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Hyperelastic model for large deformation analyses of 3D interlock composite preforms
Composites Science and Technology 72 (2012) 1352–1360
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Hyperelastic model for large deformation analyses of 3D interlock composite preforms A. Charmetant, J.G. Orliac, E. Vidal-Sallé ⇑, P. Boisse Université de Lyon, LaMCoS, INSA-Lyon, F-69621 Villeurbanne, France
a r t i c l e
i n f o
Article history: Received 14 February 2012 Received in revised form 4 May 2012 Accepted 6 May 2012 Available online 15 May 2012 Keywords: A. Fabrics/textiles B. Non-linear behaviour C. Anisotropy C. Finite element analysis Hyperelasticity
a b s t r a c t A hyperelastic constitutive law is proposed to describe the mechanical behaviour of 3D layer to layer angle interlock composite reinforcements. The objective of this model is to simulate shaping of thick textile preforms for RTM processes. After the identification of the independent deformation modes of initially orthotropic reinforcements, a strain energy potential is built up based on strain invariants representative to those modes assuming an additive composition of them. The parameters of the proposed constitutive model are identified using standard and specific mechanical tests performed on a 3D interlock material. Then, the model is validated on forming simulations on a single curve and double curve shapes. Three point bending tests on thick interlock reinforcements have been analysed experimentally and numerically. The specific transformation of cross sections is depicted by the proposed hyperelastic model. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The need of energy saving is the first reason of the increasing use of lightweight structures. Among them, composite structures are widely used. In the transportation industry like for example in the aeronautics, this increase passes through the use of composite materials in critical parts such as turbine blades or central wing boxes [1]. Laminated composites with 2D layered reinforcements have been successfully used for several decades for aircraft construction [2], civil engineering [3] for example. Nevertheless application of 2D laminated composites is restricted by their low resistance to delamination cracking due to their limited interlaminar fracture toughness. In order to overcome this difficulty, composite preforms with 3D architecture have been proposed. Among them, layer to layer angle interlocks are among the most interesting [4–6]. Nevertheless the achievement of the forming process that leads to an optimal preform is difficult. To avoid relying on time consuming trial-and-error techniques numerical tools for predicting the reinforcement deformation during forming are presently under development. In the last decades, some effort has been produced to develop algorithms for the simulation of textile reinforcement forming [7]. The first methods used to simulate the draping process are kinematical models more known as fishnet algorithms [8,9]. Those methods are efficient and need a small numerical effort. But their main drawback is their incapability to account for the mechanical behaviour of the studied ⇑ Corresponding author. Tel.: +33 4 72 43 81 46; fax: +33 4 72 43 85 25. E-mail address: [email protected] (E. Vidal-Sallé). 0266-3538/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2012.05.006
reinforcement and for the loads applied on the preform. That is the reason why mechanically based methods have been developed afterwards. Most of them are based on the finite element method in which the textile reinforcement material can be considered as a discrete assembly of yarns [10–15] or as a continuous material which constitutive behaviour accounts for the specific textile behaviour. Several models have been built for 2D continuous fibre woven or knitted reinforcements. They frequently use hypo-elastic approaches [16–20]. It has been shown [21] that Green–Naghdi and Jaumann classically used objective derivatives (and available in most finite element codes) cannot be used for textile materials which exhibit strong anisotropic behaviour. Some hyperelastic models have been developed for thin (2D) textile materials. Those models, written in the initial configuration do not need that objective derivative [22,23]. The deformation modes of interlock reinforcements are specific. The interlock weaving creates a bond between the warp and weft yarn layers. There are however some possible sliding otherwise the forming would be impossible. The deformation is mainly guided by the very large tension stiffness of the yarns. Nevertheless the other rigidities (shear, compaction) are important in the case of loadings without yarn tension. This is the case of the bending test analysed in Section 5. The preforms can be very thick in some places which lead to choose a three-dimensional modelling as it is done in the present paper. In addition it will be shown in Section 5.2 that the bending deformation does not verify the Kirchhoff plate kinematics. Few models are available for layer-to-layer angle interlocks. One of them [24] uses semi-discrete hexahedral finite elements
A. Charmetant et al. / Composites Science and Technology 72 (2012) 1352–1360
made of segment yarns immerged in a continuous medium This model is efficient in most interlock reinforcement forming simulations but some difficulties has been highlighted to simulate the kinematics of bending tests (such as those shown below). The present paper proposes a hyperelastic model for layer to layer angle interlock thick reinforcements. This model aims to describe properly all the deformation modes of interlock preforms in particular for large bending. Based on an approach of the same nature as the one used by the authors for the modelling of a single fibrous yarn [25], this paper proposes a new hyperelastic constitutive law for 3D layer to layer angle interlock preforms. In the hyperelastic framework for initially orthotropic materials, the proposed model is described based on the identified elementary deformation modes. The strain energy potential is defined for those deformation modes and the identification procedure is given. In a second part, comparisons between experiments and simulations are presented and discussed in the case of the forming of a hemispherical thick preform and of three point bending.
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I1 ¼ TrðCÞ ¼ k21 þ k22 þ k23 1 ðTrðCÞ2 TrðC 2 ÞÞ ¼ ðk1 k2 Þ2 þ ðk1 k3 Þ2 þ ðk2 k3 Þ2 2 I3 ¼ DetðCÞ ¼ ðk1 k2 k3 Þ2 I2 ¼
ð5Þ
and where
I4i ¼ C : M i ¼ M i C Mi ; I4ij ¼ M i C Mj and I5i ¼ C 2 : M i ¼ Mi C 2 Mi ði ¼ 1; 3Þ
ð6Þ
are the mixed invariants corresponding to the structural tensors. 2.3. Uncoupling assumption It is assumed here that the contribution of each deformation mode is independent from the others, which allows writing the strain energy density function as the summation of n strain energy density functions. Then, the derivative of w can be written as: n @w X @wi @Ii ¼ @C @Ii @C i¼1
ð7Þ
where Ii is the strain invariant representative to the ith deformation mode and n the total number of independent deformation modes.
