Computer Simulation of Dynamic Phenomena

Mark L. Wilkins3540630708, 9783540630708

Preferred finite difference schemes in one, two, and three space dimensions are described for solving the three fundamental equations of mechanics (conservation of mass, conservation of momentum, and conservation of energy). Models of the behavior of materials provide the closure to the three fundamentals equations for applications to problems in compressible fluid flow and solid mechanics. The use of Lagrange coordinates permits the history of mass elements to be followed where the integrated effects of plasticity and external loads change the material physical properties. Models of fracture, including size effects, are described. The detonation of explosives is modelled following the Chapman–Jouget theory with equations of state for the detonation products derived from experiments. An equation-of-state library for solids and explosives is presented with theoretical models that incorporate experimental data from the open literature. The versatility of the simulation programs is demonstrated by applications to the calculations of surface waves from an earthquake to the shock waves from supersonic flow and other examples.

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Table of contents :
Preface……Page 7
Table of Contents……Page 9
Notation……Page 15
1. Elements of Fluid Mechanics ……Page 19
1.1.4 Equation of State ……Page 20
1.2 Solutions to the Fundamental Equations……Page 21
1.3.1 Sound Speed……Page 22
1.3.2 Speed of Discontinuity Propagation……Page 23
1.3.3 Characteristics……Page 24
1.3.4 Shock Waves……Page 26
1.4.1 Conservation of Mass……Page 27
1.4.3 Conservation of Energy……Page 28
1.5 Rayleigh Line……Page 29
1.6.1 Calculation of Shock Speed……Page 31
1.6.3 Calculation of Volume Behind the Shock……Page 32
1.6.5 Reflection of a Uniform Shock……Page 33
1.6.6 Conditions Behind the First Reflected Shock from a Fixed Boundary……Page 34
1.8 Elastic-Plastic Waves……Page 35
1.10.1 Experimental Methods……Page 38
1.10.2 Relation of the Free Surface Velocity to the Shock Particle Velocity in a Solid……Page 40
1.10.3 Form of the Equation of State for Solids……Page 41
1.10.4 Detonation Pressure Measurement……Page 43
2.1 Von Neumann Finite Difference Scheme……Page 45
2.2.1 Generalized Artificial Viscosity……Page 46
2.2.2 Applications of the Generalized Artificial Viscosity in One Space Dimension……Page 47
2.3.2 Von Neumann Stability Analysis……Page 50
2.4.1 Integral Definition of a Derivative……Page 51
2.4.4 Continuity Equation……Page 52
2.6 Finite Difference Scheme for Double Operators in Two Dimensions……Page 53
2.7 Grid Stabilization……Page 54
3.1.1 Hooke’s Law……Page 55
3.2 Plastic Flow Region……Page 57
3.2.1 Yield Strength……Page 59
3.2.2 Von Mises Yield Condition……Page 61
3.2.3 Plastic Strain……Page 63
3.2.4 Tresca Yield Condition……Page 64
3.3 Flow Stress……Page 66
3.3.2 A General Form of Strain Hardening……Page 68
3.4.1 Maxwell Solid……Page 70
3.4.2 Dislocation Theory……Page 71
3.4.3 Flow Stress Measurements……Page 75
3.5 Upper Yield Point……Page 77
3.7 Hydrostatic Pressure Equation of State……Page 78
3.8 Modeling Fracture……Page 80
3.8.1 Fracture Toughness Testing……Page 83
3.8.2 Spallation……Page 85
3.8.4 Strain Damage……Page 86
3.8.5 Damage in Elastic Regime……Page 87
3.8.6 Computer Simulation of Fracture……Page 88
3.8.7 Damage in Plastic Regime……Page 89
3.9 Equation of State of Explosive Detonation Products……Page 93
3.9.1 Numerical Calculation of a Detonation……Page 97
