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An analysis of longitudinal residual stresses in EN AW-5083 alloy strips as a function of cold-rolling process parameters

  • Nebojša Tadić EMAIL logo , Mitar Mišović and Žarko Radović
Published/Copyright: August 30, 2023
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Journal of the Mechanical Behavior of Materials
From the journal Journal of the Mechanical Behavior of Materials Volume 32 Issue 1

Abstract

Various process nonhomogeneities in the cold rolling lead to an uneven distribution of deformation across the strip cross-section, resulting in the induction of residual stresses. This study investigates the longitudinal residual stresses in cold-rolled EN AW-5083 aluminum alloy strips using the finite element method (FEM) to achieve reliable predictions. The impacts of process parameters, including the reduction ratio, coefficient of contact friction, and front and back tensions, were analyzed. Changes in residual stresses, depending on the process parameters, were determined by the distribution of linear and shear strains, as well as the strain hardening conditions at the exit part of the deformation zone. An increase in the reduction ratio from 20 to 50%, as well as an increase in the friction coefficient from 0.1 to 0.2, resulted in decreased stress values. The residual stresses on the strip surface, determined by the experimental deflection method, were consistent with the results obtained by FEM simulation. Under the impact of back and/or front tensions, there is a reduction in longitudinal residual stresses, with the front tension exerting the greatest influence. The research results show that the FEM is a reliable tool for predicting residual stresses in cold-rolled strips.

Keywords: cold rolling; residual stress; finite elements method; deflection method; reduction ratio; friction coefficient; front tension; back tension

1 Introduction

The residual stresses arise during the cold rolling of sheets and strips due to non-homogenous deformation [1]. Due to the geometric conditions of sheet and strip rolling, a plane state of residual stresses is created, with the main stresses occurring in the longitudinal (rolling direction) and transverse directions [2,3].

The presence of residual stresses can cause various defects in the flatness of the strip [4,5,6]. Also, the total stress of the elements exposed to an external load, due to the superimposition with the residual stresses, can reach a high value and cause damage or reduce the reliability of the structural elements [7,8]. The shape and characteristics of cold-rolled strips are dominantly determined by longitudinal residual stresses. In order to accurately predict and effectively control the impact of longitudinal residual stresses on the appearance and properties of cold-rolled strips, it is imperative to determine the stress distribution and corresponding values across the cross-section of the strip. This entails investigating their dependence on the rolling process parameters [7].

Today, a variety of experimental techniques are available for measuring residual stresses. The selection of a proper measurement method depends on the capabilities of the measuring technique to adapt to the shape of the element, the state of the material, and the stress components to be measured [8,9]. Neutron and X-ray diffraction techniques are used for the experimental measurement of residual stress in strips and plates [10,11]. However, the accuracy of the results obtained from these measurement techniques is limited. This is due to the influence of stresses that occur during the cutting and chemical polishing of samples as well as the crystallographic texture and errors in the material constants used when calculating residual stresses. Also, experimental techniques are used based on the measurement of elastic effects that occur due to the disturbance of stress balance after the removal of metal layers from the strip surface [11,12,13,14]. These methods are based on linking the model of equivalent external load and the model of the assumed distribution of residual stresses across the cross-section of the element. In the study of Abvabi et al. [15], an inverse approach was used in which the residual stress profile across the thickness of the sheet was obtained based on experimental bending. These results are comparable to the results obtained by the X-ray diffraction method with successive removal of metal layers. In the study of Milenin et al. [16], the development and experimental validation of the residual stress model in hot-rolled steel strips based on the elastic–plastic behavior of the material was investigated. Based on a simplified representation of the stress tensor, a numerical–analytical model of the distribution of thermal residual stresses along the width of the hot-rolled strip is proposed. Different values on the upper and lower surfaces of the strip are the result of different winding and unwinding conditions of the coils. In the study of Kumar et al. [17], surface residual stresses in hot-rolled flat steel strips were measured using the strain gauge method after longitudinal laser cutting of the strips. The stresses were measured in the critical zones where characteristic defects of the strip shape develop after cutting.

