Numerical Study on the Forming Behaviour of Multilayer Sheets

: Nowadays, composite materials are playing an increasingly important role in material forming processes because they can combine remarkable physical and mechanical properties with relatively low weight. The main objective of this work is to study the forming behaviour of multilayer sheets by finite element analysis. The possibility of replacing the composite by a single equivalent material, with a plastic behaviour similar to that of the composite, was also numerically analysed. This study focuses on two three-layer sheets, each composed of two metallic outer layers and a core of polymeric material; on one of the sheets, the outer layers are steel and, on the other, aluminium. Numerical simulations of the bulge test and of the deep drawing of a U-channel profile and a square cup were used to evaluate the behaviour of the multi-layer sheets and their equivalent materials. The influences of the difference of the mechanical properties of the constituent materials and some geometrical parameters of the deep-drawing process on the plastic behaviour, namely the curves of force vs . displacement of the punch and the strain and stress distributions, were evaluated. The possibility of using the bulge test to characterize the behaviour of the composite was also analysed.


Introduction
Multi-layer sheets are composed of two or more layers, becoming increasingly popular in the aerospace and automotive industries [1]. They can also be called sandwich materials, which, as the name suggests, are mostly composed of two thin outer layers and one thick core. In general, the thin outer layers present relatively high strength and stiffness, when compared to the thick lightweight core. Composite materials, such as multi-layer sheets, are highly sought-after because they can achieve similar performance when compared to traditional materials, combined with a reduced weight. This is especially important in the aforementioned industries, where the need for greater efficiency and lower greenhouse gas emissions is ever increasing.
The performance of multi-layer sheets is governed by the volume fractions and material properties of each layer, leading to desirable features, such as low weight, high mechanical strength, ductility, corrosion resistance and good thermal and electrical properties simultaneously [2]. Multilayer sheets can be joined by manufacturing methods such as explosive welding [2], cold or hot rolling [1][2][3] and adhesive bond [1]. In this context, a poor multilayer sheet manufacturing associated with the strong dissimilarity in the mechanical behaviour between the layers and/or the volume fraction of each layer can lead to various failure modes, namely under bending loading conditions. Examples of failure modes include: (i) delamination, i.e., layer separation that occurs when the bond is relatively weak; (ii) skin wrinkling on the outer layers under compressive loading; (iii) the outer layers' failure under tensile or compressive loading; and (iv) core failure, due to shear stresses [4]. In this regard, it is crucial to investigate the mechanical properties and suitability of multi-layer metal sheets in forming processes involving large plastic deformations. A number of studies on the forming and mechanical behaviour of multi-layer sheets can be found in the literature, which include extensive investigation on the behaviour of metal-polymer-metal sandwich sheets under stretching, bending and deep-drawing loading conditions [5][6][7][8][9][10]. However, these studies, being mainly experimental, need to be complemented by in-depth stress analysis, which can be performed numerically, to better understand the forming behaviour of multilayer sheets.
In this work, a numerical simulation study is carried out in order to evaluate the forming behaviour of multi-layer sheets. Firstly, numerical simulations of the circular bulge test were performed for two composites, each composed of two outer metal layers and a core of polymeric material. This study allowed to define equivalent materials, i.e., single sheet materials whose plastic behaviour is similar to that of the composites. Afterwards, numerical simulations of the forming processes of U-channel profile and square cup were carried out for both the composites and the equivalent materials.

Materials
The study focuses on two multilayer sheets with a total thickness of 1.6 mm, each having two metallic outer layers, 0.3 mm thick, and a polymeric core, 1.0 mm thick. Two different mechanical behaviours were considered for the outer layers: one is an interstitial free steel that will be refer as "Composite 1", and the other is an AA1060-O aluminium alloy that will be refer as "Composite 2". In both composites, the mechanical behaviour of the polymeric core is considered typical of a polyethylene.
The constitutive model adopted for these materials assumes that: (i) the elastic behaviour is isotropic and described by the generalised Hooke's law; (ii) the plastic behaviour is described by the von Mises yield criterion with isotropic hardening and an associated flow rule. For the metallic materials (outer layers), the isotropic hardening behaviour was modelled by Swift's law: where represents the yield stress and its evolution with the equivalent plastic strain, ̅ ; K, and n are the material parameters; = is the initial yield stress. For Composite 1, the hardening law parameters of the outer layers were determined from the bulge test results of pressure vs. pole height results by Miranda et al. [8], using the inverse analysis procedure proposed by Pereira et al. [11], although simplified for the case of isotropy. For the metallic outer layers of Composite 2, the hardening law parameters were determined from the experimental tensile test results by Liu and Xue [12], using least-squares regression. The Swift's law parameters of the metallic outer layers of the sandwich materials are given in Table 1; this table also shows the elastic parameters of Hooke's law (Young's modulus, E, and Poisson's ratio, ν) assigned to both metallic materials. Regarding the polymeric core layer of both composites, Miranda et al. [8] and Liu and Xue [12] characterized the stress-strain behaviour of polyethylene cores using tensile tests, which is similar in both cases. This behaviour is chosen for the core material of the Composites 1 and 2, in the current work. Considering the shape of the tensile curve of the polyethylene, it was modelled by a saturation of Voce's law: where , and are the material parameters. In this context, the Voce's law parameters of the core of both composites were assessed from the tensile test results by Liu and Xue [12], using leastsquares regression. Table 2 shows these parameters as well as the elastic parameters of the core. Table 2. Material parameters of the polymeric core of both composites (E and ν are from [12]).

