ON SHEAR-FLOW LOCALIZATION WITH TRACTION-CONTROLLED BOUNDARIES

The interactive roles of inertia and material viscosity as regards the evolution of inhomogeneous plastic flow are analyzed. The analysis is presented in the context of the dynamic, one-dimensional simple shear of a thermo-viscoplastic material subjected to traction-controlled bo undaries. Existence and uniqueness questions of an exact homogeneous s olution for this initial boundary-value problem are investigated, The breakdown of the so-called quasi-static homogeneous solution is relate d to the onset of localization. We introduce a dimensionless number, c alled the deformation number, and denoted by R(D), as the ratio of ine rtial to viscous stresses. Characterization of a given deformation as being dynamic is shown to be related to large values of R(D) instead o f simply high rates of applied loading. A model problem is formulated in order to illustrate the basic features of solutions for this class of deformations. An exact solution is derived for the model problem as well as a solution based on matched asymptotic expansions. It is show n, based on the model problem and fully non-linear finite difference s olutions, that plastic deformation localizes within narrow bands in th e neighborhood of the boundaries. The shear band thickness is inversel y proportional to the square root of the deformation number. The role of material viscosity concerning the introduction of a length scale to dynamic deformations of rate-dependent solids is illustrated.