List of Figures

1. Illustration of the fracture process associated with the Titan IV SRMU grain collapse accident (taken from Chang et al., (1994)).
2. CVFE concept showing one 4-node cohesive element between two linear-strain triangular volumetric elements. The cohesive element is shown in its deformed configuration. In its undeformed configuration is has no thickness and the adjacent nodes are superposed.
3. Bilinear cohesive failure law for the pure tensile or mode I ( , left) and pure shear or mode II ( $\Delta _n = 0$, right) cases. An unloading and reloading path is also shown in the mode I case.
4. Coupled cohesive failure model described by Equation 2.4; variation of normal (top) and shear(bottom) cohesive tractions with respect to normal ($\Delta _n$) and tangential ($\Delta _t$) displacement jumps.
5. Time step defined by: (a) element size or (b) element type.
6. Subcycling region distribution.
7. Time step assignment.
8. (a) Standard 1-D mesh. (b) 1-D mesh with inserted cohesive node.
9. 2-D Cohesive element representation.
10. 2-D cohesive element insertion: (a) proposed edge for cohesive insertion, (b) inserted cohesive element, (c) ``criss-crossed'' cohesive element.
11. Connectivity update of nodes and elements.
12. Common 2-D insertion cases.
13. Illustrative example of three cohesive element insertions using Cases #2 and #3 in Figure 2.11.
14. Illustration of insertion Case #5 in Figure 2.11.
15. 1-D ``blind'' insertion test problem.
16. 1-D ``blind'' insertion test problem: evolution of the displacement jump across the cohesive element (i.e., between nodes $2$ & $4$ in Figure 2.14) resulting from a cohesive element insertion at time 0, $1000\Delta t$ ($33.3$ $s$), $2000\Delta t$ ($66.6$ $s$).
17. Schematic representation of a damped 1-D cohesive element.
18. Effect of cohesive damping: evolution of the displacement jump across the cohesive element for the simple 1-D test problem shown in Figure 2.14 and resulting from ``blind'' cohesive element insertion with damping at time 0, $1000\Delta t$ ($33.3$ $s$) and $2000\Delta t$ ($66.6$ $s$).
19. 1-D cohesive element pre-stretching concept.
20. 1-D cohesive element pre-stretching concept, with the pre-stretch applied equally on the two nodes.
21. 2-D separation contributions from neighboring cohesive elements.
22. 1-D test problem separation oscillations of nodes $2$ & $4$ (Figure 2.14) resulting from insertion with pre-stretching at time 0, $1000\Delta t$ ($33.3$ $s$), $2000\Delta t$ ($66.6$ $s$).
23. Bounding box method.
24. Multiple active cohesive regions.
25. Stress-based insertion results for simple angled case.
26. Simple CVFE mesh.
27. Partitioned CVFE mesh.
28. Reference problem in 1-D.
29. x-t diagram in 1-D.
30. Analytical solution for displacement , velocity $v$ and stress $\sigma$ in the middle of the beam for the 1-D wave problem described in Figure 3.1.
31. Subcycling test described by Smolinski (1989) by Case $C$.
32. Velocity profile of node $\char93 5$ with subcycling parameter ${\it m}=10$.
33. Velocity profile of node $\char93 15$ with subcycling parameter ${\it m}=10$.
34. Velocity profile of node $\char93 25$ with subcycling parameter ${\it m}=10$.
35. Subcycling effect on (a) displacements and (b) velocities at node $5$.
36. Test case used to get timing results for subcycling.
37. Nodes 12 through 20 are made cohesive.
38. Velocity profile of node $\char93 15$ resulting from blind insertion at the $0th$ time step ($0.0$ $s$).
39. Velocity profile of node $\char93 15$ resulting from blind insertion at the $2500th$ time step ($6.25$ $s$).
40. Velocity profile of node $\char93 15$ resulting from blind insertion at the $5000th$ time step ($12.5$ $s$).
41. Velocity profile of node $\char93 15$ resulting from blind insertion at the $10000th$ time step ($25.0$ $s$).
42. Velocity profile of node $\char93 15$ resulting from blind insertion with damping at the $5000th$ time step ($12.5$ $s$).
43. Velocity profile of node $\char93 15$ resulting from blind insertion with damping at the $10000th$ time step ($25.0$ $s$).
44. Velocity profile of node $\char93 15$ resulting from insertion with pre-stretching at the $5000th$ time step ($12.5$ $s$).
45. Velocity profile of node $\char93 15$ resulting from insertion with pre-stretching at the $10000th$ time step ($25.0$ $s$).
46. Cohesive separation for node $\char93 15$ resulting from blind insertion at $10000th$ time step ($25.0$ $s$)
47. Cohesive separation for node $\char93 15$ resulting from insertion with pre-stretching at $10000th$ time step ($25.0$ $s$)
48. Test case for dynamic cohesive node insertion with pre-stretching.
49. Test case for dynamic insertion with subcycling.
50. Velocity profile of node $400$ of dynamic cohesive node insertion at time step $150000$ ($3750$ $s$) with nodal subcycling using ${\it m}=10$.
51. Cohesive element distribution.
52. Nodal displacements of a random node ahead of the notch for a problem with an equal region ratio () with subcycling parameters of ${\it m} = 1, 4, 10, 16$ and $20$.