Material
Hooke  Figure 1 compares the hardening curves of the constituent materials of Composites 1 and 2, obtained from the constitutive parameters indicated in Table 1 (steel and aluminium alloy) and Table  2 (polymer). The dissimilarity in the hardening behaviour of the constituent materials of Composite 1 is much greater than in Composite 2, which allows the analysis of its influence on the mechanical behaviour of the composites.

Numerical Models
The numerical simulations were carried out with the in-house finite element code DD3IMP (Deep Drawing 3D Implicit Finite Element Code), developed and optimized for the simulation of sheet metal forming processes [13]. The forming tools were considered rigid and their surfaces were described by Bézier surfaces or Nagata patches [14]. The blanks were discretized with eight-node hexahedral solid elements combined with a selective reduced integration technique to avoid volumetric locking. The metallic layers are assumed to remain perfectly bonded with the polymeric core during the loading. The contact with friction was described by Coulomb's law with a friction coefficient of 0.02 for the circular bulge test and 0.144 for the other examples [15,16].

Circular Bulge Test
The circular bulge test proves to be an accurate tool for characterizing the mechanical behaviour of sheet materials up to large deformations. The data obtained from the bulge test provides complementary information to those of the tensile curve that can be used for identifying the parameters of the constitutive model, for example. A numerical model of the circular bulge test was built in order to perform the study on the plastic behaviour of multilayer sheets. The geometry of the selected tools was based on the bulge test apparatus described by Santos et al. [17], which was also used to obtain the experimental results by Miranda et al. [8] for the steel sheet. Figure 2 schematically shows the geometry of the bulge test, where RM = 75 mm is the die radius, RC = 13 mm is the die profile radius, RD = 95 mm is the radius of the central part of the draw bead and RS = 100 mm is the initial blank radius of the circular sheet. Only one quarter of the test geometry was modelled due to material and geometric symmetries. To further simplify the numerical model, the draw bead geometry was neglected and its effect was replaced by a boundary condition that restricts the radial displacement of the nodes placed at a distance equal to RD from the centre of the circular sheet. The numerical simulations assumed an incremental increase of the pressure applied to the inner surface of the sheet. The in-plane discretisation of the blank sheets was based on a previous work by Pereira et al. [11], and each layer of material presents two layers of elements through the thickness (i.e., six elements through the total sheet thickness).

Sheet Metal Forming Processes: U-Channel and Square Cup
The two deep-drawing examples selected are based on the benchmarks proposed under the Numisheet '93 conference: U-channel (2D) and the square cup [16]. Both are representative of sheet metal forming industrial processes, but the process parameters were adapted taking into account the thickness of the composite sheet and its material properties. Figure 3 schematically represents the typical setup for the bending and the deep drawing processes under study. The numerical simulation of both forming processes is composed of three phases: in phase one, the blank holder compresses the sheet against the die; in phase two, the displacement (d) is imposed to the punch, with d = 30 mm in case of the U-channel profile and d = 40 mm for the square cup (see Figure 3a), while the blank holder force (BHF) remains constant; finally, in the third phase, the forming tools are removed, with the subsequent springback.  For the U-channel forming process, various combinations of process parameters were tested in both Composite 1 and Composite 2, concerning the blank holder force (BHF), die radius (r) and clearance between the punch and the die (c). The selected values for the process parameters are shown in Table 3. For the square cup forming process, the die radius, r, is kept always equal to 5 mm and two values of clearance between the punch and the die, c, were studied: 2 mm and 4 mm; moreover, two values of the blank holder force were used: BHF = 9.8 kN and 19.6 kN, for the Composite 1, and BHF = 2.45 kN and 4.9 kN, for the Composite 2. Regarding the U-channel forming process, a section of the blank with in-plane dimensions of 75 × 35 mm was discretized with just one element in width and 150 elements in length. Note that only one element is adopted along the width direction because the boundary conditions adopted guarantee a plane strain state along this direction. In case of the square cup forming process, the initial surface dimensions of the sheet are 75 × 75 mm and was discretized with 40 elements in length and width. In both processes, the mesh has six layers of elements through the thickness, two for each layer of material. Due to material and geometric symmetries of the processes, only half of the Uchannel and a quarter of the square cup were simulated.

