# Finite Element Modeling of ECAP-Linex Combined Process of Severe Plastic Deformation

Authors: Evgeniy Panin; Irina Volokitina; Andrey Volokitin; Abdrakhman Naizabekov; Gulzhainat Akhmetova; Sergey Lezhnev; Andrey Tolkushkin; Aibol Esbolat; Rozina Yordanova
Journal: Modelling and Simulation in Engineering
 (2023)

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## Abstract

The paper presents theoretical studies of a new deformation process combining the stages of equal-channel angular pressing (ECAP) and the “Linex” scheme. For correct finite element modeling of the process, a technique with sequential input for the calculation of conveyor links is presented. To analyze the efficiency of metal processing, the main parameters of the stress–strain state are considered: equivalent strain, equivalent stress, and average hydrostatic pressure, as well as the deformation force on the main elements of the combined process: pulley, matrix, and conveyor link. To analyze the resulting deformation forces, the stages of pressing in a matrix and compression by a chain conveyor were separately considered. Equations for determining the forces acting on the drive pulley, ECA matrix, and the chain element link were obtained. Comparison of values showed that the force values in the calculation and simulation have a high level of convergence. In all three considered details, the difference value did not exceed 10%. The variational modeling allowed to determine the optimal values of the main technological and geometric parameters of the process.


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## Body

## 1. Introduction

Over the past three decades, a large number of metal forming methods have been developed and investigated, allowing to obtain blanks with an ultrafine-grained structure. These methods are based on various schemes of shear or alternating strains. Processes representing a simultaneous combination of shear and alternating strains are a separate category. All these processes make it possible to implement a special type of pressure treatment, which is called “severe plastic deformation”.High-pressure torsion is one of the oldest methods for obtaining bulk ultrafine-grained and nanostructured samples [1–3]. The samples obtained by this method have the shape of disks. The sample is clamped between the punch and the caliper and compressed under an applied pressure of several GPa. When the caliper rotates, the surface friction forces cause the sample to deform according to the shear pattern. The bulk of the material is deformed under quasi-hydrostatic compression under the action of applied pressure and pressure from the outer layers of the sample. As a result, despite the high degree of strain, the deformable sample does not collapse. In this case, the deformation of the sample has a radial inhomogeneity, which can be minimized by a large number of revolutions. Using the method of torsion under high pressure in various materials, it was possible to obtain a structure with a grain size of up to 20 nm. However, the prospects for using high-pressure torsion as an industrial method have significant disadvantages due to the small size of the workpieces being processed and low tool resistance due to high loads. This fact seriously narrows the practical application of this method and actually limits it to laboratory conditions.The method of equal-channel angular pressing (ECAP) is devoid of many of these disadvantages and allows you to obtain samples of square or rectangular cross section with a homogeneous ultrafine-grained structure with a grain size of 100–200 nm and does not require complex equipment. The method consists in pushing the workpiece through the angular channel of the matrix and implements a simple shift scheme. The technology of ECAP and its various variations is considered in [4–7]. Among the new directions in ECAP is the processing of hard-to-form materials. Experimental and theoretical modeling of the mechanics of ECAP, associated with studies of the stress–strain state, contact stresses, and friction conditions, made it possible to design tooling for obtaining large-sized blanks from various metals, such as copper, titanium, tungsten, and aluminum, as well as in various alloys based on them [8–11].In addition to the considered SPD processes, which make it possible to obtain small-sized blanks, severe plastic deformation processes are actively developing, which make it possible to process massive blanks. These methods are based on the intensification of shear and alternating strains in forging processes [12–16]. As a result, the initial billets in the form of ingots receive a high level of processing, which leads to intensive grinding of grain throughout the cross section. And the use of new deforming tool designs due to intensive shear strain allows to reduce energy consumption compared to the use of classic flat strikers.Despite the fact that the processes of severe plastic deformation are an effective way of grinding the metal structure [17–21], most of these methods remain used only in laboratory conditions. The main disadvantages of these technologies are the lack of continuity or the inability to process long workpieces. Attempts have been made to circumvent these limitations by developing combined processes, where two or more discrete processes are combined [22–26]. These methods have proven themselves well both in terms of the efficiency of metal processing and the deformation productivity. Therefore, the development of new combined deformation processes is one of the most promising areas in metal forming.In the work [27], new concepts of combined metal forming processes were proposed, one of which is the “ECAP-Linex” combined process (Figure 1). This method is designed for continuous pressing of non-ferrous metals and alloys, its key difference from the classical Linex process will be the possibility of deformation without significantly changing the initial dimensions of the workpiece.Figure 1
“ECAP-Linex” combined process. 1: Movable belt blocks; 2: ECAP matrix, 3: blank, 4: idle pulleys, 5: drive pulleys; 6: fixed locking blocks.Deformation in this device is carried out as follows. The workpiece is fed to the device, where movable chain blocks grab the workpiece and push it through the channels of the fixed matrix. Each chain gripping block is clad with two pulleys, one of which is idle, and the other is driven by an electric motor. Due to this, the chain gripping blocks are set in motion. The horizontal forming of the chain gripping blocks is created due to their movement along the workpiece and fixed locking blocks that perform a clamping role.The most important stage before the practical implementation of any deformation process is developed is its theoretical study, which is usually carried out for preliminary assessment of the emerging energy power parameters. By adjusting their values, it is possible to achieve conditions for the stable course of the deformation process, i.e., such conditions under which the deformation will occur without forced stops caused by the jamming of the workpiece in the tool.The purpose of this work is finite element modeling of a new deformation scheme “ECAP-Linex” with an assessment of the parameters of the stress–strain state and the deformation force under various process conditions.


## 2. Methodology for the Model Constructing of ECAP-Linex Process

When modeling the deformation process by finite element method (FEM), the researcher has much wider opportunities to study the parameters of the process. In particular, it becomes possible to study various parameters at any point of the workpiece and the tool, to analyze their values for exceeding the permissible limits, which makes it possible to assess the possibility of various defects on the workpiece or the probability of breakage of the deforming tool. It is also possible to carry out variational modeling, i.e., constructing a series of identical models in which one or more parameters change. After evaluating the emerging parameters of the stress–strain state and the deformation force, it is possible to determine the most optimal geometric and technological parameters of the process.To simulate the capture of metal by a conveyor, to find the force generated by the chain conveyor, it is necessary to consider in more detail the area where the workpiece receives compression. Here, two rational schemes of chain movement along the locking block are possible—angular and radial (Figure2).Figure 2
Compression section of chain conveyor: (a) with angular joint; (b) with radial joint.
(a)(b)The most optimal option would be a radial design, since in this case, there will be no lower corner in the contact zone of the workpiece and the tool. When the chain elements move along the fixed locking element, they hit an angle, and clamps will form on the workpiece surface, in the radial version, such clamps will be minimal or completely absent, which depends on the width of the chain element links. At the same time, it should be noted that, in both cases, the curve lengths bounding the deformation zone are commensurate, the difference in their lengths is about 0.5%. Therefore, for the calculation convenience, it is possible to take the shape of this deformation zone for a rolling-type shape formed by rolls.When creating a FEM model of this process, it is necessary to correctly set the speed parameters of the deforming elements. According to the principle of the Linex process [28], chain elements receive movement from rotating pulleys, when passing along the contour of fixed blocks, they grab the workpiece, compress it, and push it through the channels of the matrix. Since the linear velocity of the chain element links will be equal to the linear velocity on the surface of the rotating pulleys, it is most appropriate to simulate the movement of chain elements as follows. The rotating pulley is created with the curvature radius of the fixed block (green zone in Figure 2(b)). At the exit from the vertical axis of pulley rotation, the shaped elements of the matrix are located (Figure 3). On the upper face of the matrix, the horizontal line of which corresponds to the lower point level of the pulley radius, single links are created sequentially (the length of the links should be small, for a given radius of the pulley 50 mm, it was adopted 5 mm). The links are given in a linear velocity of movement. Taking into account the chosen rotation speed of the pulleys of 15 rpm (1.57 rad/s) and a radius of 50 mm, the linear speed of the links will be 78.5 mm/s.Figure 3
Arrangement of elements in the FEM model.The following geometric and technological parameters were used to create the basic model of ECAP-Linex process:h1: workpiece height, mm;b1: workpiece width, mm;l1: length of the first channel in the matrix, mm;l2: length of the second channel in the matrix, mm;l3: length of the third channel in the matrix, mm;μ1: friction coefficient in the deformation zone;μ2: friction coefficient in the matrix;σS: metal resistance to plastic deformation (can be taken as yield strength), MPa;φ: angle of intersection of matrix channels, deg.The following values were accepted:R=50mm, b1=10mm, h1=10mm, Δh=3mm, μ1=0,7, μ2=0,05, l1=30mm, l2=20mm, l3=15mm, σS=60MPa, φ=140°.To increase the calculation speed, it was decided to use horizontal symmetry, i.e.,1/2 of the thickness of the workpiece was modeled. According to this condition, the initial blank had a width of 9 mm, a height of 6.5 mm, and a length of 75 mm. The workpiece volume was divided into 45,000 finite elements with a volume difference factor of 3, i.e., the largest element by volume was three times larger than the smallest.


