2D Deformation and Creep
Response of Articular Cartilage
By: Mikhail Yakhnis & Robert Zhang
Motivation
• Articular cartilage
• transfers load between bones
• enables smooth motion along joints
• Cartilage has limited capacity for self repair
• Applications: biomaterials, prosthetics, biomedical devices
http://nigelhartnett.onlinemedical.com.au/images/articular%20knee%20injury.jpg
Problem Description
• Consider cartilage in an unconfined compression under constant load F
• Analyze the 2D elastic deformation over time
F
Compression plate
Articular
Cartilage
Frictionless
Supports
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
http://www.allsealsinc.com/allseals/Orings/maxwell.gif
Mathematical Model for Cartilage
• We chose the Kelvin-Voigt model to focus on the creep response
• The constitutive equation is
𝑑𝜀
𝜎 = 𝐷𝜀 + 𝜂
𝑑𝑡
Mechanical Analogue of Kelvin-Voigt Model
http://en.wikipedia.org/wiki/File:Kelvin_Voigt_diagram.svg
Assumptions for Model
F
• Conditions
•
•
•
•
•
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
L
c
• Properties
B3
• c = 0.1m; L = 0.125m
• Constant cross-sectional area A
• Isotropic elasticity*
1 𝜈
𝐸
𝜈 1
*𝐷 =
1−𝜈 2
0 0
0
0
1−𝜈
2
B4
B2
y
x
B1
Experimental Data
• 𝐻𝐴 = 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, 1996. Print.
Derivation of Weak Form
• By definition, stress 𝜎 =
𝐹
𝐴
• Strain can be rewritten as gradient of displacement u
𝜀𝑥𝑥
𝜀 = 𝛻𝑢 = 𝜀𝑦𝑦
𝜀𝑥𝑦
• Our constitutive equation (in strong form) becomes
𝑑
𝐹 = 𝐴 𝐷 𝛻𝑢 + 𝜂
𝛻𝑢
𝑑𝑡
Derivation of Weak Form
(1) Take the gradient of the force equation (which equals zero)
(2) Multiply by an arbitrary displacement 𝑤
𝐴
2𝑢
𝑑𝛻
𝐷(𝛻 2 𝑢) + 𝜂
𝑤 𝑑Ω = 0
𝑑𝑡
(3) Integrate by parts to induce symmetry of 𝑢 and 𝑤
𝑑 𝛻𝑢
𝐹𝑜 𝑤 + 𝜂𝐴
𝑤
𝑑𝑡
−𝐴
Γ
𝑑𝛻𝑢
𝐷𝛻𝑢 𝛻𝑤 + 𝜂
𝛻𝑤 𝑑Ω = 0
𝑑𝑡
Decoupling a Transient Problem
• We can decouple the formulation and assume the time and spatial variations are separate
𝑢 𝑥, 𝑡 ≈ 𝑢𝑛𝑒 𝑥, 𝑡 = 𝑛𝑗=1 𝑢𝑗𝑒 (𝑡) 𝑁𝑗𝑒 (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
𝐹𝑜
[𝑁]𝑇 +𝜂𝐴
𝑢𝑛−1 − 𝑢𝑛−2
𝐵
∆𝑡
𝑇
𝑁
(𝑥𝑥,𝑦𝑦)
𝑇
Γ
= 𝐴∫ 𝐷 𝐵
𝑇
𝐸 𝐵 𝑢𝑛 + 𝜂 𝐵
𝑇
Reddy, J. N.. "Time-Dependent Problems." An introduction to nonlinear finite element analysis. Oxford: Oxford University Press, 2004. . Print.
