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**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, bMechanical Engineering PhD program, cBiomedical Engineering PhD program,

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**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” Because of this advantage, instrumented indents are commonly used... Especially at the small scale... i.e. nanoindentation

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**A Real Nanoindentation Experiment**

1. Load Application 2. Indentation 3. Load Removal Here is a schematic of a real nanoindentation experiment. The instrument applies a load, the indenter penetrates the surface, and then the load is removed. The collected load-displacement data is usually plotted like so (load on y vs. displacement on x). After some critical load, the material plastically deforms (that is doesn’t return to its initial displacement), but elastic properties can still be calculated from a linear fit to the initial unloading. Alternatively, you can try to to stay below the critical load for plasticity.

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**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: For contact between 2 different materials, the reduced modulus is a function of the materials’ Young’s moduli. Sneddon’s equation (originally derived in 1948, and applied to nanoindentation in the early 90s) relates the reduced modulus to a measured contact stiffness and contact area Sneddon, 1948 W.C. Oliver and G.M. Pharr (1992).

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**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 2D Axisymmetric P Ei, vi Es, vs h w The analytical solution for indentation was derived for... In practice, this is not possible, so we want to model it using FEA and see how close we can approximate the analytical solution by varying dimensional parameters and boundary conditions.

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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.

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**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.

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**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 ANSYS® Academic Research, Release 14.5, Help System, Introduction to Contact Guide, ANSYS, Inc.

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**Nonlinear Finite Element Approach**

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

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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

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**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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**Too high of a pressure increases the error**

Sneddon’s equation is derived for purely elastic, small deformations. We chose a pressure of 1e8 based on experimental results. As the pressure increases, we will eventually have large deformations that should result in plasticity. Therefore, we expect the error to increase after some critical pressure. Too high of a pressure increases the error

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**Dimension of material Too small of a sample increases the error**

Sneddon’s equation is derived assuming contact with a “semi-infinite half space.” As the material size decreases, we will eventually have a geometry that cannot be accurately approximated as “infinite.” Therefore, we expect to see an increase in the error below some critical material size. Too small of a sample increases the error

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**Testing a high modulus material increases the error**

Different modulus Sneddon’s equation was initially derived for contact by an “perfectly rigid” indenter. The concept of reduced modulus was introduced as a correction for indenters with some finite stiffness. As we increase the stiffness of the material being indented, eventually the ratio between the stiffness of the material and that of the indenter becomes too high to be accurately corrected. Therefore, we expect the magnitude of the error to increase with increasing material modulus. 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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