高等固体力学（上册）

作者简介

《高等固体力学》是作者多年来在为清华大学研究生开设“高等固体力学”(原“固体本构关系”)课程及有关讲座的基础上，经逐年积累更新后编写而成。书中全面系统地阐述了固体本构关系，并扩充了应用性的内容，涉及国内外各种前沿理论和作者的研究成果。本书分上下两册出版，上册主要介绍小变形弹塑性本构关系？连续介质力学概述？大变形弹性本构关系及应用？大变形弹塑性本构关系。书末附有张量分析简介和ABAQUS理论基础，各章末附有习题？提示或解答。下册讨论介绍固体力学近二十年来几个活跃的研究领域。



《高等固体力学》可作为力学？材料等专业研究生教材，也可供相关专业的教师与科研人员参考。本书由黄克智、黄永刚编著。

书籍目录

目 录

上 册

第 1章 小变形弹塑性本构关系 ······································································1

1.1经典弹塑性本构关系 ·········································································1

1.2 J2流动理论 ······················································································· 13

1.2.1各向同性硬化 ······································································· 13

1.2.2 混合硬化 ··············································································· 16

1.3 J2形变理论及其与 J2流动理论（各向同性硬化）的比较 ············ 27

1.3.1 J2形变理论 ··········································································· 27

1.3.2 J2形变理论与 J2流动理论的比较 ······································· 33

1.4奇异屈服面塑性理论 ······································································· 35

1.4.1 Sanders理论 ········································································· 35

1.4.2 Koiter理论············································································ 41

1.5 Tresca流动理论（混合硬化） ························································ 49

1.6塑性基本假设 ··················································································· 63

1.6.1 Drucker假设········································································· 64

1.6.2 Ilyushin假设········································································· 68

1.6.3 对 J2形变理论的重新评价··················································· 70

1.7 J2角点理论 ······················································································· 74

1.7.1塑性应变率势 ······································································· 74

1.7.2 W p()80..为凸函数的条件························································

1.7.3逆塑性本构关系 ··································································· 88

1.7.4 J2角点理论 ··········································································· 93

1.7.5应变率势理论 ······································································· 98

1.8压力敏感及塑性膨胀模型 ····························································· 102 习题 1 ······································································································ 107

第 2章 连续介质力学概述 ·········································································· 117

2.1变形几何 ························································································· 117

2.1.1 F的极分解 ········································································· 121

2.1.2线元、面元与体元的变换 ·················································· 126

2.1.3 Hill应变度量与 Seth应变度量 ········································· 129

高等固体力学（上册）

2.1.4应变张量通过位移矢量表示 ·············································· 131

2.1.5在参考构形 R与即时构形 r中梯度运算的转换关系 ········ 134

2.2变形运动学 ····················································································· 138

2.2.1速度梯度、变形率、旋率 ·················································· 138

2.2.2 各种旋率 ············································································· 145

2.2.3 Hill应变度量、 Seth应变度量的率 ··································· 147

2.3应力理论 ························································································· 152

2.3.1 Cauchy应力，第一类与第二类 P-K应力························· 152

2.3.2 动量方程 ············································································· 157

2.3.3 变形功率 ············································································· 161

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2.3.4 与E，En功共轭的应力度量··········································· 162

2.4质量与能量的守恒或平衡律 ·························································· 164

2.4.1质量守恒律 ········································································· 165

2.4.2机械能平衡律 ····································································· 166

2.4.3能量平衡律 ········································································· 167

2.4.4熵不等式，熵平衡律 ························································· 168

2.5本构理论的客观性原理 ································································· 170

2.5.1 客观量 ················································································· 171

2.5.2张量的客观率（或客观导数） ·········································· 180

2.5.3本构理论的客观性原理 ····················································· 183

2.6 Lagrange嵌入（或随体）曲线坐标，张量的转移 ······················ 187

2.6.1 Lagrange嵌入曲线坐标系 ················································· 187

2.6.2张量的转移 ········································································· 191

2.6.3张量的四个客观导数 ························································· 195

2.6.4 Lagrange嵌入曲线坐标 XA与 Euler曲线坐标 xi··············· 198

2.7小变形弹塑性本构关系形式上的推广 ·········································· 199

2.7.1弹性本构关系（率形式） ·················································· 200

2.7.2各向同性硬化 Prandtl-Reuss弹塑性本构方程 ·················· 202

2.7.3 混合硬化 ············································································· 203

