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Abstract

Foldover is a common, yet complex, transient boundary condition in several ‘metal forming processes’ including the barrelling compression test (BCT). The onset of foldover during BCT is comparable to that of necking in the tensile test. However, its role in flow curve identification has been ignored in the current literature. Methods to detect and measure its onset and growth during the processes are also not available. We propose three methods to identify the onset of foldover and measure its progress with deformation. These are conducted together with case studies of BCT to have a fundamental understanding of the onset and growth of foldover in the test and more general cases. Experimental and numerical methods are developed to accomplish this based on (1) direct measurement using tracing points, (2) energy-based and (3) geometry-based indirect measurements. These outline the required supplementary data and allow proper detection of the onset. In the first method, the migration of tracing points from the free surface to the platen-sample interface, simulated numerically, is monitored and measured visually. The second method employs the load-stroke data to calculate the total power input to the deforming system. In the third method, a criterion has been derived to detect the onset of foldover by monitoring the geometry of the sample profile during the test and comparing it to a reference quadratic profile. The observations and outputs of each method are compared and discussed.

Keywords

Compression test foldover onset flow curve finite element modelling virtual experiment

Introduction

The barrelling compression test (BCT), also known as upsetting, uniaxial or axisymmetric compression, is a basic bench-scale metal forming process and it is extensively used to design the large scale and applied metal forming processes. The test is essential to indirectly study and measure the stress-strain behaviour, for example, to study dynamic recrystallisation.^1,2 However, only a few indirect measurement methods exist with serious limitations.^3,4 The erroneous BCT based flow curves have been perceived in numerous published works as measured values rather than computed values. To find more reliable and detailed solutions to the test, much more theoretical-analytical developments on the unwanted effects, for example, foldover and friction, are needed. The test’s complex boundary condition will be studied here.

As the BCT sample deforms, its boundary condition changes via partial migration of the sample’s free surface to the platen-material contact area, that is, foldover. Several unwanted effects, such as friction induced barrelling,⁵ heterogeneous deformation and foldover, drive the test into a multiaxial mode. These effects complicate the interpretation and transformation of the test results to the real forming processes and thus eventually invalidate the physical simulation. Practically, the test cannot be performed in a uniaxial mode to simulate high strains over a wide range of test conditions³; thus, it becomes impossible to avoid unwanted effects. Foldover increases with friction and deformation and is practically inevitable in many forming scenarios. One notes that the onset of BCT’s foldover and the tensile test’s necking are similar as they cause triaxiality and significant complexity in the calculation of flow stress from the measured data. Therefore, it is essential to partition the test data into those of pre and post foldover by detecting the onset of foldover.

To the authors’ best knowledge, there is no work on the identification of foldover’s onset or a criterion to measure its progress post-onset. Early works include those of Avitzur and Kohser⁶ and Kohser and Avitzur⁷ which did not propose a method to detect the onset of foldover. It is a common assumption in many earlier studies of the compression test that the foldover starts immediately along with the onset of deformation and that its final measured value grows homogeneously during the test. Consequently, in the existing literature, foldover has been only measured at the final stage of the test. This is an inevitable consequence of correlating the foldover inversely to an overall barrelled profile’s radius. As such, a finite radius and its corresponding estimated foldover form immediately when the test starts and the radius’ reduction with the test’s progress, increases the foldover. Other works include Ettouney and Stelson,⁸ Tan,⁹ Hou and Ståhlberg,¹⁰ Akata and Çetinarslan¹¹ and Fan and Dong.¹² Ettouney and Stelson⁸ model was established on two unjustified assumptions: (1) to assume a fixed total sum of free and contact surfaces of the sample, and (2) to trace the profile points on the sample using a superposition principle to estimate the foldover. A simplistic geometrical study on foldover was presented by Sivaprasad and Davies,¹³ Altinbalik and Çan¹⁴ and Lin and Lin¹⁵ presented two analyses of the test which comprised the effect without detecting its onset or quantifying it. Akata and Çetinarslan¹¹ employed similar assumptions in their work.

A numerical-analytical model was developed by Yao and Mei.² Although Fan et al.¹² included foldover in their study, they did not consider the influence of this phenomenon on the test data or results.

Due to the close correlation between foldover and friction, a closer examination of the migration phenomena (foldover) and a summary of the available works on both are given here. Friction between the platens and the workpiece during metal forming can be indirectly measured during the test to simulate industrial processes. Tan¹⁶ proposed a dynamic friction model in which the friction depends on both strain rate and normal pressure and employed the model to predict contact stresses in disc-upsetting. The model, however, does not account for barrelling and assumes a cylindrical profile which limits its applicability to real tests and processes. The barrelling was included in a frictional model developed by Avitzur.¹⁷ The model was extended first by Ebrahimi and Najafizadeh.¹⁸ They proposed the first ‘geometry based indirect measurement of friction during the test’. Numerical studies by Solhjoo¹⁹ revealed that the method was unable to calculate the friction factor correctly. The work was reinvented by Solhjoo and Khoddam²⁰ focusing on indirect measurement of the barrelling during the test to improve the accuracy of the method by introducing more suitable barrelling parameters. Khoddam et al.²¹ proposed an energy-based approach to evaluate closed-form solutions of the test. Khoddam et al.⁵ utilised a proposed kinematic solution²² in an energy conservation framework and developed a closed-form solution for friction factor during the test. They found that their identified friction factor is valid before the onset of foldover.

Friction and deformation both promote the onset of foldover.²³ Friction between the specimen and platens restricts the expansion of the contact area, causes the free surfaces to come convex and results in barrelling. If the reduction in height of the sample reaches a certain magnitude, the convex free surfaces fold over and come into contact with the platens. The friction factor can be identified with reasonable accuracy before foldover based on the barrelled profile⁵ but it changes during the deformation and foldover which adds to the problem’s complexity.

The current work aims to propose and explore different methods to detect the onset of the foldover and its quantification during the test. It proposes three methods considering (1) direct measurement using tracing points, (2) energy-based and (3) geometry-based indirect measurements to detect the onset of foldover and to quantify the event in an average sense.