2. Anisotropic hyperelastic constitutive equation 2.1. Hyperelasticity framework
3. Deformation modes of 3D interlock reinforcements The existence of a potential energy (per undeformed unit volume), function of the gradient strain tensor F can be assumed [26] which satisfies the principle of material indifference. This strain energy density is a function that can be written as a function of the strain tensor
C : w ¼ wðCÞ with C ¼ F T :F
ð1Þ
Function w must be mathematically isotropic. The dependence of the strain energy potential on the material privileged directions can be introduced explicitly using any scalar, vectorial, or tensorial functions relative to any anisotropy. For that, the strain energy density function must use structural tensors representative of the anisotropy of the material.Finally, the hyperelastic constitutive equation derives from the strain energy density function and the second Piola Kirchhoff stress tensor S is obtained. The Cauchy stress tensor can then be calculated from S using the deformation gradient tensor F:
1 @w T F J @C
r¼ F
ð2Þ
Hypo-elastic constitutive equations for woven materials with several fibre directions can be built [20] but they require either the use of strong assumptions or the use of complex tools of differential geometry [30]. In addition, they generally do not allow recovering after a closed loop loading path [23]. An alternative that avoids these drawbacks is to develop hyperelastic models for nonlinear anisotropic materials, using structural tensors [27,31]. The assumption made here is that the anisotropic directions are initially orthogonal, making the material initially orthotropic. Due to the large in-plane shear that can exhibit a woven preform during forming, the material does not remain orthotropic and becomes anisotropic. In a previous paper [25], the authors have proposed a transversely isotropic hyperelastic model able to describe the mechanical behaviour of a single yarn made up of thousands of continuous fibres and considered as a 3D domain. A procedure of the same nature is used here to build a hyperelastic constitutive equation for thick interlock composite reinforcements. 3.1. Deformation modes
2.2. Orthotropy frame and orthotropic hyperelasticity Like 2D reinforcements, interlocks are made of the interlacing of warp and weft yarns which are perpendicular to each other in the reference (initial) configuration. In the present work, the homogenised material has three privileged directions: the warp direction M1, the weft direction M2 and a third direction perpendicular to the others, M3 which corresponds to the preform thickness. As mentioned in [27], the symmetry group for such kind of material is characterised by three structural tensors:
M 1 ¼ M 1 M 1 ; M2 ¼ M2 M2 and M 3 ¼ M3 M 3
where I1, I2, I3 are the invariants of C defined by:
3.2. Invariant definitions For each deformation mode, a strain invariant can be defined based on physical observations.
ð3Þ
The representation theorems [28–29] lead to write the strain energy density function of a hyperelastic law as:
worth ¼ worth ðI1 ; I2 ; I3 ; I41 ; I42 ; I43 ; I412 ; I423 ; I51 ; I52 ; I53 Þ
Six deformation modes can be identified for a 3D interlock fabric: stretch in the warp direction; stretch in the weft direction; transverse compression of the reinforcement; in-plane shear; transverse shear in the warp direction; transverse shear in the weft direction. Those deformation modes are illustrated in Fig. 1.
ð4Þ
3.2.1. Stretch invariants in warp and weft directions The above defined I41 and I42 invariants (see Eq. (6)) correspond to the square of the stretches in the warp and weft directions. Consequently, they can be used directly to characterise the stretch deformation modes and the corresponding derivatives with respect to
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 1. Deformation modes of layer to layer interlock reinforcements (a) transverse compression, (d) in-plane shear, (b and e) stretches, (c and f) longitudinal shear.
c ¼ ðM1 ; M2 Þ ðm1 ; m2 Þ
ð11Þ
As (M1, M2) is equal to 90°, one can also write:
sinðcÞ ¼
m1 m2 km1 kkm2 k
ð12Þ
Consequently, the in-plane shear invariant and its derivative can be defined by:
Fig. 2. Transverse compression of interlock reinforcement cannot be directly linked to the length of the covariant vector m3.
C:
pffiffiffiffiffiffi Iaelong ¼ lnð I4a Þ
a ¼ 1; 2 and
@Iaelong 1 Ma ¼ 2I4a @C
a ¼ 1; 2
ð8Þ
3.2.2. Transverse compression invariant A specific strain invariant must be defined for the transverse compression of the reinforcement which is not linked to the covariant vector m3 associated to the privileged direction M3 of the reference configuration: for example, for a transverse shear (see Fig. 2), vector m corresponds, in the deformed configuration, to vector M of the initial one. Vector m is calculated from:
m¼F M
ð9Þ
The covariant vector m3 direction changes but its length does not. Nevertheless, the reinforcement exhibits a transverse compression. In the constitutive model for the yarn, that deformation mode is taken into account from the ratio between the total volume change and the stretch. By extension the compression invariant is defined using the total volume change divided by the two stretches. The invariant and its derivative with respect to
1 I3 and C are : Icomp ¼ ln I41 I42 2 @Icomp 1 1 1 C 1 M1 M2 ¼ 2 I41 I42 @C
ð10Þ
3.2.3. In-plane shear invariant The in-plane shear deformation mode is characterised by the angle variation between warp and weft directions. The angle variation between the two covariant vectors m1 and m2 is defined by:
I421 Icp ¼ pffiffiffiffiffiffiffiffiffiffiffi ¼ sinðcÞ and I41 I42 @Icp 1 Icp Icp M1 M2 ¼ pffiffiffiffiffiffiffiffiffiffiffi ðM 1 M 2 þ M 2 M1 Þ @C 2I41 2I42 2 I41 I42
ð13Þ
3.2.4. Transverse shear invariants In a similar way, one can define transverse shear using similar invariants:
I4a3 Icta ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ¼ sinðca3 Þða ¼ 1; 2Þ I4a I43
ð14Þ
and their derivatives:
@Icta 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ðMa M 3 þ M 3 Ma Þ @C 2 I4a I43 Icta Icta Ma M3 ða ¼ 1; 2Þ 2I4a 2I43
ð15Þ
4. Strain energy potential identification 4.1. Studied reinforcement The material used in the present study has been designed by SNECMA. Its thickness is large enough to be representative in terms of mechanical behaviour. The warp and weft yarns are both made up of 48,000 carbon fibres (48 K). The architecture and the dimension of the weaving is shown in Fig. 3 by X-ray tomography images of the material. The material is nearly balanced so that some strain energy density functions are assumed to be identical for both yarn directions. 4.2. Parameter identification 4.2.1. Tension in warp and weft directions Because the interlock reinforcement is nearly balanced, the mechanical behaviour in tension is supposed to be identical in warp and weft directions. A tensile test is realised on a 35 275 mm sample (see Fig. 4). Two distinct parts can be identified: a first part with a low stiffness corresponding to a decrease of
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wielong
0 2 i i 0 1K 2 elong ðIelong Þ ifIelong 6 Ielong ¼ 2 0 0 i 1 0 1 K elong ðIi elong Ielong Þ þ 2 K elong Ielong Ielong 2
ifIielong > I0elong ð16Þ
The corresponding second Piola–Kirchhoff stress tensors are (without summation):
Sielong ¼ 2
@wielong
@C 0 i 0 1 2K elong Ielong ifIelong 6 Ielong ¼ Mi I4i 2K elong ðIelong I0 Þ þ K 0 I0 elong elong elong
ifIelong > I0elong
ð17Þ
4.2.2. Transverse compression A compression test has been realised on a 50 50 mm sample. Experimental and numerical results are shown in Fig. 5. The strain energy density function is assumed to be:
" wcomp ¼ K comp
Fig. 3. Studied interlock.
1
Icomp
!p p
I0comp
Icomp I0comp
# 1
ð18Þ
Coefficient I0comp corresponds to the theoretical threshold value of Icomp for a reinforcement compression equal to 100%. It can be determined from the mass mepr and initial volume V0 of the specimen and qcarb, the mass density of the bulk carbon material.