In experimental research, it is still not easy to obtain the exact stress distribution in thin strips. In most cases, the stresses were tested only at the strip surface. Consequently, the finite element method (FEM) is a useful tool for examining the distribution of residual stresses at the cross-section of rolled strips. However, there are a limited number of studies on the theoretical prediction of residual stresses. This is partly because the FEM was used mainly to analyze the rolling process for a rigid plastic body. In the study of Dixit and Dixit [18], an analytical FE approach for an elastic–plastic body was developed. The values and distribution of residual stress across thicknesses in cold-rolled strips were analyzed. Tensile stresses were obtained in the surface layers and compressive stresses in the central layers of the strip. However, the obtained results were not experimentally verified. In the study of Hattori et al. [19], the impact of the geometric parameter of the deformation zone l d/h m (the ratio of the zone length and the mean strip thickness) on the distribution of residual stresses across the thickness of the strip was investigated using the FEM. The obtained results are well correlated with the stresses on the strip surface but not with the stresses on the strip axis. The stress distribution obtained by the X-ray diffraction method shows a qualitative agreement with the results of the FEM but the absolute values differ. In the study of Mehner et al. [20], the impact of the initial state of residual stresses in hot- and cold-rolled steel strips on the final state after the final cold rolling was investigated. Both the FEM and the X-ray diffraction method were used. It was shown that the initial state has no significant impact on the final state of residual stresses, but it depends on the state of the material, the roll-pass sequence, and the reduction ratio.

Generally speaking, there are still a limited number of more in-depth analyses of the impact of the rolling process parameters on the values and distribution of residual stresses across strip thicknesses. This especially applies to the typical range of the reduction ratios, which is characteristic of industrial conditions. In addition, in a significant number of studies, residual stresses obtained by FEM simulation do not agree well with experimental values. Also, in the available studies, the impact of front and back tensions as standard process parameters of cold rolling of sheets and strips on the values and distribution of residual stresses has not been analyzed.

This study used FEM simulation to examine the values and distribution of longitudinal residual stresses across the strip thickness, for different reduction ratio levels and coefficients of contact friction. The stress values at the strip’s surface were compared with the experimentally determined stresses using the deflection method. FEM simulation was also used to examine the effects of back and front strip tensions on longitudinal residual stresses.

The research was conducted on EN AW-5083 aluminum alloy strips. The EN AW-5083 alloy is widely used for the production of structural elements in the maritime industry, automotive industry, aircraft industry, pressure vessels, etc. Cold rolling is one of the final operations performed on a significant number of these elements. The stability and properties of these elements in subsequent technological operations and exploitation can be conditioned to a considerable extent by the presence of residual stresses. Therefore, the reliable prediction of residual stresses depending on the parameters of the rolling process is extremely important.

2 Material and experiment

The commercial aluminum alloy EN AW-5083 was selected for testing, and the chemical composition is presented in Table 1, while its mechanical properties in the initial soft annealed state are shown in Table 2. The initial material used for experimental rolling was a sheet with a thickness of 1.28 ± 0.01 mm.

Table 1

Chemical composition of EN AW-5083 alloy

Chemical composition (mass %)
Mn Mg Cu Si Fe Zn Al
0.42 4.23 0.015 0.13 0.26 0.02 rest
Table 2

Mechanical properties of EN AW-5083 alloy

Mechanical properties
R p0.2 (MPa) R m (MPa) A (%) E (Gpa)
134.7 289.7 22.86 70

Cold rolling was carried out by individual passes in a laboratory duo-rolling stand without the use of strip tension. The diameter of the rollers is 125 mm. The duo-rolling stand has only a peripheral roller velocity of 0.173 m/s. This velocity belongs to the lower part of the spectrum of the velocities of cold-rolling sheets and strips under industrial conditions. The rollers and strips were manually lubricated with hydraulic oil.

To investigate strain hardening during the rolling process, wide strips were rolled at reduction ratios (ε) from 10 to 50%, and then samples of the strips were cut for tension testing. Tension testing was performed using a universal tensile testing machine HECKERT FP100, with a force range of up to 10 kN. The yield strength (R p0.2), tensile strength (R m), and elongation at break (A) were tested (Table 3).

Table 3

Mechanical properties of cold-rolled strips depending on reduction ratios

ε (%) 0 10 15.5 20.1 25.4 29.8 40.2 50
R p0.2 (MPa) 134.72 221.71 254.33 270.10 280.24 290.31 306.79 321.92
R m (MPa) 289.66 303.62 310.78 318.12 326.54 332.53 345.82 359.46
A (%) 22.86 12.42 9.47 7.92 6.83 6.16 5.50 5.17

For experimental measurements of residual stresses using the deflection method, 20 mm strips were rolled at reduction ratios from 20 to 50%. Under industrial conditions, strips with a reduction ratio lower than 20% are rarely rolled due to the economy of the process. Also, strips with a reduction ratio greater than 50% are rarely rolled due to the tendency for edge cracks to develop. The strips’ widths (b 0) and thicknesses (h 0), as well as the reduction ratios, were selected with the aim of achieving plane strain conditions [2]. To obtain plane strain conditions, the values of strip spreading (Δb) during the rolling process were experimentally determined (Table 4). All obtained spread values were within the range of up to 1.55%, i.e., lower than 2%, which are the boundary conditions of deformation considered as a plane task, along with the condition of b 0/h 0 >10 [21,22,23]. For the deflection test, 84 mm-long samples were cut from the rolled strips. This choice of sample length is limited by the possibility of obtaining a completely flat strip during laboratory rolling.

