Modelling of anisotropic -like material during the thermoforming process

Giovanni Lelli*1, Massimo Pinsaglia2, Ernesto di Maio2 1Alcantara S.p.A. (Application Development Center), 2University of Naples "Federico II" (Department of Materials and Production Engineering) *Corresponding author: Strada di Vagno, 13, I-05035 Nera Montoro, TR, ITALY, [email protected]

Abstract: Physical and mechanical studies of The appearance and tactile feel of the Alcantara® have shown very pronounced material is similar to that of suede, making it anisotropic nonlinear features. Using suitable for a large number of luxury constitutive equations borrowed from the applications, including furniture, fashion & modelling of biological tissues like tendons accessories, contract and automotive. In and/or arteries under the form of hyperelastic particular, the most important market sector is free-energy functions, a good representation of currently represented by the premium such mechanical features can be obtained. In automotive segment, where Alcantara® particular, a combination between the represents a new category of which it is the optimization module and a modified unique product, though many imitations have hyperelastic model in COMSOL been attempted. Indeed, its characteristics Multiphysics® can be used to determine the (mechanical resistance, breathing ability, macroscopic mechanical parameters and to enduring life, easy care, colour fastness, apply them to the simulation of the behaviour resistance to wrinkling) make Alcantara® the of Alcantara® when it undergoes high- best alternative to for applications like temperature processes like thermoforming. seating, dash trimming and headliners for many premium OEM automotive suppliers. Keywords: Thermoforming, Hyperelastic, Although many manufacturing processes Anisotropic of automotive components are still based on traditional techniques, some large-scale and 1. Introduction high-rate productions require a more industrial approach. An example is represented by Alcantara® is a trade name given to a thermoforming of car headliners. composite material used to cover surfaces and Thermoforming is a manufacturing process forms in a variety of applications. The material where a thermoplastic sheet is heated to a was developed in the early 1970s by Miyoshi malleable forming temperature, formed to a Okamoto, a scientist working for the Japanese specific shape in a mould, trimmed to create a chemical company Toray Industries. In 1972, a usable product and finally cooled. The driving joint-venture between the Italian chemical force to stretch the sheet into or onto a mould group and Toray Industries gave rise to is provided either by vacuum or by a Alcantara S.p.A. countermould. Alcantara® is created via the combination Alcantara® is not able to hold the shape by of an advanced spinning process (producing itself after a thermoforming cycle, so it must very low denier bi-component "islands-in-the- be always combined with a thermoplastic sea" ) and chemical and production backing (usually, a felt made by glass and processes (needle punching, impregnation, thermoplastic polymer fibres is used for the extraction, splitting, buffing, dyeing, finishing manufacturing of car headliners). This can be etc.) which interact with each other. Alcantara achieved by sticking the product on the S.p.A. has the only European integrated backing using