Stability approach to testing the constitutive relations of superplasticity

Abstract

An approach to testing the suitability of the constitutive relations for describing superplasticity is proposed based on an analysis of the stability conditions of uniform extension of samples with respect to small perturbations of the current configuration. The necessity of accounting for the partial preservation of the local topology in varying the equations of motion is shown using a simple (nonlinear viscous) model of a superplastic medium.Formulation of the Problem. At present, the superplasticity phenomenon -the development of abnormally large irreversible strains at small loads -has been found for many polycrystalline materials of definite structure in standard tests (for example, in uniaxial tension) [1]. The complexity of this phenomenon makes it important to define and classify this effect in rational exact terms of theoretical physics in a broad sense and to develop procedures for testing the suitability of the constitutive relations describing superplastic deformation modes. The present paper deals with the well-known but insufficiently developed approach [2, 3] based on the characteristic property of superplastic deformation modes -the stability of the current body configuration with respect to small or finite perturbations. Below, we study uniaxial tension because of the simplicity of this process and because of the significant effect of free-surface perturbations on the behavior of the sample. An analysis is made of the stability of uniform deformation with respect to small perturbations. This yields "conservative" superplasticity conditions which do not rule out stable nonuniform deformation modes.To formulate the flow stability problem for a superplastic medium, whose rheology is dominated by a nonlinear viscous (with infinitesimal memory) component [1], it is necessary to analyze existing approaches. For this problem, the approach based on the Eulerian method of describing motion seems to be the most suitable for media that do not possess memory (or have infinitesimal memory) yields results inconsistent to experimental data. At the same time, the Lagrangian approach, which is usually employed to describe the behavior of materials with slowly decaying memory, provides calculation results in good agreement with experimental data. It has been established that the superplasticity mode is due (in a very narrow range of parameters) to the abnormally high sensitivity to the strain rate (in uniaxial macrouniform tension tests, the degree of sensitivity is determined by the parameter m = d ln σ/d ln ξ, where σ and ξ are the tensile stress and the rate deformation). For high-temperature creep of metals, m = 1/40-1/3, whereas in the case of superplasticity, it can reach values approximately equal to unity. We consider the following example. At the current time, let a rectangular sample in the shape of a strip of length l and width b be extended uniformly at a ratel. In addition, the sample is incompressible, i.e., ξ =l/l = −ḃ/b > 0. The medium is considered nonlinear viscous. Examining uniform perturbation...

Self-similar shapes of the free boundary of a nonlinear-viscous band under uniaxial tension

“…As in [5], the nonlinear rheological relations of an incompressible Reiner-Rivlin fluid are taken as the constitutive relations. The evolution of axially symmetric small perturbations of the free boundaries is investigated by the method of small parameter, which is the ratio of the perturbation amplitude to the band width.…”

“…Evolution of the free surface of a solid cylindrical specimen under uniaxial tension (indium-lead alloy, hot deformation, controlled parameter of the true strain rate of the order of 10 −7 sec −1 ) [4]: εs = 0 (1), 0.032 (2), 0.095 (3), 0.159 (4), 0.222 (5), 0.254 (6), 0.317 (7), 0.349 (8), 0.413 (9), 0.476 (10), and 0.508 (11).…”