Comparison of Laplacian based explicit and implicit gradient elasticity with reference to one-dimensional problems in statics and dynamics

Abstract

It has been shown previously, that Laplacian based explicit and implicit gradient elasticity models can be derived as particular cases of Mindlin’s gradient elasticity and Mindlin’s micro-structured (or equivalently Eringen’s micromorphic) elastic materials. Also, they can be established as counterparts of viscoelastic solids, in view of an analogy to linear viscoelasticity based on mechanical elements. The present paper aims to compare responses predicted by the gradient elasticity counterpart of the three-parameter viscoelastic solid with those predicted by the gradient elasticity counterpart of the viscoelastic Kelvin model and by the classical elasticity. The comparison is mainly based on closed form solutions of one-dimensional problems in statics and dynamics. Specifically, the analysis includes static loading, axial vibrations and tension step pulse loading of a bar.