1 Elements of Fluid Mechanics .................................. 1
1.1 Fundamental Equations ................................... 2
1.1.1 Equation of Motion ............................... 2
1.1.2 Continuity Equation .............................. 2
1.1.3 Energy Equation .................................. 2
1.1.4 Equation of State ................................ 2
1.2 Solutions to the Fundamental Equations .................. 3
1.3 Propagation of Discontinuities .......................... 4
1.3.1 Sound Speed ...................................... 4
1.3.2 Speed of Discontinuity Propagation ............... 5
1.3.3 Characteristics .................................. 6
1.3.4 Shock Waves ...................................... 8
1.4 Derivation of the Hugoniot Relations .................... 9
1.4.1 Conservation of Mass ............................. 9
1.4.2 Conservation of Momentum ........................ 10
1.4.3 Conservation of Energy .......................... 10
1.5 Rayleigh Line .......................................... 11
1.6 Applications of Hugoniot Equations to a Perfect Gas .... 13
1.6.1 Calculation of Shock Speed ...................... 13
1.6.2 Calculation of Shock Pressure ................... 14
1.6.3 Calculation of Volume Behind the Shock .......... 14
1.6.4 Graphical Representation ........................ 15
1.6.5 Reflection of a Uniform Shock ................... 15
1.6.6 Conditions Behind the First Reflected Shock
from a Fixed Boundary ........................... 16
1.7 Detonation Waves ....................................... 17
1.8 Elastic-Plastic Waves .................................. 17
1.9 Units and Orders of Magnitude .......................... 20
1.10 Measurements to Obtain Equation of State Data .......... 20
1.10.1 Experimental Methods ............................ 20
1.10.2 Relation of the Free Surface Velocity
to the Shock Particle Velocity in a Solid ....... 22
1.10.3 Form of the Equation of State for Solids ........ 23
1.10.4 Detonation Pressure Measurement ................. 25
2 Numerical Techniques ........................................ 27
2.1 Von Neumann Finite Difference Scheme ................... 27
2.1.1 Time Centering .................................. 28
2.1.2 Space Centering ................................. 28
2.2 Artificial Viscosity ................................... 28
2.2.1 Generalized Artificial Viscosity ................ 28
2.2.2 Applications of the Generalized Artificial
Viscosity in One Space Dimension ................ 29
2.3 Stability Conditions ................................... 32
2.3.1 Courant Condition ............................... 32
2.3.2 Von Neumann Stability Analysis .................. 32
2.4 Finite Difference Scheme in Two Dimensions ............. 33
2.4.1 Integral Definition of a Derivative ............. 33
2.4.2 Integration Paths ............................... 34
2.4.3 Properties of the Integration Scheme ............ 34
2.4.4 Continuity Equation ............................. 34
2.5 Finite Difference Scheme in Three Dimensions ........... 35
2.6 Finite Difference Scheme for Double Operators in Two
Dimensions ............................................. 35
2.7 Grid Stabilization ..................................... 36
3 Modeling the Behavior of Materials .......................... 37
3.1 Introduction ........................................... 37
3.1.1 Hooke's Law ..................................... 37
3.1.2 Rigid Body Rotation ............................. 39
3.2 Plastic Flow Region .................................... 39
3.2.1 Yield Strength .................................. 41
3.2.2 Von Mises Yield Condition ....................... 43
3.2.3 Plastic Strain .................................. 45
3.2.4 Tresca Yield Condition .......................... 46
3.3 Flow Stress ............................................ 48
3.3.1 Strain Hardening ................................ 50
3.3.2 A General Form of Strain Hardening .............. 50
3.4 Rate Dependent Yield Models ............................ 52
3.4.1 Maxwell Solid ................................... 52
3.4.2 Dislocation Theory .............................. 53
3.4.3 Flow Stress Measurements ........................ 57
3.5 Upper Yield Point ...................................... 59
3.6 Nonhomogeneous Properties .............................. 60