either a resin or a thermoplastic manufacturing cycle of ultra-micro-fibrous adhesive. non-woven : 3 production units (from Depending on the complexity of the final raw materials to the final products), each shape, the moulding process can be carried out supplying the following, are present in the either in one step (i.e., the covering and the plant. backing are shaped together) or in two steps From a technical standpoint, the resulting (i.e., the backing is pre-formed and placed in product can be defined as a composite material the mould, then the covering is heated and in which the reinforcement is a non-woven glued on it, so to avoid wrinkles near small-ray structure of ultra-microfibers in a curvatures). In any case, if the covering is porous polyurethane matrix. stretched far beyond its elastic limits (usually next to small hollows or reliefs) the material Moreover, to check the predicting ability can either break or tend to a release of residual of the model, a reference mould was built, in stresses which manifest itself in the form of a the shape of a paraboloid with an undulate detaching from the backing. For this reason, a basis (Figure 2). Standard Alcantara® with a trial-and-error approach is usually applied, 10×10 mm grid printed on its surface, using either “torture moulds” (to check if the combined with an Acryl-Butadiene-Styrene material is able to withstand the most critical 2mm-thick sheet by means of a thermoplastic deformations expected for the final shape of adhesive polyurethane-based film, was used to the manufactured part) or at worst smoothing get a thermoformed sample for the direct the sharpest edges of the final mould if some measurement of local deformations. The unexpected problem occurs. Obviously, this moulding was performed through a one-step approach is very expensive and time- process: firstly, the materials (held by a consuming, so a software tool able to highlight 650×340 mm frame throughout the entire the critical points and to predict whether the process) are heated from the covering side by covering is able to withstand high local IR lamps, then both sides of the mould move deformations or not can be very useful and towards one another to give the structure its cost-effective. final shape. After cooling and ejecting the finished part from the mould, local 2. Experimental deformations can eventually be measured by comparing the final dimensions of each As past experiences proved that this quadrangle on the surface with the initial material is characterized by different square (Figure 3). mechanical performances in warp and weft directions, it is worth pointing out that, although the microstructure of Alcantara® was represented by a three-dimensional entanglement of PET ultra-microfibers surrounded by a PU porous matrix, a simplification has been made, assuming that the overall mechanical behaviour could be described by an orthotropic model. Hence, tensile uniaxial tests based on the ASTM D638–I standard were performed on samples cut along 0° (warp, longitudinal), 90° (weft, transverse) and 45° (diagonal) directions, using an INSTRON 5565 Figure 2. Reference lab-scale mould (paraboloid) dynamometer equipped with a climate for one-step thermoforming chamber. For the sake of simplicity, as thermoforming of Alcantara® car parts is carried out at temperatures around 90°C, this value has been chosen for tests (Figure 1).

