Continuum Damage Mechanics and Numerical Applications

Cover......Page 1
Advanced Topics in Science and Technology in China......Page 2
Continuum Damage\rMechanics and Numerical\rApplications......Page 4
ISBN 9783642047077......Page 5
Preface......Page 6
Table of Contents......Page 10
1 Introduction......Page 20
References......Page 29
2.1 Development of Damage Mechanics......Page 34
2.2.1 Different Damage Definitions due to Different Measurements......Page 36
2.2.2 Damage Described by Micro-cracks and Macro-cracks......Page 40
2.2.5 Damage State Described by Ductile Void Growth......Page 43
2.3.1 Constitutive Relations for Damaged Materials......Page 46
2.3.3 Constitutive Models for Ductile Damage......Page 48
2.3.4 Constitutive Models for Damage due to Super-Plastic Void Growth......Page 50
2.3.5 Constitutive Models for Creep Damage......Page 51
2.3.6 Constitutive Models for Anisotropic Damage......Page 52
2.4.1 Kinetic Behaviors due to Micro-Structural Changes......Page 54
2.4.2 Creep Damage Growths......Page 55
2.4.3 Damage Evolution due to Cavity Nucleation and Growth......Page 56
2.4.4 Damage Evolution due to Super-Plastic Void Growth......Page 57
2.4.5 Brittle and Ductile Damage Growth......Page 59
2.4.6 Fatigue Damage Growths......Page 60
References......Page 62
3.2 Isotropic Damage Variable......Page 78
3.3 Concept of Effective Stress......Page 79
3.4.1 Hypothesis of Strain Equivalence......Page 80
3.4.2 Hypothesis of Stress Equivalence......Page 82
3.4.3 Hypothesis of Elastic Energy Equivalence......Page 83
3.4.3.1 Hypothesis of Elastic Strain Energy Equivalence......Page 84
3.4.3.2 Hypothesis of Complementary Energy Equivalence......Page 85
3.4.4 Damage Variables Based on the Two Hypotheses......Page 87
3.5 Thermodynamic Aspects......Page 89
3.5.1 First and Second Laws of Thermodynamics......Page 90
3.5.2 Thermodynamic Potential and Dissipation Inequality......Page 91
3.5.3 Dissipation Potential and Dual Relationship......Page 93
3.6 Damage Strain Energy Release Rate......Page 94
3.7 Isotropic Damage Model of Double Scalar Variables......Page 104
3.7.1 Alternative Approach of Isotropic Damage Variables......Page 105
3.7.2 Different Forms of Elastic Damaged Stress-Strain Relations......Page 106
3.7.3.1 Effective Lame Constants of Damaged Materials......Page 107
3.7.3.2 Damage Parameters in Deferent Presentations......Page 109
3.7.3.3 Damage Effective Tensor with Double Scalar Damage Variables......Page 112
3.7.3.4 Comparison with Model of Single Isotropic Damage Variable......Page 114
3.7.4 Strain Energy Release Rate with Double Scalar Damage Variables......Page 115
3.7.5.1 Isotropic Damage due to Cracks......Page 121
3.7.5.2 Isotropic Damage due to Voids......Page 122
3.7.5.3 Case Study for Tensile Specimens of Al 2024T3......Page 124
3.7.6.1 The Alternative Aspect of Damage Variables......Page 127
3.7.6.2 The Corresponding Damage Evolution Model......Page 130
3.7.6.3 Coupled Constitutive Equations Corresponding to Damage......Page 132
3.7.6.4 Conditions of Admissibility for Two Scalar Damage Effective Tensors......Page 133
3.8 Generalized Theory of Isotropic Damage Mechanics......Page 138
3.8.1 Modelling of Generalized Damage Constitutive......Page 139
3.8.2 Discussion and Analysis of Generalized Damage Model......Page 143
3.8.3 Aspects of Damage Effective Functions......Page 144
3.8.4 Dissipative Potential and Damage Evolution for Generalized Theory......Page 149
References......Page 151
4.1 Introduction......Page 154
