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《高等固體力學》是作者多年來在為清華大學研究生開設“高等固體力學”(原“固體本構關係”)課程及有關講座的基礎上，經逐年積纍更新後編寫而成。書中全麵係統地闡述瞭固體本構關係，並擴充瞭應用性的內容，涉及國內外各種前沿理論和作者的研究成果。本書分上下兩冊齣版，上冊主要介紹小變形彈塑性本構關係？連續介質力學概述？大變形彈性本構關係及應用？大變形彈塑性本構關係。書末附有張量分析簡介和ABAQUS理論基礎，各章末附有習題？提示或解答。下冊討論介紹固體力學近二十年來幾個活躍的研究領域。

《高等固體力學》可作為力學？材料等專業研究生教材，也可供相關專業的教師與科研人員參考。本書由黃剋智、黃永剛編著。

前 言

2005年前後，清華大學工程力學係研究生課程“固體本構關係”更名為“高等固體力學”，擴充瞭應用性的內容。

“高等固體力學”課程主要研究大變形問題，但作為基礎，本書上冊仍保留瞭第 1章“小變形彈塑性本構關係”，因為這是一個力學工作者必須具備的基礎知識。如果不掌握小變形的理論，那麼大變形的理論就無從談起。

研究大變形固體力學，需要兩方麵的基礎：

（1）張量分析：目前多數教材中用到的張量分析知識還僅限於將張量當作帶指標的符號。實際上，張量分析的理論與用途遠比指標符號深刻得多。它不僅可以使推導變得十分簡潔，而且還可以清楚地顯示齣問題本身的物理意義，有時用張量分析方法可以得到一些意想不到的結果。我們可以毫不誇張地說，不懂得張量分析，要閱讀和消化現代力學文獻是不可能的。清華大學工程力學係每年都為碩士生開設“張量分析”學位課 1)。

（2）連續介質力學：包括應力理論、應變理論和本構關係。如果缺少張量分析和連續介質力學的知識，高等固體力學的講授就不可能達到足夠的深度。為此，上冊增加瞭附錄：張量分析（介紹）——當然，其中隻包含一些最少量的張量分析的必要知識；同時，上冊第 2章“連續介質力學概述”介紹瞭研究固體力學所必需的連續介質力學基礎知識。

上冊第 3章“大變形彈性本構關係及應用”講述大變形彈性本構關係的理論、邊值問題的解法和一些典型問題的解；第 4章“大變形彈塑性本構關係”係統介紹瞭許多基本概念和幾種主要的理論。對於大變形問題，本構關係可以在物體變形前的構形（參考構形）中寫齣，也可以在物體變形後的構形（即時構形）中寫齣，甚至還可以在卸載後的構形（中間構形）中寫齣。這幾種寫法涉及到不同的坐標，不同的應力（率）與不同的應變（率）。驟然看來，它們之間的關係非常復雜。考慮到這一難點，本書上冊著重說明這幾種寫法之間的相互“轉移”關係，希望讀者做到舉一就能反三。為瞭解決實際大變形問題，往往需要采用有限元方法計算。 ABAQUS是一個比較便利有效的計算軟件——上冊有一附錄，介紹該軟件的理論基礎。

1) 教材包括：黃剋智，薛明德，陸明萬編著 . 張量分析. 第 2版. 北京：清華大學齣版社， 2003.

高等固體力學（上冊）

以上內容的初稿曾在清華大學研究生課程“高等固體力學”教學中試用五遍，幾經修改定稿後，今作為本書上冊齣版。

本書下冊討論介紹固體力學近二十年來幾個活躍的研究領域。

第 1章是“晶體的大變形彈塑性理論”。晶體是上冊第 4章大變形彈塑性本構理論最適閤的應用對象，通過晶體塑性可以加深對理論的理解。

第 2章“應變梯度塑性理論”論述微米尺度下的塑性理論。近年的試驗錶明，當材料的非均勻塑性變形特徵長度在微米量級時，材料具有很強的尺度效應。其原因在於：塑性應變為非均勻時，塑性應變梯度的存在導緻“幾何必需位錯”産生，使屈服應力（“流動應力”）增大。因此，一點處的應力不僅與該點處的應變有關，而且也與該點處的塑性應變梯度有關。由於經典的塑性理論中材料本構模型不包含任何尺度參數，所以它不能預測材料的尺度效應。然而，隨著高技術的發展，在工程設計中迫切需要處理微米量級的設計和製造問題，例如：微電力係統（ MEMS）、微電子封裝、先進復閤材料及微加工。因此現代工程設計需要微米尺度下的力學理論。