Methodology

Virtual experimental setups for BCT

Further to the analytical limitations of the foldover models, the commercially available compression test machines collect very limited types of data which are inadequate to study foldover. For example, the instantaneous changes of the sample’s free surface, the barrelled profile, are not recorded by commonly available test machines. To avoid the outlined analytical and experimental limitations, a virtual reality experiment is utilised in this work. It includes a finite element (FE) model of the test and some in-house post-processing computer codes. The FE model was developed, solved and post-processed using Deform™ 3D software by Scientific Forming Technologies Corporation (SFTC). Details of the model will be presented later in section “FE simulation strategy” of this article. Also, in-house post-processing computer codes were developed to read the output of the FE solutions and to utilise them to detect and measure the foldover. Contrary to the available oversimplified closed-form solutions of the test, the employed FE model provides details on the instantaneous profile of the sample during the test. The virtual setup enables the acquisition of the pseudo data including the sample’s discretised geometry (mesh), key nodal computations at the nodes and key elemental computations at each element centroid. Also, the heterogeneous distribution of effective strain, strain rate and stress within the sample, computed by the numerical solution, is used as the input for the methods which are proposed in this work to calculate foldover.

Alternatively, in a real experimental setup, one has to measure the sample’s deforming profile using a high precision visual data acquisition and processing system and measure the sample geometry instantaneously to build an accurate FE model. The measured and calculated data should then be augmented in a real-time fashion. The augmented reality setup enables indirect measurement of friction (see Khoddam et al.⁵ for details).

Indirect measurement of foldover based on data from the virtual setup is presented in this work.

An experimentally verified virtual setup

The FE solution, employed in the virtual setup, is typically considered a reference solution in many complex problems including friction during compression tests and foldover in the current study. However, it is essential to experimentally verify ‘the finite element-based setup’ or ‘the virtual setup’ outlined in section “Virtual experimental setup for BCT” to minimise the risk of employing a numerical solution that does not fully represent the experimental conditions.

We will be utilising the BCT’s experimental data in a special way here. While the typically available experimental load-displacement data are not adequate for foldover and barrelling studies, the data can be used to verify a detailed virtual FE model which can eliminate the limitations of the available test solutions. The verification also confirms that the FE model is constructed correctly and therefore fulfils a prerequisite to having a valid and representative solution. Before describing the experiments and verifying the virtual setup, which will be presented in section “Results”, let’s introduce some new terms for a better understanding of our methodology.

Foldover terminology

It is critical to note the resemblance of necking and foldover during the tensile and compression tests and that both events do not start immediately with deformation. Foldover occurs after increasing the sample profile’s barrelling when the sample’s fillet radius at the edge of its free and friction surfaces increases to a certain limit. One has to distinguish between the fillet radius and the overall profile (free surface) of the sample which might be represented with a much larger radius. Tracing points and the profile are introduced here to help with the above understandings. Foldover has to be understood qualitatively and quantitatively before it can be employed to model the material’s flow around a tight corner, flash forming, or similar highly localised deformations. The test chosen in this work helps us to better understand the complex phenomenon and its findings are transferable to several forming scenarios in open and closed die forging

Tracing points and the profile

In the current literature, a BCT sample is machined with tiny groves at its top plane to allow an experimental tracing of its surface points and migration of the free surface points to the contact zone. In this work, we utilise a similar concept, called tracing points, for a numerical study of foldover. A tracing point is defined as a virtual material point that represents the location and motion of the material point; it is traced sequentially as the sample deforms.

Figure 1(a) to (c) illustrate a deformation scenario in which the foldover occurs (with foldover); three tracing points ( $t$ , $q$ and $s$ ) are used to show the migration of tracing point $t$ to the platen-sample interface. Figure 1(d) to (f) demonstrate the same tracing points in a deformation scenario in which the foldover doesn’t occur (no foldover). This classification is essential for a better understanding of the event. During a ‘no-foldover scenario’ the tracing point $t$ remains in the free surface throughout the entire deformation; three steps of the deformation scenario are shown in Figure 1(d) to (f).

Figure 1.

Three tracing points $t, q$ and $s$ in a representative quarter of the sample at three sequential time frames are shown with and without foldover at the top and bottom rows, respectively. (a) shows the onset of foldover. (b) and (c) show the development and finalisation of the foldover, respectively. In the no-foldover scenario (d–f), the corner tracing point $t$ remains shared between the profile and interface during the three steps. (c) and (f) show the final step, with a sample height of $H_{f}$ , for each scenario with ( $ς = ς_{f}$ ) and without folding ( $ς = 0$ ), respectively.

The sample’s profile or simply profile is defined here as the loci of the tracing points which define the sample’s free surface. Points $t$ , $q$ and $s$ in Figure 1(a), (d), (e) and (f) belong to the profile.

Figure 1(a) to (c) show the onset, an intermediate development and completion of foldover, respectively. Deformation parameters associated with the onset are shown with subscript $i$ (initial) and those with the final (completion) step are shown with subscripts $f$ . The sample’s height and the foldover are shown as $H$ and $ς$ , respectively. We define the instantaneous depth of the migration, $ς$ as a measure of foldover. It is defined as the radial displacement of the point $t$ between the step $H_{i}$ at the onset and a typical step $H$ at $H < H_{i}$ , ( $ς \neq 0$ ). Also, it is assumed that the sample’s top plane diameters at the onset and the final steps are $d_{i}$ and $d_{f}$ , respectively (not shown in the figure). The final foldover $ς_{f}$ , corresponding to the last deformation step, is defined here as:

ς_{f} = 0.5 (d_{f} - d_{i})

(1)

The key inputs to finding the onset are $H_{i}$ and $d_{i}$ where foldover, $ς$ , is zero. Mathematically speaking: point $t$ is shared between the sample’s free surface and its interface with the platen until before the onset ( $ς = 0$ ). The point slips from the profile zone into the ‘interface only zone’ just after the onset and its shared position is filled with its next point which used to be a part of the profile. An expression for $ς$ will be formulated based on the observations obtained using the virtual experiment in section “Virtual experimental setups for BCT”.

The foldover is a boundary condition change that has to be properly considered in a solution to the problem. This challenging task can be performed only once the onset has been detected. Before presenting the methods to detect and quantify the onset of foldover, the virtual setup, including its FE model, is briefly summarised next.