## 3. Results and Discussion

### 3.1. Form Change and Stress–Strain State Analysis

It was found that the process proceeds stably with the above parameters (Figure4).Figure 4
Calculated model at the final stage.At the same time, gaps of 0.5 mm were deliberately made between the links to assess the possible leakage of metal into them at the time of compression during the passage of both channel joints. In this case, no signs of metal flowing into the gaps were found. This is a consequence of the difference in the kinematics of the process under consideration from the usual ECAP in this matrix. When the metal hits any of the joints, the workpiece experiences backpressure from the matrix and begins to decompress, filling the entire space of the matrix channel (Figure5(a)). At the first junction, when entering from the first channel into the second inclined one, an identical pattern of shape change is observed in both processes. However, at the second joint in the ECAP-Linex process, the metal receives an additional share of the pushing force from the coupling with the links. Therefore, due to the tension here, the joint angle is not filled (Figure 5(b)).Figure 5
Filling the corners of the joint with metal: (a) the first joint; (b) the second joint.
(a)(b)When gaps were set of 1 mm, small metal influxes with a height of 0.2–0.4 mm were formed (Figure6). Therefore, it is recommended not to exceed the gap size of 0.5 mm.Figure 6
Initial stage of metal flowing into the gaps.To analyze the stress–strain state, the following parameters were selected: equivalent strain, equivalent stress, and average hydrostatic pressure. The first two parameters show the intensity of stress and strain development, and for classical metal forming methods, this is often enough. However, the ECAP-Linex can be attributed to a group of combined processes, where the workpiece is first rolled in the deformation zone of the drive pulleys, then it is pressed in the matrix segments, simultaneously being pulled out of it by the chain element links. This indicates a rather complex deformation scheme, so it would also be advisable to consider the average hydrostatic pressure, which shows the values of tensile and compressive stresses.All three of the above parameters include the main components of strains and stresses in the following form:(1)εEQV=23ε1−ε22+ε2−ε32+ε3−ε12,(2)σEQV=12σ1−σ22+σ2−σ32+σ3−σ12,(3)σAHP=σ1+σ2+σ33,where ε1, ε2, and ε3 are the main strains; σ1, σ2, and σ3 are the main stresses.It can be seen from equations (1)–(3) that the equivalent strain and stress will always take a positive value, since they are root expressions. The average hydrostatic pressure can be either positive or negative, depending on the values of the main components of the stress tensor.Before starting the analysis, it is necessary to determine the stage of the process at which the stress–strain state parameters will be studied. The most rational solution would be the stage at which the maximum values of the backpressure of the matrix arise. This will allow to evaluate not only the numerical values of all parameters but also the possibility of the deformation process under these conditions. Therefore, the final stage of deformation was chosen for analysis, when the workpiece is located in all three channels of the matrix, the front end coming out of it.Considering the equivalent strain (Figure7), it can be noted that with this deformation scheme, the workpiece receives a strain increase in three stages: when compressed by pulleys and when passing two joints in the matrix. When analyzing the distribution of equivalent stress, it may seem that this process cannot proceed stably, since the stress level in the matrix exceeds the stresses in the center of deformation of the pulleys in terms of its value and extent. However, here it is necessary to take into account the cross-sectional area in which these stresses act—it is 17% larger in the deformation zone of the pulleys. Therefore, a level of force is created here that exceeds the backpressure of the matrix. When analyzing the average hydrostatic pressure, it is necessary to forcibly set the scale in such a way that there is a zero mark on it. This will make it easy to distinguish between stretching and compression zones by color scheme. From Figure 7(b), it can be noted that the entire volume of the workpiece is in a state of compression. A separate stretching zone is formed on the outer inclined face, which is the result of pulling the workpiece out of the matrix by links. It is this factor that leads to incomplete filling of the second joint with metal (Figure 5(b)).Figure 7
Stress–strain state parameters of ECAP-Linex process: (a) equivalent strain; (b) equivalent stress; (c) average hydrostatic pressure.
(a)(b)(c)To numerically evaluate these parameters, it is advisable to use the “Point Tracking” tool, which allows you to obtain the values of all parameters simultaneously when passing a given area of the workpiece through all deformation zones. In this case, an effective way will be to simultaneously evaluate the parameters on the surface of the workpiece and in the central section, which will make it possible to assess the level of uniformity of the distribution of parameters.To fulfill these conditions, two points were selected—the first on the workpiece surface and the second in the central section. Both points had the same coordinates on theX- and Y-axes—15 mm from the front end of the workpiece and the middle of its width, and differed only in their location in the vertical Z-plane (Figure 8).Figure 8
Place of tracking points.Figure9 shows a graph of the equivalent strain accumulation. Conventionally, it can be divided into four sections: I—the compression section of the drive pulleys; II—the section between the pulleys deformation zone and the first channel joint in the matrix, characterizes the input channel of the matrix; III—the section between the two channel joints in the matrix, characterizes the inclined channel of the matrix; IV—the section between the second channel joint in the matrix and the output from it, characterizes the output channel of the matrix. At the same time, the difference in the level of strain accumulation along the thickness of the workpiece is clearly visible. In the first zone, due to the development of an advance in the deformation zone, the difference is about 20%. In the second zone, this difference remains unchanged, which indicates that, in this section, the influence of the contact of the workpiece with the links of the chain conveyor is extremely insignificant—due to the rectilinear movement in the first channel, the decompression factor is completely absent.Figure 9
Graph of the equivalent strain development.In the third zone, after passing the first junction of the channels, the difference in strain levels begins to grow sharply. This is due to the fact that when passing the junction of the channels, the metal first completely fills the cavity of the matrix and then begins to decompress in the vertical direction, leading to tighter contact with the conveyor links. As a result, due to the increased adhesion of them, the surface layers receive a higher level of strain. In the central layer, after passing the junction of the channels, the level of strain remains at a constant level. In the fourth zone, the strain development is identical to the third zone. As a result, after one deformation cycle, a strain of 0.9 develops in the central zone, and on the surface, the strain level is about 1.34, which is 49% higher.Analyzing the graph of equivalent stress (Figure10), it can be noted that in the first zone, during compression in the pulleys, stress develops as the height of the workpiece decreases in the deformation zone. At the same time, the difference in the stress values along the thickness of the workpiece is small and is about 12% (90 MPa on the surface and 80 MPa in the center). In the second section, there is a drop in the stress level to an average of 25 MPa—since there is no strain development, the stress arises only from the friction of the metal with the rectilinear walls of the matrix. As a result, the stress level in this zone is almost the same with the thickness of the workpiece.Figure 10
Graph of the equivalent stress development.In the third zone, when passing the junction of the channels, the stress value increases sharply along the entire thickness of the workpiece up to 105 MPa. At the same time, after overcoming the joint area (short blue borders on the graph), the stress level on the surface becomes higher than in the center (118 MPa on the surface and 105 MPa in the center), which is the result of increased adhesion of the metal to the conveyor links. In the fourth zone, this effect is repeated, the stresses increase when the second joint passes up to 120 MPa, equalizing at the exit from its area. Further, stresses remain at the same level, since the output channel, like the first input channel, has a rectilinear structure, and here the stress arises only from friction against the walls of the matrix and the conveyor links. However, due to the higher level of adhesion, the stress level is much higher here than in the second zone.When considering the graph of the average hydrostatic pressure development (Figure11), it was found that in the first zone, compressive stresses develop in the central layers (approximately −10/−15MPa), while tensile stresses first act on the surface when capturing the metal (about 25 MPa), then during the steady rolling process, there are compressive stresses reaching −75 MPa. At the entrance to the matrix (the boundary of the second zone), a sharp jump of tensile stresses occurs on the surface, which is the result of free broadening during rolling in pulleys shaped like a smooth barrel. Because of this effect, the width of the workpiece is intentionally made smaller than the width of the matrix channel. However, after the matrix cavity is filled in width, the stresses decrease to the level of −10/−15MPa, which indicates the state of compression in the entire second zone. At the same time, the stress level here is low, since, as already established, there is no strain development, and the stress arises only from the friction of the metal against the rectilinear walls of the matrix.Figure 11
Graph of the average hydrostatic pressure development.When passing the first junction of the channels, the level of compressive stresses along the entire thickness of the workpiece increases to −40 MPa, then when moving in an inclined channel, the nature of stresses along the thickness changes. Tensile stresses of up to 60 MPa grow on the surface of the workpiece due to the adhesion of the metal to the conveyor links and the corresponding forced movement. In the central layers along the entire length of the inclined channel, the level of compressive stresses increases to −95 MPa. This effect is the result of the preform being pressed in the channel and with ECAP in a conventional matrix, such a stress level will act over the entire section of the workpiece. However, due to the presence of moving conveyor links, there are differences in the kinematics of movement on the workpiece surface, which leads to an increase in tensile stresses. When passing through the second junction of the channels, an increase in compressive stresses to –112 MPa occurs again, and after entering the rectilinear output channel, the stress level along the cross section almost equalizes and decreases to about –10 MPa in the center and 15 MPa on the surface. Here, the tensile stresses on the surface slightly exceed the compressive stresses in the center, since after decompressing in an inclined channel, the metal of the workpiece has an increased adhesion level to the moving links of the conveyor.


