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Non-uniform straining of rate-sensitive materials with non-monotone yield stress. (English) Zbl 0899.35008

Summary: The problem of shearing of a ductile material exhibiting strain-rate sensitivity is analyzed here. The material considered has a strain softening response, after a certain amount of plastic flow, where it is expected that the non-uniform straining destablilizes the process. We first establish the nice behavior of the dynamic plastic solution in the region of positive slopes of the yield stress. The cases of loading and unloading are analyzed separately. Emphasis is given on the parameters that influence the critical time. We present analytical steady strain solutions to illustrate the shear banding once the threshold of strain, corresponding to the maximum of yield stress, is exceeded. Moreover, we use a linear perturbation method to show that the time-dependent homogeneous solution is not stable after this threshold.

MSC:

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
35B40 Asymptotic behavior of solutions to PDEs
76E30 Nonlinear effects in hydrodynamic stability
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