On strong ellipticity for implicit and strain-limiting theories of elasticity

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. This paper 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.