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TECHNICAL PAPERS

A Nonlinear Generalized Maxwell Fluid Model for Viscoelastic Materials

[+] Author and Article Information
D. T. Corr

Departments of Orthopedic Surgery and Mechanical Engineering, University of Wisconsin, G5/332 Clinical Sciences Center, 600 Highland Avenue, Madison, WI 53792-3228

M. J. Starr

Department of Engineering Physics, University of Wisconsin, 538 Engineering Research Building, 1500 Engineering Drive, Madison, WI 53706

R. Vanderby

Departments of Orthopedic Surgery, Mechanical Engineering and Biomedical Engineering, University of Wisconsin, G5/332 Clinical Sciences Center, 600 Highland Avenue, Madison, WI 53792-3228

T. M. Best

Departments of Family Medicine and Orthopedic Surgery, University of Wisconsin Medical School, 621 Science Drive, Madison, WI 53711

J. Appl. Mech 68(5), 787-790 (Apr 26, 2001) (4 pages) doi:10.1115/1.1388615 History: Received September 20, 2000; Revised April 26, 2001
Copyright © 2001 by ASME
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References

Lockett, F. J., 1972, Nonlinear Viscoelastic Solids, Academic Press, London.
Forcinito,  M., Epstein,  M., and Herzog,  W., 1998, “Can a Rheological Muscle Model Predict Force Depression/Enhancement?” J. Biomech., 31, pp. 1093–1099.
Lakes, R. S., 1999, Viscoelastic Solids, CRC Press, Boca Raton, FL.

Figures

Grahic Jump Location
The proposed nonlinear model, consisting of a parallel arrangement of a linear spring (k1) and a second-order spring (k2), in series with a linear dashpot (c).
Grahic Jump Location
Parameter sensitivity illustrated via force-time plots of numerical solutions (hollow squares), positive root (thick gray line), and negative root closed-form solutions (black line). (a) Three decades of linear spring stiffness, k1=0.1, 1, 10 N/cm (α=10 cm/s,k2=1 N/cm2,c=1 N⋅s/cm). As k1 increases the slope increases and the nonlinearity of the low load region becomes less prominent. Variation of the linear stiffness (k1) has no effect on the peak load value, however, increasing k1 shortens the time needed to reach it. (b) Three decades of second-order spring stiffness, k2=0.1, 1, 10 N/cm2 (α=10 cm/s,k1=1 N/cm,c=1 N⋅s/cm). As k2 increases the slope increases and the nonlinearity of the low load region becomes more prominent. Variation of k2 has no effect on the maximum load value, however, increasing k2 shortens the time needed to reach the peak load. (c) Varying displacement rates, α=5, 10, 20 cm/s (k1=1 N/cm,k2=1 N/cm2,c=1 N⋅s/cm). Note that the stiffness increases with increasing α, and the load approaches a value of c⋅α as time evolves. (d) Three decades of dashpot values, c=0.1, 1, 10 N⋅s/cm (α=10 cm/s,k1=1 N/cm,k2=1 N/cm2). An increase in the viscous damping, c, increases the value of the peak load as well as the time required to reach maximum force.
Grahic Jump Location
Force-time plot for α=10,k1=1 N/cm,k2=0.5 N/cm2,c=0.001 N⋅s/cm. Note the very close agreement between the numerical solution (thin black line) and the positive root (thick gray line) closed-form solutions until t>0.1 sec, at which the numerical solution becomes unstable.

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