On Monotonicity for Strain-Limiting Theories of Elasticity

In a previous paper, we studied 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 that paper requires that the Fréchet derivative of the response function be invertible as a fourth-order tensor. In this paper, 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.