Stress-gradient materials: an analytical exploration

Abstract : A new stress-gradient elasticity theory has recently been proposed by Forest & Sab (Mech. Res. Comm. 40:16--25, 2012), and further analyzed by Forest, Legoll & Sab (J. Elas., submitted). This new model makes the assumption that the complementary energy is a function of the local stress and (the deviatoric part of) its first gradient. Its derivation (including boundary conditions) relies on a rigorous variational approach. However, the resulting set of equations is rather complex in general. In order to better understand this model, it is proposed as a first step to consider a subclass of stress-gradient materials, assuming material isotropy and restricting to only one internal material length. In this talk, we propose an analytical exploration of this simplified model. We will produce the closed-form solution to Eshelby's spherical inhomogeneity problem. From the analysis of the corresponding local fields, we will show that stress- and strain-gradient elasticity theories are not in duality, as one would be tempted to think. This elementary solution will then be used to extend the micromechanical model of Mori and Tanaka to stress gradient materials. In particular, we will show that (contrary to strain-gradient materials) a decrease of the internal material lenth tends to stiffen the material.