A Geometric Algorithm to Evaluate the Thickness Distribution of Stretched Sheets through Finite Element Analysis

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This paper presents a geometric algorithm to evaluate the thickness distribution through finite element modelling (FEM) of a sheet obtained through a forming process.

Abstract

Industry 4.0 aims to digitalize the manufacturing process to increase the productivity and the product quality of plants. A fundamental aspect of the digitalized manufacturing processes is the simulation of the manufacturing process in order to develop its virtual representation, known as digital twin, whose purposes may be monitoring, and control. Algorithms to elaborate the simulated data in order to improve the control of the manufacturing process are very important and they need to be developed. Sheet metal forming is a widely used process to manufacture parts with a high production rate and a low cost. The thinning of the stretched sheet needs to be controlled in detail, because it is strongly connected with the product quality. This work presents a simulation model and a geometric algorithm to evaluate the thickness distribution of a sheet stretched through a forming process. In order to accurately evaluate the thickness trend, a geometric algorithm was proposed which, on the basis of the position of the nodes of the internal and external surface of the sheet, was able to evaluate the thickness value. It enables finding of the minimum value of the stretched sheet thickness. The geometric algorithm was slightly modified, in a second step of the work, to experimentally evaluate the thickness trend of a sheet stretched by a forming process; it was applied to the measurement points obtained through a coordinate measurement machine on the inner and outer surfaces of the sheet. The numerical–experimental comparison of the results shows the appropriateness of the proposed algorithm for numerical data.

1. Introduction

Industry 4.0 aims to digitalize the manufacturing process to improve working conditions, to create new business models and to increase the productivity and the product quality of the plants. It tends to advanced production systems, i.e., interconnected and modular systems, that allow flexibility and performance. A fundamental aspect of the digitalized manufacturing processes is the simulation of the manufacturing process in order to develop its virtual representation, known as digital twin, whose purposes may be monitoring, and control [1]. The simulation purpose is to reproduce the behaviour of physical objects in a virtual space in order to plan or optimize products or production plants. Algorithms to elaborate the simulated data in order to improve the control of the manufacturing process are very important, and they need to be developed.

Sheet metal forming is a widely used process to manufacture parts with a high production rate and a low cost. A metal sheet is formed once it is put between a punch and a die, whose geometries are negative to one another.

To assess the quality of the metal sheet forming process, the material formability, the spring-back that changes the dimensions of the part and the minimization of the dimensional variations due to the lubrication, the material properties, and the thickness variation all need to be taken into account. A framework is discussed Ceretti et al. [2], that allows to simulate, through a finite element model, the influence of these process parameters on the sheet-stamping process in order to organize and filter raw data.

The use of materials lighter than steel, for example aluminium and magnesium alloys, reduces the component weight and, in some fields such as the automotive industry, increases the fuel economy. However, these metal alloys have a reduced formability when compared with that of steel and produce greater problems in terms of both spring-back and sheet metal breakage.

To overcome these manufacturing problems, some authors used hydraulic actuators to control the force on the blank-holder [3], to reduce processing times [4] and to prevent defects [5,6].

The complexity of sheet metal forming is connected with the large plastic deformations involved in the process. The geometry of the dies, the friction in the sheet metal–die interface, and the force on the blank-holder mainly influence the process. An unsuitable evaluation of these manufacturing parameters can reduce the quality of the part being worked, because it gives rise to wrinkles, tears, excessive spring-back and sheet thinning.

Traditionally, to design a sheet metal forming process trial-and-error methods were used exclusively. This approach was expensive and time-consuming. Nowadays, finite element analysis, able to simulate the forming process and to take into account all the manufacturing parameters, allows a faster and less expensive design than the traditional method.

The reliability of the results, costs, benefits, and times required to perform numerical simulations for industrial examples of sheet metal forming that predict fracture, final sheet thickness, wrinkling, loads, spring-back and residual stresses in the product was evaluated by Gantar et al. [7].

Finite element analysis allows the consideration of the effects of both the spring-back and the material formability in plastic deformation processes [8]. The sheet formability is typically due to the stretching of the sheet over a punch. Complex shapes may be planned by finite element analysis and obtained, such as double curved surfaces [9,10]. A simple method to predict the spring-back of an aeronautical panel is presented by Yan and Klappka [11].The deformation of small circles marked on the sheet surface is measured, whereas there is the tear, and the forming limit curve is tracked. Forming limit diagrams (FLD) are widely used as a forecasting method to avoid damage of sheet material in plastic deformation processes.