4.2.1 Re-expression of Lemaitre\'s Model......Page 155
4.2.2 Damage Evolution Equations......Page 157
4.2.3 Evaluated Damage Variables by Different Hypothesis Models......Page 161
4.3.1 Basic Equations of Elasto-Plasticity for Isotropic Damaged Materials......Page 163
4.3.2 Static Elasto-Plastic Damage Model without Damage Growth......Page 165
4.3.3 Elasto-Plastic Model with Damage Growth......Page 166
4.3.4 Nonlinear Kinetic Evolution Equations of Elasto-Plastic Damage......Page 168
4.3.5 Model of Combined Dissipation Potential......Page 171
4.4.1 Damage-Plastic Potential Functions......Page 174
4.4.2 Damage-Plastic Yield Function......Page 176
4.4.3.3 Modification of Mohr-Coulomb Yield Criterion......Page 178
4.4.3.4 Modification of Drucker-Prager Yield Criterion......Page 179
4.4.4.1 Basic Expressions for Three-Dimensional Problems......Page 180
4.4.4.2 Basic Expressions for Two-Dimensional Problems......Page 183
4.4.4.3 Formulations for Numerical Computation......Page 184
4.5.1 Simplified Damage Constitutive Model......Page 187
4.5.2 Upper Bound on Damage of Structures......Page 188
4.6.1 Gradual Constitutive Relation Coupled with Double Scale Damage......Page 191
4.6.2 Damage Evolution Criterion Based on Double Scale of Damage......Page 193
4.6.3 Damage Evolution Equation- Time Type......Page 196
4.6.4 Basic Equations and Boundary Conditions for Solving Problems......Page 200
4.7.1.1 Summarize of Damage and Fracture Combination Analysis......Page 202
4.7.1.2 Damage Evolution Model of Gradual Field......Page 205
4.7.2.1 Compatibility Equation of Deformations in Gradual Field......Page 206
4.7.2.2 Compatibility Condition of Damage Evolution......Page 208
4.7.3.1 Boundary Condition of Studied Problem......Page 210
4.7.3.2 Solving Algorithm of Studied Problem......Page 211
4.8.1 Example of Bar Specimen......Page 212
4.8.2 Compression of Plastic Damage Behavior Based on Different Hypothesis......Page 215
4.9.1 Plastic Damage Analysis for Damaged Thick Walled Cylinder......Page 217
4.9.2 Analysis for Local Damage Behaviors......Page 220
4.9.3.1 Problem Illustration......Page 223
4.9.3.2 Analysis for Upper Bound of Damage......Page 225
4.9.3.3 Some Short Comments......Page 226
4.9.4.2 Shapes of Plasticity Progress Region......Page 227
4.9.4.3 Continuous Condition of Regions......Page 228
4.9.4.4 Rate of Crack Developing......Page 230
References......Page 232
5.1 Introduction......Page 236
5.2.1 Micro description of Damage on Geometry......Page 237
5.2.2 Damage Tensor Associated with One Group of Cracks......Page 240
5.2.3 Damage Tensor Associated with Multi-Groups of Cracks......Page 243
5.3.1 Three Dimensional Space......Page 244
5.3.2 Two Dimensional Space......Page 249
5.4.1 Review of Definition of Damage Variable......Page 251
5.4.2 Decomposition of Damage Variable in One Dimension......Page 252
5.4.3 Decomposition of Symmetrized Anisotropic Damage Tensor in 3-D......Page 256
5.5.1 First and Second Laws of Thermodynamics of Anisotropic Materials......Page 260
5.5.2 Thermodynamic Potential and Dissipation Inequality in Anisotropy......Page 261
5.5.3 Dissipation Potential and Dual Relationship in Anisotropy......Page 263
5.5.4 Damage Strain Energy Release Rate of Anisotropic Damage......Page 264
5.6.1 Elastic Matrix of Damaged Materials in Three Dimensions......Page 266
5.6.2 Elastic Matrix of Damaged Materials in Two Dimensions......Page 269
5.6.3 Property of Anisotropic Damage Elastic Matrix......Page 274
