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Ellipticity and Deformations with Discontinous Gradients in Finite Elastostatics


Rosakis, Phoebus (1989) Ellipticity and Deformations with Discontinous Gradients in Finite Elastostatics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/jn2t-m109.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Applied Mechanics
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mechanics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Knowles, James K.
Thesis Committee:
  • Knowles, James K. (chair)
  • Sternberg, Eli
  • Beck, James L.
  • Caughey, Thomas Kirk
  • Rice, James R.
Defense Date:29 September 1988
Record Number:CaltechETD:etd-02122007-094508
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:612
Deposited By: Imported from ETD-db
Deposited On:12 Feb 2007
Last Modified:19 Oct 2021 23:11

Thesis Files

PDF (Rosakis_p_1989.pdf) - Final Version
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