Different models to determine the forming limit curves were presented by Banabic et al. [12], Bleck et al. [13], Firat [14], Zadpoor and Sinke [15].

The stretching and bending of the metal sheet lead to a thinning of the sheet that is typically measured by means of gauges [16] or ultrasonic probes [17]. Moreover, the sheet thinning is expressed as a unique value that is the average thickness of the stretched sheet in comparison with that of the initial plane sheet. Instead, the estimation of the thickness distribution inside the stretched sheet is important to evaluate the quality of the stretched product and its strain uniformity [18,19]. In fact, a uniform distribution of the thickness inside the stretched sheet indicates a better result of the forming process because the product shows a similar strength in each of its sections. It derives from the stress and the strain distribution in the deformed sheet [20]. Moreover, the thickness distribution may be considered as the manufacturing signature of the forming process, i.e., the imprint left by the process on the sheet. In the literature, the papers are focused on different methods to measure the thickness of sheet or thin films, such as the optical instruments [21], a micrometer [22], and so on. Moreover, in the literature, the finite element simulations of the forming process achieve a value of the stretched sheet that is calculated as the distance between two nodes of the used mesh. A distribution of the thickness inside the simulated stretched sheet may give more important information about the quality of the process and enrich the simulation model. This is the motivation of the present work.

The aim of the present work was to develop a numerical model of a forming process in which to evaluate the thickness distribution of a sheet by means of a geometric algorithm, once it was stretched numerically through a finite element model using conventional displacement elements for plane strain, axisymmetric cases. For this reason, it is possible to consider only the inner and outer profiles of the stretched sheet that were discretized by a set of points; they were the nodes of the elements used to model the sheet. The geometric algorithm calculates the distance between each couple of faced points, perpendicular to the profiles. In this way, the geometric algorithm enables the calculation of the minimum value of the thickness in each section of the stretched sheet, while the traditional method that calculates the distance between two opposite nodes of the same element provides an overestimate. The algorithm may evaluate a great number of thicknesses along the stretched profile of the sheet.

To validate the developed numerical model, some experimental tests were carried out. Some sheets were stretched, and the points of the obtained profiles were measured by a coordinate measuring machine (CMM).

CMM is widely used to inspect mechanical parts by means of a contact probe that exercises 0.1 N pressure on the part. A knowledge-based system was developed by Moroni et al. [23] to automatically configurate the touch probe used for the measurement. The coordinate measuring machine is a general-purpose device suitable to control free form surfaces, as demonstrated by Corrado et al. [24], even if it is needed to accurately choose the fixturing system, as discussed by Polini et al. [25]. CMM was used in this study to acquire a set of points on the external and internal surfaces of each stretched sheet for different steps of the forming process.

The geometric algorithm was applied to profiles of the stretched sheet obtained by the experimental approach too. A good agreement was found between numerical and experimental results. The geometric algorithm was developed in the MATLAB^® software package.

Once the sheet thickness trend has been obtained, which represents the manufacturing signature of the forming process inside a stretched sheet, it may be used inside a tolerance analysis approach to estimate the geometric deviations of assemblies constituted by sheets, such as those of automotive or aerospace applications.

The paper is organized as follows: the next paragraph presents the used material, the developed numerical model, and the algorithm to evaluate the thickness trend of the sheet stretched numerically. Therefore, the experimental tests are described together with the application of the geometric algorithm to the stretched sheets too. Finally, the thickness trends due to the numerical and experimental approaches are compared.

2. Materials and Methods

The developed finite element model is completely general. However, to explain it, the material considered in this study was the AA 6060 alloy. It is constituted of aluminium, magnesium, and silicon. Its chemical composition is the following: Al—0.6%, Si—0.3%, Fe—0.1%, Mn—0.6%, Mg—0.1%, Cu—0.15%, Zn—0.05%, Cr—0.1%Ti. It has a high resistance to corrosion. The considered sheet had a thickness of 1.03 mm, while the blank holder had a radius of 83 mm.

2.1. Finite Element Model of Forming Process

The numerical model was developed through the MSC Marc^® software package. The test consisted of stretching a sheet of AA6060 through a hemispherical punch. Considering the symmetry in geometry, load and constraints, the test was simulated by adopting 4-noded iso-parametric elements with bilinear interpolation. The developed algorithm for finite element analysis was developed for simplified problems, such as plane strain or axisymmetric elements. For this reason, it is possible to consider only the inner and outer profiles of the stretched sheet that were discretized by adopting 4-noded iso-parametric elements with bilinear interpolation (see Figure 1).

