Ellipticity and Deformations with Discontinous Gradients in Finite Elastostatics

Author: Rosakis, Phoebus

Year: 1989

Degree: Dissertation (Ph.D.)

Advisor: Knowles, James K.

Committee Members: Knowles, James K.; Sternberg, Eli; Beck, James L.; Caughey, Thomas Kirk; Rice, James R.

Option: Applied Mechanics

DOI: 10.7907/jn2t-m109

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-dimentional, 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.

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