Qualitative Distinctions in the Solutions Based on the Plasticity Theories with the Mohr-Coulomb Yield Criterion

Abstract

The following two models of the plasticity theory are considered: the model with the Mohr-Coulomb yield criterion and the classical model of the plasticity theory with a yield criterion independent of the mean stress. The deformation problem of a plastic layer enclosed between two rotating plates is studied.The plasticity theories based on the Mohr-Coulomb yield criterion are used to describe the motion of granular and loose materials and also deformation of some metallic alloys [1][2][3][4]. These models were reviewed in [4]. In the case where the angle of internal friction vanishes, most of these models reduce to the classical theory of plasticity with a yield criterion independent of the mean stress. The solutions of specific problems can, however, diverge from the corresponding solutions based on the classical theory of plasticity, and the solutions obtained by different models may differ qualitatively. In particular, this situation occurs if the law of the maximum friction is used. Alexandrov [5] compared the solutions for two processes (flow of a material through an infinite plane convergent channel and compression of a layer by two parallel rough plates) obtained by two theories of plasticity based on the Mohr-Coulomb yield criterion: Spencer's theory [2] and Hill's theory. The equations of Hill's model, which generalizes to some extent the model of [1], were given in [4]. It was shown in [5] that the solutions corresponding to these models differ qualitatively. However, the drawback of that paper was that the solutions constructed failed to exactly satisfy all boundary conditions. The authors [6] considered the problem that involved the law of the maximum friction and admitted a semi-analytical solution for Spencer's model, which satisfies all boundary conditions exactly. In that paper, the plane strain of a material compressed between rotating rough plates whose surfaces obeyed the law of the maximum friction was studied. The solution of this problem was obtained in [7] using Hill's model, and it was shown that the solution constructed differs qualitatively from the solution given in [6]. In contrast to [6] and [7], it is assumed in the present paper that the plates rotate in such a manner that their opening angle increases. This modification in the formulation of the problem has a substantial effect on the qualitative behavior of the solution for Hill's model. Figure 1 shows the geometry of the process. It is assumed that there is no outflow at point 0. We introduce polar coordinates r and θ. Owing to symmetry, it suffices to construct the solution in the region θ 0. The boundary conditions at the axis of symmetry where θ = 0 are σ rθ = 0;(1) v = 0.(Here σ rθ is the shear stress in the polar coordinate system. The plates rotate with an angular velocity ω > 0, as is shown in Fig. 1. Therefore, the impermeability condition at the surfaces of the plates becomes v = ωr for θ = θ 0 .

Qualitative behaviour of viscoplastic solutions in the vicinity of maximum-friction surfaces

“…Its specific form depends on the regime of friction, namely whether the material is sliding or sticking at the inner boundary. For sliding the value of the shear stress at r = a is given by (6). For sticking the circumferential velocity is prescribed:…”

“…If (21) has no solution, it is necessary to consider the regime of sliding for ρ = 1. In this case, it follows from (6), where τ f should be replaced with s r θ , and (15) that a necessary condition is…”

“…This model is used for describing the deformation of granular materials. It is interesting to mention that other models, also used for such materials, can lead to qualitatively different behaviour of the velocity field near the maximum-friction surface [5][6][7][8]. Of several models of pressure-dependent plasticity considered in these papers, the double-slip and rotation model [9], in addition to the double-shearing model, only leads to the same solution behaviour near the maximum-friction surfaces as the model of rigid perfectly plastic solids.…”

Compression of a plastic porous material between rotating plates

“…As the strain rate does not affect the material resistance, we can assume that Ω = 1 without loss of generality. Solutions of the problem considered for some models of plastically incompressible materials were obtained in [12][13][14]. A particular case of these solutions is the rigid perfectly/plastic solution.…”

“…Let us use the maximum friction law, which implies that the friction stresses reach the maximum possible values in the case of slipping [15], and singular velocity fields are formed if certain models of the material are used [13][14][15]. Moreover, the solution does not exist at higher values of the shear stress on the contact surface.…”