目錄
前言
第1章 緒論 1
1.1 混凝土受力力學(xué)行為的一般特征 1
1.1.1 單軸受拉特征 2
1.1.2 單軸受壓特征 3
1.1.3 雙軸受力特征 6
1.2 經(jīng)典力學(xué)對混凝土力學(xué)行為的反映 9
1.3 混凝土損傷力學(xué)的發(fā)展 12
1.4 本書基本觀點(diǎn)與內(nèi)容 15
1.4.1 本書基本觀點(diǎn) 15
1.4.2 本書基本內(nèi)容 17
第2章 應(yīng)力-應(yīng)變分析 20
2.1 應(yīng)力與應(yīng)變的基本概念 20
2.2 應(yīng)力張量及其不變量 21
2.2.1 應(yīng)力張量 21
2.2.2 主應(yīng)力 22
2.2.3 應(yīng)力張量的不變量 24
2.3 應(yīng)變張量及其不變量 27
2.3.1 應(yīng)變張量 27
2.3.2 應(yīng)變張量的不變量 27
2.4 應(yīng)力應(yīng)變分析基本方程 28
第3章 彈塑性力學(xué)基礎(chǔ) 31
3.1 彈性本構(gòu)關(guān)系 31
3.1.1 線彈性本構(gòu)關(guān)系 31
3.1.2 非線性彈性本構(gòu)關(guān)系 32
3.2 彈塑性本構(gòu)關(guān)系 33
3.2.1 屈服條件 34
3.2.2 硬化法則 35
3.2.3 流動法則與加/卸載準(zhǔn)則 36
3.2.4 彈塑性增量本構(gòu)模型 37
3.2.5 塑性勢函數(shù) 39
第4章 損傷力學(xué)基本原理 41
4.1 損傷力學(xué)基本概念 41
4.2 熱力學(xué)基本原理 43
4.2.1 熱力學(xué)第一定律 44
4.2.2 熱力學(xué)第二定律 45
4.2.3 等溫純力學(xué)過程 46
4.3 彈性損傷模型基本方程 48
4.3.1 基本方程 48
4.3.2 單標(biāo)量彈性損傷模型 49
4.4 單標(biāo)量彈塑性損傷本構(gòu)模型 50
4.4.1 彈塑性自由能 50
4.4.2 彈塑性損傷本構(gòu)模型 52
4.4.3 內(nèi)變量演化法則 52
4.4.4 建立彈塑性損傷本構(gòu)關(guān)系模型的一般步驟 55
4.4.5 評注 56
第5章 混凝土確定性損傷本構(gòu)關(guān)系 57
5.1 混凝土材料的受力損傷機(jī)制 57
5.2 混凝土彈性損傷本構(gòu)關(guān)系 58
5.2.1 應(yīng)力張量分解 58
5.2.2 Helmholtz自由能 60
5.2.3 雙標(biāo)量損傷本構(gòu)關(guān)系 62
5.2.4 本構(gòu)關(guān)系的全量表達(dá) 63
5.2.5 討論與注記 66
5.3 混凝土彈塑性損傷本構(gòu)關(guān)系 66
5.3.1 彈塑性Helmholtz自由能 67
5.3.2 彈塑性損傷模型 69
5.3.3 塑性變形的確立 71
5.3.4 損傷演化法則 75
5.3.5 切向剛度張量 83
第6章 混凝土隨機(jī)損傷本構(gòu)關(guān)系 85
6.1 細(xì)觀損傷機(jī)制分析 85
6.1.1 典型細(xì)觀單元損傷分析 85
6.1.2 垂直裂縫方向的正應(yīng)力作用 87
6.1.3 裂縫正交坐標(biāo)系內(nèi)的剪應(yīng)力作用 88
6.1.4 平行于裂縫方向的正應(yīng)力作用 89
6.1.5 損傷機(jī)制再討論 90
6.2 細(xì)觀隨機(jī)斷裂模型 90
6.2.1 細(xì)觀受拉隨機(jī)損傷模型 90
6.2.2 隨機(jī)損傷變量的均值與標(biāo)準(zhǔn)差 95
6.2.3 細(xì)觀受剪隨機(jī)損傷模型 98
6.2.4 一維隨機(jī)損傷本構(gòu)關(guān)系 99
6.2.5 單向受力狀態(tài)下細(xì)觀塑性變形的考慮 102
6.3 細(xì)觀損傷的實(shí)驗(yàn)建模 103
6.3.1 利用聲發(fā)射能率應(yīng)變關(guān)系的建模 103
6.3.2 直接利用實(shí)驗(yàn)平均應(yīng)力應(yīng)變關(guān)系的建模 107
6.4 混凝土彈塑性隨機(jī)損傷本構(gòu)關(guān)系 110
6.4.1 有效應(yīng)力分解與Helmholtz自由能 110
6.4.2 損傷本構(gòu)關(guān)系 111
6.4.3 損傷演化法則 113
6.4.4 經(jīng)驗(yàn)塑性變形 117
6.4.5 隨機(jī)損傷本構(gòu)關(guān)系的均值與方差 118
第7章 混凝土動力損傷本構(gòu)關(guān)系 121
7.1 基于經(jīng)驗(yàn)的黏性動力損傷模型 121
7.1.1 研究發(fā)展概況 121
7.1.2 塑性力學(xué)中的率相關(guān)效應(yīng)模型 122
7.1.3 黏彈塑性動力損傷模型 123
7.2 基于隨機(jī)Stefan效應(yīng)的動力損傷模型 125
7.2.1 Stefan效應(yīng) 125
7.2.2 應(yīng)變遲滯因子 129
7.2.3 基于隨機(jī)Stefan效應(yīng)的隨機(jī)動力損傷模型 131
7.3 基于多裂紋擴(kuò)展的動力損傷模型 133
7.3.1 高應(yīng)變率條件下的裂紋擴(kuò)展特征 133
7.3.2 高應(yīng)變率條件下的動力損傷機(jī)理 134
7.3.3 多裂紋同時(shí)擴(kuò)展的斷裂動力學(xué) 138
7.3.4 動力損傷模型 141
第8章 混凝土本構(gòu)關(guān)系的數(shù)值算法 143
8.1 基本概念 143
8.1.1 一維彈塑性損傷本構(gòu)關(guān)系基本方程 143
8.1.2 微分方程的數(shù)值算法 145
8.1.3 基于向后歐拉方法的塑性變形求解算法 146
8.1.4 注記 150
8.2 彈塑性損傷本構(gòu)模型數(shù)值算法 150
8.2.1 算子分離算法 150
8.2.2 有效應(yīng)力更新——最近點(diǎn)投影算法 152