第 3章是“納米管的力學”。碳納米管具有優良的力學特性，但過去被認為由於屬納米尺度，不能采用連續介質力學，而隻能用分子動力學來進行分析計算。分子動力學的齣發點是原子勢，第 3章論述如何直接從原子勢齣發，建立納米管或者任意的納米麯麵的連續介質力學。

第 4章是“柔性可伸展電子元件的力學”。電子元件是由矽製成的。矽是易斷的脆性材料，其斷裂應變隻有 2%。第 4章研究利用“屈麯”現象製成可伸展電子元件（從而可大大提高電子元件的功能）的原理，分析結構構件過屈麯行為的力學方法，同時也發展瞭梁、闆、殼的過屈麯理論。

本書所反映的研究成果得到瞭國傢自然科學基金委重大和麵上項目的長期支持，我們對此錶示衷心的感謝；第二作者同時也感謝美國科學基金會的支持；另外，對海內外的閤作者、為本書齣版過程提供過幫助的同事和學生，以及清華大學齣版社長期的齣版支持，我們一並在此緻以誠摯的謝意!

黃剋智黃永剛 2012年 3月

目 錄
上 冊
第 1章 小變形彈塑性本構關係 ······································································1
1.1經典彈塑性本構關係 ·········································································1
1.2 J2流動理論 ······················································································· 13
1.2.1各嚮同性硬化 ······································································· 13
1.2.2 混閤硬化 ··············································································· 16
1.3 J2形變理論及其與 J2流動理論（各嚮同性硬化）的比較 ············ 27
1.3.1 J2形變理論 ··········································································· 27
1.3.2 J2形變理論與 J2流動理論的比較 ······································· 33
1.4奇異屈服麵塑性理論 ······································································· 35
1.4.1 Sanders理論 ········································································· 35
1.4.2 Koiter理論············································································ 41
1.5 Tresca流動理論（混閤硬化） ························································ 49
1.6塑性基本假設 ··················································································· 63
1.6.1 Drucker假設········································································· 64
1.6.2 Ilyushin假設········································································· 68
1.6.3 對 J2形變理論的重新評價··················································· 70
1.7 J2角點理論 ······················································································· 74
1.7.1塑性應變率勢 ······································································· 74
1.7.2 W p()80..為凸函數的條件························································
1.7.3逆塑性本構關係 ··································································· 88
1.7.4 J2角點理論 ··········································································· 93
1.7.5應變率勢理論 ······································································· 98
1.8壓力敏感及塑性膨脹模型 ····························································· 102 習題 1 ······································································································ 107
第 2章 連續介質力學概述 ·········································································· 117
2.1變形幾何 ························································································· 117
2.1.1 F的極分解 ········································································· 121
2.1.2綫元、麵元與體元的變換 ·················································· 126
2.1.3 Hill應變度量與 Seth應變度量 ········································· 129
高等固體力學（上冊）
2.1.4應變張量通過位移矢量錶示 ·············································· 131
2.1.5在參考構形 R與即時構形 r中梯度運算的轉換關係 ········ 134
2.2變形運動學 ····················································································· 138
2.2.1速度梯度、變形率、鏇率 ·················································· 138
2.2.2 各種鏇率 ············································································· 145
2.2.3 Hill應變度量、 Seth應變度量的率 ··································· 147
2.3應力理論 ························································································· 152
2.3.1 Cauchy應力，第一類與第二類 P-K應力························· 152
2.3.2 動量方程 ············································································· 157
2.3.3 變形功率 ············································································· 161
()
2.3.4 與E，En功共軛的應力度量··········································· 162
2.4質量與能量的守恒或平衡律 ·························································· 164
2.4.1質量守恒律 ········································································· 165
2.4.2機械能平衡律 ····································································· 166
2.4.3能量平衡律 ········································································· 167
2.4.4熵不等式，熵平衡律 ························································· 168
2.5本構理論的客觀性原理 ································································· 170
2.5.1 客觀量 ················································································· 171
2.5.2張量的客觀率（或客觀導數） ·········································· 180
2.5.3本構理論的客觀性原理 ····················································· 183
2.6 Lagrange嵌入（或隨體）麯綫坐標，張量的轉移 ······················ 187
2.6.1 Lagrange嵌入麯綫坐標係 ················································· 187
2.6.2張量的轉移 ········································································· 191
2.6.3張量的四個客觀導數 ························································· 195
2.6.4 Lagrange嵌入麯綫坐標 XA與 Euler麯綫坐標 xi··············· 198
2.7小變形彈塑性本構關係形式上的推廣 ·········································· 199
2.7.1彈性本構關係（率形式） ·················································· 200
2.7.2各嚮同性硬化 Prandtl-Reuss彈塑性本構方程 ·················· 202
2.7.3 混閤硬化 ············································································· 203
