1. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 1 An adaptive method for coupled continuum-molecular simulation of crack propagation in solids Sorin Mitran mitran@amath.unc.edu http://www.amath.unc.edu/Faculty/mitran The University of North Carolina at Chapel Hill Applied Mathematics Program http://www.amath.unc.edu Incipient failureRupture Plots of density of broken bonds

2. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 2 Overview Computational description of failure Dynamic computation of constitutive relations Fully adaptive algorithm Salient Features Macroscopic continuum simulated using wave propagation algorithm Local elastic speeds determined by microscopic averaging Thermal motions identified and eliminated using principal component analysis Microscopic dynamics constrained to follow continuum wave modes Background Elastic materials fail due to breakdown of microscopic bonds Failure theories are empiric: ► need calibration ► work in calibration region First principles derivation of failure: ► predictive ► major analytical challenge

3. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 3 Related Research Quasicontinuum analysis – Tadmor, Ortiz & Phillips (Phil. Mag. A73:1529, 1996) ► static ► finite element description of continuum strain ► microscopic deformation given by finite element form functions ► no thermal effects Atomistic-continuum method – E & Huang (JCP 182:234, 2002) ► separate microscopic, continuum domains ► match microscopic phonon modes to macroscopic waves on atomistic-continuum boundaries ► finite difference method for elasticity equations ► no thermal effects Large-scale molecular dynamics – Zhou et al. (Phys. Rev. Lett. 78:479, 1997) ► accurate ► expensive ► useful to determine fracture mechanisms rather than as a practical tool

4. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 4 One-dimensional model of ductile failure in a rod Model features Progressive damage Bonds break due to combined thermal, mechanical effect Point masses connected by multiple springs Force-deformation law

5. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 5 Equations of motion Displacement from equilibrium: Continuum limit With damage Mass per lattice spacing: No damage: Continuum limit Presence of damage requires microscopic information, i.e. the number of broken springs Zero temperature limit

6. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 6 Two-dimensional model of thin shell failure 2D lattice of oscillators No damage Continuum limit With damage Continuum limit

7. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 7 Failure scenarios Bonds break under dynamic loading due to combined thermal (microscopic) and continuum motion Dynamic loadingMelting

8. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 8 Eliminating thermal behavior Microscopic dynamics Principal component analysis contains all system information Very little of the information is relevant macroscopically Coarse graining approaches: Principal modes – 32 point masses Cutoff after three decades ► Spatial averaging - homogenization ► Fourier mode elimination - RNG

9. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 9 Wave Propagation Method Used for continuum computation LeVeque (JCP 114, 1997) Method with physical interpretation in terms of eigenmode propagation Local Riemann problem update strategy

10. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 10 Elasticity Eigenmodes Transform continuum wave equation to a system of first-order PDE’s

11. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 11 Adaptive Computation For continuum levels Trial step on coarse grid determines placement of finer grids Boundary conditions for finer grids from space-time interpolation Time subcycling: more time steps (of smaller increments) are taken on fine grids Finer grid values are obtained by interpolation from coarser grid values Coarser grid values are updated by averaging over embedded fine grids Conservation ensured at coarse-fine interfaces (conservative fixups)

12. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 12 Direct Simulation Monte Carlo Full microscopic computation too expensive We need microscopic data to evaluate elastic speed and local damage Use a Monte Carlo simulation to sample configuration space Unbiased Monte Carlo simulation requires extensive sampling – too expensive ► hierarchical Monte Carlo ► bias sampling in accordance with principal components from immediately coarser level

13. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 13 Continuum-Microscopic Interaction Continuum to microscopic injection of values low frequency contribution from principal components of coarser grid level high frequency contribution from Maxwell- Boltzmann distribution of unresolvable coarser grid modes Microscopic to continuum restriction: ► Subtract contribution from thermal and minor modes ► The energy of these modes defines a “temperature” valid for current grid level ► Transport coefficient from standard statistical mechanics

14. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 14 Instantaneous Constitutive Relations For simple model considered here only constitutive relation is the dependence of elastic speed upon local damage Update of local damage: ► on each level ► after each time step ► check if displacement has increased beyond current elastic law restriction Force-deformation law

15. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 15 Two-dimensional example: Rupturing membrane Simulation parameters 50 initial molecular bonds n<5 ruptured bonds reform initial Gaussian deformation along x direction with amplitude umax for x<0.2  chosen so initial umax does not cause rupture for x>0.2  chosen so initial umax causes rupture zero-displacement boundary conditions adiabatic boundary conditions initial 32x32 grid 6 refinement levels (3 visualized) refinement ratios: [ 2 2 2 | 8 8 8 ] Continuum DSMC + PCA 16.7 million atoms Animation of density of ruptured atomic bonds

16. The Old Well 3/15/2003 AMS 2003 Spring Southeastern Sectional Meeting 16 Conclusions Algorithm for systematic reduction of degrees of freedom and computation of constitutive properties Microscopic-continuum interaction Diffusion treatment of thermal motion Local thermodynamic equilibrium instantiation of microscopic states Open Issues Choice of cutoffs Geometric bias from initial grid More realistic microscopic model Statistical convergence of DSMC method