8.2.3 譜分解方法 156
8.2.4 數(shù)值一致性切線剛度 157
8.2.5 算法流程 159
8.2.6 與通用非線性有限元分析程序的接口 161
第9章 混凝土框架結(jié)構(gòu)分析 164
9.1 梁柱構(gòu)件正截面N-M-*關(guān)系分析 164
9.1.1 條帶法 164
9.1.2 網(wǎng)格法 168
9.1.3 增量Dn-dM-d*關(guān)系 170
9.2 梁一柱單元分析 171
9.2.1 纖維梁單元增量剛度矩陣 171
9.2.2 剛域的考慮 175
9.2.3 注記：混凝土梁柱單元分析模型的發(fā)展 176
9.3 有限單元柔度法 177
9.3.1 基本思想 177
9.3.2 單元狀態(tài)確定 180
第10章 混凝土剪力墻結(jié)構(gòu)分析 182
10.1 平面應(yīng)力單元 182
10.2 四節(jié)點(diǎn)平板殼單元 185
10.3 精細(xì)化平板殼單元 189
10.3.1 考慮節(jié)點(diǎn)旋轉(zhuǎn)自由度的膜單元 189
10.3.2 八節(jié)點(diǎn)Reissner-Mindlin板單元 192
10.3.3 平板殼單元 195
10.4 鋼筋混凝土分層平板殼單元 196
10.4.1 分層平板殼單元的剛度構(gòu)成 196
10.4.2 損傷演化法則的修正 198
10.4.3 結(jié)構(gòu)非線性分析流程 200
第11章 混凝土實(shí)體結(jié)構(gòu)分析 202
11.1 一般結(jié)構(gòu)初邊值問題的有限元方法 202
11.1.1 初邊值問題的微分方程 202
11.1.2 初邊值問題的弱形式 203
11.1.3 初邊值問題的有限元控制方程 204
11.2 結(jié)構(gòu)有限元分析的數(shù)值實(shí)現(xiàn) 206
11.2.1 數(shù)值積分方法 207
11.2.2 非線性方程組的求解 207
11.2.3 負(fù)剛度問題及其處理 210
11.2.4 結(jié)構(gòu)有限元分析的基本步驟 212
11.3 混凝土結(jié)構(gòu)分析的應(yīng)變局部化問題 213
11.3.1 軟化材料的應(yīng)變局部化 213
11.3.2 材料應(yīng)變局部化對初邊值問題的影響 214
11.3.3 數(shù)值分析結(jié)果病態(tài) 215
11.4 混凝土材料非局部本構(gòu)關(guān)系 218
11.4.1 非局部材料本構(gòu)關(guān)系的物理機(jī)制 218
11.4.2 非局部材料本構(gòu)模型的基本概念 220
11.4.3 混凝土積分型非局部損傷模型 223
11.4.4 混凝土隱式梯度非局部損傷模型 226
11.4.5 相關(guān)評述 228
第12章 混凝土結(jié)構(gòu)隨機(jī)非線性分析 230
12.1 隨機(jī)性在物理系統(tǒng)中的傳播 230
12.1.1 概率守恒原理 230
12.1.2 概率守恒原理的隨機(jī)事件描述 231
12.1.3 廣義概率密度演化方程 232
12.1.4 廣義概率密度演化方程的解析解 234
12.2 混凝土結(jié)構(gòu)靜力非線性隨機(jī)反應(yīng)分析 235
12.2.1 結(jié)構(gòu)反應(yīng)的概率密度演化方程 235
12.2.2 概率密度演化方程的數(shù)值求解步驟 237
12.2.3 概率空間剖分 238
12.2.4 有限差分法 238
12.2.5 注記 241
12.3 混凝土結(jié)構(gòu)動力非線性隨機(jī)反應(yīng)分析 242
12.3.1 動力方程與廣義概率密度演化方程 242
12.3.2 結(jié)構(gòu)動力分析 244
12.3.3 差分格式的選取 246
12.3.4 地震反應(yīng)分析實(shí)例 249
參考文獻(xiàn) 252
附錄A 張量初步 281
A.1 笛卡兒張量 281
A.2 張量的運(yùn)算和性質(zhì) 283
A.3 張量分析初步 286
A.4 張量的矩陣表示 288
A.4.1 動力學(xué)Voigt表示 288
A.4.2 運(yùn)動學(xué)Voigt表示 288
附錄B 混凝土細(xì)觀斷裂應(yīng)變隨機(jī)場數(shù)值模擬 291
B.1 Karhunen-Loeve分解方法 291
B.2 隨機(jī)諧和函數(shù)方法 293
附錄C 多維隨機(jī)變量空間的剖分與選點(diǎn) 295
C.1 切球選點(diǎn)法 295
C.1.1 兩個(gè)隨機(jī)變量時(shí)的切圓選點(diǎn)方法 295
C.1.2 三個(gè)隨機(jī)變量時(shí)的切球選點(diǎn)方法 299
C.2 數(shù)論選點(diǎn)法 302
C.2.1 高維數(shù)值積分的數(shù)論方法 302
C.2.2 多維隨機(jī)變量空間的數(shù)論選點(diǎn)法 303
C.3 一般概率分布的處理 307
C.3.1 基本隨機(jī)變量的標(biāo)準(zhǔn)化 307
C.3.2 超球體篩選的推廣 307
索引 309
Contents
Preface
Chapter 1 Introduction 1
1.1 General mechanical behavior of concrete 1
1.1.1 Uniaxial tensile behavior 2
1.1.2 Uniaxial compressive behavior 3
1.1.3 Biaxial behaviors 6
1.2 Classical mechanical models of concrete 9
1.3 Historical development of concrete damage mechanics 12
1.4 Fundamental viewpoints and contents of this book 15
1.4.1 Fundamental viewpoints 15
1.4.2 Contents 17
Chapter 2 Analyses of stress and strain 20