Equivalent Material
An equivalent sheet material is now defined. The hardening behaviour of the materials equivalent to the composites was estimated using a rule of mixture approach [19], which considers the weighted average of the thickness of composite constituent materials, as follows: where and represent the hardening laws of the metallic outer layers (Swift's law) and polymeric core (Voce's law), respectively (see hardening parameters in Table 1; Table 2), and = and (1 − ) = , being tmetal and tpolymer the thicknesses of the metallic outer layers (tmetal = 2 × 0.3 mm) and inner polymeric core (tpolymer = 1 mm), respectively. Accordingly, a mixed Swift-Voce hardening law was selected to model the hardening behaviour of the equivalent materials, as follows [20]: where the values of the hardening parameters are those indicated in Tables 1 and 2 and α = = 0.375. The elastic parameters E and ν of the equivalent materials were also estimated as the thickness-weighted average of the elastic parameters of the constituent materials of each composite. Table 4 summarizes the values of the parameters that describe the elastoplastic behaviour of both equivalent materials, where materials "Equivalent 1" and "Equivalent 2" refer to single sheet materials estimated as equivalent to "Composite 1" and "Composite 2", respectively. Figure 4 illustrates the comparison of the hardening curves of the equivalent materials with those of the metallic outer layers and polymer inner layer of each composite (see Figure 1).

Circular Bulge Test
Numerical simulations of the circular bulge test were performed with both composite materials, each with a total thickness of 1.6 mm. The numerical simulations were also performed for each of the constituent materials of Composite 1 and Composite 2: (i) steel (with a total thickness of 2 × 0.3 mm  Figure 5 presents the pressure vs. pole height results for Composite 1 and Composite 2 as well as for their constituent materials ("Steel (2 mm × 0.3 mm)", "Aluminium (2 mm × 0.3 mm)" and "Polymer (1 mm)", in the figure); the experimental pressure vs. pole height results of Composite 1 ("Composite 1_Exp [8]" in Figure 5a), obtained by Miranda et al. [8], are also included. Finally, the results of pressure vs. pole height of the composite materials are presented as the sum of the results of their independent components: outer metallic layer + polymeric core results (see curves "Steel + Polymer" and "Aluminium + Polymer", in Figure 5). No differences are observed between these curves and those of the composites.
(a) (b) Numerical simulations of the bulge test were also performed considering the equivalent materials, whose parameters are presented in Table 4, and Figure 6 shows that the pressure vs. pole height curves of these equivalent materials overlap those of the corresponding composites.       Figure 9 shows the distributions of relative difference between the equivalent stress obtained for the numerical simulation of the equivalent material and the equivalent stress calculated from the mixture rule approach (Equation (3)), using the numerical results of the composite. In general, the equivalent stress distributions, obtained for the equivalent materials, obey to the rule of mixtures, which is not limited to the pole of the cup. In short, the equivalent material seems to satisfactorily reproduce the behaviour of the composites, during the bulge test, relative to the pressure vs. pole height curve and the equivalent plastic strain. Although there are substantial differences between the composite and the equivalent material in terms of equivalent stress distribution, the uniform value attained by the equivalent material follows the rule of mixtures, particularly in the region near the pole cup of the bulge test.

