Modeling of Cold Extrusion Processes Using Kinematic Trapezoidal Modules

In this paper, we analyze the features of using an axial trapezoidal module in design schemes with an adjacent lower module of various configurations. As adjacent kinematic modules, a rectangular rigid, longitudinal-transverse flow and a trapezoidal kinematic module with a rectilinear inclined boundary are chosen. It has been established that optimization in relative velocity of metal flow in the backward direction is possible only with a combination of kinematic modules with a radial component of one direction (sign) and impossible in other cases The paper shows practical realization of the authors' approach for the particular case of modeling of the process of combined radial-backward extrusion of the end-capped parts with a flange. The deviation of the theoretically obtained estimates of the force parameters of the process and shaping does not exceed 10 15%. Author-proposed computations for the powers of shearing forces can be successfully used for engineering-friendly modeling of more complex processes of the cold combined extrusion with additional taking into account the availability of metal flow in the forward, backward and radial directions. .


INTRODUCTION
Ensuring the competitiveness of products in the branches of mechanical engineering in modern conditions is inextricably linked with the development and reclamation of such effective methods of metal processing by pressure, as cold volume forging (CVF) [1,2]. Technological methods of CVF differ in a variety of possibilities and high efficiency in comparison with other processes of forming of details. Cold extrusion processes demonstrate a steady tendency not only to increase the volume of production of precision workpieces, but also to expand the range of stamped parts and materials. This is due to new methods of transverse and combined extrusion [2][3][4].
Currently, an effective theoretical method for studying cold extrusion processes is the energy method (power balance method). Within the framework of using this method, the selection of appropriate functions describing a kinematically possible velocity field (KPVF) has key significance, satisfying the boundary conditions, the condition of incompressibility of material and the condition of continuity of normal velocity component. The application of the kinematic modules method allows to describe the complex flow schemes with the help of a complex of elementary unified modules [5]. As a total estimate of the reduced pressure is the sum of the reduced pressures of the modules included in this calculation scheme. However, one of the characteristic features of use of the energy method is the presence of sufficiently labor-intensive procedures for the analysis of the force and kinematic regime of the deformation process [4,6]. Therefore, studies of the effectiveness of methods of simplifying the components of the reduced deformation pressure and the development of recommendations for the use of kinematic mules in new design schemes are relevant. This will help to eliminate the difficulties of using the method of kinematic modules and promote the introduction of cold combined extrusion processes in production.
A number of papers are devoted to theoretical, finite element and experimental studies of basic and combined schemes of cold extrusion processes [6][7][8][9][10][11][12][13][14][15][16]. In the paper [6] a finite-element method is used to study the effect of the design parameters of the die on the production of hollow brass parts by cold direct extrusion and expansion on a cone punch. In the paper [7] a finite element analysis of the radial-backward extrusion of hollow parts with a flange is carried out and the influence of friction conditions and the radius of curvature of the matrix on the force parameters of the deformation process is investigated. It was found that under certain conditions, the size of the part in height could be decreased due to the deformation process compared to the initial height of the workpiece. In the paper [8] the authors investigated the influence of gap height, mandrel radius on the formation and oscillation of the load in the process of successive radial-direct extrusion. The reliability of the results of finite element modeling is confirmed by experimentally obtained data presented in the literature. Of particular interest are the papers [9][10][11][12] based on the basis of various modifications of the energy method aimed at investigating of based and combined extrusion processes. Papers [9,10] are dedicated studying the process of combined backward-forward extrusion by the method of upper bound ele-ment technique using arbitrarily oriented triangular elements, combined with the method of finite elements (FEM). Perig (2014Perig ( , 2015 has applied Kudo's rigid block methods with one [11] and two [12] degrees of freedom to derivation of upper bound method-based estimations of energy-power parameters during angular extrusion process. In paper [13,14] the analysis of the combined radial-backward extrusion process in a conical matrix is carried out. Trapezoidal kinematic modules with a rectilinear inclined boundary that does not contain the axis of symmetry are used. However, the values for the total power are found numerically and in the explicit form were not obtained. Perig et al (2013Perig et al ( , 2019 have proposed non-linear regression-based engineering estimations for finite element method-derived numerical results for ECAE-induced deformations [15][16]. Thus, studies of cold extrusion processes are limited to finite-element analysis, theoretical calculations with using of elementary triangular, rectangular and trapezoidal non-axial modules. At the same time, the possibilities of simplifying calculations in order to obtain analytical dependences of the studied values are not fully used. Thus, the methods of constructing velocity fields and simplifying further calculations of the force parameters of the process and the shape change require clarification and development. There is also a need to develop generalizing recommendations for the use of calculations of known kinematic modules and features of their use in new computational schemes.
The purpose of the work was to analyze the features of the use of the axial trapezoidal kinematic module in the design schemes with adjacent lower modules of different configurations.