2.1 Concepts of stress and strain 20
2.2 Stress tensor and its invariants 21
2.2.1 Stress tensor 21
2.2.2 Principal stresses 22
2.2.3 Invariants of stress tensors 24
2.3 Strain tensor and its invariants 27
2.3.1 Strain tensor 27
2.3.2 Invariants of strain tensor 27
2.4 Fundamental equations of stress and strain analyses 28
Chapter 3 Fundaments of elastoplaticity mechanics 31
3.1 Elastic constitutive relations 31
3.1.1 Linear elastic model 31
3.1.2 Nonlinear elastic model 32
3.2 Elastoplastic model 33
3.2.1 Yield criteria 34
3.2.2 Hardening law 35
3.2.3 Flow rule and loading/unloading conditions 36
3.2.4 Incremental elastoplastic constitutive relations 37
3.2.5 Plastic yield function 39
Chapter 4Principles of damage mechanics 41
4.1 Fundamental concepts of damage mechanics 41
4.2 Principles of thermodynamics 43
4.2.1 First law of thermodynamics 44
4.2.2 Second law of thermodynamics 45
4.2.3 Isothermal process 46
4.3 Eastic damage model 48
4.3.1 Fundamental equations 48
4.3.2 Elastic damage model with a single scalar damage variable 49
4.4 Elastoplastic damage model with a single scalar damage variable 50
4.4.1 Free energy potentials 50
4.4.2 Constitutive relations 52
4.4.3 Evolution laws of internal variables 52
4.4.4 General procedure to develop a plastic-damage model 55
4.4.5 Remarks 56
Chapter 5 Deterministic damage model of concrete 57
5.1 Damage mechanisms of concrete 57
5.2 Elastic damage model of concrete 58
5.2.1 Split of stress tensor 58
5.2.2 Helmholtz free energy potential 60
5.2.3 Damage model with two scalar damage variables 62
5.2.4 Constitutive relations in total form 63
5.2.5 Discussions and remarks 66
5.3 Elastoplastic damage model of concrete 66
5.3.1 Helmholtz free energy potential 67
5.3.2 Elastoplastic damage model 69
5.3.3 Determination of plastic strains 71
5.3.4 Damage evolution law 75
5.3.5 Continuum tangent stiffness tensor 83
Chapter 6 Stochastic damage model of concrete 85
6.1 Mesoscale damage mechanisms 85
6.1.1 Mesoscale analysis of a representative element 85
6.1.2 Mode-I crack under normal extension 87
6.1.3 Mode-II crack under in-plane shear 88
6.1.4 Mode-I crack under tangential extension 89
6.1.5 Reconsideration of damage mechanisms 90
6.2 Mesoscopic stochastic fracture model 90
6.2.1 Stochastic damage model under tension 90
6.2.2 Mean value and standard deviation of random damage variable 95
6.2.3 Stochastic damage model under shear 98
6.2.4 Uniaxial stochastic damage model 99
6.2.5 Uniaxial plastic strains 102
6.3 Experimental modeling of mesoscopic damage 103
6.3.1 Modeling based on acoustic emission tests 103
6.3.2 Modeling based on averaged stress-strain curves 107
6.4 Stochastic elastoplastic damage model of concrete 110
