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Technical Briefs

Experimental Characterization and Finite Element Implementation of Soft Tissue Nonlinear Viscoelasticity

[+] Author and Article Information
Kevin L. Troyer, Snehal S. Shetye

 Department of Mechanical Engineering, Colorado State University, Fort Collins, CO 80523-1374

Christian M. Puttlitz1

 Department of Mechanical Engineering, School of Biomedical Engineering, Department of Clinical Sciences, Colorado State University, Fort Collins, CO 80523-1374puttlitz@engr.colostate.edu

1

Corresponding author.

J Biomech Eng 134(11), 114501 (Oct 11, 2012) (8 pages) doi:10.1115/1.4007630 History: Received April 04, 2012; Revised September 04, 2012; Posted September 25, 2012; Published October 11, 2012; Online October 11, 2012

Finite element (FE) models of articular joint structures do not typically implement the fully nonlinear viscoelastic behavior of the soft connective tissue components. Instead, contemporary whole joint FE models usually represent the transient soft tissue behavior with significantly simplified formulations that are computationally tractable. The resultant fidelity of these models is greatly compromised with respect to predictions under temporally varying static and dynamic loading regimes. In addition, models based upon experimentally derived nonlinear viscoelastic coefficients that do not account for the transient behavior during the loading event(s) may further reduce the model’s predictive accuracy. The current study provides the derivation and validation of a novel, phenomenological nonlinear viscoelastic formulation (based on the single integral nonlinear superposition formulation) that can be directly inputted into FE algorithms. This formulation and an accompanying experimental characterization technique, which incorporates relaxation manifested during the loading period of stress relaxation experiments, is compared to a previously published characterization method and validated against an independent analytical model. The results demonstrated that the static and dynamic FE approximations are in good agreement with the analytical solution. Additionally, the predictive accuracy of these approximations was observed to be highly dependent upon the experimental characterization technique. It is expected that implementation of the novel, computationally tractable nonlinear viscoelastic formulation and associated experimental characterization technique presented in the current study will greatly improve the predictive accuracy of the individual connective tissue components for whole joint FE simulations subjected to static and dynamic loading regimes.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

A five-component spring and dashpot nonlinear viscoelastic mechanical model. Nonlinearity of the spring constants E∞(ɛ), E1(ɛ), E2(ɛ), E3(ɛ), and E4(ɛ) was modeled via a quadratic polynomial (Eq. 3). The dashpots are characterized by their respective time constants τi (i = 1, 2, 3, 4).

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Figure 2

(a) A typical relaxation period curve fit (CVC method wRMSE = 2.62 kPa, 2.5t0 method wRMSE = 1.33 kPa); (b) the corresponding full (ramping and relaxation period) analytical and FE curve predictions (CVC method wRMSE = 54.21 kPa, 2.5t0 method wRMSE = 182.51 kPa); and (c) the ramping period only predictions. The finite element model closely approximated the analytical solution of both characterization methods.

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Figure 3

Average strain-dependent behavior of each moduli component with its corresponding fitted equation (Eq. 3). The data indicated that the E1(ɛ) coefficient (corresponding to τ1  = 0.1 s) was highly dependent upon the fitting technique (Table 2). Error bars represent one standard deviation.

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Figure 4

Average 3% strain amplitude cyclic behavior and the corresponding analytical and FE predictions from the two fitting techniques for (a) the first three cycles and (b) the last full cycle. Although the 2.5t0 method predictions closely approximated the loading phase of the dynamic behavior, its peak stress magnitude was out-of-phase from the average experimental data. Conversely, the CVC method predictions were within the bounds of experimental variability and in-phase with the average experimental data.

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Figure 5

Average 6% strain amplitude cyclic behavior and the corresponding analytical and FE predictions from the two fitting techniques for (a) the first three cycles and (b) the last full cycle. The 2.5t0 method predictions poorly approximated both the magnitude of the average experimental data and its phase. Conversely, the CVC method predictions were within the bounds of experimental variability and in-phase with the average experimental data.

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