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40 Load (N) Load 30 L T 20 D 10 0 Figure 3. Example of thermoformed part for the 0 10 20 30 40 50 60 70 80 90 100 110 120 130 measurement of local deformations. Backing: ABS Displacement (mm) (2 mm thick). Adhesive: Thermoplastic Figure 1. Load (N) / displacement (mm) curves at polyurethane-based film 90°C for the three directions analyzed: warp (blue), weft (red), diagonal (green). 3. Theory and Governing Equations Ψ is depending upon both the first invariant I1 and a constant commonly identified as μ: As it was said before, the microstructure of Alcantara® is characterized by a preferential 1 I  3  orientation of the fibres due to the process 2 1 sequence needed to get the final non-woven product. In particular, fibres seem randomly A fundamental aspect to be considered for distributed at a micro-scale level, but at the any material described by a hyperelastic law is same time they are clearly characterized by a the degree of volumetric compressibility. preferential orientation in the longitudinal Hyperelastic materials can be reasonably direction at a macroscopic level. Moreover, at considered either weakly compressible or even deformations lower than 10% the material uncompressible, with an equivalent Poisson shows a clearly nonlinear behaviour, which coefficient near to the upper limit 0.5. In this can be probably ascribed to the rearrangement case, the deformation energy density can be of fiber distribution during the first load stages. decoupled as a sum between a purely isochoric This implies the need to introduce nonlinear contribution  and a purely volumetric anisotropic constitutive equations. contribution U: A similar behaviour has been already observed for “soft composites” like fiber- p reinforced rubber composites (formed by cords 1    1 3   JpIU el 1  with high tensile strength reinforcing an 2  2  elastomer, as for example tires and conveyer belts) (Tuan, et al., 2007), as well as for where: biological tissues like arteries and tendons, which often exhibit anisotropic properties      XXχXF , gradientndeformatio (Gasser, et al., 2006). Therefore, a particular T  FFC , right Cauchy  Green tensor Helmholtz free-energy function was used, which allows to model a composite in which a J el  det(F), spherical (dilatatio elasticnal) matrix material is reinforced by families of volume variation fibres and the mechanical properties of this  1 3 kind of composites depend on preferred fiber  J el FF , isochoric (distortional) directions. The description of the constitutive deformation, F  1)det( model is given with respect to the reference T 2 3 undeformed configuration, under the  J el CFFC , modified right hypotheses that the deformation occurs at a Green-Cauchy tensor constant temperature (thereby any dependence on this parameter can be neglected as a first I 1 tr C)( 11 22  CCC 33, first modified invariant approximation) and that viscoelastic contributions are negligible. It was assumed that a fibre was embedded The behaviour of human arteries, as well as in a continuum and oriented towards a of many biological tissues, is characterized by direction identified by the referential unit an initial stage (“toe region”) showing very vector a0, with |a0|=1. The deformation χ maps low stiffness values, followed by a noticeable the fibre into its current configuration, where increase of the mechanical response, generally the vector described by an exponential function (Gasser, ®  et al., 2006). Even though Alcantara does not a Fa 0   jk aF 0 n jk show the same trend, a similarity can be found j from a qualitative standpoint, allowing using the same approach. defines the spatial orientation and |a| is the Typically, hyperelastic constitutive stretch in the direction of the fibres. According equations are based on the definition of a to the above definitions, the vector Helmholtz free-energy density Ψ, which can be expressed as a function of the invariants of  aFa 0 the deformation tensor according to different possible laws. The simplest approach by far is can be interpreted as the displacement of a neo-Hookean 0 represented by the model, where related to the distortional part of the deformation gradient (coinciding with a for distribution, i.e. the same parameter δ=0 is isochoric deformations). applied. μ (Shear modulus) and κ (Bulk modulus) The orientation of fibres initially described represent the constitutive neo-Hookean by the a0i vectors modifies during stretch parameters and can be determined through following the local deformation, so the instant specific mechanical tests. In particular, a orientation in each point is described by the uniaxial tensile test and a volumetric test product: should be enough; however, the second test is usually difficult to carry out, so κ can be  aFa 0ii derived from a presumptive value of the Poisson coefficient according to the following Hence, an anisotropic hyperelastic neo- equation (Love, 1944) (Mott, et al., 2008): Hookean soft-matrix composite material is univocally identified by the following 12     parameters:  213   μ → isochoric strain energy density coefficient; The above equations refer to hyperelastic γ1, γ2 → fiber deformation coefficients, one isotropic constitutive laws. To take into for each mean direction; account anisotropy one must rewrite Ψ as the δ → fiber distribution coefficient. sum of three terms (Gasser, et al., 2006), i.e. the energy density of the so-called “ground For the volumetric contribution, assuming ® matrix” (g), the energy density of each fiber (fi, that Alcantara can be treated as an almost where the i suffix indicates the i-th fiber inside uncompressible material, a value of ν = 0.499 the composite) and the volumetric has been considered, which gives κ = 499.67μ. contribution: 4. COMSOL numerical model – Results and discussion g  fi  U i 1 As it is very difficult to get a direct g  I 1  3  measure of each characteristic parameter, a 2 different approach was used based on the 1 2 reverse analysis of uniaxial tests performed at fi  i E i  1 2 90°C. Therefore, the Optimization feature has  p  been firstly activated in the simulation of  JpU el 1   2  uniaxial tensile deformations of ASTM D638-I dogbone samples cut along 0°/45°/90°