Figure 2 shows some steps of the forming process; it represents half of the examined problem. It is possible to see the sheet, the punch, and the die on which the sheet is positioned. The sheet was discretized through two rows of 166 elements (therefore, the size of each single element was 0.5 mm × 0.515 mm). The nodes on the outer edge of the sheet were constrained in such a way that they cannot move along the direction orthogonal to the symmetry axis to simulate the presence of a blank holder. The external node in contact with the die was locked along the direction of the symmetry axis. The sheet nodes that were on the symmetry axis could not move orthogonally. The punch and die were rigid.

An implicit non-linear finite element code was used to simulate the metal sheet stretching process. The Von Mises criterion and the isotropic yield criterion were used.

The constitutive law of the metal sheet was determined, starting from the results of tensile tests that are analysed in detail by Giuliano et al. [19]. The material was modelled in the plastic field through a constitutive equation expressed by the power law:

$$\sigma =K{\epsilon }^n$$

(1)

where σ and ε are the equivalent stress and strain, respectively, K is the resistance coefficient, and n is the material hardening index.

Contact among bodies is managed by a suitable routine. Furthermore, the frictions among sheet, punch and die were considered by introducing the modified Coulomb friction model. In this model the relationship between the tangential f_t and the normal f_n forces is due to:

$$f_t=\mu f_n\left(\frac{2}{\pi }\right)arctan\left(\frac{v_r}{R_{sv}}\right)$$

(2)

where v_r is the relative slip speed and R_sv is the relative slip speed under which the friction force tends to zero. Simulations were carried out with a value of the friction coefficient μ between 0 and 0.2.

2.2. Geometric Algorithm to Evaluate the Thickness of the Stretched Sheet

Once the stretched sheet was simulated, the aim was to measure its thickness in many sections; the used software allowed measuring of the distance between two nodes of each element, although not the minimum distance that may be measured perpendicularly to the profile tangent (see Figure 3). Therefore, a geometric algorithm was developed to be applied to the simulated sheet in order to obtain the minimum value of the thickness in different sections along the deformed profile.

The numerical model produced a set of points on the two semi-profiles of the two sides of simulated stretched sheet. Each semi-profile was fitted by a polynomial equation (first step in Figure 4), such as:

$$x=a_i{y}^i+a_{i-1}{y}^{i-1}+\dots +a_1{y}^1+a_0$$

(3)

where i = 50 and was sampled through 61 points (where 0 was on the symmetry axis).

A geometric algorithm was applied to calculate the distance between each couple of points on the two semi-profiles of the inner and outer surfaces, considering the following couple of facing profiles on the inner and outer surfaces:

$$x_{out}=a_{i_{out}}{y}^i+a_{i-1_{out}}{y}^{i-1}+\dots +a_{1_{out}}{y}^1+a_{0_{out}}$$

(4)

$$x_{in}=a_{i_{in}}{y}^i+a_{i-1_{in}}{y}^{i-1}+\dots +a_{1_{in}}{y}^1+a_{0_{in}}$$

(5)

For each point $P\left(x_{0_{out}},y_{0_{out}}\right)$ of the profile, the equation of the tangent line to the curve (4) at the point P was calculated:

$$x_{tang}=f^{\prime }\left(y\right)\cdot y+q$$

(6)

where $f^{\prime }\left(y_{0_{out}}\right)$ is the first derivation of Equation (4) at the point P, and q is equal to:

$$q=f\left(y_0\right)-f^{\prime }\left(y\right)\cdot y_0$$

(7)

Therefore, the line perpendicular to Equation (6) at the point P was calculated as:

$${\mathrm{x}}_{perp}=-\frac{1}{f^{\prime }\left(y\right)}\cdot y+q$$

(8)

Its intersection with the profile (5) was calculated as the point $P^{\prime }\left(x_{0_{in}},y_{0_{in}}\right)$.

The distance between the two points P and P’ was calculated to evaluate the thickness of the stretched sheet as:

$$thickness\left(x_0,y_0\right)=\sqrt{{\left(x_{0_{out}}-x_{0_{in}}\right)}^2+{\left(y_{0_{out}}-y_{0_{in}}\right)}^2}$$

(9)

The same reasoning was carried out point by point along the two profiles (4) and (5) to obtain the thickness trend.

