LARGE DEFORMATIONS AND HEATING OF ELASTOVISCOPLASTIC MATERIAL IN A CYLINDRICAL VISCOMETER

A solution for a coupled thermomechanical boundary value problem of the theory of large deformations that simultaneously takes into account the intense deformation of an elastoviscoplastic material and its heating due to such deformation and near-wall friction is constructed. The conditions for the emergence and development of a viscoplastic flow in the material of a cylindrical layer enclosed between rigid surfaces, one of which has the ability to rotate under the action of a changing given load relative to the axis of the viscometer, are calculated. At all stages of the process from its start to stop, the stress-strain state and temperature distributions in the areas of flow and reversible deformation are calculated, including flow stop, unloading and cooling, with the calculation of residual stresses and deformations. The yield strength is assumed to be temperature dependent.


INTRODUCTION
For rigid-plastic analysis [1], the one-dimensional problem of the viscoplastic flow of a material in a cylindrical viscometer is a test one [2,3]. In such a setting of a fundamentally nonlinear mathematical model of deformation, simplifications in the mathematical apparatus are achieved by the assumption of the non-deformability of the material of rigid advancing kernels and stagnant zones. Such regions of a deformable body can change only by changing their boundaries [4][5][6]. This assumption completely excludes the conservative component of the deformation process, leaving it exclusively dissipative. Accounting for the conservative part of the process requires a model account of the elastic properties of the deformable material, that is, the assumption of elastic deformation of the material in stagnant zones and advancing kernels. In this case, the problem of viscometric viscoplastic flow appears as a one-dimensional problem in the theory of large deformations of elastoviscoplastic materials [7][8][9][10][11][12][13]. In terms of setting and solving boundary value problems of this theory, the approach based on the formalism of nonequilibrium thermodynamics turned out to be especially convenient, when reversible and irreversible deformations, as thermodynamic state parameters in the process of deformation, are determined by formulating differential equations for their change (transfer) [11][12][13]. Using this approach, several solutions of theoretical problems including exact analytical ones are obtained. Some of these problems are analyzed in [12], where the problems of viscometric deformation are also considered [14][15][16][17]. Thus, in [14], the features of the imposition of axial displacement on the viscometric flow of the material are studied, in [16] the influence of lubrication of boundary surfaces, in [17] the viscoplastic flow is complicated by the presence of slippage on the walls of the viscometer.
In [18][19][20], it is on the example of solving viscometric problems in the theory of large deformations of elastoviscoplastic materials that the features of changes in the mechanisms of production of irreversible deformations at elastoplastic boundaries are discussed. On the one hand, such a boundary surface, irreversible deformations can grow due to the viscous properties of the deformable material (creep), and on the other hand, mainly due to plastic properties (flow) [21].
In this article, we generalize the solution obtained in [17]. We assume that slippage is retarded by friction and this leads to heating of the deformable material. Thus, we consider the coupled problem of heat production and irreversible deformation due to viscometric deformation and near-wall friction. As a mathematical model in the formulation and solution of this problem, we use the approaches [8,22], later successively described in [12]. It should be noted that within the framework of such a mathematical model, related thermomechanical problems have already been considered [23][24][25][26], while limiting themselves only to the rectilinear motion of an elastoviscoplastic material.

