FEM Modeling of Instrumented Indentation

1. 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 cBiomedicalEngineering PhD program, jbm299@cornell.edu, amt278@cornell.edu

2. 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”

3. A Real NanoindentationExperiment 2. Indentation 1. Load Application 3. Load Removal

4. 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: Sneddon, 1948 W.C. Oliver and G.M. Pharr (1992).

5. 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 2D Axisymmetric Ei, vi Es, vs h w

6. 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. 

7. 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. 

8. Penalty-Based Formulations Pros (+) and Cons (-) • Concept of contact stiffness knormalis 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) ANSYS® Academic Research, Release 14.5, Help System, Introduction to Contact Guide, ANSYS, Inc. 

9. 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

10. 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 knormalis the most important parameter affecting accuracy and convergence behavior • Large value of stiffness gives more accuracy, but problem may be difficult to converge • If knormalis too large, the model may oscillate, contact surfaces would bounce off each other ANSYS® Academic Research, Release 14.5, Help System, Introduction to Contact Guide, ANSYS, Inc. 

11. Nonlinear Finite Element Approach Loading Incrementation Procedure Newton-Raphson Iterative Method Becker, A.A. An Introductory Guide to Finite Element Analysis. p.109-125.

12. 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

13. 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

14. Automated Calculations

15. ANSYS Default Mesh (10 divisions) Quadrilateral Elements

16. ANSYS Default Results (-13.4% error)

17. Refined Mesh (160 divisions) Quadrilateral Elements

18. Refined Results (2.74% error)

19. Mesh Convergence The magnitude of the error converges  Now we change other parameters

20. Normal Lagrange(9.29% error, 5e-17 m penetration)

21. Augmented Lagrange(2.74% error, 1e-9 m penetration)

22. 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

23. Pressure Too high of a pressure increases the error

24. Dimension of material Too small of a sample increases the error

25. Different modulus Testing a high modulus material increases the error

26. 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

27. Questions?