高等固體力學(xué)（上冊(cè)）

內(nèi)容概要

《高等固體力學(xué)》是作者多年來(lái)在為清華大學(xué)研究生開設(shè)“高等固體力學(xué)”(原“固體本構(gòu)關(guān)系”)課程及有關(guān)講座的基礎(chǔ)上，經(jīng)逐年積累更新后編寫而成。書中全面系統(tǒng)地闡述了固體本構(gòu)關(guān)系，并擴(kuò)充了應(yīng)用性的內(nèi)容，涉及國(guó)內(nèi)外各種前沿理論和作者的研究成果。本書分上下兩冊(cè)出版，上冊(cè)主要介紹小變形彈塑性本構(gòu)關(guān)系？連續(xù)介質(zhì)力學(xué)概述？大變形彈性本構(gòu)關(guān)系及應(yīng)用？大變形彈塑性本構(gòu)關(guān)系。書末附有張量分析簡(jiǎn)介和ABAQUS理論基礎(chǔ)，各章末附有習(xí)題？提示或解答。下冊(cè)討論介紹固體力學(xué)近二十年來(lái)幾個(gè)活躍的研究領(lǐng)域。
《高等固體力學(xué)》可作為力學(xué)？材料等專業(yè)研究生教材，也可供相關(guān)專業(yè)的教師與科研人員參考。本書由黃克智、黃永剛編著。