3.7 Hydrostatic Pressure Equation of State ................. 60
3.8 Modeling Fracture ...................................... 62
3.8.1 Fracture Toughness Testing ...................... 65
3.8.2 Spallation ...................................... 67
3.8.3 Ductile Fracture ................................ 68
3.8.4 Strain Damage ................................... 68
3.8.5 Damage in Elastic Regime ........................ 69
3.8.6 Computer Simulation of Fracture ................. 70
3.8.7 Damage in Plastic Regime ........................ 71
3.9 Equation of State of Explosive Detonation Products ..... 75
3.9.1 Numerical Calculation of a Detonation ......... 79
4 Two-Dimensional Elastic-Plastic Flow ........................ 83
4.1 Fundamental Equations .................................. 83
4.1.1 Equation of Motion in x, y Coordinates with
Cylindrical Symmetry and Rotation About the
x Axis .......................................... 83
4.1.2 Conservation of Mass ............................ 84
4.1.3 First Law of Thermodynamics ..................... 84
4.1.4 Velocity Strains ................................ 84
4.1.5 Stress Deviator Tensor .......................... 85
4.1.6 Pressure Equation of State ...................... 85
4.1.7 Total Stresses .................................. 85
4.1.8 Artificial Viscosity ............................ 85
4.1.9 Von Mises Yield Condition ....................... 86
4.2 Finite Difference Equations ............................ 86
4.2.1 Mass Zoning ..................................... 86
4.2.2 Equations of Motion ............................. 87
4.2.3 Conservation of Mass ............................ 88
4.2.4 Calculation of Incremental Strain ............... 89
4.2.5 Calculation of Stresses ......................... 90
4.2.6 Von Mises Yield Condition ....................... 92
4.2.7 Equivalent Plastic Strain, εp ................... 92
4.2.8 Artificial Viscosity for Calculating Shocks ..... 93
4.2.9 Navier-Stokes Artificial Viscosity for
Stabilizing the Grid ............................ 94
4.2.10 Material Internal Energy ........................ 96
4.2.11 Calculation of Time Steps, Δtn+3/2 and Δtn+1 ...... 97
4.2.12 Energy Summations (Edit Routine) ................ 97
4.2.13 Principal Stresses (Edit Routine) ............... 98
4.2.14 Calculation of Load, L, on a Given k Line
(Edit Routine) .................................. 98
4.3 Boundary Conditions .................................... 99
4.3.1 Fixed Boundary on the x Axis .................... 99
4.3.2 Fixed Boundary on the у Axis ................... 100
4.3.3 Corner Zone on the x Axis ...................... 100
4.3.4 Corner Zone on the y Axis ...................... 101
4.3.5 Free Surfaces .................................. 102
4.3.6 Discussion ..................................... 102
4.4 Applications .......................................... 103
5 Sliding Interfaces in Two Dimensions ....................... 113
5.1 Sliding Interfaces Between Quadrilateral Lagrange
Zones ................................................. 114
5.1.1 Location of Master Points Associated with
a Given Slave Point ............................ 115
5.1.2 Calculation of the Volume of Sliding Zones
Associated with the Slave Grid ................. 115
5.1.3 Advancing a Slave Point ƒ in Time .............. 116
5.1.4 Location of Slave Points Associated with
a Given Master Point ........................... 120
5.1.5 Advancement in Time of Point j, k on
the Master Grid ................................ 121
5.1.6 Testing for Penetration of Grids ............... 123
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 .................. 124
5.1.8 Relocating Slave Points onto the Master
Surface when d < 0 ............................. 126
5.2 Intersecting Slide Lines .............................. 126
5.2.1 Acceleration of Points on the Intersection
of Two Slide Lines ............................. 126
5.2.2 Adjustment for Grid Penetration ................ 127
5.2.3 Relocation of Points when a Void Has Opened .... 127
6 Elastic-Plastic Flow in Three Space Dimensions ............. 129
6.1 Fundamental Equations ................................. 129
6.1.1 Equations of Motion ............................ 129
6.1.2 Conservation of Mass ........................... 129
6.1.3 First Law of Thermodynamics .................... 129