Figure 5 represents the trend of the sheet thickness obtained by the developed numerical model with three values of the friction coefficient: 0, 0.1 and 0.2. Figure 5 shows how the value of the friction coefficient influences the thickness trend, mainly in the region where the contact between sheet and punch is direct. In the absence of friction (μ = 0), the sheet metal material flowed undisturbed over the punch; the result was a thickness trend that decreased from the edge towards the component’s symmetry axis. In the presence of friction (μ = 0.1 and μ = 0.2) the stretching of the sheet was restrained by the presence of the punch, thus determining a thinning of the sheet in a region away from the axis of symmetry. Figure 4c is only an example, which cannot be considered for a comparison with the experimental results; the numerical simulation does not contain a model of material damage.

3. Experimental Validation and Results Discussion

Some experimental tests were planned and carried out in order to validate the developed finite element model. The experimental activity consisted of stretching some sheets of aluminium-based alloy on a punch whose shape was a hemisphere with a radius of 60 mm. Each sheet had a rectangular shape of 220 mm × 190 mm, and it was locked on the die through a blank holder with a circular hole of 83 mm radius (see Figure 6a).

The tests were carried out by means of the equipment designed at the Technology and Manufacturing Systems Laboratory of the University of Cassino (see Figure 6b). It is constituted of an interface, to acquire data, made up of a PC, a multimeter KEITHLEY 2700, and a power supply ATTEN TPR3003T (1 in Figure 6b), an interface to manage the test parameters and to translate the punch (2 in Figure 6b), a mould creeper (3 in Figure 6b), a punch and a load cell LUNITEK FT (4 in Figure 6b), a crossbar (5 in Figure 6b) actuated by a rotating screw jack (6 in Figure 6b), and an electric motor (7 in Figure 6b).

Giuliano et al. [19] further details on the experimental equipment that was used to perform the forming process on an AA6060 aluminium vessel are presented.

The sheet was fixed on the die by means of a blank holder and eight screws, using a torque wrench and imposing a tightening torque of 45 Nm.

During the tests, each sheet was stretched up to a predetermined value of the punch stroke. Three values of the punch stroke (10 mm, 20 mm and sheet breaking, as shown in Figure 7) were considered. Three sheets for each punch stroke were considered.

Each stretched sheet was oriented and locked on the coordinate measuring machine Prismo Vast MPS of Zeiss^® by means of standardized fixturing elements, as shown in Figure 8a. The inner and outer surfaces were measured by a probe with a tip radius of 3 mm, while a stylus length of 48 mm and 40 mm was used for the inner and the outer surfaces, respectively, of each stretched sheet. The part reference system was defined by measuring both the plane of the fixturing equipment on which the sheet was supported and the cylindrical surface of the sheet that was in contact with the blank holder, as shown in Figure 8b.

Therefore, five measurement profiles at 1.5 mm from each other were measured along the y and z directions of the part reference frame (PRF) in the centre of each sheet inner or outer surface in order to underline the maximum value reached by the sheet during the stretching along the x-direction in the part reference frame (see Figure 9). Each profile had a length of 150 mm and it was constituted by about 150 measurements points. Therefore, 1500 points were measured on both the inner and the outer surface of each sheet. It was planned to measure a set of profiles on each outer and inner surface, instead of the whole 3D surfaces, in order to experimentally reproduce the two semi-profiles binding the generic section of the deformed sheet obtained through the numerical model.

The developed geometric algorithm was applied to the 1500 points measured on both the inner and the outer sides of each sheet experimentally stretched. The first step (see Figure 10) chose, among the five profiles measured along a single direction on the outer surface, the one characterized by the highest value of the x-coordinate in the part reference frame (PRF), because it corresponded to the apex of the semi-spherical samples. The corresponding profile on the inner surface was identified in order to obtain a couple of profiles, one on the inner and one on the outer sides of the sheet, along the horizontal and vertical directions. Therefore, each profile was fitted by a polynomial equation [26], such as that of Equation (3), for the horizontal profiles and Equation (4) was used for the vertical profiles:

$$x=b_j{z}^j+b_{j-1}{z}^{j-1}+\dots +b_1{z}^1+b_0$$

(10)

where j = 50. Then, a set of 121 points were sampled on each polynomial equation by imposing a value of y or z, starting from 0 on the symmetry axis and sampling 60 points on both the right and on the left of the symmetry axis, and finding the corresponding values of x in Equations (3) and (10). The average of the x-coordinate of the points at the same distance from the symmetry axis was carried out in order to obtain one semi-profile on each side of the stretched sheet, constituted of 61 points by considering the 0 on the symmetry axis, to compare with that due to the numerical model. The same reasoning from Equation (4) to Equation (9) was applied to the two obtained sets of 60 points on the two sides of the manufactured stretched sheet in order to calculate the numerical thickness distribution.

