Banding and stability in plastic materials

Summary

In this paper we have examined the question ofmaterial stability. More specifically, we have probed the conditions under which materials become “unstable” in the sense that given a homogeneous stress field in a materials domain, their constitutive equation allows the onset of a discontinuous, yet compatible, strain field in a finite subdomain (a “stress band”)while the state of stress in the entire domain remains unchanged. The central conclusion is that plastic materials are unstable in all inelastic stress states.

This is in the light of the fact that our definition of instability is more stringent than that of Thomas and Hill where the stress state in the band need only satisfy conditions of continuity normal to the band interface. Detailed results of this investigation in so far as plastic (inelastic) solids are concerned, where the plastic strain rate tensor is proportional to the gradient of a plastic potential with respect to the stress tensor, are summarized below.

• (i) Plastic materials (soils included), whose stress state is an ultimate state, are unstable in the sense that conditions exist for the onset of a stress band. Contrary to inferences drawn from the current literature, the inclination of the band to the stress fieldis not determinate without specific assumptions regarding the state of strain in the band.

• (ii) An ultimate state of stress is not necessary for instability. More specifically, constitutive equations that are “unconditionally” stable, in the sense that they satisfy the usual thermodynamic inequalitiesand Drucker's stability postulate, allow instabilities to occur atany stage of the loading in theinelastic state. Again, the inclination of the stress band relative to the stress field is, in general, indeterminate in the sense of (i).

• (iii) As a corollary of (ii), banding instabilityis not a constitutive property, as is currently believed, andmaterial softening is not a prerequisite for the onset of instability.

• (iv) Very much like fracture, banding behavior lies outside the domain of present constitutive theory and “a theory of stability” — i.e., a theory of banding — is necessary to explain the unstable behavior of materials (soils and metals).

• (v) Lacking such a theory, all inelastic stress states areat risk, since the onset of instability in such states is always imminent and, at present, unpredictable.