Compression of a plastic porous material between rotating plates

Abstract

The problem of a plane flow of a rigid-plastic porous material between two rotating rough plates with no material flux through the point of their rotation and with a uniform distribution of porosity at the initial instant is considered under the assumption that the material behavior follows the cylindrical yield condition and the associated flow rule. The solution reduces to consecutive calculation of several ordinary integrals. It is demonstrated that the solution behavior depends on the angle between the plates, and the value of porosity at a certain stage of the deformation process can be equal to zero.In the classical theory of plasticity of incompressible materials, there are many analytical solutions obtained by an inverse method [1,2]. In the theory of plasticity of porous materials, such solutions seem to be lacking (even for the initial flow), except for a uniform stress-strain state and extremely simple problems with the friction stress neglected, for instance, problems of contraction of a hollow cylinder in a plane strain state (initial flow), contraction of a hollow spherical shell (initial flow), and flow through a convergent channel [3]. The classical problems of the plasticity theory [the most typical problem is the Prandtl problem of compression of a layer between rough plates (see, e.g., [1])], which have fairly simple solutions in the case of an incompressible rigid perfectly/plastic material (and their extensions to other models of incompressible materials [4]), cannot be extended to models of plastically compressible materials. In particular, attempts of problem extension were made in [5] for the Prandtl problem and in [6] for the flow through an infinitely convergent channel. Solutions, however, were obtained in none of these attempts (under the same assumptions at which the classical solutions are constructed). As the exact solutions are of interest and can play an important role in verification of numerical codes [7], the search for such solutions, especially in the presence of friction stresses, is an urgent task.We assume that the material behavior obeys the cylindrical yield condition proposed in [8] and written in the formHere σ eq is the equivalent stress, σ is the mean stress, τ s is the shear yield stress, and p s is the yield stress under hydrostatic compression. The equivalent and mean stresses are defined by the relations σ eq = 3/2 (τ ij τ ij ) 1/2 and σ = σ ij δ ij /3, where σ ij are the stress tensor components, τ ij are the stress deviator components, and δ ij is the Kronecker symbol. The quantities p s and τ s are assumed to be known functions of porosity η. As η → 0, we have p s → ∞, and quantity τ s tends to the shear yield stress of the base material. Though the yield condition (1) is extremely simple, it can describe some experimental observations [8]. Other solutions with the use of the yield condition (1) were obtained in [9,10]. Note that various flow regimes are possible if this condition is applied. Nevertheless, the regime described by the relations σ eq √ 3 τ s and...

Viscoplastic flow between hinged plates

“…Compression between rotating hinged plates has also been widely studied for a plastic material, under a range of different constitutive laws (Alexandrov & Lyamina 2003; Alexandrov & Jeng 2009; Alexandrov, Pirumov & Chesnikova 2009; Alexandrov & Miszuris 2015). A distinguished feature of the rigid plastic problem is the existence of unyielded regions in rigid body rotation adjacent to the rotating plates, for sufficiently large wedge angles.…”

Strain-rate intensity factor in compression of a layer of a plastic material between cylindrical surfaces

“…For some processes, the strain-rate intensity factor was determined in [7][8][9][10], where the solutions were obtained on the basis of the double shear model [11], which is the generalization of the model of a rigid perfectly/plastic material. Note that the qualitative behavior of the solutions obtained on the basis of the double shear model coincides with the behavior of the solutions obtained on the basis of the model of a rigid perfectly/plastic material [12,13].…”

“…With allowance for inequality (7) and the last relation in Eqs. (6), the boundary conditions (1) acquire the form…”

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

“…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.…”

“…Figure 2 shows the dependence A(θ 0 ) calculated by Eq. (14) for different values of the parameter c characterizing plastic anisotropy. Figure 3 shows the dimensionless strain rate intensity factor d = D/(ωr 1/2 ) as a function of θ 0 , which was calculated by Eq.…”