I0comp ¼
1 mepr ln 2 qcarb V 0
ð19Þ
The second Piola–Kirchhoff stress tensor is then:
Scomp ¼
pK 0comp I0comp
2 4 1
Icomp
!p1
I0comp
3
1 1 15 C 1 M1 M2 I41 I42 ð20Þ
Fig. 4. Tensile test in the interlock sample (prescribed displacement).
the yarns undulation, and a second part, for which the stiffness is larger and can be considered as constant in a first approximation. A parameter I0elong defines the threshold between those two behaviours. The associated strain energy density functions are:
4.2.3. In-plane shear The in-plane shear mechanical behaviour of the material is identified using a bias extension test. A tensile test is performed on a rectangular specimen such as the warp and weft directions of the tows are oriented initially at ±45° to the direction of the applied load (Fig. 6) [32]. The tensile force is measured as a function of the shear angle (see Fig. 6). It allows identifying the strain energy density function of Eq. (21). Like for the tensile test, two behaviours are identified. For small shear angles, the behaviour is nearly quadratic and becomes exponential for larger angles. The threshold between both behaviours is given by I0cp :
w1 ðIcp Þ if Icp 6 I0 cp cp wcp ðIcp Þ ¼ 2 wcp ðIcp Þ else
Fig. 5. Strain energy (a) and Cauchy stress for transverse compression (b).
ð21Þ
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Fig. 6. Experimental bias extension test for a thick interlock reinforcement.
with:
w1cp ðIcp Þ
¼
K cp12 I2cp
w2cp ðIcp Þ ¼ K cp21 ð1
þ K cp13 I3cp ; Icp Þp22 þ
All the material parameters identified are summarised in Table 1.
wcp20
ð22Þ
Function wcp must be a C2 class function, which implies some relations between all the above defined parameters. Finally, one can define: 0 0 þK 1 0 K cp12 ¼ K20 K cp13 ¼ K 16IK K cp21 ¼ K2p Icp ð1 I0cp Þ1þp22 0 22 cp
p22 ¼
K 1 ð23I0cp ÞK 0 I0cp I0cp ðK 0 þK 1 Þ
wcp20 ¼ w1cp ðI0cp Þ w2cp ðI0cp Þ
ð23Þ
A Levenberg–Marquardt algorithm is used to identify all the values (Table 1). For such test, the tensile strain energy is assumed to be zero because the yarn extension is negligible taking its large stiffness into account. 4.2.4. Transverse shears Transverse shear tests have been realised using an on-purpose device (see Fig. 7a and b). Shear along warp and weft direction tests have been prescribed in order to identify both strain energy density functions. Two parallel planes are fixed to the sample and impose pure transverse shear strains to the specimen. The strain field homogeneity is checked using small yarns stuck at the sample surface. Fig. 7c shows the comparison between the experimental results in warp and weft directions. The behaviour can be split into two phases: a first one during which the stiffness initially strong decreases significantly and a second one during which the stiffness is quite constant. Those curves allow assuming a strain energy density function defined by parts:
K jI jpcta ifjI j 6 I0 cta cta cta cta with 0 < pcta < 2and a ¼ 1; 2 wcta ðIcta Þ ¼ K 2 jIcta j2 þ K 1 jIcta j þ K 0 ð24Þ The stiffness continuity requirement imposes some relations between all the parameters so that:
K 0 ¼ ð1 32 pcta þ 12 ðpcta Þ2 ÞK cta ðI0cta Þpcta ; K 1 ¼ K cta pcta ð2 pcta ÞðI0cta Þpcta 1 K 2 ¼ 12 K cta pcta ðpcta 1ÞðI0cta Þpcta 2
ð25Þ
5. Comparisons between simulations and experiments 5.1. Deep drawing with a hemispherical punch In order to validate the proposed hyperelastic model, simulations are realised on deep drawing with a hemispherical punch. The device is shown in Fig. 8a. When the punch stroke reached 50 mm, the binder restricts the fabric movement. The friction behaviour is assumed to be described by a constant Coulomb coefficient of 0.2. Experimental (shown in Fig. 8b) and numerical external shapes are in good agreement. The obtained results show the capability of the proposed model to capture the kinematics of layer to layer angle interlock reinforcement during forming (see Fig. 8c and d). The shear angles on top and bottom faces of the preform differ significantly. That is well depicted by the 3D FE analysis based on the hyperelastic model. In addition the simulations give transverse compaction strains. These quantities are not provided when the simulation of the forming is based on standard shell finite elements. The compaction strains are important in the zone of the blank holder (Fig. 8d). 5.2. Bending of a thick interlock reinforcements Bending is a deformation mode which involves transverse shear and, locally, transverse compression. In this test shown Fig. 9a and d, the specimen is mainly subjected to transverse shear strains. This test highlights the specificities of the through the thickness kinematics of the interlock reinforcement. Two tests have been realised: a three point 0°/90° bending test, i.e. warp and weft directions are aligned with the sample edges; a three point ±45° bending test, i.e. the yarn directions form a 45° angle with the sample edges. Fig. 9b and e shows the comparisons between numerical and experimental results in terms of deformed shape. For both tests, the sample is a parallelepiped of 200 30 15 mm. The distance between fixed points is 116 mm. A 60 mm displacement is imposed to the mobile point. In the first hand, Fig. 9a shows that, in the case of the 0°/90° specimen, the cross sections remain nearly vertical and the interlock beam is very far from the Euler–Bernoulli assumption. The
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Fig. 7. Transverse shear test device (a and b) and identification curves respectively in the warp and weft directions (c).
Table 1 Strain invariants associated to thick interlock deformation modes and identified parameters. Deformation mode Stretch in warp and weft directions
Strain invariant pffiffiffiffiffiffi pffiffiffiffiffiffi Ielong1 ¼ ln I41 and Ielong2 ¼ lnð I42 Þ
Identified coefficients I0elong ¼ 0:0145 K 0elong ¼ 37:85 MPa
Compression
Icomp ¼ 12 ln
I3 I41 I42
K elong ¼ 816:33 MPa
K 0comp ¼ 7:57 103 MPa p = 2.85 I0comp ¼ 1:12
In-plane shear
421 Icp ¼ pIffiffiffiffiffiffiffiffiffi
K0 = 0.3 MPa
I41 I42
I0cp ¼ 4:2 103 K 1 ¼ 3:0 103 MPa Transverse shear in the warp and weft directions
413 423 Ict1 ¼ pIffiffiffiffiffiffiffiffiffi and Ict2 ¼ pIffiffiffiffiffiffiffiffiffi
I41 I43
simulation correctly depicts this vertical orientation of the cross sections (Fig. 9b). In the other hand, for a ±45° bending test, experimental results show a quasi Euler–Bernoulli behaviour. The cross sections remain perpendicular to the mid surface of the specimen. The numerical simulations are in fairly good agreement (Fig. 9e). The clearly different orientations of the cross sections for the specimen at 0°/90° and at ±45° are due to the very large stiffness of the yarns in tension. For a specimen initially oriented at 0°/ 90°, the yarns have the same direction as the specimen and are almost inextensible. That leads to vertical cross section. For a specimen initially oriented at ±45°, in-plane shear in the initially
I42 I43
K ct1 ¼ 2:67 103 MPa I0ct1 ¼ 5:96 102 ; pct1 ¼ 1:375
K ct2 ¼ 3:65 103 MPa I0ct2 ¼ 10:74 102 pct2 ¼ 1:5536
horizontal plane lead to cross sections perpendicular to the mid plane of the specimen. The proposed hyperelastic law describes fairly well this specific kinematics. Nevertheless, the shape of the numerical samples presents some discrepancies. The first point concerns the non loaded parts of the specimens (external to the supports). These parts are rather aligned with the central part of the specimen in the experiments. This is not the case in simulations. In particular the external parts remain horizontal in the numerical analysis of the 0°/90° specimen. Moreover the radius at the centre of the specimen is somewhat too small. It is clearer on the lower face of the specimen.
