Stability of extensional plastic flow

As developed in Section 10.3, the stability of extensional flow in visco-plastic solids is governed by intrinsic properties of the solid, such as its plastic resistance, its strain-hardening rate and its strain-rate-hardening rate, through the sensitivity of the plastic resistance to the strain rate. In many instances, however, the deforming bar or fiber contains imperfections that can affect or hasten localization in necks and subsequent rupture. Such perturbations of flow by imperfections and their effect on material stability in extensional flow have been of great interest. A well-defined scenario of this was conceived by Hutchinson and Obrecht (1977) and further developed by Hutchinson and Neale (1977). 

following the development of Hutchinson and Obrecht (1977), we consider a perfect bar of a non-linear incompressible viscous solid as a reasonable approximation to a rate-dependent plastic material, in comparison with an imperfect bar of the same material with periodic undulations having a wave length 1, where the radius of the imperfect bar is given as 

The following represent the responses of the perfeet and imperfeet bars respeet-ively, under the same axial load L, 

Hutchinson and Obrecht (1977) demonstrated that, with increasing amplitude of imperfection, off-axis stresses in the imperfect bar must also be considered since they affect the long-wave-length solution of eq. (10.23), requiring a modification factor/(wr, ), giving 

The long-wave-length stability expressed in eq. (10.23) also states quite interestingly that for m= 1, i.e., a Newtonian viscous bar, 

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