ON THE ANALYSIS OF A SOLID SUBJECT TO A FRONTAL PHASE TRANSFORMATION OF ITS MEDIUM

In the present paper, we give the general statement of a boundary value problem of the determination of the stress-strain state in a solid in which the process of frontal phase transforoiation of its medium occurs. Some conditions restricting generality of the statement are imposed. It is established that to pose the corresponding boundary value problem one must know not only the equations of state of the preceding and subsequent media but also the so-called "characteristic functions of the frontal phase translormalion" of the solid medium. These characteristic functions describe a medium in which two states are possible, as completely as two equations of state do. The basic (original) statement of the problem under consideration is formed by the standard complete systems of equations at all point of the solid, the standard matching conditions on the front of phase transformation, and the relationships between the preceding and subsequent parameters of the stress-strain stale on the front of transformations which are established by the characteristic functions. Along with the basic form of the boundary value problem, its constructive form "in velocities" is considered, and a step-by-step method of numerical solution of the problem is formulated. The formulation of a well-posed step-by-step procedure is regarded as a justification of the well-posedness of the problem. The introduction of the characteristic functions and construction of the step-by-step procedure form the main task of this paper. The construction of the characteristic functions and equations of state is beyond the issue of the statement of the boundary value problem. In this (main) part of the paper, the results are essentially based on the results of the paper [1] in which the problem underconsideration was completely solved in the case of viscoelastic models of the preceding and subsequent media and for a specific model of frontal phase transformation of a solid. (It seems that this specific model has not been completely understood.) Moreover, in [1], the deformations are assumed to be small. The characteristic functions, as well as the equations of state, can be defined by adopting some conjectures (models) related to these functions and equations with subsequent experimental verification and adjustment of these conjectures. The present paper deals with two models of frontal transformation of the medium and the characteristic functions implied by these models which were suggested in the papers [1] and [2]. In this discussion, the model of [2] suggested for the phase transformation of linearly elastic media is generalized to the description of the frontal phase transformation of inelastic media. Note that in the case of the frontal transition of elastic (including nonlinearly elastic) media described by the second model of transformation, and only in this case, the problem under consideration can be solved in a way simpler than that based on the boundary value problem described in the present paper and in [1]. Specifically, in this special case, the solution of the boundary value problem can be represented as a continuum of solutions to problems of matching two parts of the original solid in one of which the process of instantaneous phase transformation of the medium has occurred. It is just in this way that the problem of the frontal transformation of an elastic medium in a sphere was solved in the paper [2]. In the present paper, both models are regarded as possible and plausible speculative models that are not related directly to actual processes of phase transformations in solid media. No preference is given and can be given to any of the models. The construction of characteristic functions on the basis of the investigation of specific physical processes of phase transformation occurring in constrained conditions and corresponding to the frontal phase transitions is beyond the scope of the present paper.