4.1.1 Equation of Motion in x, y Coordinates with Cylindrical Symmetry and Rotation About the x Axis……Page 101
4.1.4 Velocity Strains……Page 102
4.1.8 Artificial Viscosity……Page 103
4.2.1 Mass Zoning……Page 104
4.2.2 Equations of Motion……Page 105
4.2.3 Conservation of Mass……Page 106
4.2.4 Calculation of Incremental Strain……Page 107
4.2.5 Calculation of Stresses……Page 108
4.2.7 Equivalent Plastic Strain……Page 110
4.2.8 Artificial Viscosity for Calculating Shocks……Page 111
4.2.9 Navier-Stokes Artificial Viscosity for Stabilizing the Grid……Page 112
4.2.10 Material Internal Energy……Page 114
4.2.12 Energy Summations (Edit Routine)……Page 115
4.2.14 Calculation of Load, L, on a Given k Line (Edit Routine)……Page 116
4.3.1 Fixed Boundary on the x Axis……Page 117
4.3.3 Corner Zone on the x Axis……Page 118
4.3.4 Corner Zone on the y Axis……Page 119
4.3.6 Discussion……Page 120
4.4 Applications……Page 121
5. Sliding Interfaces in Two Dimensions……Page 131
5.1 Sliding Interfaces Between Quadrilateral Lagrange Zones……Page 132
5.1.2 Calculation of the Volume of Sliding Zones Associated with the Slave Grid……Page 133
5.1.3 Advancing a Slave Point / in Time……Page 134
5.1.4 Location of Slave Points Associated with a Given Master Point……Page 138
5.1.5 Advancement in Time of Point j, k on the Master Grid……Page 139
5.1.6 Testing for Penetration of Grids……Page 141
5.1.7 Adjusting the Velocities of All Void Closed Points Where d < 0 and Where in the Previous Cycle the Point Was Void Open……Page 142
5.2.1 Acceleration of Points on the Intersection of Two Slide Lines 126……Page 144
5.2.3 Relocation of Points when a Void Has Opened……Page 145
6.1.3 First Law of Thermodynamics……Page 147
6.1.6 Pressure Equation of State……Page 148
6.2.1 Mass Zoning……Page 149
6.2.2 Equations of Motion……Page 151
6.2.4 Calculation of Incremental Strains……Page 154
6.2.5 Calculation of Stresses……Page 157
6.2.7 Plastic Strain……Page 158
6.2.8 Artificial Viscosity for Calculating Shocks……Page 159
6.2.9 Tensor Artificial Viscosity for Stabilizing the Grid……Page 160
6.2.10 Material Internal Energy……Page 163
6.4.1 Simple Harmonic Motion……Page 164
6.4.2 Plasticity……Page 167
7. Sliding Surfaces in Three Dimensions……Page 169
7.1 Calculational Steps to Advance in Time Grid Points on a Sliding Surface……Page 171
7.3.1 Zone Dimension Change at an Interface in Two Dimensions……Page 181
7.3.2 Zone Dimension Change of an Interface in Three Dimensions……Page 185
7.3.4 Example for a Zone Size Change of Two to One……Page 187
8. Magnetohydrodynamics of HEMP……Page 189
8.1 Finite Difference Scheme for Double Operators……Page 190
8.2.2 Electromagnetic Field Equations……Page 192
8.2.3 Energy Equation……Page 193
8.3.1 Equations of Motion……Page 194
8.3.2 Magnetic Diffusion……Page 195
8.3.3 Energy Equations……Page 197
8.3.5 Time-Step Control……Page 200
8.3.7 Sliding Interfaces……Page 201
8.3.8 Check Problems……Page 203
A. Effect of a Second Shock on the Principal Hugoniot……Page 207
B.1 Fundamental Equations……Page 209
B.2 Finite Difference Equations……Page 210
B.3 Boundary Conditions……Page 212
B.4 Opening and Closing Voids……Page 213
C. A Method for Determining the Plastic Work Hardening Function……Page 215
C.1 Application to 6061-T6 Aluminum……Page 217
D. Detonation of a High Explosive for a 7-Law Equation of State……Page 220
E. Magnetic Flux Calculation……Page 229
F. Thermal Diffusion Calculation……Page 242
G. Backward Substitution Method for Solving a System of Linear Equations of the Form AiHi+1 + BiHi + CiHi-1 = Di……Page 256
References……Page 259
Subject Index……Page 263