Experimental verification

To verify the proposed numerical virtual test setup, the compression tests were performed using a Servotest™ Thermo-Mechanical Treatment Simulator (TMTS) machine. An AISI 304 austenitic stainless-steel plate was hot rolled and its thickness was reduced to approximately 12 mm before machining the test samples. The chemical composition of the test sample was Fe–0.02C–1.6Mn–18.5Cr–8.4Ni–0.3Cu–0.4Si–0.1Mo (wt.%). Samples with nominal diameter and height of $D_{0} = 10 \pm 0.2 mm$ and $H_{0} = 15 \pm 0.2 mm$ , respectively were machined and used for the experiments. The chosen sample size eliminated their buckling during the test. While buckling is a key issue when selecting the geometry of the sample, for the sake of brevity, the issue is not discussed here in detail; the samples’ aspect ratio was chosen to comply with those commonly used sizes in the literature to minimise the risk of buckling. A thin layer of boron nitride was used as the lubricant at the tool sample interface and allowed to dry before testing. The layer thickness added an offset on the experimentally measured instantaneous sample height, $H$ by the TMTS machine. It was measured and an appropriate load-stroke adjustment was carefully made to the recorded data. Also, real dimensions of the sample (including deviations from the nominal size due to machining error) were used to construct the FE models corresponding to each experiment. Comparing the final sample’s profile at the end of the test and comparing it with several FE solutions of different friction coefficients in a trial-and-error fashion, an appropriate value of $m$ for each case was obtained for the experimental setup. Samples were deformed at $700 ° C$ with a variable crosshead velocity of $\overset{\cdot}{H} = H \dot{\bar{ε}}$ to achieve a constant average effective strain rate, $\dot{\bar{ε}}$ , in the sample during the entire deformation. The sample dimensions, effective strain rates and their $m$ values are listed in Table 1. Some of the tests were repeated twice to ensure the repeatability of the test however only the nominal cases are shown in Table 1. Average effective strain rates of $\dot{\bar{ε}} = 0.1, 1, 10, 30, 50$ and 10 $0 s^{- 1}$ were utilised for the experiments and their corresponding load-stroke data were recorded. The same experimental conditions were used to construct the FE models and to solve and obtain their numerical load stroke data corresponding to each case. To verify the numerical solutions, the experimental and numerical load-stroke were compared. The comparative verification will be presented in the ‘Result’ section.

Table 1.

Hot compression test samples, deformed at $700 ° C$ with a variable crosshead speed of $\overset{\cdot}{H} = H \dot{\bar{ε}}$ . All measured heights $H$ and diameters $D$ are in $mm$ .^*

Strain rate, $\dot{\bar{ε}}$ , s⁻¹	$H_{0}$	$D_{0}$	$H_{f}$	$D_{f}$	$m$
0.1	15.14	9.98	5.47	17.37	0.4
1	15.12	9.98	5.15	17.33	0.4
10	15.11	9.98	5.12	17.85	0.6
30	15.16	9.98	5.13	17.46	0.6
50	14.92	10.04	5.47	17.37	0.5
100	15.14	9.98	5.42	17.45	0.65

Subscripts of $0$ and $f$ are used to indicate the measured height $H$ or diameter $D .$ They correspond to the initial or final configuration of the test sample, respectively.

FE simulation strategy

As explained in section “FE simulation strategy”, the software and its FE model of BCT serve as a virtual lab to produce pseudo data. To correlate friction and geometrical parameters of the deformed sample, FE models of BCT for a series of BCT case studies were constructed and solved using SFTC-DEFORM Premier software. While the software has been optimised for plasticity problems in metal forming (e.g. forging and upsetting), it also allows some adjustment in terms of mesh size, mesh updating. The mesh updating was employed in the virtual setup very carefully as it complicates the tracing of the tracing points explained in section “Tracing points and the profile”. Interested reader can find more details on the models (e.g. the time integration scheme and element formulation) elsewhere.^3,5 A test scenario was considered with the initial sample diameter and height of $D_{0} = 10 mm$ , $H_{0} = 16 mm$ and $H_{f} =$ 8 mm with 3200 deformation steps which were recorded every 10 steps. The deformation step is related to the sample height $H$ according to $H = 16 - 8 / 3190 (step - 10)$ . The friction shear stress at the platen-sample interface is expressed as $mk$ where $m$ ( $0 \leq m \leq 1$ ) denotes the friction factor. In this study, a set of friction factors of $m =$ 0.1–1 with 0.1 increments was used. Due to the axisymmetric nature of the problem, only a quarter of each test sample and platen was modelled using the coordinate system shown in Figure 1. Also, a rigid upper anvil was assumed. During the simulations, the upper platen, the primary platen, was moved at a speed of $\overset{\cdot}{H} = 20 mm / s$ along the negative z-axis, the sample was positioned between the moving upper platen and the $r$ -axis as the symmetry axis. The previous numerical study indicated that the geometry-based indirect measurements were not sensitive to the material type, temperature or deformation heat.²⁴ In all FE simulations, the software’s flow library was used to include both strain and rate hardening behaviour of AISI-316 stainless steel to simulate a 760 $° C$ isothermal condition.

Axisymmetric iso-parametric quadrilateral elements with four nodes per element were utilised in the FE model with 384 nodes and 343 elements to discretise a quarter of the test sample as shown in Figure 1. For each simulation, a constant value of the friction factor ( $m_{FEA}$ ), was chosen and the calculated elemental and nodal results were extracted. The data comprised nodal coordinates, velocities, calculated forces and elemental connectivity matrix. Also, the von-Mises effective strains $\bar{ε}$ , strain rates $\dot{\bar{ε}}$ and stresses $\bar{σ}$ were evaluated at each element centroid and recorded.

In the second method of estimating the foldover, we monitor changes in the deformation energy before and after the folding to spot the foldover’s onset.

Method 1: Using tracing points to measure $ς$

This method relies on a FE solution of the problem and visual inspection of the tracing points migrating from the free surface to the platen-sample interface. The procedure employs the graphic post-processor of the FE solver. Further to the tracing points (shown as hollow dots in Figure 1), other graphical output can be used to detect the onset such as the velocity field at each node. Comparing the tracing points chosen on the sample’s upper and mid-planes in Figure 1, one can imagine that the velocity components in the mid-plane are less sensitive to the coordinates of the points than those at the top plane. A complex velocity field pattern is needed in the upper plane to represent the mixed boundary condition and the migration of the tracing points in the vicinity of a foldover zone.

Method 2: Deformation and traction power estimated by FE analysis

Indirect detection and measurement of foldover, presented in this method rely on the FE’s step solutions. To correlate the results, here we utilise three power terms used in the FE stiffness equations namely input power, deformation power and traction power. It is found that as the deformation and input powers grow with deformation, the onset of foldover can be detected by spotting the first abrupt slope change in the ‘deformation (and or input) power-stroke’ curve. These power terms, and their recovery based on the FE results, are outlined in some detail next.

