Micromorphic elasticity for an axisymetrically loaded cylindrical thick-walled structure under plane strain conditions

Abstract

In this paper we derive a closed-form solution for the problem of a second gradient elastic circular thick-walled cylinder subjected to some axisymmetric loading conditions and deformed in plane strain. The second gradient elastic model used is the one of Gologanu, Leblond, Perrin and Devaux proposed some years ago and used in the context of the numerical implementation of a micromorphic model for ductile fracture in porous plastic metals. The ordinary Cauchy stress and 'hyperstress' tensors are computed as a function of the displacement field which is obtained via a suitable combination of the balance equations, compatibility, stress-strain/hyperstress-gradient of the strain relationships, and appropriate boundary conditions. The first gradient solution is recovered when the second gradient effects are negligible. This study demonstrates that the first gradient elastic solution may not be accurate, particularly in elastic materials exhibiting substantial microstructure dependence. Future works will use the newly derived solution to interpret the size effects experimentally observed, for instance, in pressurised sandstone hollow cylinders.