4e0b\u7684\u5851\u6027\u53d8\u5f62\u5206\u6790<\/span>\n Plastic deformation analysis under monotonic, cyclic, and strain-path-change loading conditions<\/span>\n <\/li>\n <li>\n \u7ec7\u6784\u6f14\u5316\uff08texture evolution\uff09\u548c\u5851\u6027\u5404\u5411\u5f02\u6027\u53d1\u5c55\u7684\u5b9a\u91cf\u9884\u6d4b<\/span>\n Quantitative prediction of texture evolution and plastic anisotropy development<\/span>\n <\/li>\n <li>\n \u5c40\u90e8\u5316\u53d8\u5f62\u3001\u526a\u5207\u5e26\u5f62\u6210\u53ca\u7ec6\u89c2\u5c3a\u5ea6\u635f\u4f24\u8d77\u59cb\u7814\u7a76<\/span>\n Studies of localized deformation, shear band formation, and mesoscale damage initiation<\/span>\n <\/li>\n <li>\n \u5fae\u67f1\u538b\u7f29\u3001\u7eb3\u7c73\u538b\u75d5\u7b49\u5c0f\u5c3a\u5ea6\u5355\u6676\u5b9e\u9a8c\u7684\u6709\u9650\u5143\u5efa\u6a21<\/span>\n Finite element modeling of micro-pillar compression, nanoindentation, and other small-scale single-crystal experiments<\/span>\n <\/li>\n <\/ul>\n <div class=\"note\">\n \u6ce8\u610f\uff1a<\/strong>\u672c\u5b50\u7a0b\u5e8f\u57fa\u4e8e\u5c0f\u5f39-\u5927\u5851\u5047\u8bbe\uff0c\u5373\u5f39\u6027\u53d8\u5f62\u68af\u5ea6Fe\u63a5\u8fd1\u5355\u4f4d\u5f20\u91cf\uff08||Fe-I|| << 1\uff09\uff0c\u4f46\u5851\u6027\u53d8\u5f62Fp\u53ef\u4ee5\u4efb\u610f\u5927\u3002\u8be5\u5047\u8bbe\u5bf9\u4e8e\u5927\u591a\u6570\u91d1\u5c5e\u6676\u4f53\u5728\u4e2d\u7b49\u5e94\u53d8\u8303\u56f4\u5185\u662f\u5408\u7406\u7684\u3002<\/span>\n Note:<\/strong> This subroutine is based on the small-elastic-large-plastic assumption, where the elastic deformation gradient Fe is close to the identity tensor (||Fe-I|| << 1), but the plastic deformation Fp can be arbitrarily large. This assumption is reasonable for most metallic crystals at moderate strain ranges.<\/span>\n <\/div>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch2\">\n <h2>\n 02<\/span>\n \u672c\u6784\u7406\u8bba \/ Constitutive Theory<\/span>\n Constitutive Theory<\/span>\n <\/h2>\n\n <h3 id=\"ch2-1\">\n 2.1 \u5206\u89e3\u8fd0\u52a8\u5b66 \/ Kinematics of Decomposition<\/span>\n 2.1 Kinematics of Decomposition<\/span>\n <\/h3>\n \n \u6676\u4f53\u5851\u6027\u7406\u8bba\u91c7\u7528\u4e58\u6cd5\u5206\u89e3\uff08multiplicative decomposition\uff09\u5c06\u603b\u53d8\u5f62\u68af\u5ea6F\u5206\u89e3\u4e3a\u5f39\u6027\u90e8\u5206Fe\u548c\u5851\u6027\u90e8\u5206Fp\uff1a\n <\/p>\n \n Crystal plasticity theory employs multiplicative decomposition to separate the total deformation gradient F into elastic Fe and plastic Fp parts:\n <\/p>\n <div class=\"formula\">\n F = Fe<\/sup> \u00b7 Fp<\/sup><\/b>\n <\/div>\n \n \u5176\u4e2d\uff0c\u5851\u6027\u53d8\u5f62\u68af\u5ea6Fp\u7531\u6240\u6709\u6fc0\u6d3b\u6ed1\u79fb\u7cfb\u4e0a\u7684\u526a\u5207\u5e94\u53d8\u7d2f\u79ef\u4ea7\u751f\uff0c\u4fdd\u6301\u6676\u683c\u7ed3\u6784\u4e0d\u53d8\uff08\u7b49\u5bb9\u5851\u6027\u53d8\u5f62\uff09\uff1b\n \u5f39\u6027\u53d8\u5f62\u68af\u5ea6Fe\u63cf\u8ff0\u6676\u683c\u7684\u5f39\u6027\u7578\u53d8\u548c\u521a\u6027\u8f6c\u52a8\u3002\n \u8be5\u5206\u89e3\u5c06\u53c2\u8003\u6784\u578b\uff08reference configuration\uff09\u6620\u5c04\u5230\u4e2d\u95f4\u6784\u578b\uff08intermediate configuration\uff09\uff0c\n \u518d\u6620\u5c04\u5230\u5f53\u524d\u6784\u578b\uff08current configuration\uff09\u3002\n <\/p>\n \n Here, the plastic deformation gradient Fp is produced by the accumulation of shear strains on all active slip systems,\n preserving the crystal lattice structure (isochoric plastic deformation);\n the elastic deformation gradient Fe describes the elastic distortion and rigid rotation of the lattice.\n This decomposition maps from the reference configuration to the intermediate configuration,\n and then to the current configuration.\n <\/p>\n \n \u5851\u6027\u53d8\u5f62\u68af\u5ea6\u7684\u6f14\u5316\u65b9\u7a0b\u4e3a\uff1a\n <\/p>\n \n The evolution equation for the plastic deformation gradient is:\n <\/p>\n <div class=\"formula\">\n Ḟp<\/sup> = Lp<\/sup> \u00b7 Fp<\/sup><\/b> \n \u5176\u4e2d\u5851\u6027\u901f\u5ea6\u68af\u5ea6<\/span>\n where the plastic velocity gradient<\/span>\n Lp<\/sup> = Σα<\/sub> γ̇(α)<\/sup> s(α)<\/sup> ⊗ m(α)<\/sup><\/b>\n <\/div>\n \n \u4e0a\u5f0f\u4e2d\uff0cs(α)<\/sup>\u548cm(α)<\/sup>\u5206\u522b\u662f\u7b2cα\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411\uff08slip direction\uff09\u548c\u6ed1\u79fb\u9762\u6cd5\u5411\uff08slip plane normal\uff09\uff0c\n \u4e8c\u8005\u5b9a\u4e49\u5728\u4e2d\u95f4\u6784\u578b\u4e2d\u4e14\u4fdd\u6301\u6b63\u4ea4\u5f52\u4e00\u3002γ̇(α)<\/sup>\u4e3a\u5bf9\u5e94\u7684\u526a\u5207\u5e94\u53d8\u7387\u3002\n \u6ed1\u79fb\u65b9\u5411s\u548c\u6cd5\u5411m\u5728\u5f53\u524d\u6784\u578b\u4e2d\u7684\u6620\u50cf\u5206\u522b\u4e3aFe<\/sup>s\u548c(Fe<\/sup>)-T<\/sup>m\u3002\n <\/p>\n \n In the above equation, s(α)<\/sup> and m(α)<\/sup> are the slip direction and slip plane normal of the α-th slip system, respectively,\n defined in the intermediate configuration and maintained orthonormal.\n γ̇(α)<\/sup> is the corresponding shear strain rate.\n The images of slip direction s and normal m in the current configuration are Fe<\/sup>s and (Fe<\/sup>)-T<\/sup>m, respectively.\n <\/p>\n \n \u5f39\u6027\u53d8\u5f62\u68af\u5ea6Fe<\/sup>\u7684\u6781\u5206\u89e3\u4e3aFe<\/sup> = Re<\/sup>Ue<\/sup> = Ve<\/sup>Re<\/sup>\uff0c\n \u5176\u4e2dUe<\/sup>\u548cVe<\/sup>\u5206\u522b\u4e3a\u53f3\u3001\u5de6\u5f39\u6027\u62c9\u4f38\u5f20\u91cf\uff0cRe<\/sup>\u4e3a\u5f39\u6027\u8f6c\u52a8\u5f20\u91cf\u3002\n Green\u5f39\u6027\u5e94\u53d8\u5f20\u91cf\u5b9a\u4e49\u4e3a\uff1a\n <\/p>\n \n The polar decomposition of the elastic deformation gradient is Fe<\/sup> = Re<\/sup>Ue<\/sup> = Ve<\/sup>Re<\/sup>,\n where Ue<\/sup> and Ve<\/sup> are the right and left elastic stretch tensors, and Re<\/sup> is the elastic rotation tensor.\n The Green elastic strain tensor is defined as:\n <\/p>\n <div class=\"formula\">\n Ee<\/sup> = (1\/2)(Ue<\/sup>2<\/sup> – I) = (1\/2)(FeT<\/sup>Fe<\/sup> – I)<\/b>\n <\/div>\n \n \u5f39\u6027\u672c\u6784\u5173\u7cfb\u91c7\u7528\u5e7f\u4e49\u80e1\u514b\u5b9a\u5f8b\uff0c\u4ee5\u7b2c\u4e8c\u7c7bPiola-Kirchhoff\u5e94\u529bSe<\/sup>\uff08\u5b9a\u4e49\u5728\u4e2d\u95f4\u6784\u578b\uff09\u548cGreen\u5f39\u6027\u5e94\u53d8Ee<\/sup>\u8868\u793a\uff1a\n <\/p>\n \n The elastic constitutive relation adopts the generalized Hooke’s law, expressed in terms of the second Piola-Kirchhoff stress Se<\/sup>\n (defined on the intermediate configuration) and Green elastic strain Ee<\/sup>:\n <\/p>\n <div class=\"formula\">\n Se<\/sup> = C : Ee<\/sup><\/b> \n \u5176\u4e2dC\u4e3a\u56db\u9636\u5f39\u6027\u521a\u5ea6\u5f20\u91cf<\/span>\n where C is the fourth-order elastic stiffness tensor<\/span>\n <\/div>\n \n \u5f53\u524d\u6784\u578b\u4e2d\u7684Kirchhoff\u5e94\u529bτ\u548cCauchy\u5e94\u529bσ\u5206\u522b\u4e3a\uff1a\n <\/p>\n \n The Kirchhoff stress τ and Cauchy stress σ in the current configuration are:\n <\/p>\n <div class=\"formula\">\n τ = Fe<\/sup> Se<\/sup> FeT<\/sup> = J σ<\/b> \n \u5176\u4e2dJ = det(F)\u4e3a\u4f53\u79ef\u6bd4<\/span>\n where J = det(F) is the volume ratio<\/span>\n <\/div>\n\n <h3 id=\"ch2-2\">\n 2.2 Schmid\u5b9a\u5f8b \/ Schmid’s Law<\/span>\n 2.2 Schmid’s Law<\/span>\n <\/h3>\n \n Schmid\u5b9a\u5f8b\u662f\u6676\u4f53\u5851\u6027\u7406\u8bba\u7684\u6838\u5fc3\uff0c\u5b83\u6307\u51fa\uff1a\u6ed1\u79fb\u7cfbα\u4e0a\u7684\u5206\u89e3\u526a\u5e94\u529b\uff08resolved shear stress\uff09τ(α)<\/sup>\u7b49\u4e8eCauchy\u5e94\u529bσ\u5728\u6ed1\u79fb\u65b9\u5411-\u6ed1\u79fb\u9762\u6cd5\u5411\u6295\u5f71\u4e0a\u7684\u5206\u91cf\u3002\n \u5728\u7a0b\u5e8f\u5b9e\u73b0\u4e2d\uff0c\u66f4\u5e38\u7528\u7684\u662f\u5bf9\u79f0\u5316\u7684Schmid\u5f20\u91cfP(α)<\/sup>\uff1a\n <\/p>\n \n Schmid’s law is the core of crystal plasticity theory, stating that the resolved shear stress τ(α)<\/sup> on slip system α\n equals the component of Cauchy stress σ projected onto the slip direction-slip plane normal pair.\n In program implementation, the symmetrized Schmid tensor P(α)<\/sup> is more commonly used:\n <\/p>\n <div class=\"formula\">\n P(α)<\/sup> = (1\/2)( s(α)<\/sup> ⊗ m(α)<\/sup> + m(α)<\/sup> ⊗ s(α)<\/sup> )<\/b>\n <\/div>\n \n \u5206\u89e3\u526a\u5e94\u529b\u53ef\u8868\u793a\u4e3a\uff1a\n <\/p>\n \n The resolved shear stress can be expressed as:\n <\/p>\n <div class=\"formula\">\n τ(α)<\/sup> = σ : P(α)<\/sup><\/b>\n <\/div>\n \n \u8fd9\u91cc”:”\u8868\u793a\u53cc\u70b9\u79ef\uff08\u5f20\u91cf\u7f29\u5e76\uff09\uff0c\u5373σij<\/sub>Pij<\/sub>\u3002Schmid\u5f20\u91cfP(α)<\/sup>\u662f\u5bf9\u79f0\u5f20\u91cf\uff0c\n \u5b83\u5c06\u5e94\u529b\u7a7a\u95f4\u4e2d\u9a71\u52a8\u7b2cα\u4e2a\u6ed1\u79fb\u7cfb\u526a\u5207\u53d8\u5f62\u7684\u65b9\u5411\u63d0\u53d6\u51fa\u6765\u3002\n \u5f53τ(α)<\/sup>\u8fbe\u5230\u8be5\u6ed1\u79fb\u7cfb\u7684\u4e34\u754c\u5206\u5207\u5e94\u529b\uff08critical resolved shear stress, CRSS\uff09g(α)<\/sup>\u65f6\uff0c\u6ed1\u79fb\u5f00\u59cb\u6fc0\u6d3b\u3002\n <\/p>\n \n Here “:” denotes the double dot product (tensor contraction), i.e., σij<\/sub>Pij<\/sub>.\n The Schmid tensor P(α)<\/sup> is symmetric and extracts the direction in stress space that drives shear deformation on the α-th slip system.\n Slip activates when τ(α)<\/sup> reaches the critical resolved shear stress (CRSS) g(α)<\/sup> of that slip system.\n <\/p>\n <div class=\"note\">\n \u7269\u7406\u610f\u4e49\uff1a<\/strong>Schmid\u5b9a\u5f8b\u8868\u660e\uff0c\u65e0\u8bba\u5e94\u529b\u72b6\u6001\u591a\u4e48\u590d\u6742\uff0c\u53ea\u6709\u4f5c\u7528\u5728\u6ed1\u79fb\u9762\u548c\u6ed1\u79fb\u65b9\u5411\u4e0a\u7684\u526a\u5e94\u529b\u5206\u91cf\u624d\u80fd\u9a71\u52a8\u4f4d\u9519\u6ed1\u79fb\u3002\u8fd9\u662f\u6676\u4f53\u5851\u6027\u5404\u5411\u5f02\u6027\u7684\u6839\u672c\u6765\u6e90\u2014\u2014\u4e0d\u540c\u6ed1\u79fb\u7cfb\u5728\u4e0d\u540c\u5e94\u529b\u72b6\u6001\u4e0b\u7684\u6fc0\u6d3b\u7a0b\u5ea6\u4e0d\u540c\u3002<\/span>\n Physical meaning:<\/strong> Schmid’s law indicates that regardless of how complex the stress state is, only the shear stress component acting on the slip plane and in the slip direction can drive dislocation glide. This is the fundamental source of plastic anisotropy in crystals\u2014the activation degree of different slip systems varies under different stress states.<\/span>\n <\/div>\n\n <h3 id=\"ch2-3\">\n 2.3 \u7387\u76f8\u5173\u6d41\u52a8\u5f8b \/ Rate-Dependent Flow Law<\/span>\n 2.3 Rate-Dependent Flow Law<\/span>\n <\/h3>\n \n \u9ec4\u6c38\u521aUMAT\u91c7\u7528\u7387\u76f8\u5173\uff08viscoplastic\uff09\u6d41\u52a8\u6cd5\u5219\uff0c\u907f\u514d\u4e86\u7387\u65e0\u5173\u6a21\u578b\u4e2d\u6ed1\u79fb\u7cfb\u6fc0\u6d3b-\u975e\u6fc0\u6d3b\u5207\u6362\u5e26\u6765\u7684\u6570\u503c\u56f0\u96be\u3002\n \u6bcf\u4e2a\u6ed1\u79fb\u7cfb\u4e0a\u7684\u526a\u5207\u5e94\u53d8\u7387\u7531\u4ee5\u4e0b\u5e42\u5f8b\uff08power law\uff09\u63cf\u8ff0\uff1a\n <\/p>\n \n Huang’s UMAT employs a rate-dependent (viscoplastic) flow rule, avoiding numerical difficulties\n associated with the activation-deactivation switching of slip systems in rate-independent models.\n The shear strain rate on each slip system is described by the following power law:\n <\/p>\n <div class=\"formula\">\n γ̇(α)<\/sup> = γ̇0<\/sub> · | τ(α)<\/sup> \/ g(α)<\/sup> |n<\/sup> · sign( τ(α)<\/sup> \/ g(α)<\/sup> )<\/b>\n <\/div>\n \n \u5f0f\u4e2d\u5404\u53c2\u6570\u542b\u4e49\u5982\u4e0b\uff1a\n <\/p>\n \n The parameters in the equation are defined as follows:\n <\/p>\n <ul>\n <li>\n γ̇0<\/sub><\/b>:\n \u53c2\u8003\u526a\u5207\u7387\uff08reference shear rate\uff09\uff0c\u5355\u4f4d\u4e3as-1<\/sup>\uff0c\u901a\u5e38\u53d6\u4e0e\u5b9e\u9a8c\u5e94\u53d8\u901f\u7387\u76f8\u540c\u91cf\u7ea7<\/span>\n Reference shear rate, in s-1<\/sup>, typically of the same order as the experimental strain rate<\/span>\n <\/li>\n <li>\n n<\/b>:\n \u7387\u654f\u611f\u6307\u6570\uff08rate sensitivity exponent\uff09\uff0c\u65e0\u91cf\u7eb2\u3002n\u503c\u8d8a\u5927\uff0c\u6750\u6599\u884c\u4e3a\u8d8a\u63a5\u8fd1\u7387\u65e0\u5173\uff1bn→∞\u65f6\u9000\u5316\u4e3a\u7387\u65e0\u5173\u6a21\u578b<\/span>\n Rate sensitivity exponent, dimensionless. Larger n means behavior closer to rate-independent; n→∞ recovers the rate-independent model<\/span>\n <\/li>\n <li>\n τ(α)<\/sup><\/b>:\n \u7b2cα\u4e2a\u6ed1\u79fb\u7cfb\u4e0a\u7684\u5206\u89e3\u526a\u5e94\u529b<\/span>\n Resolved shear stress on the α-th slip system<\/span>\n <\/li>\n <li>\n g(α)<\/sup><\/b>:\n \u7b2cα\u4e2a\u6ed1\u79fb\u7cfb\u7684\u5f53\u524d\u4e34\u754c\u5206\u5207\u5e94\u529b\uff08\u5373\u5f53\u524d\u5f3a\u5ea6\/\u786c\u5316\u72b6\u6001\uff09<\/span>\n Current critical resolved shear stress (i.e., current strength\/hardening state) of the α-th slip system<\/span>\n <\/li>\n <li>\n sign(x)<\/b>:\n \u7b26\u53f7\u51fd\u6570\uff0c\u53d6\u503c\u4e3a-1\u30010\u62161<\/span>\n Sign function, taking values -1, 0, or 1<\/span>\n <\/li>\n <\/ul>\n \n \u7387\u76f8\u5173\u6a21\u578b\u7684\u4f18\u52bf\u5728\u4e8e\uff1a\u6240\u6709\u6ed1\u79fb\u7cfb\u59cb\u7ec8\u5904\u4e8e”\u90e8\u5206\u6fc0\u6d3b”\u72b6\u6001\uff0c\u4e0d\u9700\u8981\u663e\u5f0f\u7684\u6fc0\u6d3b\/\u975e\u6fc0\u6d3b\u5207\u6362\u6761\u4ef6\uff1b\n \u8fd9\u5927\u5927\u63d0\u9ad8\u4e86\u6570\u503c\u8ba1\u7b97\u7684\u7a33\u5b9a\u6027\u548c\u6536\u655b\u6027\u3002\n \u5728\u5178\u578b\u91d1\u5c5e\u4e2d\uff0cn\u503c\u8303\u56f4\u7ea6\u4e3a20-200\uff1aFCC\u91d1\u5c5en≈20-100\uff0cBCC\u91d1\u5c5en≈50-200\uff0cHCP\u91d1\u5c5en≈20-80\u3002\n n=1\u5bf9\u5e94\u7ebf\u6027\u7c98\u6027\uff08\u725b\u987f\u6d41\u4f53\uff09\uff0cn\u5f88\u5927\u65f6\u8fd1\u4f3c\u4e8e\u7406\u60f3\u5851\u6027\u3002\n <\/p>\n \n The advantage of the rate-dependent model is that all slip systems are always in a “partially active” state,\n without requiring explicit active\/inactive switching conditions;\n this greatly improves numerical stability and convergence.\n In typical metals, n ranges from about 20-200: FCC metals n≈20-100, BCC metals n≈50-200, HCP metals n≈20-80.\n n=1 corresponds to linear viscosity (Newtonian fluid), and large n approximates ideal plasticity.\n <\/p>\n\n <h3 id=\"ch2-4\">\n 2.4 \u786c\u5316\u5f8b \/ Hardening Laws<\/span>\n 2.4 Hardening Laws<\/span>\n <\/h3>\n \n \u4e34\u754c\u5206\u5207\u5e94\u529bg(α)<\/sup>\u968f\u7d2f\u79ef\u5851\u6027\u526a\u5207\u5e94\u53d8γ = Σα<\/sub> ∫|γ̇(α)<\/sup>|dt\u800c\u6f14\u5316\uff0c\n \u6f14\u5316\u65b9\u7a0b\u4e3a\uff1a\n <\/p>\n \n The critical resolved shear stress g(α)<\/sup> evolves with the accumulated plastic shear strain\n γ = Σα<\/sub> ∫|γ̇(α)<\/sup>|dt, according to:\n <\/p>\n <div class=\"formula\">\n ğ(α)<\/sup> = Σβ<\/sub> hαβ<\/sub> |γ̇(β)<\/sup>|<\/b>\n <\/div>\n \n \u5176\u4e2dhαβ<\/sub>\u4e3a\u786c\u5316\u77e9\u9635\uff0c\u5176\u5bf9\u89d2\u5143\u7d20hαα<\/sub>\u8868\u793a\u81ea\u786c\u5316\uff08self-hardening\uff09\uff0c\n \u975e\u5bf9\u89d2\u5143\u7d20hαβ<\/sub>(α≠β)\u8868\u793a\u6f5c\u786c\u5316\uff08latent hardening\uff09\uff0c\n \u63cf\u8ff0\u4e0d\u540c\u6ed1\u79fb\u7cfb\u4e4b\u95f4\u7684\u4ea4\u4e92\u5f3a\u5316\u6548\u5e94\u3002\n \u9ec4\u6c38\u521aUMAT\u63d0\u4f9b\u4e24\u79cd\u786c\u5316\u6a21\u578b\u9009\u9879\uff1a\n <\/p>\n \n where hαβ<\/sub> is the hardening matrix. The diagonal elements hαα<\/sub> represent self-hardening,\n and the off-diagonal elements hαβ<\/sub> (α≠β) represent latent hardening,\n describing the interaction strengthening effect between different slip systems.\n Huang’s UMAT provides two hardening model options:\n <\/p>\n\n <h4>(a) \u53cc\u66f2\u6b63\u5272\u786c\u5316\u6a21\u578b \/ Hyperbolic Secant Hardening Model<\/h4>\n \n \u81ea\u786c\u5316\u6a21\u91cf\u91c7\u7528\u53cc\u66f2\u6b63\u5272\uff08sech\u00b2\uff09\u5f62\u5f0f\uff1a\n <\/p>\n \n The self-hardening modulus adopts a hyperbolic secant (sech\u00b2) form:\n <\/p>\n <div class=\"formula\">\n hαα<\/sub> = h(γ) = H0<\/sub> · sech\u00b2( H0<\/sub> γ \/ (τs<\/sub> – τ0<\/sub>) )<\/b>\n <\/div>\n \n \u7d2f\u79ef\u526a\u5207\u5e94\u53d8γ\u4ece0\u589e\u52a0\u5230∞\u65f6\uff0c\u786c\u5316\u6a21\u91cfh\u4eceH0<\/sub>\u5355\u8c03\u9012\u51cf\u52300\uff0c\n \u4e34\u754c\u5206\u5207\u5e94\u529b\u4ece\u521d\u59cb\u503cτ0<\/sub>\u6e10\u8fd1\u8d8b\u8fd1\u4e8e\u9971\u548c\u503cτs<\/sub>\u3002\n \u79ef\u5206\u540e\u53ef\u5f97g(γ)\u7684\u663e\u5f0f\u8868\u8fbe\u5f0f\uff1a\n <\/p>\n \n As the accumulated shear strain γ increases from 0 to ∞, the hardening modulus h decreases monotonically from H0<\/sub> to 0,\n and the critical resolved shear stress asymptotically approaches the saturation value τs<\/sub> from the initial value τ0<\/sub>.\n Integration yields the explicit expression for g(γ):\n <\/p>\n <div class=\"formula\">\n g(γ) = τs<\/sub> + (τ0<\/sub> – τs<\/sub>) · tanh( H0<\/sub> γ \/ (τs<\/sub> – τ0<\/sub>) )<\/b>\n <\/div>\n \n \u53c2\u6570\u8bf4\u660e\uff1aH0<\/sub>\u4e3a\u521d\u59cb\u786c\u5316\u6a21\u91cf\uff08MPa\uff09\uff0cτ0<\/sub>\u4e3a\u521d\u59cbCRSS\uff08MPa\uff09\uff0cτs<\/sub>\u4e3a\u9971\u548cCRSS\uff08MPa\uff09\u3002\n \u8be5\u6a21\u578b\u7b80\u6d01\u4e14\u7269\u7406\u610f\u4e49\u660e\u786e\uff0c\u9002\u7528\u4e8e\u5927\u591a\u6570FCC\u91d1\u5c5e\u3002\n <\/p>\n \n Parameter definitions: H0<\/sub> is the initial hardening modulus (MPa), τ0<\/sub> is the initial CRSS (MPa), and τs<\/sub> is the saturation CRSS (MPa).\n This model is concise with clear physical meaning and is suitable for most FCC metals.\n <\/p>\n\n <h4>(b) Bassani-Wu\u786c\u5316\u6a21\u578b \/ Bassani-Wu Hardening Model<\/h4>\n \n Bassani\u548cWu\uff081991\uff09\u63d0\u51fa\u7684\u786c\u5316\u6a21\u578b\u8003\u8651\u4e86\u6ed1\u79fb\u7cfb\u95f4\u4ea4\u4e92\u4f5c\u7528\u7684\u5404\u5411\u5f02\u6027\u7279\u5f81\u3002\n \u81ea\u786c\u5316\u6a21\u91cfF(γ)\u4e3a\uff1a\n <\/p>\n \n The hardening model proposed by Bassani and Wu (1991) accounts for the anisotropic characteristics of interaction between slip systems.\n The self-hardening modulus F(γ) is:\n <\/p>\n <div class=\"formula\">\n F(γ) = (H0<\/sub> – Hs<\/sub>) · sech\u00b2( (τs<\/sub> – τ0<\/sub>) γ \/ (γ0<\/sub> + γ) ) + Hs<\/sub><\/b>\n <\/div>\n \n \u6f5c\u786c\u5316\u7cfb\u6570qαβ<\/sub>\u53d6\u51b3\u4e8e\u6ed1\u79fb\u7cfb\u5bf9(α,β)\u7684\u4ea4\u4e92\u7c7b\u578b\uff08\u5171\u9762\/\u5171\u5411\/\u6b63\u4ea4\/Lomer-Cottrell\u9501\u7b49\uff09\uff0c\n \u786c\u5316\u77e9\u9635\u4e00\u822c\u5199\u4f5chαβ<\/sub> = qαβ<\/sub> · hββ<\/sub>\u3002\n \u5b8c\u6574\u7684\u6f5c\u786c\u5316\u77e9\u9635\u662f12×12\u7684\u5bf9\u79f0\u77e9\u9635\uff0c\u5305\u542b7\u79cd\u4e0d\u540c\u7684\u4ea4\u4e92\u7c7b\u578b\u3002\n <\/p>\n \n The latent hardening coefficient qαβ<\/sub> depends on the interaction type of the slip system pair (α,β)\n (coplanar\/codirectional\/orthogonal\/Lomer-Cottrell lock, etc.).\n The hardening matrix is generally written as hαβ<\/sub> = qαβ<\/sub> · hββ<\/sub>.\n The full latent hardening matrix is a 12×12 symmetric matrix containing 7 different interaction types.\n <\/p>\n \n Bassani\u6a21\u578b\u7684\u4f18\u52bf\u5728\u4e8e\u80fd\u591f\u66f4\u51c6\u786e\u5730\u63cf\u8ff0FCC\u6676\u4f53\u4e2d\u4e0d\u540c\u6ed1\u79fb\u7cfb\u4e4b\u95f4\u7684\u4ea4\u4e92\u786c\u5316\u6548\u5e94\uff0c\n \u7279\u522b\u662f\u975e\u5171\u9762\u6ed1\u79fb\u7cfb\u4e4b\u95f4\u7684\u5f3a\u76f8\u4e92\u4f5c\u7528\uff08\u5982Lomer-Cottrell\u9501\uff09\u3002\n \u4f46\u8be5\u6a21\u578b\u9700\u8981\u989d\u5916\u7684\u6f5c\u786c\u5316\u53c2\u6570\uff0c\u4e14\u4ea4\u4e92\u7c7b\u578b\u5206\u7c7b\u8f83\u4e3a\u590d\u6742\u3002\n <\/p>\n \n The advantage of the Bassani model is its ability to more accurately describe the interaction hardening effects between different slip systems in FCC crystals,\n especially the strong interactions between non-coplanar slip systems (such as Lomer-Cottrell locks).\n However, this model requires additional latent hardening parameters, and the interaction type classification is relatively complex.\n <\/p>\n\n <h3 id=\"ch2-5\">\n 2.5 \u5e94\u529b\u66f4\u65b0 \/ Stress Update (Jaumann Rate)<\/span>\n 2.5 Stress Update (Jaumann Rate)<\/span>\n <\/h3>\n \n \u5728\u6709\u9650\u53d8\u5f62\u6846\u67b6\u4e0b\uff0c\u672c\u6784\u5173\u7cfb\u5fc5\u987b\u4ee5\u5ba2\u89c2\uff08objective\uff09\u7684\u5e94\u529b\u7387\u5f62\u5f0f\u8868\u8ff0\uff0c\u4ee5\u6d88\u9664\u521a\u6027\u8f6c\u52a8\u5e26\u6765\u7684\u865a\u5047\u5e94\u529b\u53d8\u5316\u3002\n \u9ec4\u6c38\u521aUMAT\u91c7\u7528Jaumann\u5e94\u529b\u7387\uff1a\n <\/p>\n \n In the finite deformation framework, the constitutive relation must be expressed in an objective stress rate form\n to eliminate spurious stress changes due to rigid body rotation.\n Huang’s UMAT adopts the Jaumann stress rate:\n <\/p>\n <div class=\"formula\">\n ∇σ = Cep<\/sup> : De<\/sup> = Cep<\/sup> : (D – Dp<\/sup>)<\/b>\n <\/div>\n \n \u5176\u4e2d∇σ = σ̇ – W·σ + σ·W\u4e3aJaumann\u5e94\u529b\u7387\uff0c\n D\u4e3a\u53d8\u5f62\u7387\uff08symmetric part of velocity gradient L\uff09\uff0c\n W\u4e3a\u65cb\u7387\uff08skew-symmetric part of L\uff09\uff0c\n Cep<\/sup>\u4e3a\u5f39\u5851\u6027\u521a\u5ea6\u5f20\u91cf\uff0cDp<\/sup>\u4e3a\u5851\u6027\u53d8\u5f62\u7387\uff1a\n <\/p>\n \n where ∇σ = σ̇ – W·σ + σ·W is the Jaumann stress rate,\n D is the rate of deformation (symmetric part of velocity gradient L),\n W is the spin (skew-symmetric part of L),\n Cep<\/sup> is the elastoplastic stiffness tensor, and Dp<\/sup> is the plastic rate of deformation:\n <\/p>\n <div class=\"formula\">\n Dp<\/sup> = Σα<\/sub> γ̇(α)<\/sup> P(α)<\/sup><\/b>\n <\/div>\n \n \u9690\u5f0f\u65f6\u95f4\u79ef\u5206\u91c7\u7528Newton-Raphson\u8fed\u4ee3\u6c42\u89e3\u6bcf\u4e2a\u65f6\u95f4\u6b65\u672b\u7684\u5e94\u529b\u548c\u72b6\u6001\u53d8\u91cf\u3002\n \u7ebf\u6027\u5316\u540e\u5f97\u5230\u4ee5Δγ(α)<\/sup>\u4e3a\u672a\u77e5\u91cf\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u901a\u8fc7LU\u5206\u89e3\u6c42\u89e3\u3002\n \u6536\u655b\u540e\u66f4\u65b0STATEV\u6570\u7ec4\u4e2d\u7684\u5e94\u529b\u3001\u5e94\u53d8\u548c\u5404\u6ed1\u79fb\u7cfb\u53d8\u91cf\u3002\n <\/p>\n \n Implicit time integration employs Newton-Raphson iteration to solve for the stress and state variables at the end of each time increment.\n Linearization yields a linear system of equations with Δγ(α)<\/sup> as unknowns, solved by LU decomposition.\n After convergence, the STATEV array is updated with stress, strain, and slip system variables.\n <\/p>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch3\">\n <h2>\n 03<\/span>\n Abaqus UMAT\u63a5\u53e3 \/ UMAT Interface<\/span>\n UMAT Interface<\/span>\n <\/h2>\n\n <h3 id=\"ch3-1\">\n 3.1 \u5b8c\u6574Fortran\u7b7e\u540d \/ Full Fortran Signature<\/span>\n 3.1 Full Fortran Signature<\/span>\n <\/h3>\n \n Abaqus\/Standard\u7684\u7528\u6237\u6750\u6599\u5b50\u7a0b\u5e8fUMAT\u5177\u6709\u56fa\u5b9a\u7684\u63a5\u53e3\u7b7e\u540d\u3002\u9ec4\u6c38\u521a\u5355\u6676\u5851\u6027\u5b50\u7a0b\u5e8f\u5728\u8be5\u6846\u67b6\u4e0b\u5b9e\u73b0\uff0c\u5b8c\u6574\u58f0\u660e\u5982\u4e0b\uff1a\n <\/p>\n \n The Abaqus\/Standard User Material subroutine (UMAT) has a fixed interface signature. Huang’s single-crystal plasticity subroutine is implemented within this framework. The complete declaration is:\n <\/p>\n <pre><code> SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,\n 1 RPL,DDSDDT,DRPLDE,DRPLDT,\n 2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME,\n 3 NDI,NSHR,NTENS,NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,\n 4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,KSTEP,KINC)\nC-----------------------------------------------------------------------\nC UMAT for single crystal plasticity\nC Developed by Yonggang Huang\nC-----------------------------------------------------------------------\n INCLUDE 'ABA_PARAM.INC'\n CHARACTER*80 CMNAME\n DIMENSION STRESS(NTENS),STATEV(NSTATV),\n 1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS),\n 2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),\n 3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3)\nC ... user coding ...\n RETURN\n END<\/code><\/pre>\n\n <h3 id=\"ch3-2\">\n 3.2 \u53d8\u91cf\u8bf4\u660e\u8868 \/ Variable Specification Table<\/span>\n 3.2 Variable Specification Table<\/span>\n <\/h3>\n \n \u4e0b\u8868\u5217\u51fa\u4e86UMAT\u63a5\u53e3\u4e2d\u4e0e\u5355\u6676\u5851\u6027\u5b9e\u73b0\u5bc6\u5207\u76f8\u5173\u7684\u8f93\u5165\/\u8f93\u51fa\u53d8\u91cf\uff1a\n <\/p>\n \n The following table lists the input\/output variables in the UMAT interface most relevant to the single-crystal plasticity implementation:\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>\n \u53d8\u91cf\u540d \/ Variable<\/span>\n Variable<\/span>\n <\/th>\n <th>\n \u7c7b\u578b \/ Type<\/span>\n Type<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td><code>STRESS(NTENS)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u5e94\u529b\u5f20\u91cf\uff08\u8f93\u5165\u4e3a\u589e\u91cf\u5f00\u59cb\u65f6\u7684\u5e94\u529b\uff0c\u8f93\u51fa\u4e3a\u589e\u91cf\u7ed3\u675f\u65f6\u7684\u5e94\u529b\uff09\uff0c\u6309Abaqus\u7ea6\u5b9a\u5b58\u50a8\u4e3aVoigt\u5411\u91cf\uff1aσ11, σ22, σ33, σ12, σ13, σ23<\/span>\n Stress tensor (input: stress at increment start; output: stress at increment end), stored as Voigt vector per Abaqus convention: σ11, σ22, σ33, σ12, σ13, σ23<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>STATEV(NSTATV)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u72b6\u6001\u53d8\u91cf\u6570\u7ec4\uff08solution-dependent state variables\uff09\uff0c\u7528\u4e8e\u5b58\u50a8\u5851\u6027\u5e94\u53d8\u3001\u7d2f\u79ef\u526a\u5207\u3001\u6ed1\u79fb\u7cfb\u5f3a\u5ea6\u7b49\u5386\u53f2\u4fe1\u606f\u3002\u5fc5\u987b\u5728\u672c\u5b50\u7a0b\u5e8f\u4e2d\u5b9a\u4e49\u66f4\u65b0\u89c4\u5219\u3002<\/span>\n Solution-dependent state variable array for storing plastic strain, accumulated shear, slip system strength, and other history information. Update rules must be defined in this subroutine.<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DDSDDE(NTENS,NTENS)<\/code><\/td>\n <td>\n \u4e8c\u7ef4\u6570\u7ec4 \/ Matrix<\/span>\n Matrix<\/span>\n <\/td>\n <td>\n \u6750\u6599Jacobian\u77e9\u9635\uff08\u4e00\u81f4\u5207\u7ebf\u6a21\u91cf\uff09\uff0c∂Δσ\/∂Δε\u3002\u8fd9\u662f\u9690\u5f0f\u6c42\u89e3\u5668\u4fdd\u8bc1\u4e8c\u9636\u6536\u655b\u7684\u5173\u952e\uff0c\u5fc5\u987b\u63d0\u4f9b\u51c6\u786e\u7684\u5f39\u5851\u6027\u5207\u7ebf\u521a\u5ea6\u3002<\/span>\n Material Jacobian matrix (consistent tangent modulus), ∂Δσ\/∂Δε. This is essential for the implicit solver to achieve quadratic convergence; accurate elastoplastic tangent stiffness must be provided.<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>STRAN(NTENS)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u589e\u91cf\u5f00\u59cb\u65f6\u7684\u603b\u5e94\u53d8\uff08engineering strain\uff0c\u526a\u5e94\u53d8\u4e0d\u9664\u4ee52\uff09<\/span>\n Total strain at increment start (engineering strain; shear strains are not divided by 2)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DSTRAN(NTENS)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u5e94\u53d8\u589e\u91cf\uff08engineering strain increment\uff09\uff0c\u7531Abaqus\u6839\u636e\u6574\u4f53\u5e73\u8861\u8fed\u4ee3\u63d0\u4f9b<\/span>\n Strain increment (engineering strain increment), provided by Abaqus based on global equilibrium iteration<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DFGRD0(3,3)<\/code><\/td>\n <td>\n \u77e9\u9635 \/ Matrix<\/span>\n Matrix<\/span>\n <\/td>\n <td>\n \u589e\u91cf\u5f00\u59cb\u65f6\u7684\u53d8\u5f62\u68af\u5ea6F\uff083×3\u77e9\u9635\uff09<\/span>\n Deformation gradient F at increment start (3×3 matrix)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DFGRD1(3,3)<\/code><\/td>\n <td>\n \u77e9\u9635 \/ Matrix<\/span>\n Matrix<\/span>\n <\/td>\n <td>\n \u589e\u91cf\u7ed3\u675f\u65f6\u7684\u53d8\u5f62\u68af\u5ea6F\uff08\u7531Abaqus\u57fa\u4e8e\u6574\u4f53\u4f4d\u79fb\u573a\u8ba1\u7b97\uff09<\/span>\n Deformation gradient F at increment end (computed by Abaqus based on global displacement field)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>PROPS(NPROPS)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u6750\u6599\u53c2\u6570\u6570\u7ec4\uff0c\u5728Abaqus\u8f93\u5165\u6587\u4ef6\u4e2d\u901a\u8fc7*Material, *User Material\u5173\u952e\u5b57\u5b9a\u4e49<\/span>\n Material property array, defined in the Abaqus input file via *Material, *User Material keywords<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>NPROPS<\/code><\/td>\n <td>\n \u6574\u6570 \/ Integer<\/span>\n Integer<\/span>\n <\/td>\n <td>\n PROPS\u6570\u7ec4\u957f\u5ea6\uff0c\u9ec4\u6c38\u521a\u7a0b\u5e8f\u901a\u5e38\u8981\u6c42NPROPS ≥ 160<\/span>\n Length of PROPS array; Huang’s program typically requires NPROPS ≥ 160<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>NSTATV<\/code><\/td>\n <td>\n \u6574\u6570 \/ Integer<\/span>\n Integer<\/span>\n <\/td>\n <td>\n STATEV\u6570\u7ec4\u957f\u5ea6\uff0c\u53d6\u51b3\u4e8e\u6ed1\u79fb\u7cfb\u6570\u91cf\u548c\u5b58\u50a8\u9700\u6c42\uff0c\u901a\u5e38NSTATV = 6 + NSLIP + NSLIP + …<\/span>\n Length of STATEV array, depending on number of slip systems and storage requirements; typically NSTATV = 6 + NSLIP + NSLIP + …<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DROT(3,3)<\/code><\/td>\n <td>\n \u77e9\u9635 \/ Matrix<\/span>\n Matrix<\/span>\n <\/td>\n <td>\n \u589e\u91cf\u65cb\u8f6c\u5f20\u91cf\uff08\u7531\u6781\u5206\u89e3\u5f97\u5230\uff09\uff0cAbaqus\u81ea\u52a8\u8ba1\u7b97\u5e76\u63d0\u4f9b<\/span>\n Incremental rotation tensor (from polar decomposition), automatically computed and provided by Abaqus<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DTIME<\/code><\/td>\n <td>\n \u53cc\u7cbe\u5ea6 \/ Double<\/span>\n Double<\/span>\n <\/td>\n <td>\n \u65f6\u95f4\u589e\u91cf\u5927\u5c0f\uff08\u5728\u7387\u76f8\u5173\u6750\u6599\u4e2d\u7528\u4e8e\u8ba1\u7b97γ̇\uff09<\/span>\n Time increment size (used to compute γ̇ in rate-dependent materials)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>PNEWDT<\/code><\/td>\n <td>\n \u53cc\u7cbe\u5ea6 \/ Double<\/span>\n Double<\/span>\n <\/td>\n <td>\n \u5efa\u8bae\u7684\u4e0b\u4e00\u589e\u91cf\u65f6\u95f4\u6bd4\u4f8b\u56e0\u5b50\uff080 < PNEWDT < 1\uff09\u3002\u82e5\u5c40\u90e8\u8fed\u4ee3\u4e0d\u6536\u655b\uff0c\u53ef\u8bbe\u7f6e\u6b64\u503c\u4f7fAbaqus\u81ea\u52a8\u51cf\u5c0f\u65f6\u95f4\u6b65\u3002<\/span>\n Suggested ratio for next increment time step (0 < PNEWDT < 1). If local iteration does not converge, set this to let Abaqus automatically reduce the time step.