A deforming BCT sample may be considered a non-conservative system within its two boundaries namely the tool-sample interface and the free surface of the sample. The energy change in the system is path-dependent due to the non-conservative nature of the process. The conservation of energy for the BCT system is mathematically stated as $Δ E_{in} = \int_{H_{0}}^{H} LdH$ . This means that the total energy change in the system $Δ E_{in}$ is caused by the work transferred across the system boundary $\int_{H_{0}}^{H} LdH$ in which $L$ is the variable compression test load. The work is completed the sample’s height reduces from $H_{0}$ to $H$ . It can be seen that the power is associated with two irreversible transformations of mechanical power into internal power namely internal deformation power and frictional power.

The total energy transmitted to the sample by the variable and non-conservative load, $L (H)$ , is lost irreversibly due to its consumption by the plastic deformation energy $Δ E_{def}$ and frictional energy $Δ E_{tr}$ at the platen-sample interface. This can be stated as:

Δ E_{def} + Δ E_{tr} = \int_{H_{0}}^{H} LdH

(2)

We note that both $Δ E_{def}$ and $Δ E_{tr}$ are irreversible. The above conservation of energy can be stated in the rate form; an expression for conservation of power can be written as:

P_{in} = \frac{1}{Δ t} \int_{H_{0}}^{H} LdH = P_{def} + P_{tr}

(3)

where $P_{in} = Δ E_{in} / Δ t$ , $P_{def} = Δ E_{def} / Δ t$ and $P_{tr} = Δ E_{tr} / Δ t$ are the input, deformation and traction powers, respectively, during the deformation period and $Δ t$ is the elapsed time since the beginning of deformation. Choosing a small-time increment, $L$ can be assumed constant and $P_{in} ≅ L \overset{\cdot}{H}$ where $\overset{\cdot}{H}$ is the instantaneous velocity of the platen along the $z$ axis. The right-hand side of equation (3) can be written as:

\int_{V} \bar{σ} \sqrt{\frac{2}{3} {\overset{\cdot}{ε}}_{ij} {\overset{\cdot}{ε}}_{ij}} dV + \int_{S} T_{i} . u_{s} dS

(4)

where the first integral is the power consumed (changed) in the system due to the plastic deformation, $P_{def}$ , and the second integral is the power associated with the frictional tractions, $P_{tr}$ . The integrand $T_{i} . u_{s}$ is the scalar product of the external traction $T_{i}$ and the material tangential velocity $u_{s}$ over the traction boundary $S$ . Also, $V$ denotes the deforming volume of the sample, $S$ is the area of that boundary where the traction stresses (friction-induced shearing stresses) develop. ${\overset{\cdot}{ε}}_{ij}$ and $\bar{σ}$ are the plastic strain rate at the material points and the effective stress (von Mises yields strength), respectively. Because of friction and plastic deformation in the system, integrations have to be taken over the actual strain path to account for the non-conservativeness in the system. The next two sections describe the recovery of the deformation and traction powers in the sample based on the FE solution.

Elemental deformation power for the FE solutions

The total sample deformation power $P_{def}$ in equation (4) is the sum of those for each element:

P_{def} = \sum_{i = 1}^{NUMEL} P_{def}^{i}

(5)

where $i$ is the element index and $NUMEL$ is the total number of elements in the BCT model. To recover the elemental power $P_{def}^{i}$ for the $i^{th}$ element, the following expression was used:

P_{def}^{i} = \int_{V_{i}} \bar{σ} \sqrt{\frac{2}{3} {\overset{\cdot}{ε}}_{jk} {\overset{\cdot}{ε}}_{jk}} d V_{i} = 2 π \sqrt{\frac{2}{3}} {\bar{σ}}_{i} {\dot{\bar{ε}}}_{i} A_{i} \bar{r}

(6)

where the element’s volume is $d V_{i} = 2 π A_{i} . {\bar{r}}_{i}$ . For the ith element, its area and centroid radius, $A_{i}$ and ${\bar{r}}_{i}$ , respectively has to be calculated. It requires the element vertices listed counter-clockwise around the element perimeter (i.e. ( $r_{1}, z_{1}$ ), ( $r_{2}, z_{2}$ ), ( $r_{3}, z_{3}$ ) and ( $r_{4}, z_{4}$ )). Based on the coordinates, both $A_{i}$ and ${\bar{r}}_{i}$ can now be identified, respectively as:

A_{i} = \frac{1}{2} (r_{4} (z_{1} - z_{3}) + r_{2} (- z_{1} + z_{3}) + (r_{1} - r_{3}) (z_{2} - z_{4}))

(7)

{\bar{r}}_{i} = 0.25 (r_{1} + r_{2} + r_{3} + r_{4})

(8)

The node numbers (indices) in equations (7) and (8) are based on the ith quadrilateral element’s local coordinate system. One needs the element connectivity matrix of the solution to correlate these indices to the global coordinate nodal numbers. This is essential to post-process an FE solution and to reproduce any calculations similar to those presented by equations (7) and (8). The FE solutions are typically reported based on the global node numbers rather than the local node numbers (1, 2, 3 and 4 for a quadrilateral element). The results have to be translated to the local coordinate before post-processing can be performed. The total deformation power, $P_{def}$ , is the sum of those for all elements: $P_{def} = \sum_{i = 1}^{NUMEL} P_{def}^{i}$ which can be found using equaiton (6).