### 3.2. Analysis of Deformation Forces

In the “ECAP-Linex” process, the key element of deformation is an equal-channel angular matrix with parallel channels. Chain conveyors perform a dual role. Firstly, they advance the workpiece along the channels of the matrix due to adhesion to the workpiece. Secondly, they deform the workpiece by some compression in height, due to which the main level of active friction force develops, contributing to the advancement of the workpiece through the channels of the matrix.Therefore, for the stable course of the deformation process according to the proposed scheme, it is necessary to comply with the conditions:(4)PCONV>PMATR,where PCONV is the force created by the chain conveyor; PMATR is the backpressure force created by the matrix.Let’s consider each of these efforts separately. To find the backpressure force created by the matrix, it is advisable to use the equation of the pressing force in this matrix, which was obtained in [29]:
(5)PPRESS=2σSμ22l1+l2b1+h1+2h12tgϕ2+tgϕ⋅h13μ2,here, the authors of [29] made the assumption that the input and output channels have the same length. If this condition is not met, equation (5) takes the form:
(6)PPRESS=2σSμ2l1+l2+l3b1+h1+2h12tgϕ2+tgϕ⋅h13μ2.This equation characterizes the theoretical maximum force that occurs when the workpiece is in all three channels of the matrix. In real conditions of pressing, it will always be smaller due to the fact that when the punch moves, the volume of metal in the first channel will constantly decrease.In the “ECAP-Linex” process being developed, only curly elements forming a channel will be present in this matrix design. There are no side walls, since their role is performed by the elements of the chain conveyor. Therefore, equation (6) in relation to the curly elements of the matrix takes the form:
(7)PMATR=2σSμ2h1l1+l2+l3+2h1tgϕ2+tgϕ3μ2.In this case, the deformation zone can be represented as follows (Figure12).Figure 12
Deformation zone in the conveyor.The sum of all forces acting in the deformation zone is determined by the equation:(8)PCOMPR=2bav∫γ1ατavRcosθdθ−2bav∫0γτavRcosθdθ−2bav∫0αpavRsinθdθ,where b1иbav is the workpiece width after compression and the average width; τavиpav are the average tangential and normal stresses; R is the curvature radius of the locking block (analogous to the roll radius); θ is the current angle; α is the capture angle; γ, γ1 are the angles characterizing the advance and lag zones, respectively.Integrate equation (8) taking into account the assumption that bav=b1:
(9)PCOMPR=2b1Rτavsinα−sinγ1−2b1Rτavsinγ−0−2b1Rpav−cosα+1.After replacing in this equation:1−cosα=α2/2, sinα=α, sinγ1=γ1, sinγ=γ, τav=pavμ=σSμ, equation (9) has the form:
(10)PCOMPR=2b1RσSμ1α−γ1−2b1RσSμ1γ−2b1RσSα22,where μ1 is the friction coefficient in the deformation zone.Final form of equation:(11)PCOMPR=2b1RσSμ1α−γ1−γ−α22μ1.It can be seen from equation (11) that under equal geometric conditions in the deformation zone, the magnitude of the compression force will depend on the values of the angles γ and γ1, which depend on the magnitude of the backpressure force created by the matrix. In the same compression angle α, there will be different zones of advance, lag, and adhesion each time. Therefore, to use equation (11), it is necessary to find the values of these angles.The equations of equilibrium of forces and moments acting in a symmetrical deformation zone during rolling on a front support, which arises due to an additional shape change in the matrix installed behind the rolls, have the form:(12)2bav∫γ1ατavRcosθdθ−2bav∫0γτavRcosθdθ−2bav∫0αpavRsinθdθ−σ1b1h1=0;(13)2bavτavR2α−γ1−2bavτavR2γ−2bavψRα×∫0αpavRcosθdθ+∫γ1ατavRsinθdθ−∫0γτavRsinθdθ−σ1b1h1R=0,where σ1 is the backpressure stress; ψ is the coefficient of the shoulder position of the resultant metal pressure on the rolls; α, γ are the capture angle and the angle characterizing the length of the advance zone.Taking into account the assumption thatbav=b1:
(14)∫γ1ατavcosθdθ+∫0γτavcosθdθ−∫0αpavsinθdθ−σ1h12R=0;(15)τavα−γ1−τavγ−ψα∫0αpavRcosθdθ+∫γ1ατavsinθdθ−∫0γτavsinθdθ−σ1h12R=0.Integrating equation (14) and replacing 1−cosα=α2/2, sinα=α, sinγ1=γ1, sinγ=γ, γ1=γadh+γ, after the transformations, the dependence for determining the angle characterizing the extent of the adhesion zone will be:
(16)γadh=α−2γ−pavα22τav−σ1h12τavR.After replacementpav/τav=1/μ1 и τav=σSμ1, equation (17) is transformed to the form:
(17)γadh=α−2γ−α22μ1−σ1h12μ1σSR.After integration and transformation of equation (15):
(18)α−γ1−γ=ψαpavτavα−cosα+cosγ1+cosγ−1+σ1h12τavR.Transformation of the resulting expression into a quadratic equation by performing substitutions similar to those used in solving equation (14):
(19)2γ2−γ2α−α2μ−σ1h12μσSR+αμ1ψ−2−α2+α34μ1−σ1h1σSR+ασ1h12μ1σSR+σ12h124μ1σS2R2α=0.One of the roots of the quadratic equation (19) will be the angle characterizing the length of the advance zone:
(20)γ=2α−α2μ1−σ1h12μ1σSR−2α−α2μ1−σ1h12μ1σSR2−8αμ11ψ−2−α2+α34μ1−σ1h1σSR+ασ1h12μ1σSR+σ12h124μ1σS2R2α4To find this angle characterizing the advance zone according to equation (19), it is necessary to first determine the coefficient of the shoulder position of the resultant ψ. To do this, consider the conditions under which equation (19) makes sense:
(21)2α−α2μ1−σ1h12μ1σSR2>8αμ11ψ−2−α2+α34μ1−σ1h1σSR+ασ1h12μ1σSR+σ12h124μ1σS2R2α;(22)1ψ−2−α2+α34μ1−σ1h1σSR+ασ1h12μ1σSR+σ12h124μ1σS2R2α>0.Solving the inequalities (21) and (22) together, the limits in which the value changes 1/ψ:
(23)μ18α2α−α2μ1−σ1h12μ1σSR2+2+α2−α34μ1+σ1h1σSR−ασ1h12μ1σSR−σ12h124μ1σS2R2α>1ψ>2+α2−α34μ1+σ1h1σSR−ασ1h12μ1σSR−σ12h124μ1σS2R2α.Assuming that the value1/ψ is in the middle part of the indicated limits:
(24)1ψ=μ116α2α−α2μ1−σ1h12μ1σSR2+2+α2−α34μ1+σ1h1σSR−ασ1h12μ1σSR−σ12h124μ1σS2R2α.Knowing the magnitude of the angles characterizing the extent of the advance and adhesion zones, the angle characterizing the lag zone can be found from the condition:(25)γtag=α−γadh−γ.However, in this combined process, the useful force pushing the workpiece through the channels of the matrix is expressed not only by equation (11). Here, in addition to the compression force, there will also be a force from the friction of the workpiece on the chain element links, since their movement is directed in the same direction as the movement of the workpiece.Therefore, the maximum possible force generated by the conveyor at the moment when the workpiece completely fills all channels of the matrix:(26)PCONV=2b1σSμ1Rα−γ1−γ−α22μ1+l1+l2+l3.The force of advancing the workpiece by one link of the chain element will be equal to:(27)P1_EL=2σSμ1b1lEL,where lEL is the single link length of the chain element.A trial calculation with the simulation data was performed. When entering the algorithm into Microsoft Excel and varying the value of the channel junction angle in the matrix from 90° to 180°, the following data were obtained (Figure13).Figure 13
Dependences of the forces of the “ECAP-Linex” process on the value of the channel junction angle.After calculating the model, the following force graphs for the pulley matrix and chain element link were obtained (Figure14).Figure 14
Force in FEM simulation: (a) pulley; (b) matrix; (c) link.
(a)(b)(c)Table1 shows the force values obtained by calculating equations and simulation. Comparison of values showed high convergence in all three cases.Table 1
Calculation and modeling forces.
Calculation force,NSimulation force,NDifference, %Pulley12542122302.48Matrix919290491.55Link420041162


### 3.3. Variational Modeling

#### 3.3.1. Initial Data for Variational Modeling

After analyzing the basic model, the task to determine the influence of technological and geometric parameters of the process on the stress–strain state and deformation force was set. To do this, it is necessary to change one geometric or technological parameter in the base model and track the change in stress–strain state parameters and deformation force. Based on the analysis of the obtained results, it will be possible to determine the most optimal values of the parameters.At the initial stage of variational modeling, it is necessary to determine the parameters to be varied. It is most advisable to choose those parameters that can be easily changed in real laboratory or production conditions. From the geometric parameters, it is most advisable to change the angle of the channels junction, this will require the installation of only new curly blocks without changing the design of the conveyor. It is irrational to change the radius of the pulley, since in this case, it will be necessary to manufacture new chain elements. It is also possible to change the lengths of the matrix channels. From the technological parameters, it is possible to change the workpiece temperature, the rotation speeds of the pulleys, and the movement of chain elements, as well as the friction coefficients in the matrix and conveyor.At the next stage, it is necessary to determine the parameter variation intervals. The values of the variable parameters should be achievable in real conditions. At the same time, if it is possible to vary in both directions, then it is necessary to change the initial value to the same value. Therefore, the following parameter values were selected:- Channel junction angle: 125° and 155°.- Lengths of the matrix channels: each channel has been increased by 10 mm (it is irrational to reduce the initial values, since this will negatively affect the strength of the tool).- Workpiece temperature: 100°C (a further increase in the heating temperature will be irrational, since this will lead to the beginning of recrystallization [30]).- The speed of the chain elements directly depends on the rotation speed of the pulleys. Therefore, the values of 5 rpm and 25 rpm were selected. Thus, the speeds of the chain elements were set to 26.1 mm/s and 130.8 mm/s.- The friction coefficients in the matrix and conveyor in the basic model already have critical values, so they can only be changed in one direction. As a result, the following values were set: 0.15 for the matrix and 0.5 for the conveyor.When analyzing these models, cases of unstable deformation were identified—in some models, conditions arose when, at a certain stage, the workpiece stopped moving in the matrix due to a too low level of active friction forces in the pulley deformation zone, or, conversely, due to a too high backpressure level in the matrix. In any case, these models should be considered unsuccessful, and the conditions for their implementation should be considered negative. These models include both models with modified friction coefficients—in both cases, the workpiece jammed at the entrance to the first joint of the channels. Based on this, it can be concluded that these two parameters are critical for the stable flow of the ECAP-Linex combined process, and the selected values of the friction coefficients on the pulleys 0.7 and in the matrix 0.05 can be considered the most optimal. Moreover, an unsuccessful case was the model with an increased length of the second inclined channel—in this case, the jamming of the workpiece occurred at the second junction of the channels. Therefore, the length of the second channel of 20 mm should not be increased.In addition to the unsuccessful models, it should be separately noted models in which no significant changes in stress–strain state or deformation force were recorded when the parameters were changed. These are models with increased input and output channel lengths. Such a “neutral” result is quite understandable, since these channels have a rectilinear direction. In the input channel, the level of back pressure is minimal, since only friction forces against the matrix walls act here. In the output channel, the level of backpressure increases sharply due to the design of the matrix, but in this combined process, it is almost completely annihilated by the moving links of the conveyor, which leads to incomplete filling of the second joint (see Figure5(b)). The remaining conditions led to an increase or decrease in the parameters of stress–strain state or deformation force, so the corresponding models were considered in detail.


#### 3.3.2. Models with Modified Channel Junction Angles in the Matrix

Figure15 shows graphs of the equivalent strain accumulation when using matrices with different channel junction angles.Figure 15
Equivalent strain in models with variable channel junction angles: (а) 125°; (b) 155°.
(a)(b)Comparing these graphs, we can say that the value of the joint angle in the matrix significantly affects the level of equivalent strain. In the first two stages, the strain levels are almost identical at both points (it is in the range of 0.3–0.35, as in the basic model). When hitting the first joint of the matrix, the metal receives a different level of shear strain, at the second joint, this effect is repeated. In a matrix with an angle of 125°, the strain level reaches 2.6 on the surface and 1.1 in the center, i.e., the difference in the values of strain in thickness reaches 236%. This suggests that, in this case, there is a significant gradient anisotropy in the cross section (the level of anisotropy is 3.3 times higher than in the basic model). In a matrix with an angle of 155°, the strain level reaches 0.7 on the surface and 0.65 in the center, i.e., the difference in the values of strain in thickness is only 7%, in this case, there is a fairly uniform distribution of strain over the cross section.It should be noted that in a matrix with an angle of 155°, this small difference persists at all stages, whereas in a matrix with an angle of 125°, the strain on the surface grows continuously (both due to shear strain at the joints and due to coupling with conveyor links), the central layers receive strain only in the joint zones, after their passage, the level of deformation does not change. The exception is stage III, where after passing the first joint, the strain level continues to grow, albeit less intensely than at the joint. This is due to the increased level of decompression and subsequent adhesion—in this case, this effect extends to the entire section of the workpiece. In a matrix with an angle of 155°, the growth of strain after passing the joints of the matrix is absent both in the center and on the surface. This means that in this case, the level of strain depends only on the shear strain, and the level of adhesion to the links is extremely small due to the low level of metal compression.When analyzing the equivalent stress graphs (Figure16) and comparing them with the basic model, the general similarity at all stages can be noted. The key difference is observed at stage III, where a small stress drop occurs in the basic model, which increases significantly in the matrix with an angle of 155°. As in the case of equivalent strain, here this effect is the result of a decrease in the level of coupling with the conveyor—in a model with an angle of 125°, where the coupling is significantly higher than the base model, this effect is absent. At the same time, it should be noted that a decrease in the coupling level affects the overall stress level—in the model with an angle of 155°, the maximum level of equivalent stresses in the matrix is 10–12% less than in the other two.Figure 16
Equivalent stress in models with variable channel junction angles: (а) 125°; (b) 155°.
(a)(b)When analyzing the graphs of average hydrostatic pressure (Figure17) and comparing them with the basic model, it was found that the first two stages are almost identical. At stage III, in a model with an angle of 125°, the same effect is observed as in the basic model—tensile stresses reaching 105 MPa act on the surface, and compressive stresses reaching −115 MPa act in the central zone. In the model with an angle of 155°, due to the reduced coupling with the conveyor, the level of average hydrostatic pressure along the thickness of the workpiece is almost the same. Moreover, the stress level is much lower here—compressive stresses reach −56 MPa, while the maximum tensile stresses reach only 18 MPa.Figure 17
Average hydrostatic pressure in models with variable channel junction angles: (а) 125°; (b) 155°.
(a)(b)The values of forces arising on the pulley matrix and conveyor link were also considered (Figure18). Table 2 shows the values obtained by calculating and modeling. As in the basic model, the comparison of values showed high convergence.Figure 18
Force in models with different channel junction angles in the matrix: (a) 125° pulley; (b) 155° pulley; (c) 125° matrix; (d) 155° matrix; (e) 125° link; (f) 155° link.
(a)(b)(c)(d)(e)(f)Table 2
Forces for ECAP-Linex process at different joint angles in the matrix.
Calculation force,NSimulation force,NDifference, %125°Pulley11583112602.78Matrix1015997933.73Link420041531.13155°Pulley13190128602.56Matrix853983781.92Link420039825.47There was also a clear dependence of the error level for the conveyor link on the angle value. This is the result of the fact that the force calculation equation does not take into account the possible level of decompression and finds the maximum possible force when the metal completely fills all cavities and corners of the joints. In real conditions, with an increase in the angle, the level of decompression decreases, which leads to a decrease in force and an increase in error.