5.7.1 Principal Damage Effective Matrix in Different Symmetrization Schemes......Page 276
5 .7.1.1 Damage Effective Matrix Based on Symmetrization Scheme I......Page 277
5.7.1.2 Damage Effective Matrix Based on Symmetrization Scheme II......Page 278
5.7.1.3 Damage Effective Matrix Based on Symmetrization Scheme III......Page 280
5.7.1.4 Damage Effective Matrix Based on Unsymmetrization Scheme......Page 282
5.7.1.5 Inverse of Damage Effective Transformations for Different Schemes......Page 283
5.7.2.1 Principal Damages as Eigenvalues of Second Order Damage Tensor......Page 285
5.7.2.2 Principal Damage Directions as Eigenvactors of Second Order Damage Tensor......Page 287
5.7.2.3 Property of Principal Damage Directions with respect to Damage and Stress......Page 289
5.7.2.4 Rotational Effect of Two-Dimensional Damage Coordinate System......Page 292
5.7.2.5 Damage Effective Matrix Expressed by Second Order Damage Tensor......Page 293
5.8.1 Overview of the Topic......Page 298
5.8.2 Modification of [Ψ] Based on Different Symmetrization Models......Page 299
5.8.3 Different Forms of Damage Strain Energy Release Rate......Page 300
5.8.4.2 Conclusions......Page 307
5.9.1 Review of Symmetrization Models......Page 308
5.9.2 Effects of Symmetrization on Net-Stress Tensor......Page 310
5.9.3 Influence of Symmetrization on Deviatiom Net-Stress Tensor......Page 311
5.9.4 Effects of Symmetrization on Net-Stress Invariant......Page 314
5.9.5 Effects of Symmetrization on Net Principal Stresses and Directions......Page 317
5.9 .6 Effects of Symmetrization on Damage Constitutive Relations......Page 322
5.10.1.1 Damage Evolution Equation Based on Effective Stress State......Page 326
5.10.1.2 Damage Evolution Equation Based on Damaged Strain Energy Release Rate......Page 328
5.10.1.3 Average Integration Scheme for Damage Evolution Equations......Page 329
5.11.1 Stiffness Matrix of Anisotropic Elastic Damage Model in F.E.M.......Page 330
5.11.2 Numerical Verifying for Elastic Damage Constitutive Relationship......Page 331
5.11.3 Numerical Verifying for Symmetrization Comments......Page 333
5.12.1 Anisotropic Damage Analysis for Excavation of Underground Cavern......Page 341
5.12.2 Damage Mechanics Analysis for Stability of Crag Rock Slope......Page 347
5.12.2.1 Summary of Modeling and Geological Conditions......Page 348
5.12.2.2 Traditional Finite Element Analysis......Page 349
5.12.2.3 Finite Element Analysis for Damage Mechanics......Page 350
5.12.2.4 Comparison of Results......Page 356
5.12.3.1 Introduction of Objective Statements......Page 359
5.12.3.2 Some Results from Model Test of Koyna Dam and Correlation Analysis......Page 361
5.12.3.3 Damage Analysis for Practical Koyna Dam in 1967 Earthquake......Page 365
References......Page 372
6.1 Introduction and Objective......Page 376
6.2.1.1 State Equations......Page 378
6.2.1.2 Rate Equations......Page 379
6.2.1.3 Complementary Free Energy Density......Page 382
6.2.1.4 Dissipation Potential......Page 384
6.2.2 General Constitutive Relationship of Brittle Damage Materials......Page 386
6.2.2.1 Constitutive Relationship in the Form of Whole Quantities......Page 387
6.2.2.2 Damage Development Force......Page 390
6.3.1 Dissipation Potential and Effective Concepts due to Damage......Page 391
6.3.2 Effective Operations for Anisotropic Damage......Page 392
6.3.3 Progressive Unilateral Character of Damage......Page 395
6.3.4.1 Expression of Thermodynamic Potential......Page 397
