Cohesive Zone Modeling of Dynamic Fracture: Adaptive Mesh Refinement and Coarsening

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1 Cohesive Zone Modeling of Dynamic Fracture: Adaptive Mesh Refinement and Coarsening Glaucio H. Paulino 1, Kyoungsoo Park 2, Waldemar Celes 3, Rodrigo Espinha 3 1 Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign 2 School of Civil and Environmental Engineering Yonsei University, Seoul, Korea 7/25/ Department of Computer Science Pontifical Catholic University of Rio de Janeiro (PUC-Rio)

2 Outline Cohesive Zone Modeling Potential-Based Cohesive Model Adaptive Topological Operators Computational Examples Mode I pre-defined crack path Mixed-mode fracture problem Micro-Branching Summary 7/25/2011 2

3 Cohesive Zone Modeling T n Voids and micro-cracks Macro-crack tip max Macro-crack tip Constitutive Relationship of Cohesive Fracture Non-potential based-model vs. Potential based-model Computational Methods Cohesive surface elements, enrichment functions, embedded discontinuities 7/25/2011 3

4 Previous Potentials for Fracture Needleman A. (1987) Polynomial potential / linear shear interaction Needleman A. (1990) Exponential potential / periodic dependence Beltz G.E. and Rice J.R. (1991) Generalized the potential (Exponential + Sinusoid) Xu X.P. and Needleman A. (1993) Exponential potential (Exponential + Exponential) Needleman A. 1987, A continuum model for void nucleation by inclusion debonding, Journal of Applied Mechanics, 54, Needleman A. 1990, An analysis of tensile decohesion along an interface, Journal of the Mechanics and Physics of Solid, 3, Beltz GE and Rice JR, 1991, Dislocation nucleation versus cleavage decohesion at crack tip, Modeling the Deformation of Crystalline Solids, Xu XP and Needleman, 1993, Void nucleation by inclusion debonding in a crystal matrix, Modeling Simulation Material Science Engineering, 1, /25/2011 4

5 Xu X.P. and Needleman A. (1993) Xu XP and Needleman, 1993, Void nucleation by inclusion debonding in a crystal matrix, Modeling Simulation Material Science Engineering, 1, Cohesive Relationship Normal direction: Exponential Tangential direction: Exponential Boundary Condition exp( / ) Tn B t n C t n n t 2 2 Tt An exp( t / t ) t T,0d T 0, n 0 n n n d t 0 t t t C 0 0 lim Tt n, t 0 Tt n t lim T, 0 n n n t n lim, 0 t lim T, 0 t n n t Introduce additional condition Δ n * instead of T T n n t lim *, 0 t lim, 0 t n n t 7/25/2011 5

6 100 N / m n 200 N / m t max max 30MPa 40MPa r n n 0 Mode I lim T, 0 t n n t Mode II 6

7 Remarks Previous Potential Boundary condition is not symmetric Vague fracture parameter, r (or Δ n *) Okay if fracture energies are the same Complete separation at infinity Could not control initial slope Large compliance Proposed Potential Expressed by a single function Different fracture energy : Different cohesive strength : Different cohesive relation: Different initial slope :, n t n,, t max, max lim T, 0 t n n t 7/25/2011 7

8 Boundary Conditions Various material behavior, i.e. Plateau-type 1, 2 Brittle material, 2 Quasi-brittle material, 2 7/25/2011 8

9 PPR: Unified Mixed Mode Potential Energy Constants: Γ n and Γ t Exponents: m and n Fracture energy Cohesive strength Cohesive interaction shape Initial slope Characteristic length scales: δ n and δ t Shape parameters : α and β K. Park, GH. Paulino, JR. Roesler, 2009, A unified potential-based cohesive model of mixed-mode fracture, Journal of the Mechanics and Physics of Solids 57, /25/2011 9

10 Extension for the EXTRINSIC Model Correct Limit Procedure Limit of initial slope indicators in the potential Energy constants: Γ n and Γ t Characteristic length scales: δ n and δ t Shape parameters: α and β Exclude elastic behavior Extrinsic model Consider different fracture energy: Describe different cohesive strength: Represent various cohesive shape:, 7/25/ n, t max, max

11 100 N/ m 200 N/ m n max 40MPa 5 t max 30MPa 1.3 Mode I Mode II 11

12 Topological Operators Nodal Perturbation Edge-Swap G.H. Paulino, K. Park, W. Celes, and R. Espinha, 2010, Adaptive dynamic cohesive fracture simulation using edgeswap and nodal perturbation operators, International Journal for Numerical Methods in Engineering 84, /25/

13 Topological Operators Edge-Split Adaptive mesh refinement based on a priori knowledge Vertex-Removal (or Edge-Collapse) Adaptive mesh coarsening based on a posteriori error estimation, i.e. root mean square of strain error K. Park, G.H. Paulino, W. Celes, and R. Espinha, 2010, Adaptive mesh refinement and coarsening for cohesive dynamic fracture, International Journal for Numerical Methods in Engineering (in press). 7/25/

14 Adaptive Mesh Coarsening Coarsening Criterion Local Update 7/21/

15 Mode I Pre-defined Crack Propagation Predefined path E v ρ G I T max 3.24 GPa kg/m 3 N/m 324 MPa 4k mesh 7/25/

16 Computational Results Uniform Mesh Refinement Adaptive Mesh Refinement 400x40 mesh grid 100x10 mesh grid Element size: 5μm Element size: 20~5μm elements, nodes 4448 elements, 9147 nodes 7/25/

17 Computational Results (AMR) 7/25/

18 Computational Results (AMR+C) 7/25/

19 50 mm 75 mm 200 mm 100 mm Mixed-Mode Crack Propagation Kalthoff-Winkler s Experiments 100 mm v 0 25 mm v 0 50 mm 100 mm Kalthoff, J. F., Winkler, S., Failure mode transition at high rates of shear loading. International Conference on Impact Loading and Dynamic Behavior of Materials 1, /25/

20 Finite Element Mesh Initial Discretization Animations (FE Mesh & Strain energy) 7/25/

21 Computational Results Crack Path Crack Velocity 7/25/

22 Micro-Branching Experiment Sharon E, Fineberg J. Microbranching instability and the dynamic fracture of brittle materials. Physical Review B 1996; 54(10): ε 0 4 mm 16 mm 1 mm 7/25/

23 7/25/

24 Computational Results ε 0 = ε 0 = ε 0 = /25/

25 Computational Results Crack Velocity Energy Evolution (ε 0 =0.015) 7/25/

26 Summary The potential-based constitutive model Adaptive operators Nodal perturbation, Edge-swap Edge-split, Vertex-removal Effective and efficient computational framework to simulate physical phenomena associated with fracture. The computational results of the adaptive mesh refinement and coarsening is consistent with the results of the uniform mesh refinement. 7/25/

27 QUESTIONS? 7/25/