作者簡(jiǎn)介

前 言
2005年前后，清華大學(xué)工程力學(xué)系研究生課程“固體本構(gòu)關(guān)系”更名為“高等固體力學(xué)”，擴(kuò)充了應(yīng)用性的內(nèi)容。
“高等固體力學(xué)”課程主要研究大變形問(wèn)題，但作為基礎(chǔ)，本書上冊(cè)仍保留了第 1章“小變形彈塑性本構(gòu)關(guān)系”，因?yàn)檫@是一個(gè)力學(xué)工作者必須具備的基礎(chǔ)知識(shí)。如果不掌握小變形的理論，那么大變形的理論就無(wú)從談起。
研究大變形固體力學(xué)，需要兩方面的基礎(chǔ)：
（1）張量分析：目前多數(shù)教材中用到的張量分析知識(shí)還僅限于將張量當(dāng)作帶指標(biāo)的符號(hào)。實(shí)際上，張量分析的理論與用途遠(yuǎn)比指標(biāo)符號(hào)深刻得多。它不僅可以使推導(dǎo)變得十分簡(jiǎn)潔，而且還可以清楚地顯示出問(wèn)題本身的物理意義，有時(shí)用張量分析方法可以得到一些意想不到的結(jié)果。我們可以毫不夸張地說(shuō)，不懂得張量分析，要閱讀和消化現(xiàn)代力學(xué)文獻(xiàn)是不可能的。清華大學(xué)工程力學(xué)系每年都為碩士生開設(shè)“張量分析”學(xué)位課 1)。
（2）連續(xù)介質(zhì)力學(xué)：包括應(yīng)力理論、應(yīng)變理論和本構(gòu)關(guān)系。如果缺少?gòu)埩糠治龊瓦B續(xù)介質(zhì)力學(xué)的知識(shí)，高等固體力學(xué)的講授就不可能達(dá)到足夠的深度。為此，上冊(cè)增加了附錄：張量分析（介紹）——當(dāng)然，其中只包含一些最少量的張量分析的必要知識(shí)；同時(shí)，上冊(cè)第 2章“連續(xù)介質(zhì)力學(xué)概述”介紹了研究固體力學(xué)所必需的連續(xù)介質(zhì)力學(xué)基礎(chǔ)知識(shí)。
上冊(cè)第 3章“大變形彈性本構(gòu)關(guān)系及應(yīng)用”講述大變形彈性本構(gòu)關(guān)系的理論、邊值問(wèn)題的解法和一些典型問(wèn)題的解；第 4章“大變形彈塑性本構(gòu)關(guān)系”系統(tǒng)介紹了許多基本概念和幾種主要的理論。對(duì)于大變形問(wèn)題，本構(gòu)關(guān)系可以在物體變形前的構(gòu)形（參考構(gòu)形）中寫出，也可以在物體變形后的構(gòu)形（即時(shí)構(gòu)形）中寫出，甚至還可以在卸載后的構(gòu)形（中間構(gòu)形）中寫出。這幾種寫法涉及到不同的坐標(biāo)，不同的應(yīng)力（率）與不同的應(yīng)變（率）。驟然看來(lái)，它們之間的關(guān)系非常復(fù)雜。考慮到這一難點(diǎn)，本書上冊(cè)著重說(shuō)明這幾種寫法之間的相互“轉(zhuǎn)移”關(guān)系，希望讀者做到舉一就能反三。為了解決實(shí)際大變形問(wèn)題，往往需要采用有限元方法計(jì)算。 ABAQUS是一個(gè)比較便利有效的計(jì)算軟件——上冊(cè)有一附錄，介紹該軟件的理論基礎(chǔ)。
1) 教材包括：黃克智，薛明德，陸明萬(wàn)編著 . 張量分析. 第 2版. 北京：清華大學(xué)出版社， 2003.
高等固體力學(xué)（上冊(cè)）
以上內(nèi)容的初稿曾在清華大學(xué)研究生課程“高等固體力學(xué)”教學(xué)中試用五遍，幾經(jīng)修改定稿后，今作為本書上冊(cè)出版。
本書下冊(cè)討論介紹固體力學(xué)近二十年來(lái)幾個(gè)活躍的研究領(lǐng)域。
第 1章是“晶體的大變形彈塑性理論”。晶體是上冊(cè)第 4章大變形彈塑性本構(gòu)理論最適合的應(yīng)用對(duì)象，通過(guò)晶體塑性可以加深對(duì)理論的理解。
第 2章“應(yīng)變梯度塑性理論”論述微米尺度下的塑性理論。近年的試驗(yàn)表明，當(dāng)材料的非均勻塑性變形特征長(zhǎng)度在微米量級(jí)時(shí)，材料具有很強(qiáng)的尺度效應(yīng)。其原因在于：塑性應(yīng)變?yōu)榉蔷鶆驎r(shí)，塑性應(yīng)變梯度的存在導(dǎo)致“幾何必需位錯(cuò)”產(chǎn)生，使屈服應(yīng)力（“流動(dòng)應(yīng)力”）增大。因此，一點(diǎn)處的應(yīng)力不僅與該點(diǎn)處的應(yīng)變有關(guān)，而且也與該點(diǎn)處的塑性應(yīng)變梯度有關(guān)。由于經(jīng)典的塑性理論中材料本構(gòu)模型不包含任何尺度參數(shù)，所以它不能預(yù)測(cè)材料的尺度效應(yīng)。然而，隨著高技術(shù)的發(fā)展，在工程設(shè)計(jì)中迫切需要處理微米量級(jí)的設(shè)計(jì)和制造問(wèn)題，例如：微電力系統(tǒng)（ MEMS）、微電子封裝、先進(jìn)復(fù)合材料及微加工。因此現(xiàn)代工程設(shè)計(jì)需要微米尺度下的力學(xué)理論。
第 3章是“納米管的力學(xué)”。碳納米管具有優(yōu)良的力學(xué)特性，但過(guò)去被認(rèn)為由于屬納米尺度，不能采用連續(xù)介質(zhì)力學(xué)，而只能用分子動(dòng)力學(xué)來(lái)進(jìn)行分析計(jì)算。分子動(dòng)力學(xué)的出發(fā)點(diǎn)是原子勢(shì)，第 3章論述如何直接從原子勢(shì)出發(fā)，建立納米管或者任意的納米曲面的連續(xù)介質(zhì)力學(xué)。
第 4章是“柔性可伸展電子元件的力學(xué)”。電子元件是由硅制成的。硅是易斷的脆性材料，其斷裂應(yīng)變只有 2%。第 4章研究利用“屈曲”現(xiàn)象制成可伸展電子元件（從而可大大提高電子元件的功能）的原理，分析結(jié)構(gòu)構(gòu)件過(guò)屈曲行為的力學(xué)方法，同時(shí)也發(fā)展了梁、板、殼的過(guò)屈曲理論。
本書所反映的研究成果得到了國(guó)家自然科學(xué)基金委重大和面上項(xiàng)目的長(zhǎng)期支持，我們對(duì)此表示衷心的感謝；第二作者同時(shí)也感謝美國(guó)科學(xué)基金會(huì)的支持；另外，對(duì)海內(nèi)外的合作者、為本書出版過(guò)程提供過(guò)幫助的同事和學(xué)生，以及清華大學(xué)出版社長(zhǎng)期的出版支持，我們一并在此致以誠(chéng)摯的謝意!
黃克智黃永剛 2012年 3月

