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Similarity and bifurcation in unstable viscoplastic shear. (English) Zbl 0663.73023

The linear stability of perturbed simple shear in a ductile plastic material which can strain-soften is analyzed on a finite time interval. An asymptotic study is made of the influence of two small dimensionless coefficients, which are shown to be connected by a similarity parameter. The first coefficient is a dimensionless group of parameters which can be interpreted as the ratio of the momentum flux to the plastic flow stress, while the second coefficient gives the scale of the viscous damping. A criterion is derived for the minimum strain rate where the early-time behavior changes from oscillatory to exponential.

MSC:

74C99 Plastic materials, materials of stress-rate and internal-variable type
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
35B32 Bifurcations in context of PDEs
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
35B20 Perturbations in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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