6.3.4.2 Damage Criterion......Page 399
6.3.4.3 Applied Examples for Composite Materials......Page 401
6.4.1 Introduction and Objective......Page 402
6.4.2.1 Aspects of Mean Field Theory for Brittle Damaged Materials......Page 404
6.4.2.2 Dilute Concentration (or Taylor\'s) Model of Brittle Damage......Page 405
6.4.2.4 Differential Scheme of Self-consistent Model......Page 406
6.4.3 Strain Energy due to Presence of a Single Slit......Page 407
6.4.4.1 Influence of Cracks Induced Anisotropy......Page 412
6.4.4.2 Case Study for Two Systems of Aligned Slits......Page 414
6.4.4.3 Case Study of Fan Type Slits......Page 419
6.4.5.1 Aspects of Percolation Theory......Page 422
6.4.5.2 Lattice Percolation Models......Page 423
6.4.5.3 Continuum Percolation Models......Page 424
6.4.6.2 Doubly Periodic Rectangular Array of Aligned Slits......Page 432
6.5.1 Relationship between Damage and Porosity......Page 434
6.5.2.1 Effective Concepts Due to Damage and Pore-Pressure......Page 436
6.5.2.2 Brittle Damage Model Based on Effective Shear Strength......Page 439
6.5.2.3 Brittle Damage Model Based on Effective Cohesive Strength......Page 442
6.5.3 Influence of Damage on Shear Strength of Porous Media......Page 446
6.6.1 Aspects of Brittle of Crack-Jointed Rock Mass......Page 453
6.6.2.1 Failure Model of Rock Mass......Page 454
6.6.2.2 Effects of Single Crack in Rock on the Stiffness of Rock Mass......Page 455
6.6.2.3 Constitutive Equations of Rock Mass with Multi-Cracks......Page 458
6.6.3.1 Simple Formulations of Cn and Cs......Page 461
6.6.3.2 Theoretical Modeling of Pressure Conductive Coefficients Cn......Page 462
6.6.3.3 Engineering Modeling of Shear Conductive Coefficients Cs......Page 465
6.6.4.1 2-D Constitutive Model for Compression-Shear Stress State......Page 466
6.6.4.2 3-D Constitutive Model for Compression-Shear Stress State......Page 468
6.6.5.1 Constitution of Initial Damage for Jointed Rock Mass......Page 471
6.6.5.2 Constitution of Additional Damage for Jointed Rock Mass......Page 472
6.6.6 Brittle Elasto-plastic Damage Model for Jointed Rock Mass......Page 474
(A) Problems in Three Gorges Project......Page 476
(B) Parameters for Computation......Page 477
(C) Analysis for Computed Results After Excavation Phase V......Page 478
References......Page 482
7.1 Introduction......Page 488
7.2.1 Characteristic of Anisotropic Failure......Page 489
7.2.2 Model for Modified Hill\'s Criterion......Page 491
7.2.3 Model for Modified Hoffman\'s Criterion......Page 494
7.3.1 Influence of Orientation on Hill\'s Model......Page 498
7.3.2 Influence of Orientation on Hoffman\'s Model......Page 502
7.4 Anisotropic Damage Strain Energy Release Rate......Page 506
7.5 Anisotropic Damage Elasto-plastic Theory......Page 509
7.5.1 Elasto-plastic Equations without Damage Growth......Page 510
7.5.2 Elasto-plastic Equations with Damage Growth......Page 512
7.5.3 Equivalent Principle of Damage State......Page 515
7.6 Anisotropic Hardening Model......Page 516
7.7 Anisotropic Elasto-plastic Damage Equations for Numerical Analysis......Page 524
7.8.1.1 Effective Transformation Tensors......Page 529
7.8.1.2 Concept of Effective Back-Stress Tensor......Page 532
7.8.2.1 Concept of Effective Elastic Strain......Page 533
7.8.2.2 Effective Plastic Strain Increment......Page 534
7.8.3.1 Evolutional Equations of Damaged Materials......Page 539
7.8.3.2 Co-rotational Plastic Deformation of Damaged Materials......Page 542
7.8.3.3 Coupling of Damage and Plastic Deformations......Page 543
