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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
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)
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Explicit MPM
Momentum formulation
Plane axisymmetric formulation
Isotropic linear elasticity
Natural no-slip contact between material points
D. Sulsky, S. Zhou, and H.L. Schreyer, Application of a particle-in-cell method to solid
mechanics, Comput. Phys. 87 (1995) 236-252
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
D. Sulsky, Z. Chen, and H.L. Schreyer, A particle method for history-dependent materials, 4
Comput. Methods Appl. Mech.. 118 (1994) 179-196.
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
P
4 E
P
2
3
1




R 

3
2
force of indenter
 displaceme nt of indenter
E elastic modulus of material
 Poisson ' s ratio of material
R radius of spherical indenter
spherical indenter
R
a

elastic material
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MPM Contact Mechanics Simulation
  1010 kg / m 3
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isotropic elastic material,
4 uniform quad meshes
4 material points per element
slip at grid boundary
velocity prescribed to rigid material points (indenter)
E  0.073 GPa
  0 .4
R  4 cm
Vind  1 m / s
CFL  0.5
axis of symmetry
spherical indenter
sample
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MPM Indentation Simulation Results for a
Uniform Quad Mesh
 
P  
R
*
3
2
P
P 
4 E
2
R 

2
 3 1 

*
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Locally Resolved Quad Mesh for
MPM Indentation Simulation
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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
  1010
•
•
•
•
•
isotropic elastic material
grid: 8520 4 node quad elements
4 material points per element
slip at grid boundary
velocity prescribed to rigid material points
kg / m 3
E  0.073 GPa
  0 .4
R  1 cm
Vind  1 m / s
CFL  0.25
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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
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