Calculation of friction traction

The total sample traction power $P_{tr}$ in equation (3) is the sum of those for each element:

P_{tr} = \sum_{i = 1}^{NUMEL} P_{tr}^{i} (only for the elements at the interface)

(9)

Assuming a von Mises yield behaviour for the sample, the traction power loss, $P_{tr}$ , can be calculated using the following expression for the friction traction (shearing stress) between a platen and the sample:

T_{i} = \frac{2 mk}{π} (\tan^{- 1} \frac{| u_{s} (r) |}{u_{0}}) = \frac{2 m \bar{σ}}{\sqrt{3}} (\tan^{- 1} \frac{u_{s} (r)}{u_{0}})

(10)

where $m$ is the friction factor and $k$ is material yield strength in shear. The velocity-dependent friction term in equation (10) is chosen to capture a ‘neutral point’ at the friction surface during disk compression and was first proposed by Kobayashi and Chen.²⁵ More details on the velocity dependant friction model can be found in Chen and Kobayashi²⁵ or Pietrzyk and Lenard.²⁶ For the cylindrical coordinate system shown in Figure 1, the material velocity, $u_{s} (r)$ , at the traction boundary $S$ , can be interpolated based on its nodal velocities $v_{1}$ and $v_{2}$ as:

u_{s} (r) = v_{2} + \frac{(r_{2} - r) (v_{2} - v_{1})}{r_{2} - r_{1}} (where, z = 0.5 H)

(11)

Changing the variable, $dS = 2 π rdr$ , and replacing the values, one can find the elemental traction power $P_{tr}^{i}$ as:

P_{tr}^{i} = \int_{S} T_{i} . u_{s} dS

(12)

Simpson’s 3 points numerical integration was used to calculate the elemental traction power. A plot of the deformation and traction powers against $H$ is used to detect the onset of foldover; the first abrupt change in either of the plots and its corresponding $H_{i}$ indicate the onset.

Method 3: Quadratic profile

This method is based on the comparison between the sample’s profile identified by the FE solution (virtually measured) and a reference quadratic profile. This imaginary quadratic profile was first suggested by Schey et al.²⁷ to simplify the foldover and later reinvented by Khoddam et al.⁵ to model the barrelling. The third method suggests that foldover does not start automatically. It starts and grows only when the fillet radius at the top edge of the sample’s free surface blunts to a certain limit. This is when the identified sample profile (from FE analysis or experimentally measured) deviates locally from the imaginary quadratic profile most significantly at its top edge. The technique is illustrated in Figure 2 where the FE profile is shown using a solid black line and the quadratic profile with a dashed red line. Based on observations from many numerical solutions of the BCT, it was found that at the onset of the foldover, a small fillet is formed at the ‘top edge nodes’ which are shared by the free and interface surfaces. A sample top edge node is shown as $T_{1}$ in Figure 2. The quadratic profile (defined in section “Method 3: Quadratic profile”) cannot accommodate the fillet. Consequently, upon the foldover onset, the difference between the FE and quadratic profiles grows locally, in the vicinity of the shared top edge node. We note that the sample’s barrelled profile deviates from a virtual quadratic profile once the foldover starts and the deviation grows with the development of the foldover.

Figure 2.

Profile tracing points, $T_{1}$ to $T_{23}$ , to compare the FE and Quadratic profiles. Virtual and quadratic profiles, predicted by the FE and the quadratic models, respectively. Small fillet formation near $T_{1}$ and $T_{2}$ is visible.

Based on numerical studies of the results for the second method, a criterion has been developed to detect the onset which can be stated as:

The onset occurs when the error between the real profile and its quadratic representative in the vicinity of the sample’s upper edge (vicinity of the shared node $T_{1}$ ), exceeds 3%. In this work, the FE profile was used to calculate the error. Upon a proper upgrade of the existing test rigs and availability of the instantaneous real profile, the rig can be used to calculate the error. A mathematical expression to calculate the virtual profile error compared to its quadratic counterpart is presented in section “Method 3: Quadratic profile”.

Virtual quadratic profile

The virtual quadratic profile, shown using a dashed red line in Figure 2, is a second-order curve that represents the variation of the sample’s radius as a function of $z$ . Its radius, $R (z)$ varies from $0.5 D$ in the sample’s mid-plane to $0.5 d$ at the sample’s top-plane. The profile radius becomes:

R (z) = 0.5 D - \frac{2 (D - d) z^{2}}{H^{2}}

(13)

Incompressibility principle between the sample’s initial and current geometry can be used to correlate the barrelled sample’s mid-plane diameter $D$ and its top-plane diameter $d$ . Then, the correlation becomes:

d = \frac{\sqrt{5} \sqrt{H (9 D_{0}^{2} H_{0} - 4 D^{2} H)} - 2 DH}{3 H}

(14)

where $D_{0}$ and $H_{0}$ are the initial sample diameter and height, respectively.

Quadratic profile error

Given the cylindrical coordinate shown in Figure 1, each point on the sample profile can be located based on its radius and height which are presented by $r$ and $z$ coordinates, respectively. To evaluate the goodness of fit between the virtual profile and its corresponding FE profile (virtually measured profile), we have to compare the FE profile radii’s ( $r$ coordinates of $T_{1}$ to $T_{23}$ nodes in Figure 2; $r_{1}$ to $r_{23}$ ) with those of their corresponding points on the virtual quadratic profile. The latter will be represented as ${\hat{r}}_{1}$ to ${\hat{r}}_{23} .$ Let $N_{prf}$ presents the number of the nodes that belong to the FE profile (black, solid line in Figure 2 with $N_{prf} = 23$ ). Comparing the FE profile radii and their corresponding quadratic profile radii, the quadratic profile error, $ψ (i)$ , for the $i^{th}$ ‘profile node number’ is defined as:

ψ (i) = \frac{100 δ_{i}}{S_{Dev}}

(15)

and

S_{Dev} = \sqrt{\frac{\sum_{i = 1}^{N_{prf}} {(δ_{i} - \bar{δ})}^{2}}{N_{prf} - 1}}

(16)

where $S_{Dev}$ , $δ_{i} = r_{i} - {\hat{r}}_{i}$ and $\bar{δ}$ are standard deviation, the quadratic profile error at the $i^{th}$ node and the mean value of the total differences, respectively.

Results

We verify the FE models next by comparing the load-stroke results generated by the FE models with the experimental measurements described earlier in this work.

The results relevant to the proposed methods 1, 2 and 3 are presented in the upcoming sections. Also, to verify the numerically calculated foldover, the predicted foldover value by the second method is compared in this section with that measured experimentally by Ettouney and Stelson⁸ and Tan.⁹

Based on the virtual experimental setup explained earlier, most notably the FE models and the dedicated computer codes, and based on the three methods which were discussed and developed in this work, test scenarios were performed and their data was processed. The results on the virtual detection of foldover onset are obtained and presented here. These are based on the described computational conditions and the developed theories presented earlier in this article.

Verification of the FE models

The TMTS experimental setup, shown in Figure 3, was used in this work to obtain the load-stroke data,²⁸ similar to all commercially available compression machines known to the authors, is incapable of measuring barrelling data during the test. Therefore the estimated friction coefficients $m$ presented in Table 1 should be used very carefully; the barrelling data are essential to estimate an instantaneous coefficient during the test.²⁴

Fig. 3.