7.8.4 Application of Anisotropic Gurson Plastic Damage Model to Void Growth......Page 546
7.8.5 Corotational Effective Spin Tensor......Page 549
7.9.1 Configuration of Deformation and Damage......Page 550
7.9.2 Description of Damage Tensors......Page 551
7.9.3 Corresponding Damage Effective Tensor for Symmetrized Model II......Page 552
7.9.4.1 Deformation Gradient and Finite Strain......Page 554
7.9.4.2 Damage Configurations of Deformation Gradient and Finite Strain......Page 556
7.9.4.3 De formation Gradients Due to Fictitious Damage......Page 565
7.9.4.4 ecomposition of Elasto-plastic Finite Strains Coupled with Damage......Page 566
7.9.5 Thermodynamic Description of Finite Strain Damage......Page 568
7.9.6 Damage Behavior of Elasto-plastic Finite Deformation......Page 574
7.10.1 Perforated Specimen......Page 575
7.10.2 Cracked Plate Subjected to Tension......Page 578
7.11.1 Plasticity of Gurson\'s Yield Criterion......Page 583
7.11.2 An Application of Hill Quadratic Anisotropic Yield Criterion......Page 584
7.11.3 Anisotropic Gurson\'s Plastic Model Based on Hill\'s Failure Criterion......Page 587
7.11.4.1 Finite Element Modeling for Voids Growth......Page 594
7.11.4.2 Numerical Results......Page 597
References......Page 605
8.1 Introduction......Page 608
8.2.1 General Thermodynamics Framework......Page 613
8.2.2 Stated Equivalence of Thermodynamic Entropy......Page 617
8.2.3 Visco-elasticity with Temperature Coupled to Damage......Page 619
8.2.3.1 General Anisotropie Behavior with Temperature Dependence......Page 620
8.2.3.2 General Model of Thermo-visco-elastic Constitutive......Page 621
8.2.3.3 Applicability of Double Scalar Isotropic Visco-eastic Damage......Page 625
8.2.3.4 Evolution Model of Double Scalar Isotropic Visco-elastic Damage......Page 628
8.3.1 About Visco-plastic Damage......Page 629
8.3.2.2 Differential Form of Visco-plastic Damage System Equations......Page 631
8.3.3.1 Basis of Asymptotic Expansion for Non-homogeneous Integral......Page 636
8.3.3.2 Recursive Integration Method for Visco-plastic Damage......Page 638
8.3.4 Outline of Visco-plastic Damage Equations and Algorithm......Page 643
8.3.4.2 Summary of System Equations in Integral Form......Page 644
8.3.4.4 Algorithm of Asymptotic Integration for Visco-plastic Damage......Page 645
8.4.1.1 Strain Rate of Creep Damage......Page 646
8.4.1.2 Rupture Time of Creep Damage......Page 647
8.4.2.1 Rate Models of Strain and Damage......Page 649
8.4.2.2 Constitutive Model of Creep Damage......Page 650
8.4.2.3 Determination of Scalar Coefficients......Page 652
8.4.3.1 Tensorial Generalization of Creep Law Including Damage......Page 654
8.5.1 Generalized Principle of Minimum Dissipative Energy......Page 656
8.5.2.1 Application of Principle to Visco-elasto-plastic Damage Problems......Page 660
8.5.2.2 Constitutive Model of Visco-elasto-plastic Damaged Materials......Page 661
8.5.2.3 Evolutionary Model of Visco-elasto-plastic Damage Mechanics......Page 663
8.5.3.1 Finite Element Model of Visco-elasto-plastic Dynamic Damage Problems......Page 665
8.5.3.2 Numerical Algorithm for Visco-elasto-plastic Dynamic Damage Problems......Page 667
8.6.1 Preferences of Variational Principles......Page 669
8.6.2.1 Description of Boundary Value Problem with Visco-elastic Damage......Page 670
8.6.2.2 Generalized Variational Principles I......Page 672
8.6.2.3 Generalized Variational Principles II......Page 673
8.6.2.4 Generalized Potential Energy Principle......Page 675