BASIC RELATIONS OF THE MATHEMATICAL MODEL
As the thermodynamic parameters of the state of the body in the process of its deformation, along with temperature (or entropy distribution density ), we accept reversible (thermoelastic) deformations and irreversible deformations . The latter may be creep strains or plastic flow strains; they can be determined both by the viscous properties of the deformable material and by the plastic ones. For them, we assume that there can be deformation processes in which irreversible deformations are unchanged. The adoption of such an idealization for the deformation process makes it possible to determine irreversible and reversible deformations by the following [11,12] differential equations for their change (transfer) The transfer equations (2.1) are written in the system of independent Euler space variables and via the introduced notation In (2.1) and (2.2) are the displacement and velocity vectors; are the tensors of reversible and irreversible strains; I is the unit tensor of the second rank; are the temperature and temperature of the free state (room temperature), respectively; α is the coefficient of linear expansion. The first equality in (2.1) introduces an objective time derivative of the tensor, which has an important property. With its help, the interdependence between reversible and irreversible deformations in the process of deformation, which ensures the geometric consistency of the kinematics of the process, is set. When the source in the transport equation for irreversible deformations p is equal to zero, then and p is a constant tensor.
Therefore, when the rates of irreversible deformations are equal to zero, the deformations p are unchanged, and the equality serves to calculate the components of the tensor p, which change in the same way as with rigid displacements of the body. When the non-linear additive is equal to zero, the introduced objective derivative coincides with the Zarembo-Yauman derivative.
For total Almansi deformations, it follows from (2.1) and (2.2) that [12] (2. 3) It follows from (2.3) that the introduced tensor is only the principal linear part of the reversible strain tensor . As the thermodynamic potential of the conservative part of the deformation process, we set the free energy as , and introduce the thermoelastic potential as the dependence . Here, is the internal energy distribution density. It is assumed that the free energy does not depend on irreversible deformations. The last assumption is hypothetical, however, it sig- nificantly simplifies the mathematical model by separating the conservative and dissipative components of the deformation process. In the mechanics of deformable materials with complex properties, the adoption of this statement is common [8,10,11,27]; as a rule, it is accepted when developing numerical calculation methods [28]. With such a division of the deformation process into conservative and dissipative components, the law of conservation of energy implies [12] the Murnaghan formula and the entropy balance equation Murnaghan's formula (2.4) is written for the case of a mechanically incompressible material; where p is the unknown hydrostatic pressure. In the entropy balance equation (2.5), the q and J vectors are heat and entropy flows. Wherein Further, the deformable material is considered not only mechanically incompressible, but also isotropic, for which and . For , we accept the dependence following from the expansion of this function in a Taylor series with respect to the free state of the deformable material at room temperature (2.6) In (2.6) is the shear modulus, are higher order elastic moduli, are thermomechanical constants. Specific dependences of stresses on temperature and reversible strains can be obtained if we substitute (2.6) into (2.4).
Substituting (2.6) into (2.5) allows us to write the heat equation in the form Here, is the coefficient of thermal diffusivity. In the region of deformation preceding the viscoplastic flow, and in the region of unloading, one should set in (2.7). This means that the generation of irreversible deformations in such parts of the deformation region in the form of creep deformations is neglected. Then for such regions and for the flow region , that is, the source in the transfer equation (2.1) coincides with the plastic strain rate tensor.
As a plastic potential, we set the following generalization [29,30] of the Tresca-Saint-Venant yield criterion Here, are the principal values of the tensors, η is the coefficient of viscous resistance to plastic flow, and k is the yield strength measured in tests for pure shear. The dependence of the yield stress on temperature is taken as [31] (2.9) In (2.9) T m is the melting temperature of the deformable material, k 0 is the yield point of the material at room temperature. Irreversible (plastic) strain rates are related to stresses by the associated plastic flow law Adding to the relations written here the equilibrium equation , we obtain a closed system of equations for quasistationary elastoviscoplastic deformation.

STATEMENT OF THE PROBLEM. REVERSIBLE DEFORMATION AT THE BEGINNING OF THE PROCESS
Let the elastoviscoplastic material fill the area between the cylindrical surfaces and r = R ( ). The surface is assumed to be fixed, and the surface r = R is allowed to rotate around the axis of the cylinder under the action of an external load. Therefore, the boundary conditions of the problem are relations written in cylindrical coordinates (3.1) We assume that until the time t = 0, there are no deformations in the cylindrical layer, the temperature is equal to room temperature , the initial compression is uniform = const. We consider the features of deformation for increasing, constant and decreasing shear stress on the outer surface r = R Initially, after the start of the process, the deformable material is held in the layer due to dry friction and is deformed reversibly where δ is the static friction coefficient. The geometry of the problem and the condition of incompressibility of the material define the kinematics of the medium Here, is an unknown dependent variable, namely, the central angle of twist of the material during its deformation. Substituting (2.6) into (2.4), for the considered initial reversible deformation, we obtain (3.5) When drawing (3.5), we limited ourselves to terms up to the third order in reversible (elastic) deformations. For the calculation method, this is not essential, however, it allows avoiding the cumbersome expressions. Often truly reversible deformations can be considered so small that their third order can be neglected in comparison with the first and second ones.
If we substitute (3.4) and (3.5) into the equilibrium equations (quasi-static case) (3.6) and integrate the result, we get The solution found is valid up to a certain point in time . At , the boundary condition (3.3) is violated or a viscoplastic flow begins. For , the slip begins earlier than the onset of the vis-  where ξ is the viscous friction constant. When (3.8) is fulfilled, the heating of the material begins due to near-wall friction on the surface : Conditions (3.9) assume that the surface r = R is thermally insulated; is the heat production constant due to friction; heating of the material due to the thermomechanical assosiation of reversible deformation with temperature is neglected. (we consider the coupling coefficient to be equal to zero). From (2.4) and (2.5), in this case we have the dependencies At , the elastoplastic boundary separates from the boundary surface and moves along the material of the layer, leaving behind a region of viscoplastic flow .