The Servotest TMTS experimental setup.²⁸

Figure 4 compares the experimental and numerical load stroke data as explained in section “Virtual experimental setups for BCT”. Contrary to the flow stress data, the comparison of different load stroke data is not straightforward as they depend on the sample geometry. Since the sample geometries are slightly different, as can be seen from Table 1, the plots in Figure 4 are presented separately; not overlaid.

Figure 4.

Comparison of experimental (solid lines) and virtual (dotted lines) load stroke data. (a) to (f) correspond to the data sets with an average effective strain rate of $\dot{\bar{ε}} = 0.1, 1, 10, 30, 50$ and $100 s^{- 1}$ , respectively.

Nevertheless, as will be shown in Figure 6, the deformation power is not greatly influenced by friction. Therefore, while the coefficient factor plays a significant role in data reduction, it has a minor role in the load stroke curve.

The compared sets are shown in Figure 4(a) to (f) with six average effective strains of $\dot{\bar{ε}} = 0.1, 1, 10, 30, 50$ and $100 s^{- 1}$ , respectively. They all have a similar shape and present the same trend. In all cases, the graphs are ‘N-shaped’ and the compared graphs coincide in the middle, deviate on both sides of the coincident point monotonically and the virtual load is always larger before the point and smaller after the point.

The difference between the virtual load-stroke (FE based) and the experimental load-stroke set in each case pools from several different sources. These include the oversimplified post-processing of the compression test data based on the existing methods which were extensively discussed in the ‘Introduction’. One notes that the flow data library in all commercially available finite element programs relies on the simplistic solutions of the test.⁴ Given the limitations in the available data post-processing/reduction techniques, the sets comply reasonably well in all cases which justifies the use of the virtual setup as the best available method to benchmark the data conversion techniques.

Method 1; Tracing points and physical appearance

Figure 5(a) and its zoom view (shown as Figure 5(d)) show the BCT sample at its initial undeformed mesh ( $H = 16 mm$ ). Using the software graphical post-processor, nine tracing points $T_{1}$ to $T_{9}$ have been added to the profile nodes starting at the node shared between the free surface and platen-sample interface. Figure 5(b) and its zoom view shown as Figure 5(e) show the final step of the solution, step 3200 ( $H = 8 mm$ ), all nine tracing points have been migrated from the profile to the platen-sample interface. This presents a complete foldover for the chosen tracing points. Figure 5(c) shows an intermediate solution step of 2550 in which only four tracing points have migrated to the interface which suggests a partial foldover compared to the last step. Given available controls in the post processor, one can choose any desirable solution step to assess the level of foldover at that particular deformation step.

Figure 5.

Use of nine tracing points to identify the onset and progression of foldover with deformation (step numbers: 10 – undeformed, 2550 – intermediate and 3200 – final deformation step): the position of (a) tracking points on the initial undeformed mesh, (b) the tracking points on the fully deformed mesh, (c) the tracking points on an intermediate deformation step. (d) and (e) are the insets of (a) and (b), respectively.

Close observation of the FE model and measurement of $d_{f}$ and $d_{i}$ based on method 1 enabled monitoring of the deforming mesh and identifying the migration of the tracing points at the nodal points from the free surface to the contact area. These together with $H_{i}$ and $ς_{f}$ .were measured carefully and the measured values were found to be in close agreement with those obtained by the second method (see Table 2).

Table 2.

The onset of foldover, $H_{i}$ , and measured final foldover, $ς_{f}$ ( $H_{f} = 8 mm$ and $H_{0} = 16 mm$ for all cases). For friction coefficients of $m = 0.1$ and $0.2$ , foldover did not occur (the symbols with a subscript $i$ denote their values at the onset).

$m$	$H_{i}$	$\begin{array}{l} ς_{f} = 0.5 \\ (d_{f} - d_{i}) \end{array}$	$d_{f}$	$d_{i}$	$ε_{i} = \ln (\frac{H_{0}}{H_{i}})$
0.1	–	0	–	–	–
0.2	–	0	–	–	–
0.3	8.73	0.31	12.66	12.04	0.61
0.4	9.48	0.56	12.54	11.42	0.52
0.5	9.93	0.81	12.67	11.06	0.48
0.6	10.28	0.83	12.46	10.81	0.44
0.7	10.60	0.84	12.31	10.62	0.41
0.8	10.73	0.86	12.21	10.50	0.40
0.9	10.85	0.87	12.14	10.40	0.39
1.0	10.98	0.89	12.11	10.33	0.38

The visually identified $H_{i}$ , $ς_{f}$ , $d_{f}$ and $d_{i}$ using the first method agrees well with those reported in this table.

Method 2: Monitoring deformation energy and load changes

The FE model described earlier, was solved for the chosen computational conditions. Also, in house computer codes were developed and utilised in this work to read the FE solutions and calculate the total power input $P_{in}$ , deformation power $P_{def}$ and traction power $P_{tr}$ according to equations (3), (6) and (12), respectively. The power-deformation plots for the chosen computation conditions with friction factors of $m = 0.1$ to $m = 1.0$ are shown in Figure 6(a) to (e). Each of the plots shows an overlay of two powers-deformation graphs only for two distinct friction factors (e.g. $m = 0.3$ and $0.8$ in Figure 6(c)) for better contrast.

Figure 6.

Variation of input power, $P_{in}$ and deformation power, $P_{def}$ with sample’s height, $H$ for different values of friction factor, $m .$ Onset of foldover according to the either power is shown in (a) to (e). Variations of traction power, $P_{tr}$ is shown in (f). Comparative values for each onset is given in Table 2.

The first abrupt slope change of the calculated power with deformation is used in this method as the criterion to detect the onset. The change for each $H - Power$ curve is represented by its corresponding sample height $H_{i}$ , indicates the onset of the given test scenario and are shown in Figure 6 (e.g. $H_{i} = 10.6 mm$ or $ε = \ln (H_{0} / H) = 0.411$ for $m = 0.7$ is shown in Figure 6(b)). These are also summarised in Table 2 for convenience and comparison. To calculate the final foldover $ς_{f}$ , the top-plane diameter at the onset $d_{i}$ and at the final step $d_{f}$ were also extracted from the FE results and can be found in Table 2. This allows the calculation of the foldover for any deformation step.

