Specific features of solving the problem of compression of an orthotropic plastic material between rotating plates

Abstract

An instantaneous flow with compression of a wedge-shaped layer of a rigid-plastic orthotropic material between rotating plates is considered under the assumption that the principal axes of anisotropy are rays emanating from the wedge angle and lines orthogonal to them and that the maximum friction law is valid on the plate surfaces. The solution is reduced to quadratures, and its asymptotic analysis is performed. It is found that the solution is singular near the friction surface in the general case, and conditions at which the singularity disappears are given. It is demonstrated that a rigid area can arise near the friction surface. The behavior of the resultant solution near the friction surfaces is compared with the behavior of known solutions for other models of rigid-plastic materials.In some models of rigid-plastic materials, the solutions are singular near the maximum friction surfaces (in particular, some derivatives of the projections of the velocity vector and the stress tensor components tend to infinity near such surfaces). The general asymptotic analysis of the behavior of solutions was performed in [1] for an arbitrary flow of an ideal rigid-plastic material satisfying an arbitrary smooth yield condition independent of the mean stress, in [2] for an axisymmetric flow of a material satisfying the Tresca yield condition, in [3] for a plane flow, and in [4] for an axisymmetric flow of a material satisfying the double shear model (the model itself is described in [5]). In all cases, the equivalent strain rate (second invariant of the strain rate tensor) near the maximum friction surface during slipping was found to tend to infinity in an inverse proportion to the square root from the distance to this surface. Solutions of problems for various material models show that this property of the velocity field depends on the material model [6-10]. The singular behavior of the velocity field allows new theories to be proposed to describe the changes in the material properties in a thin layer near the friction surfaces [11,12].A solution of a model problem of an instantaneous flow of a wedge-shaped layer of a rigid-plastic orthotropic material satisfying the model [13] between rotating plates with the maximum friction law on the plate surfaces is obtained in the present work. Solutions of this problem for different material models make it possible to find some specific features of these solutions, including their singular character [7,8,14]. Other solutions of model problems within the framework of the anisotropic plasticity theory were obtained in [15,16], where the asymptotic analysis of solutions in the vicinity of the maximum friction surfaces was not performed.Let us consider an instantaneous plane flow of a plastic orthotropic material compressed between two plates rotating with an angular velocity ω with respect to the point O (Fig. 1). Assuming that there is no drain at the point O, we introduce a polar coordinate system (r, θ) with the origin at the point O. As θ = 0 is considered as the axis of symmetr...