An approach for selecting general kinematically admissible velocity fields for axisymmetric forging problems is outlined. The approach accounts for the existence of a rigid zone at frictional interfaces and for the singular behavior of real velocity fields in the vicinity of maximum friction surfaces. The plastic work rate for a material obeying the von Mises yield criterion and its associated flow rule is expressed in terms of one arbitrary function of a single argument, its derivative, and anti-derivative. An upper bound solution for constrained forging is given. Comparison with an available upper bound solution is made and it is shown that the new kinematically admissible velocity field results in a more accurate solution at high friction. Other problems that can be treated are briefly discussed.
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 .
The main objective of the present paper is to provide a simple analytical solution for describing the expansion of a two-layer tube under plane-strain conditions for its subsequent use in the preliminary design of hydroforming processes. Each layer’s constitutive equations are an arbitrary pressure-independent yield criterion, its associated plastic flow rule, and an arbitrary hardening law. The elastic portion of strain is neglected. The method of solution is based on two transformations of space variables. Firstly, a Lagrangian coordinate is introduced instead of the Eulerian radial coordinate. Then, the Lagrangian coordinate is replaced with the equivalent strain. The solution reduces to ordinary integrals that, in general, should be evaluated numerically. However, for two hardening laws of practical importance, these integrals are expressed in terms of special functions. Three geometric parameters for the initial configuration, a constitutive parameter, and two arbitrary functions classify the boundary value problem. Therefore, a detailed parametric analysis of the solution is not feasible. The illustrative example demonstrates the effect of the outer layer’s thickness on the pressure applied to the inner radius of the tube.
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