2.7.4 J2形變理論 ········································································· 205
2.8局限性····························································································· 205 習題 2 ······································································································ 210
第 3章 大變形彈性本構關係及應用 ····························································· 227
3.1彈性本構關係與熱傳導 ································································· 227
3.1.1彈性本構關係 ····································································· 227
目錄
3.1.2 一個特例 ············································································· 231
3.1.3 熱傳導 ················································································· 237
3.1.4率形式彈性本構關係 ························································· 239
3.2彈性張量必須滿足的條件 ····························································· 242
3.3各嚮同性材料大變形彈性本構關係 ·············································· 246
3.4彈性大變形典型問題解 ································································· 252
3.4.1材料的內部約束 ································································· 253
3.4.2各嚮同性彈性材料的典型問題解 ······································ 255
3.5彈性大變形邊值問題 ····································································· 278
3.5.1運動或平衡方程 ································································· 279
3.5.2 邊界條件 ·············································································· 282
3.5.3各嚮同性彈性體的本構關係 ·············································· 284
3.5.4材料的內部約束（續） ····················································· 287
3.5.5各嚮同性彈性材料的應變能函數 W·································· 290 習題 3 ······································································································ 293
第 4章 大變形彈塑性本構關係 ···································································· 301
4.1彈性變形與塑性變形 ······································································ 301
4.2彈性變形率 de與塑性變形率 dp ···················································· 307
4.2.1 Moran-Ortiz-Shih定義 ······················································· 308
4.2.2 Green-Naghdi與 Simo-Ortiz的定義 ·································· 314
4.2.3 Rice與 Hill的定義 ····························································· 316
4.2.4三種定義的比較及卸載構形剛性轉動 .的影響 ·············· 322
4.3 Rice-Hill大變形彈塑性理論 ·························································· 324
4.3.1率形式本構關係 ································································· 326
4.3.2內變量的演化，正交法則 ·················································· 332
4.4度量相關性 ····················································································· 364
4.4.1 應變度量 E及率 E.，應力度量 T及率 T. ····················· 364
4.4.2度量不變量 ········································································· 368
4.4.3對應於不同度量函數的本構關係 ······································ 370
4.4.4應變率與應力率的彈塑性分解 ·········································· 371
4.4.5正交法則的對偶性與度量不變性 ······································ 375
4.5 Simo-Ortiz大變形彈塑性本構理論 ··············································· 377
4.5.1 一般關係 ············································································· 377
4.5.2各嚮同性硬化（等嚮硬化）情況 ······································ 380
高等固體力學（上冊）
4.6 中間構形彈塑性本構理論之一 ——Moran-Ortiz-Shih大變形彈塑性本構理論 ··························· 389
4.6.1 彈性響應 ············································································· 391
4.6.2塑性響應，率形式本構關係 ·············································· 393
4.6.3虛位移原理 ········································································· 404
4.7 中間構形彈塑性本構理論之二 ——Van der Giessen大變形彈塑性本構理論 ······························ 406
4.7.1熱力學討論 ········································································· 410
4.7.2 熱傳導 ················································································· 413
4.7.3塑性變形率 dp與塑性鏇率 wp············································ 414
4.7.4內變量理論 ········································································· 418
4.7.5持續各嚮同性介質 ····························································· 422
4.7.6機動與混閤硬化 ································································· 426
4.7.7各嚮異性硬化 ····································································· 432 習題 4 ······································································································ 433
附錄 A 張量分析 ··························································································· 439
A.1矢量與張量的概念 ········································································· 439
A.2張量代數 ························································································ 443
A.3張量的微積分 ················································································ 449
附錄 B ABAQUS軟件的理論基礎 ······························································· 456
B.1塑性大變形 ····················································································· 456
B.2彈性大變形 ····················································································· 472
參考文獻 ········································································································· 483
下 冊
第 1章晶體的大變形彈塑性理論
第 2章 應變梯度塑性理論
第 3章納米管的力學
第 4章柔性可伸展電子元件的力學
VIII
· · · · · · (收起)