<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>NOEL, NPT<\/code><\/td>\n <td>\n \u6574\u6570 \/ Integer<\/span>\n Integer<\/span>\n <\/td>\n <td>\n \u5355\u5143\u7f16\u53f7\u548c\u79ef\u5206\u70b9\u7f16\u53f7\uff0c\u7528\u4e8e\u8c03\u8bd5\u548c\u5b9a\u4f4d<\/span>\n Element number and integration point number, for debugging and localization<\/span>\n <\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch3-3\">\n 3.3 STATEV\u5e03\u5c40 \/ State Variable Layout Table<\/span>\n 3.3 State Variable Layout Table<\/span>\n <\/h3>\n \n STATEV\u6570\u7ec4\u662fUMAT\u5b50\u7a0b\u5e8f\u7684\u6838\u5fc3\u6570\u636e\u7ed3\u6784\uff0c\u7528\u4e8e\u5728\u589e\u91cf\u6b65\u4e4b\u95f4\u4f20\u9012\u548c\u4fdd\u5b58\u6750\u6599\u7684\u5386\u53f2\u72b6\u6001\u3002\n \u9ec4\u6c38\u521aUMAT\u7684STATEV\u5e03\u5c40\u5982\u4e0b\uff08\u4ee5FCC 12\u6ed1\u79fb\u7cfb\u4e3a\u4f8b\uff09\uff1a\n <\/p>\n \n The STATEV array is the core data structure of the UMAT subroutine, used to pass and preserve material history states between increments.\n The STATEV layout in Huang’s UMAT is as follows (for FCC 12 slip systems as an example):\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>\n \u7d22\u5f15\u8303\u56f4 \/ Index Range<\/span>\n Index Range<\/span>\n <\/th>\n <th>\n \u5185\u5bb9 \/ Content<\/span>\n Content<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>1 – 6<\/td>\n <td>\n \u5851\u6027\u5e94\u53d8 \/ Plastic strain<\/span>\n Plastic strain<\/span>\n <\/td>\n <td>\n \u7d2f\u79ef\u5851\u6027\u5e94\u53d8\u5206\u91cf εp<\/sup>11<\/sub>, εp<\/sup>22<\/sub>, εp<\/sup>33<\/sub>, γp<\/sup>12<\/sub>, γp<\/sup>13<\/sub>, γp<\/sup>23<\/sub>\uff08\u5de5\u7a0b\u526a\u5e94\u53d8\uff09<\/span>\n Accumulated plastic strain components εp<\/sup>11<\/sub>, εp<\/sup>22<\/sub>, εp<\/sup>33<\/sub>, γp<\/sup>12<\/sub>, γp<\/sup>13<\/sub>, γp<\/sup>23<\/sub> (engineering shear strains)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>7 – 18<\/td>\n <td>\n \u7d2f\u79ef\u526a\u5207\u5e94\u53d8 \/ Accumulated shear<\/span>\n Accumulated shear<\/span>\n <\/td>\n <td>\n \u5404\u6ed1\u79fb\u7cfb\u7d2f\u79ef\u526a\u5207\u5e94\u53d8 γ(α)<\/sup> = ∫|γ̇(α)<\/sup>|dt\uff0cα = 1,…,12<\/span>\n Accumulated shear strain on each slip system γ(α)<\/sup> = ∫|γ̇(α)<\/sup>|dt, α = 1,…,12<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>19 – 30<\/td>\n <td>\n \u5f53\u524d\u6ed1\u79fb\u7cfb\u5f3a\u5ea6 \/ Current slip strength<\/span>\n Current slip strength<\/span>\n <\/td>\n <td>\n \u5404\u6ed1\u79fb\u7cfb\u5f53\u524d\u4e34\u754c\u5206\u5207\u5e94\u529b g(α)<\/sup>\uff0cα = 1,…,12<\/span>\n Current critical resolved shear stress g(α)<\/sup> for each slip system, α = 1,…,12<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>31 – 39<\/td>\n <td>\n \u5f39\u6027\u53d8\u5f62\u68af\u5ea6 \/ Elastic def. grad.<\/span>\n Elastic def. grad.<\/span>\n <\/td>\n <td>\n Fe<\/sup>\u76849\u4e2a\u5206\u91cf\uff083×3\u77e9\u9635\u6309\u5217\u4f18\u5148\u5b58\u50a8\uff09<\/span>\n 9 components of Fe<\/sup> (3×3 matrix stored column-major)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>40 – 48<\/td>\n <td>\n \u5851\u6027\u53d8\u5f62\u68af\u5ea6 \/ Plastic def. grad.<\/span>\n Plastic def. grad.<\/span>\n <\/td>\n <td>\n Fp<\/sup>\u76849\u4e2a\u5206\u91cf\uff083×3\u77e9\u9635\u6309\u5217\u4f18\u5148\u5b58\u50a8\uff09<\/span>\n 9 components of Fp<\/sup> (3×3 matrix stored column-major)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>49 – 54<\/td>\n <td>\n \u5e94\u529b \/ Stress<\/span>\n Stress<\/span>\n <\/td>\n <td>\n \u5b58\u50a8Cauchy\u5e94\u529b\u5206\u91cf\uff0c\u4fbf\u4e8e\u540e\u5904\u7406\u8f93\u51fa<\/span>\n Cauchy stress components stored for post-processing output<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>55+<\/td>\n <td>\n \u5176\u4ed6\u8f85\u52a9\u53d8\u91cf \/ Aux. variables<\/span>\n Aux. variables<\/span>\n <\/td>\n <td>\n \u6676\u4f53\u53d6\u5411\u3001Euler\u89d2\u3001\u65cb\u8f6c\u77e9\u9635\u5206\u91cf\u7b49\uff08\u5177\u4f53\u53d6\u51b3\u4e8ePROPS\u914d\u7f6e\uff09<\/span>\n Crystal orientation, Euler angles, rotation matrix components, etc. (depending on PROPS configuration)<\/span>\n <\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <div class=\"note\">\n \u91cd\u8981\u63d0\u793a\uff1a<\/strong>STATEV\u6570\u7ec4\u5728Abaqus\u4e2d\u901a\u8fc7*Depvar\u5173\u952e\u5b57\u5b9a\u4e49\u5927\u5c0f\u3002\u5fc5\u987b\u786e\u4fddNSTATV\u8db3\u591f\u5bb9\u7eb3\u6240\u6709\u9700\u8981\u5b58\u50a8\u7684\u72b6\u6001\u53d8\u91cf\uff0c\u5426\u5219\u4f1a\u53d1\u751f\u6570\u7ec4\u8d8a\u754c\u9519\u8bef\u6216\u72b6\u6001\u4fe1\u606f\u4e22\u5931\u3002<\/span>\n Important:<\/strong> The STATEV array size is defined in Abaqus via the *Depvar keyword. Ensure NSTATV is large enough to hold all state variables; otherwise array overflow or state loss will occur.<\/span>\n <\/div>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch4\">\n <h2>\n 04<\/span>\n \u6750\u6599\u53c2\u6570 \/ Material Parameters<\/span>\n Material Parameters<\/span>\n <\/h2>\n\n \n \u9ec4\u6c38\u521aUMAT\u901a\u8fc7PROPS\u6570\u7ec4\u4f20\u5165\u5168\u90e8\u6750\u6599\u53c2\u6570\u3002PROPS\u7684\u7d22\u5f15\u5e03\u5c40\u7ecf\u8fc7\u7cbe\u5fc3\u8bbe\u8ba1\uff0c\n \u5c06\u5f39\u6027\u5e38\u6570\u3001\u6ed1\u79fb\u7cfb\u5b9a\u4e49\u3001\u6676\u4f53\u53d6\u5411\u3001\u7387\u76f8\u5173\u53c2\u6570\u3001\u786c\u5316\u53c2\u6570\u548c\u79ef\u5206\u63a7\u5236\u53c2\u6570\u5206\u533a\u5b58\u50a8\u3002\n \u4ee5\u4e0b\u5404\u8282\u8be6\u7ec6\u5217\u51fa\u6bcf\u4e2aPROPS\u7d22\u5f15\u5bf9\u5e94\u7684\u53c2\u6570\u542b\u4e49\u3002\n <\/p>\n \n Huang’s UMAT receives all material parameters through the PROPS array. The PROPS index layout is carefully designed,\n partitioning elastic constants, slip system definitions, crystal orientation, rate-dependent parameters, hardening parameters, and integration control parameters.\n The following sections detail the meaning of each PROPS index.\n <\/p>\n\n <h3 id=\"ch4-1\">\n 4.1 \u5f39\u6027\u5e38\u6570 PROPS(1-21) \/ Elastic Constants<\/span>\n 4.1 Elastic Constants PROPS(1-21)<\/span>\n <\/h3>\n \n \u5f39\u6027\u5e38\u6570\u533a\u5b9a\u4e49\u6676\u4f53\u6750\u6599\u7684\u5404\u5411\u5f02\u6027\u5f39\u6027\u884c\u4e3a\u3002\u5bf9\u4e8e\u7acb\u65b9\u6676\u7cfb\uff0c\u53ea\u9700\u89813\u4e2a\u72ec\u7acb\u5e38\u6570\uff08C11, C12, C44\uff09\uff0c\n \u4f46\u7a0b\u5e8f\u63a5\u53e3\u9884\u7559\u4e86\u5b8c\u657421\u4e2a\u53c2\u6570\u4ee5\u652f\u6301\u4e00\u822c\u5404\u5411\u5f02\u6027\u6750\u6599\uff08\u4e09\u659c\u6676\u7cfb\uff09\u3002\n <\/p>\n \n The elastic constant section defines the anisotropic elastic behavior of the crystal material.\n For cubic crystals, only 3 independent constants (C11, C12, C44) are needed,\n but the program interface reserves a full 21 parameters to support general anisotropic materials (triclinic).\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>1<\/td><td>C11<\/td><td>C11<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC11\uff08MPa\uff09<\/span>Elastic stiffness matrix component C11 (MPa)<\/span><\/td><\/tr>\n <tr><td>2<\/td><td>C12<\/td><td>C12<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC12\uff08MPa\uff09<\/span>Elastic stiffness matrix component C12 (MPa)<\/span><\/td><\/tr>\n <tr><td>3<\/td><td>C13<\/td><td>C13<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC13\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC13=C12<\/span>Elastic stiffness matrix component C13 (MPa); C13=C12 for cubic<\/span><\/td><\/tr>\n <tr><td>4<\/td><td>C14<\/td><td>C14<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC14\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC14=0<\/span>Elastic stiffness matrix component C14 (MPa); C14=0 for cubic<\/span><\/td><\/tr>\n <tr><td>5<\/td><td>C15<\/td><td>C15<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC15\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC15=0<\/span>Elastic stiffness matrix component C15 (MPa); C15=0 for cubic<\/span><\/td><\/tr>\n <tr><td>6<\/td><td>C16<\/td><td>C16<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC16\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC16=0<\/span>Elastic stiffness matrix component C16 (MPa); C16=0 for cubic<\/span><\/td><\/tr>\n <tr><td>7<\/td><td>C22<\/td><td>C22<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC22\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC22=C11<\/span>Elastic stiffness matrix component C22 (MPa); C22=C11 for cubic<\/span><\/td><\/tr>\n <tr><td>8<\/td><td>C23<\/td><td>C23<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC23\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC23=C12<\/span>Elastic stiffness matrix component C23 (MPa); C23=C12 for cubic<\/span><\/td><\/tr>\n <tr><td>9<\/td><td>C24<\/td><td>C24<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC24\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC24=0<\/span>Elastic stiffness matrix component C24 (MPa); C24=0 for cubic<\/span><\/td><\/tr>\n <tr><td>10<\/td><td>C25<\/td><td>C25<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC25\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC25=0<\/span>Elastic stiffness matrix component C25 (MPa); C25=0 for cubic<\/span><\/td><\/tr>\n <tr><td>11<\/td><td>C26<\/td><td>C26<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC26\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC26=0<\/span>Elastic stiffness matrix component C26 (MPa); C26=0 for cubic<\/span><\/td><\/tr>\n <tr><td>12<\/td><td>C33<\/td><td>C33<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC33\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC33=C11<\/span>Elastic stiffness matrix component C33 (MPa); C33=C11 for cubic<\/span><\/td><\/tr>\n <tr><td>13<\/td><td>C34<\/td><td>C34<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC34\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC34=0<\/span>Elastic stiffness matrix component C34 (MPa); C34=0 for cubic<\/span><\/td><\/tr>\n <tr><td>14<\/td><td>C35<\/td><td>C35<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC35\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC35=0<\/span>Elastic stiffness matrix component C35 (MPa); C35=0 for cubic<\/span><\/td><\/tr>\n <tr><td>15<\/td><td>C36<\/td><td>C36<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC36\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC36=0<\/span>Elastic stiffness matrix component C36 (MPa); C36=0 for cubic<\/span><\/td><\/tr>\n <tr><td>16<\/td><td>C44<\/td><td>C44<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC44\uff08MPa\uff09<\/span>Elastic stiffness matrix component C44 (MPa)<\/span><\/td><\/tr>\n <tr><td>17<\/td><td>C45<\/td><td>C45<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC45\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC45=0<\/span>Elastic stiffness matrix component C45 (MPa); C45=0 for cubic<\/span><\/td><\/tr>\n <tr><td>18<\/td><td>C46<\/td><td>C46<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC46\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC46=0<\/span>Elastic stiffness matrix component C46 (MPa); C46=0 for cubic<\/span><\/td><\/tr>\n <tr><td>19<\/td><td>C55<\/td><td>C55<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC55\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC55=C44<\/span>Elastic stiffness matrix component C55 (MPa); C55=C44 for cubic<\/span><\/td><\/tr>\n <tr><td>20<\/td><td>C56<\/td><td>C56<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC56\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC56=0<\/span>Elastic stiffness matrix component C56 (MPa); C56=0 for cubic<\/span><\/td><\/tr>\n <tr><td>21<\/td><td>C66<\/td><td>C66<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC66\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC66=C44<\/span>Elastic stiffness matrix component C66 (MPa); C66=C44 for cubic<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch4-2\">\n 4.2 \u6ed1\u79fb\u7cfb\u5b9a\u4e49 PROPS(25-56) \/ Slip System Definition<\/span>\n 4.2 Slip System Definition PROPS(25-56)<\/span>\n <\/h3>\n \n \u6ed1\u79fb\u7cfb\u5b9a\u4e49\u533a\u7528\u4e8e\u6307\u5b9a\u6bcf\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411s\u548c\u6ed1\u79fb\u9762\u6cd5\u5411m\uff08\u5728\u6676\u4f53\u5750\u6807\u7cfb\u4e2d\uff09\u3002\n \u6bcf\u7ec4\u6ed1\u79fb\u7cfb\u9700\u89813\u4e2a\u65b9\u5411\u5206\u91cf\u548c3\u4e2a\u6cd5\u5411\u5206\u91cf\u3002\u5bf9\u4e8eFCC\u7ed3\u6784\uff0c\u901a\u5e38\u5b9a\u4e4912\u4e2a\u6ed1\u79fb\u7cfb\uff08{111}<110>\uff09\u3002\n <\/p>\n \n The slip system definition section specifies the slip direction s and slip plane normal m (in crystal coordinates) for each slip system.\n Each slip system requires 3 direction components and 3 normal components. For FCC, 12 slip systems ({111}<110>) are typically defined.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>22<\/td><td>NSLIP<\/td><td>NSLIP<\/td><td>\u6ed1\u79fb\u7cfb\u603b\u6570\uff08\u6574\u6570\uff0c\u5982FCC=12\uff09<\/span>Total number of slip systems (integer, e.g., FCC=12)<\/span><\/td><\/tr>\n <tr><td>23<\/td><td>NSLIP1<\/td><td>NSLIP1<\/td><td>\u7b2c\u4e00\u65cf\u6ed1\u79fb\u7cfb\u6570\u91cf\uff08\u5982FCC\u4e2d\u4e3a12\uff09<\/span>Number of first-family slip systems (e.g., 12 for FCC)<\/span><\/td><\/tr>\n <tr><td>24<\/td><td>NSLIP2<\/td><td>NSLIP2<\/td><td>\u7b2c\u4e8c\u65cf\u6ed1\u79fb\u7cfb\u6570\u91cf\uff08\u5982\u6709\u9700\u8981\uff09<\/span>Number of second-family slip systems (if needed)<\/span><\/td><\/tr>\n <tr><td>25-27<\/td><td>s1(1-3)<\/td><td>Slip dir. 1<\/td><td>\u7b2c1\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411\u5206\u91cf(sx,sy,sz)<\/span>Slip direction components (sx,sy,sz) for slip system 1<\/span><\/td><\/tr>\n <tr><td>28-30<\/td><td>m1(1-3)<\/td><td>Slip normal 1<\/td><td>\u7b2c1\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u9762\u6cd5\u5411\u5206\u91cf(mx,my,mz)<\/span>Slip plane normal components (mx,my,mz) for slip system 1<\/span><\/td><\/tr>\n <tr><td>31-33<\/td><td>s2(1-3)<\/td><td>Slip dir. 2<\/td><td>\u7b2c2\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411\u5206\u91cf<\/span>Slip direction components for slip system 2<\/span><\/td><\/tr>\n <tr><td>34-36<\/td><td>m2(1-3)<\/td><td>Slip normal 2<\/td><td>\u7b2c2\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u9762\u6cd5\u5411\u5206\u91cf<\/span>Slip plane normal components for slip system 2<\/span><\/td><\/tr>\n <tr><td colspan=\"4\" style=\"text-align:center\">\n … \u4f9d\u6b64\u7c7b\u63a8\uff0c\u6bcf6\u4e2aPROPS\u5b9a\u4e49\u4e00\u4e2a\u6ed1\u79fb\u7cfb …<\/span>\n … and so on; each slip system occupies 6 PROPS entries …<\/span>\n <\/td><\/tr>\n <tr><td>55-57<\/td><td>s12(1-3)<\/td><td>Slip dir. 12<\/td><td>\u7b2c12\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411\u5206\u91cf<\/span>Slip direction components for slip system 12<\/span><\/td><\/tr>\n <tr><td>58-60<\/td><td>m12(1-3)<\/td><td>Slip normal 12<\/td><td>\u7b2c12\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u9762\u6cd5\u5411\u5206\u91cf<\/span>Slip plane normal components for slip system 12<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n \n FCC\u6676\u4f53\u5178\u578b\u768412\u4e2a\u6ed1\u79fb\u7cfb\u4e3a{111}<110>\u65cf\uff0c\u5206\u4e3a4\u4e2a\u6ed1\u79fb\u9762\uff0c\u6bcf\u4e2a\u9762\u67093\u4e2a\u6ed1\u79fb\u65b9\u5411\u3002\n \u4ee5\u4e0b\u4e3a\u6676\u4f53\u5750\u6807\u7cfb\u4e2d\u7684\u6807\u51c6\u5b9a\u4e49\uff08\u65e0\u9700\u5f52\u4e00\u5316\uff0c\u7a0b\u5e8f\u5185\u90e8\u4f1a\u81ea\u52a8\u5904\u7406\uff09\uff1a\n <\/p>\n \n The typical 12 slip systems for FCC crystals belong to the {111}<110> family, divided into 4 slip planes with 3 slip directions each.\n The following is the standard definition in crystal coordinates (normalization is handled internally by the program):\n <\/p>\n\n <pre><code>C FCC {111}<110> slip systems (standard definition)\nC Plane 1 (111): directions [1-10], [10-1], [01-1]\nC Plane 2 (1-11): directions [110], [101], [0-11]\nC Plane 3 (-111): directions [110], [10-1], [011]\nC Plane 4 (11-1): directions [1-10], [101], [011]<\/code><\/pre>\n\n <h3 id=\"ch4-3\">\n 4.3 \u6676\u4f53\u53d6\u5411 PROPS(57-70) \/ Crystal Orientation<\/span>\n 4.3 Crystal Orientation PROPS(57-70)<\/span>\n <\/h3>\n \n \u6676\u4f53\u53d6\u5411\u533a\u5b9a\u4e49\u6676\u4f53\u5750\u6807\u7cfb\u4e0e\u5168\u5c40\uff08\u8bd5\u6837\uff09\u5750\u6807\u7cfb\u4e4b\u95f4\u7684\u65cb\u8f6c\u5173\u7cfb\u3002\n \u901a\u5e38\u91c7\u7528Bunge Euler\u89d2(φ1, Φ, φ2)\u6216\u65b9\u5411\u4f59\u5f26\u77e9\u9635\u8868\u793a\u3002\n <\/p>\n \n The crystal orientation section defines the rotation relationship between the crystal coordinate system and the global (sample) coordinate system.\n Bunge Euler angles (φ1, Φ, φ2) or the direction cosine matrix are commonly used.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>57-59<\/td><td>Euler\u89d21-3<\/td><td>Euler angles<\/td><td>Bunge Euler\u89d2 φ1, Φ, φ2\uff08\u5355\u4f4d\uff1a\u5ea6\uff09<\/span>Bunge Euler angles φ1, Φ, φ2 (in degrees)<\/span><\/td><\/tr>\n <tr><td>60-62<\/td><td>\u6676\u4f53\u8f741<\/td><td>Crystal axis 1<\/td><td>\u5168\u5c40\u5750\u6807\u7cfb\u4e2d\u6676\u4f53[100]\u8f74\u7684\u65b9\u5411\u4f59\u5f26<\/span>Direction cosines of crystal [100] axis in global coordinates<\/span><\/td><\/tr>\n <tr><td>63-65<\/td><td>\u6676\u4f53\u8f742<\/td><td>Crystal axis 2<\/td><td>\u5168\u5c40\u5750\u6807\u7cfb\u4e2d\u6676\u4f53[010]\u8f74\u7684\u65b9\u5411\u4f59\u5f26<\/span>Direction cosines of crystal [010] axis in global coordinates<\/span><\/td><\/tr>\n <tr><td>66-68<\/td><td>\u6676\u4f53\u8f743<\/td><td>Crystal axis 3<\/td><td>\u5168\u5c40\u5750\u6807\u7cfb\u4e2d\u6676\u4f53[001]\u8f74\u7684\u65b9\u5411\u4f59\u5f26<\/span>Direction cosines of crystal [001] axis in global coordinates<\/span><\/td><\/tr>\n <tr><td>69<\/td><td>\u53d6\u5411\u8f93\u5165\u6a21\u5f0f<\/td><td>Orientation mode<\/td><td>0=\u4f7f\u7528Euler\u89d2\uff0c1=\u4f7f\u7528\u65b9\u5411\u4f59\u5f26\u77e9\u9635<\/span>0=use Euler angles, 1=use direction cosine matrix<\/span><\/td><\/tr>\n <tr><td>70<\/td><td>\u53d6\u5411\u66f4\u65b0\u6807\u5fd7<\/td><td>Orientation update<\/td><td>1=\u5728\u53d8\u5f62\u8fc7\u7a0b\u4e2d\u66f4\u65b0\u6676\u4f53\u53d6\u5411\uff08\u8003\u8651\u6676\u683c\u8f6c\u52a8\uff09\uff0c0=\u56fa\u5b9a\u53d6\u5411<\/span>1=update crystal orientation during deformation (lattice rotation), 0=fixed orientation<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch4-4\">\n 4.4 \u7387\u76f8\u5173\u53c2\u6570 PROPS(73-96) \/ Rate-Dependent Parameters<\/span>\n 4.4 Rate-Dependent Parameters PROPS(73-96)<\/span>\n <\/h3>\n \n \u7387\u76f8\u5173\u53c2\u6570\u533a\u63a7\u5236\u5e42\u5f8b\u6d41\u52a8\u6cd5\u5219\u7684\u884c\u4e3a\u3002\u5bf9\u4e8e\u591a\u65cf\u6ed1\u79fb\u7cfb\uff0c\u6bcf\u65cf\u53ef\u4ee5\u72ec\u7acb\u6307\u5b9a\u7387\u76f8\u5173\u53c2\u6570\u3002\n <\/p>\n \n The rate-dependent parameter section controls the behavior of the power-law flow rule. For multiple families of slip systems, each family can have independent rate-dependent parameters.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>71<\/td><td>GAMMA0_1<\/td><td>GAMMA0_1<\/td><td>\u7b2c1\u65cf\u6ed1\u79fb\u7cfb\u7684\u53c2\u8003\u526a\u5207\u7387 γ̇0<\/sub>\uff08s-1<\/sup>\uff09<\/span>Reference shear rate γ̇0<\/sub> for 1st family (s-1<\/sup>)<\/span><\/td><\/tr>\n <tr><td>72<\/td><td>AM_1<\/td><td>AM_1<\/td><td>\u7b2c1\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u654f\u611f\u6307\u6570\u5012\u6570 1\/n<\/span>Inverse rate sensitivity exponent 1\/n for 1st family<\/span><\/td><\/tr>\n <tr><td>73<\/td><td>N_1<\/td><td>N_1<\/td><td>\u7b2c1\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u654f\u611f\u6307\u6570 n<\/span>Rate sensitivity exponent n for 1st family<\/span><\/td><\/tr>\n <tr><td>74<\/td><td>GAMMA0_2<\/td><td>GAMMA0_2<\/td><td>\u7b2c2\u65cf\u6ed1\u79fb\u7cfb\u7684\u53c2\u8003\u526a\u5207\u7387\uff08\u5982\u6709\uff09<\/span>Reference shear rate for 2nd family (if applicable)<\/span><\/td><\/tr>\n <tr><td>75<\/td><td>AM_2<\/td><td>AM_2<\/td><td>\u7b2c2\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u654f\u611f\u6307\u6570\u5012\u6570 1\/n<\/span>Inverse rate sensitivity exponent 1\/n for 2nd family<\/span><\/td><\/tr>\n <tr><td>76<\/td><td>N_2<\/td><td>N_2<\/td><td>\u7b2c2\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u654f\u611f\u6307\u6570 n<\/span>Rate sensitivity exponent n for 2nd family<\/span><\/td><\/tr>\n <tr><td>77-84<\/td><td>\u6d41\u52a8\u5f8b\u53c2\u6570<\/td><td>Flow params<\/td><td>\u7b2c3-4\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u76f8\u5173\u53c2\u6570\uff08\u9884\u7559\uff09<\/span>Rate-dependent parameters for 3rd-4th families (reserved)<\/span><\/td><\/tr>\n <tr><td>85<\/td><td>TEMP_REF<\/td><td>TEMP_REF<\/td><td>\u53c2\u8003\u6e29\u5ea6\uff08K\uff09\uff0c\u7528\u4e8e\u70ed\u6fc0\u6d3b\u6a21\u578b<\/span>Reference temperature (K) for thermal activation model<\/span><\/td><\/tr>\n <tr><td>86<\/td><td>TEMP_CUR<\/td><td>TEMP_CUR<\/td><td>\u5f53\u524d\u6e29\u5ea6\uff08K\uff09<\/span>Current temperature (K)<\/span><\/td><\/tr>\n <tr><td>87-96<\/td><td>\u9884\u7559<\/td><td>Reserved<\/td><td>\u4e3a\u540e\u7eed\u6269\u5c55\u9884\u7559<\/span>Reserved for future extensions<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch4-5\">\n 4.5 \u786c\u5316\u53c2\u6570 PROPS(97-144) \/ Hardening Parameters<\/span>\n 4.5 Hardening Parameters PROPS(97-144)<\/span>\n <\/h3>\n \n \u786c\u5316\u53c2\u6570\u533a\u5305\u542b\u81ea\u786c\u5316\u548c\u6f5c\u786c\u5316\u7684\u5168\u90e8\u53c2\u6570\u3002\u6839\u636e\u9009\u62e9\u7684\u786c\u5316\u6a21\u578b\u7c7b\u578b\uff08PROPS\u4e2d\u7684\u6807\u5fd7\uff09\uff0c\u8fd9\u4e9b\u53c2\u6570\u7684\u89e3\u91ca\u6709\u6240\u4e0d\u540c\u3002\n <\/p>\n \n The hardening parameter section contains all parameters for self-hardening and latent hardening. Their interpretation depends on the selected hardening model type (flag in PROPS).\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>97<\/td><td>\u786c\u5316\u6a21\u578b\u7c7b\u578b<\/td><td>Hardening model<\/td><td>0=\u53cc\u66f2\u6b63\u5272\u786c\u5316\uff0c1=Bassani\u786c\u5316\uff0c2=\u5176\u4ed6<\/span>0=Hyperbolic secant, 1=Bassani, 2=Other<\/span><\/td><\/tr>\n <tr><td>98<\/td><td>TAU0<\/td><td>TAU0<\/td><td>\u521d\u59cb\u4e34\u754c\u5206\u5207\u5e94\u529b τ0<\/sub>\uff08MPa\uff09<\/span>Initial critical resolved shear stress τ0<\/sub> (MPa)<\/span><\/td><\/tr>\n <tr><td>99<\/td><td>TAUS<\/td><td>TAUS<\/td><td>\u9971\u548c\u4e34\u754c\u5206\u5207\u5e94\u529b τs<\/sub>\uff08MPa\uff09<\/span>Saturation critical resolved shear stress τs<\/sub> (MPa)<\/span><\/td><\/tr>\n <tr><td>100<\/td><td>H0<\/td><td>H0<\/td><td>\u521d\u59cb\u786c\u5316\u6a21\u91cf H0<\/sub>\uff08MPa\uff09<\/span>Initial hardening modulus H0<\/sub> (MPa)<\/span><\/td><\/tr>\n <tr><td>101<\/td><td>HS<\/td><td>HS<\/td><td>Bassani\u6a21\u578b\u4e2d\u7684\u9971\u548c\u786c\u5316\u6a21\u91cf Hs<\/sub>\uff08MPa\uff09<\/span>Saturation hardening modulus Hs<\/sub> in Bassani model (MPa)<\/span><\/td><\/tr>\n <tr><td>102<\/td><td>GAMMA0_HARD<\/td><td>GAMMA0_HARD<\/td><td>Bassani\u6a21\u578b\u4e2d\u7684\u7279\u5f81\u526a\u5207\u5e94\u53d8 γ0<\/sub><\/span>Characteristic shear strain γ0<\/sub> in Bassani model<\/span><\/td><\/tr>\n <tr><td>103<\/td><td>Q1<\/td><td>Q1<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q1\uff08\u5171\u9762\u6ed1\u79fb\u7cfb\u95f4\uff09<\/span>Latent hardening coefficient q1 (coplanar)<\/span><\/td><\/tr>\n <tr><td>104<\/td><td>Q2<\/td><td>Q2<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q2\uff08\u5171\u5411\u6ed1\u79fb\u7cfb\u95f4\uff09<\/span>Latent hardening coefficient q2 (codirectional)<\/span><\/td><\/tr>\n <tr><td>105<\/td><td>Q3<\/td><td>Q3<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q3\uff08\u6b63\u4ea4\u6ed1\u79fb\u7cfb\u95f4\uff09<\/span>Latent hardening coefficient q3 (orthogonal)<\/span><\/td><\/tr>\n <tr><td>106<\/td><td>Q4<\/td><td>Q4<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q4\uff08Hirth\u9501\uff09<\/span>Latent hardening coefficient q4 (Hirth lock)<\/span><\/td><\/tr>\n <tr><td>107<\/td><td>Q5<\/td><td>Q5<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q5\uff08Lomer-Cottrell\u9501\uff09<\/span>Latent hardening coefficient q5 (Lomer-Cottrell lock)<\/span><\/td><\/tr>\n <tr><td>108<\/td><td>Q6<\/td><td>Q6<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q6\uff08\u4ea4\u53c9\u6ed1\u79fb\u76f8\u5173\uff09<\/span>Latent hardening coefficient q6 (cross-slip related)<\/span><\/td><\/tr>\n <tr><td>109<\/td><td>Q7<\/td><td>Q7<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q7\uff08glissile\u9501\uff09<\/span>Latent hardening coefficient q7 (glissile lock)<\/span><\/td><\/tr>\n <tr><td>110-144<\/td><td>\u9884\u7559\/\u6269\u5c55<\/td><td>Reserved<\/td><td>\u4e3a\u591a\u79cd\u6ed1\u79fb\u7cfb\u3001\u591a\u9636\u6bb5\u786c\u5316\u3001\u6e29\u5ea6\u6548\u5e94\u7b49\u9884\u7559<\/span>Reserved for multiple slip systems, multi-stage hardening, thermal effects, etc.<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch4-6\">\n 4.6 \u79ef\u5206\u63a7\u5236 PROPS(145-160) \/ Integration Control<\/span>\n 4.6 Integration Control PROPS(145-160)<\/span>\n <\/h3>\n \n \u79ef\u5206\u63a7\u5236\u53c2\u6570\u5f71\u54cd\u9690\u5f0fNewton-Raphson\u8fed\u4ee3\u7684\u6570\u503c\u884c\u4e3a\uff0c\u5305\u62ec\u6536\u655b\u5bb9\u5dee\u3001\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u3001\n \u81ea\u52a8\u65f6\u95f4\u6b65\u957f\u63a7\u5236\u7b56\u7565\u7b49\u3002\n <\/p>\n \n Integration control parameters affect the numerical behavior of the implicit Newton-Raphson iteration,\n including convergence tolerance, maximum iteration count, and automatic time step control strategy.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>145<\/td><td>ITMAX<\/td><td>ITMAX<\/td><td>\u6700\u5927Newton-Raphson\u8fed\u4ee3\u6b21\u6570\uff08\u9ed8\u8ba450\uff09<\/span>Maximum Newton-Raphson iterations (default 50)<\/span><\/td><\/tr>\n <tr><td>146<\/td><td>TOL<\/td><td>TOL<\/td><td>\u76f8\u5bf9\u6536\u655b\u5bb9\u5dee\uff08\u9ed8\u8ba41e-6\uff09<\/span>Relative convergence tolerance (default 1e-6)<\/span><\/td><\/tr>\n <tr><td>147<\/td><td>TOL2<\/td><td>TOL2<\/td><td>\u7edd\u5bf9\u6536\u655b\u5bb9\u5dee\uff08\u9ed8\u8ba41e-10\uff09<\/span>Absolute convergence tolerance (default 1e-10)<\/span><\/td><\/tr>\n <tr><td>148<\/td><td>DGMAX<\/td><td>DGMAX<\/td><td>\u5355\u6b65\u6700\u5927\u5141\u8bb8\u526a\u5207\u5e94\u53d8\u589e\u91cf\u9650\u5236<\/span>Maximum allowable shear strain increment per step<\/span><\/td><\/tr>\n <tr><td>149<\/td><td>ITERMOD<\/td><td>ITERMOD<\/td><td>\u8fed\u4ee3\u6a21\u5f0f\uff1a0=\u6807\u51c6NR\uff0c1=\u7ebf\u641c\u7d22\u589e\u5f3a<\/span>Iteration mode: 0=standard NR, 1=line-search enhanced<\/span><\/td><\/tr>\n <tr><td>150<\/td><td>OUTPUT<\/td><td>OUTPUT<\/td><td>\u8f93\u51fa\u63a7\u5236\u6807\u5fd7\uff1a0=\u6700\u5c0f\u8f93\u51fa\uff0c1=\u6807\u51c6\uff0c2=\u8be6\u7ec6\u8c03\u8bd5<\/span>Output control flag: 0=minimal, 1=standard, 2=verbose debug<\/span><\/td><\/tr>\n <tr><td>151-160<\/td><td>\u9884\u7559<\/td><td>Reserved<\/td><td>\u4e3a\u81ea\u9002\u5e94\u6b65\u957f\u63a7\u5236\u3001\u5b50\u6b65\u79ef\u5206\u7b49\u9884\u7559<\/span>Reserved for adaptive step control, sub-stepping, etc.<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch5\">\n <h2>\n 05<\/span>\n \u4ee3\u7801\u7ed3\u6784 \/ Code Structure<\/span>\n Code Structure<\/span>\n <\/h2>\n\n <h3 id=\"ch5-1\">\n 5.1 \u5b50\u7a0b\u5e8f\u8c03\u7528\u5173\u7cfb\u56fe \/ Subroutine Call Graph<\/span>\n 5.1 Subroutine Call Graph<\/span>\n <\/h3>\n \n \u9ec4\u6c38\u521aUMAT\u7531\u4e3b\u7a0b\u5e8fUMAT\u548c\u591a\u4e2a\u8f85\u52a9\u5b50\u7a0b\u5e8f\u7ec4\u6210\uff0c\u5404\u5b50\u7a0b\u5e8f\u627f\u62c5\u660e\u786e\u7684\u804c\u8d23\u3002\n \u4ee5\u4e0b\u4e3a\u8c03\u7528\u5173\u7cfb\u56fe\uff1a\n <\/p>\n \n Huang’s UMAT consists of the main UMAT subroutine and multiple auxiliary subroutines, each with a well-defined responsibility.