A new formulation and 𝒞0‐implementation of dynamically consistent gradient elasticity

Abstract

AbstractIn his 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 𝒞0‐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 second‐order 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. Copyright © 2007 John Wiley & Sons, Ltd.