Utilization of a damaged surface for the description of creep and long-term strength in the case of complex loading

When solving boundary value problems in the mechanics of pressure shaping of noncompact materials, it is necessary to take into account irreversible deformations of volume compression (extension). A. Coulomb [1] considered the limit state pyramid in the theory of soil pressure. For plastic compressible materials, yield conditions in the form of Coulomb's pyramid bounded by a plane of constant hydrostatic pressure (the Coulomb-Mohr) pyramid were used in [2, 3]. Von Mises and Schlechter proposed a yield condition expressed by a circular cone in the space of stresses bounded by a plane of constant hydrostatic pressure (von MisesSchlechter's cone) [1]. This condition was considered in [3, 5]. The said yield conditions are of piecewise smooth type. Some models of ideal compressible plastic media with piecewise smooth yield surfaces were investigated by D. D. Ivlev and G. I. Bykovtsev [6], who showed that the relations of such models can be regarded as a theory of a hardening plastic bodies with singular load surfaces [6]. R. Green proposed a yield condition of elliptic type [7]. This condition is expressed in terms of an ellipsoid of revolution in the space of principal stresses, the symmetry axis of the ellipse coinciding with the hydrostatic axis. There is also a yield condition of cylindrical type [3]. This condition, in contrast to those mentioned above, involves no dilatancy relation between the characteristics of volume and shear strains [3]. Some boundary value problems in the mechanics of shaping of noncompact materials were solved in [3, 5, 8-14], mostly on the basis of elliptic yield conditions or those with the Coulomb-Mohr pyramid or the von Mises-Schlechter cone. A number of problems were solved in [3] with the help of cylindrical yield conditions. In the present paper, cylindrical yield conditions are used for solving boundary value problems in the mechanics of plastic flow of porous media in the cases where dilatancy is absent or its effect is insignificant. For instance, no compacting is observed in the zone of slippage on the delivery side [8] during rolling or in the matrix opening during extrusion of a porous material [9]. Note also that yield conditions of cylindrical type simplify the solution of boundary value problems with discontinuous fields of velocities and strains. In such cases, cylindrical yield conditions are convenient and effective for the evaluation of energy-force parameters of the processes of plastic deformation of noncompact materials and are useful for calculations of porosity variation. For the numerical solution of boundary value problems, the equation of the yield surface should be supplemented by relations between the compression yield stress, the shear yield stress, and the current porosity. A review of most common relation of this type can be found in [15]. In general, the relations given in the present paper allow us to take into account the geometry of the pores. As an illustration of these relations, combined with the cylindrical yield conditions, we solve the problem of double-action compacting of a porous axially symmetric blank, and we also use kinematically admissible velocity fields with strong discontinuity surfaces for solving boundary value problems related to lateral semicontinuous extrusion of a porous mass.