Table 4

Spreading values of rolled strips depending on reduction ratios

ε (%) 20 30 40 50
b 0 (mm) 20
b 1 (mm) 20.18 20.24 20.29 20.31
Δb = (b 1 – b 0)·100/b 0 (%) 0.90 1.20 1.45 1.55

3 FEM rolling simulation

In this study, the commercial software DEFORM was employed to simulate the rolling process and analyze longitudinal residual stresses. The software offers both 2D and 3D simulation capabilities for rolling processes. Generally, 3D simulation allows for the analysis of residual stress distribution across both the thickness and width of the strip. When utilizing 3D modeling, the elastic deformation of the rollers must be considered, as it plays a crucial role in inducing inhomogeneous deformation and distribution of residual stresses across the width of the strip. However, the DEFORM software package does not support the deformation of bodies with rotational motion [24]. Consequently, employing 3D modeling with the roller as a rigid body would lead to constant values of longitudinal residual stresses along the direction of the strip width. Additionally, the objective of this research was to analyze the distribution of longitudinal residual stresses across the thickness of the strip. Furthermore, the simulation results for the stresses on the surface of the strip needed to be compared with the results of stress measurements obtained through the deflection method. The deflection method, where layers from one side of the strip are removed, does not allow the determination of stress distribution across the width of the strip, and it is assumed to be constant. Due to the limitations mentioned above and the research objectives, the decision was made to opt for 2D simulation in the rolling process modeling.

Figure 1 provides an initial schematic representation of the 2D simulation of a cold-rolling process. Since rolling is symmetrical relative to the longitudinal x-axis, the figure shows the upper half of the strip’s thickness and the top roller. The lower symmetrical part of the strip and the bottom roller is disregarded by setting a boundary condition that states that the velocity of displacement of the nodes of the finite element mesh on the x-axis equals zero in the direction of the y-axis.

Figure 1 Scheme of 2D simulation of the rolling process.
Figure 1

Scheme of 2D simulation of the rolling process.

The roller and strip dimensions, peripheral roller velocity, and strip reduction ratios are the same as those used in the experimental rolling. The pusher moves in the direction of the x-axis at 10% the peripheral velocity of the roller and its force is active only before while the rolls grasp the strip.

The roller and pusher are rigid bodies. The strip is an elastic–plastic body that strain hardens according to the following equation:

(1) K f = 332.12 ( ln ( h 0 / h 1 ) ) 0.1322 ,

which is derived from an experimental evaluation of yield stress (R p0.2) by tensile testing of previously rolled strips at reduction ratios between 10 and 50% (Table 3). K f is the yield stress, and h 0 and h 1 are the thickness of the initial and rolled strips, respectively. The modulus of elasticity for the strip’s material is E = 70 GPa and Poisson’s ratio is ν = 0.33.

A single-layer mesh of square finite elements (each with 4 nodes) was chosen for the strip, which is fully suitable for simulating planar tasks of the deformation process [25]. In the case of thin bodies, the necessary minimum number of elements is 4 [24]. The mesh density was carefully selected to satisfy convergence conditions concerning the mean value of the contact stress (roll pressure). To achieve this, the mesh underwent an iterative refinement process by adjusting the number of finite elements per strip thickness from 4 to 10. This refinement continued until the contact stresses between consecutive iterations exhibited differences of approximately 1%. Eventually, a mesh with 8 square finite elements covering half of the strip’s thickness was chosen. The dimensions of these finite elements amount to 0.08 mm. When conducting rolling simulations with reduction ratios ranging from 20 to 50% on contact surfaces with lengths spanning from 4.284 to 6.746 mm, a total of 46 to 55 finite elements were arranged, respectively. This mesh density has proven to be sufficient to accurately represent the curvature of the contact surface and ensure the precision of the simulation across the entire deformation zone.

The choice of contact friction conditions on the surface between the rollers and the strip is of special importance for the rolling process. Contact friction plays an active role as it allows the rollers to grip the strip. In conditions of sliding friction, which is characteristic of the industrial rolling of sheets and strips, three areas can be distinguished in the deformation zone. At the entrance part of the deformation zone, where the peripheral velocity of the rollers is higher than the strip velocity (so-called backward slip zone), shear forces of contact friction act in the direction of rolling. In the central part of the deformation zone, where the peripheral velocity of the rollers and the strip velocity are approximately equal (so-called sticking zone), the contact friction forces change direction. In the exit part of the deformation zone, where the strip velocity is higher than the peripheral velocity of the roller (so-called forward slip zone), friction forces have the opposite direction to the direction of rolling. This complex distribution of friction forces along the contact surface has a great impact on the character of metal flow in the deformation zone, and thus, on the appearance of residual stresses. The quantitative indicator of friction conditions in plastomechanics is the coefficient of friction. An analysis and simulation of the process is primarily performed on the basis of the constant value of the friction coefficient. In this manuscript, the simulation was performed with the adopted constant values of the Coulomb friction coefficient (μ) of 0.1 and 0.2. Reviews of the literature suggest that friction coefficient values in this range are most commonly used for comparable conditions of rolling with lubrication by hydraulic oil [26,27,28].