#### 3.3.3. Model with Elevated Workpiece Temperature

When heating aluminum alloy 6061–100°C, its yield strength is slightly reduced to about 55 MPa. Figure19 shows the graphs of stress–strain state parameters when using a workpiece with an elevated temperature.Figure 19
Stress–strain state parameters in the model with elevated workpiece temperature: (a) equivalent strain; (b) equivalent stress; (c) average hydrostatic pressure.
(a)(b)(c)Analysis of the graphs of stress–strain state parameters in this model showed that the increased temperature of the workpiece leads to a decrease in the adhesion level of metal to the conveyor, which is reflected in a slight decrease in the difference of equivalent strain in the workpiece thickness. Therefore, if in the basic model, the difference was 49%, then in this model, it was 44% (with a deformation level of 1.33 on the surface and 0.92 in the center). The distribution of the equivalent stress has a similar appearance to the previously considered models with different channel junction angles—differences are observed only at stages III and IV. Here, at stage III, there is a stress drop, which is the result of a reduced adhesion level, which leads to the separation of stress zones at the joints of the matrix, whereas the model with an angle of 125° has a single stress zone along the entire length of the inclined channel. In terms of their values, the level of equivalent stresses here is almost the same as in the basic model, which is the result of a slight decrease in the yield strength at a given temperature.The distribution of the average hydrostatic pressure has a similar character—the main differences are observed at stages III and IV. At an elevated temperature of the workpiece at stage III, there are no tensile stresses in the surface layers—compressive stresses act here throughout the thickness of the workpiece, although the level of compressive stresses in the center is much higher than on the surface. At stage IV, due to the rectilinear channel, even a reduced level of adhesion is sufficient for the development of tensile stresses in the surface layers. At the same time, the overall stress level is 25–30% lower than the base value.A comparison of the resulting forces (Figure20 and Table 3) with the base values showed that an increase in the workpiece temperature leads to a decrease in force on all three parts by about 10–12%.Figure 20
Force at elevated workpiece temperature: (a) pulley; (b) matrix; (c) link.
(a)(b)(c)Table 3
Forces for ECAP-Linex process at elevated workpiece temperature.
Calculation force,NSimulation force,NDifference, %Temperature 100°СPulley11497108905.27Matrix842685421.35Link385036914.13


#### 3.3.4. Models with Modified Rotation Speeds of Pulleys and Movement Speeds of Chain Elements

Changing the rotation speed of the pulleys and, accordingly, the movement speed of the conveyor chain elements leads to a change in the strain rate. When considering the equivalent strain (Figure21), it was noted that in the central zone of the workpiece, the strain level with varying speed characteristics of the ECAP-Linex process is almost unchanged and is in the range of 0.9–0.95. On the surface, the strain level also remains constant in the range of 1.3–1.35.Figure 21
Equivalent strain with varying kinematic parameters: (a) 5 rpm (26.1 mm/s); (b) 25 rpm (130.8 mm/s).
(a)(b)When considering the equivalent stress (Figure22), it was noted that in the central zone of the workpiece at stage III, there is a drop in the level of equivalent stresses. The same drop is observed on the surface at a reduced speed of 5 rpm (26.1 mm/s). With an increase in speed characteristics, the drop in the level of equivalent stresses on the surface disappears. According to the numerical values, the maximum stress level in all cases, including the base, remains unchanged (about 120 MPa).Figure 22
Equivalent stress at variable kinematic parameters: (a) 5 rpm (26.1 mm/s); (b) 25 rpm (130.8 mm/s).
(a)(b)When analyzing the average hydrostatic pressure (Figure23), it should be noted approximately the same level of tensile and compressive stresses in the first two stages. At the third stage, at a reduced speed of 5 rpm (26.1 mm/s), there is a complete absence of tensile stresses both in the center and on the surface. An increase in speed characteristics leads to an increase in the influence of metal contact on the conveyor links, which leads to an increase in tensile stresses in the inclined channel after passing the zone of the first joint.Figure 23
Average hydrostatic pressure with varying kinematic parameters: (a) 5 rpm (26.1 mm/s); (b) 25 rpm (130.8 mm/s).
(a)(b)When analyzing the forces and comparing them with the calculated values (Figure24 and Table 4), it is necessary to take into account the fact that all equations for calculating forces do not contain speed components. Therefore, for all speed characteristics, the values of the forces in the calculation will be the same. A comparison of their model data showed that the effect of the strain rate on the force in the ECAP-Linex process does not have a pronounced dependence. This conclusion is supported by two facts:Figure 24
Force at various kinematic parameters: (a) pulley 5 rpm (26.1 mm/s); (b) pulley 25 rpm (130.8 mm/s); (c) matrix 5 rpm (26.1 mm/s); (d) matrix 25 rpm (130.8 mm/s); (e) link 5 rpm (26.1 mm/s); (f) link 25 rpm (130.8 mm/s).
(a)(b)(c)(d)(e)(f)Table 4
Forces for ECAP-Linex process at various kinematic parameters.
Calculation force,NSimulation force,NDifference, %5 rpm (26.1 mm/s)Pulley12542120803.68Matrix919295553.8Link420039376.2625 rpm (130.8 mm/s)Pulley12542119304.87Matrix919290691.34Link420038258.93- The effect of the strain rate on the force of ECAP has not been proven by either theoretical or experimental studies [21].- It is known from the rolling theory that the strain rate has the least effect on the force compared to temperature and compression.Usually, a significant influence of the speed on the deformation resistance and rolling force is fixed when it changes several tens of times. Taking into account the fact that in these models the speed changed by five times, the influence of this parameter at the stage of compression by pulleys is insignificant. Due to the absence of speed components in the equations, an increase in the difference level was recorded when comparing the values, but its maximum level does not exceed 10%. Therefore, the equations are quite suitable for calculating forces at different speeds.


