Metrikine A.V.

On causality of the
gradient elasticity models 
Journal of Sound and Vibration
297 (2006) 727742 
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 onedimensional (1D)
secondorder 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
highorder 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 secondgradient 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’s model for
the bending motion and the Mindlin’s model for the
longitudinal motion of a bar. 