4 Results and discussion

4.1 FEM analysis of residual stresses

The analysis of the conditions for the occurrence of residual stresses was carried out using contour diagrams for the components of linear (ε x ), shear (γ xy ), and effective strains (ε eff). Figures 2–4 show diagrams for the selected cases (reduction ratios of 20 and 50%; friction coefficients of 0.1 and 0.2).

Figure 2 Contour diagrams of linear strain ε x for selected reduction ratios and friction coefficients: (a) ε = 20%, μ = 0.1; (b) ε = 20%, μ = 0.2; (c) ε = 50%, μ = 0.1; and (d) ε = 50%, μ = 0.2.
Figure 2

Contour diagrams of linear strain ε x for selected reduction ratios and friction coefficients: (a) ε = 20%, μ = 0.1; (b) ε = 20%, μ = 0.2; (c) ε = 50%, μ = 0.1; and (d) ε = 50%, μ = 0.2.

Figure 3 Contour diagrams of shear deformation γ xy for selected reduction ratios and friction coefficients: (a) ε = 20%, μ = 0.1; (b) ε = 20%, μ = 0.2; (c) ε = 50%, μ = 0.1; and (d) ε = 50%, μ = 0.2.
Figure 3

Contour diagrams of shear deformation γ xy for selected reduction ratios and friction coefficients: (a) ε = 20%, μ = 0.1; (b) ε = 20%, μ = 0.2; (c) ε = 50%, μ = 0.1; and (d) ε = 50%, μ = 0.2.

Figure 4 The contour diagrams of the effective strain ε eff for the selected reduction ratios and friction coefficients: (a) ε = 20%, μ = 0.1; (b) ε = 20%, μ = 0.2; (c) ε = 50%, μ = 0.1; and (d) ε = 50%, μ = 0.2.
Figure 4

The contour diagrams of the effective strain ε eff for the selected reduction ratios and friction coefficients: (a) ε = 20%, μ = 0.1; (b) ε = 20%, μ = 0.2; (c) ε = 50%, μ = 0.1; and (d) ε = 50%, μ = 0.2.

The contour diagrams for the linear strain ε x (Figure 2a–d) show that the deformation starts first on the surface, which occurs before the rollers grip the strip. Then, it spreads over the entire thickness of the strip within the roll gap. The shape of the iso-lines shows that this relation along the deformation zone can change several times depending on the reduction ratio and the coefficient of friction. However, in all selected cases, at the exit part of the roller gap, the final values for ε x are first achieved at the strip axis and only then at the surface. This results in the occurrence of tensile stresses in the surface layers, and compressive stresses in the central layers of the strip.

Contour diagrams of shear strain (Figure 3a–d) show that at the entrance to the deformation zone, an area is formed with negative values for γ xy . This is due to the fact that internal shear strains and shear strains are caused by friction on the surface of the strip in opposite directions [19]. In all the selected cases, the absolute values for γ xy decrease from the strip surface towards its axis. At the exit part of the deformation zone, the final shear strain is achieved first on the strip surface and only then on the strip axis. The values for γ xy increase from the axis toward the surface of the strip. The greater the reduction ratio and the coefficient of friction, the greater the gradient. We can get the impression that this distribution for γ xy at the exit part of the deformation zone cancels the negative values of γ xy at the entrance part. This ultimately indicates that lower residual stress values can be expected for higher values of reduction ratio and friction coefficient.

The shape of the iso-lines on the contour diagrams for the effective strain (ε eff) (Figure 4a–d) show that at the entrance and central part of the roller gap the distribution of ε ef is dominantly determined by the distribution of ε x . At the exit part of the roller gap, the final values for ε eff are achieved first on the surface, and then on the strip axis. The values for ε eff decrease from the surface toward the strip axis. The shear strain component γ xy also has these characteristics at the exit from the roller gap. This leads to the conclusion that the distribution of ε eff at the exit part of the deformation zone is predominantly conditioned by the distribution of γ xy . This is supported by an additional analysis of the distribution of the linear strain component in the direction of the y-axis (ε y ), which along the entire deformation zone is identical to the distribution of ε x but with a negative sign in accordance with the principle of continuity.