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Fig. 8. Geometrical characteristics of deep drawing with hemispherical punch (a); preform shape after forming (b) In-plane shear angles (c) and compression ratio (d) for 50 mm punch stroke.
Fig. 9. Comparison between experiment and simulation for 3 point bending tests: experiment for 0/90° (a) and ±45° (d) fibre orientation; simulation with the proposed model for 0/90° (b) and ±45° (e) fibre orientation; simulation with added beams for 0/90°(c) and ±45° (f) fibre orientation.
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5.3. Local bending stiffness – Possible improvements The numerical simulations of thick interlock deformations presented in Sections 5.1 and 5.2 are in fairly good agreement with the experiments. In particular the different orientations of the cross sections after deformations that change with the yarn orientations are depicted by the simulations. Nevertheless in the case of specimen with yarns at 0°/90°, the position of the unloaded parts of the beam (external to the supports) is not properly depicted. They remain horizontal when they are aligned with the central part of the specimen in the experiment. In addition, the radii of curvature in the central part of the beam are smaller in the simulation than in the experiment with a larger amplitude in the ±45° case. These two points can be related to fibre bending stiffness. Each fibre has a local bending stiffness that resists to a curvature and contributes to aligning the parts of the beam external to the supports and to increasing the central radius of curvature. The hyperelastic model presented in the previous sections falls within the framework of classical continuum mechanics (respecting Cauchy’s axioms). There is no local microstructure that permits to introduce volume couples. For this, generalised continuum models are necessary [33]. A simple way to take the fibre stiffness into account is to superimpose a set of finite element beams in the warp and weft directions on the continuous hyperelastic material. The bending stiffness of these beams enables to model those fibres within the interlock reinforcement. The deformed shapes obtained in this way in the three point bending test of thick interlocks are in better agreement with the experiment (Fig 9c and f). In particular the external parts of the beam with yarns at 0°/90° are in continuity with the internal one and the central radius of curvature are in agreement with the experiment. The goal of this approach is to show that the microstructure and in particular the bending stiffness of the fibres plays a role in the mechanical behaviour of thick interlocks. The classical continuum mechanics (Cauchy) does not comprise this microstructure and there is no applied couple in volume [33]. For an advanced mechanical behaviour model of thick interlock reinforcement, one possible way of improvement could be to use mechanics of generalised continua (Cosserat continua or second-gradient theory [34,35]). Large deformation of thick interlocks is an interesting field of application of these approaches. Nevertheless they are complex in theory and above all the necessary corresponding experiments are difficult to implement. Concerning the simulations of interlock preform forming, the hyperelastic model presented in the present paper is probably a good compromise between simplicity and accuracy. 6. Conclusion In the present paper, a hyperelastic anisotropic constitutive equation has been proposed to model the large deformations of layer to layer angle interlock reinforcements. Six deformation modes have been identified and related to six strain invariants based on the structural tensors representative of an initially orthotropic material. A strain energy density function, built by the addition on six strain energy functions is defined and identified from simple mechanical tests. The proposed constitutive law has shown its ability to depict the mechanical behaviour of thick interlock reinforcements during forming. These interlock materials have shown a specific behaviour of the cross sections in bending. The orientations of these cross sections in the deformed beam depend on the initial directions of warp and weft yarns. This is correctly depicted by the proposed hyperelastic model. Nevertheless a complete description of the thick interlock behaviour in bending needs to step outside the framework of standard continuum mechanics of Cauchy. It has been shown that taking local bending stiffness into account improves the
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deformed shape. Some works are now in progress to develop continuum approaches within generalised continuum mechanics. Acknowledgement The support of SNECMA company (Groupe SAFRAN) is gratefully acknowledged. References [1] Yong JC. Composites: application and assessment of market. ICCM18 conference, Jeju, Korea, Plenary lecture; 2011 [CD-ROM]. [2] Pora J. Composite materials in the airbus A380 – from history to future. ICCM13 conference, Beijing, China, Plenary lecture; 2001 [CD-ROM]. [3] Douthe C, Baverel O, Caron JF. Form-finding of a grid shell in composite materials. J Int Assoc Shell Spatial Struct 2006;47:53–62. [4] Mouritz AP, Bannister MK, Falzon PJ, Leong KH. Review of applications for advanced three-dimensional fibre textile composites. Composites Part A 1999;30:1445–61. [5] Lomov SV, Gusakov A, Huysmans G, Prodromou AG, Verpoest I. Textile geometry preprocessor for meso-mechanical models of woven composites. Compos Sci Technol 2000;60(11):2083–95. [6] Tong L, Mouritz AP, Bannister MK. 3D Fibre reinforced polymer composites. Elsevier Science; 2002. [7] Boisse P. Finite element analysis of composite forming. In: Long AC, editor. Composite forming technologies. Cambridge: Woodhead Publishing; 2007. p. 46–79. [8] Van Der Weën F. Algorithms for draping fabrics on double-curved surfaces. Int J Numer Methods Eng 1991;31:1415–26. [9] Long AC, Rudd CD. A simulation of reinforcement deformation during the production of preform for liquid moulding processes. Int Mech J Eng Manuf 1994;208:269–78. [10] Creech G, Pickett AK. Meso-modelling of non-crimp fabric composites for coupled drape and failure analysis. J Mater Sci 2006;41:6725–36. [11] Ben Boubaker B, Haussy B, Ganghoffer JF. Discrete models of woven structures. Macrosc Approach, Compos B 2007;38(4):498–505. [12] Skordos AA, Monroy Aceves C, Sutcliffe MPF. A simplified rate dependent model of forming and wrinkling of pre-impregnated woven composites. Composites Part A 2007;38:1318–30. [13] Potluri P, Parlak I, Ramgulam R, Sagar TV. Analysis of tow deformations in textile preforms subjected to forming forces. Compos Sci Technol 2006;66:297–305. [14] Jauffrès D, Sherwood JA, Morris CD, Chen J. Discrete mesoscopic modeling for the simulation of woven-fabric reinforcement forming. Int J Mater Form 2010;3(Suppl. 2):1205–16. [15] Gatouillat S, Vidal-Sallé E, Boisse P. Advantages of the meso/macro approach for the simulation of fibre composite reinforcements. Int J Mater Forming 2010;3(Suppl. 1):643–6. [16] Dong L, Lekakou C, Bader MG. Processing of composites: simulations of the draping of the fabrics with updated material behaviour law. J Compos Mater 2001;35(2):38–63. [17] Yu WR, Pourboghrat F, Chung K, Zamploni M, Kang TJ. Non-orthogonal constitutive equation for woven fabric reinforced thermoplastic composites. Composites A 2002;33:1095–105. [18] Peng X, Cao J. A continuum mechanics-based non-orthogonal constitutive model for woven composite fabrics’. Composites A 2005;36:859–74. [19] Yu WR, Harrison P, Long A. Finite element forming simulation for non-crimp fabrics using a non-orthogonal constitutive equation. Composites: Part A 2005;36:1079–93. [20] Khan MA, Mabrouki T, Vidal-Sallé E, Boisse P. Numerical and experimental analyses of woven composite reinforcement forming using a hypoelastic behaviour. Application to the double dome benchmark. J Mater Process Technol 2010;210:378–88. [21] Badel P, Vidal-Sallé E, Boisse P. Large deformation analysis of fibrous materials using rate constitutive equations. Comput Struct 2008;86(11–12): 1164–75. [22] Balzani D, Neff P, Schröder J, Holzapfel GA. A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int J Solids Struct 2006;42:4352–71. [23] Aimène Y, Vidal–Sallé E, Hagège B, Sidoroff F, Boisse P. A hyperelastic approach for composite reinforcement large deformation analysis. J Compos Mater 2010;44(1):5–26. [24] De Luycker E, Morestin F, Boisse P, Marsal D. Simulation of 3D interlock composite performing. Compos Struct 2009;88:615–23. [25] Charmetant A, Vidal-Sallé E, Boisse P. Hyperelastic modelling for mesoscopic analyses of composite reinforcements. Compos Sci Technol 2011;71: 1623–31. [26] Ogden RW. Non-linear elastic deformations. New York: Wiley and Sons; 1984. [27] Boehler JP. Lois de comportement anisotrope des milieux continus. J Méc 1978;17:153–70. [28] Quanshui Z, Boehler JP. Tensor function representations as applied to formulating constitutive laws for clinotropic materials. Acta Mech Sin 1999;10(4):336–48.
