On Strong Ellipticity and Monotonicity for Implicit and Strain-Limiting Theories of Elasticity

Abstract

There has been increasing interest in the archival literature devoted to the study of implicit constitutive theories for non-dissipative materials generalizing the classical Green and Cauchy notions of elasticity, and for the special case of strain limiting models for which strains remain bounded, even infinitesimal, while stresses can become arbitrarily large. The first main part of this dissertation addresses the question of strong ellipticity for several classes of these models. A general approach for studying strong ellipticity for implicit theories is introduced and it is noted that there is a close connection between the questions of strong ellipticity and the existence of an equivalent Cauchy elastic formulation. For most of the models studied to date, it is shown that strong ellipticity holds if the Green-St.Venant strain is small enough, whereas it fails to hold for large strain. The large strain failure of strong ellipticity is generally associated with extreme compression.

Note that in the first main part of this dissertation, we study strong ellipticity for explicit strain-limiting theories of elasticity where the Green-St. Venant strain tensor is defined as a nonlinear response function of the second Piola-Kirchhoff stress tensor. The approach to strong ellipticity studied in the first main part of this dissertation requires that the Fréchet derivative of the response function be invertible as a fourth-order tensor. In the second main part of this dissertation, a weaker convexity notion is introduced in the case that the Fréchet derivative of the response function either fails to exist or is not invertible. We generalize the classical notion of monotonicity to a class of nonlinear strain-limiting models. It is shown that the generalized monotonicity holds for sufficiently small Green-St. Venant strains and fails (through demonstration by counterexample) when the small strain constraint is relaxed.