Constitutively admissible gradients of plastic deformation in isotropic solids

The relatively new field of “gradient plasticity” is concerned with the modeling of length scale effects observed in plastically deformed solids. The older subject of inhomogeneity in materially uniform bodies, inaugurated by Noll, offers the possibility of constructing such models in terms of inhomogeneity fields. In the case of materially uniform crystalline solids, for example, inhomogeneity is described by the torsion tensor—equivalently, the “geometrically necessary” dislocation density—induced by plastic deformation. This is the unique descriptor of length-scale effects modeled by constitutive functions involving both the plastic deformation and its gradient. In isotropic solids, inhomogeneity is characterized by the Riemann tensor induced by the plastic metric. This involves the plastic deformation together with its first and second gradients. However, in the case of isotropy, it is an open question as to whether or not inhomogeneity furnishes a complete model of length scale effects described by functions of the plastic deformation and its gradients. In this note, we show that such functions are ultimately expressible in terms of the Riemann tensor alone, and hence, that scale effects are modeled by these functions via the inhomogeneity field.