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TECHNICAL PAPERS: Soft Tissue

# Dynamic Response of Immature Bovine Articular Cartilage in Tension and Compression, and Nonlinear Viscoelastic Modeling of the Tensile Response

[+] Author and Article Information
Seonghun Park

Columbia University, Departments of Mechanical Engineering and Biomedical Engineering, 500 W. 120th st., MC 4703, New York, NY 10027

Gerard A. Ateshian

Columbia University, Departments of Mechanical Engineering and Biomedical Engineering, 500 W. 120th st., MC 4703, New York, NY 10027ateshian@columbia.edu

J Biomech Eng 128(4), 623-630 (Jan 04, 2006) (8 pages) doi:10.1115/1.2206201 History: Received August 18, 2005; Revised January 04, 2006

## Abstract

Very limited information is currently available on the constitutive modeling of the tensile response of articular cartilage and its dynamic modulus at various loading frequencies. The objectives of this study were to (1) formulate and experimentally validate a constitutive model for the intrinsic viscoelasticity of cartilage in tension, (2) confirm the hypothesis that energy dissipation in tension is less than in compression at various loading frequencies, and (3) test the hypothesis that the dynamic modulus of cartilage in unconfined compression is dependent upon the dynamic tensile modulus. Experiment 1: Immature bovine articular cartilage samples were tested in tensile stress relaxation and cyclical loading. A proposed reduced relaxation function was fitted to the stress-relaxation response and the resulting material coefficients were used to predict the response to cyclical loading. Adjoining tissue samples were tested in unconfined compression stress relaxation and cyclical loading. Experiment 2: Tensile stress relaxation experiments were performed at varying strains to explore the strain-dependence of the viscoelastic response. The proposed relaxation function successfully fit the experimental tensile stress-relaxation response, with $R2=0.970±0.019$ at 1% strain and $R2=0.992±0.007$ at 2% strain. The predicted cyclical response agreed well with experimental measurements, particularly for the dynamic modulus at various frequencies. The relaxation function, measured from 2% to 10% strain, was found to be strain dependent, indicating that cartilage is nonlinearly viscoelastic in tension. Under dynamic loading, the tensile modulus at $10Hz$ was $∼2.3$ times the value of the equilibrium modulus. In contrast, the dynamic stiffening ratio in unconfined compression was $∼24$. The energy dissipation in tension was found to be significantly smaller than in compression (dynamic phase angle of $16.7±7.4deg$ versus $53.5±12.8deg$ at $10−3Hz$). A very strong linear correlation was observed between the dynamic tensile and dynamic compressive moduli at various frequencies $(R2=0.908±0.100)$. The tensile response of cartilage is nonlinearly viscoelastic, with the relaxation response varying with strain. A proposed constitutive relation for the tensile response was successfully validated. The frequency response of the tensile modulus of cartilage was reported for the first time. Results emphasize that fluid-flow dependent viscoelasticity dominates the compressive response of cartilage, whereas intrinsic solid matrix viscoelasticity dominates the tensile response. Yet the dynamic compressive modulus of cartilage is critically dependent upon elevated values of the dynamic tensile modulus.

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## Figures

Figure 1

Schematic of the custom-designed testing apparatus

Figure 2

Representative tensile stress-relaxation responses for a typical specimen from Experiment 1, showing the first and second tests at 1% and 2% strain. The solid curves represent the curve-fitted theoretical stress responses.

Figure 3

Mean and standard deviation of (a) E+Y, (b) α, (c) β, and (d) τ from tensile stress-relaxation tests at various applied strains. Experiment 1: ▴ for ‖ samples, ▵ for ⊥ samples. Experiment 2: ◆ for ‖ samples, ∎ for ⊥ samples. Additional symbols denote significant statistical differences against corresponding values at 2% (∗), 4% (†), and 6% (×) strain levels (p<0.05).

Figure 4

Tensile stress-strain response for a typical specimen at various loading frequencies (Experiment 1). Symbols represent the experimental response and solid curves represent the theoretical prediction, using material properties curve fitted from the corresponding stress-relaxation experiment for this specimen.

Figure 5

Experimental dynamic moduli ∣G*∣ (a) and phase angles ∠G* (b) in tension and compression, and corresponding theoretical predictions (tensile case), as a function of loading frequency. Tensile results are presented for ‖ samples only, as ⊥ samples exhibit a virtually identical response. Symbols denote significant statistical differences between dynamic tension and compression (#), and between experimental results and theoretical predictions (∴) (p<0.05).

Figure 6

Representative tensile stress-relaxation responses for a typical specimen from Experiment 2, showing the first of two tests at each of 2%, 4%, 6%, 8%, and 10% strain. The solid curves represent the curve-fitted theoretical stress responses.

Figure 7

Linear regression of unconfined compression equilibrium stress versus strain responses (mean and standard deviation over all cylindrical disks in Experiment 1). The slope represents the compressive equilibrium Young’s modulus, E−Y.

Figure 8

Parametric plot of compressive ∣G*∣ versus tensile ∣G*∣ for ‖ samples, at various frequencies (closed symbols). Open symbol represents the equilibrium response. A very similar correlation is observed with ⊥ samples. Means and standard deviations of the correlation coefficients are reported in the text.

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