In general, nonhomogeneous deformation is present throughout the entire cross-section of the strip. This means that in addition to operating stresses, additional stresses also occur. From the point of view of the aim of this article, the additional stresses at the exit from the deformation zone are significant because they remain permanently accumulated in the strip and represent residual stresses. The analysis carried out shows that it is very difficult to precisely quantify the influence of process parameters on the conditions for the formation of residual stresses. In that case, the analysis of longitudinal stresses after reaching the steady state of deformation during rolling presents itself as a valid approach. Accordingly, the stresses were analyzed at the free end of the strip in a zone far enough from the deformation zone where the rolling force is completely unloaded. Figure 5(a) and (b) shows the contour diagrams for the selected examples of the longitudinal stresses at the cross-section of the strip for different process parameters. The positions at the free end of the strips are indicated, where stress values were extracted which no longer change but remain permanently in the strip. They represent the final state of the longitudinal residual stresses.

Figure 5 The contour diagrams of longitudinal stresses for selected reduction ratio and friction coefficient: (a) ε = 20%, μ = 0.1 and (b) ε = 20%, μ = 0.2.
Figure 5

The contour diagrams of longitudinal stresses for selected reduction ratio and friction coefficient: (a) ε = 20%, μ = 0.1 and (b) ε = 20%, μ = 0.2.

The values and distribution of longitudinal residual stresses across the strip thickness were analyzed depending on the reduction ratio and friction coefficient. Due to the symmetrical rolling conditions, the results are shown on one-half of the rolled strip’s thickness (h 1/2) (Figure 6).

Figure 6 Distribution of residual stresses across the strip’s thickness (normalized thickness) for different values of the reduction ratio and friction coefficient: (a) ε = 20%, (b) ε = 30%, (c) ε = 40%, and (d) ε = 50%.
Figure 6

Distribution of residual stresses across the strip’s thickness (normalized thickness) for different values of the reduction ratio and friction coefficient: (a) ε = 20%, (b) ε = 30%, (c) ε = 40%, and (d) ε = 50%.

Under all configurations of process parameters, tensile stresses occurred on the strip’s surface, where they reached their maximum, while compressive stresses occurred on the strip’s longitudinal axis. This is in accordance with the distribution of linear strain ε x at the exit part of the deformation zone (Figure 2a–d). In qualitative terms, the stress distributions are significantly consistent with the results of a previous study [11]. The neutral axes are located approximately between 0.5 and 0.65 (h/(h 1/2)). The stress distribution is asymmetric, with an exception of a strip rolled at the reduction ratio of 20% and friction coefficient of 0.2, which has an approximately symmetrical stress distribution (Figure 6a). On the whole, surface stresses are in the interval between 43.5 and 96.5 MPa, while the stresses on the strip’s axis are lower, ranging between −27.5 and −65.4 MPa.

As the reduction ratio increases, the values of residual stresses decrease across the strip’s thickness. This is because, with higher reduction rates, the inner layers along the strip’s thickness are deformed more than the surface layers [29]. This is also consistent with the development of linear strain ε x at the exit part of the deformation zone. This reduces the impact of contact friction on the development of internal shear strains across the thickness of the strip and on the induction of residual stresses.

Increasing the friction coefficient from 0.1 to 0.2 results in lower values of residual stresses. This coincides with the results obtained in a previous study [19]. Namely, at the entrance of the deformation zone, the shear strain due to contact friction occurs in the direction of rolling, while at the exit of the deformation zone, it has the opposite direction to the direction of rolling [30]. However, as already emphasized, the internal shear strain in the layers across the strip thickness has the opposite direction to the shear strain on the contact surface. Thus, at the entrance of the deformation zone, the shear strain γ xy has negative values, while at the exit, it has positive values. With an increase in the coefficient of friction, the values and the gradient for γ xy at the exit from the deformation zone increase (Figure 3a–d). This largely cancels the effects of γ xy from the entrance into the deformation zone, resulting in lower residual stress values with an increase in the friction coefficient. The least differences in stress distribution along the strip’s thickness caused by the friction coefficient were observed at the reduction ratio of 50% (Figure 6d).

In general, the presence of tensile stresses in the surface layers of the strip and compressive stresses in the central layers are determined by the distribution of linear strain at the exit part of the deformation zone. Meanwhile, the values of residual stresses and their variations depending on process parameters are determined by the gradient and ratio between the distribution of linear strain and the distribution of shear strain at the exit part of the deformation zone.