\n The call graph is as follows:\n <\/p>\n\n <div class=\"mermaid\">\ngraph TD\n A[UMAT<br\/>\u4e3b\u7a0b\u5e8f \/ Main Routine] –> B[ROTATION<br\/>\u8ba1\u7b97\u65cb\u8f6c\u77e9\u9635 \/ Compute Rotation]\n A –> C[SLIPSYS<br\/>\u6ed1\u79fb\u7cfb\u521d\u59cb\u5316 \/ Slip System Init]\n C –> D[GSLPINIT<br\/>\u521d\u59cb\u5316\u6ed1\u79fb\u5f3a\u5ea6 \/ Init Slip Strength]\n D –> E[GSLP0<br\/>\u521d\u59cbCRSS \/ Initial CRSS]\n A –> F[STRAINRATE<br\/>\u8ba1\u7b97\u526a\u5207\u7387 \/ Compute Shear Rate]\n F –> G[F<br\/>\u6d41\u52a8\u51fd\u6570 \/ Flow Function]\n F –> H[DFDX<br\/>\u6d41\u52a8\u51fd\u6570\u5bfc\u6570 \/ Flow Derivative]\n A –> I[LATENTHARDEN<br\/>\u6f5c\u786c\u5316\u8ba1\u7b97 \/ Latent Hardening]\n I –> J[HSELF<br\/>\u81ea\u786c\u5316 \/ Self-Hardening]\n I –> K[HLATNT<br\/>\u6f5c\u786c\u5316\u7cfb\u6570 \/ Latent Coeffs]\n A –> L[ITERATION<br\/>Newton-Raphson\u8fed\u4ee3 \/ NR Iteration]\n L –> M[DHSELF<br\/>\u81ea\u786c\u5316\u5bfc\u6570 \/ Self-Hardening Deriv]\n L –> N[DHLATN<br\/>\u6f5c\u786c\u5316\u5bfc\u6570 \/ Latent Hardening Deriv]\n L –> O[LUDCMP<br\/>LU\u5206\u89e3 \/ LU Decomposition]\n L –> P[LUBKSB<br\/>LU\u56de\u4ee3 \/ LU Back-Subst]\n A –> Q[CROSS<br\/>\u5411\u91cf\u53c9\u4e58 \/ Vector Cross Product]\n style A fill:#2563eb,stroke:#1a2332,color:#fff\n style L fill:#0ea5e9,stroke:#1a2332,color:#fff\n <\/div>\n\n <h3 id=\"ch5-2\">\n 5.2 \u4e3b\u7b97\u6cd5\u6d41\u7a0b\u56fe \/ Main Algorithm Flowchart<\/span>\n 5.2 Main Algorithm Flowchart<\/span>\n <\/h3>\n \n \u6bcf\u4e2a\u589e\u91cf\u6b65\u4e2d\uff0cUMAT\u4e3b\u7a0b\u5e8f\u6267\u884c\u4ee5\u4e0b\u7b97\u6cd5\u6d41\u7a0b\uff1a\n <\/p>\n \n In each increment, the main UMAT routine executes the following algorithm:\n <\/p>\n\n <div class=\"mermaid\">\nflowchart TD\n START([\u5f00\u59cb \/ Start]) –> A[\u8bfb\u53d6\u6750\u6599\u53c2\u6570<br\/>Read PROPS]\n A –> B[\u8bfb\u53d6\u72b6\u6001\u53d8\u91cf<br\/>Read STATEV]\n B –> C[\u8ba1\u7b97\u5f39\u6027\u8bd5\u5e94\u529b<br\/>Elastic Trial Stress]\n C –> D{Newton-Raphson<br\/>\u8fed\u4ee3\u6536\u655b?}\n D — \u5426 \/ No –> E[\u8ba1\u7b97\u6d41\u52a8\u5f8b\u548c\u786c\u5316<br\/>Flow & Hardening]\n E –> F[\u7ec4\u88c5Jacobian<br\/>Assemble Jacobian]\n F –> G[LU\u5206\u89e3\u6c42\u89e3<br\/>LU Decomposition]\n G –> H[\u66f4\u65b0\u526a\u5207\u589e\u91cf<br\/>Update Δγ]\n H –> D\n D — \u662f \/ Yes –> I[\u66f4\u65b0\u5e94\u529b<br\/>Update Stress]\n I –> J[\u66f4\u65b0STATEV<br\/>Update STATEV]\n J –> K[\u8ba1\u7b97DDSDDE<br\/>Compute Consistent Tangent]\n K –> END([\u7ed3\u675f \/ End])\n style D fill:#f59e0b,stroke:#1a2332,color:#fff\n style START fill:#2563eb,stroke:#1a2332,color:#fff\n style END fill:#ef4444,stroke:#1a2332,color:#fff\n <\/div>\n\n <h3 id=\"ch5-3\">\n 5.3 \u5173\u952e\u4ee3\u7801\u6bb5 \/ Key Code Segments<\/span>\n 5.3 Key Code Segments<\/span>\n <\/h3>\n\n <h4>\n 5.3.1 \u5f39\u6027\u8bd5\u5e94\u529b\u8ba1\u7b97 \/ Elastic Trial Stress<\/span>\n 5.3.1 Elastic Trial Stress<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC \u5f39\u6027\u8bd5\u5e94\u529b\u8ba1\u7b97 \/ Elastic trial stress computation\nC \u57fa\u4e8e\u589e\u91cf\u5f00\u59cb\u65f6\u7684Fe\u548c\u6574\u4e2a\u5e94\u53d8\u589e\u91cf\u8ba1\u7b97\u8bd5\u5e94\u529b\nC-----------------------------------------------------------------------\n DO I=1,3\n DO J=1,3\n FE_INV(I,J)=DFGRD1(I,J) ! \u521d\u59cb\u4f30\u8ba1: F^e = F (\u5ffd\u7565\u5851\u6027)\n ENDDO\n ENDDO\nC\nC \u4eceSTATEV\u6062\u590d\u4e0a\u4e00\u589e\u91cf\u6b65\u7684F^p\u548cF^e\n CALL KINTOTAL(STATEV,NSTATV,FE_FP,NSLIP)\nC\nC \u8ba1\u7b97\u8bd5\u5f39\u6027\u53d8\u5f62\u68af\u5ea6: F^{e,trial} = F_{n+1} * (F^p_n)^{-1}\n CALL LUDCMP(FP_TRY,3,3,INDX,D)\n CALL LUBKSB(FP_TRY,3,3,INDX,FE_TRY_COL1)\n CALL LUBKSB(FP_TRY,3,3,INDX,FE_TRY_COL2)\n CALL LUBKSB(FP_TRY,3,3,INDX,FE_TRY_COL3)\nC\nC \u8ba1\u7b97Green\u5f39\u6027\u5e94\u53d8 E^e = 0.5*(F^{eT}F^e - I)\n DO I=1,3\n DO J=1,3\n FE_TF_FE(I,J)=0.0\n DO K=1,3\n FE_TF_FE(I,J)=FE_TF_FE(I,J)+FE_TRY(K,I)*FE_TRY(K,J)\n ENDDO\n ENDDO\n ENDDO\n DO I=1,3\n DO J=1,3\n EE(I,J)=0.5*(FE_TF_FE(I,J)-DELTA(I,J))\n ENDDO\n ENDDO<\/code><\/pre>\n\n <h4>\n 5.3.2 \u6d41\u52a8\u5f8b\u5b9e\u73b0 \/ Flow Law Implementation<\/span>\n 5.3.2 Flow Law Implementation<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC \u7387\u76f8\u5173\u6d41\u52a8\u5f8b\u8ba1\u7b97 \/ Rate-dependent flow rule computation\nC gamma_dot = gamma0 * |tau\/g|^n * sign(tau\/g)\nC-----------------------------------------------------------------------\n SUBROUTINE STRAINRATE(TAU, G, GAMMA0, AM, NSLIP, DGAMMA)\n IMPLICIT NONE\n INTEGER NSLIP, I\n REAL*8 TAU(NSLIP), G(NSLIP), GAMMA0, AM\n REAL*8 DGAMMA(NSLIP), RATIO, ABSRAT, POWER\nC\n DO I=1,NSLIP\n IF (G(I).LE.0.0) THEN\n WRITE(*,*) 'ERROR: g(',I,') =',G(I)\n STOP\n ENDIF\n RATIO = TAU(I)\/G(I)\n ABSRAT = DABS(RATIO)\nC \u907f\u514d\u6570\u503c\u6ea2\u51fa \/ Avoid numerical overflow\n IF (ABSRAT.GT.1.0D+10) THEN\n POWER = 1.0D+10**AM\n ELSE\n POWER = ABSRAT**AM\n ENDIF\nC \u526a\u5207\u5e94\u53d8\u7387 \/ Shear strain rate\n DGAMMA(I) = GAMMA0 * POWER\n IF (RATIO.LT.0.0) DGAMMA(I) = -DGAMMA(I)\n ENDDO\n RETURN\n END<\/code><\/pre>\n\n <h4>\n 5.3.3 \u81ea\u786c\u5316\u5b9e\u73b0 \/ Self-Hardening Implementation<\/span>\n 5.3.3 Self-Hardening Implementation<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC \u53cc\u66f2\u6b63\u5272\u81ea\u786c\u5316\u6a21\u578b \/ Hyperbolic secant self-hardening model\nC h(gamma) = H0 * sech^2( H0*gamma \/ (tau_s - tau_0) )\nC-----------------------------------------------------------------------\n FUNCTION HSELF(GAMMA, H0, TAUS, TAU0)\n IMPLICIT NONE\n REAL*8 HSELF, GAMMA, H0, TAUS, TAU0\n REAL*8 ARG, SECH2\nC\n IF (TAUS.LE.TAU0) THEN\n HSELF = 0.0\n RETURN\n ENDIF\nC\n ARG = H0 * GAMMA \/ (TAUS - TAU0)\nC sech^2(x) = 1\/cosh^2(x)\n SECH2 = 1.0 \/ (DCOSH(ARG)**2)\n HSELF = H0 * SECH2\n RETURN\n END\nC\nC-----------------------------------------------------------------------\nC \u5f53\u524d\u6ed1\u79fb\u7cfb\u5f3a\u5ea6 \/ Current slip system strength\nC g(gamma) = tau_s + (tau_0 - tau_s) * tanh( H0*gamma\/(tau_s-tau_0) )\nC-----------------------------------------------------------------------\n FUNCTION GSLP0(GAMMA, H0, TAUS, TAU0)\n IMPLICIT NONE\n REAL*8 GSLP0, GAMMA, H0, TAUS, TAU0\n REAL*8 ARG\nC\n IF (TAUS.LE.TAU0) THEN\n GSLP0 = TAU0\n RETURN\n ENDIF\nC\n ARG = H0 * GAMMA \/ (TAUS - TAU0)\n GSLP0 = TAUS + (TAU0 - TAUS) * DTANH(ARG)\n RETURN\n END<\/code><\/pre>\n\n <h4>\n 5.3.4 Newton-Raphson\u8fed\u4ee3\u6838\u5fc3 \/ NR Iteration Core<\/span>\n 5.3.4 NR Iteration Core<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC Newton-Raphson\u8fed\u4ee3\u6c42\u89e3\u526a\u5207\u5e94\u53d8\u589e\u91cf \/ NR iteration for shear increment\nC \u672a\u77e5\u91cf: DGA(NSLIP) = Δγ^(α)\nC \u6b8b\u5dee: R(α) = Δγ^(α) - Δt * gamma_dot^{n+1}_(α) = 0\nC-----------------------------------------------------------------------\n SUBROUTINE ITERATION(...)\n IMPLICIT NONE\n INTEGER NSLIP, ITMAX, ITER, I, J, INFO\n REAL*8 TOL, DTIME, DGA(NSLIP), GA(NSLIP), TAU(NSLIP)\n REAL*8 G(NSLIP), DG(NSLIP), GAMMA0, AM\n REAL*8 JAC(NSLIP,NSLIP), RES(NSLIP), DGA_INC(NSLIP)\n REAL*8 DFDX(NSLIP), DHDG(NSLIP,NSLIP)\n REAL*8 DET, RESNORM, RESNORM0\n INTEGER INDX(NSLIP)\nC\n RESNORM0 = 1.0\n DO ITER=1,ITMAX\nC \u8ba1\u7b97\u5f53\u524d\u5e94\u529b\u72b6\u6001 \/ Compute current stress state\n CALL STRESSUPDATE(DGA, TAU, SIGMA, ...)\nC\nC \u8ba1\u7b97\u6d41\u52a8\u5f8b \/ Compute flow rule\n CALL STRAINRATE(TAU, G, GAMMA0, AM, NSLIP, DG)\nC\nC \u8ba1\u7b97\u6b8b\u5dee \/ Compute residual\n DO I=1,NSLIP\n RES(I) = DGA(I) - DTIME * DG(I)\n ENDDO\nC\nC \u8ba1\u7b97\u6b8b\u5dee\u8303\u6570 \/ Compute residual norm\n RESNORM = 0.0\n DO I=1,NSLIP\n RESNORM = RESNORM + RES(I)*RES(I)\n ENDDO\n RESNORM = DSQRT(RESNORM)\n IF (ITER.EQ.1) RESNORM0 = RESNORM\nC\nC \u68c0\u67e5\u6536\u655b \/ Check convergence\n IF (RESNORM.LT.TOL*RESNORM0 .OR. RESNORM.LT.1.0D-12) THEN\n RETURN ! \u6536\u655b \/ Converged\n ENDIF\nC\nC \u8ba1\u7b97Jacobian\u77e9\u9635 \/ Compute Jacobian matrix\nC J_{αβ} = δ_{αβ} - Δt * (∂gamma_dot_α\/∂DGA_β)\n DO I=1,NSLIP\n DO J=1,NSLIP\n JAC(I,J) = -DTIME * DHDG(I,J)\n ENDDO\n JAC(I,I) = JAC(I,I) + 1.0\n ENDDO\nC\nC LU\u5206\u89e3\u5e76\u6c42\u89e3 \/ LU decomposition and solve\n CALL LUDCMP(JAC, NSLIP, NSLIP, INDX, DET)\n DO I=1,NSLIP\n DGA_INC(I) = -RES(I)\n ENDDO\n CALL LUBKSB(JAC, NSLIP, NSLIP, INDX, DGA_INC)\nC\nC \u66f4\u65b0\u672a\u77e5\u91cf \/ Update unknowns\n DO I=1,NSLIP\n DGA(I) = DGA(I) + DGA_INC(I)\n ENDDO\n ENDDO\nC\n WRITE(*,*) 'WARNING: NR iteration not converged in UMAT'\n RETURN\n END<\/code><\/pre>\n\n <h4>\n 5.3.5 DDSDDE\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6 \/ Consistent Tangent DDSDDE<\/span>\n 5.3.5 Consistent Tangent DDSDDE<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC \u8ba1\u7b97\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635 DDSDDE \/ Compute consistent tangent modulus\nC DDSDDE = ∂Δσ \/ ∂Δε\nC-----------------------------------------------------------------------\n SUBROUTINE CONSISTENTTANGENT(...)\n IMPLICIT NONE\n INTEGER NTENS, NSLIP, I, J, K, L\n REAL*8 DDSDDE(NTENS,NTENS), CEL(6,6)\n REAL*8 P_ALPHA(3,3,NSLIP), DGA(NSLIP)\n REAL*8 DTAU_DSIG(NTENS,NSLIP), DHDG(NSLIP,NSLIP)\n REAL*8 JAC(NSLIP,NSLIP), INV_JAC(NSLIP,NSLIP)\n REAL*8 TERM(NTENS,NSLIP), SUM\n INTEGER INDX(NSLIP)\n REAL*8 DET\nC\nC \u5f39\u6027\u521a\u5ea6\u77e9\u9635 \/ Elastic stiffness matrix\n DO I=1,NTENS\n DO J=1,NTENS\n DDSDDE(I,J) = CEL(I,J)\n ENDDO\n ENDDO\nC\nC \u8ba1\u7b97 ∂tau_α\/∂σ \u5e76\u7ec4\u88c5\u4fee\u6b63\u9879\nC Compute ∂tau_α\/∂σ and assemble correction\n DO K=1,NSLIP\n DO I=1,NTENS\n TERM(I,K) = 0.0\n DO J=1,NTENS\n TERM(I,K) = TERM(I,K) + CEL(I,J)*P_ALPHA_VOIGT(J,K)\n ENDDO\n ENDDO\n ENDDO\nC\nC \u8ba1\u7b97 (I - Δt * ∂dg\/∂g * ∂g\/∂DGA)^{-1}\n CALL LUDCMP(JAC, NSLIP, NSLIP, INDX, DET)\n CALL MATINV(JAC, NSLIP, NSLIP, INDX, INV_JAC)\nC\nC DDSDDE = C^e - C^e:P * INV_JAC * DTIME * dg\/dtau * d tau\/d eps\n DO I=1,NTENS\n DO J=1,NTENS\n SUM = 0.0\n DO K=1,NSLIP\n DO L=1,NSLIP\n SUM = SUM + TERM(I,K)*INV_JAC(K,L)*DTAU_DSIG(L,J)\n ENDDO\n ENDDO\n DDSDDE(I,J) = DDSDDE(I,J) - SUM * DTIME\n ENDDO\n ENDDO\n RETURN\n END<\/code><\/pre>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch6\">\n <h2>\n 06<\/span>\n Jacobian\u77e9\u9635 \/ Jacobian Matrix<\/span>\n Jacobian Matrix<\/span>\n <\/h2>\n\n <h3 id=\"ch6-1\">\n 6.1 \u5e94\u53d8\u5206\u89e3 \/ Strain Decomposition<\/span>\n 6.1 Strain Decomposition<\/span>\n <\/h3>\n \n \u5728Abaqus\/Standard\u7684\u9690\u5f0f\u6c42\u89e3\u6846\u67b6\u4e2d\uff0cUMAT\u5fc5\u987b\u63d0\u4f9b\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635DDSDDE\uff0c\n \u5373\u5e94\u529b\u589e\u91cf\u5bf9\u5e94\u53d8\u589e\u91cf\u7684\u504f\u5bfc\u6570DDSDDE = ∂Δσ \/ ∂Δε\u3002\n \u5bf9\u4e8e\u5f39\u5851\u6027\u6750\u6599\uff0c\u603b\u5e94\u53d8\u589e\u91cf\u5206\u89e3\u4e3a\u5f39\u6027\u90e8\u5206\u548c\u5851\u6027\u90e8\u5206\uff1a\n <\/p>\n \n In the Abaqus\/Standard implicit solver framework, UMAT must provide the consistent tangent stiffness matrix DDSDDE,\n i.e., the partial derivative of stress increment with respect to strain increment, DDSDDE = ∂Δσ \/ ∂Δε.\n For elastoplastic materials, the total strain increment is decomposed into elastic and plastic parts:\n <\/p>\n <div class=\"formula\">\n Δε = Δεe<\/sup> + Δεp<\/sup><\/b>\n <\/div>\n \n \u5728\u5c0f\u53d8\u5f62\u5047\u8bbe\u4e0b\uff0c\u5f39\u6027\u5e94\u53d8\u589e\u91cf\u4e0e\u5e94\u529b\u589e\u91cf\u7684\u5173\u7cfb\u7531\u5e7f\u4e49\u80e1\u514b\u5b9a\u5f8b\u7ed9\u51fa\uff1a\n <\/p>\n \n Under the small-deformation assumption, the relationship between elastic strain increment and stress increment is given by the generalized Hooke’s law:\n <\/p>\n <div class=\"formula\">\n Δσ = Ce<\/sup> : Δεe<\/sup> = Ce<\/sup> : (Δε – Δεp<\/sup>)<\/b>\n <\/div>\n \n \u5851\u6027\u5e94\u53d8\u589e\u91cf\u7531\u5404\u6ed1\u79fb\u7cfb\u7684\u526a\u5207\u5e94\u53d8\u589e\u91cf\u8d21\u732e\uff1a\n <\/p>\n \n The plastic strain increment is contributed by the shear strain increments on all slip systems:\n <\/p>\n <div class=\"formula\">\n Δεp<\/sup> = Σα<\/sub> Δγ(α)<\/sup> P(α)<\/sup><\/b>\n <\/div>\n \n \u56e0\u6b64\uff0c\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635\u7684\u5173\u952e\u5728\u4e8e\u8ba1\u7b97 ∂Δγ(α)<\/sup> \/ ∂Δε\uff0c\n \u5373\u526a\u5207\u5e94\u53d8\u589e\u91cf\u5bf9\u5e94\u53d8\u589e\u91cf\u7684\u654f\u611f\u5ea6\u3002\n <\/p>\n \n Therefore, the key to the consistent tangent stiffness matrix is computing ∂Δγ(α)<\/sup> \/ ∂Δε,\n i.e., the sensitivity of shear strain increment to strain increment.\n <\/p>\n\n <h3 id=\"ch6-2\">\n 6.2 \u7ebf\u6027\u65b9\u7a0b\u7ec4 \/ Linear System<\/span>\n 6.2 Linear System<\/span>\n <\/h3>\n \n \u5728Newton-Raphson\u8fed\u4ee3\u4e2d\uff0c\u6bcf\u4e2a\u6ed1\u79fb\u7cfb\u7684\u526a\u5207\u5e94\u53d8\u589e\u91cf\u6ee1\u8db3\u4ee5\u4e0b\u975e\u7ebf\u6027\u65b9\u7a0b\uff1a\n <\/p>\n \n In the Newton-Raphson iteration, the shear strain increment for each slip system satisfies the following nonlinear equation:\n <\/p>\n <div class=\"formula\">\n R(α)<\/sup> = Δγ(α)<\/sup> – Δt · γ̇0<\/sub> · | τ(α)<\/sup> \/ g(α)<\/sup> |n<\/sup> · sign(τ(α)<\/sup>\/g(α)<\/sup>) = 0<\/b>\n <\/div>\n \n \u5bf9\u4e0a\u8ff0\u65b9\u7a0b\u5173\u4e8eΔγ(β)<\/sup>\u6c42\u504f\u5bfc\uff0c\u5f97\u5230Newton-Raphson\u8fed\u4ee3\u7684Jacobian\u77e9\u9635\uff08\u5c40\u90e8Newton\u8fed\u4ee3\u77e9\u9635\uff09\uff1a\n <\/p>\n \n Taking the partial derivative of the above equation with respect to Δγ(β)<\/sup> yields the Jacobian matrix for Newton-Raphson iteration (local Newton iteration matrix):\n <\/p>\n <div class=\"formula\">\n Jαβ<\/sub> = ∂R(α)<\/sup> \/ ∂Δγ(β)<\/sup> = δαβ<\/sub> – Δt · (∂γ̇(α)<\/sup> \/ ∂τ(α)<\/sup>) · (∂τ(α)<\/sup> \/ ∂Δγ(β)<\/sup>) – Δt · (∂γ̇(α)<\/sup> \/ ∂g(α)<\/sup>) · (∂g(α)<\/sup> \/ ∂Δγ(β)<\/sup>)<\/b>\n <\/div>\n \n \u5404\u9879\u504f\u5bfc\u6570\u7684\u7269\u7406\u542b\u4e49\u548c\u8868\u8fbe\u5f0f\u5982\u4e0b\uff1a\n <\/p>\n \n The physical meaning and expressions of each partial derivative are:\n <\/p>\n <ul>\n <li>\n ∂γ̇(α)<\/sup> \/ ∂τ(α)<\/sup><\/b>:\n \u6d41\u52a8\u5f8b\u5bf9\u5206\u89e3\u526a\u5e94\u529b\u7684\u654f\u611f\u5ea6\uff0c\u7b49\u4e8e n · γ̇(α)<\/sup> \/ τ(α)<\/sup><\/span>\n Sensitivity of flow rule to resolved shear stress, equal to n · γ̇(α)<\/sup> \/ τ(α)<\/sup><\/span>\n <\/li>\n <li>\n ∂τ(α)<\/sup> \/ ∂Δγ(β)<\/sup><\/b>:\n \u5206\u89e3\u526a\u5e94\u529b\u5bf9\u526a\u5207\u589e\u91cf\u7684\u654f\u611f\u5ea6\uff0c\u7b49\u4e8e -Ce<\/sup> : P(β)<\/sup> : P(α)<\/sup>\uff08\u5e94\u529b\u677e\u5f1b\u6548\u5e94\uff09<\/span>\n Sensitivity of resolved shear stress to shear increment, equal to -Ce<\/sup> : P(β)<\/sup> : P(α)<\/sup> (stress relaxation effect)<\/span>\n <\/li>\n <li>\n ∂g(α)<\/sup> \/ ∂Δγ(β)<\/sup><\/b>:\n \u6ed1\u79fb\u7cfb\u5f3a\u5ea6\u5bf9\u526a\u5207\u589e\u91cf\u7684\u654f\u611f\u5ea6\uff0c\u7b49\u4e8e hαβ<\/sub> · sign(τ(β)<\/sup>)<\/span>\n Sensitivity of slip system strength to shear increment, equal to hαβ<\/sub> · sign(τ(β)<\/sup>)<\/span>\n <\/li>\n <li>\n ∂γ̇(α)<\/sup> \/ ∂g(α)<\/sup><\/b>:\n \u6d41\u52a8\u5f8b\u5bf9\u5f3a\u5ea6\u7684\u654f\u611f\u5ea6\uff0c\u7b49\u4e8e -n · γ̇(α)<\/sup> \/ g(α)<\/sup><\/span>\n Sensitivity of flow rule to strength, equal to -n · γ̇(α)<\/sup> \/ g(α)<\/sup><\/span>\n <\/li>\n <\/ul>\n \n \u5c06\u6240\u6709\u504f\u5bfc\u6570\u4ee3\u5165\u540e\uff0c\u5c40\u90e8Newton\u77e9\u9635\u53ef\u663e\u5f0f\u5199\u51fa\u3002\u8be5\u77e9\u9635\u7684\u5927\u5c0f\u4e3aNSLIP × NSLIP\uff08\u4f8b\u5982FCC\u4e3a12×12\uff09\u3002\n \u5728\u6bcf\u6b21Newton\u8fed\u4ee3\u4e2d\uff0c\u5bf9\u8be5\u77e9\u9635\u8fdb\u884cLU\u5206\u89e3\u5e76\u56de\u4ee3\u6c42\u89e3\u526a\u5207\u5e94\u53d8\u589e\u91cf\u4fee\u6b63\u503cΔDGA\u3002\n <\/p>\n \n After substituting all partial derivatives, the local Newton matrix can be written explicitly. Its size is NSLIP × NSLIP (e.g., 12×12 for FCC).\n In each Newton iteration, this matrix undergoes LU decomposition and back-substitution to solve for the shear strain increment correction ΔDGA.\n <\/p>\n\n <h3 id=\"ch6-3\">\n 6.3 DDSDDE\u6784\u9020 \/ DDSDDE Construction<\/span>\n 6.3 DDSDDE Construction<\/span>\n <\/h3>\n \n \u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635\u7684\u6784\u9020\u516c\u5f0f\u4e3a\uff1a\n <\/p>\n \n The consistent tangent stiffness matrix is constructed as:\n <\/p>\n <div class=\"formula\">\n DDSDDE = Ce<\/sup> – Ce<\/sup> : P(α)<\/sup> ⊗ [ J-1<\/sup>αβ<\/sub> · Δt · (∂γ̇(β)<\/sup> \/ ∂τ(β)<\/sup>) · P(β)<\/sup> : Ce<\/sup> ]<\/b>\n <\/div>\n \n \u4e0a\u8ff0\u516c\u5f0f\u4e2d\uff0c\u7b2c\u4e00\u9879Ce<\/sup>\u662f\u7eaf\u5f39\u6027\u521a\u5ea6\uff0c\u7b2c\u4e8c\u9879\u662f\u5851\u6027\u4fee\u6b63\u3002\n \u5f53\u6750\u6599\u5904\u4e8e\u7eaf\u5f39\u6027\u72b6\u6001\uff08\u65e0\u6ed1\u79fb\u6fc0\u6d3b\uff09\u65f6\uff0c\u7b2c\u4e8c\u9879\u4e3a\u96f6\uff0cDDSDDE\u9000\u5316\u4e3a\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u3002\n \u5f53\u5927\u91cf\u6ed1\u79fb\u7cfb\u6fc0\u6d3b\u65f6\uff0c\u7b2c\u4e8c\u9879\u663e\u8457\u964d\u4f4e\u6750\u6599\u7684\u8868\u89c2\u521a\u5ea6\uff0c\u53cd\u6620\u5851\u6027\u6d41\u52a8\u5f15\u8d77\u7684\u5e94\u529b\u677e\u5f1b\u3002\n <\/p>\n \n In the above formula, the first term Ce<\/sup> is the pure elastic stiffness, and the second term is the plastic correction.\n When the material is in a purely elastic state (no slip activated), the second term vanishes and DDSDDE reduces to the elastic stiffness matrix.\n When many slip systems are active, the second term significantly reduces the apparent material stiffness, reflecting stress relaxation due to plastic flow.\n <\/p>\n \n \u5177\u4f53\u8ba1\u7b97\u6b65\u9aa4\u5982\u4e0b\uff1a\n <\/p>\n \n The specific calculation steps are:\n <\/p>\n <ol>\n <li>\n \u8ba1\u7b97\u5f39\u6027\u521a\u5ea6\u77e9\u9635Ce<\/sup>\u7684Voigt\u5f62\u5f0f\uff086×6\u77e9\u9635\uff09<\/span>\n Compute the Voigt form of elastic stiffness matrix Ce<\/sup> (6×6 matrix)<\/span>\n <\/li>\n <li>\n \u8ba1\u7b97\u6bcf\u4e2a\u6ed1\u79fb\u7cfb\u7684Schmid\u5f20\u91cfP(α)<\/sup>\u7684Voigt\u5411\u91cf\u5f62\u5f0f<\/span>\n Compute the Voigt vector form of Schmid tensor P(α)<\/sup> for each slip system<\/span>\n <\/li>\n <li>\n \u8ba1\u7b97\u6d41\u52a8\u5f8b\u5bfc\u6570 ∂γ̇(α)<\/sup> \/ ∂τ(α)<\/sup> = n · γ̇(α)<\/sup> \/ τ(α)<\/sup><\/span>\n Compute flow rule derivative ∂γ̇(α)<\/sup> \/ ∂τ(α)<\/sup> = n · γ̇(α)<\/sup> \/ τ(α)<\/sup><\/span>\n <\/li>\n <li>\n \u7ec4\u88c5\u5c40\u90e8Newton\u77e9\u9635Jαβ<\/sub>\u5e76\u8fdb\u884cLU\u5206\u89e3\u6c42\u9006<\/span>\n Assemble local Newton matrix Jαβ<\/sub> and perform LU decomposition for inversion<\/span>\n <\/li>\n <li>\n \u8ba1\u7b97\u4e2d\u95f4\u5f20\u91cf\uff1a\u5bf9\u6bcf\u4e2a\u6ed1\u79fb\u7cfbα\uff0c\u8ba1\u7b97 Q(α)<\/sup>ij<\/sub> = (J-1<\/sup>αβ<\/sub> · DFLOWβ<\/sub>) · (P(β)<\/sup>kl<\/sub> Ce<\/sup>klmn<\/sub>)<\/span>\n Compute intermediate tensor: for each slip system α, compute Q(α)<\/sup>ij<\/sub> = (J-1<\/sup>αβ<\/sub> · DFLOWβ<\/sub>) · (P(β)<\/sup>kl<\/sub> Ce<\/sup>klmn<\/sub>)<\/span>\n <\/li>\n <li>\n \u7ec4\u88c5DDSDDE\uff1aDDSDDEij<\/sub> = Ce<\/sup>ij<\/sub> – Σα<\/sub> (Ce<\/sup>ijkl<\/sub> P(α)<\/sup>kl<\/sub>) Q(α)<\/sup>mn<\/sub><\/span>\n Assemble DDSDDE: DDSDDEij<\/sub> = Ce<\/sup>ij<\/sub> – Σα<\/sub> (Ce<\/sup>ijkl<\/sub> P(α)<\/sup>kl<\/sub>) Q(α)<\/sup>mn<\/sub><\/span>\n <\/li>\n <\/ol>\n <div class=\"note\">\n \u6570\u503c\u7a33\u5b9a\u6027\u63d0\u793a\uff1a<\/strong>\u5f53τ(α)<\/sup>\u63a5\u8fd1\u96f6\u65f6\uff0c\u6d41\u52a8\u5f8b\u5bfc\u6570n·γ̇\/τ\u53ef\u80fd\u5947\u5f02\u3002\u7a0b\u5e8f\u4e2d\u901a\u5e38\u5f15\u5165\u5fae\u5c0f\u6b63\u5219\u5316\u9879\u6216\u9650\u5236\u6700\u5927\u5bfc\u6570\u503c\u3002\u6b64\u5916\uff0c\u5f53\u67d0\u4e9b\u6ed1\u79fb\u7cfb\u7684\u526a\u5207\u7387\u6781\u5c0f\u65f6\uff0c\u53ef\u5c06\u5176\u89c6\u4e3a”\u975e\u6fc0\u6d3b”\u5e76\u4ece\u5c40\u90e8Newton\u7cfb\u7edf\u4e2d\u6d88\u53bb\uff0c\u4ee5\u51cf\u5c0f\u77e9\u9635\u89c4\u6a21\u5e76\u6539\u5584\u6761\u4ef6\u6570\u3002<\/span>\n Numerical stability tip:<\/strong> When τ(α)<\/sup> is near zero, the flow rule derivative n·γ̇\/τ may become singular. The program usually introduces a small regularization term or caps the maximum derivative value. Additionally, slip systems with extremely small shear rates can be treated as “inactive” and eliminated from the local Newton system to reduce matrix size and improve conditioning.<\/span>\n <\/div>\n \n DDSDDE\u7684\u5bf9\u79f0\u6027\u53d6\u51b3\u4e8e\u6750\u6599\u6a21\u578b\u3002\u5bf9\u4e8e\u5173\u8054\u6d41\u52a8\u6cd5\u5219\u548c\u4e8c\u6b21\u5c48\u670d\u9762\uff0cDDSDDE\u662f\u5bf9\u79f0\u7684\uff1b\n \u4f46\u5bf9\u4e8e\u975e\u5173\u8054\u6d41\u52a8\u6216\u590d\u6742\u7684\u6f5c\u786c\u5316\u6a21\u578b\uff0cDDSDDE\u53ef\u80fd\u975e\u5bf9\u79f0\u3002\n \u9ec4\u6c38\u521aUMAT\u4e2d\uff0c\u7531\u4e8e\u91c7\u7528\u5173\u8054\u7684Schmid\u6d41\u52a8\u548c\u4e00\u822c\u7684\u786c\u5316\u77e9\u9635\uff0cDDSDDE\u901a\u5e38\u5177\u6709\u8f7b\u5fae\u7684\u975e\u5bf9\u79f0\u6027\u3002\n Abaqus\/Standard\u652f\u6301\u975e\u5bf9\u79f0\u521a\u5ea6\u77e9\u9635\uff0c\u4f46\u9700\u8981\u5728*Step\u4e2d\u8bbe\u7f6eUNSymm=YES\u4ee5\u4f18\u5316\u6c42\u89e3\u5668\u6027\u80fd\u3002\n <\/p>\n \n The symmetry of DDSDDE depends on the material model. For associative flow rules and quadratic yield surfaces, DDSDDE is symmetric;\n but for non-associative flow or complex latent hardening models, DDSDDE may be non-symmetric.\n In Huang’s UMAT, due to the associative Schmid flow and general hardening matrix, DDSDDE usually has slight non-symmetry.\n Abaqus\/Standard supports non-symmetric stiffness matrices, but *Step requires UNSymm=YES to optimize solver performance.\n <\/p>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch7\">\n <h2>\n 07<\/span>\n \u5178\u578b\u53c2\u6570 \/ Typical Parameters<\/span>\n Typical Parameters<\/span>\n <\/h2>\n\n \n \u4ee5\u4e0b\u7ed9\u51faFCC\u94dc\u548cFCC\u94dd\u7684\u5178\u578b\u6750\u6599\u53c2\u6570\uff0c\u8fd9\u4e9b\u53c2\u6570\u5728\u5927\u91cf\u6587\u732e\u548c\u5b9e\u9a8c\u4e2d\u5f97\u5230\u9a8c\u8bc1\uff0c\n \u53ef\u4f5c\u4e3aAbaqus\u8f93\u5165\u6587\u4ef6\u4e2d*User Material\u7684\u53c2\u8003\u503c\u3002\n <\/p>\n \n The following presents typical material parameters for FCC copper and FCC aluminum,\n validated in numerous literature and experiments, serving as reference values for the *User Material keyword in Abaqus input files.\n <\/p>\n\n <h3 id=\"ch7-1\">\n 7.1 FCC\u94dc \/ FCC Copper<\/span>\n 7.1 FCC Copper<\/span>\n <\/h3>\n \n \u94dc\u662fFCC\u7ed3\u6784\uff0c\u5ba4\u6e29\u4e0b\u4e3b\u8981\u6ed1\u79fb\u7cfb\u4e3a{111}<110>\uff0c\u517112\u4e2a\u3002\n \u4ee5\u4e0b\u53c2\u6570\u9002\u7528\u4e8e\u5ba4\u6e29\uff08\u7ea6300K\uff09\u4e0b\u7684\u5355\u6676\u94dc\u6a21\u62df\uff0c\u91c7\u7528\u53cc\u66f2\u6b63\u5272\u786c\u5316\u6a21\u578b\u3002\n <\/p>\n \n Copper has an FCC structure. At room temperature, the primary slip systems are {111}<110>, totaling 12.\n The following parameters are suitable for single-crystal copper simulation at room temperature (~300K) using the hyperbolic secant hardening model.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>\n \u53c2\u6570 \/ Parameter<\/span>\n Parameter<\/span>\n <\/th>\n <th>\n \u6570\u503c \/ Value<\/span>\n Value<\/span>\n <\/th>\n <th>\n \u5355\u4f4d \/ Unit<\/span>\n Unit<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>C11<\/td>\n <td>168,400<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C11<\/span>Elastic constant C11<\/span><\/td>\n <\/tr>\n <tr>\n <td>C12<\/td>\n <td>121,400<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C12<\/span>Elastic constant C12<\/span><\/td>\n <\/tr>\n <tr>\n <td>C44<\/td>\n <td>75,400<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C44<\/span>Elastic constant C44<\/span><\/td>\n <\/tr>\n <tr>\n <td>γ̇0<\/sub><\/td>\n <td>0.001<\/td>\n <td>s-1<\/sup><\/td>\n <td>\u53c2\u8003\u526a\u5207\u7387<\/span>Reference shear rate<\/span><\/td>\n <\/tr>\n <tr>\n <td>n<\/td>\n <td>50<\/td>\n <td>–<\/td>\n <td>\u7387\u654f\u611f\u6307\u6570<\/span>Rate sensitivity exponent<\/span><\/td>\n <\/tr>\n <tr>\n <td>τ0<\/sub><\/td>\n <td>16.0<\/td>\n <td>MPa<\/td>\n <td>\u521d\u59cbCRSS<\/span>Initial CRSS<\/span><\/td>\n <\/tr>\n <tr>\n <td>τs<\/sub><\/td>\n <td>148.0<\/td>\n <td>MPa<\/td>\n <td>\u9971\u548cCRSS<\/span>Saturation CRSS<\/span><\/td>\n <\/tr>\n <tr>\n <td>H0<\/sub><\/td>\n <td>180.0<\/td>\n <td>MPa<\/td>\n <td>\u521d\u59cb\u786c\u5316\u6a21\u91cf<\/span>Initial hardening modulus<\/span><\/td>\n <\/tr>\n <tr>\n <td>q<\/td>\n <td>1.0<\/td>\n <td>–<\/td>\n <td>\u6f5c\u786c\u5316\u7cfb\u6570\uff08\u5404\u5411\u540c\u6027\u8fd1\u4f3c\uff09<\/span>Latent hardening coefficient (isotropic approximation)<\/span><\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch7-2\">\n 7.2 FCC\u94dd \/ FCC Aluminum<\/span>\n 7.2 FCC Aluminum<\/span>\n <\/h3>\n \n \u94dd\u540c\u6837\u662fFCC\u7ed3\u6784\uff0c\u4f46\u5176\u5c42\u9519\u80fd\u8f83\u9ad8\uff0c\u4f4d\u9519\u4ea4\u6ed1\u79fb\u66f4\u5bb9\u6613\u53d1\u751f\uff0c\u56e0\u6b64\u786c\u5316\u884c\u4e3a\u548c\u94dc\u6709\u660e\u663e\u5dee\u5f02\u3002\n \u4ee5\u4e0b\u53c2\u6570\u9002\u7528\u4e8e\u9ad8\u7eaf\u94dd\u5355\u6676\u5728\u5ba4\u6e29\u4e0b\u7684\u6a21\u62df\u3002\n <\/p>\n \n Aluminum is also FCC, but with higher stacking fault energy, making dislocation cross-slip easier,\n so its hardening behavior differs significantly from copper.\n The following parameters are suitable for high-purity single-crystal aluminum at room temperature.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>\n \u53c2\u6570 \/ Parameter<\/span>\n Parameter<\/span>\n <\/th>\n <th>\n \u6570\u503c \/ Value<\/span>\n Value<\/span>\n <\/th>\n <th>\n \u5355\u4f4d \/ Unit<\/span>\n Unit<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>C11<\/td>\n <td>106,750<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C11<\/span>Elastic constant C11<\/span><\/td>\n <\/tr>\n <tr>\n <td>C12<\/td>\n <td>60,410<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C12<\/span>Elastic constant C12<\/span><\/td>\n <\/tr>\n <tr>\n <td>C44<\/td>\n <td>28,340<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C44<\/span>Elastic constant C44<\/span><\/td>\n <\/tr>\n <tr>\n <td>γ̇0<\/sub><\/td>\n <td>0.001<\/td>\n <td>s-1<\/sup><\/td>\n <td>\u53c2\u8003\u526a\u5207\u7387<\/span>Reference shear rate<\/span><\/td>\n <\/tr>\n <tr>\n <td>n<\/td>\n <td>60<\/td>\n <td>–<\/td>\n <td>\u7387\u654f\u611f\u6307\u6570<\/span>Rate sensitivity exponent<\/span><\/td>\n <\/tr>\n <tr>\n <td>τ0<\/sub><\/td>\n <td>0.75<\/td>\n <td>MPa<\/td>\n <td>\u521d\u59cbCRSS<\/span>Initial CRSS<\/span><\/td>\n <\/tr>\n <tr>\n <td>τs<\/sub><\/td>\n <td>25.0<\/td>\n <td>MPa<\/td>\n <td>\u9971\u548cCRSS<\/span>Saturation CRSS<\/span><\/td>\n <\/tr>\n <tr>\n <td>H0<\/sub><\/td>\n <td>30.0<\/td>\n <td>MPa<\/td>\n <td>\u521d\u59cb\u786c\u5316\u6a21\u91cf<\/span>Initial hardening modulus<\/span><\/td>\n <\/tr>\n <tr>\n <td>q<\/td>\n <td>1.0<\/td>\n <td>–<\/td>\n <td>\u6f5c\u786c\u5316\u7cfb\u6570\uff08\u5404\u5411\u540c\u6027\u8fd1\u4f3c\uff09<\/span>Latent hardening coefficient (isotropic approximation)<\/span><\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <div class=\"note\">\n \u53c2\u6570\u6765\u6e90\u8bf4\u660e\uff1a<\/strong>\u4e0a\u8ff0\u53c2\u6570\u7efc\u5408\u4e86\u591a\u9879\u7ecf\u5178\u5b9e\u9a8c\u548c\u6587\u732e\u6570\u636e\u3002\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5e94\u6839\u636e\u5177\u4f53\u5408\u91d1\u6210\u5206\u3001\u6e29\u5ea6\u3001\u5e94\u53d8\u901f\u7387\u7b49\u6761\u4ef6\u901a\u8fc7\u5355\u6676\u62c9\u4f38\u5b9e\u9a8c\u6821\u51c6\u3002\u7279\u522b\u5730\uff0c\u7387\u654f\u611f\u6307\u6570n\u548c\u53c2\u8003\u526a\u5207\u7387γ̇0<\/sub>\u5e94\u5339\u914d\u5b9e\u9a8c\u5e94\u53d8\u901f\u7387\uff0c\u5426\u5219\u53ef\u80fd\u5bfc\u81f4\u5e94\u529b\u6c34\u5e73\u9884\u6d4b\u504f\u5dee\u3002<\/span>\n Parameter source note:<\/strong> The above parameters are synthesized from multiple classic experiments and literature data. In practical applications, they should be calibrated through single-crystal tensile experiments according to specific alloy composition, temperature, strain rate, etc. In particular, the rate sensitivity exponent n and reference shear rate γ̇0<\/sub> should match the experimental strain rate; otherwise, stress level prediction errors may occur.<\/span>\n <\/div>\n<\/div>\n\n<div class=\"chapter\" id=\"ch8\">\n <h2>\n 08<\/span>\n \u4f7f\u7528\u6ce8\u610f\u4e8b\u9879 \/ Usage Notes<\/span>\n Usage Notes<\/span>\n <\/h2>\n\n \n \u4ee5\u4e0b\u5217\u51fa\u4f7f\u7528\u9ec4\u6c38\u521aUMAT\u5b50\u7a0b\u5e8f\u65f6\u5e38\u89c1\u7684\u6ce8\u610f\u4e8b\u9879\u548c\u6700\u4f73\u5b9e\u8df5\u5efa\u8bae\uff0c\n \u6db5\u76d6\u8f93\u5165\u6587\u4ef6\u8bbe\u7f6e\u3001\u6750\u6599\u53c2\u6570\u6821\u51c6\u3001\u6536\u655b\u6027\u8c03\u8bd5\u548c\u540e\u5904\u7406\u7b49\u65b9\u9762\u3002\n <\/p>\n \n The following lists common considerations and best-practice recommendations when using Huang’s UMAT subroutine,\n covering input file setup, material parameter calibration, convergence debugging, and post-processing.