S. E. Aleksandrov

Compression of a viscoelastic layer between rough parallel plates

If the maximal friction law is applied, then some generalizations of the Prandtl solution for the compression of a plastic layer between rough plates do not exist. In particular, this pertains to the viscoplastic solutions obtained earlier. In the present paper, we show that these solutions do not exist because of the properties of the model material and introduce a model for which this solution can be constructed. The obtained solution is singular. In particular, the equivalent strain rate tends to infinity as the friction surface is approached, and its asymptotic behavior exactly coincides with that arising in the classical solution. The obtained solution is illustrated by numerical examples, which, in particular, show that an extremely thin boundary layer may arise near the friction surfaces.

Solution of the thermoelastoplastic problem for a thin disk of plastically compressible material subjected to thermal loading

Elastoplastic solutions for thin plates and disks are sensitive to loading and plasticity conditions [1–5]. The plasticity condition for a number of metal materials depends on the mean stress [6–8]. In this case, when using the associated flow rule, plastic deformations do not satisfy the incompressibility condition, which is commonly accepted in statements of boundary-value problems for thin elastoplastic plates and disks [9–13]. It is of interest to determine the effect of plastic compressibility on the behavior of solutions for such structures. In this paper, a hollow disk in a rigid container subjected to a uniform temperature field is considered. The plasticity condition proposed in [14] is accepted. A general study of the set of equations including this plasticity condition and the associated flow rule was performed in [15]. The solution under the Mises plasticity condition was obtained in [1].

Strain rate intensity factors for a plastic mass flow between two conical surfaces

The concept of strain rate intensity factor was introduced in [1], where the asymptotic expansion of the velocity field in a perfectly rigid-plastic material was obtained near the maximum friction surface, which is determined by the condition that the specific friction forces on this surface are equal to the simple shear yield strength. In particular, it was shown in this paper that near the maximum friction surface the equivalent strain rate (the second invariant of the strain rate tensor) tends to infinity inversely proportional to the square root of the distance to this surface. We note that the same result was obtained in the case of plane flow in [2]. The strain rate intensity factor is defined to be the coefficient of the leading singular number in the series expansion of the equivalent strain rate near the maximum friction surface. It was shown in [3] that there is a sufficiently complete formal analogy between the strain rate intensity factor and the stress intensity factor in mechanics of cracks [4]. In [5], it was suggested to use the concept of strain rate intensity factor to estimate the thickness of the layer near the friction surface where one should take into account viscosity effects. (Thus, this is an intensive strain layer formed as a result of a very large equivalent strain rate.) Therefore, the problem of calculating the strain rate intensity factor in specific processes is topical in the development of the general concept based on the use of the strain rate intensity factor and its applications in the theory of metal forming processes. These factors have already been calculated for several processes such as plane upsetting and drawing [3]. In the present paper, we calculate the distribution of the strain rate intensity factor in a plastic mass flow through an infinite converging channel formed by two conical surfaces on which the law of maximum friction acts (Fig. 1). A specific characteristic of this problem is the existence of two maximum friction surfaces and, accordingly, two distributions of the strain rate intensity factor. Since, according to the theory [5], the strain rate intensity factor is related to the thickness of the intensive strain layer near the friction surface, the solution of this problem may serve as a starting point for experimental confirmations of the theory. Note that the intensive strain layer thickness can be determined experimentally without any difficulties [6, 7] and the flow in an infinite channel of the shape under study can successfully model the tube drawing process [8].

Compression of an axisymmetric layer on a rigid mandrel in creep

An approximate solution describing the compression of an axisymmetric layer ofmaterial on a rigid mandrel under the equations of the creep theory is constructed. The constitutive equation is introduced so that the equivalent stress tends to a finite value as the equivalent strain rate tends to infinity. Such a constitutive equation leads to a qualitatively different asymptotic behavior of the solution near the mandrel surface, on which the maximum friction law is satisfied, compared with the well-known solution for the creep model based on the power-law relationship between the equivalent stress and the equivalent strain rate. It is shown that the solution existence depends on the value of one of the parameters contained in the constitutive equations. If the solution exists, then the equivalent strain rate tends to infinity as the maximum friction surface is approached, and the qualitative asymptotic behavior of the solution depends on the value of the same parameter. There is a range of variation of this parameter for which the qualitative behavior of the equivalent strain rate near the maximum friction surface coincides with the behavior of the same variable in ideally rigid-plastic solutions.