The residual stresses on the strip’s surface and axis, where they reach their highest tensile and compressive values, respectively, can be taken as quantitative indicators of the level of residual stress in thin strips. Changes in residual stresses on the strip’s surface and axis depending on the reduction ratios and friction coefficients are shown in Figure 7. The values of surface stress for the friction coefficient of 0.1 and all reduction ratios are approximately 1.5 times higher than those for the friction coefficient of 0.2 (Figure 7a). At high reduction ratio values (40 and 50%), residual surface stresses are stabilized, with maximum differences between them not exceeding 3 MPa, for the same friction coefficients. This may be a consequence of the strain hardening model (Eq. (1)), i.e., a small increase in the yield stress in the area where reduction ratios are higher. As a result, there are fewer pronounced differences in the mode of the metal flows.

Figure 7 Residual stress on the strip’s surface (a) and axis (b).
Figure 7

Residual stress on the strip’s surface (a) and axis (b).

On the other hand, the values of residual stress on the strip’s axis change somewhat differently (Figure 7b), which is possibly caused by the stabilization of their distribution along the strip’s thickness. The values of compressive residual stress on the strip’s axis for the friction coefficient of 0.1 are higher compared to those for the friction coefficient of 0.2, with a maximum difference of up to 21 MPa (ε = 30%).

4.2 Experimental determination of residual stresses

The longitudinal residual stresses were determined using the deflection method, which is based on disturbing the stress equilibrium and measuring the resulting elastic lines (bends). The deflection method was developed based on the basic principles of experimental methods with material destruction [8,9]. The complete procedure for the application of the method and the derivation of equations for the calculation of residual stresses is presented in previous studies [12,13]. Therefore, only the main steps of the experimental–analytical procedure are presented below.

The basic criteria and conditions for applying the deflection method for determining residual stresses in rolled strips are the following:

  1. The rolled strip must be entirely flat, indicating a symmetrical balance of residual stresses in the upper and lower halves of the strip.

  2. The elastic bending (elastic line) caused by disturbing the stress balance must be measured with great precision, and an accurate analytical model of the elastic line must be derived.

  3. The equivalent external load causing such bending is determined using the elastic line model.

  4. By equating the equivalent external moment load with the resulting moment load from residual stresses (after disturbing their balance), a relation for calculating longitudinal residual stresses on the strip surface is derived.

The stress equilibrium in flat-rolled strips was disturbed by removing layers of metal on one side by etching in a 20% NaOH solution at room temperature, while the rest of the sample was protected. The average thickness of the removed layers was 0.257 mm. As a result of the disturbed stress equilibrium, the strip elastically bends, exhibiting symmetry with respect to the transverse axis. The appearance of the strip after rolling and removal of layers is shown in Figure 8. After the disturbance of stress equilibrium, the measured elastic line follows the shape of the bent sample. The model of the elastic line plays a crucial role in calculating residual stresses and serves as the initial data for the analysis. To construct this model, deflection measurements were taken at regular intervals of 5 mm along the length of the strip. The deflection measurements were performed using a comparator with a precision of 0.05 mm. In order to derive the final relation for the calculation of residual stresses, one must know the model of their distribution across the strip thickness (assumed linear symmetrical distribution). Additionally, knowledge of the equivalent external moment load corresponding to the given elastic line model is required. Detailed research showed that the measured elastic line can very accurately be described by the elastic line of the cantilever bend. This also defines the equivalent external load model. As previously emphasized, a relation for calculating longitudinal residual stresses on the strip surface is derived from the condition of equality between the equivalent external moment load and the resulting moment load arising from residual stresses. The derived final relations for the elastic line, distribution model, and values of the longitudinal residual stresses are shown in Table 5. In the case of successive removal of layers (Δ changes from zero to half the strip’s thickness), elastic lines completely retain the same characteristics. This means that the residual stresses can be precisely represented by the model equation (2) given in Table 5.

Figure 8 The appearance of the rolled strip: (a) initial state and (b) after the removal of layer.
Figure 8

The appearance of the rolled strip: (a) initial state and (b) after the removal of layer.

Table 5

Schematics and equations for the elastic line and residual stresses

Scheme of stress distribution through the thickness of strip
The equation for stress distribution [12,13]
σ x = σ s 2 y h ‒ 1
(2)
The equation for the elastic line [12,13]
f = a · ( x / l ) 2
(3)
The equation for surface stress [12,13]
σ s = 8 E ( h ‒ Δ / 2 ) 3 [ 6 Δ h ‒ 6 Δ 2 + Δ 3 / h ] f max l 2
(4)

y – position by the strip thickness; h – half the thickness of the rolled strip (h 1/2); σ x , σ s – longitudinal stress along the cross-section and surface, respectively; f – the strip bend; a – coefficient; x – coordinate in the length direction of the strip; l – half of the measuring length of the strip; Δ – thickness of the removed layer; E – modulus of elasticity; f max – maximum bend.