To correlate the foldover and sample height, we note that $ς$ is zero for $H > H_{i}$ and grows nonlinearly between $H_{f} \leq H \leq H_{i}$ . To accommodate both behaviours in one single mathematical expression of $ς$ , the following model is proposed here:

ς = ς_{f} {(\frac{H - H_{i}}{H_{f} - H_{i}})}^{2} (for H_{f} \leq H \leq H_{i})

(17)

ς = 0 (for H > H_{i})

(18)

The top horizontal axis in Figure 6 is an average uniaxial strain $ε = \ln (H_{0} / H)$ which is also included in Table 2. The average strain simply corresponds to a given sample height compared to its undeformed value. It represents the average deformation in the sample at the onset.

Figure 6(f) shows the traction power $P_{tr}$ for different friction factors. It is clear that by growing the friction factor, slippage at the interface reduces which results in a reduction in $P_{tr}$ with friction factor $m$ . For a fixed value of $m$ , the traction power $P_{tr}$ increases with deformation. This can be attributed to the work hardening and elongating the platen-sample interface. While in Figure 6(f), there are some abrupt slope changes in $P_{tr} - H$ curves (more noticeably for larger friction factors), the changes seem less indicative of the onset when compared with those of $P_{in} - H$ or $P_{tr} - H$ curves (Figure 6(a)–(e)).

Method 3: Deviation of the quadratic and actual profiles at the edge

The results obtained by the FE step solutions explained in section “FE simulation strategy” are used as the inputs for a dedicated computer code which was prepared in this work to implement the third method. The code generates the radial position of the tracing points on the FE profile and the quadratic profile and stores them in two separate arrays, respectively. The latter is generated using equations (13) and (14). Quadratic error $ψ (i)$ is calculated based on equation (15) and $ψ (i) - i$ curves are constructed and presented in Figure 7(a) to (f). Each curve corresponds to a given value of $m$ , four insets are added to each curve for better clarity and distinction and easier comparison.

Figure 7.

Quadratic profile error $ψ (i)$ versus the profile node number as a function of friction factor $m$ and sample height $H .$ The illustration in (a) shows the profile node numbering and compares the two profiles. (b) to (f) compare the quadratic profile errors for $m = 0.1$ , $0.3$ , $0.5$ , $0.7$ and $0.9$ , respectively.

An important conclusion from the presented analyses and results in this work is that the flow curve identified based on the current scenarios is only reliable for effective strains smaller than $ε_{i}$ in Tables 2 and 3. This is better understood by noting that the traditional flow curve identification methods are incapable of incorporating foldover in their analyses.

Table 3.

The onset of foldover, $H_{i}$ , and measured final foldover, $ς_{f}$ based on method 2 (see Figure 2) ( $H_{f} = 8 mm$ and $H_{0} = 16 mm$ for all cases). For friction coefficients of $m = 0.1$ and $0.2$ , foldover did not occur.

$m$	Foldover’s onset	$H_{i}$	$ς_{f}$	$d_{f}$	$d_{i}$	$ε_{i} = \ln (\frac{H_{0}}{H_{i}})$
0.1	$ψ (i)$ < 0.5, no foldover	–	0	–	–	–
0.3	$H = 8 - 9$ , Figure 6(c)	8.50	0.43	12.66	11.80	0.63
0.5	$H = 9 - 10$ , Figure 6(d)	9.50	0.90	12.67	10.86	0.52
0.7	$H = 11 - 12$ , Figure 6(e)	11.50	1.14	12.31	10.02	0.33
0.9	$H = 11 - 12$ , Figure 6(f)	11.50	1.19	12.14	9.75	0.33

Comparing experimental and numerical foldover

Experimental measurement of foldover

Existing analytical and experimental studies of foldover rely on two simplifying assumptions:

The barrelled profile is homogenous; it can be represented fully with a circular profile.

The foldover starts immediately with deformation; even all available numerical simulations only consider the final foldover and assume that it starts from the beginning of deformation (see e.g. Ettouney and Stelson⁸)

Some of these studies exclude the friction coefficient specifically as a parameter in their models. However, depending on the forming parameters, including the friction coefficient, a deformation scenario may complete with zero foldover; for example, Figure 1(d) to (f), or the foldover may be postponed and only starts at a certain level of deformation (Figure 1(b)). Therefore, an inclusive/comprehensive model of foldover should accommodate all possibilities. A mathematical expression of foldover, $ς,$ is presented later in this work in which the foldover can start at $H_{i}$ . It accounts for instantaneous development and growth measured from its onset.

Our mathematical model, presented as the second method, is compared with the published experimental data available on foldover in Ettouney and Stelson⁸ and Tan,⁹ where the experimental data only provide $ς_{f} = 0.5 (d_{f} - d_{i})$ . In their work, only the final value of the foldover was reported after completing the test and its previous grow was assumed to be homogeneous and linear and that the foldover immediately starts with deformation.

FE simulation of the experimental results

A dedicated numeric model corresponding to the experiment was constructed in this work to verify the validity of our numerical FE simulations and their predicted final foldover. The sample made of Al6061-T6 was deformed at 40 $° C$ with a crosshead speed of 10 mm/s. The geometric parameters of the test simulation and other main simulation parameters are summarised in Table 4.

Table 4.

Summary of the FE represented in the experiments Ettouney and Stelson⁸ and Tan.⁹

$m$	$H_{0}, mm$	$D_{0}, mm$	$H_{f}, mm$	$H_{i}, mm$
0.26*	17.5	12.68	8	10.97

Equivalent to a friction coefficient of $μ = 0.12$ .⁹

Upon availability of a dedicated experimental setup in future works, more representative experimental data may be collected and compared with our proposed model. Therefore, in the current experimental verification, shown in Figure 8, only the measured $ς_{f}$ is comparable with a real experiment in which foldover starts only at $H_{i} \geq H$ . The results also show the un-realistic homogeneous growth of foldover in the experimental study by Ettouney and Stelson.⁸

Figure 8.

Comparison of experimental results from Ettouney and Stelson⁸ and Tan⁹ with FE simulations presented in Table 3 and that represented by equation (17).

Discussions and conclusions

The discrepancy between the verification data can be explained by noting that the flow curve employed in the FE model was included in the software library. However, the flow curves available in such libraries are typically constructed using the common flow curve identification methods which are very primitive at this stage; this is one of the key arguments in the current work. Having said that, the comparison is close enough for the verification purpose and indicates that the FE model is representative enough for benchmarking purposes and to generate the reference data to be used in the virtual setup.

