On the role of inhomogeneities in the deformation of elastic bodies. (English) Zbl 1059.74015

Summary: It is quite common to approximate “mildly” inhomogeneous bodies as homogeneous bodies belonging to a certain constitutive class in view of the simplification that such an approximation provides. In this study, we investigate the consequences of such an assumption, and we show that it is clearly inappropriate for many classes of inhomogeneous bodies. We choose specific boundary value problems to illustrate the fact that we could be grossly in error, both qualitatively and quantitatively, with regard to local measures such as stresses and strains. In the examples considered, we find that, for global quantities such as applied forces and moments, the error could be significant. Not only could the material parameters found from, say an extension test and torsion test, which neglect the inhomogeneity of the body, be quite different from one that incorporates the inhomogeneity but also the values for the material parameter in the homogenized approximation gleaned from these different experiments could be different. In the process of elucidating our thesis, we investigate an important class of deformations, which in view of the paucity of boundary value problems that have been solved for nonlinear inhomogeneous solids is worth documenting in its own right.


74E05 Inhomogeneity in solid mechanics
74B20 Nonlinear elasticity
Full Text: DOI


[1] [4] Ericksen, J.L.: Deformations possible in every compressible isotropic perfectly elastic material . J. Math. Phys., 34, 126-128 (1955). · Zbl 0064.42105
[2] [5] Ericksen, J.L.: Deformations possible in every isotropic incompressible perfectly elastic body . Z. Angew. Math. Phys., 5, 466-486 (1954). · Zbl 0059.17509
[3] [6] Carroll, M.M.: Controllable deformations of incompressible simple materials . J. Elasticity, 5, 515-525 (1967).
[4] [7] Wineman, A.S. Universal deformations of incompressible simple materials. University of Michigan Technical Report, Ann Arbor (1967).
[5] [8] Fosdick, R.L.: Dynamically possible motions of incompressible isotropic simple materials . Arch. Ration. Mech. Anal., 29(4), 272 (1968). · Zbl 0164.27305
[6] [9] McLeod J.B. and Rajagopal, K.R.: Inhomogeneous non-unidirectional deformations of a wedge of a non-linearly elastic material . Arch. Ration. Mech. Anal., 147(3) 179-196 (1999). · Zbl 0991.74015
[7] [10] Treloar, L.R.G.: The elasticity of a network of long chain molecules - II . Trans. Faraday Soc., 39(9-10), 241-246 (1943).
[8] [11] Fung, Y.C.: Elasticity of soft tissues in elongation . Am. J. Physiol., 213, 1532-1544 (1967).
[9] [12] Corless, R.M., Ginnet, G.H., Hare, D.E.G., Jeffrey, D.J., and Knuth, D.E.: On the Lambert W function . Adv. Comput. Math., 5, 329-359 (1996). · Zbl 0863.65008
[10] [14] Saravanan, U. and Rajagopal K.R.: A comparison of the response of inhomogeneous isotropic elastic cylindrical and spherical shells and their homogenized counterparts . J. Elasticity, in press. · Zbl 1156.74352
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.