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Research Papers

# The Viscoelastic Standard Nonlinear Solid Model: Predicting the Response of the Lumbar Intervertebral Disk to Low-Frequency Vibrations

[+] Author and Article Information
Kevin M. Groth1

The Kevin P. Granata Musculoskeletal Biomechanics Laboratory, Department of Mechanical Engineering, School of Biomedical Engineering and Science, Virginia Polytechnic Institute and State University, 100 Randolph Hall (0238), Blacksburg, VA 24061kgroth@vt.edu

Kevin P. Granata

The Kevin P. Granata Musculoskeletal Biomechanics Laboratory, Department of Engineering Science and Mechanics, School of Biomedical Engineering and Science, Virginia Polytechnic Institute and State University, 100 Randolph Hall (0238), Blacksburg, VA 24061

1

Corresponding author.

J Biomech Eng 130(3), 031005 (Apr 22, 2008) (6 pages) doi:10.1115/1.2904464 History: Received January 10, 2007; Revised October 17, 2007; Published April 22, 2008

## Abstract

Due to the mathematical complexity of current musculoskeletal spine models, there is a need for computationally efficient models of the intervertebral disk (IVD). The aim of this study is to develop a mathematical model that will adequately describe the motion of the IVD under axial cyclic loading as well as maintain computational efficiency for use in future musculoskeletal spine models. Several studies have successfully modeled the creep characteristics of the IVD using the three-parameter viscoelastic standard linear solid (SLS) model. However, when the SLS model is subjected to cyclic loading, it underestimates the load relaxation, the cyclic modulus, and the hysteresis of the human lumbar IVD. A viscoelastic standard nonlinear solid (SNS) model was used to predict the response of the human lumbar IVD subjected to low-frequency vibration. Nonlinear behavior of the SNS model was simulated by a strain-dependent elastic modulus on the SLS model. Parameters of the SNS model were estimated from experimental load deformation and stress-relaxation curves obtained from the literature. The SNS model was able to predict the cyclic modulus of the IVD at frequencies of $0.01Hz$, $0.1Hz$, and $1Hz$. Furthermore, the SNS model was able to quantitatively predict the load relaxation at a frequency of $0.01Hz$. However, model performance was unsatisfactory when predicting load relaxation and hysteresis at higher frequencies ($0.1Hz$ and $1Hz$). The SLS model of the lumbar IVD may require strain-dependent elastic and viscous behavior to represent the dynamic response to compressive strain.

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## Figures

Figure 3

Stress relaxation of the IVD. The constant parameters (E1 and μ) for the Kelvin body of the SNS model were determined by least squares regression with the experimental stress-relaxation response of a lumbar IVD. Note that the experimental values used were estimated from graphical results reported by Holmes and Hukins.

Figure 4

Cyclic relaxation of the IVD models subjected to a frequency of 0.01Hz. Note the increase in peak-to-peak force of the SNS model compared to the SLS model.

Figure 1

SNS model. The arrangement of the standard model consists of a Kelvin body in series with a spring. The SNS model in the present study differs from the SLS model in that it replaces the series spring’s constant modulus with a strain-dependent modulus.

Figure 2

Instantaneous elastic response of the IVD. (a) Relationship for how the elastic modulus E2(e) changes with axial compressive strain e was found through a linear least squares regression. (b) The IVD experiences a nonlinear stress when subjected to compressive strain input. The nonlinear stress is due to an increase in stiffness as the intervertebral joint (IVJ) is compressed.

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