## 3.1. Form Change and Stress–Strain State Analysis

It was found that the process proceeds stably with the above parameters (Figure4).Figure 4
Calculated model at the final stage.At the same time, gaps of 0.5 mm were deliberately made between the links to assess the possible leakage of metal into them at the time of compression during the passage of both channel joints. In this case, no signs of metal flowing into the gaps were found. This is a consequence of the difference in the kinematics of the process under consideration from the usual ECAP in this matrix. When the metal hits any of the joints, the workpiece experiences backpressure from the matrix and begins to decompress, filling the entire space of the matrix channel (Figure5(a)). At the first junction, when entering from the first channel into the second inclined one, an identical pattern of shape change is observed in both processes. However, at the second joint in the ECAP-Linex process, the metal receives an additional share of the pushing force from the coupling with the links. Therefore, due to the tension here, the joint angle is not filled (Figure 5(b)).Figure 5
Filling the corners of the joint with metal: (a) the first joint; (b) the second joint.
(a)(b)When gaps were set of 1 mm, small metal influxes with a height of 0.2–0.4 mm were formed (Figure6). Therefore, it is recommended not to exceed the gap size of 0.5 mm.Figure 6
Initial stage of metal flowing into the gaps.To analyze the stress–strain state, the following parameters were selected: equivalent strain, equivalent stress, and average hydrostatic pressure. The first two parameters show the intensity of stress and strain development, and for classical metal forming methods, this is often enough. However, the ECAP-Linex can be attributed to a group of combined processes, where the workpiece is first rolled in the deformation zone of the drive pulleys, then it is pressed in the matrix segments, simultaneously being pulled out of it by the chain element links. This indicates a rather complex deformation scheme, so it would also be advisable to consider the average hydrostatic pressure, which shows the values of tensile and compressive stresses.All three of the above parameters include the main components of strains and stresses in the following form:(1)εEQV=23ε1−ε22+ε2−ε32+ε3−ε12,(2)σEQV=12σ1−σ22+σ2−σ32+σ3−σ12,(3)σAHP=σ1+σ2+σ33,where ε1, ε2, and ε3 are the main strains; σ1, σ2, and σ3 are the main stresses.It can be seen from equations (1)–(3) that the equivalent strain and stress will always take a positive value, since they are root expressions. The average hydrostatic pressure can be either positive or negative, depending on the values of the main components of the stress tensor.Before starting the analysis, it is necessary to determine the stage of the process at which the stress–strain state parameters will be studied. The most rational solution would be the stage at which the maximum values of the backpressure of the matrix arise. This will allow to evaluate not only the numerical values of all parameters but also the possibility of the deformation process under these conditions. Therefore, the final stage of deformation was chosen for analysis, when the workpiece is located in all three channels of the matrix, the front end coming out of it.Considering the equivalent strain (Figure7), it can be noted that with this deformation scheme, the workpiece receives a strain increase in three stages: when compressed by pulleys and when passing two joints in the matrix. When analyzing the distribution of equivalent stress, it may seem that this process cannot proceed stably, since the stress level in the matrix exceeds the stresses in the center of deformation of the pulleys in terms of its value and extent. However, here it is necessary to take into account the cross-sectional area in which these stresses act—it is 17% larger in the deformation zone of the pulleys. Therefore, a level of force is created here that exceeds the backpressure of the matrix. When analyzing the average hydrostatic pressure, it is necessary to forcibly set the scale in such a way that there is a zero mark on it. This will make it easy to distinguish between stretching and compression zones by color scheme. From Figure 7(b), it can be noted that the entire volume of the workpiece is in a state of compression. A separate stretching zone is formed on the outer inclined face, which is the result of pulling the workpiece out of the matrix by links. It is this factor that leads to incomplete filling of the second joint with metal (Figure 5(b)).Figure 7
Stress–strain state parameters of ECAP-Linex process: (a) equivalent strain; (b) equivalent stress; (c) average hydrostatic pressure.
(a)(b)(c)To numerically evaluate these parameters, it is advisable to use the “Point Tracking” tool, which allows you to obtain the values of all parameters simultaneously when passing a given area of the workpiece through all deformation zones. In this case, an effective way will be to simultaneously evaluate the parameters on the surface of the workpiece and in the central section, which will make it possible to assess the level of uniformity of the distribution of parameters.To fulfill these conditions, two points were selected—the first on the workpiece surface and the second in the central section. Both points had the same coordinates on theX- and Y-axes—15 mm from the front end of the workpiece and the middle of its width, and differed only in their location in the vertical Z-plane (Figure 8).Figure 8
Place of tracking points.Figure9 shows a graph of the equivalent strain accumulation. Conventionally, it can be divided into four sections: I—the compression section of the drive pulleys; II—the section between the pulleys deformation zone and the first channel joint in the matrix, characterizes the input channel of the matrix; III—the section between the two channel joints in the matrix, characterizes the inclined channel of the matrix; IV—the section between the second channel joint in the matrix and the output from it, characterizes the output channel of the matrix. At the same time, the difference in the level of strain accumulation along the thickness of the workpiece is clearly visible. In the first zone, due to the development of an advance in the deformation zone, the difference is about 20%. In the second zone, this difference remains unchanged, which indicates that, in this section, the influence of the contact of the workpiece with the links of the chain conveyor is extremely insignificant—due to the rectilinear movement in the first channel, the decompression factor is completely absent.Figure 9
Graph of the equivalent strain development.In the third zone, after passing the first junction of the channels, the difference in strain levels begins to grow sharply. This is due to the fact that when passing the junction of the channels, the metal first completely fills the cavity of the matrix and then begins to decompress in the vertical direction, leading to tighter contact with the conveyor links. As a result, due to the increased adhesion of them, the surface layers receive a higher level of strain. In the central layer, after passing the junction of the channels, the level of strain remains at a constant level. In the fourth zone, the strain development is identical to the third zone. As a result, after one deformation cycle, a strain of 0.9 develops in the central zone, and on the surface, the strain level is about 1.34, which is 49% higher.Analyzing the graph of equivalent stress (Figure10), it can be noted that in the first zone, during compression in the pulleys, stress develops as the height of the workpiece decreases in the deformation zone. At the same time, the difference in the stress values along the thickness of the workpiece is small and is about 12% (90 MPa on the surface and 80 MPa in the center). In the second section, there is a drop in the stress level to an average of 25 MPa—since there is no strain development, the stress arises only from the friction of the metal with the rectilinear walls of the matrix. As a result, the stress level in this zone is almost the same with the thickness of the workpiece.Figure 10
Graph of the equivalent stress development.In the third zone, when passing the junction of the channels, the stress value increases sharply along the entire thickness of the workpiece up to 105 MPa. At the same time, after overcoming the joint area (short blue borders on the graph), the stress level on the surface becomes higher than in the center (118 MPa on the surface and 105 MPa in the center), which is the result of increased adhesion of the metal to the conveyor links. In the fourth zone, this effect is repeated, the stresses increase when the second joint passes up to 120 MPa, equalizing at the exit from its area. Further, stresses remain at the same level, since the output channel, like the first input channel, has a rectilinear structure, and here the stress arises only from friction against the walls of the matrix and the conveyor links. However, due to the higher level of adhesion, the stress level is much higher here than in the second zone.When considering the graph of the average hydrostatic pressure development (Figure11), it was found that in the first zone, compressive stresses develop in the central layers (approximately −10/−15MPa), while tensile stresses first act on the surface when capturing the metal (about 25 MPa), then during the steady rolling process, there are compressive stresses reaching −75 MPa. At the entrance to the matrix (the boundary of the second zone), a sharp jump of tensile stresses occurs on the surface, which is the result of free broadening during rolling in pulleys shaped like a smooth barrel. Because of this effect, the width of the workpiece is intentionally made smaller than the width of the matrix channel. However, after the matrix cavity is filled in width, the stresses decrease to the level of −10/−15MPa, which indicates the state of compression in the entire second zone. At the same time, the stress level here is low, since, as already established, there is no strain development, and the stress arises only from the friction of the metal against the rectilinear walls of the matrix.Figure 11
Graph of the average hydrostatic pressure development.When passing the first junction of the channels, the level of compressive stresses along the entire thickness of the workpiece increases to −40 MPa, then when moving in an inclined channel, the nature of stresses along the thickness changes. Tensile stresses of up to 60 MPa grow on the surface of the workpiece due to the adhesion of the metal to the conveyor links and the corresponding forced movement. In the central layers along the entire length of the inclined channel, the level of compressive stresses increases to −95 MPa. This effect is the result of the preform being pressed in the channel and with ECAP in a conventional matrix, such a stress level will act over the entire section of the workpiece. However, due to the presence of moving conveyor links, there are differences in the kinematics of movement on the workpiece surface, which leads to an increase in tensile stresses. When passing through the second junction of the channels, an increase in compressive stresses to –112 MPa occurs again, and after entering the rectilinear output channel, the stress level along the cross section almost equalizes and decreases to about –10 MPa in the center and 15 MPa on the surface. Here, the tensile stresses on the surface slightly exceed the compressive stresses in the center, since after decompressing in an inclined channel, the metal of the workpiece has an increased adhesion level to the moving links of the conveyor.


## 3.2. Analysis of Deformation Forces

In the “ECAP-Linex” process, the key element of deformation is an equal-channel angular matrix with parallel channels. Chain conveyors perform a dual role. Firstly, they advance the workpiece along the channels of the matrix due to adhesion to the workpiece. Secondly, they deform the workpiece by some compression in height, due to which the main level of active friction force develops, contributing to the advancement of the workpiece through the channels of the matrix.Therefore, for the stable course of the deformation process according to the proposed scheme, it is necessary to comply with the conditions:(4)PCONV>PMATR,where PCONV is the force created by the chain conveyor; PMATR is the backpressure force created by the matrix.Let’s consider each of these efforts separately. To find the backpressure force created by the matrix, it is advisable to use the equation of the pressing force in this matrix, which was obtained in [29]:
(5)PPRESS=2σSμ22l1+l2b1+h1+2h12tgϕ2+tgϕ⋅h13μ2,here, the authors of [29] made the assumption that the input and output channels have the same length. If this condition is not met, equation (5) takes the form:
(6)PPRESS=2σSμ2l1+l2+l3b1+h1+2h12tgϕ2+tgϕ⋅h13μ2.This equation characterizes the theoretical maximum force that occurs when the workpiece is in all three channels of the matrix. In real conditions of pressing, it will always be smaller due to the fact that when the punch moves, the volume of metal in the first channel will constantly decrease.In the “ECAP-Linex” process being developed, only curly elements forming a channel will be present in this matrix design. There are no side walls, since their role is performed by the elements of the chain conveyor. Therefore, equation (6) in relation to the curly elements of the matrix takes the form:
(7)PMATR=2σSμ2h1l1+l2+l3+2h1tgϕ2+tgϕ3μ2.In this case, the deformation zone can be represented as follows (Figure12).Figure 12
Deformation zone in the conveyor.The sum of all forces acting in the deformation zone is determined by the equation:(8)PCOMPR=2bav∫γ1ατavRcosθdθ−2bav∫0γτavRcosθdθ−2bav∫0αpavRsinθdθ,where b1иbav is the workpiece width after compression and the average width; τavиpav are the average tangential and normal stresses; R is the curvature radius of the locking block (analogous to the roll radius); θ is the current angle; α is the capture angle; γ, γ1 are the angles characterizing the advance and lag zones, respectively.Integrate equation (8) taking into account the assumption that bav=b1:
(9)PCOMPR=2b1Rτavsinα−sinγ1−2b1Rτavsinγ−0−2b1Rpav−cosα+1.After replacing in this equation:1−cosα=α2/2, sinα=α, sinγ1=γ1, sinγ=γ, τav=pavμ=σSμ, equation (9) has the form:
(10)PCOMPR=2b1RσSμ1α−γ1−2b1RσSμ1γ−2b1RσSα22,where μ1 is the friction coefficient in the deformation zone.Final form of equation:(11)PCOMPR=2b1RσSμ1α−γ1−γ−α22μ1.It can be seen from equation (11) that under equal geometric conditions in the deformation zone, the magnitude of the compression force will depend on the values of the angles γ and γ1, which depend on the magnitude of the backpressure force created by the matrix. In the same compression angle α, there will be different zones of advance, lag, and adhesion each time. Therefore, to use equation (11), it is necessary to find the values of these angles.The equations of equilibrium of forces and moments acting in a symmetrical deformation zone during rolling on a front support, which arises due to an additional shape change in the matrix installed behind the rolls, have the form:(12)2bav∫γ1ατavRcosθdθ−2bav∫0γτavRcosθdθ−2bav∫0αpavRsinθdθ−σ1b1h1=0;(13)2bavτavR2α−γ1−2bavτavR2γ−2bavψRα×∫0αpavRcosθdθ+∫γ1ατavRsinθdθ−∫0γτavRsinθdθ−σ1b1h1R=0,where σ1 is the backpressure stress; ψ is the coefficient of the shoulder position of the resultant metal pressure on the rolls; α, γ are the capture angle and the angle characterizing the length of the advance zone.Taking into account the assumption thatbav=b1:
(14)∫γ1ατavcosθdθ+∫0γτavcosθdθ−∫0αpavsinθdθ−σ1h12R=0;(15)τavα−γ1−τavγ−ψα∫0αpavRcosθdθ+∫γ1ατavsinθdθ−∫0γτavsinθdθ−σ1h12R=0.Integrating equation (14) and replacing 1−cosα=α2/2, sinα=α, sinγ1=γ1, sinγ=γ, γ1=γadh+γ, after the transformations, the dependence for determining the angle characterizing the extent of the adhesion zone will be:
(16)γadh=α−2γ−pavα22τav−σ1h12τavR.After replacementpav/τav=1/μ1 и τav=σSμ1, equation (17) is transformed to the form:
(17)γadh=α−2γ−α22μ1−σ1h12μ1σSR.After integration and transformation of equation (15):
(18)α−γ1−γ=ψαpavτavα−cosα+cosγ1+cosγ−1+σ1h12τavR.Transformation of the resulting expression into a quadratic equation by performing substitutions similar to those used in solving equation (14):
(19)2γ2−γ2α−α2μ−σ1h12μσSR+αμ1ψ−2−α2+α34μ1−σ1h1σSR+ασ1h12μ1σSR+σ12h124μ1σS2R2α=0.One of the roots of the quadratic equation (19) will be the angle characterizing the length of the advance zone:
(20)γ=2α−α2μ1−σ1h12μ1σSR−2α−α2μ1−σ1h12μ1σSR2−8αμ11ψ−2−α2+α34μ1−σ1h1σSR+ασ1h12μ1σSR+σ12h124μ1σS2R2α4To find this angle characterizing the advance zone according to equation (19), it is necessary to first determine the coefficient of the shoulder position of the resultant ψ. To do this, consider the conditions under which equation (19) makes sense:
(21)2α−α2μ1−σ1h12μ1σSR2>8αμ11ψ−2−α2+α34μ1−σ1h1σSR+ασ1h12μ1σSR+σ12h124μ1σS2R2α;(22)1ψ−2−α2+α34μ1−σ1h1σSR+ασ1h12μ1σSR+σ12h124μ1σS2R2α>0.Solving the inequalities (21) and (22) together, the limits in which the value changes 1/ψ:
(23)μ18α2α−α2μ1−σ1h12μ1σSR2+2+α2−α34μ1+σ1h1σSR−ασ1h12μ1σSR−σ12h124μ1σS2R2α>1ψ>2+α2−α34μ1+σ1h1σSR−ασ1h12μ1σSR−σ12h124μ1σS2R2α.Assuming that the value1/ψ is in the middle part of the indicated limits:
(24)1ψ=μ116α2α−α2μ1−σ1h12μ1σSR2+2+α2−α34μ1+σ1h1σSR−ασ1h12μ1σSR−σ12h124μ1σS2R2α.Knowing the magnitude of the angles characterizing the extent of the advance and adhesion zones, the angle characterizing the lag zone can be found from the condition:(25)γtag=α−γadh−γ.However, in this combined process, the useful force pushing the workpiece through the channels of the matrix is expressed not only by equation (11). Here, in addition to the compression force, there will also be a force from the friction of the workpiece on the chain element links, since their movement is directed in the same direction as the movement of the workpiece.Therefore, the maximum possible force generated by the conveyor at the moment when the workpiece completely fills all channels of the matrix:(26)PCONV=2b1σSμ1Rα−γ1−γ−α22μ1+l1+l2+l3.The force of advancing the workpiece by one link of the chain element will be equal to:(27)P1_EL=2σSμ1b1lEL,where lEL is the single link length of the chain element.A trial calculation with the simulation data was performed. When entering the algorithm into Microsoft Excel and varying the value of the channel junction angle in the matrix from 90° to 180°, the following data were obtained (Figure13).Figure 13
Dependences of the forces of the “ECAP-Linex” process on the value of the channel junction angle.After calculating the model, the following force graphs for the pulley matrix and chain element link were obtained (Figure14).Figure 14
Force in FEM simulation: (a) pulley; (b) matrix; (c) link.
(a)(b)(c)Table1 shows the force values obtained by calculating equations and simulation. Comparison of values showed high convergence in all three cases.Table 1
Calculation and modeling forces.
Calculation force,NSimulation force,NDifference, %Pulley12542122302.48Matrix919290491.55Link420041162