2.7.4 J2形变理论 ········································································· 205

2.8局限性····························································································· 205 习题 2 ······································································································ 210

第 3章 大变形弹性本构关系及应用 ····························································· 227

3.1弹性本构关系与热传导 ································································· 227

3.1.1弹性本构关系 ····································································· 227

目录

3.1.2 一个特例 ············································································· 231

3.1.3 热传导 ················································································· 237

3.1.4率形式弹性本构关系 ························································· 239

3.2弹性张量必须满足的条件 ····························································· 242

3.3各向同性材料大变形弹性本构关系 ·············································· 246

3.4弹性大变形典型问题解 ································································· 252

3.4.1材料的内部约束 ································································· 253

3.4.2各向同性弹性材料的典型问题解 ······································ 255

3.5弹性大变形边值问题 ····································································· 278

3.5.1运动或平衡方程 ································································· 279

3.5.2 边界条件 ·············································································· 282

3.5.3各向同性弹性体的本构关系 ·············································· 284

3.5.4材料的内部约束（续） ····················································· 287

3.5.5各向同性弹性材料的应变能函数 W·································· 290 习题 3 ······································································································ 293

第 4章 大变形弹塑性本构关系 ···································································· 301

4.1弹性变形与塑性变形 ······································································ 301

4.2弹性变形率 de与塑性变形率 dp ···················································· 307

4.2.1 Moran-Ortiz-Shih定义 ······················································· 308

4.2.2 Green-Naghdi与 Simo-Ortiz的定义 ·································· 314

4.2.3 Rice与 Hill的定义 ····························································· 316

4.2.4三种定义的比较及卸载构形刚性转动 .的影响 ·············· 322

4.3 Rice-Hill大变形弹塑性理论 ·························································· 324

4.3.1率形式本构关系 ································································· 326

4.3.2内变量的演化，正交法则 ·················································· 332

4.4度量相关性 ····················································································· 364

4.4.1 应变度量 E及率 E.，应力度量 T及率 T. ····················· 364

4.4.2度量不变量 ········································································· 368

4.4.3对应于不同度量函数的本构关系 ······································ 370

4.4.4应变率与应力率的弹塑性分解 ·········································· 371

4.4.5正交法则的对偶性与度量不变性 ······································ 375

4.5 Simo-Ortiz大变形弹塑性本构理论 ··············································· 377

4.5.1 一般关系 ············································································· 377

4.5.2各向同性硬化（等向硬化）情况 ······································ 380

高等固体力学（上册）

4.6
中间构形弹塑性本构理论之一 ——Moran-Ortiz-Shih大变形弹塑性本构理论 ··························· 389

4.6.1 弹性响应 ············································································· 391

4.6.2塑性响应，率形式本构关系 ·············································· 393

4.6.3虚位移原理 ········································································· 404

4.7
中间构形弹塑性本构理论之二 ——Van der Giessen大变形弹塑性本构理论 ······························ 406

4.7.1热力学讨论 ········································································· 410

4.7.2 热传导 ················································································· 413

4.7.3塑性变形率 dp与塑性旋率 wp············································ 414

4.7.4内变量理论 ········································································· 418

4.7.5持续各向同性介质 ····························································· 422

4.7.6机动与混合硬化 ································································· 426

4.7.7各向异性硬化 ····································································· 432 习题 4 ······································································································ 433

附录 A 张量分析 ··························································································· 439

A.1矢量与张量的概念 ········································································· 439

A.2张量代数 ························································································ 443

A.3张量的微积分 ················································································ 449

附录 B ABAQUS软件的理论基础 ······························································· 456

B.1塑性大变形 ····················································································· 456

B.2弹性大变形 ····················································································· 472

参考文献 ········································································································· 483

下 册

第 1章晶体的大变形弹塑性理论

第 2章 应变梯度塑性理论

第 3章纳米管的力学

第 4章柔性可伸展电子元件的力学

VIII

内容概要

前 言

2005年前后，清华大学工程力学系研究生课程“固体本构关系”更名为“高等固体力学”，扩充了应用性的内容。

“高等固体力学”课程主要研究大变形问题，但作为基础，本书上册仍保留了第 1章“小变形弹塑性本构关系”，因为这是一个力学工作者必须具备的基础知识。如果不掌握小变形的理论，那么大变形的理论就无从谈起。