VISCOPLASTIC FLOW
In the time following the moment , the deformation region is divided into two parts. In one of them , reversible deformation continues (region I), and in the other one , a viscoplastic flow develops (region II). The kinematic dependences of the accepted mathematical model  Thus, for the temperature distribution over the deformable layer, we have the system of equations (4.5) and (4.6) with boundary and initial conditions (4.7) The initial condition for is taken as the temperature distribution at the time of the onset of plastic flow that is calculated by solving the problem of reversible deformation with a heat source due to friction on the surface . For the numerical implementation of problem (4.5)-(4.7), two grids are introduced with respect to the variable r, which change with time: in the thermoelastic region and in the viscoplastic flow region . At each time step , the grid changes due to the movement of the elastoplastic boundary. The temperature values for the newly constructed grid at the previous time step are found by interpolation.
Using the temperature distribution found, the rates of irreversible (plastic) deformations and the rates of total and elastic (4.1) deformations are calculated (4.3). By integrating the distribution of strain rates at (1 ) 1 = ( ).  2   II  II  II  II  II  rr  r  II  II   II   m   a t  lt  r  r l l at at k r r By integrating the first of the equilibrium equations (3.6), we determine the function of the additional hydrostatic pressure and, consequently, the diagonal components of the stress tensor.

DEFORMATION UNDER CONSTANT AND DECREASING LOAD
From the moment of time , the shear stress on the surface r = R is constant: , . However, the domain of viscoplastic flow continues its development. The heat conduction equations (4.5) and (4.6) remain valid for the regions of reversible and irreversible deformation, respectively. The boundary conditions for these equations take the form in the entire cylindrical layer. However, the process of equalizing the temperature across the material due to thermal conductivity continues. It does not stop even after the point in time at Figure 1 shows the change in the elastoplastic boundary (solid line) during the entire deformation process; for comparison, the dashed line shows the change in the elastoplastic boundary without taking into account temperature effects; Figures 2 and 3 show the distribution of relative temperature and angle of rotation at different moments of time. The calculations have been carried out in dimensionless variables the calculation results are given for the following values of the constants 6. CONCLUSION The constructed numerical-analytical solution of the coupled large-strain problem on the viscometric deformation of an elastoviscoplastic material is another example of the usefulness of using differential equations for the transfer of reversible and irreversible strain tensors for this purpose. Consistent adherence to the formalism of thermodynamics helps not only in formulating but also in solving the problems of the theory.
Let us note some qualitative features of the deformation process, which are determined by taking into account the heating of the material. The development of the viscoplastic flow region occurs with noticeable differences from the isothermal case (Fig. 1). Only under conditions of an active loading process such a development is the same, however, at a constant load, an increase in the viscoplastic flow region is noticeable due to the coupling of the problem and an increase in temperature in the deformable material due to the accumulated heat during friction and a continuing increase in temperature due to irreversible deformation. The movements of the unloading elastoplastic boundaries during unloading are completely different. In the isothermal case, this boundary reaches the fixed surface (viscoplastic flow region disappears) and only then does the material adhere to this immovable surface. When temperature effects are  taken into account, the viscoplastic flow region decreases more slowly. In this case, slippage initially ends and only then viscoplastic flow stops. The values of the angle of rotation are significantly different (Fig. 3).
In the isothermal case, they are much smaller at the same load.
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