## 3.3. Variational Modeling

### 3.3.1. Initial Data for Variational Modeling

After analyzing the basic model, the task to determine the influence of technological and geometric parameters of the process on the stress–strain state and deformation force was set. To do this, it is necessary to change one geometric or technological parameter in the base model and track the change in stress–strain state parameters and deformation force. Based on the analysis of the obtained results, it will be possible to determine the most optimal values of the parameters.At the initial stage of variational modeling, it is necessary to determine the parameters to be varied. It is most advisable to choose those parameters that can be easily changed in real laboratory or production conditions. From the geometric parameters, it is most advisable to change the angle of the channels junction, this will require the installation of only new curly blocks without changing the design of the conveyor. It is irrational to change the radius of the pulley, since in this case, it will be necessary to manufacture new chain elements. It is also possible to change the lengths of the matrix channels. From the technological parameters, it is possible to change the workpiece temperature, the rotation speeds of the pulleys, and the movement of chain elements, as well as the friction coefficients in the matrix and conveyor.At the next stage, it is necessary to determine the parameter variation intervals. The values of the variable parameters should be achievable in real conditions. At the same time, if it is possible to vary in both directions, then it is necessary to change the initial value to the same value. Therefore, the following parameter values were selected:- Channel junction angle: 125° and 155°.- Lengths of the matrix channels: each channel has been increased by 10 mm (it is irrational to reduce the initial values, since this will negatively affect the strength of the tool).- Workpiece temperature: 100°C (a further increase in the heating temperature will be irrational, since this will lead to the beginning of recrystallization [30]).- The speed of the chain elements directly depends on the rotation speed of the pulleys. Therefore, the values of 5 rpm and 25 rpm were selected. Thus, the speeds of the chain elements were set to 26.1 mm/s and 130.8 mm/s.- The friction coefficients in the matrix and conveyor in the basic model already have critical values, so they can only be changed in one direction. As a result, the following values were set: 0.15 for the matrix and 0.5 for the conveyor.When analyzing these models, cases of unstable deformation were identified—in some models, conditions arose when, at a certain stage, the workpiece stopped moving in the matrix due to a too low level of active friction forces in the pulley deformation zone, or, conversely, due to a too high backpressure level in the matrix. In any case, these models should be considered unsuccessful, and the conditions for their implementation should be considered negative. These models include both models with modified friction coefficients—in both cases, the workpiece jammed at the entrance to the first joint of the channels. Based on this, it can be concluded that these two parameters are critical for the stable flow of the ECAP-Linex combined process, and the selected values of the friction coefficients on the pulleys 0.7 and in the matrix 0.05 can be considered the most optimal. Moreover, an unsuccessful case was the model with an increased length of the second inclined channel—in this case, the jamming of the workpiece occurred at the second junction of the channels. Therefore, the length of the second channel of 20 mm should not be increased.In addition to the unsuccessful models, it should be separately noted models in which no significant changes in stress–strain state or deformation force were recorded when the parameters were changed. These are models with increased input and output channel lengths. Such a “neutral” result is quite understandable, since these channels have a rectilinear direction. In the input channel, the level of back pressure is minimal, since only friction forces against the matrix walls act here. In the output channel, the level of backpressure increases sharply due to the design of the matrix, but in this combined process, it is almost completely annihilated by the moving links of the conveyor, which leads to incomplete filling of the second joint (see Figure5(b)). The remaining conditions led to an increase or decrease in the parameters of stress–strain state or deformation force, so the corresponding models were considered in detail.


### 3.3.2. Models with Modified Channel Junction Angles in the Matrix

Figure15 shows graphs of the equivalent strain accumulation when using matrices with different channel junction angles.Figure 15
Equivalent strain in models with variable channel junction angles: (а) 125°; (b) 155°.
(a)(b)Comparing these graphs, we can say that the value of the joint angle in the matrix significantly affects the level of equivalent strain. In the first two stages, the strain levels are almost identical at both points (it is in the range of 0.3–0.35, as in the basic model). When hitting the first joint of the matrix, the metal receives a different level of shear strain, at the second joint, this effect is repeated. In a matrix with an angle of 125°, the strain level reaches 2.6 on the surface and 1.1 in the center, i.e., the difference in the values of strain in thickness reaches 236%. This suggests that, in this case, there is a significant gradient anisotropy in the cross section (the level of anisotropy is 3.3 times higher than in the basic model). In a matrix with an angle of 155°, the strain level reaches 0.7 on the surface and 0.65 in the center, i.e., the difference in the values of strain in thickness is only 7%, in this case, there is a fairly uniform distribution of strain over the cross section.It should be noted that in a matrix with an angle of 155°, this small difference persists at all stages, whereas in a matrix with an angle of 125°, the strain on the surface grows continuously (both due to shear strain at the joints and due to coupling with conveyor links), the central layers receive strain only in the joint zones, after their passage, the level of deformation does not change. The exception is stage III, where after passing the first joint, the strain level continues to grow, albeit less intensely than at the joint. This is due to the increased level of decompression and subsequent adhesion—in this case, this effect extends to the entire section of the workpiece. In a matrix with an angle of 155°, the growth of strain after passing the joints of the matrix is absent both in the center and on the surface. This means that in this case, the level of strain depends only on the shear strain, and the level of adhesion to the links is extremely small due to the low level of metal compression.When analyzing the equivalent stress graphs (Figure16) and comparing them with the basic model, the general similarity at all stages can be noted. The key difference is observed at stage III, where a small stress drop occurs in the basic model, which increases significantly in the matrix with an angle of 155°. As in the case of equivalent strain, here this effect is the result of a decrease in the level of coupling with the conveyor—in a model with an angle of 125°, where the coupling is significantly higher than the base model, this effect is absent. At the same time, it should be noted that a decrease in the coupling level affects the overall stress level—in the model with an angle of 155°, the maximum level of equivalent stresses in the matrix is 10–12% less than in the other two.Figure 16
Equivalent stress in models with variable channel junction angles: (а) 125°; (b) 155°.
(a)(b)When analyzing the graphs of average hydrostatic pressure (Figure17) and comparing them with the basic model, it was found that the first two stages are almost identical. At stage III, in a model with an angle of 125°, the same effect is observed as in the basic model—tensile stresses reaching 105 MPa act on the surface, and compressive stresses reaching −115 MPa act in the central zone. In the model with an angle of 155°, due to the reduced coupling with the conveyor, the level of average hydrostatic pressure along the thickness of the workpiece is almost the same. Moreover, the stress level is much lower here—compressive stresses reach −56 MPa, while the maximum tensile stresses reach only 18 MPa.Figure 17
Average hydrostatic pressure in models with variable channel junction angles: (а) 125°; (b) 155°.
(a)(b)The values of forces arising on the pulley matrix and conveyor link were also considered (Figure18). Table 2 shows the values obtained by calculating and modeling. As in the basic model, the comparison of values showed high convergence.Figure 18
Force in models with different channel junction angles in the matrix: (a) 125° pulley; (b) 155° pulley; (c) 125° matrix; (d) 155° matrix; (e) 125° link; (f) 155° link.
(a)(b)(c)(d)(e)(f)Table 2
Forces for ECAP-Linex process at different joint angles in the matrix.
Calculation force,NSimulation force,NDifference, %125°Pulley11583112602.78Matrix1015997933.73Link420041531.13155°Pulley13190128602.56Matrix853983781.92Link420039825.47There was also a clear dependence of the error level for the conveyor link on the angle value. This is the result of the fact that the force calculation equation does not take into account the possible level of decompression and finds the maximum possible force when the metal completely fills all cavities and corners of the joints. In real conditions, with an increase in the angle, the level of decompression decreases, which leads to a decrease in force and an increase in error.


