No abstract

No abstract

Gradient elasticity theories can be used to simulate dispersive wave propagation as it occurs in heterogeneous materials. Compared to the second-order partial differential equations of classical elasticity, in its most general format gradient elasticity also contains fourth-order spatial, temporal as well as mixed spatial temporal derivatives. The inclusion of the various higher-order terms has been motivated through arguments of causality and asymptotic accuracy, but for numerical implementations it is also important that standard discretization tools\ud can be used for the interpolation in space and the integration in time. In this paper, we will formulate four different simplifications of the general gradient elasticity theory. We will study the dispersive properties of the models, their causality according to Einstein and their behavior in simple initial/boundary value problems

SUMMARYIn this article a special form of gradient elasticity is presented that can be used to describe wave dispersion. This new format of gradient elasticity is an appropriate dynamic extension of the earlier static counterpart of the gradient elasticity theory advocated in the early 1990s by Aifantis and co-workers. In order to capture dispersion of propagating waves, both higher-order inertia and higher-order stiffness contributions are included, a fact which implies (and is denoted as) dynamic consistency. The two higher-order terms are accompanied by two associated length scales. To facilitate finite element implementations, the model is rewritten such that C 0 -continuity of the interpolation is sufficient. An auxiliary displacement field is introduced which allows the original fourth-order equations to be split into two coupled sets of secondorder equations. Positive-definiteness of the kinetic energy requires that the inertia length scale is larger than the stiffness length scale. The governing equations, boundary conditions and the discretized system of equations are presented. Finally, dispersive wave propagation in a one-dimensional bar is considered in a numerical example.

No abstract