Presentation on theme: "2D Deformation and Creep Response of Articular Cartilage"— Presentation transcript:
12D Deformation and Creep Response of Articular Cartilage By: Mikhail Yakhnis & Robert Zhang
2Motivation Articular cartilage transfers load between bonesenables smooth motion along jointsCartilage has limited capacity for self repairApplications: biomaterials, prosthetics, biomedical devices
3Problem DescriptionConsider cartilage in an unconfined compression under constant load FAnalyze the 2D elastic deformation over timeArticular CartilageFCompression plateFrictionless Supports
4Material Background Cartilage 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
5Mathematical Model for Cartilage We chose the Kelvin-Voigt model to focus on the creep responseThe constitutive equation is𝜎=𝐷𝜀+𝜂 𝑑𝜀 𝑑𝑡Mechanical Analogue of Kelvin-Voigt Model
6Assumptions for Model Conditions Properties F L c B3 B4 B2 B1 Constant force F normal to boundary B3No gravity (body force)2D, plane stress*Confined in y-direction along B1 and B3Confined in x-direction along B4Propertiesc = 0.1m; L = 0.125mConstant cross-sectional area AIsotropic elasticity* * 𝐷= 𝐸 1−𝜈 𝜈 0 𝜈 −𝜈 2B3B2B1B4xy
7Experimental Data 𝐻 𝐴 =7𝑒5 𝑃𝑎 (Aggregate Modulus) 𝐸=3.37𝑒5 𝑃𝑎 𝜈=0.396 Data Book on Mechanical Properties of Living Cells, Tissues, and Organs /. Tokyo ; New York : Springer, Print.
8Derivation of Weak Form By definition, stress 𝜎= 𝐹 𝐴Strain can be rewritten as gradient of displacement u𝜀=𝛻𝑢= 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝜀 𝑥𝑦Our constitutive equation (in strong form) becomes 𝐹=𝐴 𝐷 𝛻𝑢+𝜂 𝑑 𝑑𝑡 𝛻𝑢
9Derivation of Weak Form (1) Take the gradient of the force equation (which equals zero) (2) Multiply by an arbitrary displacement 𝑤 𝐴 𝐷( 𝛻 2 𝑢)+𝜂 𝑑 𝛻 2 𝑢 𝑑𝑡 𝑤 𝑑Ω =0 (3) Integrate by parts to induce symmetry of 𝑢 and 𝑤 𝐹 𝑜 𝑤+𝜂𝐴 𝑑 𝛻𝑢 𝑑𝑡 𝑤 Γ −𝐴 𝐷𝛻𝑢 𝛻𝑤+𝜂 𝑑𝛻𝑢 𝑑𝑡 𝛻𝑤 𝑑Ω =0
10Decoupling a Transient Problem We can decouple the formulation and assume the time and spatial variations are separate𝑢 𝑥,𝑡 ≈ 𝑢 𝑛 𝑒 𝑥,𝑡 = 𝑗=1 𝑛 𝑢 𝑗 𝑒 (𝑡) 𝑁 𝑗 𝑒 (𝑥)where 𝑢 is a function of time only and basis function N is function of spaceThe weak differential equation rewritten in matrix form is𝐹 𝑜 [𝑁] 𝑇 +𝜂𝐴 𝑢 𝑛−1 − 𝑢 𝑛−2 ∆𝑡 𝐵 𝑇 (𝑥𝑥,𝑦𝑦) 𝑁 𝑇 Γ =𝐴∫ 𝐷 𝐵 𝑇 𝐸 𝐵 𝑢 𝑛 +𝜂 𝐵 𝑇 𝐵 𝑢 𝑛−1 − 𝑢 𝑛−2 ∆𝑡 𝑑ΩReddy, J. N.. "Time-Dependent Problems." An introduction to nonlinear finite element analysis. Oxford: Oxford University Press, Print.
12Modeling Creep in MATLAB Changes in Preprocessor.mProvide initial displacementDefine time stepAdjust boundary conditionsChanges in Assemble.mAssemble the damping matrix [C]Changes in NodalSoln.mAdd initial condition, damping, time inputsModify reaction force and displacement equations
13Modeling Creep in MATLAB Discussion:MATLAB result converges toward experimental data farther away from initial time10% error at 6 secondsMATLAB model reaches equilibrium faster than experimental data
15Modeling Creep in ANSYS A variety of models are availableDifferences include suitability for primaryand secondary creepUsually of the form 𝜀 𝑐𝑟 = 𝑓 1 𝜎 𝑓 2 𝜀 𝑓 3 𝑡 𝑓 4 (𝑇)ExamplesStrain Hardening: 𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝜀 𝐶 3 𝑒 − 𝐶 4 /𝑇Time Hardening: 𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑡 𝐶 3 𝑒 − 𝐶 4 /𝑇ANSYS Advanced Nonlinear Materials: Lecture 3 – Rate Dependent Creep
16Considerations 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?
17Parameters in the ANSYS Model Experimental data provides aggregate modulus and Poisson’s ratioYoung’s Modulus can be derived from𝐻 𝐴 = 𝐸 1−𝜈 1+𝜈 1−2𝜈The solution for time-dependent strain in the K-V model is𝜀 𝑡 = 𝜎 𝑜 𝐸 (1− 𝑒 −𝜆𝑡 )We can use the Modified Exponential Function in ANSYS𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑟 𝑒 −𝑟𝑡𝑟= 𝐶 5 𝜎 𝐶 3 𝑒 − 𝐶 4 /𝑇where 𝐶 2 =1, 𝐶 3 = 𝐶 4 =0; we can solve for 𝐶 1 and 𝐶 5ANSYS Advanced Nonlinear Materials: Lecture 3 – Rate Dependent Creep
18ANSYS Results – Creep Response Short Term Response – 30 SecondsLong Term Response – 3000 Seconds
20Comparison of ANSYS and Experiment Result:Theoretical Model-Based ANSYS data tends to overshoot experimental dataError is between 30% to 40% per data pointExperimental-based model performs betterDiscussion: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 Hardening𝜖 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑡 𝐶 𝑒 − 𝐶 4 𝑇 𝐶 𝐶 5 𝜎 𝐶 6 𝑡 𝑒 − 𝐶 7 𝑇
21ANSYS Model: Mesh and Time Refinement Mesh [Nodes]Time [s]Base Case805Between 0.1 and 900Refinement15747Between 1e-4 and 1e-2Time% Difference w.r.t. Base Case-MeshMesh and Time1-0.4590.000-0.4702-0.3670.025-0.2024-0.2941.4471.1456-0.2672.0081.7338-0.2552.2612.001102.7322.3842.136
22Sensitivity Analysis Recall the creep model: 𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑟 𝑒 −𝑟𝑡 𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑟 𝑒 −𝑟𝑡𝑟= 𝐶 5 𝜎 𝐶 3 𝑒 − 𝐶 4 /𝑇We varied each non-zero model constant by 50%* to perform a rudimentary sensitivity analysis:TimeBase CaseCase C1Difference %Case C2Case C513.60E-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 instead
232D Deformation and Creep Response of Articular Cartilage By: DJ Mikey Mike & Big Rob ZhangThank you for listening. Questions?