6.4.1 Splits of effective stress and Helmholtz free energy potential 110
6.4.2 Damage constitutive relations 111
6.4.3 Damage evolution law 113
6.4.4 Heuristic modeling of plastic strains 117
6.4.5 Mean value and standard deviation 118
Chapter 7 Dynamic damage model of concrete 121
7.1 Heuristic viscodamage model 121
7.1.1 Literature review 121
7.1.2 Rate-dependent plastic model 122
7.1.3 Viscoplastic dynamic damage model 123
7.2 Models based on stochastic Stefan effect 125
7.2.1 Stefan effect 125
7.2.2 Strain lag factor 129
7.2.3 Dynamic damage model 131
7.3 Dynamic damage model based on multicrack propgation 133
7.3.1 Crack patterns under high strain rate 133
7.3.2 Dynamic damage mechanisms under high strain rate 134
7.3.3 Fracture dynamics of multicrack propagation 138
7.3.4 Dynamic damage model 141
Chapter 8 Numerical implementation of constitutive models 143
8.1 Fundamental concepts 143
8.1.1 Elastoplastic damage model in 1-D 143
8.1.2 Numerical algorithm for solving differential equations 145
8.1.3 Numerical algorithm based on backward Euler method 146
8.1.4 Remarks 150
8.2 Numerical algorithm for elasoplastic damage model 150
8.2.1 0perator split method 150
8.2.2 Closest point projection method for effective stress updating 152
8.2.3 Spectral decomposition method 156
8.2.4 Algorithmic consistent tangent moduli 157
8.2.5 Flow chart of the numerical implementation 159
8.2.6 Interface to general nonlinear finite element program 161
Chapter 9 Nonlinear analysis of reinforced concrete frame structure 164
9.1 Analysis of cross-section behavior 164
9.1.1 Strip method 164
9.1.2 Fiber method 168
9.1.3 Incremental cross-section behavior 170
9.2 Beam-column element 171
9.2.1 Incremental stiffness matrix of fiber element 171
9.2.2 Rigid domain 175
9.2.3 Historic development of RC beam-column elements: a review l-f 6
9.3 Forced-based element 177
9.3.1 Fundamental equations 177
9.3.2 State determination of force-based elements 180
Chapter 10 Nonlinear analysis of reinforced concrete shearwall structures 182
10.1 Plane stress element 182
10.2 4-noded plane shell element 185
10.3 Refined plane shell element 189
10.3.1 Membrane element with nodal rotation 189
10.3.2 8-noded Reinssner-Mindlin plate element 192
10.3.3 Refined plane shell element 195
10.4 Laminate plane shell element for reinforced concrete 196
10.4.1 Stiffness matrix of laminate plane shell element 196
10.4.2 Modified damage evolution laws 198
10.4.3 Flowcharts for structural nonlinear analysis 200
Chapter 11 Nonlinear analysis of concrete solid structures 202
11.1 Finite element method for initial boundary value problem 202
11.1.1 Strong form 202
11.1.3 Finite element governing equations 204
11.2 Finite element implementation of structural nonlinear analysis 206
