On a class of Gibbs potential-based nonlinear elastic models with small strains
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2014, Springer-Verlag Wien. The aim of this paper is to show that it is possible to develop a fully internally consistent model for an isotropic elastic body that has nonlinear response relations but with small strains, if one chooses to begin from a Gibbs potential (or complementary energy) rather than the classical Helmholtz potential (or strain energy). We demonstrate the practical applicability of such a model and also compare the results of exact solutions (internal pressurization of a circular cylinder and the bending of a rectangular block) for incompressible materials undergoing finite deformations with corresponding exact solutions based on small deformations. Surprisingly, the results show quite remarkable agreement up to maximum strains of nearly 2030%. Given that most applications of nonlinear elastic materials are in this range, the results have broad implications for the ability to consistently use small deformation nonlinear theories for a wide class of applications than hitherto considered. We also show that the nonlinear compatibility equations play a key role in determining the applicability of this approach. In particular, we show that for 3-D bodies whose dimensions are of comparable orders of magnitude (i.e., excluding rod-like or shell-like bodies), the finite deformation compatibility condition ensures that small strains and strain gradients imply small relative rotations and hence small deformations. For thin bodies (rods, plates and shells), it is well-known result that small strains do not lead to small deformations due to the possibility of large rotation gradients thus leading to other possible approximations.