directions with respect to the selvedge (Figure E i represents a Green-Lagrange strain-like 4). quantity which characterizes the strain in the direction of the mean orientation a0i of the i-th family of fibres: D (45°) T (90°)

i  IE 1   I  131 4i L (0°) I 4i 00 ii :  aaCaa 00 ii ij Cij , ji where δ is a dispersion parameter related to the spatial distribution of fibres (the same for each Figure 4. Simulation of the deformation (color data family of fibres, ranging from δ=1/3, spherical set: Von Mises stress) of ASTM D638-I dogbone symmetry, to δ=0, ideal alignment) and I 4i is samples cut along the 0°/45°/90° directions. a tensor invariant equal to the square of the stretch in the direction of a0i. determined from A comparison between experimental and histological data of the tissue. A basic numerical data in the common range of assumption in the present work is that both deformations for thermoforming applications families of fibres have the same (ideal) is reported in Figure 5. 80 A plot of the final deformed configuration 70 L_real is reported in Figure 7. 60 T_real L_sim 50 T_sim 40 Load (N) Load 30 20 10 0 0 10 20 30 40 50 60 70 80 Displacement (mm)

Figure 5. Experimental vs. numerical load/displacement curves for L and T uniaxial tests.

As a fair agreement was found for uniaxial configurations, the model was extended to the 3D simulation of thermoforming with the pilot Figure 7. Deformed shape of the fabric. mould shown in Figure 2. To reduce the number of variables without 7. Conclusions and future work compromising the completeness of the model, the geometry was cut through its two main Starting from considerations about both the symmetry planes. Boundary conditions were microstructure and the macroscopic behaviour set as follows: of Alcantara® products, a nonlinear anisotropic → Fabric fixed at the external boundaries; mechanical model has been proposed and → Top mould (female) fixed; verified. The approach was based on → Bottom mould (male) moving at a constitutive equations similar to those of “soft constant speed; matrix composites” like fiber-reinforced → Contact between the upper side of the elastomers or biological tissues (e.g. arteries, fabric and the internal surface of the female tendons). In particular, the proposed model is mould, as well as between the lower side of the based on the definition of a Helmholtz free- fabric and the external surface of the male energy density as a function of different mould; contributions coming from both matrix and → Very high friction coefficient between fibres (for the sake of simplicity, the material the lower side of the fabric and the external has been approximate with an orthotropic surface of the male mould, to simulate the structure, with only two mean fiber effect of the glue (acting instantaneously at the orientations parallel to longitudinal and time of contact). transverse directions). The initial geometry is shown in Figure 6. The constitutive parameters were determined through a combination of the Optimization module with Structural Mechanics, by imposing the minimum square difference between real and numerical load/displacement curves for three dogbone ASTM D638-I samples cut along 0°/45°/90° angles with respect to the selvedge, which underwent uniaxial tensile tests at 90°C. The model showed a fair ability to predict the real behaviour of Alcantara® under plane stress conditions, so it was applied to the simulation of thermoforming within a pilot mould. Even in this case, a reasonable agreement between real and numerical results was found. The present work describes a first attempt

to predict the mechanical behaviour of Figure 6. Initial configuration for the simulation of Alcantara® when it undergoes conditions of the thermoforming process (pilot mould). high temperature and pressure. Obviously, the above mentioned model can be enriched from many points of view, for example by using more complex hyperelastic laws, or by taking into account the temperature dependence of material characteristic parameters (to be able to describe non-isothermal processes). If a good predicting ability will be demonstrated, this will allow the application of the model to real case studies, like thermoforming of car headliners.

8. References

1. Gasser, T.C., Ogden, R.W. and Holzapfel, G.A. 2006. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. Journal of the Royal Society Interface. 2006, 3, pp. 15-35. 2. Love, A.E.H. 1944. A Treatise on the Mathematical Theory of Elasticity, 4th edition. New York : Dover Publications, 1944. 3. Mott, P.H., Dorgan, J.R. and Roland, C.M. 2008. The bulk modulus and Poisson’s ratio of "incompressible" materials. Journal of Sound and Vibration. 2008, 312, pp. 572–575. 4. Tuan, H.S. and Marvalova, B. 2007. Constitutive material model of fiber- reinforced composites at finite strains in COMSOL Multiphysics. 2007.

9. Acknowledgements

The authors are very grateful to Ing. Gian Luigi Zanotelli and Ing. Daniele Panfiglio of COMSOL s.r.l. (Brescia, Italy) for their valued contribution to this work.