𝑢𝑛−1 − 𝑢𝑛−2
𝐵
𝑑Ω
∆𝑡
Displacement Equation for Creep Response
𝐾 = 𝐵
• At each time step n
𝑢𝑛−1 − 𝑢𝑛−2
𝑇
𝐹𝑜 [𝑁] +𝜂𝐴
𝐵
∆𝑡
= 𝐴∫
𝑇
𝐵
𝑇
𝑁
𝑇
𝐶 =𝜂 𝐵
𝑇
𝐷 𝐵 𝑢𝑛 + 𝜂 𝐵
𝑇
𝐷 𝐵
𝑇
𝐵
𝑢𝑛−1 − 𝑢𝑛−2
𝐵
𝑑Ω
∆𝑡
• The equation for 𝑢𝑛 becomes
𝑢𝑛 =
[𝐶]−1
𝐹𝑜
𝐾 𝑢𝑛−1 −
𝑁
𝐴
𝑇
𝑢𝑛−1 − 𝑢𝑛−2
+𝜂
𝐵
∆𝑡
𝑇
𝑁
𝑇
Γ
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
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
Modeling Creep in MATLAB
Short Time Creep Response Using Various Time Steps
0.02
0.02
0.018
0.018
0.016
0.016
0.014
0.014
0.012
0.012
Strain (m/m)
Strain (m/m)
Short Time Creep Response Using
Various Mesh Sizes
0.01
0.008
0.01
0.008
0.006
0.006
0.004
0.004
tstep = 1.00s
tstep = 0.50s
tstep = 0.25s
0.002
5x10
10x20
20x40
0.002
0
0
0
2
4
6
8
10
Time (sec)
12
14
16
18
20
0
5
10
Time (sec)
15
20
Modeling Creep in ANSYS
• A variety of models are available
• Differences include suitability for primary
and secondary creep
• Usually of the form 𝜀𝑐𝑟 = 𝑓1 𝜎 𝑓2 𝜀 𝑓3 𝑡 𝑓4 (𝑇)
• Examples
1. Strain Hardening: 𝜀𝑐𝑟 = 𝐶1 𝜎 𝐶2 𝜀 𝐶3 𝑒 −𝐶4 /𝑇
2. Time Hardening: 𝜀𝑐𝑟 = 𝐶1 𝜎 𝐶2 𝑡 𝐶3 𝑒 −𝐶4/𝑇
ANSYS Advanced Nonlinear Materials: Lecture 3 – Rate Dependent Creep
http://www.ansys-blog.com/wp-content/uploads/2012/06/Three-Types-of-Creep.png
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?
Parameters in the ANSYS Model
• Experimental data provides aggregate modulus and Poisson’s ratio
• Young’s Modulus can be derived from
𝐸 1−𝜈
𝐻𝐴 =
1 + 𝜈 1 − 2𝜈
• The solution for time-dependent strain in the K-V model is
𝜎𝑜
𝜀 𝑡 = (1 − 𝑒 −𝜆𝑡 )
𝐸
• We can use the Modified Exponential Function in ANSYS
𝜀𝑐𝑟 = 𝐶1 𝜎 𝐶2 𝑟𝑒 −𝑟𝑡
𝑟 = 𝐶5 𝜎 𝐶3 𝑒 −𝐶4 /𝑇
where 𝐶2 = 1, 𝐶3 = 𝐶4 = 0; we can solve for 𝐶1 and 𝐶5
ANSYS Advanced Nonlinear Materials: Lecture 3 – Rate Dependent Creep
ANSYS Results – Creep Response
Short Term Response – 30 Seconds
Long Term Response – 3000 Seconds
Animation of Deformation in ANSYS
Comparison of ANSYS and Experiment
Result:
Experimental vs. Simulation Short Time Creep Response
0.03
• Theoretical Model-Based ANSYS data tends to
overshoot experimental data
Strain (m/m)
0.025
0.02
• Error is between 30% to 40% per data point
0.015
• Experimental-based model performs better
0.01
Discussion:
0.005
Experimental Data
• Results demonstrate the limitations of ANSYS models
ANSYS Theoretical Based Results
ANSYS Exp Based Model
0
0
2
4
6
8
10
12
• A combined primary-secondary model is ideal
Time (s)
Primary +
Secondary Time
Hardening
𝜖𝑐𝑟 =
𝐶4
𝐶1 𝜎 𝐶2 𝑡 𝐶3+1 𝑒 − 𝑇
𝐶3 + 1
• Long term response in ANSYS is not accurate
𝐶7
+ 𝐶5 𝜎 𝐶6 𝑡𝑒 − 𝑇
• Function models primary response
ANSYS Model: Mesh and Time Refinement
Simulation Refinement for Short Time Deformation Response
6.50E-03
Mesh [Nodes]
Time [s]
Base Case
805
Between 0.1 and 900
Refinement
15747
Between 1e-4 and 1e-2
6.00E-03
Time
Strain (m/m)
5.50E-03
-
Mesh
Time
Mesh and Time
1
-0.459
0.000
-0.470
Base Case
2
-0.367
0.025
-0.202
Mesh Refined
4
-0.294
1.447
1.145
Time Refined
6
-0.267
2.008
1.733
Mesh and Time Refined
8
-0.255
2.261
2.001
10
2.732
2.384
2.136
5.00E-03
4.50E-03
4.00E-03
3.50E-03
3.00E-03
0
2
4
6
Time (s)
8
% Difference w.r.t. Base Case
10
12
Sensitivity Analysis
• Recall the creep model:
𝜀𝑐𝑟 = 𝐶1 𝜎 𝐶2 𝑟𝑒 −𝑟𝑡
𝑟 = 𝐶5 𝜎 𝐶3 𝑒 −𝐶4/𝑇
• We varied each non-zero model constant by 50%* to perform a
rudimentary sensitivity analysis:
Time
Base Case
Case C1
Difference %
Case C2
Difference %
Case C5
Difference %
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
2D Deformation and Creep
Response of Articular Cartilage
By: DJ Mikey Mike & Big Rob Zhang
Thank you for listening. Questions?