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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.

MSC:
74E05 Inhomogeneity in solid mechanics
74B20 Nonlinear elasticity
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