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)
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) 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.
5
indentersphericalofradiusR
materialofratiosPoisson
materialofulusmodelasticE
indenterofntdisplaceme
indenterofforceP
'
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
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
5.0
/1
4
4.0
073.0
/1010 3
CFL
smV
cmR
GPaE
mkg
ind
7
MPM Indentation Simulation Results for aUniform Quad Mesh
2
2
*
134
REP
P
2
3
*
R
P
8
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
9
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
10
Comparison of Numerical & Analytical Solution
11
Comparison of Numerical & Analytical Solution (zoom in)
12
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