FEM Modeling of Instrumented
Indentation
MAE 5700: Finite Element Analysis for Mechanical and
Aerospace Design
Joseph Carlonia, Julia Chenb, Jonathan Mathenyc, Ashley Torresc
aMaterials Science PhD program, jdc357@cornell.edu
bMechanical Engineering PhD program, tc468@cornell.edu
cBiomedical Engineering PhD program, jbm299@cornell.edu, amt278@cornell.edu
Introduction to instrumented indentation
• A special form of indentation hardness testing where load
vs. displacement data is collected continuously
• The resulting load-displacement data can be used to
determine the plastic and elastic properties of the material
• Commonly used to test the elastic properties of a material,
especially at a small scale  “nanoindentation”
2
A Real Nanoindentation Experiment
1. Load Application
2. Indentation
3. Load Removal
3
Nanoindentation Equations
• The reduced modulus of contact between two
materials is a function of the Young’s moduli:
• Sneddon’s equation relates the reduced modulus of a
contact to the contact stiffness and contact area:
4
Sneddon, 1948
W.C. Oliver and G.M. Pharr (1992).
Motivation / Problem Statement
• Sneddon’s equation was derived for contact between a
rigid indenter and a “semi-infinite half space”
• We want to model the elastic portion of an indentation in
ANSYS so that we can vary dimensional parameters to see
how they affect the accuracy of Sneddon’s equation
P
Ei, vi
2D
Axisymmetric
Es, vs
w
h
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Solid Body Contact
•
•
•
Assume: strains are small, materials are elastic, surfaces are frictionless
Contact – is a changing-status nonlinearity. The stiffness, depends on
whether the parts are touching or separated
We establish a relationship between the two surfaces to prevent them from
passing through each other in the analysis termed, contact compatibility
ANSYS® Academic Research, Release 14.5, Help
System, Introduction to Contact Guide, ANSYS, Inc.
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Normal Lagrange Formulation
• Adds an extra degree of freedom
(contact pressure) to satisfy
contact compatibility
• Contact force is solved for
explicitly instead of using stiffness
and penetration
• Enforces zero/nearly-zero
penetration with pressure DOF
• Only applies to forces in directions
Normal to contact surface
• Direct solvers are used
ANSYS® Academic Research, Release 14.5, Help
System, Introduction to Contact Guide, ANSYS, Inc.
7
Penalty-Based Formulations
• Concept of contact stiffness knormal is used
in both
• The higher the contact stiffness, the lower
the penetration
• As long as xpenetration is small or negligible,
the solution results will be accurate
• The Augmented Lagrange method is less
sensitive to the magnitude of the contact
stiffness knormal because of λ (pressure)
Pros (+) and Cons (-)
ANSYS® Academic Research, Release 14.5, Help
System, Introduction to Contact Guide, ANSYS, Inc.
8
ANSYS Detection Method
ANSYS® Academic Research, Release 14.5, Help
System, Introduction to Contact Guide, ANSYS, Inc.
• Allows you to choose the
location of contact detection in
order to obtain convergence
• Normal Lagrange uses Nodal
Detection, resulting in fewer
points
• Pure Penalty and Augmented
Lagrange use Gauss point
detection, resulting in more
detection points
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ANSYS Contact Stiffness
•
•
Normal stiffness can be automatically adjusted during the solution to
enhance convergence at the end of each iteration
The Normal Contact Stiffness knormal is the most important parameter
affecting accuracy and convergence behavior
– Large value of stiffness gives more accuracy, but problem may be
difficult to converge
– If knormal is too large, the model may oscillate, contact surfaces
would bounce off each other
10
ANSYS® Academic Research, Release 14.5, Help
System, Introduction to Contact Guide, ANSYS, Inc.
Nonlinear Finite Element Approach
Newton-Raphson Iterative Method
Becker, A.A. An Introductory Guide to
Finite Element Analysis. p.109-125.
Loading Incrementation Procedure
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Initial problem set-up
• Materials
– Indenter- Diamond
• Young’s Modulus=1.14E12 Pa
• Poisson’s Ratio=0.07
– Tested Material- “Calcite”
• Young’s Modulus=7E10 Pa
• Poisson’s Ratio=0.3
• Both Materials Type
– Isotropic Elasticity
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Initial problem set-up
• Axisymmetric Model
• Boundary Conditions
– Fixed displacement (in x)
along axis of symmetry
– Fixed support on bottom
edge of material
• Loading
– Pressure (1E8 Pa) applied
normal to top edge of
indenter
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Automated Calculations
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ANSYS Default Mesh (10 divisions)
Quadrilateral Elements
15
ANSYS Default Results (-13.4% error)
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Refined Mesh (160 divisions)
Quadrilateral Elements
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Refined Results (2.74% error)
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Mesh Convergence
The magnitude of the error converges
 Now we change other parameters
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Normal Lagrange
(9.29% error, 5e-17 m penetration)
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Augmented Lagrange
(2.74% error, 1e-9 m penetration)
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Final setup
• Contact Type: Frictionless
– Target Body: indenter
– Contact Body: material
• Behavior: Symmetric
• Contact Formulation: Augmented
Lagrange
• Update Stiffness: Each Iteration
• Stiffness factor: 1
• Auto time step: min 1, max 10
• Weak springs: off
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Pressure
pressure
7.00%
6.11%
6.00%
5.00%
4.15%
error
4.00%
3.00%
2.74%
2.74%
2.74%
2.74%
2.74%
2.75%
2.75%
2.76%
0.7
0.8
0.9
1
2
3
4
5
2.00%
1.00%
0.00%
10
20
pressure / 1e8
Too high of a pressure increases the error
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Dimension of material
narrow to wide material
20.00%
-0.08%
2.74%
1.93%
2.24%
20.00
3.00
40.00
0.00%
error
-20.00%
-40.00%
-60.00%
-73.93%
-80.00%
-89.69%
-100.00%
1.20
2.00
10.00
size of material / size of indenter (5e-6 m)
Too small of a sample increases the error
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Different modulus
low to high modulus of material
16.00%
14.31%
14.00%
12.00%
error
10.00%
8.00%
6.00%
4.00%
2.74%
2.00%
0.00%
-1.17%
-2.00%
0.02
0.06
0.18
modulus of material / modulus of indenter (diamond: 1.14e12 Pa)
Testing a high modulus material increases the error
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Conclusion
• Indentation can be accurately modeled using
ANSYS and a well-refined mesh
• The validity of Sneddon’s equation has been
explored:
– Lower pressure  More accurate
– Larger sample  More accurate
– More compliant sample  More accurate
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Questions?
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