This paper is concerned with a possible form of material instability arising from a non-unique dependence of stress on deformation and leading to heterogeneous states of stress in otherwise homogeneous deformations. Following a discussion of certain theories and experiments that suggest this type of stress behavior, an analysis is given of some possible quasi-static bifurcations of steady-state stress fields for extensional motions of isotropic elastic solids and viscoelastic fluids. These involve extended lamellar and axisymmetric phase structures having common extension. For certain other restricted forms of non-linear elasticity, the classical Eshelby [Proc. R. Soc. London A 241 (1957) 376] theory is employed to describe dilute ellipsoidal phases having a deformation different from the globally imposed mean.