p. COMPACTION OF POROUS POWDER BODY CONSISTING OF THE ELASTIC-PLASTIC

In the development of technological processes of producing cold-pressed sintered parts of low porosity, special attention is paid to the mechanism of density variation. In powder metallurgy, a multicomponent charge consisting of plastic metals, as well as poorly compressible inclusions and compounds, is often used. Such charge can equally be attributed to the charge consisting of iron powder, cast iron and glass. In this charge, the first component (base) is ductile iron, and the other two, cast iron and glass, are elastic components. It is of some interest what kind of compaction can be obtained in this case and what resulting equations can be used to estimate the mechanics of compaction of such a powder charge. The resulting equations of compaction of porous powder bodies of iron-cast iron-glass are proposed. The analysis of the isotropic, rigid-plastic hardening material such as iron-cast iron-glass is given. When compacting such a material, the rate of energy dissipation (pressing pressure) is determined by the rate of volume and form change of the body. It is shown that the difference between compressed (cast iron and glass) and plastic compacted (iron) materials forms special mechanical properties of the matrix. Consequently, hydrostatic pressure can affect the form change of the body, and shear stresses – volume change. The results of the mathematical approach to obtaining the resulting equations of compaction of the elastic-plastic medium showed the way to build a theory of plasticity of the compacted body, which eliminates the need to take into account the type of loading surface. When accounting the loading surface, it is impossible to obtain universal equations of compaction of the porous elastic-plastic medium. It is shown that to apply the classical formulation of the model of the elastic-plastic compacted body, it is necessary to assume that the loading surface is convex-closed


Introduction
A consistent phenomenological description of the processes of formation of powders and porous bodies of the elastic-plastic medium as the most important element eliminates the choice of governing or rheological equations. For sintering and hot pressing, thanks to the works [1,2], some clarity in understanding of this issue has been achieved, while for cold molding processes characterized by plastic flow, there is no consensus about the type of governing equations. In this regard, the formation of general restrictions imposed on such equations, based on the current concepts of irreversible thermodynamics and continuum mechanics, is relevant. In this case, an approach to constructing a theory of plasticity should be used, based on setting the properties of the dissipative function [3][4][5][6]. 5. A Practical Method for Measuring Rolling Element of Rolling Bearings / Nagatomo T., Okamura Y., Takahashi K., Kigawa T., Noguchi S. // The Japan Society of Mehanical Engineers. 2010. Vol. 11. Р. 404-412. 6. Kuz'menko A. Raspredelenie nagruzki mezhdu sharikami v radial'nom podshipnike kacheniya // Problemy trybolohii. 2010

Literature review and problem statement
In [2,3], physical justification is considered and mathematical evaluation of the elastic-plastic deformation of powder compressed materials is performed. In these works, it is shown that the solution of the boundary-value problem of the process of cold compaction of the elastic-plastic medium does not depend on the type of loading surface. It is determined by dependences of axial and lateral pressure on porosity. However, the author does not consider the fact that each pressing scheme can correspond to a certain character of bending of layers parallel to the base plane prior to pressing.
The phenomenological approach to the processes of compaction of the powder medium proposed in [4] takes into account the case of forming blanks of bushing type. However, this is the simplest type of blank, on which analytical dependencies are obtained.
The question of building a mathematical model of blanks of a more complex shape remains open. In this case, it is necessary to consider not only the mechanics of compaction of the plastic component, but also interaction and joint deformation of both the plastic and elastic component, taking into account the peculiarities of the loading surface.
The solution of this problem in this context is considered in [7][8][9]. However, they do not take into account the participation of several elastic components (cast iron and glass), along with the plastic component of the medium, with different characteristics and behavior during deformation.
Therefore, there is the problem of form change during compaction of the isotropic, rigid-plastic hardening powder medium in which energy transfer rate (pressing pressure) depends on the rate of volume and form change.

The aim and objectives of the study
The aim of the work is to obtain the resulting equations describing the compaction of the porous powder body consisting of the elastic-plastic medium.
To achieve this aim, the following objectives are set: -to formulate general restrictions on governing equations, based on current concepts of irreversible thermodynamics and continuum mechanics; -to choose an approach to the construction of the elastic-plastic medium and justify the type of mathematical model of the isotropic, rigidly plastic hardening material, taking into account the rates of volume and form change of the body.