### 3.3.3. Model with Elevated Workpiece Temperature

When heating aluminum alloy 6061–100°C, its yield strength is slightly reduced to about 55 MPa. Figure19 shows the graphs of stress–strain state parameters when using a workpiece with an elevated temperature.Figure 19
Stress–strain state parameters in the model with elevated workpiece temperature: (a) equivalent strain; (b) equivalent stress; (c) average hydrostatic pressure.
(a)(b)(c)Analysis of the graphs of stress–strain state parameters in this model showed that the increased temperature of the workpiece leads to a decrease in the adhesion level of metal to the conveyor, which is reflected in a slight decrease in the difference of equivalent strain in the workpiece thickness. Therefore, if in the basic model, the difference was 49%, then in this model, it was 44% (with a deformation level of 1.33 on the surface and 0.92 in the center). The distribution of the equivalent stress has a similar appearance to the previously considered models with different channel junction angles—differences are observed only at stages III and IV. Here, at stage III, there is a stress drop, which is the result of a reduced adhesion level, which leads to the separation of stress zones at the joints of the matrix, whereas the model with an angle of 125° has a single stress zone along the entire length of the inclined channel. In terms of their values, the level of equivalent stresses here is almost the same as in the basic model, which is the result of a slight decrease in the yield strength at a given temperature.The distribution of the average hydrostatic pressure has a similar character—the main differences are observed at stages III and IV. At an elevated temperature of the workpiece at stage III, there are no tensile stresses in the surface layers—compressive stresses act here throughout the thickness of the workpiece, although the level of compressive stresses in the center is much higher than on the surface. At stage IV, due to the rectilinear channel, even a reduced level of adhesion is sufficient for the development of tensile stresses in the surface layers. At the same time, the overall stress level is 25–30% lower than the base value.A comparison of the resulting forces (Figure20 and Table 3) with the base values showed that an increase in the workpiece temperature leads to a decrease in force on all three parts by about 10–12%.Figure 20
Force at elevated workpiece temperature: (a) pulley; (b) matrix; (c) link.
(a)(b)(c)Table 3
Forces for ECAP-Linex process at elevated workpiece temperature.
Calculation force,NSimulation force,NDifference, %Temperature 100°СPulley11497108905.27Matrix842685421.35Link385036914.13


### 3.3.4. Models with Modified Rotation Speeds of Pulleys and Movement Speeds of Chain Elements

Changing the rotation speed of the pulleys and, accordingly, the movement speed of the conveyor chain elements leads to a change in the strain rate. When considering the equivalent strain (Figure21), it was noted that in the central zone of the workpiece, the strain level with varying speed characteristics of the ECAP-Linex process is almost unchanged and is in the range of 0.9–0.95. On the surface, the strain level also remains constant in the range of 1.3–1.35.Figure 21
Equivalent strain with varying kinematic parameters: (a) 5 rpm (26.1 mm/s); (b) 25 rpm (130.8 mm/s).
(a)(b)When considering the equivalent stress (Figure22), it was noted that in the central zone of the workpiece at stage III, there is a drop in the level of equivalent stresses. The same drop is observed on the surface at a reduced speed of 5 rpm (26.1 mm/s). With an increase in speed characteristics, the drop in the level of equivalent stresses on the surface disappears. According to the numerical values, the maximum stress level in all cases, including the base, remains unchanged (about 120 MPa).Figure 22
Equivalent stress at variable kinematic parameters: (a) 5 rpm (26.1 mm/s); (b) 25 rpm (130.8 mm/s).
(a)(b)When analyzing the average hydrostatic pressure (Figure23), it should be noted approximately the same level of tensile and compressive stresses in the first two stages. At the third stage, at a reduced speed of 5 rpm (26.1 mm/s), there is a complete absence of tensile stresses both in the center and on the surface. An increase in speed characteristics leads to an increase in the influence of metal contact on the conveyor links, which leads to an increase in tensile stresses in the inclined channel after passing the zone of the first joint.Figure 23
Average hydrostatic pressure with varying kinematic parameters: (a) 5 rpm (26.1 mm/s); (b) 25 rpm (130.8 mm/s).
(a)(b)When analyzing the forces and comparing them with the calculated values (Figure24 and Table 4), it is necessary to take into account the fact that all equations for calculating forces do not contain speed components. Therefore, for all speed characteristics, the values of the forces in the calculation will be the same. A comparison of their model data showed that the effect of the strain rate on the force in the ECAP-Linex process does not have a pronounced dependence. This conclusion is supported by two facts:Figure 24
Force at various kinematic parameters: (a) pulley 5 rpm (26.1 mm/s); (b) pulley 25 rpm (130.8 mm/s); (c) matrix 5 rpm (26.1 mm/s); (d) matrix 25 rpm (130.8 mm/s); (e) link 5 rpm (26.1 mm/s); (f) link 25 rpm (130.8 mm/s).
(a)(b)(c)(d)(e)(f)Table 4
Forces for ECAP-Linex process at various kinematic parameters.
Calculation force,NSimulation force,NDifference, %5 rpm (26.1 mm/s)Pulley12542120803.68Matrix919295553.8Link420039376.2625 rpm (130.8 mm/s)Pulley12542119304.87Matrix919290691.34Link420038258.93- The effect of the strain rate on the force of ECAP has not been proven by either theoretical or experimental studies [21].- It is known from the rolling theory that the strain rate has the least effect on the force compared to temperature and compression.Usually, a significant influence of the speed on the deformation resistance and rolling force is fixed when it changes several tens of times. Taking into account the fact that in these models the speed changed by five times, the influence of this parameter at the stage of compression by pulleys is insignificant. Due to the absence of speed components in the equations, an increase in the difference level was recorded when comparing the values, but its maximum level does not exceed 10%. Therefore, the equations are quite suitable for calculating forces at different speeds.


## 3.3.1. Initial Data for Variational Modeling

After analyzing the basic model, the task to determine the influence of technological and geometric parameters of the process on the stress–strain state and deformation force was set. To do this, it is necessary to change one geometric or technological parameter in the base model and track the change in stress–strain state parameters and deformation force. Based on the analysis of the obtained results, it will be possible to determine the most optimal values of the parameters.At the initial stage of variational modeling, it is necessary to determine the parameters to be varied. It is most advisable to choose those parameters that can be easily changed in real laboratory or production conditions. From the geometric parameters, it is most advisable to change the angle of the channels junction, this will require the installation of only new curly blocks without changing the design of the conveyor. It is irrational to change the radius of the pulley, since in this case, it will be necessary to manufacture new chain elements. It is also possible to change the lengths of the matrix channels. From the technological parameters, it is possible to change the workpiece temperature, the rotation speeds of the pulleys, and the movement of chain elements, as well as the friction coefficients in the matrix and conveyor.At the next stage, it is necessary to determine the parameter variation intervals. The values of the variable parameters should be achievable in real conditions. At the same time, if it is possible to vary in both directions, then it is necessary to change the initial value to the same value. Therefore, the following parameter values were selected:- Channel junction angle: 125° and 155°.- Lengths of the matrix channels: each channel has been increased by 10 mm (it is irrational to reduce the initial values, since this will negatively affect the strength of the tool).- Workpiece temperature: 100°C (a further increase in the heating temperature will be irrational, since this will lead to the beginning of recrystallization [30]).- The speed of the chain elements directly depends on the rotation speed of the pulleys. Therefore, the values of 5 rpm and 25 rpm were selected. Thus, the speeds of the chain elements were set to 26.1 mm/s and 130.8 mm/s.- The friction coefficients in the matrix and conveyor in the basic model already have critical values, so they can only be changed in one direction. As a result, the following values were set: 0.15 for the matrix and 0.5 for the conveyor.When analyzing these models, cases of unstable deformation were identified—in some models, conditions arose when, at a certain stage, the workpiece stopped moving in the matrix due to a too low level of active friction forces in the pulley deformation zone, or, conversely, due to a too high backpressure level in the matrix. In any case, these models should be considered unsuccessful, and the conditions for their implementation should be considered negative. These models include both models with modified friction coefficients—in both cases, the workpiece jammed at the entrance to the first joint of the channels. Based on this, it can be concluded that these two parameters are critical for the stable flow of the ECAP-Linex combined process, and the selected values of the friction coefficients on the pulleys 0.7 and in the matrix 0.05 can be considered the most optimal. Moreover, an unsuccessful case was the model with an increased length of the second inclined channel—in this case, the jamming of the workpiece occurred at the second junction of the channels. Therefore, the length of the second channel of 20 mm should not be increased.In addition to the unsuccessful models, it should be separately noted models in which no significant changes in stress–strain state or deformation force were recorded when the parameters were changed. These are models with increased input and output channel lengths. Such a “neutral” result is quite understandable, since these channels have a rectilinear direction. In the input channel, the level of back pressure is minimal, since only friction forces against the matrix walls act here. In the output channel, the level of backpressure increases sharply due to the design of the matrix, but in this combined process, it is almost completely annihilated by the moving links of the conveyor, which leads to incomplete filling of the second joint (see Figure5(b)). The remaining conditions led to an increase or decrease in the parameters of stress–strain state or deformation force, so the corresponding models were considered in detail.


