18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS HYPERELASTIC APPROACH FOR THE SIMULATION OF WOVEN REINFORCEMENTS AT MESOSCALE A. Charmetant 1 *, E. Vidal-Sallé 1 , P. Boisse 1 1 Université de Lyon, CNRS, INSA-Lyon, LaMCoS UMR5259, F-69621, France * Corresponding author (adrien.charmetant@insa-lyon.fr) Keywords : fabric reinforced composite, mesoscale simulations, finite strain hyperelasticity k = (also force is measured). The particular case 1 1 Introduction referred to as equibiaxial traction) is of much The high specific properties and the wide range of importance for the identification of the behaviour shapes which can be processed are the principal law associated to the compaction and distortion of reasons why fabric-reinforced composites keep the yarn in the transverse plane: when both networks developing in state of the art aeronautical industries are submitted to identical elongation, the yarns cross and are generalising amongst other industries. The sections undergo large transformation. so-called RTM (Resin Transfer Moulding) process is 2.2 Picture frame test a typical process used to form fabric-reinforced parts: the dry reinforcement is first deformed to Picture frame test [2] is one of the two reference match the shape of the final part, then the polymer tests used for the characterisation of shear resistance resin is transferred by injection or infusion (different of fabric-reinforced composites. A pin-jointed frame variations of RTM process). constrains the fabric to undergo pure shear Intense research efforts concentrate on simulating deformation (fig.1). The measured force and the behaviour of dry woven reinforcements during displacement data are post processed to obtain an the first step of RTM process (preforming step): equivalent shear torque and the shear angle. The during this step, the yarns get their final shape and equivalent shear torque is computed from picture fibre density. These properties are of great interest frame experiments by the formula: ( ) since the final part directly inherits its stiffness L γ cos frame = C F L properties from them, and since they represent the ( ) s π − γ (1) 2 2cos / 4 / 2 initial conditions for the simulation of the transfer fabric step. with F the effort measured on the frame, L This information about the shape and density of the the frame L yarns lies at the scale of the woven cell. In this paper side of the frame, the side of the square of fabric fabric inside the frame and γ the angle variation a model is proposed for the unit cell of plain weave glass fabric reinforcements. A hyperelastic between warp and weft directions (i.e. the shear behaviour law is set up to describe the behaviour of angle). On the other hand, an energy-based approach the yarn, considered as a continuum at the scale of allows the calculation of this torque per unit initial the unit cell. The results of this model are compared area from the simulation: to experimental results of biaxial traction test and ɺ W picture frame test, and show good overall = C (2) s S γ concordance. ɺ u 2 Mechanical behaviour of woven reinforcements ɺ is the time-derivative of the external where W 2.1 Biaxial traction S the initial surface of the unit cell and γ ɺ work, u the time-derivative of the shear angle Biaxial traction test are non-trivial tests which are During the first phase of the shear deformation, the presented in [1]. It consists in subjecting warp and warp and weft rotate with only small deformation of weft networks of the fabric to longitudinal the yarns. When the so-called jamming angle of the deformations. The ratio between the deformations in fabric is reached, higher efforts appear as well as = ε ε k warp and weft directions is denoted / finite deformation of the cross section of the yarns. 1 2 (where 1 is the direction along which the traction
• the extension of the yarn in the direction of One aim of the model presented here is to describe I precisely the changes of geometry of the unit cell fibres ( ), ext • the compaction of the yarn in the plane of during these two phases of shear deformation, so I that permeability calculation can be performed by isotropy ( ), comp • the distortion of the yarn in the plane of using the obtained solid skeleton of the woven unit I cell. isotropy ( ), dist • the shear along fibres ( I ). 3 Behaviour law for the yarns sh These deformation modes are assumed to be 3.1 Hyperelastic materials independent, which means that no couplings between them are considered. According to this A hyperelastic material is a non-dissipative material uncoupling assumption, an energy density function whose stress tensor derivates from stored energy is associated to each invariant: the whole energy density function. If S denotes the second Piola- density function is written under the form: Kirchhoff tensor, w the stored energy density ( ) ( ) ( ) = + w C w I w I function and C the right Cauchy-Green tensor, the tot ext ext comp comp general form of the constitutive law of hyperelastic ( ) ( ) (6) + w I + w I dist dist sh sh materials is: ( ) ∂ w C The second Piola-Kirchhoff stress tensor is then = S 2 (3) ∂ C obtained by differentiation of (6) by using (3): ∂ ∂ ∂ ∂ w I w I comp comp S = ext ext + ∂ ∂ ∂ ∂ I C I C The theory of invariants [3] states that at most five ext comp invariants are needed to describe the behaviour of ∂ w ∂ I ∂ w ∂ I (7) transversally isotropic materials. The stored energy + + dist dist sh sh ∂ ∂ ∂ ∂ I C I C density function can then be defined as a function of dist sh the following invariants: ( ) ( ) = w C w I I I I I Finally, the Cauchy stress tensor is computed by use , , , , (4) 1 2 3 4 5 of the usual formula: 1 T where σ = F S F ⋅ ⋅ ( ) J (7) = I C trace , 1 ( ) ( ) ( ) 1 trace 2 = − 3.3 Identification of the behavior law I C C 2 trace , 2 (5) 2 A piecewise nonlinear behaviour of the yarn in ( ) = = = I C I C M I C 2 M extension is defined, as some types of yarn (usually det , : , : 3 4 5 glass fibres yarns) may have a stiffening behaviour at the beginning of an extension test. The parameters Amongst these invariants, the first three are the of this extension behaviour law are directly principal invariants of the right Cauchy-Green identified with an extension test on the yarn alone. tensor, and the last two are mixed invariants which As it can be seen on Fig.2, curve “Yarn”, the characterize the anisotropy of the material. The tensor M is the structural tensor built from the obtained identified behaviour matches the extension principal direction m of the transversely isotropic behaviour of the yarn very well. Because direct tests = ⊗ material: M m m on the yarn, other than extension test, are difficult to . perform on the yarn itself, the form of the strain 3.2 Physically based invariant formulation energy density function of the other deformation modes are postulated, and then identified with an In order to define a physically motivated behaviour law, different invariants ( ) I I I I inverse method using macroscopic tests on the , , , obtained ext comp dist sh fabric. as combinations of the invariants (5), are used to The behaviour law associated to the shear along describe the transformation of the yarns. The fibres is assumed to show a linear response of different deformation modes characterised by these stresses, which only requires a simple, quadratic, new invariants are respectively:
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