Accession Number : ADA201717

Title :   Ellipticity and Deformations with Discontinuous Gradients in Finite Elastostatics.

Descriptive Note : Technical rept.,

Corporate Author : CALIFORNIA INST OF TECH PASADENA DIV OF ENGINEERING AND APPLIED SCIENCE

Personal Author(s) : Rosakis, Phoebus

Report Date : SEP 1988

Pagination or Media Count : 60

Abstract : Loss of ellipticity of the equilibrium equations of finite elastostatics is closely related to the possible emergence of elastostatic shocks, i.e., deformations with discontinuous gradients. In certain situations where constitutive response functions are essentially one dimensional, such as anti-plane shear or bar theories, strong ellipticity is closely related to convexity of the elastic potential and invertibility of certain constitutive response functions. The present work addresses the analogous issues within the context of three-dimensional elastostatics of compressible but not necessarily isotropic hyperelastic materials. A certain direction dependent resolution of the deformation gradient is introduced and its existence and uniqueness for a given direction are established. The elastic potential is expressed as a function of kinematic variables arising from this resolution. Strong ellipticity is shown to be equivalent to the positive definiteness of the Hessian matrix of this function, thus sufficing for its strict convexity. The underlying variables are interpretable physically as simple shears and extensions. Their work-conjugates define a traction response mapping. It is shown that discontinuous deformation gradients are sustainable if and only if this mapping fails to be invertible. This result is explicit, in the sense that it characterizes the set of all possible piecewise homogeneous deformations, given the elastic potential function. (JHD)

Descriptors :   *DEFORMATION, *ELASTIC PROPERTIES, *RECOVERY, *SHOCK(MECHANICS), EQUATIONS, EQUILIBRIUM(GENERAL), GRADIENTS, HOMOGENEITY, KINEMATICS, MAPPING(TRANSFORMATIONS), RESPONSE, STATICS, THREE DIMENSIONAL, TRACTION.

Subject Categories : Mechanics

Distribution Statement : APPROVED FOR PUBLIC RELEASE