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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**

We chose the Kelvin-Voigt model to focus on the creep response The constitutive equation is 𝜎=𝐷𝜀+𝜂 𝑑𝜀 𝑑𝑡 Mechanical Analogue of Kelvin-Voigt Model

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**Assumptions for Model Conditions Properties F L c B3 B4 B2 B1**

Constant force F normal to boundary B3 No gravity (body force) 2D, plane stress* Confined in y-direction along B1 and B3 Confined in x-direction along B4 Properties c = 0.1m; L = 0.125m Constant cross-sectional area A Isotropic elasticity* * 𝐷= 𝐸 1−𝜈 𝜈 0 𝜈 −𝜈 2 B3 B2 B1 B4 x y

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**Experimental 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.

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**Derivation of Weak Form**

By definition, stress 𝜎= 𝐹 𝐴 Strain can be rewritten as gradient of displacement u 𝜀=𝛻𝑢= 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝜀 𝑥𝑦 Our constitutive equation (in strong form) becomes 𝐹=𝐴 𝐷 𝛻𝑢+𝜂 𝑑 𝑑𝑡 𝛻𝑢

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**Derivation 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

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**Decoupling a Transient Problem**

We can decouple the formulation and assume the time and spatial variations are separate 𝑢 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 space The weak differential equation rewritten in matrix form is 𝐹 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 𝑑Ω 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**

𝐾 = 𝐵 𝑇 𝐷 𝐵 𝐶 =𝜂 𝐵 𝑇 𝐵 At each time step n 𝐹 𝑜 [𝑁] 𝑇 +𝜂𝐴 𝑢 𝑛−1 − 𝑢 𝑛−2 ∆𝑡 𝐵 𝑇 𝑁 𝑇 =𝐴∫ 𝐵 𝑇 𝐷 𝐵 𝑢 𝑛 +𝜂 𝐵 𝑇 𝐵 𝑢 𝑛−1 − 𝑢 𝑛−2 ∆𝑡 𝑑Ω The equation for 𝑢 𝑛 becomes 𝑢 𝑛 = [𝐶] −1 𝐾 𝑢 𝑛−1 − 𝐹 𝑜 𝐴 𝑁 𝑇 +𝜂 𝑢 𝑛−1 − 𝑢 𝑛−2 ∆𝑡 𝐵 𝑇 𝑁 𝑇 Γ

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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**

A variety of models are available Differences include suitability for primary and secondary creep Usually of the form 𝜀 𝑐𝑟 = 𝑓 1 𝜎 𝑓 2 𝜀 𝑓 3 𝑡 𝑓 4 (𝑇) Examples Strain Hardening: 𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝜀 𝐶 3 𝑒 − 𝐶 4 /𝑇 Time Hardening: 𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑡 𝐶 3 𝑒 − 𝐶 4 /𝑇 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**

Experimental data provides aggregate modulus and Poisson’s ratio Young’s Modulus can be derived from 𝐻 𝐴 = 𝐸 1−𝜈 1+𝜈 1−2𝜈 The solution for time-dependent strain in the K-V model is 𝜀 𝑡 = 𝜎 𝑜 𝐸 (1− 𝑒 −𝜆𝑡 ) We can use the Modified Exponential Function in ANSYS 𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑟 𝑒 −𝑟𝑡 𝑟= 𝐶 5 𝜎 𝐶 3 𝑒 − 𝐶 4 /𝑇 where 𝐶 2 =1, 𝐶 3 = 𝐶 4 =0; we can solve for 𝐶 1 and 𝐶 5 ANSYS Advanced Nonlinear Materials: Lecture 3 – Rate Dependent Creep

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**ANSYS Results – Creep Response**

Short Term Response – 30 Seconds Long Term Response – 3000 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 𝜖 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑡 𝐶 𝑒 − 𝐶 4 𝑇 𝐶 𝐶 5 𝜎 𝐶 6 𝑡 𝑒 − 𝐶 7 𝑇

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**ANSYS Model: Mesh and Time Refinement**

Mesh [Nodes] Time [s] Base Case 805 Between 0.1 and 900 Refinement 15747 Between 1e-4 and 1e-2 Time % Difference w.r.t. Base Case - Mesh Mesh and Time 1 -0.459 0.000 -0.470 2 -0.367 0.025 -0.202 4 -0.294 1.447 1.145 6 -0.267 2.008 1.733 8 -0.255 2.261 2.001 10 2.732 2.384 2.136

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**Sensitivity Analysis Recall the creep model: 𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑟 𝑒 −𝑟𝑡**

𝜀 𝑐𝑟 = 𝐶 1 𝜎 𝐶 2 𝑟 𝑒 −𝑟𝑡 𝑟= 𝐶 5 𝜎 𝐶 3 𝑒 − 𝐶 4 /𝑇 We varied each non-zero model constant by 50%* to perform a rudimentary sensitivity analysis: Time Base Case Case C1 Difference % Case C2 Case C5 1 3.60E-03 6.64E-03 84.59 7.70E-03 114.09 6.53E-03 81.49 2 4.36E-03 7.69E-03 76.20 9.53E-03 118.54 7.39E-03 69.36 4 5.23E-03 8.92E-03 70.51 1.18E-02 124.92 8.23E-03 57.35 6 5.68E-03 9.56E-03 68.20 1.29E-02 127.25 8.55E-03 50.43 8 5.92E-03 9.89E-03 67.13 1.35E-02 128.25 8.67E-03 46.46 10 6.04E-03 1.01E-02 66.61 1.38E-02 128.74 8.71E-03 44.23 *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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