6.1.4 Velocity Strains ............................... 130
6.1.5 Stress Deviator Tensor ......................... 130
6.1.6 Pressure Equation of State ..................... 130
6.1.7 Total Stresses ................................. 131
6.1.8 Artificial Viscosity for Calculating Shocks .... 131
6.1.9 Von Mises Yield Condition ...................... 131
6.2 Finite Difference Equations for HEMP 3D .............. 131
6.2.1 Mass Zoning .................................... 131
6.2.2 Equations of Motion ............................ 133
6.2.3 Conservation of Mass ........................... 136
6.2.4 Calculation of Incremental Strains ............. 136
6.2.5 Calculation of Stresses ........................ 139
6.2.6 Von Mises Yield Condition ...................... 140
6.2.7 Plastic Strain ................................. 140
6.2.8 Artificial Viscosity for Calculating Shocks .... 141
6.2.9 Tensor Artificial Viscosity for Stabilizing
the Grid ....................................... 142
6.2.10 Material Internal Energy ....................... 145
6.2.11 Time Step Calculations ......................... 146
6.3 Boundary Conditions ................................... 146
6.4 Check Problems ........................................ 146
6.4.1 Simple Harmonic Motion ......................... 146
6.4.2 Plasticity ..................................... 149
7 Sliding Surfaces in Three Dimensions ....................... 151
7.1 Calculational Steps to Advance in Time Grid Points
on a Sliding Surface .................................. 153
7.2 Applications of Sliding Surface Routine ............... 163
7.3 Zone Dimension Change and Subcycling .................. 163
7.3.1 Zone Dimension Change at an Interface
in Two Dimensions .............................. 163
7.3.2 Zone Dimension Change of an Interface in
Three Dimensions ............................... 167
7.3.3 Subcycling with Zone Dimension Change in Two
Dimensions ..................................... 169
7.3.4 Example for a Zone Size Change of Two to One ... 169
8 Magnetohydrodynamics of HEMP ............................... 171
8.1 Finite Difference Scheme for Double Operators ......... 172
8.2 Fundamental Equations of Magnetohydrodynamics ......... 174
8.2.1 Equation of Motion ............................. 174
8.2.2 Electromagnetic Field Equations ................ 174
8.2.3 Energy Equation ................................ 175
8.2.4 Continuity Equation ............................ 176
8.2.5 Constitutive Relations ......................... 176
8.3 Difference Equations for Magnetohydrodynamics ......... 176
8.3.1 Equations of Motion ............................ 176
8.3.2 Magnetic Diffusion ............................. 177
8.3.3 Energy Equations ............................... 179
8.3.4 Continuity Equation ............................ 182
8.3.5 Time-Step Control .............................. 182
8.3.6 Boundary Conditions ............................ 183
8.3.7 Sliding Interfaces ............................. 183
8.3.8 Check Problems ................................. 185
Appendices .................................................... 189
A Effect of a Second Shock on the Principal Hugoniot ......... 189
B Finite Difference Program for One Space Dimension
and Time ................................................... 191
B.1 Fundamental Equations ................................. 191
B.2 Finite Difference Equations ........................... 192
B.3 Boundary Conditions ................................... 194
B.4 Opening and Closing Voids ............................. 195
C A Method for Determining the Plastic Work Hardening
Function ................................................... 197
C.1 Application to 6061-T6 Aluminum ....................... 199
D Detonation of a High Explosive for a γ-Law Equation
of State ................................................... 202
E Magnetic Flux Calculation .................................. 211
F Thermal Diffusion Calculation .............................. 224
G Backward Substitution Method for Solving a System of
Linear Equations of the Form AiHi+1 + BiHi + СiНi-1 = Di ..... 238
References .................................................... 241
Subject Index ................................................. 245
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