© 1992 by Oxford University Press
DEFORMATIONS OF AN ELASTIC, INTERNALLY CONSTRAINED MATERIAL PART 2: NONHOMOGENEOUS DEFORMATIONS
(
Department of Engineering Mechanics, University of Nebraska-Lincoln Lincoln, Nebraska 68588–0347, USA
Department of Mathematical Physics, University College Dublin 4
)
This is a sequel to part 1 in which a theory describing the nonlinear mechanical response of a class of homogeneous, isotropic elastic materials subject to finite deformations and characterized by Bell's constraint tr V = 3 was presented. Although Bell discovered this constraint within the context of experimental work in finite-strain plasticity, in part 1 it was examined for homogeneous deformations in the context of finite elasticity. Here the emphasis is on non-homogeneous, finite deformations for isotropic, homogeneous, elastic Bell-constrained bodies. Six families of solutions are presented. They include the bending and stretching of a rectangular block; the straightening of a cylindrical sector into a rectangular block; the bending with axialstretch of one circular cylindrical wedge into a similar wedge; the inflation, bending, extension, and azimuthal shearing of an annular wedge; the radial deformation of a spherical shell and membrane-including inflation, compression, and eversion; and, finally, the equibiaxial stretch and sinusoidal shear of a block.
Bell's constraint, tr V = 3, which has to be satisfied for every deformation, plays acentral role in the determination of which deformations are allowable. One surprising consequence of the Bell constraint is that a thick spherical shell may not be everted when the inner radius is more than half the outer radius. For each deformation the corresponding strains are determined, the equilibrium equations are solved in the absence of body forces, the corresponding stress fields are presented, and some physical results are illustrated.
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