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[29] Itskov M. On the theory of fourth-order tensors and their applications in computational mechanics. Comput Methods Appl Mech Eng 2000;189(2):419–38. [30] Rougée P. Mécanique des grandes transformations. Springer; 1997. [31] Criscione JC, Douglas AS, Hunter WC. Physically based strain invariant set for materials exhibiting transversely isotropic behavior. J Mech Phys Solids 2001;49:871–97. [32] Cao J, Akkerman R, Boisse P, Chen J, et al. Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results. Composites: Part A 2008;39:1037–53.
[33] Maugin GA, Metrikine V, editors. Mechanics of generalized continua: one hundred years after the Cosserats, Advances in mechanics and mathematics, vol. 21. Springer; 2010. [34] Forest S. Cosserat overall modeling of heterogeneous materials. Mech Res Commun 1998;25:449–54. [35] Madeo A, Dell’Isola F, Ianiro N, Sciarra G. A variational deduction of second gradient poroelasticity II: an application to the consolidation problem. J Mech Mater Struct 2008;3:607–25.
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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Hyperelastic model for large deformation analyses of 3D interlock composite preforms A. Charmetant, J.G. Orliac, E. Vidal-Sallé ⇑, P. Boisse Université de Lyon, LaMCoS, INSA-Lyon, F-69621 Villeurbanne, France
a r t i c l e
i n f o
Article history: Received 14 February 2012 Received in revised form 4 May 2012 Accepted 6 May 2012 Available online 15 May 2012 Keywords: A. Fabrics/textiles B. Non-linear behaviour C. Anisotropy C. Finite element analysis Hyperelasticity
a b s t r a c t A hyperelastic constitutive law is proposed to describe the mechanical behaviour of 3D layer to layer angle interlock composite reinforcements. The objective of this model is to simulate shaping of thick textile preforms for RTM processes. After the identification of the independent deformation modes of initially orthotropic reinforcements, a strain energy potential is built up based on strain invariants representative to those modes assuming an additive composition of them. The parameters of the proposed constitutive model are identified using standard and specific mechanical tests performed on a 3D interlock material. Then, the model is validated on forming simulations on a single curve and double curve shapes. Three point bending tests on thick interlock reinforcements have been analysed experimentally and numerically. The specific transformation of cross sections is depicted by the proposed hyperelastic model. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The need of energy saving is the first reason of the increasing use of lightweight structures. Among them, composite structures are widely used. In the transportation industry like for example in the aeronautics, this increase passes through the use of composite materials in critical parts such as turbine blades or central wing boxes [1]. Laminated composites with 2D layered reinforcements have been successfully used for several decades for aircraft construction [2], civil engineering [3] for example. Nevertheless application of 2D laminated composites is restricted by their low resistance to delamination cracking due to their limited interlaminar fracture toughness. In order to overcome this difficulty, composite preforms with 3D architecture have been proposed. Among them, layer to layer angle interlocks are among the most interesting [4–6]. Nevertheless the achievement of the forming process that leads to an optimal preform is difficult. To avoid relying on time consuming trial-and-error techniques numerical tools for predicting the reinforcement deformation during forming are presently under development. In the last decades, some effort has been produced to develop algorithms for the simulation of textile reinforcement forming [7]. The first methods used to simulate the draping process are kinematical models more known as fishnet algorithms [8,9]. Those methods are efficient and need a small numerical effort. But their main drawback is their incapability to account for the mechanical behaviour of the studied ⇑ Corresponding author. Tel.: +33 4 72 43 81 46; fax: +33 4 72 43 85 25. E-mail address: [email protected] (E. Vidal-Sallé). 0266-3538/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2012.05.006
reinforcement and for the loads applied on the preform. That is the reason why mechanically based methods have been developed afterwards. Most of them are based on the finite element method in which the textile reinforcement material can be considered as a discrete assembly of yarns [10–15] or as a continuous material which constitutive behaviour accounts for the specific textile behaviour. Several models have been built for 2D continuous fibre woven or knitted reinforcements. They frequently use hypo-elastic approaches [16–20]. It has been shown [21] that Green–Naghdi and Jaumann classically used objective derivatives (and available in most finite element codes) cannot be used for textile materials which exhibit strong anisotropic behaviour. Some hyperelastic models have been developed for thin (2D) textile materials. Those models, written in the initial configuration do not need that objective derivative [22,23]. The deformation modes of interlock reinforcements are specific. The interlock weaving creates a bond between the warp and weft yarn layers. There are however some possible sliding otherwise the forming would be impossible. The deformation is mainly guided by the very large tension stiffness of the yarns. Nevertheless the other rigidities (shear, compaction) are important in the case of loadings without yarn tension. This is the case of the bending test analysed in Section 5. The preforms can be very thick in some places which lead to choose a three-dimensional modelling as it is done in the present paper. In addition it will be shown in Section 5.2 that the bending deformation does not verify the Kirchhoff plate kinematics. Few models are available for layer-to-layer angle interlocks. One of them [24] uses semi-discrete hexahedral finite elements
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made of segment yarns immerged in a continuous medium This model is efficient in most interlock reinforcement forming simulations but some difficulties has been highlighted to simulate the kinematics of bending tests (such as those shown below). The present paper proposes a hyperelastic model for layer to layer angle interlock thick reinforcements. This model aims to describe properly all the deformation modes of interlock preforms in particular for large bending. Based on an approach of the same nature as the one used by the authors for the modelling of a single fibrous yarn [25], this paper proposes a new hyperelastic constitutive law for 3D layer to layer angle interlock preforms. In the hyperelastic framework for initially orthotropic materials, the proposed model is described based on the identified elementary deformation modes. The strain energy potential is defined for those deformation modes and the identification procedure is given. In a second part, comparisons between experiments and simulations are presented and discussed in the case of the forming of a hemispherical thick preform and of three point bending.