\n <\/p>\n\n <h3 id=\"ch8-1\">\n 8.1 Abaqus\u8f93\u5165\u6587\u4ef6\u8bbe\u7f6e \/ Input File Setup<\/span>\n 8.1 Input File Setup<\/span>\n <\/h3>\n <ul>\n <li>\n *User Material\u5173\u952e\u5b57\uff1a<\/strong>\u5fc5\u987b\u5728\u8f93\u5165\u6587\u4ef6\u4e2d\u901a\u8fc7*User Material, constants=NPROPS\u5b9a\u4e49PROPS\u6570\u7ec4\u957f\u5ea6\uff0c\u5e76\u786e\u4fdd*Depvar\u7684NSTATV\u8db3\u591f\u5927\u4ee5\u5bb9\u7eb3\u6240\u6709\u72b6\u6001\u53d8\u91cf\u3002\u5bf9\u4e8eFCC 12\u6ed1\u79fb\u7cfb\uff0c\u5efa\u8baeNSTATV >= 70\u3002<\/span>\n *User Material keyword:<\/strong> The PROPS array length must be defined via *User Material, constants=NPROPS, and *Depvar’s NSTATV must be large enough for all state variables. For FCC 12 slip systems, NSTATV >= 70 is recommended.<\/span>\n <\/li>\n <li>\n UNSymm\u9009\u9879\uff1a<\/strong>\u7531\u4e8e\u6676\u4f53\u5851\u6027\u7684\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635\u901a\u5e38\u8f7b\u5fae\u975e\u5bf9\u79f0\uff0c\u5efa\u8bae\u5728*Step\u4e2d\u4f7f\u7528UNSymm=YES\u4ee5\u542f\u7528\u975e\u5bf9\u79f0\u6c42\u89e3\u5668\uff0c\u63d0\u9ad8\u6536\u655b\u901f\u5ea6\u548c\u7a33\u5b9a\u6027\u3002<\/span>\n UNSymm option:<\/strong> Because the consistent tangent stiffness matrix of crystal plasticity is usually slightly non-symmetric, it is recommended to use UNSymm=YES in *Step to enable the non-symmetric solver, improving convergence speed and stability.<\/span>\n <\/li>\n <li>\n \u65f6\u95f4\u6b65\u957f\u63a7\u5236\uff1a<\/strong>\u6676\u4f53\u5851\u6027UMAT\u5bf9\u65f6\u95f4\u6b65\u957f\u654f\u611f\u3002\u5efa\u8bae\u5728\u521d\u59cb\u52a0\u8f7d\u9636\u6bb5\u4f7f\u7528\u8f83\u5c0f\u7684\u65f6\u95f4\u6b65\uff0c\u5f85\u5851\u6027\u53d8\u5f62\u7a33\u5b9a\u540e\u53ef\u9002\u5ea6\u589e\u5927\u3002\u53ef\u901a\u8fc7*Static\u4e2d\u7684INITIAL\u548cMAXIMUM\u53c2\u6570\u63a7\u5236\u3002<\/span>\n Time step control:<\/strong> Crystal plasticity UMAT is sensitive to time step size. It is recommended to use small time steps in the initial loading stage, and moderately increase after plastic deformation stabilizes. Control via INITIAL and MAXIMUM in *Static.<\/span>\n <\/li>\n <li>\n \u5355\u5143\u7c7b\u578b\u9009\u62e9\uff1a<\/strong>\u63a8\u8350\u4f7f\u7528C3D8R\uff088\u8282\u70b9\u516d\u9762\u4f53\u51cf\u7f29\u79ef\u5206\u5355\u5143\uff09\u6216C3D8\uff08\u5b8c\u5168\u79ef\u5206\uff09\u3002\u907f\u514d\u4f7f\u7528\u7ebf\u6027\u4e09\u89d2\u5f62\/\u56db\u9762\u4f53\u5355\u5143\u6a21\u62df\u5927\u5e94\u53d8\u5c40\u90e8\u5316\uff0c\u56e0\u4e3a\u5b83\u4eec\u5728\u4e0d\u53ef\u538b\u7f29\u5851\u6027\u53d8\u5f62\u4e2d\u4f1a\u4ea7\u751f\u4f53\u79ef\u9501\u5b9a\u3002<\/span>\n Element type selection:<\/strong> C3D8R (8-node hexahedron with reduced integration) or C3D8 (full integration) are recommended. Avoid linear triangular\/tetrahedral elements for large-strain localization, as they suffer volumetric locking in incompressible plastic deformation.<\/span>\n <\/li>\n <li>\n \u6c99\u6f0f\u63a7\u5236\uff1a<\/strong>\u4f7f\u7528C3D8R\u5355\u5143\u65f6\uff0c\u5fc5\u987b\u914d\u5408*Hourglass Stiffness\u6216*Section Controls\u8bbe\u7f6e\u5408\u7406\u7684\u6c99\u6f0f\u521a\u5ea6\uff0c\u4ee5\u6291\u5236\u96f6\u80fd\u6a21\u5f0f\u3002<\/span>\n Hourglass control:<\/strong> When using C3D8R elements, *Hourglass Stiffness or *Section Controls must be set with reasonable hourglass stiffness to suppress zero-energy modes.<\/span>\n <\/li>\n <\/ul>\n\n <h3 id=\"ch8-2\">\n 8.2 \u6750\u6599\u53c2\u6570\u6821\u51c6 \/ Parameter Calibration<\/span>\n 8.2 Parameter Calibration<\/span>\n <\/h3>\n <ul>\n <li>\n \u5f39\u6027\u5e38\u6570\uff1a<\/strong>C11\u3001C12\u3001C44\u5e94\u4ece\u8d85\u58f0\u5171\u632f\u6216\u5355\u6676\u5f39\u6027\u5b9e\u9a8c\u83b7\u53d6\u3002\u6ce8\u610fAbaqus\u4e2d\u5e94\u529b\u5355\u4f4d\u4e3aMPa\uff0c\u5f39\u6027\u5e38\u6570\u4e5f\u5fc5\u987b\u4ee5MPa\u8f93\u5165\u3002<\/span>\n Elastic constants:<\/strong> C11, C12, C44 should be obtained from ultrasonic resonance or single-crystal elastic experiments. Note that Abaqus uses MPa for stress, so elastic constants must also be entered in MPa.<\/span>\n <\/li>\n <li>\n \u521d\u59cbCRSS tau0\uff1a<\/strong>\u53ef\u901a\u8fc7\u5355\u6676\u521d\u59cb\u5c48\u670d\u5e94\u529b\u7ed3\u5408Schmid\u56e0\u5b50\u4f30\u7b97\u3002\u5bf9\u4e8e\u4e0d\u540c\u53d6\u5411\u7684\u5355\u6676\u8bd5\u6837\uff0ctau0\u5e94\u552f\u4e00\uff0c\u800c\u5b8f\u89c2\u5c48\u670d\u5e94\u529b\u968f\u53d6\u5411\u53d8\u5316\u3002<\/span>\n Initial CRSS tau0:<\/strong> Can be estimated from single-crystal initial yield stress combined with the Schmid factor. For single-crystal specimens of different orientations, tau0 should be unique, while macroscopic yield stress varies with orientation.<\/span>\n <\/li>\n <li>\n \u7387\u654f\u611f\u6307\u6570n\uff1a<\/strong>\u5e94\u901a\u8fc7\u4e0d\u540c\u5e94\u53d8\u901f\u7387\u4e0b\u7684\u5355\u6676\u62c9\u4f38\u5b9e\u9a8c\u786e\u5b9a\u3002n\u503c\u5bf9\u6d41\u52a8\u5e94\u529b\u6c34\u5e73\u6709\u663e\u8457\u5f71\u54cd\uff0c\u4f46\u5bf9\u786c\u5316\u6a21\u91cf\u5f71\u54cd\u8f83\u5c0f\u3002\u82e5\u5b9e\u9a8c\u6570\u636e\u4e0d\u8db3\uff0c\u53ef\u53c2\u8003\u6587\u732e\u4e2d\u540c\u79cd\u6750\u6599\u7684\u63a8\u8350\u503c\u3002<\/span>\n Rate sensitivity exponent n:<\/strong> Should be determined from single-crystal tensile experiments at different strain rates. n significantly affects flow stress level but has minor effect on hardening modulus. If experimental data is insufficient, refer to recommended values in the literature for the same material.<\/span>\n <\/li>\n <li>\n \u786c\u5316\u53c2\u6570\uff1a<\/strong>H0\u3001taus\u548ctau0\u5e94\u901a\u8fc7\u5355\u6676\u5e94\u529b-\u5e94\u53d8\u66f2\u7ebf\u7684\u5851\u6027\u6bb5\u62df\u5408\u786e\u5b9a\u3002\u5efa\u8bae\u4f7f\u7528\u975e\u7ebf\u6027\u6700\u5c0f\u4e8c\u4e58\u6cd5\u5bf9\u5b9e\u9a8c\u6570\u636e\u8fdb\u884c\u4f18\u5316\u62df\u5408\u3002<\/span>\n Hardening parameters:<\/strong> H0, taus, and tau0 should be determined by fitting the plastic portion of single-crystal stress-strain curves. Nonlinear least-squares optimization is recommended for fitting experimental data.<\/span>\n <\/li>\n <li>\n \u6f5c\u786c\u5316\u7cfb\u6570q\uff1a<\/strong>\u5404\u5411\u540c\u6027\u8fd1\u4f3c\u4e0b\u53ef\u53d6q=1.0\u3002\u82e5\u8981\u66f4\u7cbe\u786e\u63cf\u8ff0\uff0c\u9700\u8981\u901a\u8fc7\u53cc\u6ed1\u79fb\u6216\u591a\u6ed1\u79fb\u5b9e\u9a8c\u6807\u5b9aBassani\u6a21\u578b\u4e2d\u76847\u4e2a\u6f5c\u786c\u5316\u7cfb\u6570\u3002<\/span>\n Latent hardening coefficient q:<\/strong> In isotropic approximation, q=1.0 can be used. For more accurate description, the 7 latent hardening coefficients in the Bassani model need to be calibrated through double-slip or multi-slip experiments.<\/span>\n <\/li>\n <li>\n \u6676\u4f53\u53d6\u5411\uff1a<\/strong>\u5355\u6676\u6a21\u62df\u4e2d\u53d6\u5411\u53c2\u6570\u81f3\u5173\u91cd\u8981\u3002\u5efa\u8bae\u901a\u8fc7EBSD\u5b9e\u9a8c\u6d4b\u91cf\u5b9e\u9645\u8bd5\u6837\u7684Euler\u89d2\uff0c\u800c\u975e\u4ec5\u4f9d\u8d56\u540d\u4e49\u53d6\u5411\u3002\u5373\u4f7f\u5fae\u5c0f\u7684\u53d6\u5411\u504f\u5dee\u4e5f\u53ef\u80fd\u5bfc\u81f4\u5b8f\u89c2\u5e94\u529b-\u5e94\u53d8\u54cd\u5e94\u7684\u663e\u8457\u5dee\u5f02\u3002<\/span>\n Crystal orientation:<\/strong> Orientation parameters are crucial in single-crystal simulations. It is recommended to measure the Euler angles of actual specimens via EBSD, rather than relying solely on nominal orientations. Even small orientation deviations can cause significant differences in macroscopic stress-strain response.<\/span>\n <\/li>\n <\/ul>\n\n <h3 id=\"ch8-3\">\n 8.3 \u6536\u655b\u6027\u8c03\u8bd5 \/ Convergence Debugging<\/span>\n 8.3 Convergence Debugging<\/span>\n <\/h3>\n <ul>\n <li>\n \u65f6\u95f4\u6b65\u8fc7\u5927\uff1a<\/strong>\u82e5\u51fa\u73b0”Too many attempts made for this increment”\u9519\u8bef\uff0c\u6700\u5e38\u89c1\u7684\u539f\u56e0\u662f\u65f6\u95f4\u6b65\u8fc7\u5927\u5bfc\u81f4\u5c40\u90e8Newton\u8fed\u4ee3\u4e0d\u6536\u655b\u3002\u53ef\u901a\u8fc7\u51cf\u5c0f\u6700\u5927\u65f6\u95f4\u6b65\u6216\u542f\u7528\u81ea\u52a8\u6b65\u957f\u63a7\u5236\u89e3\u51b3\u3002<\/span>\n Time step too large:<\/strong> If “Too many attempts made for this increment” appears, the most common cause is excessive time step size causing local Newton iteration divergence. Reduce maximum time step or enable automatic step control.<\/span>\n <\/li>\n <li>\n \u5947\u5f02Jacobian\uff1a<\/strong>\u5f53\u591a\u4e2a\u6ed1\u79fb\u7cfb\u540c\u65f6\u6fc0\u6d3b\u4e14\u786c\u5316\u77e9\u9635\u63a5\u8fd1\u5947\u5f02\u65f6\uff0cLU\u5206\u89e3\u53ef\u80fd\u5931\u8d25\u3002\u53ef\u901a\u8fc7\u589e\u52a0\u6536\u655b\u5bb9\u5deeTOL\u6216\u5728\u6d41\u52a8\u5f8b\u4e2d\u5f15\u5165\u6b63\u5219\u5316\u6765\u6539\u5584\u3002<\/span>\n Singular Jacobian:<\/strong> When multiple slip systems activate simultaneously and the hardening matrix is nearly singular, LU decomposition may fail. Increase convergence tolerance TOL or introduce regularization in the flow rule.<\/span>\n <\/li>\n <li>\n \u8d1f\u526a\u5207\u7387\uff1a<\/strong>\u82e5\u67d0\u4e9b\u6ed1\u79fb\u7cfb\u7684\u5206\u89e3\u526a\u5e94\u529b\u4e3a\u8d1f\u4e14\u7edd\u5bf9\u503c\u5f88\u5927\uff0c\u6d41\u52a8\u5f8b\u53ef\u80fd\u4ea7\u751f\u4e0d\u5408\u7406\u7684\u8d1f\u526a\u5207\u7387\u3002\u7a0b\u5e8f\u4e2d\u5e94\u6709\u7b26\u53f7\u5904\u7406\u903b\u8f91\uff0c\u786e\u4fdd\u7269\u7406\u5408\u7406\u6027\u3002<\/span>\n Negative shear rate:<\/strong> If resolved shear stress on some slip systems is negative with large magnitude, the flow rule may produce unreasonable negative shear rates. The program should have sign-handling logic to ensure physical consistency.<\/span>\n <\/li>\n <li>\n \u68c0\u67e5STATEV\u521d\u59cb\u5316\uff1a<\/strong>STATEV\u7684\u521d\u59cb\u503c\u5fc5\u987b\u6b63\u786e\u8bbe\u7f6e\uff0c\u7279\u522b\u662f\u5f39\u6027\u53d8\u5f62\u68af\u5ea6Fe\u5e94\u521d\u59cb\u5316\u4e3a\u5355\u4f4d\u77e9\u9635\uff0cFp\u4e5f\u521d\u59cb\u5316\u4e3a\u5355\u4f4d\u77e9\u9635\uff0cg\u521d\u59cb\u5316\u4e3atau0\u3002\u4efb\u4f55\u9519\u8bef\u7684\u521d\u503c\u90fd\u53ef\u80fd\u5bfc\u81f4\u7b2c\u4e00\u6b65\u5373\u53d1\u6563\u3002<\/span>\n Check STATEV initialization:<\/strong> Initial STATEV values must be set correctly, especially Fe and Fp initialized to identity matrices, and g initialized to tau0. Any incorrect initial value may cause divergence in the first step.<\/span>\n <\/li>\n <li>\n PNEWDT\u7684\u4f7f\u7528\uff1a<\/strong>\u5f53\u5c40\u90e8\u8fed\u4ee3\u8d85\u8fc7ITMAX\u4ecd\u672a\u6536\u655b\u65f6\uff0c\u5e94\u5728UMAT\u4e2d\u8bbe\u7f6ePNEWDT < 1\uff08\u59820.5\uff09\uff0c\u544a\u77e5Abaqus\u51cf\u5c0f\u5f53\u524d\u589e\u91cf\u6b65\u5e76\u91cd\u8bd5\u3002\u8fd9\u662f\u4e00\u79cd\u6709\u6548\u7684\u81ea\u9002\u5e94\u6b65\u957f\u7b56\u7565\u3002<\/span>\n Use of PNEWDT:<\/strong> When local iteration exceeds ITMAX without convergence, set PNEWDT < 1 (e.g., 0.5) in UMAT to instruct Abaqus to reduce the current increment and retry. This is an effective adaptive step strategy.<\/span>\n <\/li>\n <li>\n \u8f93\u51fa\u8c03\u8bd5\u4fe1\u606f\uff1a<\/strong>\u8bbe\u7f6eOUTPUT=2\u53ef\u542f\u7528\u8be6\u7ec6\u8c03\u8bd5\u8f93\u51fa\uff0c\u5305\u62ec\u6bcf\u6b21Newton\u8fed\u4ee3\u7684\u6b8b\u5dee\u8303\u6570\u3001\u526a\u5207\u589e\u91cf\u548c\u5e94\u529b\u72b6\u6001\u3002\u8fd9\u5bf9\u4e8e\u5b9a\u4f4d\u53d1\u6563\u539f\u56e0\u975e\u5e38\u6709\u5e2e\u52a9\u3002<\/span>\n Output debug information:<\/strong> Setting OUTPUT=2 enables verbose debug output, including residual norm, shear increments, and stress state for each Newton iteration. This is very helpful for locating divergence causes.<\/span>\n <\/li>\n <\/ul>\n\n <h3 id=\"ch8-4\">\n 8.4 \u540e\u5904\u7406\u4e0e\u7ed3\u679c\u9a8c\u8bc1 \/ Post-Processing and Verification<\/span>\n 8.4 Post-Processing and Verification<\/span>\n <\/h3>\n <ul>\n <li>\n \u5e94\u529b-\u5e94\u53d8\u66f2\u7ebf\uff1a<\/strong>\u5c06Abaqus\u8f93\u51fa\u7684\u5b8f\u89c2\u5e94\u529b-\u5e94\u53d8\u66f2\u7ebf\u4e0e\u5355\u6676\u5b9e\u9a8c\u6570\u636e\u8fdb\u884c\u5bf9\u6bd4\uff0c\u9a8c\u8bc1\u6a21\u578b\u9884\u6d4b\u7684\u57fa\u672c\u529b\u5b66\u884c\u4e3a\u662f\u5426\u6b63\u786e\u3002<\/span>\n Stress-strain curve:<\/strong> Compare the macroscopic stress-strain curve from Abaqus with single-crystal experimental data to verify whether the basic mechanical behavior predicted by the model is correct.<\/span>\n <\/li>\n <li>\n \u6ed1\u79fb\u7cfb\u6fc0\u6d3b\u5206\u6790\uff1a<\/strong>\u901a\u8fc7\u540e\u5904\u7406STATEV\u4e2d\u5b58\u50a8\u7684\u7d2f\u79ef\u526a\u5207\u5e94\u53d8\uff0c\u53ef\u4ee5\u8bc6\u522b\u54ea\u4e9b\u6ed1\u79fb\u7cfb\u88ab\u6fc0\u6d3b\u53ca\u5176\u76f8\u5bf9\u6d3b\u52a8\u5f3a\u5ea6\u3002\u8fd9\u5bf9\u4e8e\u7406\u89e3\u53d8\u5f62\u673a\u5236\u548c\u7ec7\u6784\u6f14\u5316\u81f3\u5173\u91cd\u8981\u3002<\/span>\n Slip system activation analysis:<\/strong> By post-processing accumulated shear strains stored in STATEV, one can identify which slip systems are activated and their relative activity levels. This is crucial for understanding deformation mechanisms and texture evolution.<\/span>\n <\/li>\n <li>\n \u7ec7\u6784\u9a8c\u8bc1\uff1a<\/strong>\u5982\u679c\u542f\u7528\u4e86\u53d6\u5411\u66f4\u65b0\uff08PROPS(70)=1\uff09\uff0c\u5e94\u63d0\u53d6\u53d8\u5f62\u540e\u7684\u6676\u4f53\u53d6\u5411\u5e76\u4e0e\u5b9e\u9a8cEBSD\u7ec7\u6784\u6570\u636e\u8fdb\u884c\u5bf9\u6bd4\uff0c\u8bc4\u4f30\u7ec7\u6784\u6f14\u5316\u9884\u6d4b\u7684\u51c6\u786e\u6027\u3002<\/span>\n Texture verification:<\/strong> If orientation update is enabled (PROPS(70)=1), extract deformed crystal orientations and compare with experimental EBSD texture data to assess the accuracy of texture evolution prediction.<\/span>\n <\/li>\n <li>\n \u7f51\u683c\u6536\u655b\u6027\uff1a<\/strong>\u8fdb\u884c\u7f51\u683c\u654f\u611f\u6027\u5206\u6790\uff0c\u786e\u4fdd\u7ed3\u679c\u4e0d\u4f9d\u8d56\u4e8e\u79bb\u6563\u5316\u7a0b\u5ea6\u3002\u6676\u4f53\u5851\u6027\u6a21\u62df\u4e2d\uff0c\u5e94\u53d8\u5c40\u90e8\u5316\u5e26\uff08shear band\uff09\u7684\u5bbd\u5ea6\u901a\u5e38\u53d7\u7f51\u683c\u5c3a\u5bf8\u5f71\u54cd\uff0c\u5e94\u4f7f\u7528\u8db3\u591f\u7ec6\u5bc6\u7684\u7f51\u683c\u3002<\/span>\n Mesh convergence:<\/strong> Perform mesh sensitivity analysis to ensure results do not depend on discretization level. In crystal plasticity simulations, the width of strain localization bands (shear bands) is usually affected by mesh size; sufficiently fine meshes should be used.<\/span>\n <\/li>\n <li>\n \u80fd\u91cf\u5b88\u6052\u68c0\u67e5\uff1a<\/strong>\u9a8c\u8bc1\u5851\u6027\u8017\u6563\u529f\u4e0e\u5916\u90e8\u529f\u7684\u5173\u7cfb\uff0c\u786e\u4fdd\u6570\u503c\u5b9e\u73b0\u6ca1\u6709\u80fd\u91cf\u9519\u8bef\u3002Abaqus\u8f93\u51fa\u7684\u80fd\u91cf\u5386\u53f2\uff08ALLIE\u3001ALLKE\u3001ALLPD\u7b49\uff09\u53ef\u7528\u4e8e\u6b64\u76ee\u7684\u3002<\/span>\n Energy conservation check:<\/strong> Verify the relationship between plastic dissipation work and external work to ensure the numerical implementation has no energy errors. Abaqus energy history outputs (ALLIE, ALLKE, ALLPD, etc.) can be used for this purpose.<\/span>\n <\/li>\n <\/ul>\n\n <h3 id=\"ch8-5\">\n 8.5 \u5e38\u89c1\u9519\u8bef\u4e0e\u89e3\u51b3\u65b9\u6848 \/ Common Errors and Solutions<\/span>\n 8.5 Common Errors and Solutions<\/span>\n <\/h3>\n <table>\n <thead>\n <tr>\n <th>\n \u9519\u8bef\u73b0\u8c61 \/ Error<\/span>\n Error Symptom<\/span>\n <\/th>\n <th>\n \u53ef\u80fd\u539f\u56e0 \/ Cause<\/span>\n Possible Cause<\/span>\n <\/th>\n <th>\n \u89e3\u51b3\u65b9\u6848 \/ Solution<\/span>\n Solution<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>\u589e\u91cf\u6b65\u591a\u6b21\u5c1d\u8bd5\u540e\u4ecd\u5931\u8d25<\/span>Increment fails after many attempts<\/span><\/td>\n <td>\u65f6\u95f4\u6b65\u8fc7\u5927\u6216\u6750\u6599\u53c2\u6570\u4e0d\u5408\u7406<\/span>Time step too large or material parameters unreasonable<\/span><\/td>\n <td>\u51cf\u5c0f\u6700\u5927\u65f6\u95f4\u6b65\uff0c\u68c0\u67e5\u6750\u6599\u53c2\u6570\u6570\u91cf\u7ea7<\/span>Reduce max time step, check parameter magnitudes<\/span><\/td>\n <\/tr>\n <tr>\n <td>\u5e94\u529b\u8f93\u51fa\u4e3aNaN\u6216Inf<\/span>Stress output is NaN or Inf<\/span><\/td>\n <td>\u9664\u96f6\u9519\u8bef\u6216\u6307\u6570\u6ea2\u51fa<\/span>Division by zero or exponent overflow<\/span><\/td>\n <td>\u68c0\u67e5tau\u548cg\u662f\u5426\u4e3a\u6b63\uff0c\u9650\u5236\u5e42\u5f8b\u6307\u6570\u8303\u56f4<\/span>Ensure tau and g are positive, limit power-law exponent range<\/span><\/td>\n <\/tr>\n <tr>\n <td>STATEV\u503c\u5f02\u5e38\u589e\u5927<\/span>STATEV values abnormally large<\/span><\/td>\n <td>\u72b6\u6001\u53d8\u91cf\u672a\u6b63\u786e\u521d\u59cb\u5316<\/span>State variables not properly initialized<\/span><\/td>\n <td>\u5728*Initial Conditions\u4e2d\u6b63\u786e\u8bbe\u7f6eSTATEV\u521d\u503c<\/span>Set correct STATEV initial values in *Initial Conditions<\/span><\/td>\n <\/tr>\n <tr>\n <td>\u7ed3\u679c\u4e0e\u5b9e\u9a8c\u5dee\u5f02\u5de8\u5927<\/span>Results differ greatly from experiment<\/span><\/td>\n <td>\u53d6\u5411\u53c2\u6570\u9519\u8bef\u6216\u5355\u4f4d\u4e0d\u4e00\u81f4<\/span>Orientation parameters wrong or unit inconsistency<\/span><\/td>\n <td>\u6838\u5bf9Euler\u89d2\u5355\u4f4d\u548c\u6ed1\u79fb\u7cfb\u5b9a\u4e49\uff0c\u7edf\u4e00\u5e94\u529b\u5355\u4f4d\u4e3aMPa<\/span>Check Euler angle units and slip system definitions, unify stress units to MPa<\/span><\/td>\n <\/tr>\n <tr>\n <td>\u8fd0\u884c\u901f\u5ea6\u6781\u6162<\/span>Extremely slow execution<\/span><\/td>\n <td>\u65f6\u95f4\u6b65\u8fc7\u5c0f\u6216Newton\u8fed\u4ee3\u6b21\u6570\u8fc7\u591a<\/span>Time step too small or too many Newton iterations<\/span><\/td>\n <td>\u589e\u5927\u6536\u655b\u5bb9\u5deeTOL\uff0c\u964d\u4f4eITMAX\uff0c\u6216\u68c0\u67e5\u786c\u5316\u53c2\u6570\u662f\u5426\u5bfc\u81f4\u75c5\u6001\u95ee\u9898<\/span>Increase TOL, reduce ITMAX, or check if hardening parameters cause ill-conditioning<\/span><\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n<\/div>\n\n\n<div class=\"chapter\" id=\"refs\">\n <h2>\n Ref<\/span>\n \u53c2\u8003\u6587\u732e \/ References<\/span>\n References<\/span>\n <\/h2>\n \n \u4ee5\u4e0b\u5217\u51fa\u9ec4\u6c38\u521aUMAT\u5b50\u7a0b\u5e8f\u76f8\u5173\u7684\u6838\u5fc3\u7406\u8bba\u6587\u732e\uff0c\u6309\u53d1\u8868\u65f6\u95f4\u6392\u5e8f\uff1a\n <\/p>\n \n The following lists core theoretical references related to Huang’s UMAT subroutine, in chronological order:\n <\/p>\n <div class=\"ref-item\">\n [1] Peirce D, Asaro RJ, Needleman A. Material rate dependence and localized deformation in crystalline solids.\n Acta Metallurgica<\/em>, 1983, 31(12): 1951-1976.\n <\/div>\n <div class=\"ref-item\">\n [2] Peirce D, Shih CF, Needleman A. A tangent modulus method for rate dependent solids.\n Computers & Structures<\/em>, 1984, 18(5): 875-887.\n <\/div>\n <div class=\"ref-item\">\n [3] Asaro RJ, Needleman A. Texture development and strain hardening in rate dependent polycrystals.\n Acta Metallurgica<\/em>, 1985, 33(6): 923-953.\n <\/div>\n <div class=\"ref-item\">\n [4] Harren SV, Deve HE, Asaro RJ. Shear band formation in plane strain compression.\n Acta Metallurgica<\/em>, 1988, 36(9): 2435-2480.\n <\/div>\n <div class=\"ref-item\">\n [5] Bassani JL, Wu TY. Latent hardening in single crystals. II. Analytical characterization and predictions.\n Proceedings of the Royal Society A<\/em>, 1991, 435(1893): 21-41.\n <\/div>\n <div class=\"ref-item\">\n [6] Huang Y. A user-material subroutine incorporating single crystal plasticity in the ABAQUS finite element program.\n Harvard University Report MECH-178<\/em>, 1991.\n <\/div>\n <div class=\"ref-item\">\n [7] Kalidindi SR, Bronkhorst CA, Anand L. Crystallographic texture evolution in bulk deformation processing of FCC metals.\n Journal of the Mechanics and Physics of Solids<\/em>, 1992, 40(3): 537-569.\n <\/div>\n <div class=\"ref-item\">\n [8] Kocks UF, Tome CN, Wenk HR. Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties<\/em>.\n Cambridge University Press, 1998.\n <\/div>\n <div class=\"ref-item\">\n [9] Roters F, Eisenlohr P, Hantcherli L, et al. Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications.\n Acta Materialia<\/em>, 2010, 58(4): 1152-1211.\n <\/div>\n <div class=\"ref-item\">\n [10] Dunne FPE, Petrinic N. Introduction to Computational Plasticity<\/em>. Oxford University Press, 2005.\n <\/div>\n<\/div>\n\n\n<div class=\"chapter\" id=\"appendix\">\n <h2>\n App<\/span>\n \u7b26\u53f7\u8868 \/ Notation<\/span>\n Notation<\/span>\n <\/h2>\n \u672c\u6587\u6863\u4e2d\u4f7f\u7528\u7684\u5173\u952e\u6570\u5b66\u7b26\u53f7\u53ca\u5176\u542b\u4e49\u5982\u4e0b\u8868\u6240\u793a\uff1a<\/p>\n The key mathematical symbols used in this document and their meanings are listed in the following table:<\/p>\n <table>\n <thead>\n <tr>\n <th>\u7b26\u53f7 \/ Symbol<\/span>Symbol<\/span><\/th>\n <th>\u542b\u4e49 \/ Meaning<\/span>Meaning<\/span><\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>F<\/b><\/td><td>\u603b\u53d8\u5f62\u68af\u5ea6 \/ Total deformation gradient<\/span>Total deformation gradient<\/span><\/td><\/tr>\n <tr><td>Fe<\/sup><\/b><\/td><td>\u5f39\u6027\u53d8\u5f62\u68af\u5ea6 \/ Elastic deformation gradient<\/span>Elastic deformation gradient<\/span><\/td><\/tr>\n <tr><td>Fp<\/sup><\/b><\/td><td>\u5851\u6027\u53d8\u5f62\u68af\u5ea6 \/ Plastic deformation gradient<\/span>Plastic deformation gradient<\/span><\/td><\/tr>\n <tr><td>σ<\/b><\/td><td>Cauchy\u5e94\u529b\u5f20\u91cf \/ Cauchy stress tensor<\/span>Cauchy stress tensor<\/span><\/td><\/tr>\n <tr><td>τ<\/b><\/td><td>Kirchhoff\u5e94\u529b\u5f20\u91cf \/ Kirchhoff stress tensor<\/span>Kirchhoff stress tensor<\/span><\/td><\/tr>\n <tr><td>Se<\/sup><\/b><\/td><td>\u7b2c\u4e8c\u7c7bPiola-Kirchhoff\u5e94\u529b \/ 2nd P-K stress<\/span>2nd Piola-Kirchhoff stress<\/span><\/td><\/tr>\n <tr><td>Ee<\/sup><\/b><\/td><td>Green\u5f39\u6027\u5e94\u53d8 \/ Green elastic strain<\/span>Green elastic strain<\/span><\/td><\/tr>\n <tr><td>γ̇(α)<\/sup><\/b><\/td><td>\u7b2cα\u6ed1\u79fb\u7cfb\u526a\u5207\u5e94\u53d8\u7387 \/ Shear strain rate on slip system α<\/span>Shear strain rate on slip system α<\/span><\/td><\/tr>\n <tr><td>τ(α)<\/sup><\/b><\/td><td>\u5206\u89e3\u526a\u5e94\u529b \/ Resolved shear stress<\/span>Resolved shear stress<\/span><\/td><\/tr>\n <tr><td>g(α)<\/sup><\/b><\/td><td>\u4e34\u754c\u5206\u5207\u5e94\u529b\uff08\u6ed1\u79fb\u5f3a\u5ea6\uff09\/ CRSS (slip strength)<\/span>Critical resolved shear stress (slip strength)<\/span><\/td><\/tr>\n <tr><td>P(α)<\/sup><\/b><\/td><td>Schmid\u5f20\u91cf \/ Schmid tensor<\/span>Schmid tensor<\/span><\/td><\/tr>\n <tr><td>s(α)<\/sup><\/b><\/td><td>\u6ed1\u79fb\u65b9\u5411 \/ Slip direction<\/span>Slip direction<\/span><\/td><\/tr>\n <tr><td>m(α)<\/sup><\/b><\/td><td>\u6ed1\u79fb\u9762\u6cd5\u5411 \/ Slip plane normal<\/span>Slip plane normal<\/span><\/td><\/tr>\n <tr><td>hαβ<\/sub><\/b><\/td><td>\u786c\u5316\u77e9\u9635 \/ Hardening matrix<\/span>Hardening matrix<\/span><\/td><\/tr>\n <tr><td>Ce<\/sup><\/b><\/td><td>\u5f39\u6027\u521a\u5ea6\u5f20\u91cf \/ Elastic stiffness tensor<\/span>Elastic stiffness tensor<\/span><\/td><\/tr>\n <tr><td>DDSDDE<\/b><\/td><td>\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635 \/ Consistent tangent modulus<\/span>Consistent tangent modulus<\/span><\/td><\/tr>\n <tr><td>STATEV<\/b><\/td><td>\u72b6\u6001\u53d8\u91cf\u6570\u7ec4 \/ State variable array<\/span>State variable array<\/span><\/td><\/tr>\n <tr><td>PROPS<\/b><\/td><td>\u6750\u6599\u53c2\u6570\u6570\u7ec4 \/ Material property array<\/span>Material property array<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n<\/div>\n\n<\/div>\n\n<footer style=\"text-align:center; 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{"id":261,"date":"2026-06-29T21:14:44","date_gmt":"2026-06-29T13:14:44","guid":{"rendered":"https:\/\/numsimlab.com\/?p=261"},"modified":"2026-06-29T21:18:53","modified_gmt":"2026-06-29T13:18:53","slug":"%e6%99%b6%e4%bd%93%e5%a1%91%e6%80%a7%e5%8a%9b%e5%ad%a6huang-umat%e5%ad%90%e7%a8%8b%e5%ba%8f%e8%af%a6%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/numsimlab.com\/?p=261","title":{"rendered":"\u6676\u4f53\u5851\u6027\u529b\u5b66Huang UMAT\u5b50\u7a0b\u5e8f\u8be6\u89e3"},"content":{"rendered":"\n\n\n\n\n\n\nHuangUMAT\u5355\u6676\u5851\u6027\u5b50\u7a0b\u5e8f\u8be6\u89e3<\/title>\n<title data-lang=\"en\" style=\"display:none\">Huang’s UMAT Crystal Plasticity Subroutine<\/title>\n<style>\n:root {\n --bg: #f5f7fa;\n --bg2: #eef1f6;\n --ink: #1a2332;\n --muted: #5a6a7e;\n --rule: #d0d7e2;\n --accent: #2563eb;\n --accent2: #0ea5e9;\n --code-bg: #1a2332;\n --code-fg: #e2e8f0;\n}\n* { box-sizing: border-box; margin: 0; padding: 0; }\nhtml { scroll-behavior: smooth; }\nbody {\n 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1.3rem; }\n .toc ul { columns: 1; }\n table { font-size: 0.82rem; }\n th, td { padding: 6px 8px; }\n}\n.hljs-keyword { color: #c678dd; }\n.hljs-string { color: #98c379; }\n.hljs-number { color: #d19a66; }\n.hljs-comment { color: #5c6370; font-style: italic; }\n.hljs-function { color: #61afef; }\n.hljs-params { color: #e06c75; }\n.hljs-built_in { color: #e5c07b; }\n<\/style>\n<script>\nfunction switchLanguage(lang) {\n document.documentElement.lang = lang;\n document.querySelectorAll('[data-lang]').forEach(function(el) {\n if (el.tagName.toLowerCase() === 'title') {\n if (el.getAttribute('data-lang') === lang) el.style.display = '';\n else el.style.display = 'none';\n return;\n }\n if (el.getAttribute('data-lang') === lang) {\n el.style.display = '';\n } else {\n el.style.display = 'none';\n }\n });\n document.querySelectorAll('.lang-bar button').forEach(function(btn) {\n btn.classList.toggle('active', btn.getAttribute('data-set-lang') === lang);\n });\n if (lang === 'en') document.body.classList.add('en-mode');\n else document.body.classList.remove('en-mode');\n if (window.mermaid) {\n mermaid.initialize({ startOnLoad: true, theme: 'neutral', securityLevel: 'loose' });\n }\n}\ndocument.addEventListener('DOMContentLoaded', function() {\n switchLanguage('zh');\n});\n<\/script>\n<\/head>\n<body>\n\n<div class=\"lang-bar\">\n <button data-set-lang=\"zh\" onclick=\"switchLanguage('zh')\" class=\"active\">\n \u4e2d\u6587<\/span>\n Chinese<\/span>\n <\/button>\n <button data-set-lang=\"en\" onclick=\"switchLanguage('en')\">\n English<\/span>\n English<\/span>\n <\/button>\n<\/div>\n\n<div class=\"container\">\n\n\n<div class=\"cover\">\n <h1>\n Huang UMAT \u5355\u6676\u5851\u6027\u5b50\u7a0b\u5e8f\u8be6\u89e3<\/span>\n Detailed Explanation of Huang’s UMAT Crystal Plasticity Subroutine<\/span>\n <\/h1>\n <h2>\n Crystal Plasticity Finite Element Method (CPFEM) \/ \u6676\u4f53\u5851\u6027\u6709\u9650\u5143\u65b9\u6cd5<\/span>\n Crystal Plasticity Finite Element Method (CPFEM)<\/span>\n <\/h2>\n <div class=\"subtitle\">\n \u57fa\u4e8e Abaqus\/Standard \u9690\u5f0f\u6c42\u89e3\u5668<\/span>\n For Abaqus\/Standard Implicit Solver<\/span>\n <\/div>\n <div class=\"meta\">\n \u5b8c\u6574\u672c\u6784\u7406\u8bba\u3001UMAT\u63a5\u53e3\u3001\u4ee3\u7801\u7ed3\u6784\u4e0e\u5b9e\u73b0\u7ec6\u8282<\/span>\n Complete Constitutive Theory, UMAT Interface, Code Structure and Implementation Details<\/span>\n <\/div>\n<\/div>\n\n\n\n\n<div class=\"toc\" id=\"toc\">\n <h2>\n \u76ee\u5f55 \/ Table of Contents<\/span>\n Table of Contents<\/span>\n <\/h2>\n <ul>\n <li><a href=\"#ch1\">\n Chapter 1 \u6982\u8ff0 \/ Overview<\/span>\n Chapter 1 Overview<\/span>\n <\/a><\/li>\n <li><a href=\"#ch2\">\n Chapter 2 \u672c\u6784\u7406\u8bba \/ Constitutive Theory<\/span>\n Chapter 2 Constitutive Theory<\/span>\n <\/a><\/li>\n <li><a href=\"#ch3\">\n Chapter 3 Abaqus UMAT\u63a5\u53e3 \/ UMAT Interface<\/span>\n Chapter 3 Abaqus UMAT Interface<\/span>\n <\/a><\/li>\n <li><a href=\"#ch4\">\n Chapter 4 \u6750\u6599\u53c2\u6570 \/ Material Parameters<\/span>\n Chapter 4 Material Parameters<\/span>\n <\/a><\/li>\n <li><a href=\"#ch5\">\n Chapter 5 \u4ee3\u7801\u7ed3\u6784 \/ Code Structure<\/span>\n Chapter 5 Code Structure<\/span>\n <\/a><\/li>\n <li><a href=\"#ch6\">\n Chapter 6 Jacobian\u77e9\u9635 \/ Jacobian Matrix<\/span>\n Chapter 6 Jacobian Matrix<\/span>\n <\/a><\/li>\n <li><a href=\"#ch7\">\n Chapter 7 \u5178\u578b\u53c2\u6570 \/ Typical Parameters<\/span>\n Chapter 7 Typical Parameters<\/span>\n <\/a><\/li>\n <li><a href=\"#ch8\">\n Chapter 8 \u4f7f\u7528\u6ce8\u610f\u4e8b\u9879 \/ Usage Notes<\/span>\n Chapter 8 Usage Notes<\/span>\n <\/a><\/li>\n <li><a href=\"#refs\">\n \u53c2\u8003\u6587\u732e \/ References<\/span>\n References<\/span>\n <\/a><\/li>\n <\/ul>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch1\">\n <h2>\n 01<\/span>\n \u6982\u8ff0 \/ Overview<\/span>\n Overview<\/span>\n <\/h2>\n\n <h3 id=\"ch1-1\">\n 1.1 \u672c\u6784\u6a21\u578b\u7b80\u4ecb \/ Constitutive Model Introduction<\/span>\n 1.1 Constitutive Model Introduction<\/span>\n <\/h3>\n \n Huang\uff08Yonggang Huang\uff09\u6559\u6388\u5f00\u53d1\u7684UMAT\u5355\u6676\u5851\u6027\u5b50\u7a0b\u5e8f\u662f\u6676\u4f53\u5851\u6027\u6709\u9650\u5143\u65b9\u6cd5\uff08CPFEM\uff09\u9886\u57df\u6700\u5e7f\u6cdb\u4f7f\u7528\u7684\u5f00\u6e90\u5b9e\u73b0\u4e4b\u4e00\u3002\n \u8be5\u5b50\u7a0b\u5e8f\u57fa\u4e8eAbaqus\/Standard\u7528\u6237\u6750\u6599\u5b50\u7a0b\u5e8f\uff08UMAT\uff09\u63a5\u53e3\uff0c\u91c7\u7528\u9690\u5f0f\u65f6\u95f4\u79ef\u5206\u65b9\u6848\uff0c\n \u80fd\u591f\u51c6\u786e\u6a21\u62df\u5355\u6676\u53ca\u591a\u6676\u6750\u6599\u5728\u590d\u6742\u52a0\u8f7d\u6761\u4ef6\u4e0b\u7684\u5851\u6027\u53d8\u5f62\u884c\u4e3a\u3002\n \u8be5\u6a21\u578b\u8003\u8651\u4e86\u6676\u4f53\u7684\u5404\u5411\u5f02\u6027\u5f39\u6027\u3001\u6676\u4f53\u5b66\u6ed1\u79fb\u7cfb\u4e0a\u7684\u526a\u5207\u53d8\u5f62\uff0c\u4ee5\u53ca\u4f4d\u9519\u4ea4\u4e92\u4f5c\u7528\u5bfc\u81f4\u7684\u786c\u5316\u6548\u5e94\u3002\n <\/p>\n \n The UMAT single-crystal plasticity subroutine developed by Professor Yonggang Huang\n is one of the most widely used open-source implementations in the field of Crystal Plasticity Finite Element Method (CPFEM).