Interaction of deformation waves and localization phenomena in inelastic solids

Abstract

The main objective of this paper is the investigation of the interaction and reflection of elastic–viscoplastic waves which can lead to localization phenomena in solids. The rate type constitutive structure for an elastic–viscoplastic material with thermomechanical coupling is developed. An adiabatic inelastic flow process is considered. The Cauchy problem is investigated and the conditions for well-posedness are examined. Discussion of fundamental features of rate-dependent plastic medium is presented. This medium has dissipative and dispersive properties. Mathematical analysis of the evolution problem (the dynamical initial-boundary value problem) is presented. The dispersion property implies that in the viscoplastic medium any initial disturbance can break up into a system of group of oscillations or wavelets. On the other hand, the dissipation property causes the amplitude of a harmonic wavetrain to decay with time. In the evolution problem considered in such dissipative and dispersive medium, the stress and deformation due to wave reflections and interactions are not uniformly distributed, and this kind of heterogeneity can lead to strain localization in the absence of geometrical or material imperfections. Since the rate-independent plastic response is obtained as the limit case, when the relaxation time T m tends to zero, the theory of viscoplasticity offers the regularization procedure for the numerical solution of the dynamical initial-boundary value problems with localization of plastic deformation. Numerical examples are presented for a steel bar axisymmetric specimen subjected to tension, with the controlled displacements imposed at one or two opposite sides with different velocities. Two cases of the initial-boundary conditions are considered; (A) symmetric (double side) tension of the specimen which results in symmetric pattern of deformations; (B) asymmetric (single side) tension of the specimen with the opposite side fixed, which leads to non-symmetric deformation. For both cases of boundary conditions a set of examples is computed with different initial velocities changing between 0.5 and 20 m/s. The final states are defined by prescribed value of the total elongation of a specimen. In the numerical examples the attention is focused on the investigation of the interactions and reflections of waves and on the location of localization of plastic deformation. The distribution of plastic equivalent strain, temperature and vector plots of velocities represents the results. The computations are performed using the industrial finite element program ABAQUS (explicit method).