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I1 ¼ TrðCÞ ¼ k21 þ k22 þ k23 1 ðTrðCÞ2 TrðC 2 ÞÞ ¼ ðk1 k2 Þ2 þ ðk1 k3 Þ2 þ ðk2 k3 Þ2 2 I3 ¼ DetðCÞ ¼ ðk1 k2 k3 Þ2 I2 ¼
ð5Þ
and where
I4i ¼ C : M i ¼ M i C Mi ; I4ij ¼ M i C Mj and I5i ¼ C 2 : M i ¼ Mi C 2 Mi ði ¼ 1; 3Þ
ð6Þ
are the mixed invariants corresponding to the structural tensors. 2.3. Uncoupling assumption It is assumed here that the contribution of each deformation mode is independent from the others, which allows writing the strain energy density function as the summation of n strain energy density functions. Then, the derivative of w can be written as: n @w X @wi @Ii ¼ @C @Ii @C i¼1
ð7Þ
where Ii is the strain invariant representative to the ith deformation mode and n the total number of independent deformation modes.
2. Anisotropic hyperelastic constitutive equation 2.1. Hyperelasticity framework
3. Deformation modes of 3D interlock reinforcements The existence of a potential energy (per undeformed unit volume), function of the gradient strain tensor F can be assumed [26] which satisfies the principle of material indifference. This strain energy density is a function that can be written as a function of the strain tensor
C : w ¼ wðCÞ with C ¼ F T :F
ð1Þ
Function w must be mathematically isotropic. The dependence of the strain energy potential on the material privileged directions can be introduced explicitly using any scalar, vectorial, or tensorial functions relative to any anisotropy. For that, the strain energy density function must use structural tensors representative of the anisotropy of the material.Finally, the hyperelastic constitutive equation derives from the strain energy density function and the second Piola Kirchhoff stress tensor S is obtained. The Cauchy stress tensor can then be calculated from S using the deformation gradient tensor F:
1 @w T F J @C
r¼ F
ð2Þ
Hypo-elastic constitutive equations for woven materials with several fibre directions can be built [20] but they require either the use of strong assumptions or the use of complex tools of differential geometry [30]. In addition, they generally do not allow recovering after a closed loop loading path [23]. An alternative that avoids these drawbacks is to develop hyperelastic models for nonlinear anisotropic materials, using structural tensors [27,31]. The assumption made here is that the anisotropic directions are initially orthogonal, making the material initially orthotropic. Due to the large in-plane shear that can exhibit a woven preform during forming, the material does not remain orthotropic and becomes anisotropic. In a previous paper [25], the authors have proposed a transversely isotropic hyperelastic model able to describe the mechanical behaviour of a single yarn made up of thousands of continuous fibres and considered as a 3D domain. A procedure of the same nature is used here to build a hyperelastic constitutive equation for thick interlock composite reinforcements. 3.1. Deformation modes
2.2. Orthotropy frame and orthotropic hyperelasticity Like 2D reinforcements, interlocks are made of the interlacing of warp and weft yarns which are perpendicular to each other in the reference (initial) configuration. In the present work, the homogenised material has three privileged directions: the warp direction M1, the weft direction M2 and a third direction perpendicular to the others, M3 which corresponds to the preform thickness. As mentioned in [27], the symmetry group for such kind of material is characterised by three structural tensors:
M 1 ¼ M 1 M 1 ; M2 ¼ M2 M2 and M 3 ¼ M3 M 3
where I1, I2, I3 are the invariants of C defined by:
3.2. Invariant definitions For each deformation mode, a strain invariant can be defined based on physical observations.
ð3Þ
The representation theorems [28–29] lead to write the strain energy density function of a hyperelastic law as:
worth ¼ worth ðI1 ; I2 ; I3 ; I41 ; I42 ; I43 ; I412 ; I423 ; I51 ; I52 ; I53 Þ
Six deformation modes can be identified for a 3D interlock fabric: stretch in the warp direction; stretch in the weft direction; transverse compression of the reinforcement; in-plane shear; transverse shear in the warp direction; transverse shear in the weft direction. Those deformation modes are illustrated in Fig. 1.
ð4Þ
3.2.1. Stretch invariants in warp and weft directions The above defined I41 and I42 invariants (see Eq. (6)) correspond to the square of the stretches in the warp and weft directions. Consequently, they can be used directly to characterise the stretch deformation modes and the corresponding derivatives with respect to
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 1. Deformation modes of layer to layer interlock reinforcements (a) transverse compression, (d) in-plane shear, (b and e) stretches, (c and f) longitudinal shear.
c ¼ ðM1 ; M2 Þ ðm1 ; m2 Þ
ð11Þ
As (M1, M2) is equal to 90°, one can also write:
sinðcÞ ¼
m1 m2 km1 kkm2 k
ð12Þ
Consequently, the in-plane shear invariant and its derivative can be defined by:
Fig. 2. Transverse compression of interlock reinforcement cannot be directly linked to the length of the covariant vector m3.