Strain rate intensity factors for a plastic material layer compressed between cylindrical surfaces

For some models of rigid-plastic bodies, the strain rate fields turn out to be singular near the maximum friction surfaces. In particular, the equivalent strain rate (the second invariant of the strain rate tensor) tends to infinity when approaching such frictions surfaces. The coefficient multiplying the leading singular term in the series expansion of the equivalent strain rate near the maximum friction surfaces is called the strain rate intensity factor. This coefficient occurs in several models predicting the development of intensive plastic deformation layers near friction surfaces and in equations describing the change in the material structure in such layers. In the present paper, the solution is constructed for the compression of a layer of a plastic material obeying the double shear model between cylindrical surfaces on each of which the maximum friction law holds. The dependence of two strain rate intensity factors on the material and process parameters is calculated and analyzed.

Riemann method for the plane strain of a homogeneous porous plastic material

The system of static equations describing the stress state in a homogeneous porous plastic material obeying the pyramidal yield criterion is studied under plane strain conditions. It is shown that determining the curvature radii of the characteristics amounts to solving the telegraph equation. Thus, it is expedient to construct the net of characteristics by the Riemann method, which is widely used to solve boundary value problems in the classical theory of plasticity of incompressible materials. These solutions can directly be generalized to the considered porous material model.

Motion of a rigid bar in a rigid-viscoplastic medium: The influence of the model type on the solution behavior

The paper deals with rigid-plastic materials satisfying the vonMises plasticity condition under the assumption that their yield point in pure shear depends only on the equivalent strain rate (rigid-viscoplastic material models). The rigid-viscoplastic models are classified by the yield point behavior in pure shear as the equivalent strain rate tends to zero or infinity. All in all, four classes of rigid-viscoplastic material models are distinguished. For each of these classes of material models, the solution is constructed for the translational motion of an axisymmetric rigid bar along its symmetry axis in a rigid-plastic medium. It is assumed that the maximum friction law acts on the surface of contact between the bar and the rigid-viscoplastic medium. It is shown that the solutions provided by models of different classes qualitatively differ from each other. A qualitative comparison with experimental results known in the literature is carried out. It is shown that predicting the formation of an intensive plastic deformation layer near the friction surface, which is observed in experiments, is possible if the rigid-viscoplastic model contains the saturation stress (the stress, bounded in magnitude, to which the yield point in pure shear tends as the equivalent strain rate tends to infinity).

Study of compression settlement of a three-layer rigid-plastic strip between parallel plates

The process of compression settlement of a three-layer strip between parallel plates is investigated under the plane strain conditions. The inner layer of the strip is assumed to be made of a rigid-plastic hardening material, and the two outer layers are assumed to be ideally rigid-plastic. The boundary value problem has two symmetry axes. It is assumed that the strip thickness is much less than its width. The boundary conditions at the strip edge and at the center are satisfied in integral form. Two friction regimes, i.e., sliding and adhesion, are possible on the surface of contact between the strip and the plates and on the interface between the layers. It is shown that the general structure of the solution depends on the regimes realized at the moment. In particular, one of the layers can remain rigid at a certain stage of the deformation process. The differential equations are stated which permit exactly determining the conditions of the friction regime change and the state of each layer (rigid or plastic); these equations must be solved numerically. For some values of parameters of the boundary value problem, the velocity field is singular near one or both surfaces of friction. In these cases, it is necessary to calculate the strain rate intensity coefficient whose value probably controls the process of formation of a narrow layer with strongly changed properties near the corresponding surface of friction.

Stress-strain state in an elastoplastic cylindrical tube with free ends. I. General solution

We obtain a general solution for the stress-strain state in an elastoplastic tube whose ends are stress-free. The tube is subjected to internal and external pressures which can vary in time rather arbitrarily. But it is assumed that the radius of the elastoplastic boundary does not decrease during the entire deformation process. The tube material obeys a yield condition depending on the mean stress. The corresponding yield surface has the shape of a cone in the space of principal stresses. The theory of plastic flow is used. The plastic potential is taken in the form of the von Mises condition. Thus, the associated plastic flow law is not satisfied, and the material is plastically incompressible. Numerical methods are only needed for successively solving several transcendental equations and calculating ordinary integrals.

Generalization of the Prandtl solution to the case of axisymmetric deformation of materials obeying the double shear model

A semianalytic solution of the problem on the compression of an annular layer of a plastic material obeying the double shear model on a cylindrical mandrel is obtained. The approximate statement of boundary conditions, which cannot be satisfied exactly in the framework of the constructed solution, is based on the same assumptions as the statement of the classical plasticity problem of compression of a material layer between rough plates (Prandtl’s problem). It is assumed that the maximum friction law is satisfied on the inner surface of the layer. The solution is singular near this surface. The strain rate intensity factor is calculated, and its dependence on the process and material parameters is shown.