A nonlinear diffusion equation arising in the study of soft tissue
Author:
Mark H. Holmes
Journal:
Quart. Appl. Math. 41 (1983), 209-220
MSC:
Primary 73P05; Secondary 76Z05, 92A09
DOI:
https://doi.org/10.1090/qam/719505
MathSciNet review:
719505
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Abstract: A nonlinear diffusion equation which describes the deformation of a soft fluid-filled tissue is studied. The nonlinearity in the problem arises from the permeability of the elastic phase which for a number of tissues, such as articular cartilage, is strongly dependent on the strain. Moreover, for most tissues the exact dependence is not known, and so the functional dependence of the permeability on the strain is not determined until after the problem is solved. The approach uses perturbation methods for the diffusive boundary layers that occur in the problem and similarity solutions to solve the reduced problems. Once the solution is obtained, the permeability function is determined and some of the limitations of the present model of soft tissue are discussed.
R. M. Bowen, Theory of mixtures, in Continuum physics (A. E. Erigen, ed.), Academic Press, New York, 1976, pp. 1–127
R. E. Craine, A. E. Green, and P. M. Naghdi, A mixture of viscous elastic materials with different constituent temperatures, Q. J. Appl. Math. 23, 171–184 (1970)
W. M. Lai and V. C. Mow, Drag-induced compression of articular cartilage during a permeation experiment, Biorheology 17, 110–123 (1980)
W. M. Lai, V. C. Mow, and V. Roth, Effects of a nonlinear strain-dependent permeability and rate of compression on the stress behavior of articular cartilage, J. Biomech. Eng. 103K, 61–66 (1981)
V. C. Mow, S. C. Kuei, W. M. Lai, and C. G. Armstrong, Biphasic creep and stress relaxation of articular cartilage: theory and experiments, J. Biomech. Eng. 102K, 73–84 (1980)
J. J. Shi, K. K. Rajagopal, and A. S. Wineman, Application of the theory of interacting continua to the diffusion of a fluid through a non-linear elastic media, Int. J. Eng. Sci. 19, 871–889 (1981)
R. M. Bowen, Theory of mixtures, in Continuum physics (A. E. Erigen, ed.), Academic Press, New York, 1976, pp. 1–127
R. E. Craine, A. E. Green, and P. M. Naghdi, A mixture of viscous elastic materials with different constituent temperatures, Q. J. Appl. Math. 23, 171–184 (1970)
W. M. Lai and V. C. Mow, Drag-induced compression of articular cartilage during a permeation experiment, Biorheology 17, 110–123 (1980)
W. M. Lai, V. C. Mow, and V. Roth, Effects of a nonlinear strain-dependent permeability and rate of compression on the stress behavior of articular cartilage, J. Biomech. Eng. 103K, 61–66 (1981)
V. C. Mow, S. C. Kuei, W. M. Lai, and C. G. Armstrong, Biphasic creep and stress relaxation of articular cartilage: theory and experiments, J. Biomech. Eng. 102K, 73–84 (1980)
J. J. Shi, K. K. Rajagopal, and A. S. Wineman, Application of the theory of interacting continua to the diffusion of a fluid through a non-linear elastic media, Int. J. Eng. Sci. 19, 871–889 (1981)
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Article copyright:
© Copyright 1983
American Mathematical Society