C:
pffiffiffiffiffiffi Iaelong ¼ lnð I4a Þ
a ¼ 1; 2 and
@Iaelong 1 Ma ¼ 2I4a @C
a ¼ 1; 2
ð8Þ
3.2.2. Transverse compression invariant A specific strain invariant must be defined for the transverse compression of the reinforcement which is not linked to the covariant vector m3 associated to the privileged direction M3 of the reference configuration: for example, for a transverse shear (see Fig. 2), vector m corresponds, in the deformed configuration, to vector M of the initial one. Vector m is calculated from:
m¼F M
ð9Þ
The covariant vector m3 direction changes but its length does not. Nevertheless, the reinforcement exhibits a transverse compression. In the constitutive model for the yarn, that deformation mode is taken into account from the ratio between the total volume change and the stretch. By extension the compression invariant is defined using the total volume change divided by the two stretches. The invariant and its derivative with respect to
1 I3 and C are : Icomp ¼ ln I41 I42 2 @Icomp 1 1 1 C 1 M1 M2 ¼ 2 I41 I42 @C
ð10Þ
3.2.3. In-plane shear invariant The in-plane shear deformation mode is characterised by the angle variation between warp and weft directions. The angle variation between the two covariant vectors m1 and m2 is defined by:
I421 Icp ¼ pffiffiffiffiffiffiffiffiffiffiffi ¼ sinðcÞ and I41 I42 @Icp 1 Icp Icp M1 M2 ¼ pffiffiffiffiffiffiffiffiffiffiffi ðM 1 M 2 þ M 2 M1 Þ @C 2I41 2I42 2 I41 I42
ð13Þ
3.2.4. Transverse shear invariants In a similar way, one can define transverse shear using similar invariants:
I4a3 Icta ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ¼ sinðca3 Þða ¼ 1; 2Þ I4a I43
ð14Þ
and their derivatives:
@Icta 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ðMa M 3 þ M 3 Ma Þ @C 2 I4a I43 Icta Icta Ma M3 ða ¼ 1; 2Þ 2I4a 2I43
ð15Þ
4. Strain energy potential identification 4.1. Studied reinforcement The material used in the present study has been designed by SNECMA. Its thickness is large enough to be representative in terms of mechanical behaviour. The warp and weft yarns are both made up of 48,000 carbon fibres (48 K). The architecture and the dimension of the weaving is shown in Fig. 3 by X-ray tomography images of the material. The material is nearly balanced so that some strain energy density functions are assumed to be identical for both yarn directions. 4.2. Parameter identification 4.2.1. Tension in warp and weft directions Because the interlock reinforcement is nearly balanced, the mechanical behaviour in tension is supposed to be identical in warp and weft directions. A tensile test is realised on a 35 275 mm sample (see Fig. 4). Two distinct parts can be identified: a first part with a low stiffness corresponding to a decrease of
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wielong
0 2 i i 0 1K 2 elong ðIelong Þ ifIelong 6 Ielong ¼ 2 0 0 i 1 0 1 K elong ðIi elong Ielong Þ þ 2 K elong Ielong Ielong 2
ifIielong > I0elong ð16Þ
The corresponding second Piola–Kirchhoff stress tensors are (without summation):
Sielong ¼ 2
@wielong
@C 0 i 0 1 2K elong Ielong ifIelong 6 Ielong ¼ Mi I4i 2K elong ðIelong I0 Þ þ K 0 I0 elong elong elong
ifIelong > I0elong
ð17Þ
4.2.2. Transverse compression A compression test has been realised on a 50 50 mm sample. Experimental and numerical results are shown in Fig. 5. The strain energy density function is assumed to be:
" wcomp ¼ K comp
Fig. 3. Studied interlock.
1
Icomp
!p p
I0comp
Icomp I0comp
# 1
ð18Þ
Coefficient I0comp corresponds to the theoretical threshold value of Icomp for a reinforcement compression equal to 100%. It can be determined from the mass mepr and initial volume V0 of the specimen and qcarb, the mass density of the bulk carbon material.
I0comp ¼
1 mepr ln 2 qcarb V 0
ð19Þ
The second Piola–Kirchhoff stress tensor is then:
Scomp ¼
pK 0comp I0comp
2 4 1
Icomp
!p1
I0comp
3
1 1 15 C 1 M1 M2 I41 I42 ð20Þ
Fig. 4. Tensile test in the interlock sample (prescribed displacement).
the yarns undulation, and a second part, for which the stiffness is larger and can be considered as constant in a first approximation. A parameter I0elong defines the threshold between those two behaviours. The associated strain energy density functions are:
4.2.3. In-plane shear The in-plane shear mechanical behaviour of the material is identified using a bias extension test. A tensile test is performed on a rectangular specimen such as the warp and weft directions of the tows are oriented initially at ±45° to the direction of the applied load (Fig. 6) [32]. The tensile force is measured as a function of the shear angle (see Fig. 6). It allows identifying the strain energy density function of Eq. (21). Like for the tensile test, two behaviours are identified. For small shear angles, the behaviour is nearly quadratic and becomes exponential for larger angles. The threshold between both behaviours is given by I0cp :
w1 ðIcp Þ if Icp 6 I0 cp cp wcp ðIcp Þ ¼ 2 wcp ðIcp Þ else
Fig. 5. Strain energy (a) and Cauchy stress for transverse compression (b).
ð21Þ
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Fig. 6. Experimental bias extension test for a thick interlock reinforcement.
with:
w1cp ðIcp Þ
¼
K cp12 I2cp
w2cp ðIcp Þ ¼ K cp21 ð1
þ K cp13 I3cp ; Icp Þp22 þ
All the material parameters identified are summarised in Table 1.
wcp20
ð22Þ
Function wcp must be a C2 class function, which implies some relations between all the above defined parameters. Finally, one can define: 0 0 þK 1 0 K cp12 ¼ K20 K cp13 ¼ K 16IK K cp21 ¼ K2p Icp ð1 I0cp Þ1þp22 0 22 cp
p22 ¼
K 1 ð23I0cp ÞK 0 I0cp I0cp ðK 0 þK 1 Þ
wcp20 ¼ w1cp ðI0cp Þ w2cp ðI0cp Þ
ð23Þ
A Levenberg–Marquardt algorithm is used to identify all the values (Table 1). For such test, the tensile strain energy is assumed to be zero because the yarn extension is negligible taking its large stiffness into account. 4.2.4. Transverse shears Transverse shear tests have been realised using an on-purpose device (see Fig. 7a and b). Shear along warp and weft direction tests have been prescribed in order to identify both strain energy density functions. Two parallel planes are fixed to the sample and impose pure transverse shear strains to the specimen. The strain field homogeneity is checked using small yarns stuck at the sample surface. Fig. 7c shows the comparison between the experimental results in warp and weft directions. The behaviour can be split into two phases: a first one during which the stiffness initially strong decreases significantly and a second one during which the stiffness is quite constant. Those curves allow assuming a strain energy density function defined by parts:
K jI jpcta ifjI j 6 I0 cta cta cta cta with 0 < pcta < 2and a ¼ 1; 2 wcta ðIcta Þ ¼ K 2 jIcta j2 þ K 1 jIcta j þ K 0 ð24Þ The stiffness continuity requirement imposes some relations between all the parameters so that:
K 0 ¼ ð1 32 pcta þ 12 ðpcta Þ2 ÞK cta ðI0cta Þpcta ; K 1 ¼ K cta pcta ð2 pcta ÞðI0cta Þpcta 1 K 2 ¼ 12 K cta pcta ðpcta 1ÞðI0cta Þpcta 2
ð25Þ
5. Comparisons between simulations and experiments 5.1. Deep drawing with a hemispherical punch In order to validate the proposed hyperelastic model, simulations are realised on deep drawing with a hemispherical punch. The device is shown in Fig. 8a. When the punch stroke reached 50 mm, the binder restricts the fabric movement. The friction behaviour is assumed to be described by a constant Coulomb coefficient of 0.2. Experimental (shown in Fig. 8b) and numerical external shapes are in good agreement. The obtained results show the capability of the proposed model to capture the kinematics of layer to layer angle interlock reinforcement during forming (see Fig. 8c and d). The shear angles on top and bottom faces of the preform differ significantly. That is well depicted by the 3D FE analysis based on the hyperelastic model. In addition the simulations give transverse compaction strains. These quantities are not provided when the simulation of the forming is based on standard shell finite elements. The compaction strains are important in the zone of the blank holder (Fig. 8d). 5.2. Bending of a thick interlock reinforcements Bending is a deformation mode which involves transverse shear and, locally, transverse compression. In this test shown Fig. 9a and d, the specimen is mainly subjected to transverse shear strains. This test highlights the specificities of the through the thickness kinematics of the interlock reinforcement. Two tests have been realised: a three point 0°/90° bending test, i.e. warp and weft directions are aligned with the sample edges; a three point ±45° bending test, i.e. the yarn directions form a 45° angle with the sample edges. Fig. 9b and e shows the comparisons between numerical and experimental results in terms of deformed shape. For both tests, the sample is a parallelepiped of 200 30 15 mm. The distance between fixed points is 116 mm. A 60 mm displacement is imposed to the mobile point. In the first hand, Fig. 9a shows that, in the case of the 0°/90° specimen, the cross sections remain nearly vertical and the interlock beam is very far from the Euler–Bernoulli assumption. The
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Fig. 7. Transverse shear test device (a and b) and identification curves respectively in the warp and weft directions (c).