Deep Drawing of a U-Channel Profile
Numerical simulations of the U-channel profile were performed to assess the behaviour of the composite materials and compared it to the respective equivalent materials, using the combinations of process parameters indicated in Table 3. First, the results of the evolution of the punch force vs. its displacement are analysed. Afterwards, the analysis focuses on the strain and stress distributions in the sheet thickness, after a punch displacement of 30 mm. Figure 10 shows the results of the evolution of the punch force, normalized with the blank width, vs. its displacement for the Composites 1 and 2 and their corresponding equivalent materials, for the combinations of process parameters presented in Table 3. In general, the curves of the composites are similar to those of their equivalent materials, except in case A of Composite 1 from Table 3 (see Figure  10a); in this context, increasing the BHF value from 0.28 kN/mm (case A-see Table 3) to 1.12 kN/mm (cases B to E-see Table 3) leads to a better correspondence between the force vs. displacement results of each composite and their equivalent materials. The curves of the composites and their corresponding equivalent materials are not significantly altered by variations in the clearance and in the die radius, as can be seen by comparing the Figure 10b-e. Therefore, the analyses carried out in the following section focus mainly on Cases A and B.   Figure 11 shows strain and stress distributions in the sheet thickness for the Composite 1 and the equivalent material, respectively for cases A and B (see Composite 1 in Table 3), presenting the distributions of equivalent plastic strain ( ̅ ), Green-Lagrange shear strain ( ) and equivalent stress ( ). In general, the difference of mechanical behaviour between the layers of the Composite 1 produces a stress gradient through-thickness, as expected, but also in strain, the highest values of which are located in the polymeric core in the vertical wall region. In case of the equivalent material, there are almost no strain and stress gradients across the thickness, for which the maximum values of equivalent plastic strain and equivalent stress are clearly smaller than in the composite. The closeness observed between the distributions of equivalent plastic strain and shear strain in the core layer of the composite indicate that shear strains are prevalent, in which the shear strain magnitudes and gradients at the vertical wall certainly favour core failure and failure by delamination; the equivalent materials show negligible shear strain values.

Force vs. Displacement
The increase of the blank holder force, from 0.28 kN/mm (case A-see Figure 11a) to 1.12 kN/mm (case B-see Figure 11b), leads to a decrease of about 25% in the maximum value of equivalent plastic strain, and subsequent reduction of the strain gradient in the composite. On the contrary, the equivalent material shows a strain distribution nearly identical for both values of blank holder force, in particular the maximum equivalent plastic strain is not significantly dissimilar. In this context, the diverse process conditions defined for cases A and B, which give rise to these differences in the strain distribution in the composite, are also reflected in the punch force vs. displacement results: for the case A (see Figure 10a), the equivalent material curve has a noticeably higher force level than the composite, which does not occur for case B (Figure 10b).
Although we do not find it necessary to show figures, it is worth to point out the effect of changing the die radius and the clearance between the punch and the die in the composite. The increase of the die radius, from 5 mm to 8 mm, leads to a strong reduction of the maximum values of the equivalent plastic strain (from 0.717 to 0.328) and shear strain (from 0.619 to 0.264), and, consequently, the strain gradients through thickness are reduced; on the other hand, the increase of the die radius does not have a significant effect on the equivalent stress results. Changing the clearance between the punch and the die, from 1.6 mm to 2.0 mm, does not have a significant effect on the stress and strain distributions. In addition, to check for any potential issues that may arise from the parameters of the hardening law, namely the hardening coefficient of Swift's law, a lower value was considered for the metallic layers of Composite 1 (n equal to 0.10, in contrast to 0.239 for the steel). This material was chosen because its relatively low hardening coefficient is potentially more prone to plastic instabilities. However, the results are quite similar in terms of equivalent plastic strain and shear strain distributions, with no impact on the conclusions.
In summary, the process conditions influence the distribution of equivalent plastic strain through the thickness, which can reach relatively high values in the inner polymeric layer, due to the redundant shear deformation that occurs. In the cases analysed, the increase in the blank holder force from 0.28 kN/mm to 1.12 kN/mm led to a less heterogeneous strain distribution. It should be noted that high values of shear strain can lead to core failure [4]. Moreover, strong strain gradients, between the core and the outer layers, can cause delamination failures, as was observed in the case of the threepoint bending test [5]; this depends strongly on the geometrical parameters of the test, such as the space between the counter rolls and the punch.  Figure 12 shows strain and stress distributions in the sheet thickness of the Composite 2 and the equivalent material, respectively for cases A and B (see Composite 2 in Table 3), presenting the distributions of equivalent plastic strain, Green-Lagrange shear strain and equivalent stress. In all the cases, the through-thickness strain and stress gradients are smoother than those found in Composite 1 and present lower values of equivalent plastic strain and equivalent stress. The equivalent plastic strain and shear strain distributions are more homogeneous when compared with the corresponding cases A and B of Composite 1, where the levels of shear strain are noticeably lower than those of equivalent plastic strain; this suggests that shear strain has a less relevant role than in Composite 1. In this context, the BHF has less influence on the results of Composite 2 than of Composite 1. Finally, it should be noted that the strain distribution of the composite is relatively close to that of the corresponding equivalent material; this is because the difference in the mechanical behaviour of the constituent materials of the Composite 2 is small, when compared to Composite 1 (see Figure 4).