## 3.3.2. Models with Modified Channel Junction Angles in the Matrix

Figure15 shows graphs of the equivalent strain accumulation when using matrices with different channel junction angles.Figure 15
Equivalent strain in models with variable channel junction angles: (а) 125°; (b) 155°.
(a)(b)Comparing these graphs, we can say that the value of the joint angle in the matrix significantly affects the level of equivalent strain. In the first two stages, the strain levels are almost identical at both points (it is in the range of 0.3–0.35, as in the basic model). When hitting the first joint of the matrix, the metal receives a different level of shear strain, at the second joint, this effect is repeated. In a matrix with an angle of 125°, the strain level reaches 2.6 on the surface and 1.1 in the center, i.e., the difference in the values of strain in thickness reaches 236%. This suggests that, in this case, there is a significant gradient anisotropy in the cross section (the level of anisotropy is 3.3 times higher than in the basic model). In a matrix with an angle of 155°, the strain level reaches 0.7 on the surface and 0.65 in the center, i.e., the difference in the values of strain in thickness is only 7%, in this case, there is a fairly uniform distribution of strain over the cross section.It should be noted that in a matrix with an angle of 155°, this small difference persists at all stages, whereas in a matrix with an angle of 125°, the strain on the surface grows continuously (both due to shear strain at the joints and due to coupling with conveyor links), the central layers receive strain only in the joint zones, after their passage, the level of deformation does not change. The exception is stage III, where after passing the first joint, the strain level continues to grow, albeit less intensely than at the joint. This is due to the increased level of decompression and subsequent adhesion—in this case, this effect extends to the entire section of the workpiece. In a matrix with an angle of 155°, the growth of strain after passing the joints of the matrix is absent both in the center and on the surface. This means that in this case, the level of strain depends only on the shear strain, and the level of adhesion to the links is extremely small due to the low level of metal compression.When analyzing the equivalent stress graphs (Figure16) and comparing them with the basic model, the general similarity at all stages can be noted. The key difference is observed at stage III, where a small stress drop occurs in the basic model, which increases significantly in the matrix with an angle of 155°. As in the case of equivalent strain, here this effect is the result of a decrease in the level of coupling with the conveyor—in a model with an angle of 125°, where the coupling is significantly higher than the base model, this effect is absent. At the same time, it should be noted that a decrease in the coupling level affects the overall stress level—in the model with an angle of 155°, the maximum level of equivalent stresses in the matrix is 10–12% less than in the other two.Figure 16
Equivalent stress in models with variable channel junction angles: (а) 125°; (b) 155°.
(a)(b)When analyzing the graphs of average hydrostatic pressure (Figure17) and comparing them with the basic model, it was found that the first two stages are almost identical. At stage III, in a model with an angle of 125°, the same effect is observed as in the basic model—tensile stresses reaching 105 MPa act on the surface, and compressive stresses reaching −115 MPa act in the central zone. In the model with an angle of 155°, due to the reduced coupling with the conveyor, the level of average hydrostatic pressure along the thickness of the workpiece is almost the same. Moreover, the stress level is much lower here—compressive stresses reach −56 MPa, while the maximum tensile stresses reach only 18 MPa.Figure 17
Average hydrostatic pressure in models with variable channel junction angles: (а) 125°; (b) 155°.
(a)(b)The values of forces arising on the pulley matrix and conveyor link were also considered (Figure18). Table 2 shows the values obtained by calculating and modeling. As in the basic model, the comparison of values showed high convergence.Figure 18
Force in models with different channel junction angles in the matrix: (a) 125° pulley; (b) 155° pulley; (c) 125° matrix; (d) 155° matrix; (e) 125° link; (f) 155° link.
(a)(b)(c)(d)(e)(f)Table 2
Forces for ECAP-Linex process at different joint angles in the matrix.
Calculation force,NSimulation force,NDifference, %125°Pulley11583112602.78Matrix1015997933.73Link420041531.13155°Pulley13190128602.56Matrix853983781.92Link420039825.47There was also a clear dependence of the error level for the conveyor link on the angle value. This is the result of the fact that the force calculation equation does not take into account the possible level of decompression and finds the maximum possible force when the metal completely fills all cavities and corners of the joints. In real conditions, with an increase in the angle, the level of decompression decreases, which leads to a decrease in force and an increase in error.


## 3.3.3. Model with Elevated Workpiece Temperature

When heating aluminum alloy 6061–100°C, its yield strength is slightly reduced to about 55 MPa. Figure19 shows the graphs of stress–strain state parameters when using a workpiece with an elevated temperature.Figure 19
Stress–strain state parameters in the model with elevated workpiece temperature: (a) equivalent strain; (b) equivalent stress; (c) average hydrostatic pressure.
(a)(b)(c)Analysis of the graphs of stress–strain state parameters in this model showed that the increased temperature of the workpiece leads to a decrease in the adhesion level of metal to the conveyor, which is reflected in a slight decrease in the difference of equivalent strain in the workpiece thickness. Therefore, if in the basic model, the difference was 49%, then in this model, it was 44% (with a deformation level of 1.33 on the surface and 0.92 in the center). The distribution of the equivalent stress has a similar appearance to the previously considered models with different channel junction angles—differences are observed only at stages III and IV. Here, at stage III, there is a stress drop, which is the result of a reduced adhesion level, which leads to the separation of stress zones at the joints of the matrix, whereas the model with an angle of 125° has a single stress zone along the entire length of the inclined channel. In terms of their values, the level of equivalent stresses here is almost the same as in the basic model, which is the result of a slight decrease in the yield strength at a given temperature.The distribution of the average hydrostatic pressure has a similar character—the main differences are observed at stages III and IV. At an elevated temperature of the workpiece at stage III, there are no tensile stresses in the surface layers—compressive stresses act here throughout the thickness of the workpiece, although the level of compressive stresses in the center is much higher than on the surface. At stage IV, due to the rectilinear channel, even a reduced level of adhesion is sufficient for the development of tensile stresses in the surface layers. At the same time, the overall stress level is 25–30% lower than the base value.A comparison of the resulting forces (Figure20 and Table 3) with the base values showed that an increase in the workpiece temperature leads to a decrease in force on all three parts by about 10–12%.Figure 20
Force at elevated workpiece temperature: (a) pulley; (b) matrix; (c) link.
(a)(b)(c)Table 3
Forces for ECAP-Linex process at elevated workpiece temperature.
Calculation force,NSimulation force,NDifference, %Temperature 100°СPulley11497108905.27Matrix842685421.35Link385036914.13


## 3.3.4. Models with Modified Rotation Speeds of Pulleys and Movement Speeds of Chain Elements

Changing the rotation speed of the pulleys and, accordingly, the movement speed of the conveyor chain elements leads to a change in the strain rate. When considering the equivalent strain (Figure21), it was noted that in the central zone of the workpiece, the strain level with varying speed characteristics of the ECAP-Linex process is almost unchanged and is in the range of 0.9–0.95. On the surface, the strain level also remains constant in the range of 1.3–1.35.Figure 21
Equivalent strain with varying kinematic parameters: (a) 5 rpm (26.1 mm/s); (b) 25 rpm (130.8 mm/s).
(a)(b)When considering the equivalent stress (Figure22), it was noted that in the central zone of the workpiece at stage III, there is a drop in the level of equivalent stresses. The same drop is observed on the surface at a reduced speed of 5 rpm (26.1 mm/s). With an increase in speed characteristics, the drop in the level of equivalent stresses on the surface disappears. According to the numerical values, the maximum stress level in all cases, including the base, remains unchanged (about 120 MPa).Figure 22
Equivalent stress at variable kinematic parameters: (a) 5 rpm (26.1 mm/s); (b) 25 rpm (130.8 mm/s).
(a)(b)When analyzing the average hydrostatic pressure (Figure23), it should be noted approximately the same level of tensile and compressive stresses in the first two stages. At the third stage, at a reduced speed of 5 rpm (26.1 mm/s), there is a complete absence of tensile stresses both in the center and on the surface. An increase in speed characteristics leads to an increase in the influence of metal contact on the conveyor links, which leads to an increase in tensile stresses in the inclined channel after passing the zone of the first joint.Figure 23
Average hydrostatic pressure with varying kinematic parameters: (a) 5 rpm (26.1 mm/s); (b) 25 rpm (130.8 mm/s).
(a)(b)When analyzing the forces and comparing them with the calculated values (Figure24 and Table 4), it is necessary to take into account the fact that all equations for calculating forces do not contain speed components. Therefore, for all speed characteristics, the values of the forces in the calculation will be the same. A comparison of their model data showed that the effect of the strain rate on the force in the ECAP-Linex process does not have a pronounced dependence. This conclusion is supported by two facts:Figure 24
Force at various kinematic parameters: (a) pulley 5 rpm (26.1 mm/s); (b) pulley 25 rpm (130.8 mm/s); (c) matrix 5 rpm (26.1 mm/s); (d) matrix 25 rpm (130.8 mm/s); (e) link 5 rpm (26.1 mm/s); (f) link 25 rpm (130.8 mm/s).
(a)(b)(c)(d)(e)(f)Table 4
Forces for ECAP-Linex process at various kinematic parameters.
Calculation force,NSimulation force,NDifference, %5 rpm (26.1 mm/s)Pulley12542120803.68Matrix919295553.8Link420039376.2625 rpm (130.8 mm/s)Pulley12542119304.87Matrix919290691.34Link420038258.93- The effect of the strain rate on the force of ECAP has not been proven by either theoretical or experimental studies [21].- It is known from the rolling theory that the strain rate has the least effect on the force compared to temperature and compression.Usually, a significant influence of the speed on the deformation resistance and rolling force is fixed when it changes several tens of times. Taking into account the fact that in these models the speed changed by five times, the influence of this parameter at the stage of compression by pulleys is insignificant. Due to the absence of speed components in the equations, an increase in the difference level was recorded when comparing the values, but its maximum level does not exceed 10%. Therefore, the equations are quite suitable for calculating forces at different speeds.


## 4. Conclusion

In this work, finite element modeling of ECAP-Linex process was carried out. For the correct simulation of the process, a method with sequential input of the conveyor links into the calculation was presented. In accordance with the study objectives, to analyze the efficiency of metal processing, the main parameters of the stress–strain state were considered: equivalent strain, equivalent stress, and average hydrostatic pressure, as well as the deformation force on the main elements of the combined process: pulley, matrix, and conveyor link. To analyze the resulting deformation forces, the stages of pressing in a matrix and compression by a chain conveyor were separately considered. The obtained equations were used during the trial calculation. Verification of the obtained values of equations using computer simulation by the FEM in the Deform program showed that the values of the forces in the calculation and modeling have a high level of convergence, for all three considered details, the difference did not exceed 10%. Thus, the problem of determining the adequacy of the model and the formulas obtained was solved.The variational modeling allowed to determine the optimal values of the main technological and geometric parameters of the process. As an optimization parameter for metal forming, either the level of metal processing or the deformation force is usually considered. From this point of view, a matrix with an angle of 125° is the most optimal in terms of the metal processing level, and at the same time not recommended in terms of force. At the same time, the matrix with an angle of 155° has opposite recommendations. Therefore, in this combined process, it can be recommended to use a matrix with an angle of 140° as a golden mean. An increase in the heating temperature of the workpiece below the start point of recrystallization favorably affects the reduction of force parameters with an almost unchanged metal processing level. The change in speed characteristics has no pronounced positive or negative properties, except for a decrease in the level of tensile stresses in the inclined channel of the matrix at a reduced speed. Therefore, any of the considered high-speed options can be recommended. The friction coefficients in the matrix and the conveyor must have the limit values set in the basic model (0.7 for the conveyor and 0.05 for the matrix), their change leads to a violation of the deformation stability.Similarly, it is not recommended to increase the base lengths of the matrix channels: 30 mm for the first channel, 20 mm for the second channel, and 15 mm for the third channel. An increase in the second channel leads to jamming of the workpiece, and an increase in the first and third channels will be an irrational decision, since in this case, there are no significant changes in stress–strain state or deformation forces. Thus, the problem of determining the most optimal values of geometric and technological parameters of the ECAP-Linex process was solved.



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*Source: 1573884-2023-03-07.xml*