Table 6 shows the experimental data for the reduction ratio, the thickness of the removed layer, and the maximum bend. In some cases, the reduction ratios deviated from the set values by a maximum of 1% in absolute terms. The calculated values for the elastic line and residual stresses on the strip surface are also presented. The low values of the mean square error (MSE) of the approximation of the elastic line confirm that it can be described with high precision by the model equation (3). This is also confirmed by the diagram in Figure 9 for the selected example.

Table 6

Experimental and calculated results of elastic lines and residual stresses

Sample no. ε (%) h 1 (mm) 2l (mm) Δ (mm) Elastic line f = a (x/l) 2 MSE f max (mm) σ s (MPa)
1 19.53 1.030 84 0.312 f = 2.303 (x/l) 2 0.0130 2.427 81.20
2 20.00 1.024 0.310 f = 2.261 (x/l) 2 0.0180 2.371 78.92
3 20.47 1.018 0.311 f = 2.456 (x/l) 2 0.0109 2.611 84.41
4 20.47 1.018 0.309 f = 2.139 (x/l) 2 0.0302 2.246 74.08
5 30.00 0.896 0.315 f = 2.739 (x/l) 2 0.0273 2.877 69.73
6 30.47 0.890 0.314 f = 2.874 (x/l) 2 0.0348 3.021 72.42
7 30.00 0.896 0.298 f = 2.387 (x/l) 2 0.0271 2.506 64.98
8 40.00 0.768 0.198 f = 2.044 (x/l) 2 0.0215 2.151 65.55
9 40.46 0.762 0.220 f = 2.059 (x/l) 2 0.0119 2.161 56.78
10 39.45 0.775 0.230 f = 2.113 (x/l) 2 0.0342 2.219 57.30
11 50.00 0.640 0.197 f = 2.739 (x/l) 2 0.0207 2.901 59.11
12 50.00 0.640 0.200 f = 2.224 (x/l) 2 0.0356 2.334 46.68
13 50.46 0.634 0.190 f = 2.129 (x/l) 2 0.0282 2.235 46.65
14 50.47 0.634 0.192 f = 2.384 (x/l) 2 0.0350 2.503 51.57
Figure 9 Measured and calculated values of the elastic line for sample 1.
Figure 9

Measured and calculated values of the elastic line for sample 1.

The calculated residual stress values from Table 4 are shown in Figure 10 as a function of the strip reduction ratio. The stress values obtained by FEM simulation are shown parallelly. Considering the dispersion areas of the experimentally determined stress values within the same range of reduction ratios, it is apparent that only intervals of residual stresses can be discussed. The obtained values of residual stresses ranged from 46.6 to 84.4 MPa, for all reduction ratios. The residual stresses on the strip’s surface decreased with increased reduction ratios, which coincides with the results obtained by FEM simulation.

Figure 10 The longitudinal residual stresses on the strip’s surface determined by the deflection method and FEM simulation.
Figure 10

The longitudinal residual stresses on the strip’s surface determined by the deflection method and FEM simulation.

If the stress values obtained by FEM simulation are viewed as intervals of values obtained for friction coefficients from 0.1 to 0.2, then these intervals of residual stresses are wider and incorporate the dispersion area of the stresses obtained through the experimental procedure (Figure 10). This suggests that the experimental rolling with manual lubrication was conducted under the conditions corresponding to a friction coefficient in the interval between 0.1 and 0.2. A comparison of the mean values of stress intervals obtained by both methods yields a maximum difference of up to 6 MPa (ε = 40%), which is an acceptable difference considering the values of residual stresses and dispersion intervals.

4.3 Strip tension effects

Changes in residual stresses in response to back and/or front tensions, as standard operating parameters in the technology of thin strip rolling, were analyzed using FEM simulation of strip rolling at a reduction ratio of 20% and friction coefficients between 0.1 and 0.2. The initial FEM simulation was stopped when a steady rolling state was achieved. The pusher was removed as it was no longer in contact with the strip and has no role in the further rolling process. Then, the back and/or front tension conditions were set, with the back tension set at 20 MPa and the front tension at 60 MPa. A new simulation was then run under a new set of conditions.

The contour diagrams of the selected example (reduction ratio, 20%; coefficient of friction, 0.1) presenting the distribution of longitudinal stresses across the cross-section of the strip depending on the tension parameters are shown in Figure 11. When applying back tension (Figure 11a), the stresses at the indicated position of the free end of the strip represent residual stresses. When applying front tension, as well as front and back tensions together, there are resultant stresses at the free end of the strip that come from residual stresses and front tension (Figure 11b and c). The resulting stresses are significantly lower than the yield stress (272 MPa, Table 3) for this strip reduction ratio. Therefore, they are in the elastic region. This means that they will relax when the front tension is removed and only residual stresses will remain in the strip. In this way, residual stresses were identified in both cases when front tension was present.

