1. Solid object break-up Ivan Dramaliev CS260, Winter’03

2. Outline Methods Linear elastic fracture method Continuous and discrete models Forces and fractures

3. Past work Viscoelastic and plastic deformations (Terzopoulos and Fleisher) Solid object breakup using a spring and mass system (Norton) Limitations: - undetermined locality of deformation - directional patterns

4. Linear elastic fracture method Contains information about the magnitude and orientation of internal stresses Object material described by a set of differential equations applied to a continuous model Computer application done using finite element discretization

5. Continuous model Define material location using a 3D vector u Deformation is specified using a function x(u) Measure local deformation using Green’s strain tensor , which is a symmetric 3x3 matrix Strain change is measured using the strain rate tensor Apply the stress tensor , which is a combination of elastic and viscous stresses, to model the material properties Assume isotropic material, which reduces the two component stress tensors to terms with four parameters describing the material’s rigidity, resistant to change in volume and dissipation of internal kinetic energy

6. Material discretization Done using finite element method Split into tetrahedra, with the four nodes of each containing the position in material coordinates, position in world coordinates and velocity in world coordinates Elements exert elastic and damping on their nodes

7. Element interaction Use element penetration to compute forces during collisions

8. Fractures Three loading modes describe propagation of cracks - opening - in-plane shear - out-of-plane shear Cracks are specified by adjacent material elements which do not share nodes

9. Internal forces For each node calculate tensile and compressive stresses These stresses then define the tensile and compressive forces exerted on each node by its adjacent elements Calculate a separation tensor from balanced tensile and compressive forces acting at each node If the largest positive eigenvalue of the separation tensor exceed the material toughness, then a fracture occurs The orientation of the fracture plane is perpendicular to the eigenvector corresponding to the “cracking” eigenvalue

10. Screenshots Taken from “Graphical Modeling and Animation of Brittle Fracture” by J. O’Brien and J. Hodgins SIGGRAPH 99, Los Angeles, CA, USA Copyright ACM 1999 0-201-48560-5/99/08