On causality of the gradient elasticity models

Abstract

This paper is concerned with causality of the gradient elasticity models of heterogeneous materials. As a rule, these models are not strictly causal since they allow an infinite speed of energy transfer by means of either propagating or transient evanescent waves. A discussion is presented in this paper of both physical and mathematical implications of this fact. This discussion is carried out employing one-dimensional (1D) second-order gradient models. A phenomenological enhancement is proposed, which makes these models causal. The main idea behind this enhancement is that a partial differential equation that governs dynamic behaviour of a causal gradient elasticity model must be of the same order with respect to spatial coordinate and with respect to time. The validity of this idea is confirmed in this paper by deriving a second-order 1D continuum model for concrete. A brief comparison is provided in conclusion of the equations of motion of the 1D second-gradient elasticity models and those used in dynamics of thin bars. It is shown that the proposed causal model corresponds to the most advanced dynamic models of such bars, namely to the Timoshenko model for the bending motion and the Mindlin model for the longitudinal motion of a bar.