\n Based on the Abaqus\/Standard User Material (UMAT) interface with an implicit time integration scheme,\n it accurately simulates the plastic deformation behavior of single-crystal and polycrystalline materials under complex loading conditions.\n The model accounts for anisotropic elasticity, crystallographic slip system shear deformation,\n and hardening effects due to dislocation interactions.\n <\/p>\n \n \u672c\u7a0b\u5e8f\u7684\u6838\u5fc3\u7279\u5f81\u5305\u62ec\uff1a\u57fa\u4e8e\u4e2d\u95f4\u6784\u578b\u7684\u4e58\u6cd5\u5206\u89e3\u8fd0\u52a8\u5b66\u6846\u67b6\u3001Schmid\u5b9a\u5f8b\u9a71\u52a8\u7684\u6ed1\u79fb\u7cfb\u6fc0\u6d3b\u5224\u636e\u3001\n \u7387\u76f8\u5173\u5e42\u5f8b\u6d41\u52a8\u6cd5\u5219\u3001\u53ef\u9009\u62e9\u7684\u786c\u5316\u6a21\u578b\uff08\u53cc\u66f2\u6b63\u5272\u786c\u5316\u6216Bassani-Wu\u786c\u5316\uff09\uff0c\n \u4ee5\u53ca\u5b8c\u6574\u7684\u9690\u5f0f\u5e94\u529b\u66f4\u65b0\u7b97\u6cd5\u548c\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635\uff08Jacobian\uff09\u8ba1\u7b97\u3002\n <\/p>\n \n Key features of this subroutine include: multiplicative decomposition kinematics based on the intermediate configuration,\n Schmid’s law driven slip system activation criterion, rate-dependent power-law flow rule,\n selectable hardening models (hyperbolic secant hardening or Bassani-Wu hardening),\n and a complete implicit stress update algorithm with consistent tangent stiffness matrix (Jacobian) computation.\n <\/p>\n\n <h3 id=\"ch1-2\">\n 1.2 \u7406\u8bba\u57fa\u7840 \/ Theoretical Foundation<\/span>\n 1.2 Theoretical Foundation<\/span>\n <\/h3>\n \n \u9ec4\u6c38\u521aUMAT\u7684\u7406\u8bba\u57fa\u7840\u4e3b\u8981\u6765\u6e90\u4e8ePeirce\u3001Asaro\u548cNeedleman\uff081983\uff09\u4ee5\u53caPeirce\u3001Shih\u548cNeedleman\uff081984\uff09\u7684\u7ecf\u5178\u5de5\u4f5c\u3002\n \u8fd9\u4e9b\u6587\u732e\u5efa\u7acb\u4e86\u6676\u4f53\u5851\u6027\u7406\u8bba\u5728\u73b0\u4ee3\u6709\u9650\u5143\u6846\u67b6\u4e0b\u7684\u6570\u5b66\u8868\u8ff0\uff0c\u5305\u62ec\uff1a\n <\/p>\n \n The theoretical foundation of Huang’s UMAT is primarily derived from the seminal works of\n Peirce, Asaro and Needleman (1983) and Peirce, Shih and Needleman (1984).\n These papers established the mathematical formulation of crystal plasticity theory within the modern finite element framework, including:\n <\/p>\n <ul>\n <li>\n \u603b\u53d8\u5f62\u68af\u5ea6F\u7684\u4e58\u6cd5\u5206\u89e3\u4e3a\u5f39\u6027\u90e8\u5206Fe\u548c\u5851\u6027\u90e8\u5206Fp<\/span>\n Multiplicative decomposition of the total deformation gradient F into elastic Fe and plastic Fp parts<\/span>\n <\/li>\n <li>\n \u57fa\u4e8eSchmid\u5206\u89e3\u7684\u5206\u89e3\u5e94\u529b\uff08resolved shear stress\uff09\u6982\u5ff5<\/span>\n The concept of resolved shear stress based on Schmid decomposition<\/span>\n <\/li>\n <li>\n \u7387\u76f8\u5173\u6d41\u52a8\u6cd5\u5219\uff0c\u907f\u514d\u7387\u65e0\u5173\u6a21\u578b\u7684\u975e\u552f\u4e00\u6027\u548c\u6570\u503c\u4e0d\u7a33\u5b9a\u6027<\/span>\n Rate-dependent flow rule to avoid non-uniqueness and numerical instability of rate-independent models<\/span>\n <\/li>\n <li>\n \u81ea\u786c\u5316\uff08self-hardening\uff09\u548c\u6f5c\u786c\u5316\uff08latent hardening\uff09\u7684\u7edf\u4e00\u63cf\u8ff0<\/span>\n Unified description of self-hardening and latent hardening<\/span>\n <\/li>\n <\/ul>\n \n \u540e\u7eedAsaro\u548cNeedleman\uff081985\uff09\u4ee5\u53caHarren\u3001Deve\u548cAsaro\uff081988\uff09\u7684\u5de5\u4f5c\u8fdb\u4e00\u6b65\u5b8c\u5584\u4e86\u6676\u4f53\u5851\u6027\u7406\u8bba\u5728\u6709\u9650\u53d8\u5f62\u6761\u4ef6\u4e0b\u7684\u6570\u503c\u5b9e\u73b0\u3002\n \u9ec4\u6c38\u521a\u5728\u5176\u5b50\u7a0b\u5e8f\u4e2d\u6574\u5408\u4e86\u8fd9\u4e9b\u7406\u8bba\u6210\u679c\uff0c\u5e76\u9488\u5bf9Abaqus\/Standard\u7684\u9690\u5f0f\u6c42\u89e3\u5668\u8fdb\u884c\u4e86\u7b97\u6cd5\u4f18\u5316\u3002\n <\/p>\n \n Subsequent works by Asaro and Needleman (1985) and Harren, Deve and Asaro (1988) further refined\n the numerical implementation of crystal plasticity theory under finite deformation conditions.\n Huang integrated these theoretical achievements into his subroutine and optimized the algorithm\n specifically for the Abaqus\/Standard implicit solver.\n <\/p>\n\n <h3 id=\"ch1-3\">\n 1.3 \u9002\u7528\u8303\u56f4 \/ Scope<\/span>\n 1.3 Scope<\/span>\n <\/h3>\n \n \u672c\u5b50\u7a0b\u5e8f\u9002\u7528\u4e8e\u4ee5\u4e0b\u7814\u7a76\u573a\u666f\uff1a\n <\/p>\n \n This subroutine is applicable to the following research scenarios:\n <\/p>\n <ul>\n <li>\n FCC\u3001BCC\u548cHCP\u6676\u4f53\u7ed3\u6784\u7684\u5355\u6676\u548c\u591a\u6676\u91d1\u5c5e\u6750\u6599\u7684\u529b\u5b66\u884c\u4e3a\u6a21\u62df<\/span>\n Mechanical behavior simulation of single-crystal and polycrystalline metals with FCC, BCC, and HCP crystal structures<\/span>\n <\/li>\n <li>\n \u5355\u8c03\u52a0\u8f7d\u3001\u5faa\u73af\u52a0\u8f7d\u53ca\u5e94\u53d8\u8def\u5f84\u53d8\u5316\u6761\u4ef6\u4e0b\u7684\u5851\u6027\u53d8\u5f62\u5206\u6790<\/span>\n Plastic deformation analysis under monotonic, cyclic, and strain-path-change loading conditions<\/span>\n <\/li>\n <li>\n \u7ec7\u6784\u6f14\u5316\uff08texture evolution\uff09\u548c\u5851\u6027\u5404\u5411\u5f02\u6027\u53d1\u5c55\u7684\u5b9a\u91cf\u9884\u6d4b<\/span>\n Quantitative prediction of texture evolution and plastic anisotropy development<\/span>\n <\/li>\n <li>\n \u5c40\u90e8\u5316\u53d8\u5f62\u3001\u526a\u5207\u5e26\u5f62\u6210\u53ca\u7ec6\u89c2\u5c3a\u5ea6\u635f\u4f24\u8d77\u59cb\u7814\u7a76<\/span>\n Studies of localized deformation, shear band formation, and mesoscale damage initiation<\/span>\n <\/li>\n <li>\n \u5fae\u67f1\u538b\u7f29\u3001\u7eb3\u7c73\u538b\u75d5\u7b49\u5c0f\u5c3a\u5ea6\u5355\u6676\u5b9e\u9a8c\u7684\u6709\u9650\u5143\u5efa\u6a21<\/span>\n Finite element modeling of micro-pillar compression, nanoindentation, and other small-scale single-crystal experiments<\/span>\n <\/li>\n <\/ul>\n <div class=\"note\">\n \u6ce8\u610f\uff1a<\/strong>\u672c\u5b50\u7a0b\u5e8f\u57fa\u4e8e\u5c0f\u5f39-\u5927\u5851\u5047\u8bbe\uff0c\u5373\u5f39\u6027\u53d8\u5f62\u68af\u5ea6Fe\u63a5\u8fd1\u5355\u4f4d\u5f20\u91cf\uff08||Fe-I|| << 1\uff09\uff0c\u4f46\u5851\u6027\u53d8\u5f62Fp\u53ef\u4ee5\u4efb\u610f\u5927\u3002\u8be5\u5047\u8bbe\u5bf9\u4e8e\u5927\u591a\u6570\u91d1\u5c5e\u6676\u4f53\u5728\u4e2d\u7b49\u5e94\u53d8\u8303\u56f4\u5185\u662f\u5408\u7406\u7684\u3002<\/span>\n Note:<\/strong> This subroutine is based on the small-elastic-large-plastic assumption, where the elastic deformation gradient Fe is close to the identity tensor (||Fe-I|| << 1), but the plastic deformation Fp can be arbitrarily large. This assumption is reasonable for most metallic crystals at moderate strain ranges.<\/span>\n <\/div>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch2\">\n <h2>\n 02<\/span>\n \u672c\u6784\u7406\u8bba \/ Constitutive Theory<\/span>\n Constitutive Theory<\/span>\n <\/h2>\n\n <h3 id=\"ch2-1\">\n 2.1 \u5206\u89e3\u8fd0\u52a8\u5b66 \/ Kinematics of Decomposition<\/span>\n 2.1 Kinematics of Decomposition<\/span>\n <\/h3>\n \n \u6676\u4f53\u5851\u6027\u7406\u8bba\u91c7\u7528\u4e58\u6cd5\u5206\u89e3\uff08multiplicative decomposition\uff09\u5c06\u603b\u53d8\u5f62\u68af\u5ea6F\u5206\u89e3\u4e3a\u5f39\u6027\u90e8\u5206Fe\u548c\u5851\u6027\u90e8\u5206Fp\uff1a\n <\/p>\n \n Crystal plasticity theory employs multiplicative decomposition to separate the total deformation gradient F into elastic Fe and plastic Fp parts:\n <\/p>\n <div class=\"formula\">\n F = Fe<\/sup> \u00b7 Fp<\/sup><\/b>\n <\/div>\n \n \u5176\u4e2d\uff0c\u5851\u6027\u53d8\u5f62\u68af\u5ea6Fp\u7531\u6240\u6709\u6fc0\u6d3b\u6ed1\u79fb\u7cfb\u4e0a\u7684\u526a\u5207\u5e94\u53d8\u7d2f\u79ef\u4ea7\u751f\uff0c\u4fdd\u6301\u6676\u683c\u7ed3\u6784\u4e0d\u53d8\uff08\u7b49\u5bb9\u5851\u6027\u53d8\u5f62\uff09\uff1b\n \u5f39\u6027\u53d8\u5f62\u68af\u5ea6Fe\u63cf\u8ff0\u6676\u683c\u7684\u5f39\u6027\u7578\u53d8\u548c\u521a\u6027\u8f6c\u52a8\u3002\n \u8be5\u5206\u89e3\u5c06\u53c2\u8003\u6784\u578b\uff08reference configuration\uff09\u6620\u5c04\u5230\u4e2d\u95f4\u6784\u578b\uff08intermediate configuration\uff09\uff0c\n \u518d\u6620\u5c04\u5230\u5f53\u524d\u6784\u578b\uff08current configuration\uff09\u3002\n <\/p>\n \n Here, the plastic deformation gradient Fp is produced by the accumulation of shear strains on all active slip systems,\n preserving the crystal lattice structure (isochoric plastic deformation);\n the elastic deformation gradient Fe describes the elastic distortion and rigid rotation of the lattice.\n This decomposition maps from the reference configuration to the intermediate configuration,\n and then to the current configuration.\n <\/p>\n \n \u5851\u6027\u53d8\u5f62\u68af\u5ea6\u7684\u6f14\u5316\u65b9\u7a0b\u4e3a\uff1a\n <\/p>\n \n The evolution equation for the plastic deformation gradient is:\n <\/p>\n <div class=\"formula\">\n Ḟp<\/sup> = Lp<\/sup> \u00b7 Fp<\/sup><\/b> \n \u5176\u4e2d\u5851\u6027\u901f\u5ea6\u68af\u5ea6<\/span>\n where the plastic velocity gradient<\/span>\n Lp<\/sup> = Σα<\/sub> γ̇(α)<\/sup> s(α)<\/sup> ⊗ m(α)<\/sup><\/b>\n <\/div>\n \n \u4e0a\u5f0f\u4e2d\uff0cs(α)<\/sup>\u548cm(α)<\/sup>\u5206\u522b\u662f\u7b2cα\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411\uff08slip direction\uff09\u548c\u6ed1\u79fb\u9762\u6cd5\u5411\uff08slip plane normal\uff09\uff0c\n \u4e8c\u8005\u5b9a\u4e49\u5728\u4e2d\u95f4\u6784\u578b\u4e2d\u4e14\u4fdd\u6301\u6b63\u4ea4\u5f52\u4e00\u3002γ̇(α)<\/sup>\u4e3a\u5bf9\u5e94\u7684\u526a\u5207\u5e94\u53d8\u7387\u3002\n \u6ed1\u79fb\u65b9\u5411s\u548c\u6cd5\u5411m\u5728\u5f53\u524d\u6784\u578b\u4e2d\u7684\u6620\u50cf\u5206\u522b\u4e3aFe<\/sup>s\u548c(Fe<\/sup>)-T<\/sup>m\u3002\n <\/p>\n \n In the above equation, s(α)<\/sup> and m(α)<\/sup> are the slip direction and slip plane normal of the α-th slip system, respectively,\n defined in the intermediate configuration and maintained orthonormal.\n γ̇(α)<\/sup> is the corresponding shear strain rate.\n The images of slip direction s and normal m in the current configuration are Fe<\/sup>s and (Fe<\/sup>)-T<\/sup>m, respectively.\n <\/p>\n \n \u5f39\u6027\u53d8\u5f62\u68af\u5ea6Fe<\/sup>\u7684\u6781\u5206\u89e3\u4e3aFe<\/sup> = Re<\/sup>Ue<\/sup> = Ve<\/sup>Re<\/sup>\uff0c\n \u5176\u4e2dUe<\/sup>\u548cVe<\/sup>\u5206\u522b\u4e3a\u53f3\u3001\u5de6\u5f39\u6027\u62c9\u4f38\u5f20\u91cf\uff0cRe<\/sup>\u4e3a\u5f39\u6027\u8f6c\u52a8\u5f20\u91cf\u3002\n Green\u5f39\u6027\u5e94\u53d8\u5f20\u91cf\u5b9a\u4e49\u4e3a\uff1a\n <\/p>\n \n The polar decomposition of the elastic deformation gradient is Fe<\/sup> = Re<\/sup>Ue<\/sup> = Ve<\/sup>Re<\/sup>,\n where Ue<\/sup> and Ve<\/sup> are the right and left elastic stretch tensors, and Re<\/sup> is the elastic rotation tensor.\n The Green elastic strain tensor is defined as:\n <\/p>\n <div class=\"formula\">\n Ee<\/sup> = (1\/2)(Ue<\/sup>2<\/sup> – I) = (1\/2)(FeT<\/sup>Fe<\/sup> – I)<\/b>\n <\/div>\n \n \u5f39\u6027\u672c\u6784\u5173\u7cfb\u91c7\u7528\u5e7f\u4e49\u80e1\u514b\u5b9a\u5f8b\uff0c\u4ee5\u7b2c\u4e8c\u7c7bPiola-Kirchhoff\u5e94\u529bSe<\/sup>\uff08\u5b9a\u4e49\u5728\u4e2d\u95f4\u6784\u578b\uff09\u548cGreen\u5f39\u6027\u5e94\u53d8Ee<\/sup>\u8868\u793a\uff1a\n <\/p>\n \n The elastic constitutive relation adopts the generalized Hooke’s law, expressed in terms of the second Piola-Kirchhoff stress Se<\/sup>\n (defined on the intermediate configuration) and Green elastic strain Ee<\/sup>:\n <\/p>\n <div class=\"formula\">\n Se<\/sup> = C : Ee<\/sup><\/b> \n \u5176\u4e2dC\u4e3a\u56db\u9636\u5f39\u6027\u521a\u5ea6\u5f20\u91cf<\/span>\n where C is the fourth-order elastic stiffness tensor<\/span>\n <\/div>\n \n \u5f53\u524d\u6784\u578b\u4e2d\u7684Kirchhoff\u5e94\u529bτ\u548cCauchy\u5e94\u529bσ\u5206\u522b\u4e3a\uff1a\n <\/p>\n \n The Kirchhoff stress τ and Cauchy stress σ in the current configuration are:\n <\/p>\n <div class=\"formula\">\n τ = Fe<\/sup> Se<\/sup> FeT<\/sup> = J σ<\/b> \n \u5176\u4e2dJ = det(F)\u4e3a\u4f53\u79ef\u6bd4<\/span>\n where J = det(F) is the volume ratio<\/span>\n <\/div>\n\n <h3 id=\"ch2-2\">\n 2.2 Schmid\u5b9a\u5f8b \/ Schmid’s Law<\/span>\n 2.2 Schmid’s Law<\/span>\n <\/h3>\n \n Schmid\u5b9a\u5f8b\u662f\u6676\u4f53\u5851\u6027\u7406\u8bba\u7684\u6838\u5fc3\uff0c\u5b83\u6307\u51fa\uff1a\u6ed1\u79fb\u7cfbα\u4e0a\u7684\u5206\u89e3\u526a\u5e94\u529b\uff08resolved shear stress\uff09τ(α)<\/sup>\u7b49\u4e8eCauchy\u5e94\u529bσ\u5728\u6ed1\u79fb\u65b9\u5411-\u6ed1\u79fb\u9762\u6cd5\u5411\u6295\u5f71\u4e0a\u7684\u5206\u91cf\u3002\n \u5728\u7a0b\u5e8f\u5b9e\u73b0\u4e2d\uff0c\u66f4\u5e38\u7528\u7684\u662f\u5bf9\u79f0\u5316\u7684Schmid\u5f20\u91cfP(α)<\/sup>\uff1a\n <\/p>\n \n Schmid’s law is the core of crystal plasticity theory, stating that the resolved shear stress τ(α)<\/sup> on slip system α\n equals the component of Cauchy stress σ projected onto the slip direction-slip plane normal pair.\n In program implementation, the symmetrized Schmid tensor P(α)<\/sup> is more commonly used:\n <\/p>\n <div class=\"formula\">\n P(α)<\/sup> = (1\/2)( s(α)<\/sup> ⊗ m(α)<\/sup> + m(α)<\/sup> ⊗ s(α)<\/sup> )<\/b>\n <\/div>\n \n \u5206\u89e3\u526a\u5e94\u529b\u53ef\u8868\u793a\u4e3a\uff1a\n <\/p>\n \n The resolved shear stress can be expressed as:\n <\/p>\n <div class=\"formula\">\n τ(α)<\/sup> = σ : P(α)<\/sup><\/b>\n <\/div>\n \n \u8fd9\u91cc”:”\u8868\u793a\u53cc\u70b9\u79ef\uff08\u5f20\u91cf\u7f29\u5e76\uff09\uff0c\u5373σij<\/sub>Pij<\/sub>\u3002Schmid\u5f20\u91cfP(α)<\/sup>\u662f\u5bf9\u79f0\u5f20\u91cf\uff0c\n \u5b83\u5c06\u5e94\u529b\u7a7a\u95f4\u4e2d\u9a71\u52a8\u7b2cα\u4e2a\u6ed1\u79fb\u7cfb\u526a\u5207\u53d8\u5f62\u7684\u65b9\u5411\u63d0\u53d6\u51fa\u6765\u3002\n \u5f53τ(α)<\/sup>\u8fbe\u5230\u8be5\u6ed1\u79fb\u7cfb\u7684\u4e34\u754c\u5206\u5207\u5e94\u529b\uff08critical resolved shear stress, CRSS\uff09g(α)<\/sup>\u65f6\uff0c\u6ed1\u79fb\u5f00\u59cb\u6fc0\u6d3b\u3002\n <\/p>\n \n Here “:” denotes the double dot product (tensor contraction), i.e., σij<\/sub>Pij<\/sub>.\n The Schmid tensor P(α)<\/sup> is symmetric and extracts the direction in stress space that drives shear deformation on the α-th slip system.\n Slip activates when τ(α)<\/sup> reaches the critical resolved shear stress (CRSS) g(α)<\/sup> of that slip system.\n <\/p>\n <div class=\"note\">\n \u7269\u7406\u610f\u4e49\uff1a<\/strong>Schmid\u5b9a\u5f8b\u8868\u660e\uff0c\u65e0\u8bba\u5e94\u529b\u72b6\u6001\u591a\u4e48\u590d\u6742\uff0c\u53ea\u6709\u4f5c\u7528\u5728\u6ed1\u79fb\u9762\u548c\u6ed1\u79fb\u65b9\u5411\u4e0a\u7684\u526a\u5e94\u529b\u5206\u91cf\u624d\u80fd\u9a71\u52a8\u4f4d\u9519\u6ed1\u79fb\u3002\u8fd9\u662f\u6676\u4f53\u5851\u6027\u5404\u5411\u5f02\u6027\u7684\u6839\u672c\u6765\u6e90\u2014\u2014\u4e0d\u540c\u6ed1\u79fb\u7cfb\u5728\u4e0d\u540c\u5e94\u529b\u72b6\u6001\u4e0b\u7684\u6fc0\u6d3b\u7a0b\u5ea6\u4e0d\u540c\u3002<\/span>\n Physical meaning:<\/strong> Schmid’s law indicates that regardless of how complex the stress state is, only the shear stress component acting on the slip plane and in the slip direction can drive dislocation glide. This is the fundamental source of plastic anisotropy in crystals\u2014the activation degree of different slip systems varies under different stress states.<\/span>\n <\/div>\n\n <h3 id=\"ch2-3\">\n 2.3 \u7387\u76f8\u5173\u6d41\u52a8\u5f8b \/ Rate-Dependent Flow Law<\/span>\n 2.3 Rate-Dependent Flow Law<\/span>\n <\/h3>\n \n \u9ec4\u6c38\u521aUMAT\u91c7\u7528\u7387\u76f8\u5173\uff08viscoplastic\uff09\u6d41\u52a8\u6cd5\u5219\uff0c\u907f\u514d\u4e86\u7387\u65e0\u5173\u6a21\u578b\u4e2d\u6ed1\u79fb\u7cfb\u6fc0\u6d3b-\u975e\u6fc0\u6d3b\u5207\u6362\u5e26\u6765\u7684\u6570\u503c\u56f0\u96be\u3002\n \u6bcf\u4e2a\u6ed1\u79fb\u7cfb\u4e0a\u7684\u526a\u5207\u5e94\u53d8\u7387\u7531\u4ee5\u4e0b\u5e42\u5f8b\uff08power law\uff09\u63cf\u8ff0\uff1a\n <\/p>\n \n Huang’s UMAT employs a rate-dependent (viscoplastic) flow rule, avoiding numerical difficulties\n associated with the activation-deactivation switching of slip systems in rate-independent models.\n The shear strain rate on each slip system is described by the following power law:\n <\/p>\n <div class=\"formula\">\n γ̇(α)<\/sup> = γ̇0<\/sub> · | τ(α)<\/sup> \/ g(α)<\/sup> |n<\/sup> · sign( τ(α)<\/sup> \/ g(α)<\/sup> )<\/b>\n <\/div>\n \n \u5f0f\u4e2d\u5404\u53c2\u6570\u542b\u4e49\u5982\u4e0b\uff1a\n <\/p>\n \n The parameters in the equation are defined as follows:\n <\/p>\n <ul>\n <li>\n γ̇0<\/sub><\/b>:\n \u53c2\u8003\u526a\u5207\u7387\uff08reference shear rate\uff09\uff0c\u5355\u4f4d\u4e3as-1<\/sup>\uff0c\u901a\u5e38\u53d6\u4e0e\u5b9e\u9a8c\u5e94\u53d8\u901f\u7387\u76f8\u540c\u91cf\u7ea7<\/span>\n Reference shear rate, in s-1<\/sup>, typically of the same order as the experimental strain rate<\/span>\n <\/li>\n <li>\n n<\/b>:\n \u7387\u654f\u611f\u6307\u6570\uff08rate sensitivity exponent\uff09\uff0c\u65e0\u91cf\u7eb2\u3002n\u503c\u8d8a\u5927\uff0c\u6750\u6599\u884c\u4e3a\u8d8a\u63a5\u8fd1\u7387\u65e0\u5173\uff1bn→∞\u65f6\u9000\u5316\u4e3a\u7387\u65e0\u5173\u6a21\u578b<\/span>\n Rate sensitivity exponent, dimensionless. Larger n means behavior closer to rate-independent; n→∞ recovers the rate-independent model<\/span>\n <\/li>\n <li>\n τ(α)<\/sup><\/b>:\n \u7b2cα\u4e2a\u6ed1\u79fb\u7cfb\u4e0a\u7684\u5206\u89e3\u526a\u5e94\u529b<\/span>\n Resolved shear stress on the α-th slip system<\/span>\n <\/li>\n <li>\n g(α)<\/sup><\/b>:\n \u7b2cα\u4e2a\u6ed1\u79fb\u7cfb\u7684\u5f53\u524d\u4e34\u754c\u5206\u5207\u5e94\u529b\uff08\u5373\u5f53\u524d\u5f3a\u5ea6\/\u786c\u5316\u72b6\u6001\uff09<\/span>\n Current critical resolved shear stress (i.e., current strength\/hardening state) of the α-th slip system<\/span>\n <\/li>\n <li>\n sign(x)<\/b>:\n \u7b26\u53f7\u51fd\u6570\uff0c\u53d6\u503c\u4e3a-1\u30010\u62161<\/span>\n Sign function, taking values -1, 0, or 1<\/span>\n <\/li>\n <\/ul>\n \n \u7387\u76f8\u5173\u6a21\u578b\u7684\u4f18\u52bf\u5728\u4e8e\uff1a\u6240\u6709\u6ed1\u79fb\u7cfb\u59cb\u7ec8\u5904\u4e8e”\u90e8\u5206\u6fc0\u6d3b”\u72b6\u6001\uff0c\u4e0d\u9700\u8981\u663e\u5f0f\u7684\u6fc0\u6d3b\/\u975e\u6fc0\u6d3b\u5207\u6362\u6761\u4ef6\uff1b\n \u8fd9\u5927\u5927\u63d0\u9ad8\u4e86\u6570\u503c\u8ba1\u7b97\u7684\u7a33\u5b9a\u6027\u548c\u6536\u655b\u6027\u3002\n \u5728\u5178\u578b\u91d1\u5c5e\u4e2d\uff0cn\u503c\u8303\u56f4\u7ea6\u4e3a20-200\uff1aFCC\u91d1\u5c5en≈20-100\uff0cBCC\u91d1\u5c5en≈50-200\uff0cHCP\u91d1\u5c5en≈20-80\u3002\n n=1\u5bf9\u5e94\u7ebf\u6027\u7c98\u6027\uff08\u725b\u987f\u6d41\u4f53\uff09\uff0cn\u5f88\u5927\u65f6\u8fd1\u4f3c\u4e8e\u7406\u60f3\u5851\u6027\u3002\n <\/p>\n \n The advantage of the rate-dependent model is that all slip systems are always in a “partially active” state,\n without requiring explicit active\/inactive switching conditions;\n this greatly improves numerical stability and convergence.\n In typical metals, n ranges from about 20-200: FCC metals n≈20-100, BCC metals n≈50-200, HCP metals n≈20-80.\n n=1 corresponds to linear viscosity (Newtonian fluid), and large n approximates ideal plasticity.\n <\/p>\n\n <h3 id=\"ch2-4\">\n 2.4 \u786c\u5316\u5f8b \/ Hardening Laws<\/span>\n 2.4 Hardening Laws<\/span>\n <\/h3>\n \n \u4e34\u754c\u5206\u5207\u5e94\u529bg(α)<\/sup>\u968f\u7d2f\u79ef\u5851\u6027\u526a\u5207\u5e94\u53d8γ = Σα<\/sub> ∫|γ̇(α)<\/sup>|dt\u800c\u6f14\u5316\uff0c\n \u6f14\u5316\u65b9\u7a0b\u4e3a\uff1a\n <\/p>\n \n The critical resolved shear stress g(α)<\/sup> evolves with the accumulated plastic shear strain\n γ = Σα<\/sub> ∫|γ̇(α)<\/sup>|dt, according to:\n <\/p>\n <div class=\"formula\">\n ğ(α)<\/sup> = Σβ<\/sub> hαβ<\/sub> |γ̇(β)<\/sup>|<\/b>\n <\/div>\n \n \u5176\u4e2dhαβ<\/sub>\u4e3a\u786c\u5316\u77e9\u9635\uff0c\u5176\u5bf9\u89d2\u5143\u7d20hαα<\/sub>\u8868\u793a\u81ea\u786c\u5316\uff08self-hardening\uff09\uff0c\n \u975e\u5bf9\u89d2\u5143\u7d20hαβ<\/sub>(α≠β)\u8868\u793a\u6f5c\u786c\u5316\uff08latent hardening\uff09\uff0c\n \u63cf\u8ff0\u4e0d\u540c\u6ed1\u79fb\u7cfb\u4e4b\u95f4\u7684\u4ea4\u4e92\u5f3a\u5316\u6548\u5e94\u3002\n \u9ec4\u6c38\u521aUMAT\u63d0\u4f9b\u4e24\u79cd\u786c\u5316\u6a21\u578b\u9009\u9879\uff1a\n <\/p>\n \n where hαβ<\/sub> is the hardening matrix. The diagonal elements hαα<\/sub> represent self-hardening,\n and the off-diagonal elements hαβ<\/sub> (α≠β) represent latent hardening,\n describing the interaction strengthening effect between different slip systems.\n Huang’s UMAT provides two hardening model options:\n <\/p>\n\n <h4>(a) \u53cc\u66f2\u6b63\u5272\u786c\u5316\u6a21\u578b \/ Hyperbolic Secant Hardening Model<\/h4>\n \n \u81ea\u786c\u5316\u6a21\u91cf\u91c7\u7528\u53cc\u66f2\u6b63\u5272\uff08sech\u00b2\uff09\u5f62\u5f0f\uff1a\n <\/p>\n \n The self-hardening modulus adopts a hyperbolic secant (sech\u00b2) form:\n <\/p>\n <div class=\"formula\">\n hαα<\/sub> = h(γ) = H0<\/sub> · sech\u00b2( H0<\/sub> γ \/ (τs<\/sub> – τ0<\/sub>) )<\/b>\n <\/div>\n \n \u7d2f\u79ef\u526a\u5207\u5e94\u53d8γ\u4ece0\u589e\u52a0\u5230∞\u65f6\uff0c\u786c\u5316\u6a21\u91cfh\u4eceH0<\/sub>\u5355\u8c03\u9012\u51cf\u52300\uff0c\n \u4e34\u754c\u5206\u5207\u5e94\u529b\u4ece\u521d\u59cb\u503cτ0<\/sub>\u6e10\u8fd1\u8d8b\u8fd1\u4e8e\u9971\u548c\u503cτs<\/sub>\u3002\n \u79ef\u5206\u540e\u53ef\u5f97g(γ)\u7684\u663e\u5f0f\u8868\u8fbe\u5f0f\uff1a\n <\/p>\n \n As the accumulated shear strain γ increases from 0 to ∞, the hardening modulus h decreases monotonically from H0<\/sub> to 0,\n and the critical resolved shear stress asymptotically approaches the saturation value τs<\/sub> from the initial value τ0<\/sub>.\n Integration yields the explicit expression for g(γ):\n <\/p>\n <div class=\"formula\">\n g(γ) = τs<\/sub> + (τ0<\/sub> – τs<\/sub>) · tanh( H0<\/sub> γ \/ (τs<\/sub> – τ0<\/sub>) )<\/b>\n <\/div>\n \n \u53c2\u6570\u8bf4\u660e\uff1aH0<\/sub>\u4e3a\u521d\u59cb\u786c\u5316\u6a21\u91cf\uff08MPa\uff09\uff0cτ0<\/sub>\u4e3a\u521d\u59cbCRSS\uff08MPa\uff09\uff0cτs<\/sub>\u4e3a\u9971\u548cCRSS\uff08MPa\uff09\u3002\n \u8be5\u6a21\u578b\u7b80\u6d01\u4e14\u7269\u7406\u610f\u4e49\u660e\u786e\uff0c\u9002\u7528\u4e8e\u5927\u591a\u6570FCC\u91d1\u5c5e\u3002\n <\/p>\n \n Parameter definitions: H0<\/sub> is the initial hardening modulus (MPa), τ0<\/sub> is the initial CRSS (MPa), and τs<\/sub> is the saturation CRSS (MPa).\n This model is concise with clear physical meaning and is suitable for most FCC metals.\n <\/p>\n\n <h4>(b) Bassani-Wu\u786c\u5316\u6a21\u578b \/ Bassani-Wu Hardening Model<\/h4>\n \n Bassani\u548cWu\uff081991\uff09\u63d0\u51fa\u7684\u786c\u5316\u6a21\u578b\u8003\u8651\u4e86\u6ed1\u79fb\u7cfb\u95f4\u4ea4\u4e92\u4f5c\u7528\u7684\u5404\u5411\u5f02\u6027\u7279\u5f81\u3002\n \u81ea\u786c\u5316\u6a21\u91cfF(γ)\u4e3a\uff1a\n <\/p>\n \n The hardening model proposed by Bassani and Wu (1991) accounts for the anisotropic characteristics of interaction between slip systems.\n The self-hardening modulus F(γ) is:\n <\/p>\n <div class=\"formula\">\n F(γ) = (H0<\/sub> – Hs<\/sub>) · sech\u00b2( (τs<\/sub> – τ0<\/sub>) γ \/ (γ0<\/sub> + γ) ) + Hs<\/sub><\/b>\n <\/div>\n \n \u6f5c\u786c\u5316\u7cfb\u6570qαβ<\/sub>\u53d6\u51b3\u4e8e\u6ed1\u79fb\u7cfb\u5bf9(α,β)\u7684\u4ea4\u4e92\u7c7b\u578b\uff08\u5171\u9762\/\u5171\u5411\/\u6b63\u4ea4\/Lomer-Cottrell\u9501\u7b49\uff09\uff0c\n \u786c\u5316\u77e9\u9635\u4e00\u822c\u5199\u4f5chαβ<\/sub> = qαβ<\/sub> · hββ<\/sub>\u3002\n \u5b8c\u6574\u7684\u6f5c\u786c\u5316\u77e9\u9635\u662f12×12\u7684\u5bf9\u79f0\u77e9\u9635\uff0c\u5305\u542b7\u79cd\u4e0d\u540c\u7684\u4ea4\u4e92\u7c7b\u578b\u3002\n <\/p>\n \n The latent hardening coefficient qαβ<\/sub> depends on the interaction type of the slip system pair (α,β)\n (coplanar\/codirectional\/orthogonal\/Lomer-Cottrell lock, etc.).\n The hardening matrix is generally written as hαβ<\/sub> = qαβ<\/sub> · hββ<\/sub>.\n The full latent hardening matrix is a 12×12 symmetric matrix containing 7 different interaction types.\n <\/p>\n \n Bassani\u6a21\u578b\u7684\u4f18\u52bf\u5728\u4e8e\u80fd\u591f\u66f4\u51c6\u786e\u5730\u63cf\u8ff0FCC\u6676\u4f53\u4e2d\u4e0d\u540c\u6ed1\u79fb\u7cfb\u4e4b\u95f4\u7684\u4ea4\u4e92\u786c\u5316\u6548\u5e94\uff0c\n \u7279\u522b\u662f\u975e\u5171\u9762\u6ed1\u79fb\u7cfb\u4e4b\u95f4\u7684\u5f3a\u76f8\u4e92\u4f5c\u7528\uff08\u5982Lomer-Cottrell\u9501\uff09\u3002\n \u4f46\u8be5\u6a21\u578b\u9700\u8981\u989d\u5916\u7684\u6f5c\u786c\u5316\u53c2\u6570\uff0c\u4e14\u4ea4\u4e92\u7c7b\u578b\u5206\u7c7b\u8f83\u4e3a\u590d\u6742\u3002\n <\/p>\n \n The advantage of the Bassani model is its ability to more accurately describe the interaction hardening effects between different slip systems in FCC crystals,\n especially the strong interactions between non-coplanar slip systems (such as Lomer-Cottrell locks).\n However, this model requires additional latent hardening parameters, and the interaction type classification is relatively complex.\n <\/p>\n\n <h3 id=\"ch2-5\">\n 2.5 \u5e94\u529b\u66f4\u65b0 \/ Stress Update (Jaumann Rate)<\/span>\n 2.5 Stress Update (Jaumann Rate)<\/span>\n <\/h3>\n \n \u5728\u6709\u9650\u53d8\u5f62\u6846\u67b6\u4e0b\uff0c\u672c\u6784\u5173\u7cfb\u5fc5\u987b\u4ee5\u5ba2\u89c2\uff08objective\uff09\u7684\u5e94\u529b\u7387\u5f62\u5f0f\u8868\u8ff0\uff0c\u4ee5\u6d88\u9664\u521a\u6027\u8f6c\u52a8\u5e26\u6765\u7684\u865a\u5047\u5e94\u529b\u53d8\u5316\u3002\n \u9ec4\u6c38\u521aUMAT\u91c7\u7528Jaumann\u5e94\u529b\u7387\uff1a\n <\/p>\n \n In the finite deformation framework, the constitutive relation must be expressed in an objective stress rate form\n to eliminate spurious stress changes due to rigid body rotation.\n Huang’s UMAT adopts the Jaumann stress rate:\n <\/p>\n <div class=\"formula\">\n ∇σ = Cep<\/sup> : De<\/sup> = Cep<\/sup> : (D – Dp<\/sup>)<\/b>\n <\/div>\n \n \u5176\u4e2d∇σ = σ̇ – W·σ + σ·W\u4e3aJaumann\u5e94\u529b\u7387\uff0c\n D\u4e3a\u53d8\u5f62\u7387\uff08symmetric part of velocity gradient L\uff09\uff0c\n W\u4e3a\u65cb\u7387\uff08skew-symmetric part of L\uff09\uff0c\n Cep<\/sup>\u4e3a\u5f39\u5851\u6027\u521a\u5ea6\u5f20\u91cf\uff0cDp<\/sup>\u4e3a\u5851\u6027\u53d8\u5f62\u7387\uff1a\n <\/p>\n \n where ∇σ = σ̇ – W·σ + σ·W is the Jaumann stress rate,\n D is the rate of deformation (symmetric part of velocity gradient L),\n W is the spin (skew-symmetric part of L),\n Cep<\/sup> is the elastoplastic stiffness tensor, and Dp<\/sup> is the plastic rate of deformation:\n <\/p>\n <div class=\"formula\">\n Dp<\/sup> = Σα<\/sub> γ̇(α)<\/sup> P(α)<\/sup><\/b>\n <\/div>\n \n \u9690\u5f0f\u65f6\u95f4\u79ef\u5206\u91c7\u7528Newton-Raphson\u8fed\u4ee3\u6c42\u89e3\u6bcf\u4e2a\u65f6\u95f4\u6b65\u672b\u7684\u5e94\u529b\u548c\u72b6\u6001\u53d8\u91cf\u3002\n \u7ebf\u6027\u5316\u540e\u5f97\u5230\u4ee5Δγ(α)<\/sup>\u4e3a\u672a\u77e5\u91cf\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u901a\u8fc7LU\u5206\u89e3\u6c42\u89e3\u3002\n \u6536\u655b\u540e\u66f4\u65b0STATEV\u6570\u7ec4\u4e2d\u7684\u5e94\u529b\u3001\u5e94\u53d8\u548c\u5404\u6ed1\u79fb\u7cfb\u53d8\u91cf\u3002\n <\/p>\n \n Implicit time integration employs Newton-Raphson iteration to solve for the stress and state variables at the end of each time increment.\n Linearization yields a linear system of equations with Δγ(α)<\/sup> as unknowns, solved by LU decomposition.\n After convergence, the STATEV array is updated with stress, strain, and slip system variables.\n <\/p>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch3\">\n <h2>\n 03<\/span>\n Abaqus UMAT\u63a5\u53e3 \/ UMAT Interface<\/span>\n UMAT Interface<\/span>\n <\/h2>\n\n <h3 id=\"ch3-1\">\n 3.1 \u5b8c\u6574Fortran\u7b7e\u540d \/ Full Fortran Signature<\/span>\n 3.1 Full Fortran Signature<\/span>\n <\/h3>\n \n Abaqus\/Standard\u7684\u7528\u6237\u6750\u6599\u5b50\u7a0b\u5e8fUMAT\u5177\u6709\u56fa\u5b9a\u7684\u63a5\u53e3\u7b7e\u540d\u3002\u9ec4\u6c38\u521a\u5355\u6676\u5851\u6027\u5b50\u7a0b\u5e8f\u5728\u8be5\u6846\u67b6\u4e0b\u5b9e\u73b0\uff0c\u5b8c\u6574\u58f0\u660e\u5982\u4e0b\uff1a\n <\/p>\n \n The Abaqus\/Standard User Material subroutine (UMAT) has a fixed interface signature. Huang’s single-crystal plasticity subroutine is implemented within this framework. The complete declaration is:\n <\/p>\n <pre><code> SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,\n 1 RPL,DDSDDT,DRPLDE,DRPLDT,\n 2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME,\n 3 NDI,NSHR,NTENS,NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,\n 4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,KSTEP,KINC)\nC-----------------------------------------------------------------------\nC UMAT for single crystal plasticity\nC Developed by Yonggang Huang\nC-----------------------------------------------------------------------\n INCLUDE 'ABA_PARAM.INC'\n CHARACTER*80 CMNAME\n DIMENSION STRESS(NTENS),STATEV(NSTATV),\n 1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS),\n 2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),\n 3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3)\nC ... user coding ...