Energy Dissipation on Transition from Reversible to Irreversible Deformation

Abstract

In the mechanics of a deformable solid, explanations of the transition from reversible to irreversible deformation lack adequate mathematical basis. In the present work, the observable phenomena are described on the basis of energy principles in mechanics. Two models are considered. The first provides a two-stage description of deformation that is uniform over the volume, with linear extension of a uniform sample that has isotropic properties. In the first stage, the familiar Lagrange equations of motion are employed. The relation between the longitudinal and transverse strains is determined by Poisson’s ratio. In the second stage, after the critical state has been reached, the deformation remains uniform; the equations of motion are similar to those in the first stage, but the relation between the longitudinal and transverse strains is different, facilitating the restoration of the initial particle volume. Decrease is noted in the energy of the particles determined by their change in volume and shape. The excess energy is liberated to the surroundings as heat. In the second model, the deforming body is assumed to be an ideal rigid–plastic material. The initial undeformed sample passes to the plastic state when the tangential stress reaches the critical value. The position of the shear planes is determined from the extremal principles of plasticity theory. The most likely motion is slip along planes whose normal is inclined at 45° to the axis of maximum normal stress. On account of the change in stress state after the formation of primary slip bands, successive formation of several other families of slip planes is possible. Shear in the second and then the third and subsequent families will require successively less energy. However, it is impossible for several families of slip planes to exist simultaneously, since decrease in the forces stops the slipping along the initial plane. Heat sources on the slip planes lead to energy dissipation and decrease in the forces. The further development of deformation calls for increase in the forces to the critical level corresponding to initiation of the first stage. Both models are consistent with experimental observations of irreversible deformation. In particular, in static extension during plane deformation, sample failure most often occurs at an angle of around 21°.