Figure 11a shows the experimental profiles (inner and outer) of the deformed sheets for a stroke of the punch up to 10 mm, 20 mm and breaking. Inner and outer profiles of the stretched sheet were measured (see Figure 11a) to obtain the thickness along the sheet (see Figure 11b).

Figure 11b reports the experimental trend of the sheet thickness for the three different forming steps. Figure 11b underlines how the minimum value of the sheet thickness moves away from the symmetry axis of the spherical surface; it is due to the presence of friction between the sheet and punch. In fact, friction slows the sliding of the material on the punch by moving the minimum thickness value away the axis of symmetry as increasing as the stroke of the punch. This demonstrates the correct working of the proposed geometric algorithm.

Figure 12 shows the comparison between the two sheet thickness trends due to the numerical model and the experimental tests. It should be noted that a greater discrepancy between the results is where contact occurs between sheet and punch. In particular, the numerical thickness variation due to a friction coefficient μ = 0.1 is the nearest to that obtained experimentally.

Table 1. Comparison between numerical and experimental values of thickness.

Punch Stroke	Percentage Difference $\left|\left(\mathit{t}{\mathit{h}}_{\mathit{n}\mathit{u}\mathit{m}}-\mathit{t}{\mathit{h}}_{\mathit{e}\mathit{x}\mathit{p}}\right)/\mathit{t}{\mathit{h}}_{\mathit{e}\mathit{x}\mathit{p}}\right| \times 100 (\%)$
	μ = 0	μ = 0.1	μ = 0.2
≈10 mm	2.25	2.21	2.18
≈20 mm	8.69	4.01	5.37

shows the maximum value of the percentage difference in absolute value $\left|\left(t{h}_{num}-t{h}_{nexp}\right)/t{h}_{exp}\right|\ast 100$ between the thickness values due to numerical (th_num) and experimental (th_ex) approaches along the 61 points on the inner and outer surfaces of the stretched sheet. This percentage difference ranges between a minimum of 4%, when µ = 0.1 and the punch stroke is equal to 20 mm, and a maximum of 9%, when µ = 0 and the punch stroke is equal to 20 mm. This maximum value is probably due to the model adopted to describe the plastic behaviour of the material. However, the difference between numerical and experimental results is so low that it can be considered negligible.

Figure 13 shows the curves related to the thickness results versus the distance from the symmetry axis due to traditional node-to-node method and the proposed geometric algorithm. The difference between the two methods depends on the dimensions of the mesh used to discretize the part (fine or coarse), the deformation depth and the friction coefficient.

4. Conclusions

Algorithms to elaborate the simulated data in order to improve the control of the manufacturing process are very important and they need to be developed. Sheet forming is a widely used process to manufacture at low cost. The thinning of the stretched sheet needs to be kept under control, because it is strongly connected with the product quality. This work presents a simulation model and a geometric algorithm to evaluate the thickness distribution of a sheet, once stretched over a punch. It was developed for the numerical model because it allows measuring of the minimum distance that may be measured perpendicularly to the profile tangent, whereas the traditional methods calculate the distance between two opposite nodes of the mesh. The minimum distance is the critical parameter characterizing the sheet thinning.

A coordinate measuring machine, which is a general-purpose device suitable to control free form surfaces, was used in this paper to validate the simulation model. It acquires a set of points on the outer and inner surfaces of a stretched sheet for different steps of the forming process. The proposed geometric algorithm was modified to be applied to calculate the AA6060 sheet thickness from the measurement points on a generic 2D section of the sheet stretched over a hemispherical punch.

The comparison between numerical and experimental results showed a good agreement. A greater discrepancy between the results was noted where contact occurs between the sheet and punch. In particular, the maximum value of the percentage difference module between the thickness values due to numerical and experimental approaches along the 61 points on the inner and outer surfaces of the stretched sheet was calculated. This percentage difference ranges between a minimum of 4%, when µ = 0.1 and the punch stroke was equal to 20 mm, and a maximum of 9%, when µ = 0 and the punch stroke was equal to 20 mm. This maximum value is probably due to the model adopted to describe the plastic behaviour of the material.

The obtained results show the sheet thickness trend, the manufacturing signature left by the forming process on a stretched sheet; this be used to estimate the geometric deviations of the assemblies constituted by sheets, such as those of automotive of aerospace applications.