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2D Deformation and Creep Response of Articular Cartilage By: Mikhail Yakhnis & Robert Zhang

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Motivation Articular cartilage transfers load between bones enables smooth motion along joints Cartilage has limited capacity for self repair Applications: biomaterials, prosthetics, biomedical devices

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Problem Description Consider cartilage in an unconfined compression under constant load F Analyze the 2D elastic deformation over time Articular Cartilage F Compression plate Frictionless Supports

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Material Background Cartilage often modeled as a viscoelastic material Viscous and elastic by superposition Elasticity and viscosity can be linear or nonlinear Established models: Kelvin-Voigt, Maxwell, Standard-Linear Solid

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Mathematical Model for Cartilage Mechanical Analogue of Kelvin-Voigt Model

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Assumptions for Model B3 B2 B1 B4 x y F c L

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Experimental Data Data Book on Mechanical Properties of Living Cells, Tissues, and Organs /. Tokyo ; New York : Springer, Print.

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Derivation of Weak Form

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Decoupling a Transient Problem Reddy, J. N.. "Time-Dependent Problems." An introduction to nonlinear finite element analysis. Oxford: Oxford University Press, Print.

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Displacement Equation for Creep Response

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Modeling Creep in MATLAB Changes in Preprocessor.m Provide initial displacement Define time step Adjust boundary conditions Changes in Assemble.m Assemble the damping matrix [C] Changes in NodalSoln.m Add initial condition, damping, time inputs Modify reaction force and displacement equations

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Modeling Creep in MATLAB Discussion: MATLAB result converges toward experimental data farther away from initial time 10% error at 6 seconds MATLAB model reaches equilibrium faster than experimental data

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Modeling Creep in MATLAB

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Modeling Creep in ANSYS ANSYS Advanced Nonlinear Materials: Lecture 3 – Rate Dependent Creep

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Considerations for ANSYS Model What experimental data is available to us? Can we fit the experimental data to the model? Can we use the built-in Mechanical APDL curve fitting procedure? Is there more emphasis on primary creep or secondary creep? Does the model satisfy our constitutive equation?

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Parameters in the ANSYS Model ANSYS Advanced Nonlinear Materials: Lecture 3 – Rate Dependent Creep

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ANSYS Results – Creep Response Long Term Response – 3000 Seconds Short Term Response – 30 Seconds

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Animation of Deformation in ANSYS

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Comparison of ANSYS and Experiment Result: Theoretical Model-Based ANSYS data tends to overshoot experimental data Error is between 30% to 40% per data point Experimental-based model performs better Discussion: Results demonstrate the limitations of ANSYS models A combined primary-secondary model is ideal Long term response in ANSYS is not accurate Function models primary response Primary + Secondary Time Hardening

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ANSYS Model: Mesh and Time Refinement Time % Difference w.r.t. Base Case -MeshTimeMesh and Time Mesh [Nodes]Time [s] Base Case805Between 0.1 and 900 Refinement15747Between 1e-4 and 1e-2

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Sensitivity Analysis TimeBase CaseCase C1Difference %Case C2Difference %Case C5Difference % 13.60E E E E E E E E E E E E E E E E E E E E E E E E *The simulation did not converge at C2 +50% so C2 +10% was used instead

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2D Deformation and Creep Response of Articular Cartilage By: DJ Mikey Mike & Big Rob Zhang Thank you for listening. Questions?

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