8.6.3.1 Description of Visco-elastic Damage in Timoshenko Beam......Page 677
8.6.3.2 Application of Generalized Variational Principle to Timoshenko Beam......Page 678
8.7.1.1 Description of Visco-elastic Damage Behavior in Studied Materials......Page 681
8.7.1.2 Description of Experimental Results......Page 683
8.7.1.3 Discussion of Remarks......Page 686
8.7.2.2 Numerical Observation of Uncoupled Model......Page 687
8.7.2.3 Numerical Observation in Coupled Case......Page 691
8.7.3 Numerical Studies of Visco-plastic Damage Behavior in Simple Structures......Page 692
8.7.3.2 Application to Plate with a Central Circular Hole......Page 693
8.8.1.1 Description of Effects Due to Localization Approach......Page 696
8.8.1.2 Model of Damage Analysis for Creep Crack Problem......Page 697
8.8.1.3 Effects of Localization Approach in Creep Damage Crack Problem......Page 698
8.8.2.1 Numerical Modeling of Damage Localization in Uniform Stress Field......Page 700
8.8.2.2 Observation of Stress Sensitivity of Damage Evolution......Page 703
8.8.2.3 Effects of Stress Sensitivity and Related Mesh-dependence......Page 704
8.8.3.1 Non-Local Damage Model......Page 706
8.8.3.2 Stress Limitation Method......Page 708
8.8.3.3 Modification of Damage Evolution Equation......Page 709
8.9.1.1 Specification of Studies......Page 711
8.9.1.2 Filtration and Calibration of Formulations to be Used......Page 713
8.9.1.3 Finite Element Modeling for Filtrated Formulations......Page 714
8.9.2.1 Analyzed Conditions......Page 715
8.9.2.2 Numerical Results and Comparison......Page 717
8.9.3.1 Mechanism and Numerical Model......Page 718
8.9.3.2 Numerical Analysis and Results Discussion......Page 721
8.9.4.1 Project Description of Longtan Concrete Gravity Dam System......Page 723
8.9.4.2 Topography and Geological Materials of Dam System......Page 724
8.9.4.3 Numerical Model of Earthquake of Dam System......Page 725
8.9.4.4 Visco-elasto-plastic Damage Analysis for Eathquike of Longtan Gravity Dam......Page 727
8.9.4.5 Safety Assessment of Longtan Gravity Dam under Earthquake......Page 733
References......Page 734
9.1 Introduction......Page 742
9.2.1 Basic Equations of Dynamic Evolutional System......Page 743
9.2.2 Variation Principle of Dynamic Evolutional Continuous System......Page 746
9.2.3 Unified Description of Dynamic Evolutionary Continuous System......Page 747
9.2.4 Hamilton-Jacobi-Bellman Equations for Dynamic Evolutionary System......Page 749
9.2.5 Schemes of Numerical Solutions......Page 752
9.3.1 Damage Growth Equations......Page 753
9.3.2 Concept of Damage Propagation......Page 756
9.4.1 Governing Equations of Motion for Anisotropic Damaged Structures......Page 757
9.4.2 Finite Element Discretization of Dynamic Damaged Body......Page 758
9.4.3.1 Numerical Integration Scheme of Damage Growth......Page 760
9.4.3.2 Determinant of parameters A and n.......Page 762
9.4.4 Damping for Damaged Materials......Page 765
9.5.1 Introduction of Wave with Damage......Page 768
9.5.2 Wave Propagation Characters in Damaged Media......Page 769
9.5.2.1 Divisional Model of Damaged Media......Page 770
9.5.2.2 Basic Theory of Wave Propagation in Damage Media......Page 771
9.5.2.4 One Dimensional Wave in Inhomogeneous Damaged Media......Page 775
9.5.2.5 Solution of Wave in Damaged Media by Time Track Transformation......Page 779
9.5.3.1 Example for One Dimensional Harmonic Wave in Damaged Media......Page 780
9.5.3.2 Example of Application of Time Track Transformation [9-65]......Page 788
