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FEM Modeling of Instrumented Indentation MAE 5700: Finite Element Analysis for Mechanical and Aerospace Design Joseph Carloni a, Julia Chen b, Jonathan Matheny c, Ashley Torres c a Materials Science PhD program, jdc357@cornell.edu b Mechanical Engineering PhD program, tc468@cornell.edu c Biomedical Engineering PhD program, jbm299@cornell.edu, amt278@cornell.edu Slide 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 2 Slide 3 A Real Nanoindentation Experiment 1. Load Application 2. Indentation 3. Load Removal 3 Slide 4 Nanoindentation Equations The reduced modulus of contact between two materials is a function of the Youngs moduli: Sneddons 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). Slide 5 Motivation / Problem Statement Sneddons 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 Sneddons equation 2D Axisymmetric P E i, v i E s, v s h w 5 Slide 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. 6 Slide 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. 7 Slide 8 Penalty-Based Formulations Concept of contact stiffness k normal is used in both The higher the contact stiffness, the lower the penetration As long as x penetration is small or negligible, the solution results will be accurate The Augmented Lagrange method is less sensitive to the magnitude of the contact stiffness k normal because of (pressure) Pros (+) and Cons (-) ANSYS Academic Research, Release 14.5, Help System, Introduction to Contact Guide, ANSYS, Inc. 8 Slide 9 ANSYS Detection Method 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 ANSYS Academic Research, Release 14.5, Help System, Introduction to Contact Guide, ANSYS, Inc. 9 Slide 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 k normal 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 k normal is 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. 10 Slide 11 Nonlinear Finite Element Approach Newton-Raphson Iterative Method Loading Incrementation Procedure Becker, A.A. An Introductory Guide to Finite Element Analysis. p.109-125. 11 Slide 12 Initial problem set-up Materials Indenter- Diamond Youngs Modulus=1.14E12 Pa Poissons Ratio=0.07 Tested Material- Calcite Youngs Modulus=7E10 Pa Poissons Ratio=0.3 Both Materials Type Isotropic Elasticity 12 Slide 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 13 Slide 14 Automated Calculations 14 Slide 15 ANSYS Default Mesh (10 divisions) Quadrilateral Elements 15 Slide 16 ANSYS Default Results (-13.4% error) 16 Slide 17 Refined Mesh (160 divisions) Quadrilateral Elements 17 Slide 18 Refined Results (2.74% error) 18 Slide 19 Mesh Convergence The magnitude of the error converges Now we change other parameters 19 Slide 20 Normal Lagrange (9.29% error, 5e-17 m penetration) 20 Slide 21 Augmented Lagrange (2.74% error, 1e-9 m penetration) 21 Slide 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 22 Slide 23 Pressure Too high of a pressure increases the error 23 Slide 24 Dimension of material Too small of a sample increases the error 24 Slide 25 Different modulus Testing a high modulus material increases the error 25 Slide 26 Conclusion Indentation can be accurately modeled using ANSYS and a well-refined mesh The validity of Sneddons equation has been explored: Lower pressure More accurate Larger sample More accurate More compliant sample More accurate 26 Slide 27 Questions? 27