Solution of a Hertzian Contact Mechanics Problem Using the

Material Point MethodJason Sanchez

Department of Mechanical EngineeringUniversity 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)

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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)

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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) 236-252

2. 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) 423-445

3. D. Sulsky, Z. Chen, and H.L. Schreyer, A particle method for history-dependent materials , Comput. Methods Appl. Mech.. 118 (1994) 179-196.

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indentersphericalofradiusR

materialofratiosPoisson

materialofulusmodelasticE

indenterofntdisplaceme

indenterofforceP

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Hertzian Contact Mechanics Betweena 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

2

3

213

4

R

EP

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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

5.0

/1

4

4.0

073.0

/1010 3

CFL

smV

cmR

GPaE

mkg

ind

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MPM Indentation Simulation Results for aUniform Quad Mesh

2

2

*

134

REP

P

2

3

*

R

P

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Locally Resolved Quad Mesh forMPM Indentation Simulation

• 8520 elements• Resolved elements: dx = dy = 0.0185 cm• Coarse elements: dx = dy = 0.1667 cm• Best uniform grid simulation results correspond to 72000 elements with dx = dy = 0.03 cm

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MPM Contact Mechanics Simulation With Locally Resolved Mesh

• isotropic elastic material• grid: 8520 4 node quad elements• 4 material points per element• slip at grid boundary• velocity prescribed to rigid material points 25.0

/1

1

4.0

073.0

/1010 3

CFL

smV

cmR

GPaE

mkg

ind

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Comparison of Numerical & Analytical Solution

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Comparison of Numerical & Analytical Solution (zoom in)

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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