9.5.3.3 Example of Transient Wave Propagation in Damaged Media......Page 791
9.5.4.1 Illustration of Damage Development Due to Wave......Page 796
9.5.4.2 Modeling Kinematical Waves......Page 798
9.5.4.3 Ogin\'s Model......Page 802
9.5.5.1 Essential Aspects of Damage Wave......Page 803
9.5.5.2 Thermodynamics Basis of Damage Wave......Page 804
9.5.5.3 Governing Equations of Damage Wave......Page 809
9.5.5.4 One-dimensional Solutions of Damage Wave......Page 811
9.6.1.1 A Brief Introduction of Dynamic Damage Behavior and Required Equations......Page 816
9.6.1.2 Behavior of Elastic Constitution of Damaged Material......Page 817
9.6.1.3 Behavior of Damping Matrix of Damaged Structure......Page 818
9.6.2 Response of Damaged Simple Structure under Dynamic Loading......Page 819
9.6.3 Lagrangian F E Analysis for Dynamics of Damaged Deep Beam......Page 827
9.6.4 Damage Evolution in Deep Beam during Dynamic Response......Page 831
9.6.5.1 Influence of Damage on Frequency of Damaged Structure......Page 839
9.6.5.2 Influence of Damage on Damping Ratio......Page 842
9.6.5.3 Influence of Damage on Magnification Factor......Page 846
9.6.5.4 Influence of Damage on Phase Angle......Page 850
9.7 Dynamic Damage Analysis for Brittle Rock and Its Application......Page 851
9.7.1 Purpose of Brittle Rock Dynamic Damage Studies......Page 852
9.7.2.1 Test of Wave Propagation in Jointed Samples......Page 853
9.7.2.2 Modeling Wave Propagation in Jointed Rock Masses......Page 854
9.7.3.1 Dynamic Damage Model of Micro-jointed Rock Mass......Page 858
9.7.3.2 Damping Matrix of Jointed Damage Materials......Page 859
9.7.3.3 Example of Numerical Application......Page 860
9.7.4 Impact Response Behavior of Dynamic Damaged Brittle Rock......Page 861
9.7.4.1 Relation between Attenuation Coefficient and Damage Energy of Sound Wave......Page 862
9.7.4.2 Modeling of Damage Evolution Equations......Page 863
9.7.5 Example of Numerical Applications and Validation......Page 864
9.7.5.2 Discussion and Analysis of Simulated Results......Page 865
9.7.6.1 Fragmentation Concepts of Brittle Rock......Page 868
9.7.6.2 Fragmentation due to Damage Evolution......Page 869
9.7.6.3 Description of Fragmentation Behaviour......Page 870
9.7.6.4 Determination of Material Parameters......Page 871
9.7.6.5 Analysis for Fragmentation of Brittle Rock Due to Dynamic Damage......Page 873
9.8.1.1 Objective Studies of Arch Dam Earthquake......Page 877
9.8.1.2 Dynamic Explication for Damage of Arch Dams Due to Earthquake......Page 879
9.8.1.3 Dynamic Damage Behaviours in Constitutive Relations......Page 882
9.8.1.4 Algorithm of Time Integration for Dynamic Equations......Page 887
9.8.1.5 Computer Implementation......Page 889
9.8.1.6 Numerical Results for Arch Dam Responses......Page 890
9.8.2.1 Purpose of Arch Dam Explosive Damage Analysis......Page 899
9.8.2.2 Estimation of Initial Damage States......Page 900
9.8.2.3 Modeling of Objective Brittle Dynamic Damage......Page 901
9.8.2.4 Results Analysed for Explosived Brittle Damage in Arch Dam......Page 907
9.8.2.5 Discussion of Results......Page 910
9.8.3.2 Material Damage and Destruction......Page 911
9.8.3.3 Constitutive Equations for Materials during Penetration......Page 912
9.8.3.4 Fundamental Equations and Technique of Numerical Computation......Page 913
9.8.3.5 Calculation Results and Discussion......Page 914
9.8.3.6 Conclusions and Expectations......Page 918
References......Page 919
Index......Page 930