Localization phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings

Abstract

The main objective of the paper is the investigation of localization phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings. Recent experimental observations for cycle fatigue damage mechanics at high temperature and dynamic loadings of metals suggest that the intrinsic microdamage process does very much dependent on the strain rate and the wave shape effects and is mostly developed in the regions where the plastic deformation is localized. The micro damage kinetics interacts with thermal and load changes to make failure of solids a highly rate, temperature and history dependent, nonlinear process. A general constitutive model of elasto-viscoplastic damaged polycrystalline solids is developed within the thermodynamic framework of the rate type covariance structure with finite set of the internal state variables. A set of the internal state variables is assumed and interpreted such that the theory developed takes account of the effects as follows: (i) plastic non-normality; (ii) plastic strain induced anisotropy (kinematic hardening); (iii) softening generated by microdamage mechanisms (nucleation, growth and coalescence of microcracks); (iv) thermomechanical coupling (thermal plastic softening and thermal expansion); (v) rate sensitivity; (vi) plastic spin. To describe suitably the time and temperature dependent effects observed experimentally and the accumulation of the plastic deformation and damage during dynamic cyclic loading process the kinetics of micro damage and the kinematic hardening law have been modified. The relaxation time is used as a regularization parameter. By assuming that the relaxation time tends to zero, the rate independent elasticplastic response can be obtained. The viscoplastic regularization procedure assures the stable integration algorithm by using the finite difference method. Particular attention is focused on the well-posedness of the evolution problem (the initial-boundary value problem) as well as on its numerical solutions. The Lax-Richtmyer equivalence theorem is formulated and conditions under which this theory is valid are examined. Utilizing the finite difference method for regularized elasto-viscoplastic model, the numerical investigation of the three-dimensional dynamic adiabatic deformation in a particular body under cyclic loading condition is presented. Particular examples have been considered, namely dynamic, adiabatic and isothermal, cyclic loading processes for a thin steel plate with small rectangular hole located in the centre. To the upper edge of the plate the normal and parallel displacements are applied while the lower edge is supported rigidly. Both these displacements change in time cyclically. Small two asymmetric regions which undergo significant deformations and temperature rise have been determined. Their evolution until occurrence of final fracture has been simulated. The accumulation of damage and equivalent plastic deformation on each considered cycle has been obtained. It has been found that this accumulation distinctly depends on the wave shape of the assumed loading cycle.