書籍目錄

目 錄
上 冊(cè)
第 1章 小變形彈塑性本構(gòu)關(guān)系 ······································································1
1.1經(jīng)典彈塑性本構(gòu)關(guān)系 ·········································································1
1.2 J2流動(dòng)理論 ······················································································· 13
1.2.1各向同性硬化 ······································································· 13
1.2.2 混合硬化 ··············································································· 16
1.3 J2形變理論及其與 J2流動(dòng)理論（各向同性硬化）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(dòng)理論的比較 ······································· 33
1.4奇異屈服面塑性理論 ······································································· 35
1.4.1 Sanders理論 ········································································· 35
1.4.2 Koiter理論············································································ 41
1.5 Tresca流動(dòng)理論（混合硬化） ························································ 49
1.6塑性基本假設(shè) ··················································································· 63
1.6.1 Drucker假設(shè)········································································· 64
1.6.2 Ilyushin假設(shè)········································································· 68
1.6.3 對(duì) J2形變理論的重新評(píng)價(jià)··················································· 70
1.7 J2角點(diǎn)理論 ······················································································· 74
1.7.1塑性應(yīng)變率勢(shì) ······································································· 74
1.7.2 W p()80..為凸函數(shù)的條件························································
1.7.3逆塑性本構(gòu)關(guān)系 ··································································· 88
1.7.4 J2角點(diǎn)理論 ··········································································· 93
1.7.5應(yīng)變率勢(shì)理論 ······································································· 98
1.8壓力敏感及塑性膨脹模型 ····························································· 102 習(xí)題 1 ······································································································ 107
第 2章 連續(xù)介質(zhì)力學(xué)概述 ·········································································· 117
2.1變形幾何 ························································································· 117
2.1.1 F的極分解 ········································································· 121
2.1.2線元、面元與體元的變換 ·················································· 126
2.1.3 Hill應(yīng)變度量與 Seth應(yīng)變度量 ········································· 129
高等固體力學(xué)（上冊(cè)）
2.1.4應(yīng)變張量通過(guò)位移矢量表示 ·············································· 131
2.1.5在參考構(gòu)形 R與即時(shí)構(gòu)形 r中梯度運(yùn)算的轉(zhuǎn)換關(guān)系 ········ 134
2.2變形運(yùn)動(dòng)學(xué) ····················································································· 138
2.2.1速度梯度、變形率、旋率 ·················································· 138
2.2.2 各種旋率 ············································································· 145
2.2.3 Hill應(yīng)變度量、 Seth應(yīng)變度量的率 ··································· 147
2.3應(yīng)力理論 ························································································· 152
2.3.1 Cauchy應(yīng)力，第一類與第二類 P-K應(yīng)力························· 152
2.3.2 動(dòng)量方程 ············································································· 157
2.3.3 變形功率 ············································································· 161
()
2.3.4 與E，En功共軛的應(yīng)力度量··········································· 162
2.4質(zhì)量與能量的守恒或平衡律 ·························································· 164
2.4.1質(zhì)量守恒律 ········································································· 165
2.4.2機(jī)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(gòu)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(dǎo)數(shù)） ·········································· 180
2.5.3本構(gòu)理論的客觀性原理 ····················································· 183
2.6 Lagrange嵌入（或隨體）曲線坐標(biāo)，張量的轉(zhuǎn)移 ······················ 187
2.6.1 Lagrange嵌入曲線坐標(biāo)系 ················································· 187
2.6.2張量的轉(zhuǎn)移 ········································································· 191
2.6.3張量的四個(gè)客觀導(dǎo)數(shù) ························································· 195
2.6.4 Lagrange嵌入曲線坐標(biāo) XA與 Euler曲線坐標(biāo) xi··············· 198
2.7小變形彈塑性本構(gòu)關(guān)系形式上的推廣 ·········································· 199
2.7.1彈性本構(gòu)關(guān)系（率形式） ·················································· 200
2.7.2各向同性硬化 Prandtl-Reuss彈塑性本構(gòu)方程 ·················· 202
2.7.3 混合硬化 ············································································· 203
2.7.4 J2形變理論 ········································································· 205