Strain and Stress Distribution for Composite 2
The influence of the clearance between the die and the punch and the influence of the die radius are qualitatively similar to that described for the Composite 1, although less important.

Deep Drawing of a Square Cup
Numerical simulations of the square cup forming process were performed to assess the behaviour of the composite materials and compare it to that of the respective equivalent materials. This forming process presents a more complex symmetry than the U-channel profile, offering a greater variety of strain paths, simultaneously (i.e., depending of the point location). First, the punch force vs. displacement results are analysed for two values of clearance, c, between the punch and the die: 2 mm and 4 mm. Two levels of blank holder force (BHF) were considered for each composite: (i) 9.8 kN and 19.6 kN, for the Composite 1; (ii) 2.45 N and 4.9 N, for Composite 2. Afterwards, the analysis focuses on the equivalent plastic strain and Green-Lagrange shear strain distributions in the sheet thickness, at the end of the forming process, which occurs after a punch displacement of 40 mm; the flange contours and flange draw-in are also analysed.

Force vs. Displacement
The results of punch force vs. displacement for the two composites and their corresponding equivalent materials, for the above-mentioned two values of clearance and two levels of BHF, are shown in Figure 13. The comparison between the curves of the composite and equivalent materials reveals identical behaviours during the deep drawing of a square cup, for both values of c and both levels of BHF, as was already the case for the U-channel profile, when the BHF value is high enough.  Figure 14) and 4.9 kN (Composite 2, Figure 15), where the clearance between the punch and the die is equal to 2 mm. Concerning the strain distributions in surface, there is not significant difference between the composites and the equivalent materials; this is also the case for BHF equal to 9.8 kN (Composite 1) and 2.45 kN (Composite 2), although not shown in the figures. However, there are differences through thickness, in case of Composite 1, although these differences are not noticeable in the case of Composite 2. This is related to the occurrence of shear deformation in the case of Composite 1 (see Figure 14) that does not occur in Composite 2 (see Figure 15), because in the latter case the difference in mechanical properties between the polymeric core and the outer metal layers is smaller than in the Composite 1 (see Figure  4). Finally, Figure 16   material, highlighting relatively small differences between the contours of composites and their equivalent materials (less than 5%). For the clearance between the punch and the die, c, equal to 4 mm (figures not shown), the results are quite similar to those of c = 2 mm, concerning the equivalent plastic strain and the Green-Lagrange shear strain, as well as the flange contour and draw-in.

Conclusions
The present work allowed the investigation of the plastic behaviour of sandwich composites, made up of two outer layers of a metal, steel or aluminium, and a polymeric core, with recourse to numerical simulation of bulge test and deep-drawing processes of a U-channel profile and a square cup. Equivalent materials, representing a single material with the same thickness and behaviour in tension defined by a rule of mixtures approach, were considered for comparison during the analysis of the behaviour of the sandwich composites.
The bulge test results showed that the pressure vs. pole height curve of the composite can be obtained by the weighted sum of the corresponding curves of the constituent layers. The respective equivalent material seems to reproduce well the behaviour of the composite in relation to the equivalent plastic strain; therefore, differences between both materials with regard to equivalent stress were observed, which, however, follows the rule of mixtures for the composite.
Under certain forming process conditions of the U-channel profile, a high equivalent plastic strain can occur in the composite core in contrast to the outer layers; this is due to the occurrence of shear stresses in the inner polymer layer, which can promote core failure and/or delamination. For example, the blank holder force is an influential parameter in the magnitude of the strain gradient through thickness of the composite. When the difference between the mechanical behaviour of the constituent materials of the composite decreases, the strain gradient in the thickness direction of the composite becomes smoother and approximates that of the corresponding equivalent material.
In order to test the forming process of a component with more complex geometry than the Uchannel profile, the analysis of the deep-drawing of a square cup was performed. For this forming process, the conclusions are qualitatively the same as for the U-channel profile. Moreover, the analysis of the strain distributions in surface does not show significant difference between the composites and the equivalent materials, whatever the BHF value. The differences are mainly observed through thickness in the case of Composite 1, for which shear strain occurs; the differences are barely noticeable in the case of Composite 2, for which there is almost no shear strain.
Finally, the curves of force vs. displacement of the punch of the composites are similar to those of their equivalent materials, for both composites whatever the processes conditions, except in one case of the U-channel of Composite 1, where the BHF has a relatively small value.