53. Nodal displacements of a random node for the $2:1$ region ratio with subcycling parameters of ${\it m} = 1, 4, 10, 16$ and $20$.
54. Nodal displacements of a random node for the $4.5:1$ region ratio with subcycling parameters of ${\it m} = 1, 4, 10$ and $16$.
55. Percent time savings vs region ratio for various subcycling parameters.
56. Simple 2-D meshes with three cohesive elements inserted along (a) "horizontal" and (b) "mixed" interfaces.
57. Normalized average stress levels for the volumetric elements of the middle cohesive element. Vertical lines at the $0th$ ($0.0$ $s$), $2500th$ ($1.0$ $s$) and $5000th$ ($2.0$ $s$) time step represent dynamic insertion times.
58. Normalized separation of the tracking node for "horizontal" blind insertion at the $0th$ ($0.0$ $s$), $2500th$ ($1.0$ $s$) and $5000th$ ($2.0$ $s$) time step.
59. Normalized separation of the tracking node for "mixed" blind insertion at the $0th$ ($0.0$ $s$), $2500th$ ($1.0$ $s$) and $5000th$ ($2.0$ $s$) time step.
60. Normalized separation of the tracking node for "horizontal" blind insertion with damping at the $0th$ ($0.0$ $s$), $2500th$ ($1.0$ $s$ with $\eta = 3.8$) and $5000th$ ($2.0$ $s$ with $\eta = 4.4$) time step.
61. Normalized separation of the tracking node for "mixed" blind insertion with damping at the $0th$ ($0.0$ $s$), $2500th$ ($1.0$ $s$ with $\eta = 2.4$) and $5000th$ ($2.0$ $s$ with $\eta = 4.3$) time step.
62. Normalized separation of the tracking node for ``horizontal'' insertion with with pre-stretching at the $0th$ ($0.0$ $s$), $2500th$ ($1.0$ $s$) and $5000th$ ($2.0$ $s$) time step.
63. Normalized separation of the tracking node for ``mixed'' insertion with with pre-stretching at the $0th$ ($0.0$ $s$), $2500th$ ($1.0$ $s$) and $5000th$ ($2.0$ $s$) time step.
64. Effect of blind insertion vs pre-stretching on the amplitude of the traction oscillations for increasing stress insertion levels, for the ``horizontal'' and ``mixed'' cases.
65. Schematic of a mode I crack problem.
66. Number of cohesive elements present in the domain over time for bounding box insertion.
67. Mode I case crack tip distance versus time.
68. Mode I reference case with cohesive elements present from the beginning of the simulation (10x exaggeration).
69. Mode I bounding box solution (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).
70. Schematic representation of L-angle test specimen with boundary conditions.
71. Number of cohesive elements present in the domain over time for various stress insertion levels.
72. L-angle case crack tip distance versus time, for various stress insertion levels.
73. L-angle reference case with cohesive elements present from the beginning of the simulation (10x exaggeration).
74. L-angle case with stress based cohesive element insertion for a 15% stress level (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).
75. L-angle case with stress based cohesive element insertion for a 30% stress level (10x exaggeration).
76. L-angle case with stress based cohesive element insertion for a 45% stress level (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).
77. Schematic representation of interface test specimen with boundary conditions.
78. Deformation after 5000 time steps for stress insertion of 45% (10x exaggeration).
79. Deformation after 17500 time steps for stress insertion of 45% (10x exaggeration).
80. Deformation after 25500 time steps for stress insertion of 45% (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).
81. Schematic representation of interface test specimen with boundary conditions.
82. Crack length history for a weak 60 degree interface.
83. Crack speed history for a weak 60 degree interface.
84. Mode I crack trapped along the weak Loctite-384 interface for a 45% stress-based insertion (no exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).
85. Mode I crack trapped along the strong Weldon-100 interface for a 45% stress based insertion (no exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).
86. Close-up of crack region along a weak interface (no exaggeration).
87. Close-up of crack region along a strong interface (no exaggeration).
88. L-angle reference case with cohesive elements inserted every (a) 100 time steps (b) 500 time steps at the 30% stress level (10x exaggeration).
89. L-angle reference case with cohesive elements inserted every (a) 1000 time steps (b) 10000 time steps at the 30% stress level (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).
90. Number of cohesive elements present over time for insertion intervals of 100, 500, 1000, 5000 and 10000.
91. Close-up of the three intervals presented in Figure 4.40.
92. Schematic of a mode I crack problem using nodal subcycling and dynamic stress insertion of $45\%$.
93. Region ratio over time for the subcycling solutions with $m = 6, 10$ and $14$.
94. Reference solution with cohesive elements present everywhere in the domain at the beginning of the simulation. No subcycling is used (10x exaggeration).
95. (a) Solution having only dynamic insertion at 45% of the local stress with no subcycling. (b) Combined dynamic insertion with subcycling, $m = 6$ (10x exaggeration).
96. (a) Combined dynamic insertion with subcycling, $m = 10$ (b) Combined dynamic insertion with subcycling, $m = 14$ (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).
97. Speedup results for L-angle case using 1, 2, 4, and 6 processors.
98. Adaptive crack propagation.
99. 11th element broken into 7 pieces.
100. Velocity profile for new node #26 after element break up at time step 10000.