2.8局限性····························································································· 205 習(xí)題 2 ······································································································ 210
第 3章 大變形彈性本構(gòu)關(guān)系及應(yīng)用 ····························································· 227
3.1彈性本構(gòu)關(guān)系與熱傳導(dǎo) ································································· 227
3.1.1彈性本構(gòu)關(guān)系 ····································································· 227
目錄
3.1.2 一個(gè)特例 ············································································· 231
3.1.3 熱傳導(dǎo) ················································································· 237
3.1.4率形式彈性本構(gòu)關(guān)系 ························································· 239
3.2彈性張量必須滿足的條件 ····························································· 242
3.3各向同性材料大變形彈性本構(gòu)關(guān)系 ·············································· 246
3.4彈性大變形典型問(wèn)題解 ································································· 252
3.4.1材料的內(nèi)部約束 ································································· 253
3.4.2各向同性彈性材料的典型問(wèn)題解 ······································ 255
3.5彈性大變形邊值問(wèn)題 ····································································· 278
3.5.1運(yùn)動(dòng)或平衡方程 ································································· 279
3.5.2 邊界條件 ·············································································· 282
3.5.3各向同性彈性體的本構(gòu)關(guān)系 ·············································· 284
3.5.4材料的內(nèi)部約束（續(xù)） ····················································· 287
3.5.5各向同性彈性材料的應(yīng)變能函數(shù) W·································· 290 習(xí)題 3 ······································································································ 293
第 4章 大變形彈塑性本構(gòu)關(guān)系 ···································································· 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(gòu)形剛性轉(zhuǎn)動(dòng) .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(gòu)關(guān)系 ································································· 326
4.3.2內(nèi)變量的演化，正交法則 ·················································· 332
4.4度量相關(guān)性 ····················································································· 364
4.4.1 應(yīng)變度量 E及率 E.，應(yīng)力度量 T及率 T. ····················· 364
4.4.2度量不變量 ········································································· 368
4.4.3對(duì)應(yīng)于不同度量函數(shù)的本構(gòu)關(guān)系 ······································ 370
4.4.4應(yīng)變率與應(yīng)力率的彈塑性分解 ·········································· 371
4.4.5正交法則的對(duì)偶性與度量不變性 ······································ 375
4.5 Simo-Ortiz大變形彈塑性本構(gòu)理論 ··············································· 377
4.5.1 一般關(guān)系 ············································································· 377
4.5.2各向同性硬化（等向硬化）情況 ······································ 380
高等固體力學(xué)（上冊(cè)）
4.6	中間構(gòu)形彈塑性本構(gòu)理論之一 ——Moran-Ortiz-Shih大變形彈塑性本構(gòu)理論 ··························· 389
4.6.1 彈性響應(yīng) ············································································· 391
4.6.2塑性響應(yīng)，率形式本構(gòu)關(guān)系 ·············································· 393
4.6.3虛位移原理 ········································································· 404
4.7	中間構(gòu)形彈塑性本構(gòu)理論之二 ——Van der Giessen大變形彈塑性本構(gòu)理論 ······························ 406
4.7.1熱力學(xué)討論 ········································································· 410
4.7.2 熱傳導(dǎo) ················································································· 413
4.7.3塑性變形率 dp與塑性旋率 wp············································ 414
4.7.4內(nèi)變量理論 ········································································· 418
4.7.5持續(xù)各向同性介質(zhì) ····························································· 422
4.7.6機(jī)動(dòng)與混合硬化 ································································· 426
4.7.7各向異性硬化 ····································································· 432 習(xí)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(shù) ························································································ 443
A.3張量的微積分 ················································································ 449
附錄 B ABAQUS軟件的理論基礎(chǔ) ······························································· 456
B.1塑性大變形 ····················································································· 456
B.2彈性大變形 ····················································································· 472
參考文獻(xiàn) ········································································································· 483
下 冊(cè)
第 1章晶體的大變形彈塑性理論
第 2章 應(yīng)變梯度塑性理論
第 3章納米管的力學(xué)
第 4章柔性可伸展電子元件的力學(xué)
VIII

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