PRESENTATION OF THE MAIN MATERIAL
The first key stage of using the energy method can be considered the selection of functions (sets of functions) that describe KPVF. No less important is the second stage -the stage of searching of tools to simplify the components of the energy equation, which will allow to obtain the given pressure in an analytical form. This applies primarily to computational schemes containing trapezoidal and triangular modules with curvilinear boundaries, when trapezoid kinematic modules with a curvilinear sloped boundary are used.
The question of using the kinematic parameters of the process as variable with the possibility of further optimization of these parameters remains open. The first attempts to solve the problem of simplifying the calculations of the power of deformation forces the possibility of using linearization of the intensity of deformation rate was discussed [17,18].
Another technique to simplify calculations of a reduced pressure in analytical form is searching for a family of functions describing the necessary inclined curvilinear boundary of the kinematic module that allows to obtain the power of deformation, friction and shear forces [19]. This method is based on the use of limit functions as solutions to some differential equation. In the future, this allows us to obtain an expression of the intensity of deformation rates in the form acceptable for further integration over the region of the kinematic module. Another method of simplifying the reduced pressure components of the reduced pressure is the use of upper estimates of the power of deformation forces according to the Cauchy-Bunyakovsky formulas or cubature formulas [20,21]. It may also be necessary to simplify the power of shear and friction forces taking into account the KPVF for modules with curvilinear boundaries. Obtaining the function of the reduced pressure in an analytical form will allow optimization for the selected variable parameters. Taking into account the perspectivity for the introduction of combined extrusion processes in the production, it is necessary, first of all, to analyze kinematic modules with some degrees of freedom of metal flow.
As an internal kinematic module 1, an axial trapezoidal module with an inclined line is widely used (Figure 1).

