2D Deformation and Creep of Articular Cartilage

2D Deformation and Creep Response of Articular CartilageBy: Mikhail Yakhnis & Robert ZhangMotivationArticular cartilage transfers load between bones enables smooth motion along joints

Cartilage has limited capacity for self repairApplications: biomaterials, prosthetics, biomedical devices

http://nigelhartnett.onlinemedical.com.au/images/articular%20knee%20injury.jpg2Problem DescriptionConsider cartilage in an unconfined compression under constant load FAnalyze the 2D elastic deformation over time

Articular CartilageFCompression plateFrictionless SupportsMaterial BackgroundCartilage often modeled as a viscoelastic material Viscous and elastic by superpositionElasticity and viscosity can be linear or nonlinearEstablished models: Kelvin-Voigt, Maxwell, Standard-Linear Solid

http://www.allsealsinc.com/allseals/Orings/maxwell.gif4Mathematical Model for Cartilage

Mechanical Analogue of Kelvin-Voigt Modelhttp://en.wikipedia.org/wiki/File:Kelvin_Voigt_diagram.svg5Assumptions for ModelB3B2B1B4xyFcLExperimental Data

Data Book on Mechanical Properties of Living Cells, Tissues, and Organs /. Tokyo ; New York : Springer, 1996. Print. Derivation of Weak Form Derivation of Weak Form Decoupling a Transient Problem Reddy, J. N.. "Time-Dependent Problems." An introduction to nonlinear finite element analysis. Oxford: Oxford University Press, 2004. . Print.10Displacement Equation for Creep ResponseModeling Creep in MATLABChanges in Preprocessor.mProvide initial displacementDefine time step Adjust boundary conditions

Changes in Assemble.mAssemble the damping matrix [C]

Changes in NodalSoln.mAdd initial condition, damping, time inputsModify reaction force and displacement equations

Modeling Creep in MATLABDiscussion:MATLAB result converges toward experimental data farther away from initial time 10% error at 6 seconds

MATLAB model reaches equilibrium faster than experimental dataModeling Creep in MATLABModeling Creep in ANSYS

ANSYS Advanced Nonlinear Materials: Lecture 3 Rate Dependent Creephttp://www.ansys-blog.com/wp-content/uploads/2012/06/Three-Types-of-Creep.png

15Considerations for ANSYS ModelWhat 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?Parameters in the ANSYS ModelANSYS Advanced Nonlinear Materials: Lecture 3 Rate Dependent Creep17ANSYS Results Creep ResponseLong Term Response 3000 SecondsShort Term Response 30 Seconds

Animation of Deformation in ANSYS

Comparison of ANSYS and ExperimentResult:Theoretical Model-Based ANSYS data tends to overshoot experimental dataError is between 30% to 40% per data pointExperimental-based model performs better

Discussion: Results demonstrate the limitations of ANSYS modelsA combined primary-secondary model is idealLong term response in ANSYS is not accurateFunction models primary responsePrimary + Secondary Time HardeningANSYS Model: Mesh and Time RefinementTime% Difference w.r.t. Base Case-MeshTimeMesh and Time1-0.4590.000-0.4702-0.3670.025-0.2024-0.2941.4471.1456-0.2672.0081.7338-0.2552.2612.001102.7322.3842.136Mesh [Nodes]Time [s]Base Case805Between 0.1 and 900Refinement15747Between 1e-4 and 1e-2Sensitivity AnalysisTimeBase CaseCase C1Difference %Case C2Difference %Case C5Difference %13.60E-036.64E-0384.597.70E-03114.096.53E-0381.4924.36E-037.69E-0376.209.53E-03118.547.39E-0369.3645.23E-038.92E-0370.511.18E-02124.928.23E-0357.3565.68E-039.56E-0368.201.29E-02127.258.55E-0350.4385.92E-039.89E-0367.131.35E-02128.258.67E-0346.46106.04E-031.01E-0266.611.38E-02128.748.71E-0344.23*The simulation did not converge at C2 +50% so C2 +10% was used instead2D Deformation and Creep Response of Articular CartilageBy: DJ Mikey Mike & Big Rob ZhangThank you for listening. Questions?