Vol. 5, No. 1, 2017

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Reducible and irreducible forms of stabilised gradient elasticity in dynamics

Harm Askes and Inna M. Gitman

Vol. 5 (2017), No. 1, 1–17

The continualisation of discrete particle models has been a popular tool to formulate higher-order gradient elasticity models. However, a straightforward continualisation leads to unstable continuum models. Padé approximations can be used to stabilise the model, but the resulting formulation depends on the particular equation that is transformed with the Padé approximation. In this contribution, we study two different stabilised gradient elasticity models; one is an irreducible form with displacement degrees of freedom only, and the other is a reducible form where the primary unknowns are not only displacements but also the Cauchy stresses — this turns out to be Eringen’s theory of gradient elasticity. Although they are derived from the same discrete model, there are significant differences in variationally consistent boundary conditions and resulting finite element implementations, with implications for the capability (or otherwise) to suppress crack tip singularities.

gradient elasticity, mixed formulation, length scale, nonlocal elasticity
Mathematical Subject Classification 2010
Primary: 74-XX
Received: 9 March 2016
Revised: 22 July 2016
Accepted: 26 September 2016
Published: 31 January 2017

Communicated by Francesco dell'Isola
Harm Askes
Department of Civil and Structural Engineering
University of Sheffield
S1 3JD
United Kingdom
Inna M. Gitman
Department of Mechanical Engineering
University of Sheffield
S1 3JD
United Kingdom