Table 1 Strain invariants associated to thick interlock deformation modes and identified parameters. Deformation mode Stretch in warp and weft directions
Strain invariant pffiffiffiffiffiffi pffiffiffiffiffiffi Ielong1 ¼ ln I41 and Ielong2 ¼ lnð I42 Þ
Identified coefficients I0elong ¼ 0:0145 K 0elong ¼ 37:85 MPa
Compression
Icomp ¼ 12 ln
I3 I41 I42
K elong ¼ 816:33 MPa
K 0comp ¼ 7:57 103 MPa p = 2.85 I0comp ¼ 1:12
In-plane shear
421 Icp ¼ pIffiffiffiffiffiffiffiffiffi
K0 = 0.3 MPa
I41 I42
I0cp ¼ 4:2 103 K 1 ¼ 3:0 103 MPa Transverse shear in the warp and weft directions
413 423 Ict1 ¼ pIffiffiffiffiffiffiffiffiffi and Ict2 ¼ pIffiffiffiffiffiffiffiffiffi
I41 I43
simulation correctly depicts this vertical orientation of the cross sections (Fig. 9b). In the other hand, for a ±45° bending test, experimental results show a quasi Euler–Bernoulli behaviour. The cross sections remain perpendicular to the mid surface of the specimen. The numerical simulations are in fairly good agreement (Fig. 9e). The clearly different orientations of the cross sections for the specimen at 0°/90° and at ±45° are due to the very large stiffness of the yarns in tension. For a specimen initially oriented at 0°/ 90°, the yarns have the same direction as the specimen and are almost inextensible. That leads to vertical cross section. For a specimen initially oriented at ±45°, in-plane shear in the initially
I42 I43
K ct1 ¼ 2:67 103 MPa I0ct1 ¼ 5:96 102 ; pct1 ¼ 1:375
K ct2 ¼ 3:65 103 MPa I0ct2 ¼ 10:74 102 pct2 ¼ 1:5536
horizontal plane lead to cross sections perpendicular to the mid plane of the specimen. The proposed hyperelastic law describes fairly well this specific kinematics. Nevertheless, the shape of the numerical samples presents some discrepancies. The first point concerns the non loaded parts of the specimens (external to the supports). These parts are rather aligned with the central part of the specimen in the experiments. This is not the case in simulations. In particular the external parts remain horizontal in the numerical analysis of the 0°/90° specimen. Moreover the radius at the centre of the specimen is somewhat too small. It is clearer on the lower face of the specimen.
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Fig. 8. Geometrical characteristics of deep drawing with hemispherical punch (a); preform shape after forming (b) In-plane shear angles (c) and compression ratio (d) for 50 mm punch stroke.
Fig. 9. Comparison between experiment and simulation for 3 point bending tests: experiment for 0/90° (a) and ±45° (d) fibre orientation; simulation with the proposed model for 0/90° (b) and ±45° (e) fibre orientation; simulation with added beams for 0/90°(c) and ±45° (f) fibre orientation.
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5.3. Local bending stiffness – Possible improvements The numerical simulations of thick interlock deformations presented in Sections 5.1 and 5.2 are in fairly good agreement with the experiments. In particular the different orientations of the cross sections after deformations that change with the yarn orientations are depicted by the simulations. Nevertheless in the case of specimen with yarns at 0°/90°, the position of the unloaded parts of the beam (external to the supports) is not properly depicted. They remain horizontal when they are aligned with the central part of the specimen in the experiment. In addition, the radii of curvature in the central part of the beam are smaller in the simulation than in the experiment with a larger amplitude in the ±45° case. These two points can be related to fibre bending stiffness. Each fibre has a local bending stiffness that resists to a curvature and contributes to aligning the parts of the beam external to the supports and to increasing the central radius of curvature. The hyperelastic model presented in the previous sections falls within the framework of classical continuum mechanics (respecting Cauchy’s axioms). There is no local microstructure that permits to introduce volume couples. For this, generalised continuum models are necessary [33]. A simple way to take the fibre stiffness into account is to superimpose a set of finite element beams in the warp and weft directions on the continuous hyperelastic material. The bending stiffness of these beams enables to model those fibres within the interlock reinforcement. The deformed shapes obtained in this way in the three point bending test of thick interlocks are in better agreement with the experiment (Fig 9c and f). In particular the external parts of the beam with yarns at 0°/90° are in continuity with the internal one and the central radius of curvature are in agreement with the experiment. The goal of this approach is to show that the microstructure and in particular the bending stiffness of the fibres plays a role in the mechanical behaviour of thick interlocks. The classical continuum mechanics (Cauchy) does not comprise this microstructure and there is no applied couple in volume [33]. For an advanced mechanical behaviour model of thick interlock reinforcement, one possible way of improvement could be to use mechanics of generalised continua (Cosserat continua or second-gradient theory [34,35]). Large deformation of thick interlocks is an interesting field of application of these approaches. Nevertheless they are complex in theory and above all the necessary corresponding experiments are difficult to implement. Concerning the simulations of interlock preform forming, the hyperelastic model presented in the present paper is probably a good compromise between simplicity and accuracy. 6. Conclusion In the present paper, a hyperelastic anisotropic constitutive equation has been proposed to model the large deformations of layer to layer angle interlock reinforcements. Six deformation modes have been identified and related to six strain invariants based on the structural tensors representative of an initially orthotropic material. A strain energy density function, built by the addition on six strain energy functions is defined and identified from simple mechanical tests. The proposed constitutive law has shown its ability to depict the mechanical behaviour of thick interlock reinforcements during forming. These interlock materials have shown a specific behaviour of the cross sections in bending. The orientations of these cross sections in the deformed beam depend on the initial directions of warp and weft yarns. This is correctly depicted by the proposed hyperelastic model. Nevertheless a complete description of the thick interlock behaviour in bending needs to step outside the framework of standard continuum mechanics of Cauchy. It has been shown that taking local bending stiffness into account improves the
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