\n RETURN\n END<\/code><\/pre>\n\n <h3 id=\"ch3-2\">\n 3.2 \u53d8\u91cf\u8bf4\u660e\u8868 \/ Variable Specification Table<\/span>\n 3.2 Variable Specification Table<\/span>\n <\/h3>\n \n \u4e0b\u8868\u5217\u51fa\u4e86UMAT\u63a5\u53e3\u4e2d\u4e0e\u5355\u6676\u5851\u6027\u5b9e\u73b0\u5bc6\u5207\u76f8\u5173\u7684\u8f93\u5165\/\u8f93\u51fa\u53d8\u91cf\uff1a\n <\/p>\n \n The following table lists the input\/output variables in the UMAT interface most relevant to the single-crystal plasticity implementation:\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>\n \u53d8\u91cf\u540d \/ Variable<\/span>\n Variable<\/span>\n <\/th>\n <th>\n \u7c7b\u578b \/ Type<\/span>\n Type<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td><code>STRESS(NTENS)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u5e94\u529b\u5f20\u91cf\uff08\u8f93\u5165\u4e3a\u589e\u91cf\u5f00\u59cb\u65f6\u7684\u5e94\u529b\uff0c\u8f93\u51fa\u4e3a\u589e\u91cf\u7ed3\u675f\u65f6\u7684\u5e94\u529b\uff09\uff0c\u6309Abaqus\u7ea6\u5b9a\u5b58\u50a8\u4e3aVoigt\u5411\u91cf\uff1aσ11, σ22, σ33, σ12, σ13, σ23<\/span>\n Stress tensor (input: stress at increment start; output: stress at increment end), stored as Voigt vector per Abaqus convention: σ11, σ22, σ33, σ12, σ13, σ23<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>STATEV(NSTATV)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u72b6\u6001\u53d8\u91cf\u6570\u7ec4\uff08solution-dependent state variables\uff09\uff0c\u7528\u4e8e\u5b58\u50a8\u5851\u6027\u5e94\u53d8\u3001\u7d2f\u79ef\u526a\u5207\u3001\u6ed1\u79fb\u7cfb\u5f3a\u5ea6\u7b49\u5386\u53f2\u4fe1\u606f\u3002\u5fc5\u987b\u5728\u672c\u5b50\u7a0b\u5e8f\u4e2d\u5b9a\u4e49\u66f4\u65b0\u89c4\u5219\u3002<\/span>\n Solution-dependent state variable array for storing plastic strain, accumulated shear, slip system strength, and other history information. Update rules must be defined in this subroutine.<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DDSDDE(NTENS,NTENS)<\/code><\/td>\n <td>\n \u4e8c\u7ef4\u6570\u7ec4 \/ Matrix<\/span>\n Matrix<\/span>\n <\/td>\n <td>\n \u6750\u6599Jacobian\u77e9\u9635\uff08\u4e00\u81f4\u5207\u7ebf\u6a21\u91cf\uff09\uff0c∂Δσ\/∂Δε\u3002\u8fd9\u662f\u9690\u5f0f\u6c42\u89e3\u5668\u4fdd\u8bc1\u4e8c\u9636\u6536\u655b\u7684\u5173\u952e\uff0c\u5fc5\u987b\u63d0\u4f9b\u51c6\u786e\u7684\u5f39\u5851\u6027\u5207\u7ebf\u521a\u5ea6\u3002<\/span>\n Material Jacobian matrix (consistent tangent modulus), ∂Δσ\/∂Δε. This is essential for the implicit solver to achieve quadratic convergence; accurate elastoplastic tangent stiffness must be provided.<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>STRAN(NTENS)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u589e\u91cf\u5f00\u59cb\u65f6\u7684\u603b\u5e94\u53d8\uff08engineering strain\uff0c\u526a\u5e94\u53d8\u4e0d\u9664\u4ee52\uff09<\/span>\n Total strain at increment start (engineering strain; shear strains are not divided by 2)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DSTRAN(NTENS)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u5e94\u53d8\u589e\u91cf\uff08engineering strain increment\uff09\uff0c\u7531Abaqus\u6839\u636e\u6574\u4f53\u5e73\u8861\u8fed\u4ee3\u63d0\u4f9b<\/span>\n Strain increment (engineering strain increment), provided by Abaqus based on global equilibrium iteration<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DFGRD0(3,3)<\/code><\/td>\n <td>\n \u77e9\u9635 \/ Matrix<\/span>\n Matrix<\/span>\n <\/td>\n <td>\n \u589e\u91cf\u5f00\u59cb\u65f6\u7684\u53d8\u5f62\u68af\u5ea6F\uff083×3\u77e9\u9635\uff09<\/span>\n Deformation gradient F at increment start (3×3 matrix)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DFGRD1(3,3)<\/code><\/td>\n <td>\n \u77e9\u9635 \/ Matrix<\/span>\n Matrix<\/span>\n <\/td>\n <td>\n \u589e\u91cf\u7ed3\u675f\u65f6\u7684\u53d8\u5f62\u68af\u5ea6F\uff08\u7531Abaqus\u57fa\u4e8e\u6574\u4f53\u4f4d\u79fb\u573a\u8ba1\u7b97\uff09<\/span>\n Deformation gradient F at increment end (computed by Abaqus based on global displacement field)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>PROPS(NPROPS)<\/code><\/td>\n <td>\n \u6570\u7ec4 \/ Array<\/span>\n Array<\/span>\n <\/td>\n <td>\n \u6750\u6599\u53c2\u6570\u6570\u7ec4\uff0c\u5728Abaqus\u8f93\u5165\u6587\u4ef6\u4e2d\u901a\u8fc7*Material, *User Material\u5173\u952e\u5b57\u5b9a\u4e49<\/span>\n Material property array, defined in the Abaqus input file via *Material, *User Material keywords<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>NPROPS<\/code><\/td>\n <td>\n \u6574\u6570 \/ Integer<\/span>\n Integer<\/span>\n <\/td>\n <td>\n PROPS\u6570\u7ec4\u957f\u5ea6\uff0c\u9ec4\u6c38\u521a\u7a0b\u5e8f\u901a\u5e38\u8981\u6c42NPROPS ≥ 160<\/span>\n Length of PROPS array; Huang’s program typically requires NPROPS ≥ 160<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>NSTATV<\/code><\/td>\n <td>\n \u6574\u6570 \/ Integer<\/span>\n Integer<\/span>\n <\/td>\n <td>\n STATEV\u6570\u7ec4\u957f\u5ea6\uff0c\u53d6\u51b3\u4e8e\u6ed1\u79fb\u7cfb\u6570\u91cf\u548c\u5b58\u50a8\u9700\u6c42\uff0c\u901a\u5e38NSTATV = 6 + NSLIP + NSLIP + …<\/span>\n Length of STATEV array, depending on number of slip systems and storage requirements; typically NSTATV = 6 + NSLIP + NSLIP + …<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DROT(3,3)<\/code><\/td>\n <td>\n \u77e9\u9635 \/ Matrix<\/span>\n Matrix<\/span>\n <\/td>\n <td>\n \u589e\u91cf\u65cb\u8f6c\u5f20\u91cf\uff08\u7531\u6781\u5206\u89e3\u5f97\u5230\uff09\uff0cAbaqus\u81ea\u52a8\u8ba1\u7b97\u5e76\u63d0\u4f9b<\/span>\n Incremental rotation tensor (from polar decomposition), automatically computed and provided by Abaqus<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>DTIME<\/code><\/td>\n <td>\n \u53cc\u7cbe\u5ea6 \/ Double<\/span>\n Double<\/span>\n <\/td>\n <td>\n \u65f6\u95f4\u589e\u91cf\u5927\u5c0f\uff08\u5728\u7387\u76f8\u5173\u6750\u6599\u4e2d\u7528\u4e8e\u8ba1\u7b97γ̇\uff09<\/span>\n Time increment size (used to compute γ̇ in rate-dependent materials)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>PNEWDT<\/code><\/td>\n <td>\n \u53cc\u7cbe\u5ea6 \/ Double<\/span>\n Double<\/span>\n <\/td>\n <td>\n \u5efa\u8bae\u7684\u4e0b\u4e00\u589e\u91cf\u65f6\u95f4\u6bd4\u4f8b\u56e0\u5b50\uff080 < PNEWDT < 1\uff09\u3002\u82e5\u5c40\u90e8\u8fed\u4ee3\u4e0d\u6536\u655b\uff0c\u53ef\u8bbe\u7f6e\u6b64\u503c\u4f7fAbaqus\u81ea\u52a8\u51cf\u5c0f\u65f6\u95f4\u6b65\u3002<\/span>\n Suggested ratio for next increment time step (0 < PNEWDT < 1). If local iteration does not converge, set this to let Abaqus automatically reduce the time step.<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td><code>NOEL, NPT<\/code><\/td>\n <td>\n \u6574\u6570 \/ Integer<\/span>\n Integer<\/span>\n <\/td>\n <td>\n \u5355\u5143\u7f16\u53f7\u548c\u79ef\u5206\u70b9\u7f16\u53f7\uff0c\u7528\u4e8e\u8c03\u8bd5\u548c\u5b9a\u4f4d<\/span>\n Element number and integration point number, for debugging and localization<\/span>\n <\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch3-3\">\n 3.3 STATEV\u5e03\u5c40 \/ State Variable Layout Table<\/span>\n 3.3 State Variable Layout Table<\/span>\n <\/h3>\n \n STATEV\u6570\u7ec4\u662fUMAT\u5b50\u7a0b\u5e8f\u7684\u6838\u5fc3\u6570\u636e\u7ed3\u6784\uff0c\u7528\u4e8e\u5728\u589e\u91cf\u6b65\u4e4b\u95f4\u4f20\u9012\u548c\u4fdd\u5b58\u6750\u6599\u7684\u5386\u53f2\u72b6\u6001\u3002\n \u9ec4\u6c38\u521aUMAT\u7684STATEV\u5e03\u5c40\u5982\u4e0b\uff08\u4ee5FCC 12\u6ed1\u79fb\u7cfb\u4e3a\u4f8b\uff09\uff1a\n <\/p>\n \n The STATEV array is the core data structure of the UMAT subroutine, used to pass and preserve material history states between increments.\n The STATEV layout in Huang’s UMAT is as follows (for FCC 12 slip systems as an example):\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>\n \u7d22\u5f15\u8303\u56f4 \/ Index Range<\/span>\n Index Range<\/span>\n <\/th>\n <th>\n \u5185\u5bb9 \/ Content<\/span>\n Content<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>1 – 6<\/td>\n <td>\n \u5851\u6027\u5e94\u53d8 \/ Plastic strain<\/span>\n Plastic strain<\/span>\n <\/td>\n <td>\n \u7d2f\u79ef\u5851\u6027\u5e94\u53d8\u5206\u91cf εp<\/sup>11<\/sub>, εp<\/sup>22<\/sub>, εp<\/sup>33<\/sub>, γp<\/sup>12<\/sub>, γp<\/sup>13<\/sub>, γp<\/sup>23<\/sub>\uff08\u5de5\u7a0b\u526a\u5e94\u53d8\uff09<\/span>\n Accumulated plastic strain components εp<\/sup>11<\/sub>, εp<\/sup>22<\/sub>, εp<\/sup>33<\/sub>, γp<\/sup>12<\/sub>, γp<\/sup>13<\/sub>, γp<\/sup>23<\/sub> (engineering shear strains)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>7 – 18<\/td>\n <td>\n \u7d2f\u79ef\u526a\u5207\u5e94\u53d8 \/ Accumulated shear<\/span>\n Accumulated shear<\/span>\n <\/td>\n <td>\n \u5404\u6ed1\u79fb\u7cfb\u7d2f\u79ef\u526a\u5207\u5e94\u53d8 γ(α)<\/sup> = ∫|γ̇(α)<\/sup>|dt\uff0cα = 1,…,12<\/span>\n Accumulated shear strain on each slip system γ(α)<\/sup> = ∫|γ̇(α)<\/sup>|dt, α = 1,…,12<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>19 – 30<\/td>\n <td>\n \u5f53\u524d\u6ed1\u79fb\u7cfb\u5f3a\u5ea6 \/ Current slip strength<\/span>\n Current slip strength<\/span>\n <\/td>\n <td>\n \u5404\u6ed1\u79fb\u7cfb\u5f53\u524d\u4e34\u754c\u5206\u5207\u5e94\u529b g(α)<\/sup>\uff0cα = 1,…,12<\/span>\n Current critical resolved shear stress g(α)<\/sup> for each slip system, α = 1,…,12<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>31 – 39<\/td>\n <td>\n \u5f39\u6027\u53d8\u5f62\u68af\u5ea6 \/ Elastic def. grad.<\/span>\n Elastic def. grad.<\/span>\n <\/td>\n <td>\n Fe<\/sup>\u76849\u4e2a\u5206\u91cf\uff083×3\u77e9\u9635\u6309\u5217\u4f18\u5148\u5b58\u50a8\uff09<\/span>\n 9 components of Fe<\/sup> (3×3 matrix stored column-major)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>40 – 48<\/td>\n <td>\n \u5851\u6027\u53d8\u5f62\u68af\u5ea6 \/ Plastic def. grad.<\/span>\n Plastic def. grad.<\/span>\n <\/td>\n <td>\n Fp<\/sup>\u76849\u4e2a\u5206\u91cf\uff083×3\u77e9\u9635\u6309\u5217\u4f18\u5148\u5b58\u50a8\uff09<\/span>\n 9 components of Fp<\/sup> (3×3 matrix stored column-major)<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>49 – 54<\/td>\n <td>\n \u5e94\u529b \/ Stress<\/span>\n Stress<\/span>\n <\/td>\n <td>\n \u5b58\u50a8Cauchy\u5e94\u529b\u5206\u91cf\uff0c\u4fbf\u4e8e\u540e\u5904\u7406\u8f93\u51fa<\/span>\n Cauchy stress components stored for post-processing output<\/span>\n <\/td>\n <\/tr>\n <tr>\n <td>55+<\/td>\n <td>\n \u5176\u4ed6\u8f85\u52a9\u53d8\u91cf \/ Aux. variables<\/span>\n Aux. variables<\/span>\n <\/td>\n <td>\n \u6676\u4f53\u53d6\u5411\u3001Euler\u89d2\u3001\u65cb\u8f6c\u77e9\u9635\u5206\u91cf\u7b49\uff08\u5177\u4f53\u53d6\u51b3\u4e8ePROPS\u914d\u7f6e\uff09<\/span>\n Crystal orientation, Euler angles, rotation matrix components, etc. (depending on PROPS configuration)<\/span>\n <\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <div class=\"note\">\n \u91cd\u8981\u63d0\u793a\uff1a<\/strong>STATEV\u6570\u7ec4\u5728Abaqus\u4e2d\u901a\u8fc7*Depvar\u5173\u952e\u5b57\u5b9a\u4e49\u5927\u5c0f\u3002\u5fc5\u987b\u786e\u4fddNSTATV\u8db3\u591f\u5bb9\u7eb3\u6240\u6709\u9700\u8981\u5b58\u50a8\u7684\u72b6\u6001\u53d8\u91cf\uff0c\u5426\u5219\u4f1a\u53d1\u751f\u6570\u7ec4\u8d8a\u754c\u9519\u8bef\u6216\u72b6\u6001\u4fe1\u606f\u4e22\u5931\u3002<\/span>\n Important:<\/strong> The STATEV array size is defined in Abaqus via the *Depvar keyword. Ensure NSTATV is large enough to hold all state variables; otherwise array overflow or state loss will occur.<\/span>\n <\/div>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch4\">\n <h2>\n 04<\/span>\n \u6750\u6599\u53c2\u6570 \/ Material Parameters<\/span>\n Material Parameters<\/span>\n <\/h2>\n\n \n \u9ec4\u6c38\u521aUMAT\u901a\u8fc7PROPS\u6570\u7ec4\u4f20\u5165\u5168\u90e8\u6750\u6599\u53c2\u6570\u3002PROPS\u7684\u7d22\u5f15\u5e03\u5c40\u7ecf\u8fc7\u7cbe\u5fc3\u8bbe\u8ba1\uff0c\n \u5c06\u5f39\u6027\u5e38\u6570\u3001\u6ed1\u79fb\u7cfb\u5b9a\u4e49\u3001\u6676\u4f53\u53d6\u5411\u3001\u7387\u76f8\u5173\u53c2\u6570\u3001\u786c\u5316\u53c2\u6570\u548c\u79ef\u5206\u63a7\u5236\u53c2\u6570\u5206\u533a\u5b58\u50a8\u3002\n \u4ee5\u4e0b\u5404\u8282\u8be6\u7ec6\u5217\u51fa\u6bcf\u4e2aPROPS\u7d22\u5f15\u5bf9\u5e94\u7684\u53c2\u6570\u542b\u4e49\u3002\n <\/p>\n \n Huang’s UMAT receives all material parameters through the PROPS array. The PROPS index layout is carefully designed,\n partitioning elastic constants, slip system definitions, crystal orientation, rate-dependent parameters, hardening parameters, and integration control parameters.\n The following sections detail the meaning of each PROPS index.\n <\/p>\n\n <h3 id=\"ch4-1\">\n 4.1 \u5f39\u6027\u5e38\u6570 PROPS(1-21) \/ Elastic Constants<\/span>\n 4.1 Elastic Constants PROPS(1-21)<\/span>\n <\/h3>\n \n \u5f39\u6027\u5e38\u6570\u533a\u5b9a\u4e49\u6676\u4f53\u6750\u6599\u7684\u5404\u5411\u5f02\u6027\u5f39\u6027\u884c\u4e3a\u3002\u5bf9\u4e8e\u7acb\u65b9\u6676\u7cfb\uff0c\u53ea\u9700\u89813\u4e2a\u72ec\u7acb\u5e38\u6570\uff08C11, C12, C44\uff09\uff0c\n \u4f46\u7a0b\u5e8f\u63a5\u53e3\u9884\u7559\u4e86\u5b8c\u657421\u4e2a\u53c2\u6570\u4ee5\u652f\u6301\u4e00\u822c\u5404\u5411\u5f02\u6027\u6750\u6599\uff08\u4e09\u659c\u6676\u7cfb\uff09\u3002\n <\/p>\n \n The elastic constant section defines the anisotropic elastic behavior of the crystal material.\n For cubic crystals, only 3 independent constants (C11, C12, C44) are needed,\n but the program interface reserves a full 21 parameters to support general anisotropic materials (triclinic).\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>1<\/td><td>C11<\/td><td>C11<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC11\uff08MPa\uff09<\/span>Elastic stiffness matrix component C11 (MPa)<\/span><\/td><\/tr>\n <tr><td>2<\/td><td>C12<\/td><td>C12<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC12\uff08MPa\uff09<\/span>Elastic stiffness matrix component C12 (MPa)<\/span><\/td><\/tr>\n <tr><td>3<\/td><td>C13<\/td><td>C13<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC13\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC13=C12<\/span>Elastic stiffness matrix component C13 (MPa); C13=C12 for cubic<\/span><\/td><\/tr>\n <tr><td>4<\/td><td>C14<\/td><td>C14<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC14\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC14=0<\/span>Elastic stiffness matrix component C14 (MPa); C14=0 for cubic<\/span><\/td><\/tr>\n <tr><td>5<\/td><td>C15<\/td><td>C15<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC15\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC15=0<\/span>Elastic stiffness matrix component C15 (MPa); C15=0 for cubic<\/span><\/td><\/tr>\n <tr><td>6<\/td><td>C16<\/td><td>C16<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC16\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC16=0<\/span>Elastic stiffness matrix component C16 (MPa); C16=0 for cubic<\/span><\/td><\/tr>\n <tr><td>7<\/td><td>C22<\/td><td>C22<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC22\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC22=C11<\/span>Elastic stiffness matrix component C22 (MPa); C22=C11 for cubic<\/span><\/td><\/tr>\n <tr><td>8<\/td><td>C23<\/td><td>C23<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC23\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC23=C12<\/span>Elastic stiffness matrix component C23 (MPa); C23=C12 for cubic<\/span><\/td><\/tr>\n <tr><td>9<\/td><td>C24<\/td><td>C24<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC24\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC24=0<\/span>Elastic stiffness matrix component C24 (MPa); C24=0 for cubic<\/span><\/td><\/tr>\n <tr><td>10<\/td><td>C25<\/td><td>C25<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC25\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC25=0<\/span>Elastic stiffness matrix component C25 (MPa); C25=0 for cubic<\/span><\/td><\/tr>\n <tr><td>11<\/td><td>C26<\/td><td>C26<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC26\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC26=0<\/span>Elastic stiffness matrix component C26 (MPa); C26=0 for cubic<\/span><\/td><\/tr>\n <tr><td>12<\/td><td>C33<\/td><td>C33<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC33\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC33=C11<\/span>Elastic stiffness matrix component C33 (MPa); C33=C11 for cubic<\/span><\/td><\/tr>\n <tr><td>13<\/td><td>C34<\/td><td>C34<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC34\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC34=0<\/span>Elastic stiffness matrix component C34 (MPa); C34=0 for cubic<\/span><\/td><\/tr>\n <tr><td>14<\/td><td>C35<\/td><td>C35<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC35\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC35=0<\/span>Elastic stiffness matrix component C35 (MPa); C35=0 for cubic<\/span><\/td><\/tr>\n <tr><td>15<\/td><td>C36<\/td><td>C36<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC36\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC36=0<\/span>Elastic stiffness matrix component C36 (MPa); C36=0 for cubic<\/span><\/td><\/tr>\n <tr><td>16<\/td><td>C44<\/td><td>C44<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC44\uff08MPa\uff09<\/span>Elastic stiffness matrix component C44 (MPa)<\/span><\/td><\/tr>\n <tr><td>17<\/td><td>C45<\/td><td>C45<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC45\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC45=0<\/span>Elastic stiffness matrix component C45 (MPa); C45=0 for cubic<\/span><\/td><\/tr>\n <tr><td>18<\/td><td>C46<\/td><td>C46<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC46\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC46=0<\/span>Elastic stiffness matrix component C46 (MPa); C46=0 for cubic<\/span><\/td><\/tr>\n <tr><td>19<\/td><td>C55<\/td><td>C55<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC55\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC55=C44<\/span>Elastic stiffness matrix component C55 (MPa); C55=C44 for cubic<\/span><\/td><\/tr>\n <tr><td>20<\/td><td>C56<\/td><td>C56<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC56\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC56=0<\/span>Elastic stiffness matrix component C56 (MPa); C56=0 for cubic<\/span><\/td><\/tr>\n <tr><td>21<\/td><td>C66<\/td><td>C66<\/td><td>\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u5206\u91cfC66\uff08MPa\uff09\uff0c\u7acb\u65b9\u6676\u7cfb\u4e0bC66=C44<\/span>Elastic stiffness matrix component C66 (MPa); C66=C44 for cubic<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch4-2\">\n 4.2 \u6ed1\u79fb\u7cfb\u5b9a\u4e49 PROPS(25-56) \/ Slip System Definition<\/span>\n 4.2 Slip System Definition PROPS(25-56)<\/span>\n <\/h3>\n \n \u6ed1\u79fb\u7cfb\u5b9a\u4e49\u533a\u7528\u4e8e\u6307\u5b9a\u6bcf\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411s\u548c\u6ed1\u79fb\u9762\u6cd5\u5411m\uff08\u5728\u6676\u4f53\u5750\u6807\u7cfb\u4e2d\uff09\u3002\n \u6bcf\u7ec4\u6ed1\u79fb\u7cfb\u9700\u89813\u4e2a\u65b9\u5411\u5206\u91cf\u548c3\u4e2a\u6cd5\u5411\u5206\u91cf\u3002\u5bf9\u4e8eFCC\u7ed3\u6784\uff0c\u901a\u5e38\u5b9a\u4e4912\u4e2a\u6ed1\u79fb\u7cfb\uff08{111}<110>\uff09\u3002\n <\/p>\n \n The slip system definition section specifies the slip direction s and slip plane normal m (in crystal coordinates) for each slip system.\n Each slip system requires 3 direction components and 3 normal components. For FCC, 12 slip systems ({111}<110>) are typically defined.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>22<\/td><td>NSLIP<\/td><td>NSLIP<\/td><td>\u6ed1\u79fb\u7cfb\u603b\u6570\uff08\u6574\u6570\uff0c\u5982FCC=12\uff09<\/span>Total number of slip systems (integer, e.g., FCC=12)<\/span><\/td><\/tr>\n <tr><td>23<\/td><td>NSLIP1<\/td><td>NSLIP1<\/td><td>\u7b2c\u4e00\u65cf\u6ed1\u79fb\u7cfb\u6570\u91cf\uff08\u5982FCC\u4e2d\u4e3a12\uff09<\/span>Number of first-family slip systems (e.g., 12 for FCC)<\/span><\/td><\/tr>\n <tr><td>24<\/td><td>NSLIP2<\/td><td>NSLIP2<\/td><td>\u7b2c\u4e8c\u65cf\u6ed1\u79fb\u7cfb\u6570\u91cf\uff08\u5982\u6709\u9700\u8981\uff09<\/span>Number of second-family slip systems (if needed)<\/span><\/td><\/tr>\n <tr><td>25-27<\/td><td>s1(1-3)<\/td><td>Slip dir. 1<\/td><td>\u7b2c1\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411\u5206\u91cf(sx,sy,sz)<\/span>Slip direction components (sx,sy,sz) for slip system 1<\/span><\/td><\/tr>\n <tr><td>28-30<\/td><td>m1(1-3)<\/td><td>Slip normal 1<\/td><td>\u7b2c1\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u9762\u6cd5\u5411\u5206\u91cf(mx,my,mz)<\/span>Slip plane normal components (mx,my,mz) for slip system 1<\/span><\/td><\/tr>\n <tr><td>31-33<\/td><td>s2(1-3)<\/td><td>Slip dir. 2<\/td><td>\u7b2c2\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411\u5206\u91cf<\/span>Slip direction components for slip system 2<\/span><\/td><\/tr>\n <tr><td>34-36<\/td><td>m2(1-3)<\/td><td>Slip normal 2<\/td><td>\u7b2c2\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u9762\u6cd5\u5411\u5206\u91cf<\/span>Slip plane normal components for slip system 2<\/span><\/td><\/tr>\n <tr><td colspan=\"4\" style=\"text-align:center\">\n … \u4f9d\u6b64\u7c7b\u63a8\uff0c\u6bcf6\u4e2aPROPS\u5b9a\u4e49\u4e00\u4e2a\u6ed1\u79fb\u7cfb …<\/span>\n … and so on; each slip system occupies 6 PROPS entries …<\/span>\n <\/td><\/tr>\n <tr><td>55-57<\/td><td>s12(1-3)<\/td><td>Slip dir. 12<\/td><td>\u7b2c12\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u65b9\u5411\u5206\u91cf<\/span>Slip direction components for slip system 12<\/span><\/td><\/tr>\n <tr><td>58-60<\/td><td>m12(1-3)<\/td><td>Slip normal 12<\/td><td>\u7b2c12\u4e2a\u6ed1\u79fb\u7cfb\u7684\u6ed1\u79fb\u9762\u6cd5\u5411\u5206\u91cf<\/span>Slip plane normal components for slip system 12<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n \n FCC\u6676\u4f53\u5178\u578b\u768412\u4e2a\u6ed1\u79fb\u7cfb\u4e3a{111}<110>\u65cf\uff0c\u5206\u4e3a4\u4e2a\u6ed1\u79fb\u9762\uff0c\u6bcf\u4e2a\u9762\u67093\u4e2a\u6ed1\u79fb\u65b9\u5411\u3002\n \u4ee5\u4e0b\u4e3a\u6676\u4f53\u5750\u6807\u7cfb\u4e2d\u7684\u6807\u51c6\u5b9a\u4e49\uff08\u65e0\u9700\u5f52\u4e00\u5316\uff0c\u7a0b\u5e8f\u5185\u90e8\u4f1a\u81ea\u52a8\u5904\u7406\uff09\uff1a\n <\/p>\n \n The typical 12 slip systems for FCC crystals belong to the {111}<110> family, divided into 4 slip planes with 3 slip directions each.\n The following is the standard definition in crystal coordinates (normalization is handled internally by the program):\n <\/p>\n\n <pre><code>C FCC {111}<110> slip systems (standard definition)\nC Plane 1 (111): directions [1-10], [10-1], [01-1]\nC Plane 2 (1-11): directions [110], [101], [0-11]\nC Plane 3 (-111): directions [110], [10-1], [011]\nC Plane 4 (11-1): directions [1-10], [101], [011]<\/code><\/pre>\n\n <h3 id=\"ch4-3\">\n 4.3 \u6676\u4f53\u53d6\u5411 PROPS(57-70) \/ Crystal Orientation<\/span>\n 4.3 Crystal Orientation PROPS(57-70)<\/span>\n <\/h3>\n \n \u6676\u4f53\u53d6\u5411\u533a\u5b9a\u4e49\u6676\u4f53\u5750\u6807\u7cfb\u4e0e\u5168\u5c40\uff08\u8bd5\u6837\uff09\u5750\u6807\u7cfb\u4e4b\u95f4\u7684\u65cb\u8f6c\u5173\u7cfb\u3002\n \u901a\u5e38\u91c7\u7528Bunge Euler\u89d2(φ1, Φ, φ2)\u6216\u65b9\u5411\u4f59\u5f26\u77e9\u9635\u8868\u793a\u3002\n <\/p>\n \n The crystal orientation section defines the rotation relationship between the crystal coordinate system and the global (sample) coordinate system.\n Bunge Euler angles (φ1, Φ, φ2) or the direction cosine matrix are commonly used.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>57-59<\/td><td>Euler\u89d21-3<\/td><td>Euler angles<\/td><td>Bunge Euler\u89d2 φ1, Φ, φ2\uff08\u5355\u4f4d\uff1a\u5ea6\uff09<\/span>Bunge Euler angles φ1, Φ, φ2 (in degrees)<\/span><\/td><\/tr>\n <tr><td>60-62<\/td><td>\u6676\u4f53\u8f741<\/td><td>Crystal axis 1<\/td><td>\u5168\u5c40\u5750\u6807\u7cfb\u4e2d\u6676\u4f53[100]\u8f74\u7684\u65b9\u5411\u4f59\u5f26<\/span>Direction cosines of crystal [100] axis in global coordinates<\/span><\/td><\/tr>\n <tr><td>63-65<\/td><td>\u6676\u4f53\u8f742<\/td><td>Crystal axis 2<\/td><td>\u5168\u5c40\u5750\u6807\u7cfb\u4e2d\u6676\u4f53[010]\u8f74\u7684\u65b9\u5411\u4f59\u5f26<\/span>Direction cosines of crystal [010] axis in global coordinates<\/span><\/td><\/tr>\n <tr><td>66-68<\/td><td>\u6676\u4f53\u8f743<\/td><td>Crystal axis 3<\/td><td>\u5168\u5c40\u5750\u6807\u7cfb\u4e2d\u6676\u4f53[001]\u8f74\u7684\u65b9\u5411\u4f59\u5f26<\/span>Direction cosines of crystal [001] axis in global coordinates<\/span><\/td><\/tr>\n <tr><td>69<\/td><td>\u53d6\u5411\u8f93\u5165\u6a21\u5f0f<\/td><td>Orientation mode<\/td><td>0=\u4f7f\u7528Euler\u89d2\uff0c1=\u4f7f\u7528\u65b9\u5411\u4f59\u5f26\u77e9\u9635<\/span>0=use Euler angles, 1=use direction cosine matrix<\/span><\/td><\/tr>\n <tr><td>70<\/td><td>\u53d6\u5411\u66f4\u65b0\u6807\u5fd7<\/td><td>Orientation update<\/td><td>1=\u5728\u53d8\u5f62\u8fc7\u7a0b\u4e2d\u66f4\u65b0\u6676\u4f53\u53d6\u5411\uff08\u8003\u8651\u6676\u683c\u8f6c\u52a8\uff09\uff0c0=\u56fa\u5b9a\u53d6\u5411<\/span>1=update crystal orientation during deformation (lattice rotation), 0=fixed orientation<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch4-4\">\n 4.4 \u7387\u76f8\u5173\u53c2\u6570 PROPS(73-96) \/ Rate-Dependent Parameters<\/span>\n 4.4 Rate-Dependent Parameters PROPS(73-96)<\/span>\n <\/h3>\n \n \u7387\u76f8\u5173\u53c2\u6570\u533a\u63a7\u5236\u5e42\u5f8b\u6d41\u52a8\u6cd5\u5219\u7684\u884c\u4e3a\u3002\u5bf9\u4e8e\u591a\u65cf\u6ed1\u79fb\u7cfb\uff0c\u6bcf\u65cf\u53ef\u4ee5\u72ec\u7acb\u6307\u5b9a\u7387\u76f8\u5173\u53c2\u6570\u3002\n <\/p>\n \n The rate-dependent parameter section controls the behavior of the power-law flow rule. For multiple families of slip systems, each family can have independent rate-dependent parameters.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>71<\/td><td>GAMMA0_1<\/td><td>GAMMA0_1<\/td><td>\u7b2c1\u65cf\u6ed1\u79fb\u7cfb\u7684\u53c2\u8003\u526a\u5207\u7387 γ̇0<\/sub>\uff08s-1<\/sup>\uff09<\/span>Reference shear rate γ̇0<\/sub> for 1st family (s-1<\/sup>)<\/span><\/td><\/tr>\n <tr><td>72<\/td><td>AM_1<\/td><td>AM_1<\/td><td>\u7b2c1\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u654f\u611f\u6307\u6570\u5012\u6570 1\/n<\/span>Inverse rate sensitivity exponent 1\/n for 1st family<\/span><\/td><\/tr>\n <tr><td>73<\/td><td>N_1<\/td><td>N_1<\/td><td>\u7b2c1\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u654f\u611f\u6307\u6570 n<\/span>Rate sensitivity exponent n for 1st family<\/span><\/td><\/tr>\n <tr><td>74<\/td><td>GAMMA0_2<\/td><td>GAMMA0_2<\/td><td>\u7b2c2\u65cf\u6ed1\u79fb\u7cfb\u7684\u53c2\u8003\u526a\u5207\u7387\uff08\u5982\u6709\uff09<\/span>Reference shear rate for 2nd family (if applicable)<\/span><\/td><\/tr>\n <tr><td>75<\/td><td>AM_2<\/td><td>AM_2<\/td><td>\u7b2c2\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u654f\u611f\u6307\u6570\u5012\u6570 1\/n<\/span>Inverse rate sensitivity exponent 1\/n for 2nd family<\/span><\/td><\/tr>\n <tr><td>76<\/td><td>N_2<\/td><td>N_2<\/td><td>\u7b2c2\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u654f\u611f\u6307\u6570 n<\/span>Rate sensitivity exponent n for 2nd family<\/span><\/td><\/tr>\n <tr><td>77-84<\/td><td>\u6d41\u52a8\u5f8b\u53c2\u6570<\/td><td>Flow params<\/td><td>\u7b2c3-4\u65cf\u6ed1\u79fb\u7cfb\u7684\u7387\u76f8\u5173\u53c2\u6570\uff08\u9884\u7559\uff09<\/span>Rate-dependent parameters for 3rd-4th families (reserved)<\/span><\/td><\/tr>\n <tr><td>85<\/td><td>TEMP_REF<\/td><td>TEMP_REF<\/td><td>\u53c2\u8003\u6e29\u5ea6\uff08K\uff09\uff0c\u7528\u4e8e\u70ed\u6fc0\u6d3b\u6a21\u578b<\/span>Reference temperature (K) for thermal activation model<\/span><\/td><\/tr>\n <tr><td>86<\/td><td>TEMP_CUR<\/td><td>TEMP_CUR<\/td><td>\u5f53\u524d\u6e29\u5ea6\uff08K\uff09<\/span>Current temperature (K)<\/span><\/td><\/tr>\n <tr><td>87-96<\/td><td>\u9884\u7559<\/td><td>Reserved<\/td><td>\u4e3a\u540e\u7eed\u6269\u5c55\u9884\u7559<\/span>Reserved for future extensions<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch4-5\">\n 4.5 \u786c\u5316\u53c2\u6570 PROPS(97-144) \/ Hardening Parameters<\/span>\n 4.5 Hardening Parameters PROPS(97-144)<\/span>\n <\/h3>\n \n \u786c\u5316\u53c2\u6570\u533a\u5305\u542b\u81ea\u786c\u5316\u548c\u6f5c\u786c\u5316\u7684\u5168\u90e8\u53c2\u6570\u3002\u6839\u636e\u9009\u62e9\u7684\u786c\u5316\u6a21\u578b\u7c7b\u578b\uff08PROPS\u4e2d\u7684\u6807\u5fd7\uff09\uff0c\u8fd9\u4e9b\u53c2\u6570\u7684\u89e3\u91ca\u6709\u6240\u4e0d\u540c\u3002\n <\/p>\n \n The hardening parameter section contains all parameters for self-hardening and latent hardening. Their interpretation depends on the selected hardening model type (flag in PROPS).