Solution of a Hertzian Contact Mechanics Problem Using the Material Point Method Jason Sanchez Department of Mechanical Engineering University of New Mexico 18 March 2008

2 Nanoindentation Simulation of Blast Resistant Cement DTRA blast resistant concrete investigation (UNM Dept. of Civil Engineering) How well does a nanoindentation simulation reproduce experimental data for blast resistant cement? –Force vs. displacement response –Indenter impression Material modeling of blast resistant concrete at micro-scale –Isotropic material to begin –elastic-plastic constitutive model –Possibly inhomogeneous material (fibers, other particles, etc. ) Simulation method it the material point method (MPM)

3 Work Breakdown Perform a benchmark problem with MPM (Hertzian contact mechanics) Constitutive modeling –Elastic-plastic constitutive model Contact algorithm at indenter interface –compression only, friction at interface –decohesion 3D MPM Birkovitch Indentation Simulation –Parallel MPM implementation necessary (use of HPC)

4 Benchmark MPM Indentation Simulation Hertzian contact of a rigid spherical indenter contacting a isotropic elastic material Reproduce theoretical force vs. displacement response MPM Implementation (references 1-3) –Explicit MPM –Momentum formulation –Plane axisymmetric formulation –Isotropic linear elasticity –Natural no-slip contact between material points 1.D. Sulsky, S. Zhou, and H.L. Schreyer, Application of a particle-in-cell method to solid mechanics, Comput. Phys. 87 (1995) D. Sulsky and H.L. Schreyer, MPM simulation of dynamic material failure with a decohesion constitutive model, European Journal of Mechanics A/Solids. 23 (2004) D. Sulsky, Z. Chen, and H.L. Schreyer, A particle method for history-dependent materials, Comput. Methods Appl. Mech (1994)

5 Hertzian Contact Mechanics Between a Rigid Spherical Indenter and a Flat Specimen local deformations at the contact no consideration for bulk deformations or support of the bodies small strains, linear elasticity R  a elastic material spherical indenter

6 MPM Contact Mechanics Simulation isotropic elastic material, 4 uniform quad meshes 4 material points per element slip at grid boundary velocity prescribed to rigid material points (indenter) sample spherical indenter axis of symmetry

7 MPM Indentation Simulation Results for a Uniform Quad Mesh

8 Locally Resolved Quad Mesh for MPM Indentation Simulation 8520 elements Resolved elements: dx = dy = cm Coarse elements: dx = dy = cm Best uniform grid simulation results correspond to elements with dx = dy = 0.03 cm

9 MPM Contact Mechanics Simulation With Locally Resolved Mesh isotropic elastic material grid: node quad elements 4 material points per element slip at grid boundary velocity prescribed to rigid material points

10 Comparison of Numerical & Analytical Solution

11 Comparison of Numerical & Analytical Solution (zoom in)

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13 Conclusions, current, and Future work Conclusions –MPM reproduces analytical force vs. displacement results (Hertzian contact mechanics) –Highly resolved spatial mesh is necessary at indenter-material interface Constitutive model for axisymmetric analysis (current work) –plasticity –Decohesion (initiation of cracking) Contact algorithm at interface (current work) –compression only, friction at interface, decohesion 3D MPM Indentation Simulation (summer / fall 08) –Parallel MPM implementation –Incorporate locally resolved mesh generator