Figure 11 The contour diagrams of longitudinal stresses for ε = 20% and μ = 0.1 depending on strip tension: (a) back, (b) front, and (c) combined back and front.
Figure 11

The contour diagrams of longitudinal stresses for ε = 20% and μ = 0.1 depending on strip tension: (a) back, (b) front, and (c) combined back and front.

After applying back tension (Figure 12a), the stress distribution shape along the strip’s thickness remains practically unchanged compared to rolling without tension. The stress values for rolling with the friction coefficient of 0.1, and back tension applied to the strip, are lower compared to those obtained for rolling without tension, with the differences of 13 MPa on the strip’s surface, and 20 MPa on the strip’s axis. The residual stress values for rolling with the friction coefficient of 0.2 and applied back tension are only marginally different to those for rolling without back tension. Obviously, the back tension acting on the strip at the entrance to the deformation zone has little impact on the residual stresses. This is in accordance with the previously identified impact of the distribution of linear and shear strains at the exit from the deformation zone, where the state of residual stresses is finally determined.

Figure 12 The effects of tension on the residual stresses: (a) back, (b) front, and (c) combined back and front.
Figure 12

The effects of tension on the residual stresses: (a) back, (b) front, and (c) combined back and front.

As previously stated, the analysis of the effects of front tension and the combined effects of back and front tensions on stress values requires prior unloading of the rolled free end of the strip, where the superimposition of longitudinal residual stresses and external tensile load occurs. So, the identified residual stresses across the strip’s thickness have lower values compared to rolling without tension. When the front tension is applied (Figure 12b), the residual stresses are up to 46% lower on the strip’s surface and up to 56% lower on the strip’s axis, compared with rolling without tension, for both values of the friction coefficient. When rolling with both back and front tensions is applied (Figure 12c), the stress is up to 39% lower on the strip’s surface and up to 20% lower on the strip’s axis, compared with rolling without tension. The impact of front tension is expected, considering that at the exit from the deformation zone, front tension reduces the yield stress throughout the entire thickness of the strip. This reduces its change across the thickness of the strip, i.e., it becomes more homogeneous, which is inevitably reflected in residual stresses. This is supported by a subsequent check when it was found that the gradient of the effective stress at the exit from the deformation zone (at the position where the strip is still in contact with the rollers) when front tension is applied is significantly lower than when there is no strip tension (Figure 13). Therefore, on the whole, the front tension has the strongest effect on lowering the values of the induced residual stresses.

Figure 13 Contour diagrams of effective stress for ε = 20% and μ = 0.1: (a) without strip tension and (b) with front strip tension.
Figure 13

Contour diagrams of effective stress for ε = 20% and μ = 0.1: (a) without strip tension and (b) with front strip tension.

5 Conclusions

FEM simulation was used to investigate values and distribution of longitudinal residual stresses along the thickness of cold-rolled strips of aluminum alloy EN AW-5083 for different reduction ratios, friction coefficient values, and back and front tensions. The experimental measurements of the residual stresses on the strip’s surface were done using the deflection method. Based on the implemented research program and performed analyses, the following conclusions were reached:

  1. The values and gradient of longitudinal residual stresses across the strip thicknesses are determined by the ratio of the distribution of linear strains and of shear strains, as well as the conditions of strain hardening at the exit part of the deformation zone.

  2. The longitudinal residual stresses obtained by FEM simulation of cold rolling are balanced across the entire strip’s thickness. The surface stresses are tensile with a change interval between 43.5 and 96.5 MPa, while the axial stresses are compressive with a change interval between −27.5 and −65.4 MPa.

  3. Increased reduction ratios result in lower values of residual stresses.

  4. Increasing Coulomb’s friction coefficient from 0.1 to 0.2 causes residual stresses to increase by up to 1.5 times.

  5. The values of the residual stresses on the strip’s surface obtained by the deflection method range from 46.6 to 84.4 MPa. These values and their trends of change for different reduction ratios agree well with the results obtained by the simulation. Therefore, the FEM is a reliable tool for predicting residual stresses in cold-rolled strips.

  6. FEM simulation of rolling with tension shows that residual stresses decrease when tension is applied to the strip. Front tension has the strongest effect, resulting in stress decreases of up to 56%.

  1. Funding information: The authors state that no funding was involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2022-12-23
Revised: 2023-07-19
Accepted: 2023-08-01
Published Online: 2023-08-30

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/jmbm-2022-0297
Keywords for this article
cold rolling; residual stress; finite elements method; deflection method; reduction ratio; friction coefficient; front tension; back tension
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BY 4.0

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