While the first method is a convenient way to detect the onset and the growth of foldover, it is mainly a useful yet simple tool for visual inspection and verification for foldover studies. This method, however, depends on the mesh size. Refining the mesh enough, the detected onset, $H_{i}$ , approaches to a limit which is typical for a converging FE solution.

The FE model developed in this study employs an adoptive meshing to ensure the convergence of its solution. As a result, the mesh is updated when needed; and at the larger deformation steps, some tracing points may not remain a ‘FE node’ while they still belong to the profile. An example of this is shown in Figure 5(c) in which, as a result of mesh updating, the tracing points $T_{6}$ and $T_{8}$ are not the FE nodes. This, however, does not reduce the applicability of the method.

The second method relies on a detailed solution of the problem as its input to calculate the deformation power. However, as shown in Figure 6, both the deformation and input powers predict the same onset. Therefore, the calculated load-deformation can be considered a more convenient criterion to construct the detection curve and eventually the onset. One might correctly argue that by refining the mesh size, the abrupt slope change in $P_{in}$ or $P_{def} - H$ curves shrink and become eventually undetectable. Starting from a coarse mesh and reducing the mesh size, it was observed that by refining the mesh, the detected $H_{i}$ approaches to a limit. This justifies the choice of the criterion to detect the onset. The mesh size used in this research was deliberately chosen to allow a detectable $H_{i}$ which was reasonably close to the limit. Therefore, one requirement for this method is to optimise the mesh size in the model according to the above-mentioned argument. The same logic applies to the first method as well.

The proposed second technique requires some significant numerical calculations which are only achievable after running a numerical simulation of the test. One notes that the input power is proportional to the load. Therefore, the numerical results are also representative of the measured load. Based on the results shown in Figure 6, the difference between the measured load-stroke with and without foldover, demonstrated by a jump in each curve, is comparable to the load measuring errors (2%–7%). These make the second method, the least usable method among the proposed three methods here. The first two methods provide an identical set of predictions for the onset (see Table 1).

Provided the availability of the measured $D$ and $d$ (instantaneous top and mid-plane diameters), the third method is a convenient method to detect the onset. This is not currently an option with the available compression test machines. A closer look at Figure 7 indicates that while the errors are amplified for clarity, the largest error is ∼8%. For the tracing points far from the shared node $T_{1}$ , the error is always less than 0.5%. It grows very sharply with an increase in $H$ after the onset for the tracing points near the foldover zone. These include tracing points near the shared tracing point (i.e. $T_{1}$ , $T_{2}$ and $T_{3}$ ). The sharp gradient in the $ψ (i) - i$ curves with $H \geq H_{i}$ makes the onset detection quite easy and unmistakable.

A comparison of the predicted onset by the two first methods (Table 1) and that made by the third method (summarised in Table 2) shows a reasonably good agreement. However, the choice of the best method to detect the onset and implement it in the ‘test rigs of the future’ is also a matter of technology availability and how the sample’s profile changes can be measured during the test.

The experimental validation of the predicted foldover, presented in the results section, showed a very close agreement on the predicted and measured final foldover.⁸ The measured foldovers before the onset,⁸ are not comparable with the predictions in this work due to the poor assumptions in the experimental work explained earlier.

The following conclusions can be made from this work:

Contrary to the experimental and numerical observations, previous studies on foldover assume a homogeneous development of foldover throughout the test which does not recognise a foldover free stage during low friction/deformation tests; the studies correlated foldover to the barrelled profile radius and therefore predict an immediate onset of foldover with deformation. It was emphasised a need to develop methods to directly or indirectly detect foldover onset during the test and to measure its growth.

Three methods were introduced in this work to identify the foldover onset and to measure the phenomenon during the test. They demonstrate a critical need in the measurement of the sample’s profile, particularly at the top edge of the sample, to overcome the limitations of the existing compression test rigs.

It was shown that foldover can start even with a friction factor of $m = 0.3$ after an average uniaxial strain of $ε = 0.6$ . An increase in friction factor and deformation promoted the onset; for a friction factor of $m = 0.9$ , the onset was as early as $ε = 0.39$ . The third method to monitor the onset and progress of the foldover, based on the finite element and virtual quadratic profile data, was found to be the most convenient method.

Experimental data on foldover were employed and compared with their corresponding numerical predictions based on the virtual setup proposed in this work.

To make a reliable interpretation of the compression test, it is essential to partition the test data into pre- and post-triaxiality sections. The usable data, before instability, can only be defined by employing the proposed methods or developing similar methods to monitor the formation of foldover. This involves more development on both experimental and theoretical fronts to make the profile data measurement possible and usable during the compression test.

Footnotes

Acknowledgements

The authors wish to thank Dr. Thaneshan Sapanathan for his useful comments on the final versions of this work.

Handling Editor: Chenhui Liang

Author contributions

Individual contributions to the paper using the relevant CRediT roles: conceptualisation: SK, MM, BM; data curation: SK, MM; formal analysis: SK, MM; funding acquisition: PH and SK; investigation: SK, BM, MM; methodology: SK, BM, MM; project administration: SK; resources, software: PH and SK; supervision: SK; validation: SK; visualisation: SK; sample preparation and performing compression test experiments and collecting raw data: AT and SK; roles/writing – original draft: SK, BM, MM; writing – review & editing: SK, PH, BM, MM.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Shahin Khoddam

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

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A comprehensive experimental and numerical study of foldover onset and growth during the barrelling compression test

Abstract

Keywords

Introduction

Methodology

Virtual experimental setups for BCT

An experimentally verified virtual setup

Foldover terminology

Tracing points and the profile

Experimental verification

FE simulation strategy

Method 1: Using tracing points to measure ς

Method 2: Deformation and traction power estimated by FE analysis

Elemental deformation power for the FE solutions

Calculation of friction traction

Method 3: Quadratic profile

Virtual quadratic profile

Quadratic profile error

Results

Verification of the FE models

Method 1; Tracing points and physical appearance

Method 2: Monitoring deformation energy and load changes

Method 3: Deviation of the quadratic and actual profiles at the edge

Comparing experimental and numerical foldover

Experimental measurement of foldover

FE simulation of the experimental results

Discussions and conclusions

Footnotes

Acknowledgements

Author contributions

Declaration of conflicting interests

Funding

ORCID iD

Data availability

References

Method 1: Using tracing points to measure $ς$