Figure 1. Axial trapezoidal kinematic module with straight T=T 1 (z) and curvilinear T=T 2 (z) inclined borders
This module can be used with the lower adjacent of various configurations and rigid kinematic modules bordering the upper and inclined limits. In this case, the inclined boundary in the simplest case can be in the form of a straight line, in general -a curve reflecting the nature of the metal flow. This kinematic module can be used for modeling of the flow in combined extrusion processes with two degrees of metal flow in the forward and reverse directions.
We consider KPVF of module 1 in general form: Using for (1) in the simplest case the rectilinear inclined boundary of the flow interface (rectilinear inclined boundary of the flow interface), we obtain the co-relation [19]: where The inefficiency of using linearization to obtain the power of deformation forces for a curvilinear boundary of a general form is demonstrated in [13]. The use of the upper bound is also difficult because of the complexity of the integrand. Therefore, as an alternative method, the search for a "convenient" function satisfying these conditions is chosen and does not cause difficulties in further calculations.
In this regard, as a curvilinear face, we choose a function that satisfies the differential equation: Given the general solution of equation (3) and boundary conditions Thus, the main difficulties of calculations of the axial kinematic module with an inclined curvilinear boundary (4)  Of interest is the analysis of the magnitude of the power shear forces between module 1 and the possible configurations of module 2: where ( ) For the rectilinear boundary of module 1 in the form ) For a curved incli- Further calculations are carried out in relative terms We pass for dimensionless quantities For a curved inclined boundary, the optimal value 2 W shifts in the direction of decreasing this value. Thus, the use of relative velocity 2 W as a variable parameter is possible. This optimization is possible only when combining kinematic modules with a set of radial velocity components r V one direction (sign) for both adjacent modules and not possible in other cases. Indeed, for the entire volume of module 1 for any form of boundary ) and for module 2 as 2а is made the condition 0 . For combinations of module 1 and modules 2b and 2c, this condition did not meet. Under the condition of radial components of one direction (sign) in adjacent modules 1 and 2a, calculations of the combined radialbackward extrusion process are carried out ( Figure .3). In paper [19] the value of the reduced pressure p taking into account the optimization for kinematic parameter opt W 2 is obtained: Given the optimal value opt W 2 according to (8), we obtained data on the increase in the size of the semifinished product in the vertical ↑ Δ 1 l and radial → Δ 2 l directions in the course of the deformation process at the initial height 0 H : A comparative analysis of this deformation process in the presence of rectilinear and curvilinear inclined boundaries of the kinematic module 2 is carried out. The comparative analysis of theoretically and experimentally obtained data showed the reliability of the optimization results by the kinematic parameter 2 W . Overvaluation of the theoretically obtained results for the energy-power parameters of the process (R 1 =10.5 mm, R 2 =1 mm, h 1 =3 mm, H 0 =17 mm) does not exceed 10 15 % (Figure 4). Comparative analysis of increments of a semifinished product was performed for (1): R 1 =10.5 mm, R 2 =14 mm, h 1 =5 mm, H 0 =14 mm and (2): R 1 =10.5 mm, R 2 =14 mm, h 1 =3 mm, H 0 =14 mm, based on the proposed theoretical assessments with point data obtained by modeling in Qform2D/3D ( Figure 5).
The deviation of magnitude ↑ Δ 1 l throughout the entire deformation process for ) ( 2 z T T = indicates the smallest deviation compared to ) ( 1 z T T = from the point data obtained by modeling in Qform2D/3D [19].
At the same time, optimization by kinematic parameter 2 W for schemes with a disconnected deformation zones in the presence of a rigid zone of type 2b no results were obtained [22].

CONCLUSIONS
The present paper has developed the novel engineering computations for power of shearing force for the combination of the axial trapezoidal module with the lower module of common configurations. Engineering conditions of usage of kinematical parameter of outlet metal velocity as optimization parameter have been found for the first time. The authors' approach provides possibility to derive the optimized value of dimensionless extrusion pressure, related energy-power characteristics and deformation-induced plastic form change for the processes of combined extrusion with the several degrees of freedom of plastic flow.
The paper analyzes the features of using the axial trapezoidal module in the design schemes with an adjacent lower module of various configurations. As adjacent kinematic modules, a rectangular rigid, longitudinal-transverse flow and a trapezoidal kinematic module with a rectilinear inclined boundary are chosen. For each of the adjacent modules, the shear forces W . The optimal value of the quantity 2 W corresponds to the fulfillment of the condition that the power of the shear forces is equal to zero. It has been established that optimization with respect to the relative metal flow velocity in the backward direction is possible only with a combination of kinematic modules with a radial component r V of one direction (sign) and is impossible in other cases. The calculations of the combined radial -backward extrusion process r V are carried out under the condition of the same radial component direction r V in adjacent modules. The usage of the relative metal flow velocity in the backward direction as an optimization parameter was demonstrated. The comparative analysis of theoretical and experimental data and simulation results in Qform2D/3D was conducted. The deviation of the theoretically obtained estimates of the force parameters of the process and shaping does not exceed 10 -15%.
Author-proposed computations for the powers of shearing forces can be successfully used for engineering-friendly modeling of more complex processes of the cold combined extrusion with additional taking into account of availability of metal flow in the forward, backward and radial directions.