A note on the wave equation for a class of constitutive relations for nonlinear elastic bodies that are not Green elastic

Abstract

A class of constitutive relations for elastic bodies has been proposed recently, where the linearized strain tensor is expressed as a nonlinear function of the stress tensor. Considering this new type of constitutive equation, the initial boundary value problem for such elastic bodies has been expressed only in terms of the stress tensor. In this communication, this new type of nonlinear wave equation is studied for the case of a one-dimensional straight bar. Conditions for the existence of the travelling wave solutions are given and some self-similar solutions are obtained.

Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity

“…The strain-limiting models (4.2) and (4.4) correspond to the nonlinear constitutive relations proposed in [22] and [10], respectively. The model (4.3) has been recently proposed in [15]. The crucial fact about the above three cases is that we have h ′ (S) > 0 for all S. This guarantees the existence of the inverse function h −1 (ω) = g(ω) with g(0) = 0 on a suitable interval of ω.…”

Viscoelasticity with limiting strain

“…which is a higher order equation for the stress highlighting the fact that the shear stress is governed by a nonlinear hyperbolic equation. Some other wave propagation related studies also exist such as [9,26,30,32,55], and they will be investigated in more detail in Section 6 since they are dissipative. Fracture, as mentioned in the introduction, is one of the principle issues urging the necessity to understand the nonlinear relationship between the stress and the strain when the strain remains bounded, possibly infinitesimal, while the stress can become arbitrarily large.…”

“…where ℵ and ϑ are positive constants. In fact, as explained by Bustamante et al [32], relation (23) is proposed as an approximation of the expression…”

“…which was introduced and studied in [8,42], where α, β, γ and ι are constants. Clearly, as explained in [32], equation (24) models the response of an elastic body for which the strain remains small independently of the magnitude of the stress. Moreover, the relation (23) is more reasonable than (24) to study the qualitative properties of the implicit solutions.…”

“…Moreover, the relation (23) is more reasonable than (24) to study the qualitative properties of the implicit solutions. This type of relation, together with an exponential and polynomial type nonlinearities, are discussed by Bustamante et al [32] where the particular case of a one-dimensional bar is considered and boundary-value problems are analyzed in an elastic setting.…”