研究大变形固体力学，需要两方面的基础：

（1）张量分析：目前多数教材中用到的张量分析知识还仅限于将张量当作带指标的符号。实际上，张量分析的理论与用途远比指标符号深刻得多。它不仅可以使推导变得十分简洁，而且还可以清楚地显示出问题本身的物理意义，有时用张量分析方法可以得到一些意想不到的结果。我们可以毫不夸张地说，不懂得张量分析，要阅读和消化现代力学文献是不可能的。清华大学工程力学系每年都为硕士生开设“张量分析”学位课 1)。

（2）连续介质力学：包括应力理论、应变理论和本构关系。如果缺少张量分析和连续介质力学的知识，高等固体力学的讲授就不可能达到足够的深度。为此，上册增加了附录：张量分析（介绍）——当然，其中只包含一些最少量的张量分析的必要知识；同时，上册第 2章“连续介质力学概述”介绍了研究固体力学所必需的连续介质力学基础知识。

上册第 3章“大变形弹性本构关系及应用”讲述大变形弹性本构关系的理论、边值问题的解法和一些典型问题的解；第 4章“大变形弹塑性本构关系”系统介绍了许多基本概念和几种主要的理论。对于大变形问题，本构关系可以在物体变形前的构形（参考构形）中写出，也可以在物体变形后的构形（即时构形）中写出，甚至还可以在卸载后的构形（中间构形）中写出。这几种写法涉及到不同的坐标，不同的应力（率）与不同的应变（率）。骤然看来，它们之间的关系非常复杂。考虑到这一难点，本书上册着重说明这几种写法之间的相互“转移”关系，希望读者做到举一就能反三。为了解决实际大变形问题，往往需要采用有限元方法计算。 ABAQUS是一个比较便利有效的计算软件——上册有一附录，介绍该软件的理论基础。

1) 教材包括：黄克智，薛明德，陆明万编著 . 张量分析. 第 2版. 北京：清华大学出版社， 2003.

高等固体力学（上册）

以上内容的初稿曾在清华大学研究生课程“高等固体力学”教学中试用五遍，几经修改定稿后，今作为本书上册出版。

本书下册讨论介绍固体力学近二十年来几个活跃的研究领域。

第 1章是“晶体的大变形弹塑性理论”。晶体是上册第 4章大变形弹塑性本构理论最适合的应用对象，通过晶体塑性可以加深对理论的理解。

第 2章“应变梯度塑性理论”论述微米尺度下的塑性理论。近年的试验表明，当材料的非均匀塑性变形特征长度在微米量级时，材料具有很强的尺度效应。其原因在于：塑性应变为非均匀时，塑性应变梯度的存在导致“几何必需位错”产生，使屈服应力（“流动应力”）增大。因此，一点处的应力不仅与该点处的应变有关，而且也与该点处的塑性应变梯度有关。由于经典的塑性理论中材料本构模型不包含任何尺度参数，所以它不能预测材料的尺度效应。然而，随着高技术的发展，在工程设计中迫切需要处理微米量级的设计和制造问题，例如：微电力系统（ MEMS）、微电子封装、先进复合材料及微加工。因此现代工程设计需要微米尺度下的力学理论。

第 3章是“纳米管的力学”。碳纳米管具有优良的力学特性，但过去被认为由于属纳米尺度，不能采用连续介质力学，而只能用分子动力学来进行分析计算。分子动力学的出发点是原子势，第 3章论述如何直接从原子势出发，建立纳米管或者任意的纳米曲面的连续介质力学。

第 4章是“柔性可伸展电子元件的力学”。电子元件是由硅制成的。硅是易断的脆性材料，其断裂应变只有 2%。第 4章研究利用“屈曲”现象制成可伸展电子元件（从而可大大提高电子元件的功能）的原理，分析结构构件过屈曲行为的力学方法，同时也发展了梁、板、壳的过屈曲理论。

本书所反映的研究成果得到了国家自然科学基金委重大和面上项目的长期支持，我们对此表示衷心的感谢；第二作者同时也感谢美国科学基金会的支持；另外，对海内外的合作者、为本书出版过程提供过帮助的同事和学生，以及清华大学出版社长期的出版支持，我们一并在此致以诚挚的谢意!

黄克智黄永刚 2012年 3月

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