\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>97<\/td><td>\u786c\u5316\u6a21\u578b\u7c7b\u578b<\/td><td>Hardening model<\/td><td>0=\u53cc\u66f2\u6b63\u5272\u786c\u5316\uff0c1=Bassani\u786c\u5316\uff0c2=\u5176\u4ed6<\/span>0=Hyperbolic secant, 1=Bassani, 2=Other<\/span><\/td><\/tr>\n <tr><td>98<\/td><td>TAU0<\/td><td>TAU0<\/td><td>\u521d\u59cb\u4e34\u754c\u5206\u5207\u5e94\u529b τ0<\/sub>\uff08MPa\uff09<\/span>Initial critical resolved shear stress τ0<\/sub> (MPa)<\/span><\/td><\/tr>\n <tr><td>99<\/td><td>TAUS<\/td><td>TAUS<\/td><td>\u9971\u548c\u4e34\u754c\u5206\u5207\u5e94\u529b τs<\/sub>\uff08MPa\uff09<\/span>Saturation critical resolved shear stress τs<\/sub> (MPa)<\/span><\/td><\/tr>\n <tr><td>100<\/td><td>H0<\/td><td>H0<\/td><td>\u521d\u59cb\u786c\u5316\u6a21\u91cf H0<\/sub>\uff08MPa\uff09<\/span>Initial hardening modulus H0<\/sub> (MPa)<\/span><\/td><\/tr>\n <tr><td>101<\/td><td>HS<\/td><td>HS<\/td><td>Bassani\u6a21\u578b\u4e2d\u7684\u9971\u548c\u786c\u5316\u6a21\u91cf Hs<\/sub>\uff08MPa\uff09<\/span>Saturation hardening modulus Hs<\/sub> in Bassani model (MPa)<\/span><\/td><\/tr>\n <tr><td>102<\/td><td>GAMMA0_HARD<\/td><td>GAMMA0_HARD<\/td><td>Bassani\u6a21\u578b\u4e2d\u7684\u7279\u5f81\u526a\u5207\u5e94\u53d8 γ0<\/sub><\/span>Characteristic shear strain γ0<\/sub> in Bassani model<\/span><\/td><\/tr>\n <tr><td>103<\/td><td>Q1<\/td><td>Q1<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q1\uff08\u5171\u9762\u6ed1\u79fb\u7cfb\u95f4\uff09<\/span>Latent hardening coefficient q1 (coplanar)<\/span><\/td><\/tr>\n <tr><td>104<\/td><td>Q2<\/td><td>Q2<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q2\uff08\u5171\u5411\u6ed1\u79fb\u7cfb\u95f4\uff09<\/span>Latent hardening coefficient q2 (codirectional)<\/span><\/td><\/tr>\n <tr><td>105<\/td><td>Q3<\/td><td>Q3<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q3\uff08\u6b63\u4ea4\u6ed1\u79fb\u7cfb\u95f4\uff09<\/span>Latent hardening coefficient q3 (orthogonal)<\/span><\/td><\/tr>\n <tr><td>106<\/td><td>Q4<\/td><td>Q4<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q4\uff08Hirth\u9501\uff09<\/span>Latent hardening coefficient q4 (Hirth lock)<\/span><\/td><\/tr>\n <tr><td>107<\/td><td>Q5<\/td><td>Q5<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q5\uff08Lomer-Cottrell\u9501\uff09<\/span>Latent hardening coefficient q5 (Lomer-Cottrell lock)<\/span><\/td><\/tr>\n <tr><td>108<\/td><td>Q6<\/td><td>Q6<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q6\uff08\u4ea4\u53c9\u6ed1\u79fb\u76f8\u5173\uff09<\/span>Latent hardening coefficient q6 (cross-slip related)<\/span><\/td><\/tr>\n <tr><td>109<\/td><td>Q7<\/td><td>Q7<\/td><td>\u6f5c\u786c\u5316\u7cfb\u6570q7\uff08glissile\u9501\uff09<\/span>Latent hardening coefficient q7 (glissile lock)<\/span><\/td><\/tr>\n <tr><td>110-144<\/td><td>\u9884\u7559\/\u6269\u5c55<\/td><td>Reserved<\/td><td>\u4e3a\u591a\u79cd\u6ed1\u79fb\u7cfb\u3001\u591a\u9636\u6bb5\u786c\u5316\u3001\u6e29\u5ea6\u6548\u5e94\u7b49\u9884\u7559<\/span>Reserved for multiple slip systems, multi-stage hardening, thermal effects, etc.<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch4-6\">\n 4.6 \u79ef\u5206\u63a7\u5236 PROPS(145-160) \/ Integration Control<\/span>\n 4.6 Integration Control PROPS(145-160)<\/span>\n <\/h3>\n \n \u79ef\u5206\u63a7\u5236\u53c2\u6570\u5f71\u54cd\u9690\u5f0fNewton-Raphson\u8fed\u4ee3\u7684\u6570\u503c\u884c\u4e3a\uff0c\u5305\u62ec\u6536\u655b\u5bb9\u5dee\u3001\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u3001\n \u81ea\u52a8\u65f6\u95f4\u6b65\u957f\u63a7\u5236\u7b56\u7565\u7b49\u3002\n <\/p>\n \n Integration control parameters affect the numerical behavior of the implicit Newton-Raphson iteration,\n including convergence tolerance, maximum iteration count, and automatic time step control strategy.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>PROPS Index<\/th>\n <th>\n \u4e2d\u6587\u540d\u79f0<\/span>\n Name (Chinese)<\/span>\n <\/th>\n <th>\n \u82f1\u6587\u540d\u79f0 \/ English Name<\/span>\n English Name<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>145<\/td><td>ITMAX<\/td><td>ITMAX<\/td><td>\u6700\u5927Newton-Raphson\u8fed\u4ee3\u6b21\u6570\uff08\u9ed8\u8ba450\uff09<\/span>Maximum Newton-Raphson iterations (default 50)<\/span><\/td><\/tr>\n <tr><td>146<\/td><td>TOL<\/td><td>TOL<\/td><td>\u76f8\u5bf9\u6536\u655b\u5bb9\u5dee\uff08\u9ed8\u8ba41e-6\uff09<\/span>Relative convergence tolerance (default 1e-6)<\/span><\/td><\/tr>\n <tr><td>147<\/td><td>TOL2<\/td><td>TOL2<\/td><td>\u7edd\u5bf9\u6536\u655b\u5bb9\u5dee\uff08\u9ed8\u8ba41e-10\uff09<\/span>Absolute convergence tolerance (default 1e-10)<\/span><\/td><\/tr>\n <tr><td>148<\/td><td>DGMAX<\/td><td>DGMAX<\/td><td>\u5355\u6b65\u6700\u5927\u5141\u8bb8\u526a\u5207\u5e94\u53d8\u589e\u91cf\u9650\u5236<\/span>Maximum allowable shear strain increment per step<\/span><\/td><\/tr>\n <tr><td>149<\/td><td>ITERMOD<\/td><td>ITERMOD<\/td><td>\u8fed\u4ee3\u6a21\u5f0f\uff1a0=\u6807\u51c6NR\uff0c1=\u7ebf\u641c\u7d22\u589e\u5f3a<\/span>Iteration mode: 0=standard NR, 1=line-search enhanced<\/span><\/td><\/tr>\n <tr><td>150<\/td><td>OUTPUT<\/td><td>OUTPUT<\/td><td>\u8f93\u51fa\u63a7\u5236\u6807\u5fd7\uff1a0=\u6700\u5c0f\u8f93\u51fa\uff0c1=\u6807\u51c6\uff0c2=\u8be6\u7ec6\u8c03\u8bd5<\/span>Output control flag: 0=minimal, 1=standard, 2=verbose debug<\/span><\/td><\/tr>\n <tr><td>151-160<\/td><td>\u9884\u7559<\/td><td>Reserved<\/td><td>\u4e3a\u81ea\u9002\u5e94\u6b65\u957f\u63a7\u5236\u3001\u5b50\u6b65\u79ef\u5206\u7b49\u9884\u7559<\/span>Reserved for adaptive step control, sub-stepping, etc.<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch5\">\n <h2>\n 05<\/span>\n \u4ee3\u7801\u7ed3\u6784 \/ Code Structure<\/span>\n Code Structure<\/span>\n <\/h2>\n\n <h3 id=\"ch5-1\">\n 5.1 \u5b50\u7a0b\u5e8f\u8c03\u7528\u5173\u7cfb\u56fe \/ Subroutine Call Graph<\/span>\n 5.1 Subroutine Call Graph<\/span>\n <\/h3>\n \n \u9ec4\u6c38\u521aUMAT\u7531\u4e3b\u7a0b\u5e8fUMAT\u548c\u591a\u4e2a\u8f85\u52a9\u5b50\u7a0b\u5e8f\u7ec4\u6210\uff0c\u5404\u5b50\u7a0b\u5e8f\u627f\u62c5\u660e\u786e\u7684\u804c\u8d23\u3002\n \u4ee5\u4e0b\u4e3a\u8c03\u7528\u5173\u7cfb\u56fe\uff1a\n <\/p>\n \n Huang’s UMAT consists of the main UMAT subroutine and multiple auxiliary subroutines, each with a well-defined responsibility.\n The call graph is as follows:\n <\/p>\n\n <div class=\"mermaid\">\ngraph TD\n A[UMAT<br\/>\u4e3b\u7a0b\u5e8f \/ Main Routine] –> B[ROTATION<br\/>\u8ba1\u7b97\u65cb\u8f6c\u77e9\u9635 \/ Compute Rotation]\n A –> C[SLIPSYS<br\/>\u6ed1\u79fb\u7cfb\u521d\u59cb\u5316 \/ Slip System Init]\n C –> D[GSLPINIT<br\/>\u521d\u59cb\u5316\u6ed1\u79fb\u5f3a\u5ea6 \/ Init Slip Strength]\n D –> E[GSLP0<br\/>\u521d\u59cbCRSS \/ Initial CRSS]\n A –> F[STRAINRATE<br\/>\u8ba1\u7b97\u526a\u5207\u7387 \/ Compute Shear Rate]\n F –> G[F<br\/>\u6d41\u52a8\u51fd\u6570 \/ Flow Function]\n F –> H[DFDX<br\/>\u6d41\u52a8\u51fd\u6570\u5bfc\u6570 \/ Flow Derivative]\n A –> I[LATENTHARDEN<br\/>\u6f5c\u786c\u5316\u8ba1\u7b97 \/ Latent Hardening]\n I –> J[HSELF<br\/>\u81ea\u786c\u5316 \/ Self-Hardening]\n I –> K[HLATNT<br\/>\u6f5c\u786c\u5316\u7cfb\u6570 \/ Latent Coeffs]\n A –> L[ITERATION<br\/>Newton-Raphson\u8fed\u4ee3 \/ NR Iteration]\n L –> M[DHSELF<br\/>\u81ea\u786c\u5316\u5bfc\u6570 \/ Self-Hardening Deriv]\n L –> N[DHLATN<br\/>\u6f5c\u786c\u5316\u5bfc\u6570 \/ Latent Hardening Deriv]\n L –> O[LUDCMP<br\/>LU\u5206\u89e3 \/ LU Decomposition]\n L –> P[LUBKSB<br\/>LU\u56de\u4ee3 \/ LU Back-Subst]\n A –> Q[CROSS<br\/>\u5411\u91cf\u53c9\u4e58 \/ Vector Cross Product]\n style A fill:#2563eb,stroke:#1a2332,color:#fff\n style L fill:#0ea5e9,stroke:#1a2332,color:#fff\n <\/div>\n\n <h3 id=\"ch5-2\">\n 5.2 \u4e3b\u7b97\u6cd5\u6d41\u7a0b\u56fe \/ Main Algorithm Flowchart<\/span>\n 5.2 Main Algorithm Flowchart<\/span>\n <\/h3>\n \n \u6bcf\u4e2a\u589e\u91cf\u6b65\u4e2d\uff0cUMAT\u4e3b\u7a0b\u5e8f\u6267\u884c\u4ee5\u4e0b\u7b97\u6cd5\u6d41\u7a0b\uff1a\n <\/p>\n \n In each increment, the main UMAT routine executes the following algorithm:\n <\/p>\n\n <div class=\"mermaid\">\nflowchart TD\n START([\u5f00\u59cb \/ Start]) –> A[\u8bfb\u53d6\u6750\u6599\u53c2\u6570<br\/>Read PROPS]\n A –> B[\u8bfb\u53d6\u72b6\u6001\u53d8\u91cf<br\/>Read STATEV]\n B –> C[\u8ba1\u7b97\u5f39\u6027\u8bd5\u5e94\u529b<br\/>Elastic Trial Stress]\n C –> D{Newton-Raphson<br\/>\u8fed\u4ee3\u6536\u655b?}\n D — \u5426 \/ No –> E[\u8ba1\u7b97\u6d41\u52a8\u5f8b\u548c\u786c\u5316<br\/>Flow & Hardening]\n E –> F[\u7ec4\u88c5Jacobian<br\/>Assemble Jacobian]\n F –> G[LU\u5206\u89e3\u6c42\u89e3<br\/>LU Decomposition]\n G –> H[\u66f4\u65b0\u526a\u5207\u589e\u91cf<br\/>Update Δγ]\n H –> D\n D — \u662f \/ Yes –> I[\u66f4\u65b0\u5e94\u529b<br\/>Update Stress]\n I –> J[\u66f4\u65b0STATEV<br\/>Update STATEV]\n J –> K[\u8ba1\u7b97DDSDDE<br\/>Compute Consistent Tangent]\n K –> END([\u7ed3\u675f \/ End])\n style D fill:#f59e0b,stroke:#1a2332,color:#fff\n style START fill:#2563eb,stroke:#1a2332,color:#fff\n style END fill:#ef4444,stroke:#1a2332,color:#fff\n <\/div>\n\n <h3 id=\"ch5-3\">\n 5.3 \u5173\u952e\u4ee3\u7801\u6bb5 \/ Key Code Segments<\/span>\n 5.3 Key Code Segments<\/span>\n <\/h3>\n\n <h4>\n 5.3.1 \u5f39\u6027\u8bd5\u5e94\u529b\u8ba1\u7b97 \/ Elastic Trial Stress<\/span>\n 5.3.1 Elastic Trial Stress<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC \u5f39\u6027\u8bd5\u5e94\u529b\u8ba1\u7b97 \/ Elastic trial stress computation\nC \u57fa\u4e8e\u589e\u91cf\u5f00\u59cb\u65f6\u7684Fe\u548c\u6574\u4e2a\u5e94\u53d8\u589e\u91cf\u8ba1\u7b97\u8bd5\u5e94\u529b\nC-----------------------------------------------------------------------\n DO I=1,3\n DO J=1,3\n FE_INV(I,J)=DFGRD1(I,J) ! \u521d\u59cb\u4f30\u8ba1: F^e = F (\u5ffd\u7565\u5851\u6027)\n ENDDO\n ENDDO\nC\nC \u4eceSTATEV\u6062\u590d\u4e0a\u4e00\u589e\u91cf\u6b65\u7684F^p\u548cF^e\n CALL KINTOTAL(STATEV,NSTATV,FE_FP,NSLIP)\nC\nC \u8ba1\u7b97\u8bd5\u5f39\u6027\u53d8\u5f62\u68af\u5ea6: F^{e,trial} = F_{n+1} * (F^p_n)^{-1}\n CALL LUDCMP(FP_TRY,3,3,INDX,D)\n CALL LUBKSB(FP_TRY,3,3,INDX,FE_TRY_COL1)\n CALL LUBKSB(FP_TRY,3,3,INDX,FE_TRY_COL2)\n CALL LUBKSB(FP_TRY,3,3,INDX,FE_TRY_COL3)\nC\nC \u8ba1\u7b97Green\u5f39\u6027\u5e94\u53d8 E^e = 0.5*(F^{eT}F^e - I)\n DO I=1,3\n DO J=1,3\n FE_TF_FE(I,J)=0.0\n DO K=1,3\n FE_TF_FE(I,J)=FE_TF_FE(I,J)+FE_TRY(K,I)*FE_TRY(K,J)\n ENDDO\n ENDDO\n ENDDO\n DO I=1,3\n DO J=1,3\n EE(I,J)=0.5*(FE_TF_FE(I,J)-DELTA(I,J))\n ENDDO\n ENDDO<\/code><\/pre>\n\n <h4>\n 5.3.2 \u6d41\u52a8\u5f8b\u5b9e\u73b0 \/ Flow Law Implementation<\/span>\n 5.3.2 Flow Law Implementation<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC \u7387\u76f8\u5173\u6d41\u52a8\u5f8b\u8ba1\u7b97 \/ Rate-dependent flow rule computation\nC gamma_dot = gamma0 * |tau\/g|^n * sign(tau\/g)\nC-----------------------------------------------------------------------\n SUBROUTINE STRAINRATE(TAU, G, GAMMA0, AM, NSLIP, DGAMMA)\n IMPLICIT NONE\n INTEGER NSLIP, I\n REAL*8 TAU(NSLIP), G(NSLIP), GAMMA0, AM\n REAL*8 DGAMMA(NSLIP), RATIO, ABSRAT, POWER\nC\n DO I=1,NSLIP\n IF (G(I).LE.0.0) THEN\n WRITE(*,*) 'ERROR: g(',I,') =',G(I)\n STOP\n ENDIF\n RATIO = TAU(I)\/G(I)\n ABSRAT = DABS(RATIO)\nC \u907f\u514d\u6570\u503c\u6ea2\u51fa \/ Avoid numerical overflow\n IF (ABSRAT.GT.1.0D+10) THEN\n POWER = 1.0D+10**AM\n ELSE\n POWER = ABSRAT**AM\n ENDIF\nC \u526a\u5207\u5e94\u53d8\u7387 \/ Shear strain rate\n DGAMMA(I) = GAMMA0 * POWER\n IF (RATIO.LT.0.0) DGAMMA(I) = -DGAMMA(I)\n ENDDO\n RETURN\n END<\/code><\/pre>\n\n <h4>\n 5.3.3 \u81ea\u786c\u5316\u5b9e\u73b0 \/ Self-Hardening Implementation<\/span>\n 5.3.3 Self-Hardening Implementation<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC \u53cc\u66f2\u6b63\u5272\u81ea\u786c\u5316\u6a21\u578b \/ Hyperbolic secant self-hardening model\nC h(gamma) = H0 * sech^2( H0*gamma \/ (tau_s - tau_0) )\nC-----------------------------------------------------------------------\n FUNCTION HSELF(GAMMA, H0, TAUS, TAU0)\n IMPLICIT NONE\n REAL*8 HSELF, GAMMA, H0, TAUS, TAU0\n REAL*8 ARG, SECH2\nC\n IF (TAUS.LE.TAU0) THEN\n HSELF = 0.0\n RETURN\n ENDIF\nC\n ARG = H0 * GAMMA \/ (TAUS - TAU0)\nC sech^2(x) = 1\/cosh^2(x)\n SECH2 = 1.0 \/ (DCOSH(ARG)**2)\n HSELF = H0 * SECH2\n RETURN\n END\nC\nC-----------------------------------------------------------------------\nC \u5f53\u524d\u6ed1\u79fb\u7cfb\u5f3a\u5ea6 \/ Current slip system strength\nC g(gamma) = tau_s + (tau_0 - tau_s) * tanh( H0*gamma\/(tau_s-tau_0) )\nC-----------------------------------------------------------------------\n FUNCTION GSLP0(GAMMA, H0, TAUS, TAU0)\n IMPLICIT NONE\n REAL*8 GSLP0, GAMMA, H0, TAUS, TAU0\n REAL*8 ARG\nC\n IF (TAUS.LE.TAU0) THEN\n GSLP0 = TAU0\n RETURN\n ENDIF\nC\n ARG = H0 * GAMMA \/ (TAUS - TAU0)\n GSLP0 = TAUS + (TAU0 - TAUS) * DTANH(ARG)\n RETURN\n END<\/code><\/pre>\n\n <h4>\n 5.3.4 Newton-Raphson\u8fed\u4ee3\u6838\u5fc3 \/ NR Iteration Core<\/span>\n 5.3.4 NR Iteration Core<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC Newton-Raphson\u8fed\u4ee3\u6c42\u89e3\u526a\u5207\u5e94\u53d8\u589e\u91cf \/ NR iteration for shear increment\nC \u672a\u77e5\u91cf: DGA(NSLIP) = Δγ^(α)\nC \u6b8b\u5dee: R(α) = Δγ^(α) - Δt * gamma_dot^{n+1}_(α) = 0\nC-----------------------------------------------------------------------\n SUBROUTINE ITERATION(...)\n IMPLICIT NONE\n INTEGER NSLIP, ITMAX, ITER, I, J, INFO\n REAL*8 TOL, DTIME, DGA(NSLIP), GA(NSLIP), TAU(NSLIP)\n REAL*8 G(NSLIP), DG(NSLIP), GAMMA0, AM\n REAL*8 JAC(NSLIP,NSLIP), RES(NSLIP), DGA_INC(NSLIP)\n REAL*8 DFDX(NSLIP), DHDG(NSLIP,NSLIP)\n REAL*8 DET, RESNORM, RESNORM0\n INTEGER INDX(NSLIP)\nC\n RESNORM0 = 1.0\n DO ITER=1,ITMAX\nC \u8ba1\u7b97\u5f53\u524d\u5e94\u529b\u72b6\u6001 \/ Compute current stress state\n CALL STRESSUPDATE(DGA, TAU, SIGMA, ...)\nC\nC \u8ba1\u7b97\u6d41\u52a8\u5f8b \/ Compute flow rule\n CALL STRAINRATE(TAU, G, GAMMA0, AM, NSLIP, DG)\nC\nC \u8ba1\u7b97\u6b8b\u5dee \/ Compute residual\n DO I=1,NSLIP\n RES(I) = DGA(I) - DTIME * DG(I)\n ENDDO\nC\nC \u8ba1\u7b97\u6b8b\u5dee\u8303\u6570 \/ Compute residual norm\n RESNORM = 0.0\n DO I=1,NSLIP\n RESNORM = RESNORM + RES(I)*RES(I)\n ENDDO\n RESNORM = DSQRT(RESNORM)\n IF (ITER.EQ.1) RESNORM0 = RESNORM\nC\nC \u68c0\u67e5\u6536\u655b \/ Check convergence\n IF (RESNORM.LT.TOL*RESNORM0 .OR. RESNORM.LT.1.0D-12) THEN\n RETURN ! \u6536\u655b \/ Converged\n ENDIF\nC\nC \u8ba1\u7b97Jacobian\u77e9\u9635 \/ Compute Jacobian matrix\nC J_{αβ} = δ_{αβ} - Δt * (∂gamma_dot_α\/∂DGA_β)\n DO I=1,NSLIP\n DO J=1,NSLIP\n JAC(I,J) = -DTIME * DHDG(I,J)\n ENDDO\n JAC(I,I) = JAC(I,I) + 1.0\n ENDDO\nC\nC LU\u5206\u89e3\u5e76\u6c42\u89e3 \/ LU decomposition and solve\n CALL LUDCMP(JAC, NSLIP, NSLIP, INDX, DET)\n DO I=1,NSLIP\n DGA_INC(I) = -RES(I)\n ENDDO\n CALL LUBKSB(JAC, NSLIP, NSLIP, INDX, DGA_INC)\nC\nC \u66f4\u65b0\u672a\u77e5\u91cf \/ Update unknowns\n DO I=1,NSLIP\n DGA(I) = DGA(I) + DGA_INC(I)\n ENDDO\n ENDDO\nC\n WRITE(*,*) 'WARNING: NR iteration not converged in UMAT'\n RETURN\n END<\/code><\/pre>\n\n <h4>\n 5.3.5 DDSDDE\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6 \/ Consistent Tangent DDSDDE<\/span>\n 5.3.5 Consistent Tangent DDSDDE<\/span>\n <\/h4>\n <pre><code>C-----------------------------------------------------------------------\nC \u8ba1\u7b97\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635 DDSDDE \/ Compute consistent tangent modulus\nC DDSDDE = ∂Δσ \/ ∂Δε\nC-----------------------------------------------------------------------\n SUBROUTINE CONSISTENTTANGENT(...)\n IMPLICIT NONE\n INTEGER NTENS, NSLIP, I, J, K, L\n REAL*8 DDSDDE(NTENS,NTENS), CEL(6,6)\n REAL*8 P_ALPHA(3,3,NSLIP), DGA(NSLIP)\n REAL*8 DTAU_DSIG(NTENS,NSLIP), DHDG(NSLIP,NSLIP)\n REAL*8 JAC(NSLIP,NSLIP), INV_JAC(NSLIP,NSLIP)\n REAL*8 TERM(NTENS,NSLIP), SUM\n INTEGER INDX(NSLIP)\n REAL*8 DET\nC\nC \u5f39\u6027\u521a\u5ea6\u77e9\u9635 \/ Elastic stiffness matrix\n DO I=1,NTENS\n DO J=1,NTENS\n DDSDDE(I,J) = CEL(I,J)\n ENDDO\n ENDDO\nC\nC \u8ba1\u7b97 ∂tau_α\/∂σ \u5e76\u7ec4\u88c5\u4fee\u6b63\u9879\nC Compute ∂tau_α\/∂σ and assemble correction\n DO K=1,NSLIP\n DO I=1,NTENS\n TERM(I,K) = 0.0\n DO J=1,NTENS\n TERM(I,K) = TERM(I,K) + CEL(I,J)*P_ALPHA_VOIGT(J,K)\n ENDDO\n ENDDO\n ENDDO\nC\nC \u8ba1\u7b97 (I - Δt * ∂dg\/∂g * ∂g\/∂DGA)^{-1}\n CALL LUDCMP(JAC, NSLIP, NSLIP, INDX, DET)\n CALL MATINV(JAC, NSLIP, NSLIP, INDX, INV_JAC)\nC\nC DDSDDE = C^e - C^e:P * INV_JAC * DTIME * dg\/dtau * d tau\/d eps\n DO I=1,NTENS\n DO J=1,NTENS\n SUM = 0.0\n DO K=1,NSLIP\n DO L=1,NSLIP\n SUM = SUM + TERM(I,K)*INV_JAC(K,L)*DTAU_DSIG(L,J)\n ENDDO\n ENDDO\n DDSDDE(I,J) = DDSDDE(I,J) - SUM * DTIME\n ENDDO\n ENDDO\n RETURN\n END<\/code><\/pre>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch6\">\n <h2>\n 06<\/span>\n Jacobian\u77e9\u9635 \/ Jacobian Matrix<\/span>\n Jacobian Matrix<\/span>\n <\/h2>\n\n <h3 id=\"ch6-1\">\n 6.1 \u5e94\u53d8\u5206\u89e3 \/ Strain Decomposition<\/span>\n 6.1 Strain Decomposition<\/span>\n <\/h3>\n \n \u5728Abaqus\/Standard\u7684\u9690\u5f0f\u6c42\u89e3\u6846\u67b6\u4e2d\uff0cUMAT\u5fc5\u987b\u63d0\u4f9b\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635DDSDDE\uff0c\n \u5373\u5e94\u529b\u589e\u91cf\u5bf9\u5e94\u53d8\u589e\u91cf\u7684\u504f\u5bfc\u6570DDSDDE = ∂Δσ \/ ∂Δε\u3002\n \u5bf9\u4e8e\u5f39\u5851\u6027\u6750\u6599\uff0c\u603b\u5e94\u53d8\u589e\u91cf\u5206\u89e3\u4e3a\u5f39\u6027\u90e8\u5206\u548c\u5851\u6027\u90e8\u5206\uff1a\n <\/p>\n \n In the Abaqus\/Standard implicit solver framework, UMAT must provide the consistent tangent stiffness matrix DDSDDE,\n i.e., the partial derivative of stress increment with respect to strain increment, DDSDDE = ∂Δσ \/ ∂Δε.\n For elastoplastic materials, the total strain increment is decomposed into elastic and plastic parts:\n <\/p>\n <div class=\"formula\">\n Δε = Δεe<\/sup> + Δεp<\/sup><\/b>\n <\/div>\n \n \u5728\u5c0f\u53d8\u5f62\u5047\u8bbe\u4e0b\uff0c\u5f39\u6027\u5e94\u53d8\u589e\u91cf\u4e0e\u5e94\u529b\u589e\u91cf\u7684\u5173\u7cfb\u7531\u5e7f\u4e49\u80e1\u514b\u5b9a\u5f8b\u7ed9\u51fa\uff1a\n <\/p>\n \n Under the small-deformation assumption, the relationship between elastic strain increment and stress increment is given by the generalized Hooke’s law:\n <\/p>\n <div class=\"formula\">\n Δσ = Ce<\/sup> : Δεe<\/sup> = Ce<\/sup> : (Δε – Δεp<\/sup>)<\/b>\n <\/div>\n \n \u5851\u6027\u5e94\u53d8\u589e\u91cf\u7531\u5404\u6ed1\u79fb\u7cfb\u7684\u526a\u5207\u5e94\u53d8\u589e\u91cf\u8d21\u732e\uff1a\n <\/p>\n \n The plastic strain increment is contributed by the shear strain increments on all slip systems:\n <\/p>\n <div class=\"formula\">\n Δεp<\/sup> = Σα<\/sub> Δγ(α)<\/sup> P(α)<\/sup><\/b>\n <\/div>\n \n \u56e0\u6b64\uff0c\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635\u7684\u5173\u952e\u5728\u4e8e\u8ba1\u7b97 ∂Δγ(α)<\/sup> \/ ∂Δε\uff0c\n \u5373\u526a\u5207\u5e94\u53d8\u589e\u91cf\u5bf9\u5e94\u53d8\u589e\u91cf\u7684\u654f\u611f\u5ea6\u3002\n <\/p>\n \n Therefore, the key to the consistent tangent stiffness matrix is computing ∂Δγ(α)<\/sup> \/ ∂Δε,\n i.e., the sensitivity of shear strain increment to strain increment.\n <\/p>\n\n <h3 id=\"ch6-2\">\n 6.2 \u7ebf\u6027\u65b9\u7a0b\u7ec4 \/ Linear System<\/span>\n 6.2 Linear System<\/span>\n <\/h3>\n \n \u5728Newton-Raphson\u8fed\u4ee3\u4e2d\uff0c\u6bcf\u4e2a\u6ed1\u79fb\u7cfb\u7684\u526a\u5207\u5e94\u53d8\u589e\u91cf\u6ee1\u8db3\u4ee5\u4e0b\u975e\u7ebf\u6027\u65b9\u7a0b\uff1a\n <\/p>\n \n In the Newton-Raphson iteration, the shear strain increment for each slip system satisfies the following nonlinear equation:\n <\/p>\n <div class=\"formula\">\n R(α)<\/sup> = Δγ(α)<\/sup> – Δt · γ̇0<\/sub> · | τ(α)<\/sup> \/ g(α)<\/sup> |n<\/sup> · sign(τ(α)<\/sup>\/g(α)<\/sup>) = 0<\/b>\n <\/div>\n \n \u5bf9\u4e0a\u8ff0\u65b9\u7a0b\u5173\u4e8eΔγ(β)<\/sup>\u6c42\u504f\u5bfc\uff0c\u5f97\u5230Newton-Raphson\u8fed\u4ee3\u7684Jacobian\u77e9\u9635\uff08\u5c40\u90e8Newton\u8fed\u4ee3\u77e9\u9635\uff09\uff1a\n <\/p>\n \n Taking the partial derivative of the above equation with respect to Δγ(β)<\/sup> yields the Jacobian matrix for Newton-Raphson iteration (local Newton iteration matrix):\n <\/p>\n <div class=\"formula\">\n Jαβ<\/sub> = ∂R(α)<\/sup> \/ ∂Δγ(β)<\/sup> = δαβ<\/sub> – Δt · (∂γ̇(α)<\/sup> \/ ∂τ(α)<\/sup>) · (∂τ(α)<\/sup> \/ ∂Δγ(β)<\/sup>) – Δt · (∂γ̇(α)<\/sup> \/ ∂g(α)<\/sup>) · (∂g(α)<\/sup> \/ ∂Δγ(β)<\/sup>)<\/b>\n <\/div>\n \n \u5404\u9879\u504f\u5bfc\u6570\u7684\u7269\u7406\u542b\u4e49\u548c\u8868\u8fbe\u5f0f\u5982\u4e0b\uff1a\n <\/p>\n \n The physical meaning and expressions of each partial derivative are:\n <\/p>\n <ul>\n <li>\n ∂γ̇(α)<\/sup> \/ ∂τ(α)<\/sup><\/b>:\n \u6d41\u52a8\u5f8b\u5bf9\u5206\u89e3\u526a\u5e94\u529b\u7684\u654f\u611f\u5ea6\uff0c\u7b49\u4e8e n · γ̇(α)<\/sup> \/ τ(α)<\/sup><\/span>\n Sensitivity of flow rule to resolved shear stress, equal to n · γ̇(α)<\/sup> \/ τ(α)<\/sup><\/span>\n <\/li>\n <li>\n ∂τ(α)<\/sup> \/ ∂Δγ(β)<\/sup><\/b>:\n \u5206\u89e3\u526a\u5e94\u529b\u5bf9\u526a\u5207\u589e\u91cf\u7684\u654f\u611f\u5ea6\uff0c\u7b49\u4e8e -Ce<\/sup> : P(β)<\/sup> : P(α)<\/sup>\uff08\u5e94\u529b\u677e\u5f1b\u6548\u5e94\uff09<\/span>\n Sensitivity of resolved shear stress to shear increment, equal to -Ce<\/sup> : P(β)<\/sup> : P(α)<\/sup> (stress relaxation effect)<\/span>\n <\/li>\n <li>\n ∂g(α)<\/sup> \/ ∂Δγ(β)<\/sup><\/b>:\n \u6ed1\u79fb\u7cfb\u5f3a\u5ea6\u5bf9\u526a\u5207\u589e\u91cf\u7684\u654f\u611f\u5ea6\uff0c\u7b49\u4e8e hαβ<\/sub> · sign(τ(β)<\/sup>)<\/span>\n Sensitivity of slip system strength to shear increment, equal to hαβ<\/sub> · sign(τ(β)<\/sup>)<\/span>\n <\/li>\n <li>\n ∂γ̇(α)<\/sup> \/ ∂g(α)<\/sup><\/b>:\n \u6d41\u52a8\u5f8b\u5bf9\u5f3a\u5ea6\u7684\u654f\u611f\u5ea6\uff0c\u7b49\u4e8e -n · γ̇(α)<\/sup> \/ g(α)<\/sup><\/span>\n Sensitivity of flow rule to strength, equal to -n · γ̇(α)<\/sup> \/ g(α)<\/sup><\/span>\n <\/li>\n <\/ul>\n \n \u5c06\u6240\u6709\u504f\u5bfc\u6570\u4ee3\u5165\u540e\uff0c\u5c40\u90e8Newton\u77e9\u9635\u53ef\u663e\u5f0f\u5199\u51fa\u3002\u8be5\u77e9\u9635\u7684\u5927\u5c0f\u4e3aNSLIP × NSLIP\uff08\u4f8b\u5982FCC\u4e3a12×12\uff09\u3002\n \u5728\u6bcf\u6b21Newton\u8fed\u4ee3\u4e2d\uff0c\u5bf9\u8be5\u77e9\u9635\u8fdb\u884cLU\u5206\u89e3\u5e76\u56de\u4ee3\u6c42\u89e3\u526a\u5207\u5e94\u53d8\u589e\u91cf\u4fee\u6b63\u503cΔDGA\u3002\n <\/p>\n \n After substituting all partial derivatives, the local Newton matrix can be written explicitly. Its size is NSLIP × NSLIP (e.g., 12×12 for FCC).\n In each Newton iteration, this matrix undergoes LU decomposition and back-substitution to solve for the shear strain increment correction ΔDGA.\n <\/p>\n\n <h3 id=\"ch6-3\">\n 6.3 DDSDDE\u6784\u9020 \/ DDSDDE Construction<\/span>\n 6.3 DDSDDE Construction<\/span>\n <\/h3>\n \n \u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635\u7684\u6784\u9020\u516c\u5f0f\u4e3a\uff1a\n <\/p>\n \n The consistent tangent stiffness matrix is constructed as:\n <\/p>\n <div class=\"formula\">\n DDSDDE = Ce<\/sup> – Ce<\/sup> : P(α)<\/sup> ⊗ [ J-1<\/sup>αβ<\/sub> · Δt · (∂γ̇(β)<\/sup> \/ ∂τ(β)<\/sup>) · P(β)<\/sup> : Ce<\/sup> ]<\/b>\n <\/div>\n \n \u4e0a\u8ff0\u516c\u5f0f\u4e2d\uff0c\u7b2c\u4e00\u9879Ce<\/sup>\u662f\u7eaf\u5f39\u6027\u521a\u5ea6\uff0c\u7b2c\u4e8c\u9879\u662f\u5851\u6027\u4fee\u6b63\u3002\n \u5f53\u6750\u6599\u5904\u4e8e\u7eaf\u5f39\u6027\u72b6\u6001\uff08\u65e0\u6ed1\u79fb\u6fc0\u6d3b\uff09\u65f6\uff0c\u7b2c\u4e8c\u9879\u4e3a\u96f6\uff0cDDSDDE\u9000\u5316\u4e3a\u5f39\u6027\u521a\u5ea6\u77e9\u9635\u3002\n \u5f53\u5927\u91cf\u6ed1\u79fb\u7cfb\u6fc0\u6d3b\u65f6\uff0c\u7b2c\u4e8c\u9879\u663e\u8457\u964d\u4f4e\u6750\u6599\u7684\u8868\u89c2\u521a\u5ea6\uff0c\u53cd\u6620\u5851\u6027\u6d41\u52a8\u5f15\u8d77\u7684\u5e94\u529b\u677e\u5f1b\u3002\n <\/p>\n \n In the above formula, the first term Ce<\/sup> is the pure elastic stiffness, and the second term is the plastic correction.\n When the material is in a purely elastic state (no slip activated), the second term vanishes and DDSDDE reduces to the elastic stiffness matrix.\n When many slip systems are active, the second term significantly reduces the apparent material stiffness, reflecting stress relaxation due to plastic flow.\n <\/p>\n \n \u5177\u4f53\u8ba1\u7b97\u6b65\u9aa4\u5982\u4e0b\uff1a\n <\/p>\n \n The specific calculation steps are:\n <\/p>\n <ol>\n <li>\n \u8ba1\u7b97\u5f39\u6027\u521a\u5ea6\u77e9\u9635Ce<\/sup>\u7684Voigt\u5f62\u5f0f\uff086×6\u77e9\u9635\uff09<\/span>\n Compute the Voigt form of elastic stiffness matrix Ce<\/sup> (6×6 matrix)<\/span>\n <\/li>\n <li>\n \u8ba1\u7b97\u6bcf\u4e2a\u6ed1\u79fb\u7cfb\u7684Schmid\u5f20\u91cfP(α)<\/sup>\u7684Voigt\u5411\u91cf\u5f62\u5f0f<\/span>\n Compute the Voigt vector form of Schmid tensor P(α)<\/sup> for each slip system<\/span>\n <\/li>\n <li>\n \u8ba1\u7b97\u6d41\u52a8\u5f8b\u5bfc\u6570 ∂γ̇(α)<\/sup> \/ ∂τ(α)<\/sup> = n · γ̇(α)<\/sup> \/ τ(α)<\/sup><\/span>\n Compute flow rule derivative ∂γ̇(α)<\/sup> \/ ∂τ(α)<\/sup> = n · γ̇(α)<\/sup> \/ τ(α)<\/sup><\/span>\n <\/li>\n <li>\n \u7ec4\u88c5\u5c40\u90e8Newton\u77e9\u9635Jαβ<\/sub>\u5e76\u8fdb\u884cLU\u5206\u89e3\u6c42\u9006<\/span>\n Assemble local Newton matrix Jαβ<\/sub> and perform LU decomposition for inversion<\/span>\n <\/li>\n <li>\n \u8ba1\u7b97\u4e2d\u95f4\u5f20\u91cf\uff1a\u5bf9\u6bcf\u4e2a\u6ed1\u79fb\u7cfbα\uff0c\u8ba1\u7b97 Q(α)<\/sup>ij<\/sub> = (J-1<\/sup>αβ<\/sub> · DFLOWβ<\/sub>) · (P(β)<\/sup>kl<\/sub> Ce<\/sup>klmn<\/sub>)<\/span>\n Compute intermediate tensor: for each slip system α, compute Q(α)<\/sup>ij<\/sub> = (J-1<\/sup>αβ<\/sub> · DFLOWβ<\/sub>) · (P(β)<\/sup>kl<\/sub> Ce<\/sup>klmn<\/sub>)<\/span>\n <\/li>\n <li>\n \u7ec4\u88c5DDSDDE\uff1aDDSDDEij<\/sub> = Ce<\/sup>ij<\/sub> – Σα<\/sub> (Ce<\/sup>ijkl<\/sub> P(α)<\/sup>kl<\/sub>) Q(α)<\/sup>mn<\/sub><\/span>\n Assemble DDSDDE: DDSDDEij<\/sub> = Ce<\/sup>ij<\/sub> – Σα<\/sub> (Ce<\/sup>ijkl<\/sub> P(α)<\/sup>kl<\/sub>) Q(α)<\/sup>mn<\/sub><\/span>\n <\/li>\n <\/ol>\n <div class=\"note\">\n \u6570\u503c\u7a33\u5b9a\u6027\u63d0\u793a\uff1a<\/strong>\u5f53τ(α)<\/sup>\u63a5\u8fd1\u96f6\u65f6\uff0c\u6d41\u52a8\u5f8b\u5bfc\u6570n·γ̇\/τ\u53ef\u80fd\u5947\u5f02\u3002\u7a0b\u5e8f\u4e2d\u901a\u5e38\u5f15\u5165\u5fae\u5c0f\u6b63\u5219\u5316\u9879\u6216\u9650\u5236\u6700\u5927\u5bfc\u6570\u503c\u3002\u6b64\u5916\uff0c\u5f53\u67d0\u4e9b\u6ed1\u79fb\u7cfb\u7684\u526a\u5207\u7387\u6781\u5c0f\u65f6\uff0c\u53ef\u5c06\u5176\u89c6\u4e3a”\u975e\u6fc0\u6d3b”\u5e76\u4ece\u5c40\u90e8Newton\u7cfb\u7edf\u4e2d\u6d88\u53bb\uff0c\u4ee5\u51cf\u5c0f\u77e9\u9635\u89c4\u6a21\u5e76\u6539\u5584\u6761\u4ef6\u6570\u3002<\/span>\n Numerical stability tip:<\/strong> When τ(α)<\/sup> is near zero, the flow rule derivative n·γ̇\/τ may become singular. The program usually introduces a small regularization term or caps the maximum derivative value. Additionally, slip systems with extremely small shear rates can be treated as “inactive” and eliminated from the local Newton system to reduce matrix size and improve conditioning.<\/span>\n <\/div>\n \n DDSDDE\u7684\u5bf9\u79f0\u6027\u53d6\u51b3\u4e8e\u6750\u6599\u6a21\u578b\u3002\u5bf9\u4e8e\u5173\u8054\u6d41\u52a8\u6cd5\u5219\u548c\u4e8c\u6b21\u5c48\u670d\u9762\uff0cDDSDDE\u662f\u5bf9\u79f0\u7684\uff1b\n \u4f46\u5bf9\u4e8e\u975e\u5173\u8054\u6d41\u52a8\u6216\u590d\u6742\u7684\u6f5c\u786c\u5316\u6a21\u578b\uff0cDDSDDE\u53ef\u80fd\u975e\u5bf9\u79f0\u3002\n \u9ec4\u6c38\u521aUMAT\u4e2d\uff0c\u7531\u4e8e\u91c7\u7528\u5173\u8054\u7684Schmid\u6d41\u52a8\u548c\u4e00\u822c\u7684\u786c\u5316\u77e9\u9635\uff0cDDSDDE\u901a\u5e38\u5177\u6709\u8f7b\u5fae\u7684\u975e\u5bf9\u79f0\u6027\u3002\n Abaqus\/Standard\u652f\u6301\u975e\u5bf9\u79f0\u521a\u5ea6\u77e9\u9635\uff0c\u4f46\u9700\u8981\u5728*Step\u4e2d\u8bbe\u7f6eUNSymm=YES\u4ee5\u4f18\u5316\u6c42\u89e3\u5668\u6027\u80fd\u3002\n <\/p>\n \n The symmetry of DDSDDE depends on the material model. For associative flow rules and quadratic yield surfaces, DDSDDE is symmetric;\n but for non-associative flow or complex latent hardening models, DDSDDE may be non-symmetric.\n In Huang’s UMAT, due to the associative Schmid flow and general hardening matrix, DDSDDE usually has slight non-symmetry.\n Abaqus\/Standard supports non-symmetric stiffness matrices, but *Step requires UNSymm=YES to optimize solver performance.\n <\/p>\n<\/div>\n\n\n<div class=\"chapter\" id=\"ch7\">\n <h2>\n 07<\/span>\n \u5178\u578b\u53c2\u6570 \/ Typical Parameters<\/span>\n Typical Parameters<\/span>\n <\/h2>\n\n \n \u4ee5\u4e0b\u7ed9\u51faFCC\u94dc\u548cFCC\u94dd\u7684\u5178\u578b\u6750\u6599\u53c2\u6570\uff0c\u8fd9\u4e9b\u53c2\u6570\u5728\u5927\u91cf\u6587\u732e\u548c\u5b9e\u9a8c\u4e2d\u5f97\u5230\u9a8c\u8bc1\uff0c\n \u53ef\u4f5c\u4e3aAbaqus\u8f93\u5165\u6587\u4ef6\u4e2d*User Material\u7684\u53c2\u8003\u503c\u3002\n <\/p>\n \n The following presents typical material parameters for FCC copper and FCC aluminum,\n validated in numerous literature and experiments, serving as reference values for the *User Material keyword in Abaqus input files.\n <\/p>\n\n <h3 id=\"ch7-1\">\n 7.1 FCC\u94dc \/ FCC Copper<\/span>\n 7.1 FCC Copper<\/span>\n <\/h3>\n \n \u94dc\u662fFCC\u7ed3\u6784\uff0c\u5ba4\u6e29\u4e0b\u4e3b\u8981\u6ed1\u79fb\u7cfb\u4e3a{111}<110>\uff0c\u517112\u4e2a\u3002\n \u4ee5\u4e0b\u53c2\u6570\u9002\u7528\u4e8e\u5ba4\u6e29\uff08\u7ea6300K\uff09\u4e0b\u7684\u5355\u6676\u94dc\u6a21\u62df\uff0c\u91c7\u7528\u53cc\u66f2\u6b63\u5272\u786c\u5316\u6a21\u578b\u3002\n <\/p>\n \n Copper has an FCC structure. At room temperature, the primary slip systems are {111}<110>, totaling 12.\n The following parameters are suitable for single-crystal copper simulation at room temperature (~300K) using the hyperbolic secant hardening model.