Instability, softening and localization of deformation

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Abstract
Material strain softening is commonly taken as a necessary and sufficient condition for localization in deforming rocks. However, there is a wide range of experimental and theoretical information which shows that localization can occur in sands, brittle rocks and ductile metals under strain-hardening conditions. This paper aims to bring these two contrasted views together. Three separate criteria are necessary in order to understand localization behaviour. The first involves the stability of the deforming system. The second determines whether a deforming system will undergo bifurcation so as to cease deforming in a homogeneous mode and instead deform in an inhomogeneous mode such as barrelling or localization. The stability and bifurcation criteria are independent of each other since barrelling is a stable mode whereas localization is unstable. The third criterion establishes if the unstable bifurcation mode is one of localization or of some other kind. Localization may arise from the presence of vertices on the yield surface (as in the case of pressure insensitive, rate dependent metals and in brittle rocks due to the development of preferred microfractures for slip) or from the constitutive relation being such that the plastic strain-rate vector is not normal to the yield surface (as in the cases of pressure sensitive, dilatant rocks, of materials deforming by crystal-plastic processes involving dislocation cross-slip and/or climb, and of visco-plastic materials in which voids are forming due to diffusive processes). It is important to distinguish between material and system softening (or hardening) behaviour. The theory for a kinematically unconstrained shortening experiment (that is, rigid, frictionless platens) indicates that localization can occur in strain-hardening materials but the system must strain-soften from then on; that is, localization occurs at peak stress for the system even though the material may continue to harden (or soften). However, the addition of kinematic constraints (such as friction at elastic platens, a constraint to deform in plane strain or at constant volume) means that localization may occur in a system that is monotonically strain hardening. Shear zones in naturally deformed rocks show ample evidence of dilatant behaviour in that evidence for the passage of large volumes of fluid during localization is common as is the development of dilatant vein systems. As such, since shear zones are strongly constrained by the elastic and (limited) plastic response of the relatively undeformed rocks surrounding the shear zones, strain-hardening behaviour of the system is to be expected as the norm, even if the rocks within the shear zones are undergoing material strain-softening.