Metrikine A.V.


On causality of the gradient elasticity models


Journal of Sound and Vibration 297 (2006) 727-742



Summary. This paper is concerned with causality of the gradient elasticity models of heterogeneous materials. As a rule, these models are not causal since they allow an infinite speed of the energy transfer by means of either propagating or transient evanescent waves. A discussion is presented in this paper of both the physical and mathematical implications of this fact. This discussion is carried out employing one-dimensional (1D) second-order gradient models. A phenomenological enhancement is further proposed, which makes these models causal. Not only the proposed model complies with the causality requirement, it also helps to estimate the high-order elastic parameters, which are difficult to measure. This is demonstrated by using concrete as an example of the heterogeneous materials. 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 Timoshenkoís model for the bending motion and the Mindlinís model for the longitudinal motion of a bar.