Overview Class 6 (Tues, Feb 4) Equations of Elasticity De as long as mation in addition to Material Coordinates Green & Cauchy Strain Tensors Green & Cauchy Strain Tensors

Overview Class 6 (Tues, Feb 4) Equations of Elasticity De as long as mation in addition to Material Coordinates Green & Cauchy Strain Tensors Green & Cauchy Strain Tensors www.phwiki.com

Overview Class 6 (Tues, Feb 4) Equations of Elasticity De as long as mation in addition to Material Coordinates Green & Cauchy Strain Tensors Green & Cauchy Strain Tensors

Currier, Joanna, Contributing Writer/Restaurant Critic has reference to this Academic Journal, PHwiki organized this Journal Overview Class 6 (Tues, Feb 4) Begin de as long as mable models!! Background on elasticity Elastostatics: generalized 3D springs Boundary integral as long as mulation of linear elasticity (from ARTDEFO (SIGGRAPH 99)) Equations of Elasticity Full equations of nonlinear elastodynamics Nonlinearities due to geometry (large de as long as mation; rotation of local coord frame) material (nonlinear stress-strain curve; volume preservation) Simplification as long as small-strain (“linear geometry”) Dynamic in addition to quasistatic cases useful in different contexts Very stiff almost rigid objects Haptics Animation style De as long as mation in addition to Material Coordinates w: unde as long as med world/body material coordinate x=x(w): de as long as med material coordinate u=x-w: displacement vector of material point Body Frame w

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Green & Cauchy Strain Tensors 3×3 matrix describing stretch (diagonal) in addition to shear (off-diagonal) Green & Cauchy Strain Tensors 3×3 matrix describing stretch (diagonal) in addition to shear (off-diagonal) Stress Tensor Describes as long as ces acting inside an object dA (tiny area) n w

Stress Tensor Describes as long as ces acting inside an object dA (tiny area) n w Body Forces Body as long as ces follow by Green’s theorem, i.e., related to divergence of stress tensor Body Forces Body as long as ces follow by Green’s theorem, i.e., related to divergence of stress tensor

Newton’s 2nd Law of Motion Simple (finite volume) discretization w dV Newton’s 2nd Law of Motion Simple (finite volume) discretization w dV Stress-strain Relationship Still need to know this to compute anything An inherent material property

Stress-strain Relationship Still need to know this to compute anything An inherent material property Strain Rate Tensor & Damping Strain Rate Tensor & Damping

Navier’s Eqn of Linear Elastostatics Linear Cauchy strain approx. Linear isotropic stress-strain approx. Time-independent equilibrium case: Navier’s Eqn of Linear Elastostatics Linear Cauchy strain approx. Linear isotropic stress-strain approx. Time-independent equilibrium case: Material properties G,n provide easy way to specify physical behavior

Solution Techniques Many ways to approximation solutions to Navier’s ( in addition to full nonlinear) equations Will return to this later. Detour: ArtDefo paper ArtDefo – Accurate Real Time De as long as mable Objects Doug L. James, Dinesh K. Pai. Proceedings of SIGGRAPH 99. pp. 65-72. 1999. Boundary Conditions Types: Displacements u on Gu (aka Dirichlet) Tractions ( as long as ces) p on Gp (aka Neumann) Boundary Value Problem (BVP) Specify interaction with environment Boundary Integral Equation Form Directly relates u in addition to p on the boundary!

Boundary Element Method (BEM) Define ui, pi at nodes H u = G p Solving the BVP A v = z, A large, dense H u = G p H,G large & dense BIE, BEM in addition to Graphics No interior meshing Smaller (but dense) system matrices Sharp edges easy with constant elements Easy tractions ( as long as haptics) Easy to h in addition to le mixed in addition to changing BC (interaction) More difficult to h in addition to le complex inhomogeneity, non-linearity

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Currier, Joanna Contributing Writer/Restaurant Critic

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