1. Selection and justification of restrictions on governing equations
The isotropic, rigid-plastic hardening material such as "iron-cast iron-glass", the energy dissipation rate of which is determined by the rates of volume and form change is introduced into consideration.
The last two parameters are, respectively, the first invariant of the strain rate tensor ij and the second invariant of its deviator, and, therefore, expressed through ij In the future, hydrostatic pressure and shear stress intensity, which are, respectively, the first invariant of the stress tensor s ij and the second invariant of its deviator, will also play an important role. They are connected with s ij by the relations Choosing an approach to the construction of the elastic-plastic medium based on setting the properties of the dissipative function We take the following definition of an elastic-plastic body: the dissipative function D is homogeneous, of the first degree by ; ij the same function serves as a potential for the stress tensor We consider materials for which the von Mises yield criterion is valid in the following form where P, t are the stresses corresponding to the kinematic parameters and g; p 1 , t 1 are any other stresses. For isotropic material, the properties of which are specified above, the function D allows for , g, the current porosity q, as well as the parameters c k , characterizing the state of the powder particle material or the porous body framework as arguments.
By the Euler's theorem on homogeneous functions, the postulate on uniformity D by and g is expressed by the equation Let us proceed to the analysis of the corollaries of the postulate (1). From (1), taking into account what was said about the arguments of the function D, the following equation can be obtained: Since by definition the relation (4) can be given the following form In this case, the tensor identity is used We find a simple relationship between p, t and , g = , dD From (7) and (8) In order to obtain them, on the basis of (7) and (3), we simplify the expression for Assuming throughout what follows that p and t are rather smooth functions and g, we differentiate both sides of the equation (9) first with respect to , and then with respect to g. Then, using (3) and (7), we get The last two expressions can be considered as a system of first-order partial differential equations with respect to p and t. Direct check shows that its general solution is where = g / . s The equations obtained characterize the scalar properties of compacted materials.
Equations (8) and (11) are a complete system of governing equations of the plastically compacted body. If its tensor properties do not differ from those for viscous and plastically incompressible materials, scalar properties possess a known originality. In order to emphasize it, we present the equations characterizing the scalar properties of a viscous porous body.
these equations are uniquely solvable with respect to and g. At the same time, from equations (12) it is impossible to determine and g, p and t depend on their relationship. Thus, there is a cross-effect of various invariants of stress and strain rates on each other. This phenomenon is not characteristic of classical models of viscous and elastic bodies, naturally inherent in the model of a plastically compacted body.
The specified formal difference between viscous compressed and plastically compacted materials predetermines special mechanical properties of the latter: hydrostatic pressure can affect the form change, and shear stresses -volume change. This property should be associated with the dilatancy effect [7], characteristic of compacted materials.
It turns out further that the functions entering the equation (11) are not arbitrary and must satisfy a certain relation. Indeed, from (7) it follows that Given (11) Thus, whatever the governing equations (11), the functions p and t entering them must satisfy (14).
Let us proceed to the analysis of the corollaries of the postulate expressed by inequality (2). To do this, we consider two different stress states, characterized by stresses p 1 , t 1 and p 2 , t 2 .
Let the first statically admissible stress state correspond to the kinematic field, characterized by the value of the parameter s=s 1 , and the second -to s=s 2 . Then, according to (2), there is a pair of inequalities The latter inequality holds for any value of s, therefore the function p is a monotonically non-decreasing function.
Using this result, it is also possible to obtain additional information regarding t. It follows from the monotony of p that dp/ds 3 0. Therefore, on the basis of (14) and by virtue of the assumption of smoothness of the function p and t t ³ 0; d ds The resulting inequalities show that with negative values of s, t increases, and with s=0 it reaches a maximum, and with positive s -decreases. Note, however, that t is non-negative by definition. Therefore, the graph is "bell-shaped", and with an unlimited increase |s|t asymmetrically tends to zero (Fig. 1, a). The last of these circumstances imposes known limitations on the form of the function p: its graph will have two horizontal asymptotes (Fig. 1, b). The functions p and t are bounded.
The tensor properties of this model are characterized by the equation (8).
Note that such a formulation of the governing equations is close to the traditional models of a viscous and elastic body, in which the scalar equations are formulated as stress -strain rate or stress -strain ratios. It should be noted that the special provision of the theory of plasticity is justified by the fact that it was intensively developed in relation to incompressible media. Related to this is the importance of such specific concepts as yield stress, loading surface, and others. The result obtained in this work indicates the way to construct a theory of plasticity of a compacted body, which eliminates the need to formulate the type of loading surface.
At the same time, the ideas developed here do not contradict the classical ones and moreover, confirm them. To see this, we use monotonicity and smoothness of the function p and the solution of the first of the equations (11) with respect to S. Substituting the result into the second expression (11), we obtain the equation Its characteristic difference is that it does not contain deformations. Since the values of p and t correspond to plastic flow, we have the function t q c ≅ t − t q c = ( , , , ) ( , , ) 0.
It can be identified with the loading function, and the surface corresponding to it in the stress tensor space -with the loading surface. The same name is saved for the graph of the equation (15) in the coordinates of p, t. The properties of this surface with fixed q and c k are defined using inequality (2). Then, according to (15), this inequality can be written as Given (14), we have the equality As a result of substitutions (17) into the last inequality, by means of obvious transformations we find The resulting inequality shows that the graph of the function t(p) lies under the tangent. Consequently, the loading surface is convex. Since the functions p and t are bounded, we can conclude that this surface is also closed.
In order to determine the relationship between the surface of loading and the strain rate tensor, we use the fact that, due to the independence of the loading function f of s, there is the inequality 0. df df dP dP d ds dP ds d ds Given (14), as well as the definition = g / , S we get This result allows for a simple geometric interpretation. The pair of numbers , γ can be considered as vector components in the p, t plane. Then equality (18) shows that this vector is collinear to the normal K of the surface f or is orthogonal to it. Equality (18) acquires a more general meaning, if we use its parametric representation = µ , df dp g = µ t . df d Then, using (8) by simple transformations, we obtain the equation showing that in the stress tensor space, the strain rate tensor is "orthogonal" to the loading surface, that is, the associated law is true [3][4][5][6]. Thus, the classical formulation of the model of the elastic-plastic compacted body is a consequence of the previously developed representations. In this case, the loading surface (16) must be closed, convex, and the function f must satisfy the equation (18). Expressions (16) and (18) are the scalar governing equations. Tensor properties remain the same.
Note, however, that the traditional form of the model, associated with the known arbitrariness of the choice of loading surface, leads to the equations (11) only with a strictly convex surface. This can be seen on the example of the Drucker-Prager model (the generalized Coulomb-Mohr plasticity condition) [7], where p and t are bound by the linear equation t + a + = 0.
p c The law associated with such a condition of plasticity leads to the corollary ag − = 0. This corollary, together with the previous relation, does not allow one to uniquely determine p and t.
The most common example of a strictly convex loading surface is spheroid [8][9][10][11], whose equation in the р -t axes is t + = y q y q where y q ( ), φ q ( ) are the porosity functions; k is the value associated with the yield stress of the base metal.
The flow rule associated with this surface (19) leads to the equation