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>\n \u53c2\u6570 \/ Parameter<\/span>\n Parameter<\/span>\n <\/th>\n <th>\n \u6570\u503c \/ Value<\/span>\n Value<\/span>\n <\/th>\n <th>\n \u5355\u4f4d \/ Unit<\/span>\n Unit<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>C11<\/td>\n <td>168,400<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C11<\/span>Elastic constant C11<\/span><\/td>\n <\/tr>\n <tr>\n <td>C12<\/td>\n <td>121,400<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C12<\/span>Elastic constant C12<\/span><\/td>\n <\/tr>\n <tr>\n <td>C44<\/td>\n <td>75,400<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C44<\/span>Elastic constant C44<\/span><\/td>\n <\/tr>\n <tr>\n <td>γ̇0<\/sub><\/td>\n <td>0.001<\/td>\n <td>s-1<\/sup><\/td>\n <td>\u53c2\u8003\u526a\u5207\u7387<\/span>Reference shear rate<\/span><\/td>\n <\/tr>\n <tr>\n <td>n<\/td>\n <td>50<\/td>\n <td>–<\/td>\n <td>\u7387\u654f\u611f\u6307\u6570<\/span>Rate sensitivity exponent<\/span><\/td>\n <\/tr>\n <tr>\n <td>τ0<\/sub><\/td>\n <td>16.0<\/td>\n <td>MPa<\/td>\n <td>\u521d\u59cbCRSS<\/span>Initial CRSS<\/span><\/td>\n <\/tr>\n <tr>\n <td>τs<\/sub><\/td>\n <td>148.0<\/td>\n <td>MPa<\/td>\n <td>\u9971\u548cCRSS<\/span>Saturation CRSS<\/span><\/td>\n <\/tr>\n <tr>\n <td>H0<\/sub><\/td>\n <td>180.0<\/td>\n <td>MPa<\/td>\n <td>\u521d\u59cb\u786c\u5316\u6a21\u91cf<\/span>Initial hardening modulus<\/span><\/td>\n <\/tr>\n <tr>\n <td>q<\/td>\n <td>1.0<\/td>\n <td>–<\/td>\n <td>\u6f5c\u786c\u5316\u7cfb\u6570\uff08\u5404\u5411\u540c\u6027\u8fd1\u4f3c\uff09<\/span>Latent hardening coefficient (isotropic approximation)<\/span><\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <h3 id=\"ch7-2\">\n 7.2 FCC\u94dd \/ FCC Aluminum<\/span>\n 7.2 FCC Aluminum<\/span>\n <\/h3>\n \n \u94dd\u540c\u6837\u662fFCC\u7ed3\u6784\uff0c\u4f46\u5176\u5c42\u9519\u80fd\u8f83\u9ad8\uff0c\u4f4d\u9519\u4ea4\u6ed1\u79fb\u66f4\u5bb9\u6613\u53d1\u751f\uff0c\u56e0\u6b64\u786c\u5316\u884c\u4e3a\u548c\u94dc\u6709\u660e\u663e\u5dee\u5f02\u3002\n \u4ee5\u4e0b\u53c2\u6570\u9002\u7528\u4e8e\u9ad8\u7eaf\u94dd\u5355\u6676\u5728\u5ba4\u6e29\u4e0b\u7684\u6a21\u62df\u3002\n <\/p>\n \n Aluminum is also FCC, but with higher stacking fault energy, making dislocation cross-slip easier,\n so its hardening behavior differs significantly from copper.\n The following parameters are suitable for high-purity single-crystal aluminum at room temperature.\n <\/p>\n\n <table>\n <thead>\n <tr>\n <th>\n \u53c2\u6570 \/ Parameter<\/span>\n Parameter<\/span>\n <\/th>\n <th>\n \u6570\u503c \/ Value<\/span>\n Value<\/span>\n <\/th>\n <th>\n \u5355\u4f4d \/ Unit<\/span>\n Unit<\/span>\n <\/th>\n <th>\n \u8bf4\u660e \/ Description<\/span>\n Description<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>C11<\/td>\n <td>106,750<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C11<\/span>Elastic constant C11<\/span><\/td>\n <\/tr>\n <tr>\n <td>C12<\/td>\n <td>60,410<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C12<\/span>Elastic constant C12<\/span><\/td>\n <\/tr>\n <tr>\n <td>C44<\/td>\n <td>28,340<\/td>\n <td>MPa<\/td>\n <td>\u5f39\u6027\u5e38\u6570C44<\/span>Elastic constant C44<\/span><\/td>\n <\/tr>\n <tr>\n <td>γ̇0<\/sub><\/td>\n <td>0.001<\/td>\n <td>s-1<\/sup><\/td>\n <td>\u53c2\u8003\u526a\u5207\u7387<\/span>Reference shear rate<\/span><\/td>\n <\/tr>\n <tr>\n <td>n<\/td>\n <td>60<\/td>\n <td>–<\/td>\n <td>\u7387\u654f\u611f\u6307\u6570<\/span>Rate sensitivity exponent<\/span><\/td>\n <\/tr>\n <tr>\n <td>τ0<\/sub><\/td>\n <td>0.75<\/td>\n <td>MPa<\/td>\n <td>\u521d\u59cbCRSS<\/span>Initial CRSS<\/span><\/td>\n <\/tr>\n <tr>\n <td>τs<\/sub><\/td>\n <td>25.0<\/td>\n <td>MPa<\/td>\n <td>\u9971\u548cCRSS<\/span>Saturation CRSS<\/span><\/td>\n <\/tr>\n <tr>\n <td>H0<\/sub><\/td>\n <td>30.0<\/td>\n <td>MPa<\/td>\n <td>\u521d\u59cb\u786c\u5316\u6a21\u91cf<\/span>Initial hardening modulus<\/span><\/td>\n <\/tr>\n <tr>\n <td>q<\/td>\n <td>1.0<\/td>\n <td>–<\/td>\n <td>\u6f5c\u786c\u5316\u7cfb\u6570\uff08\u5404\u5411\u540c\u6027\u8fd1\u4f3c\uff09<\/span>Latent hardening coefficient (isotropic approximation)<\/span><\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n\n <div class=\"note\">\n \u53c2\u6570\u6765\u6e90\u8bf4\u660e\uff1a<\/strong>\u4e0a\u8ff0\u53c2\u6570\u7efc\u5408\u4e86\u591a\u9879\u7ecf\u5178\u5b9e\u9a8c\u548c\u6587\u732e\u6570\u636e\u3002\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5e94\u6839\u636e\u5177\u4f53\u5408\u91d1\u6210\u5206\u3001\u6e29\u5ea6\u3001\u5e94\u53d8\u901f\u7387\u7b49\u6761\u4ef6\u901a\u8fc7\u5355\u6676\u62c9\u4f38\u5b9e\u9a8c\u6821\u51c6\u3002\u7279\u522b\u5730\uff0c\u7387\u654f\u611f\u6307\u6570n\u548c\u53c2\u8003\u526a\u5207\u7387γ̇0<\/sub>\u5e94\u5339\u914d\u5b9e\u9a8c\u5e94\u53d8\u901f\u7387\uff0c\u5426\u5219\u53ef\u80fd\u5bfc\u81f4\u5e94\u529b\u6c34\u5e73\u9884\u6d4b\u504f\u5dee\u3002<\/span>\n Parameter source note:<\/strong> The above parameters are synthesized from multiple classic experiments and literature data. In practical applications, they should be calibrated through single-crystal tensile experiments according to specific alloy composition, temperature, strain rate, etc. In particular, the rate sensitivity exponent n and reference shear rate γ̇0<\/sub> should match the experimental strain rate; otherwise, stress level prediction errors may occur.<\/span>\n <\/div>\n<\/div>\n\n<div class=\"chapter\" id=\"ch8\">\n <h2>\n 08<\/span>\n \u4f7f\u7528\u6ce8\u610f\u4e8b\u9879 \/ Usage Notes<\/span>\n Usage Notes<\/span>\n <\/h2>\n\n \n \u4ee5\u4e0b\u5217\u51fa\u4f7f\u7528\u9ec4\u6c38\u521aUMAT\u5b50\u7a0b\u5e8f\u65f6\u5e38\u89c1\u7684\u6ce8\u610f\u4e8b\u9879\u548c\u6700\u4f73\u5b9e\u8df5\u5efa\u8bae\uff0c\n \u6db5\u76d6\u8f93\u5165\u6587\u4ef6\u8bbe\u7f6e\u3001\u6750\u6599\u53c2\u6570\u6821\u51c6\u3001\u6536\u655b\u6027\u8c03\u8bd5\u548c\u540e\u5904\u7406\u7b49\u65b9\u9762\u3002\n <\/p>\n \n The following lists common considerations and best-practice recommendations when using Huang’s UMAT subroutine,\n covering input file setup, material parameter calibration, convergence debugging, and post-processing.\n <\/p>\n\n <h3 id=\"ch8-1\">\n 8.1 Abaqus\u8f93\u5165\u6587\u4ef6\u8bbe\u7f6e \/ Input File Setup<\/span>\n 8.1 Input File Setup<\/span>\n <\/h3>\n <ul>\n <li>\n *User Material\u5173\u952e\u5b57\uff1a<\/strong>\u5fc5\u987b\u5728\u8f93\u5165\u6587\u4ef6\u4e2d\u901a\u8fc7*User Material, constants=NPROPS\u5b9a\u4e49PROPS\u6570\u7ec4\u957f\u5ea6\uff0c\u5e76\u786e\u4fdd*Depvar\u7684NSTATV\u8db3\u591f\u5927\u4ee5\u5bb9\u7eb3\u6240\u6709\u72b6\u6001\u53d8\u91cf\u3002\u5bf9\u4e8eFCC 12\u6ed1\u79fb\u7cfb\uff0c\u5efa\u8baeNSTATV >= 70\u3002<\/span>\n *User Material keyword:<\/strong> The PROPS array length must be defined via *User Material, constants=NPROPS, and *Depvar’s NSTATV must be large enough for all state variables. For FCC 12 slip systems, NSTATV >= 70 is recommended.<\/span>\n <\/li>\n <li>\n UNSymm\u9009\u9879\uff1a<\/strong>\u7531\u4e8e\u6676\u4f53\u5851\u6027\u7684\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635\u901a\u5e38\u8f7b\u5fae\u975e\u5bf9\u79f0\uff0c\u5efa\u8bae\u5728*Step\u4e2d\u4f7f\u7528UNSymm=YES\u4ee5\u542f\u7528\u975e\u5bf9\u79f0\u6c42\u89e3\u5668\uff0c\u63d0\u9ad8\u6536\u655b\u901f\u5ea6\u548c\u7a33\u5b9a\u6027\u3002<\/span>\n UNSymm option:<\/strong> Because the consistent tangent stiffness matrix of crystal plasticity is usually slightly non-symmetric, it is recommended to use UNSymm=YES in *Step to enable the non-symmetric solver, improving convergence speed and stability.<\/span>\n <\/li>\n <li>\n \u65f6\u95f4\u6b65\u957f\u63a7\u5236\uff1a<\/strong>\u6676\u4f53\u5851\u6027UMAT\u5bf9\u65f6\u95f4\u6b65\u957f\u654f\u611f\u3002\u5efa\u8bae\u5728\u521d\u59cb\u52a0\u8f7d\u9636\u6bb5\u4f7f\u7528\u8f83\u5c0f\u7684\u65f6\u95f4\u6b65\uff0c\u5f85\u5851\u6027\u53d8\u5f62\u7a33\u5b9a\u540e\u53ef\u9002\u5ea6\u589e\u5927\u3002\u53ef\u901a\u8fc7*Static\u4e2d\u7684INITIAL\u548cMAXIMUM\u53c2\u6570\u63a7\u5236\u3002<\/span>\n Time step control:<\/strong> Crystal plasticity UMAT is sensitive to time step size. It is recommended to use small time steps in the initial loading stage, and moderately increase after plastic deformation stabilizes. Control via INITIAL and MAXIMUM in *Static.<\/span>\n <\/li>\n <li>\n \u5355\u5143\u7c7b\u578b\u9009\u62e9\uff1a<\/strong>\u63a8\u8350\u4f7f\u7528C3D8R\uff088\u8282\u70b9\u516d\u9762\u4f53\u51cf\u7f29\u79ef\u5206\u5355\u5143\uff09\u6216C3D8\uff08\u5b8c\u5168\u79ef\u5206\uff09\u3002\u907f\u514d\u4f7f\u7528\u7ebf\u6027\u4e09\u89d2\u5f62\/\u56db\u9762\u4f53\u5355\u5143\u6a21\u62df\u5927\u5e94\u53d8\u5c40\u90e8\u5316\uff0c\u56e0\u4e3a\u5b83\u4eec\u5728\u4e0d\u53ef\u538b\u7f29\u5851\u6027\u53d8\u5f62\u4e2d\u4f1a\u4ea7\u751f\u4f53\u79ef\u9501\u5b9a\u3002<\/span>\n Element type selection:<\/strong> C3D8R (8-node hexahedron with reduced integration) or C3D8 (full integration) are recommended. Avoid linear triangular\/tetrahedral elements for large-strain localization, as they suffer volumetric locking in incompressible plastic deformation.<\/span>\n <\/li>\n <li>\n \u6c99\u6f0f\u63a7\u5236\uff1a<\/strong>\u4f7f\u7528C3D8R\u5355\u5143\u65f6\uff0c\u5fc5\u987b\u914d\u5408*Hourglass Stiffness\u6216*Section Controls\u8bbe\u7f6e\u5408\u7406\u7684\u6c99\u6f0f\u521a\u5ea6\uff0c\u4ee5\u6291\u5236\u96f6\u80fd\u6a21\u5f0f\u3002<\/span>\n Hourglass control:<\/strong> When using C3D8R elements, *Hourglass Stiffness or *Section Controls must be set with reasonable hourglass stiffness to suppress zero-energy modes.<\/span>\n <\/li>\n <\/ul>\n\n <h3 id=\"ch8-2\">\n 8.2 \u6750\u6599\u53c2\u6570\u6821\u51c6 \/ Parameter Calibration<\/span>\n 8.2 Parameter Calibration<\/span>\n <\/h3>\n <ul>\n <li>\n \u5f39\u6027\u5e38\u6570\uff1a<\/strong>C11\u3001C12\u3001C44\u5e94\u4ece\u8d85\u58f0\u5171\u632f\u6216\u5355\u6676\u5f39\u6027\u5b9e\u9a8c\u83b7\u53d6\u3002\u6ce8\u610fAbaqus\u4e2d\u5e94\u529b\u5355\u4f4d\u4e3aMPa\uff0c\u5f39\u6027\u5e38\u6570\u4e5f\u5fc5\u987b\u4ee5MPa\u8f93\u5165\u3002<\/span>\n Elastic constants:<\/strong> C11, C12, C44 should be obtained from ultrasonic resonance or single-crystal elastic experiments. Note that Abaqus uses MPa for stress, so elastic constants must also be entered in MPa.<\/span>\n <\/li>\n <li>\n \u521d\u59cbCRSS tau0\uff1a<\/strong>\u53ef\u901a\u8fc7\u5355\u6676\u521d\u59cb\u5c48\u670d\u5e94\u529b\u7ed3\u5408Schmid\u56e0\u5b50\u4f30\u7b97\u3002\u5bf9\u4e8e\u4e0d\u540c\u53d6\u5411\u7684\u5355\u6676\u8bd5\u6837\uff0ctau0\u5e94\u552f\u4e00\uff0c\u800c\u5b8f\u89c2\u5c48\u670d\u5e94\u529b\u968f\u53d6\u5411\u53d8\u5316\u3002<\/span>\n Initial CRSS tau0:<\/strong> Can be estimated from single-crystal initial yield stress combined with the Schmid factor. For single-crystal specimens of different orientations, tau0 should be unique, while macroscopic yield stress varies with orientation.<\/span>\n <\/li>\n <li>\n \u7387\u654f\u611f\u6307\u6570n\uff1a<\/strong>\u5e94\u901a\u8fc7\u4e0d\u540c\u5e94\u53d8\u901f\u7387\u4e0b\u7684\u5355\u6676\u62c9\u4f38\u5b9e\u9a8c\u786e\u5b9a\u3002n\u503c\u5bf9\u6d41\u52a8\u5e94\u529b\u6c34\u5e73\u6709\u663e\u8457\u5f71\u54cd\uff0c\u4f46\u5bf9\u786c\u5316\u6a21\u91cf\u5f71\u54cd\u8f83\u5c0f\u3002\u82e5\u5b9e\u9a8c\u6570\u636e\u4e0d\u8db3\uff0c\u53ef\u53c2\u8003\u6587\u732e\u4e2d\u540c\u79cd\u6750\u6599\u7684\u63a8\u8350\u503c\u3002<\/span>\n Rate sensitivity exponent n:<\/strong> Should be determined from single-crystal tensile experiments at different strain rates. n significantly affects flow stress level but has minor effect on hardening modulus. If experimental data is insufficient, refer to recommended values in the literature for the same material.<\/span>\n <\/li>\n <li>\n \u786c\u5316\u53c2\u6570\uff1a<\/strong>H0\u3001taus\u548ctau0\u5e94\u901a\u8fc7\u5355\u6676\u5e94\u529b-\u5e94\u53d8\u66f2\u7ebf\u7684\u5851\u6027\u6bb5\u62df\u5408\u786e\u5b9a\u3002\u5efa\u8bae\u4f7f\u7528\u975e\u7ebf\u6027\u6700\u5c0f\u4e8c\u4e58\u6cd5\u5bf9\u5b9e\u9a8c\u6570\u636e\u8fdb\u884c\u4f18\u5316\u62df\u5408\u3002<\/span>\n Hardening parameters:<\/strong> H0, taus, and tau0 should be determined by fitting the plastic portion of single-crystal stress-strain curves. Nonlinear least-squares optimization is recommended for fitting experimental data.<\/span>\n <\/li>\n <li>\n \u6f5c\u786c\u5316\u7cfb\u6570q\uff1a<\/strong>\u5404\u5411\u540c\u6027\u8fd1\u4f3c\u4e0b\u53ef\u53d6q=1.0\u3002\u82e5\u8981\u66f4\u7cbe\u786e\u63cf\u8ff0\uff0c\u9700\u8981\u901a\u8fc7\u53cc\u6ed1\u79fb\u6216\u591a\u6ed1\u79fb\u5b9e\u9a8c\u6807\u5b9aBassani\u6a21\u578b\u4e2d\u76847\u4e2a\u6f5c\u786c\u5316\u7cfb\u6570\u3002<\/span>\n Latent hardening coefficient q:<\/strong> In isotropic approximation, q=1.0 can be used. For more accurate description, the 7 latent hardening coefficients in the Bassani model need to be calibrated through double-slip or multi-slip experiments.<\/span>\n <\/li>\n <li>\n \u6676\u4f53\u53d6\u5411\uff1a<\/strong>\u5355\u6676\u6a21\u62df\u4e2d\u53d6\u5411\u53c2\u6570\u81f3\u5173\u91cd\u8981\u3002\u5efa\u8bae\u901a\u8fc7EBSD\u5b9e\u9a8c\u6d4b\u91cf\u5b9e\u9645\u8bd5\u6837\u7684Euler\u89d2\uff0c\u800c\u975e\u4ec5\u4f9d\u8d56\u540d\u4e49\u53d6\u5411\u3002\u5373\u4f7f\u5fae\u5c0f\u7684\u53d6\u5411\u504f\u5dee\u4e5f\u53ef\u80fd\u5bfc\u81f4\u5b8f\u89c2\u5e94\u529b-\u5e94\u53d8\u54cd\u5e94\u7684\u663e\u8457\u5dee\u5f02\u3002<\/span>\n Crystal orientation:<\/strong> Orientation parameters are crucial in single-crystal simulations. It is recommended to measure the Euler angles of actual specimens via EBSD, rather than relying solely on nominal orientations. Even small orientation deviations can cause significant differences in macroscopic stress-strain response.<\/span>\n <\/li>\n <\/ul>\n\n <h3 id=\"ch8-3\">\n 8.3 \u6536\u655b\u6027\u8c03\u8bd5 \/ Convergence Debugging<\/span>\n 8.3 Convergence Debugging<\/span>\n <\/h3>\n <ul>\n <li>\n \u65f6\u95f4\u6b65\u8fc7\u5927\uff1a<\/strong>\u82e5\u51fa\u73b0”Too many attempts made for this increment”\u9519\u8bef\uff0c\u6700\u5e38\u89c1\u7684\u539f\u56e0\u662f\u65f6\u95f4\u6b65\u8fc7\u5927\u5bfc\u81f4\u5c40\u90e8Newton\u8fed\u4ee3\u4e0d\u6536\u655b\u3002\u53ef\u901a\u8fc7\u51cf\u5c0f\u6700\u5927\u65f6\u95f4\u6b65\u6216\u542f\u7528\u81ea\u52a8\u6b65\u957f\u63a7\u5236\u89e3\u51b3\u3002<\/span>\n Time step too large:<\/strong> If “Too many attempts made for this increment” appears, the most common cause is excessive time step size causing local Newton iteration divergence. Reduce maximum time step or enable automatic step control.<\/span>\n <\/li>\n <li>\n \u5947\u5f02Jacobian\uff1a<\/strong>\u5f53\u591a\u4e2a\u6ed1\u79fb\u7cfb\u540c\u65f6\u6fc0\u6d3b\u4e14\u786c\u5316\u77e9\u9635\u63a5\u8fd1\u5947\u5f02\u65f6\uff0cLU\u5206\u89e3\u53ef\u80fd\u5931\u8d25\u3002\u53ef\u901a\u8fc7\u589e\u52a0\u6536\u655b\u5bb9\u5deeTOL\u6216\u5728\u6d41\u52a8\u5f8b\u4e2d\u5f15\u5165\u6b63\u5219\u5316\u6765\u6539\u5584\u3002<\/span>\n Singular Jacobian:<\/strong> When multiple slip systems activate simultaneously and the hardening matrix is nearly singular, LU decomposition may fail. Increase convergence tolerance TOL or introduce regularization in the flow rule.<\/span>\n <\/li>\n <li>\n \u8d1f\u526a\u5207\u7387\uff1a<\/strong>\u82e5\u67d0\u4e9b\u6ed1\u79fb\u7cfb\u7684\u5206\u89e3\u526a\u5e94\u529b\u4e3a\u8d1f\u4e14\u7edd\u5bf9\u503c\u5f88\u5927\uff0c\u6d41\u52a8\u5f8b\u53ef\u80fd\u4ea7\u751f\u4e0d\u5408\u7406\u7684\u8d1f\u526a\u5207\u7387\u3002\u7a0b\u5e8f\u4e2d\u5e94\u6709\u7b26\u53f7\u5904\u7406\u903b\u8f91\uff0c\u786e\u4fdd\u7269\u7406\u5408\u7406\u6027\u3002<\/span>\n Negative shear rate:<\/strong> If resolved shear stress on some slip systems is negative with large magnitude, the flow rule may produce unreasonable negative shear rates. The program should have sign-handling logic to ensure physical consistency.<\/span>\n <\/li>\n <li>\n \u68c0\u67e5STATEV\u521d\u59cb\u5316\uff1a<\/strong>STATEV\u7684\u521d\u59cb\u503c\u5fc5\u987b\u6b63\u786e\u8bbe\u7f6e\uff0c\u7279\u522b\u662f\u5f39\u6027\u53d8\u5f62\u68af\u5ea6Fe\u5e94\u521d\u59cb\u5316\u4e3a\u5355\u4f4d\u77e9\u9635\uff0cFp\u4e5f\u521d\u59cb\u5316\u4e3a\u5355\u4f4d\u77e9\u9635\uff0cg\u521d\u59cb\u5316\u4e3atau0\u3002\u4efb\u4f55\u9519\u8bef\u7684\u521d\u503c\u90fd\u53ef\u80fd\u5bfc\u81f4\u7b2c\u4e00\u6b65\u5373\u53d1\u6563\u3002<\/span>\n Check STATEV initialization:<\/strong> Initial STATEV values must be set correctly, especially Fe and Fp initialized to identity matrices, and g initialized to tau0. Any incorrect initial value may cause divergence in the first step.<\/span>\n <\/li>\n <li>\n PNEWDT\u7684\u4f7f\u7528\uff1a<\/strong>\u5f53\u5c40\u90e8\u8fed\u4ee3\u8d85\u8fc7ITMAX\u4ecd\u672a\u6536\u655b\u65f6\uff0c\u5e94\u5728UMAT\u4e2d\u8bbe\u7f6ePNEWDT < 1\uff08\u59820.5\uff09\uff0c\u544a\u77e5Abaqus\u51cf\u5c0f\u5f53\u524d\u589e\u91cf\u6b65\u5e76\u91cd\u8bd5\u3002\u8fd9\u662f\u4e00\u79cd\u6709\u6548\u7684\u81ea\u9002\u5e94\u6b65\u957f\u7b56\u7565\u3002<\/span>\n Use of PNEWDT:<\/strong> When local iteration exceeds ITMAX without convergence, set PNEWDT < 1 (e.g., 0.5) in UMAT to instruct Abaqus to reduce the current increment and retry. This is an effective adaptive step strategy.<\/span>\n <\/li>\n <li>\n \u8f93\u51fa\u8c03\u8bd5\u4fe1\u606f\uff1a<\/strong>\u8bbe\u7f6eOUTPUT=2\u53ef\u542f\u7528\u8be6\u7ec6\u8c03\u8bd5\u8f93\u51fa\uff0c\u5305\u62ec\u6bcf\u6b21Newton\u8fed\u4ee3\u7684\u6b8b\u5dee\u8303\u6570\u3001\u526a\u5207\u589e\u91cf\u548c\u5e94\u529b\u72b6\u6001\u3002\u8fd9\u5bf9\u4e8e\u5b9a\u4f4d\u53d1\u6563\u539f\u56e0\u975e\u5e38\u6709\u5e2e\u52a9\u3002<\/span>\n Output debug information:<\/strong> Setting OUTPUT=2 enables verbose debug output, including residual norm, shear increments, and stress state for each Newton iteration. This is very helpful for locating divergence causes.<\/span>\n <\/li>\n <\/ul>\n\n <h3 id=\"ch8-4\">\n 8.4 \u540e\u5904\u7406\u4e0e\u7ed3\u679c\u9a8c\u8bc1 \/ Post-Processing and Verification<\/span>\n 8.4 Post-Processing and Verification<\/span>\n <\/h3>\n <ul>\n <li>\n \u5e94\u529b-\u5e94\u53d8\u66f2\u7ebf\uff1a<\/strong>\u5c06Abaqus\u8f93\u51fa\u7684\u5b8f\u89c2\u5e94\u529b-\u5e94\u53d8\u66f2\u7ebf\u4e0e\u5355\u6676\u5b9e\u9a8c\u6570\u636e\u8fdb\u884c\u5bf9\u6bd4\uff0c\u9a8c\u8bc1\u6a21\u578b\u9884\u6d4b\u7684\u57fa\u672c\u529b\u5b66\u884c\u4e3a\u662f\u5426\u6b63\u786e\u3002<\/span>\n Stress-strain curve:<\/strong> Compare the macroscopic stress-strain curve from Abaqus with single-crystal experimental data to verify whether the basic mechanical behavior predicted by the model is correct.<\/span>\n <\/li>\n <li>\n \u6ed1\u79fb\u7cfb\u6fc0\u6d3b\u5206\u6790\uff1a<\/strong>\u901a\u8fc7\u540e\u5904\u7406STATEV\u4e2d\u5b58\u50a8\u7684\u7d2f\u79ef\u526a\u5207\u5e94\u53d8\uff0c\u53ef\u4ee5\u8bc6\u522b\u54ea\u4e9b\u6ed1\u79fb\u7cfb\u88ab\u6fc0\u6d3b\u53ca\u5176\u76f8\u5bf9\u6d3b\u52a8\u5f3a\u5ea6\u3002\u8fd9\u5bf9\u4e8e\u7406\u89e3\u53d8\u5f62\u673a\u5236\u548c\u7ec7\u6784\u6f14\u5316\u81f3\u5173\u91cd\u8981\u3002<\/span>\n Slip system activation analysis:<\/strong> By post-processing accumulated shear strains stored in STATEV, one can identify which slip systems are activated and their relative activity levels. This is crucial for understanding deformation mechanisms and texture evolution.<\/span>\n <\/li>\n <li>\n \u7ec7\u6784\u9a8c\u8bc1\uff1a<\/strong>\u5982\u679c\u542f\u7528\u4e86\u53d6\u5411\u66f4\u65b0\uff08PROPS(70)=1\uff09\uff0c\u5e94\u63d0\u53d6\u53d8\u5f62\u540e\u7684\u6676\u4f53\u53d6\u5411\u5e76\u4e0e\u5b9e\u9a8cEBSD\u7ec7\u6784\u6570\u636e\u8fdb\u884c\u5bf9\u6bd4\uff0c\u8bc4\u4f30\u7ec7\u6784\u6f14\u5316\u9884\u6d4b\u7684\u51c6\u786e\u6027\u3002<\/span>\n Texture verification:<\/strong> If orientation update is enabled (PROPS(70)=1), extract deformed crystal orientations and compare with experimental EBSD texture data to assess the accuracy of texture evolution prediction.<\/span>\n <\/li>\n <li>\n \u7f51\u683c\u6536\u655b\u6027\uff1a<\/strong>\u8fdb\u884c\u7f51\u683c\u654f\u611f\u6027\u5206\u6790\uff0c\u786e\u4fdd\u7ed3\u679c\u4e0d\u4f9d\u8d56\u4e8e\u79bb\u6563\u5316\u7a0b\u5ea6\u3002\u6676\u4f53\u5851\u6027\u6a21\u62df\u4e2d\uff0c\u5e94\u53d8\u5c40\u90e8\u5316\u5e26\uff08shear band\uff09\u7684\u5bbd\u5ea6\u901a\u5e38\u53d7\u7f51\u683c\u5c3a\u5bf8\u5f71\u54cd\uff0c\u5e94\u4f7f\u7528\u8db3\u591f\u7ec6\u5bc6\u7684\u7f51\u683c\u3002<\/span>\n Mesh convergence:<\/strong> Perform mesh sensitivity analysis to ensure results do not depend on discretization level. In crystal plasticity simulations, the width of strain localization bands (shear bands) is usually affected by mesh size; sufficiently fine meshes should be used.<\/span>\n <\/li>\n <li>\n \u80fd\u91cf\u5b88\u6052\u68c0\u67e5\uff1a<\/strong>\u9a8c\u8bc1\u5851\u6027\u8017\u6563\u529f\u4e0e\u5916\u90e8\u529f\u7684\u5173\u7cfb\uff0c\u786e\u4fdd\u6570\u503c\u5b9e\u73b0\u6ca1\u6709\u80fd\u91cf\u9519\u8bef\u3002Abaqus\u8f93\u51fa\u7684\u80fd\u91cf\u5386\u53f2\uff08ALLIE\u3001ALLKE\u3001ALLPD\u7b49\uff09\u53ef\u7528\u4e8e\u6b64\u76ee\u7684\u3002<\/span>\n Energy conservation check:<\/strong> Verify the relationship between plastic dissipation work and external work to ensure the numerical implementation has no energy errors. Abaqus energy history outputs (ALLIE, ALLKE, ALLPD, etc.) can be used for this purpose.<\/span>\n <\/li>\n <\/ul>\n\n <h3 id=\"ch8-5\">\n 8.5 \u5e38\u89c1\u9519\u8bef\u4e0e\u89e3\u51b3\u65b9\u6848 \/ Common Errors and Solutions<\/span>\n 8.5 Common Errors and Solutions<\/span>\n <\/h3>\n <table>\n <thead>\n <tr>\n <th>\n \u9519\u8bef\u73b0\u8c61 \/ Error<\/span>\n Error Symptom<\/span>\n <\/th>\n <th>\n \u53ef\u80fd\u539f\u56e0 \/ Cause<\/span>\n Possible Cause<\/span>\n <\/th>\n <th>\n \u89e3\u51b3\u65b9\u6848 \/ Solution<\/span>\n Solution<\/span>\n <\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr>\n <td>\u589e\u91cf\u6b65\u591a\u6b21\u5c1d\u8bd5\u540e\u4ecd\u5931\u8d25<\/span>Increment fails after many attempts<\/span><\/td>\n <td>\u65f6\u95f4\u6b65\u8fc7\u5927\u6216\u6750\u6599\u53c2\u6570\u4e0d\u5408\u7406<\/span>Time step too large or material parameters unreasonable<\/span><\/td>\n <td>\u51cf\u5c0f\u6700\u5927\u65f6\u95f4\u6b65\uff0c\u68c0\u67e5\u6750\u6599\u53c2\u6570\u6570\u91cf\u7ea7<\/span>Reduce max time step, check parameter magnitudes<\/span><\/td>\n <\/tr>\n <tr>\n <td>\u5e94\u529b\u8f93\u51fa\u4e3aNaN\u6216Inf<\/span>Stress output is NaN or Inf<\/span><\/td>\n <td>\u9664\u96f6\u9519\u8bef\u6216\u6307\u6570\u6ea2\u51fa<\/span>Division by zero or exponent overflow<\/span><\/td>\n <td>\u68c0\u67e5tau\u548cg\u662f\u5426\u4e3a\u6b63\uff0c\u9650\u5236\u5e42\u5f8b\u6307\u6570\u8303\u56f4<\/span>Ensure tau and g are positive, limit power-law exponent range<\/span><\/td>\n <\/tr>\n <tr>\n <td>STATEV\u503c\u5f02\u5e38\u589e\u5927<\/span>STATEV values abnormally large<\/span><\/td>\n <td>\u72b6\u6001\u53d8\u91cf\u672a\u6b63\u786e\u521d\u59cb\u5316<\/span>State variables not properly initialized<\/span><\/td>\n <td>\u5728*Initial Conditions\u4e2d\u6b63\u786e\u8bbe\u7f6eSTATEV\u521d\u503c<\/span>Set correct STATEV initial values in *Initial Conditions<\/span><\/td>\n <\/tr>\n <tr>\n <td>\u7ed3\u679c\u4e0e\u5b9e\u9a8c\u5dee\u5f02\u5de8\u5927<\/span>Results differ greatly from experiment<\/span><\/td>\n <td>\u53d6\u5411\u53c2\u6570\u9519\u8bef\u6216\u5355\u4f4d\u4e0d\u4e00\u81f4<\/span>Orientation parameters wrong or unit inconsistency<\/span><\/td>\n <td>\u6838\u5bf9Euler\u89d2\u5355\u4f4d\u548c\u6ed1\u79fb\u7cfb\u5b9a\u4e49\uff0c\u7edf\u4e00\u5e94\u529b\u5355\u4f4d\u4e3aMPa<\/span>Check Euler angle units and slip system definitions, unify stress units to MPa<\/span><\/td>\n <\/tr>\n <tr>\n <td>\u8fd0\u884c\u901f\u5ea6\u6781\u6162<\/span>Extremely slow execution<\/span><\/td>\n <td>\u65f6\u95f4\u6b65\u8fc7\u5c0f\u6216Newton\u8fed\u4ee3\u6b21\u6570\u8fc7\u591a<\/span>Time step too small or too many Newton iterations<\/span><\/td>\n <td>\u589e\u5927\u6536\u655b\u5bb9\u5deeTOL\uff0c\u964d\u4f4eITMAX\uff0c\u6216\u68c0\u67e5\u786c\u5316\u53c2\u6570\u662f\u5426\u5bfc\u81f4\u75c5\u6001\u95ee\u9898<\/span>Increase TOL, reduce ITMAX, or check if hardening parameters cause ill-conditioning<\/span><\/td>\n <\/tr>\n <\/tbody>\n <\/table>\n<\/div>\n\n\n<div class=\"chapter\" id=\"refs\">\n <h2>\n Ref<\/span>\n \u53c2\u8003\u6587\u732e \/ References<\/span>\n References<\/span>\n <\/h2>\n \n \u4ee5\u4e0b\u5217\u51fa\u9ec4\u6c38\u521aUMAT\u5b50\u7a0b\u5e8f\u76f8\u5173\u7684\u6838\u5fc3\u7406\u8bba\u6587\u732e\uff0c\u6309\u53d1\u8868\u65f6\u95f4\u6392\u5e8f\uff1a\n <\/p>\n \n The following lists core theoretical references related to Huang’s UMAT subroutine, in chronological order:\n <\/p>\n <div class=\"ref-item\">\n [1] Peirce D, Asaro RJ, Needleman A. Material rate dependence and localized deformation in crystalline solids.\n Acta Metallurgica<\/em>, 1983, 31(12): 1951-1976.\n <\/div>\n <div class=\"ref-item\">\n [2] Peirce D, Shih CF, Needleman A. A tangent modulus method for rate dependent solids.\n Computers & Structures<\/em>, 1984, 18(5): 875-887.\n <\/div>\n <div class=\"ref-item\">\n [3] Asaro RJ, Needleman A. Texture development and strain hardening in rate dependent polycrystals.\n Acta Metallurgica<\/em>, 1985, 33(6): 923-953.\n <\/div>\n <div class=\"ref-item\">\n [4] Harren SV, Deve HE, Asaro RJ. Shear band formation in plane strain compression.\n Acta Metallurgica<\/em>, 1988, 36(9): 2435-2480.\n <\/div>\n <div class=\"ref-item\">\n [5] Bassani JL, Wu TY. Latent hardening in single crystals. II. Analytical characterization and predictions.\n Proceedings of the Royal Society A<\/em>, 1991, 435(1893): 21-41.\n <\/div>\n <div class=\"ref-item\">\n [6] Huang Y. A user-material subroutine incorporating single crystal plasticity in the ABAQUS finite element program.\n Harvard University Report MECH-178<\/em>, 1991.\n <\/div>\n <div class=\"ref-item\">\n [7] Kalidindi SR, Bronkhorst CA, Anand L. Crystallographic texture evolution in bulk deformation processing of FCC metals.\n Journal of the Mechanics and Physics of Solids<\/em>, 1992, 40(3): 537-569.\n <\/div>\n <div class=\"ref-item\">\n [8] Kocks UF, Tome CN, Wenk HR. Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties<\/em>.\n Cambridge University Press, 1998.\n <\/div>\n <div class=\"ref-item\">\n [9] Roters F, Eisenlohr P, Hantcherli L, et al. Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications.\n Acta Materialia<\/em>, 2010, 58(4): 1152-1211.\n <\/div>\n <div class=\"ref-item\">\n [10] Dunne FPE, Petrinic N. Introduction to Computational Plasticity<\/em>. Oxford University Press, 2005.\n <\/div>\n<\/div>\n\n\n<div class=\"chapter\" id=\"appendix\">\n <h2>\n App<\/span>\n \u7b26\u53f7\u8868 \/ Notation<\/span>\n Notation<\/span>\n <\/h2>\n \u672c\u6587\u6863\u4e2d\u4f7f\u7528\u7684\u5173\u952e\u6570\u5b66\u7b26\u53f7\u53ca\u5176\u542b\u4e49\u5982\u4e0b\u8868\u6240\u793a\uff1a<\/p>\n The key mathematical symbols used in this document and their meanings are listed in the following table:<\/p>\n <table>\n <thead>\n <tr>\n <th>\u7b26\u53f7 \/ Symbol<\/span>Symbol<\/span><\/th>\n <th>\u542b\u4e49 \/ Meaning<\/span>Meaning<\/span><\/th>\n <\/tr>\n <\/thead>\n <tbody>\n <tr><td>F<\/b><\/td><td>\u603b\u53d8\u5f62\u68af\u5ea6 \/ Total deformation gradient<\/span>Total deformation gradient<\/span><\/td><\/tr>\n <tr><td>Fe<\/sup><\/b><\/td><td>\u5f39\u6027\u53d8\u5f62\u68af\u5ea6 \/ Elastic deformation gradient<\/span>Elastic deformation gradient<\/span><\/td><\/tr>\n <tr><td>Fp<\/sup><\/b><\/td><td>\u5851\u6027\u53d8\u5f62\u68af\u5ea6 \/ Plastic deformation gradient<\/span>Plastic deformation gradient<\/span><\/td><\/tr>\n <tr><td>σ<\/b><\/td><td>Cauchy\u5e94\u529b\u5f20\u91cf \/ Cauchy stress tensor<\/span>Cauchy stress tensor<\/span><\/td><\/tr>\n <tr><td>τ<\/b><\/td><td>Kirchhoff\u5e94\u529b\u5f20\u91cf \/ Kirchhoff stress tensor<\/span>Kirchhoff stress tensor<\/span><\/td><\/tr>\n <tr><td>Se<\/sup><\/b><\/td><td>\u7b2c\u4e8c\u7c7bPiola-Kirchhoff\u5e94\u529b \/ 2nd P-K stress<\/span>2nd Piola-Kirchhoff stress<\/span><\/td><\/tr>\n <tr><td>Ee<\/sup><\/b><\/td><td>Green\u5f39\u6027\u5e94\u53d8 \/ Green elastic strain<\/span>Green elastic strain<\/span><\/td><\/tr>\n <tr><td>γ̇(α)<\/sup><\/b><\/td><td>\u7b2cα\u6ed1\u79fb\u7cfb\u526a\u5207\u5e94\u53d8\u7387 \/ Shear strain rate on slip system α<\/span>Shear strain rate on slip system α<\/span><\/td><\/tr>\n <tr><td>τ(α)<\/sup><\/b><\/td><td>\u5206\u89e3\u526a\u5e94\u529b \/ Resolved shear stress<\/span>Resolved shear stress<\/span><\/td><\/tr>\n <tr><td>g(α)<\/sup><\/b><\/td><td>\u4e34\u754c\u5206\u5207\u5e94\u529b\uff08\u6ed1\u79fb\u5f3a\u5ea6\uff09\/ CRSS (slip strength)<\/span>Critical resolved shear stress (slip strength)<\/span><\/td><\/tr>\n <tr><td>P(α)<\/sup><\/b><\/td><td>Schmid\u5f20\u91cf \/ Schmid tensor<\/span>Schmid tensor<\/span><\/td><\/tr>\n <tr><td>s(α)<\/sup><\/b><\/td><td>\u6ed1\u79fb\u65b9\u5411 \/ Slip direction<\/span>Slip direction<\/span><\/td><\/tr>\n <tr><td>m(α)<\/sup><\/b><\/td><td>\u6ed1\u79fb\u9762\u6cd5\u5411 \/ Slip plane normal<\/span>Slip plane normal<\/span><\/td><\/tr>\n <tr><td>hαβ<\/sub><\/b><\/td><td>\u786c\u5316\u77e9\u9635 \/ Hardening matrix<\/span>Hardening matrix<\/span><\/td><\/tr>\n <tr><td>Ce<\/sup><\/b><\/td><td>\u5f39\u6027\u521a\u5ea6\u5f20\u91cf \/ Elastic stiffness tensor<\/span>Elastic stiffness tensor<\/span><\/td><\/tr>\n <tr><td>DDSDDE<\/b><\/td><td>\u4e00\u81f4\u5207\u7ebf\u521a\u5ea6\u77e9\u9635 \/ Consistent tangent modulus<\/span>Consistent tangent modulus<\/span><\/td><\/tr>\n <tr><td>STATEV<\/b><\/td><td>\u72b6\u6001\u53d8\u91cf\u6570\u7ec4 \/ State variable array<\/span>State variable array<\/span><\/td><\/tr>\n <tr><td>PROPS<\/b><\/td><td>\u6750\u6599\u53c2\u6570\u6570\u7ec4 \/ Material property array<\/span>Material property array<\/span><\/td><\/tr>\n <\/tbody>\n <\/table>\n<\/div>\n\n<\/div>\n\n<footer style=\"text-align:center; 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