Burns, Timothy J. Similarity and bifurcation in unstable viscoplastic shear. (English) Zbl 0663.73023 SIAM J. Appl. Math. 49, No. 1, 314-329 (1989). 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. Cited in 3 Documents 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 Keywords:mixed initial-boundary value problem; infinite system of initial value problems; Fourier series solution; convergence; adiabatic shear; change of type; turning point; linear stability; perturbed simple shear; ductile plastic material; strain-soften; finite time interval; asymptotic study; small dimensionless coefficients; similarity parameter; minimum strain rate PDFBibTeX XMLCite \textit{T. J. Burns}, SIAM J. Appl. Math. 49, No. 1, 314--329 (1989; Zbl 0663.73023) Full Text: DOI