11.2.1 Numerical quadrature method 207
11.2.2 Solution of nonlinear equations 207
11.2.3 Ill-condition of stiffness matrix 210
11.2.4 Flowcharts for structural finite element analysis 212
11.3 Strain localization in concrete 213
11.3.1 Strain localization in strain softening materials 213
11.3.2 Effects of strain localization on initial boundary value problems 214
11.3.3 Pathological numerical results 215
11.4 Nonlocal models of concrete 218
11.4.1 Physical mechanisms of nonlocal models 218
11.4.2 Fundamental concepts of nonlocal models 220
11.4.3 Nonlocal damage model of concrete 223
11.4.4 Implicit gradient-enhanced damage model of concrete 226
11.4.5 Remarks 228
Chapter 12 Stochastic nonlinear analysis of reinforced concrete structures 230
12.1 Propagation of randomness in physical system 230
12.1.1 Principle of preservation of probability 230
12.1.2 Random event description of preservation of probability 231
12.1.3 Generalized probability density evolution equation 232
12.1.4 Closed-form solution of generalized probability density evolution equation 234
12.2 Static stochastic nonlinear analysis of reinforced concrete
12.2.1 Probability density evolution equation of structural responses 235
12.2.2 Numerical scheme for probability density evolution equation 237
12.2.3 Partition of probability-assigned space 238
12.2.4 Finite difference method 238
12.2.5 Remarks 241
12.3 Stochastic dynamic nonlinear analysis of reinforced concrete
12.3.1 Dynamic system and generalized probability density evolution equation 242
12.3.2 Structural dynamic analysis 244
12.3.3 Selection of finite difference strategies 246
12.3.4 Numerical example of seismic analysis 249
Appendix A An introduction to tensor 281
A.1 Cartesian tensors 281
A.2 0perattions and properties of tensors 283
A.3 Preliminary tensor calculus 286
A.4 Matrix notations of tensors 288
A.4.1 Voigt notations of kinetic variables 288
A.4.2 Voigt notations of kinematic variables 288
Appendix B Numerical simulation of random field of mesoscopic fracture strains 291
B.1 Karhunen-Loeve decomposition method 291
B.2 Stochastic harmonic function method 293
Appendix C Partition of high dimensional probability-assigned space and point selection 295
C.1 Point selection method based on tangent spheres 295
C.1.1 Tangent circle method for two random variables 295
C.1.2 Tangent sphere method for three random variables 299
C.2 Point selection method based on number theory 302
C.2.1 Number theory for high-dimensional numerical integration 302
C.2.2 Number theory based point selction method for multi-random-variable space 303
C.3 Arbitrary probability distribution 307
C.3.1 Standardization of random variables 307
C.3.2 Extension of hyper-spherical sieving method 307
Index 309