Treatment of the composite fabric’s shaping using a Lagrangian formulation

Mathematical and Computer Modelling 49 (2009) 1337–1349

Contents lists available at ScienceDirect

Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm

Treatment of the composite fabric’s shaping using a Lagrangian formulation A. El Hami a,∗ , B. Radi b , A. Cherouat c a

LMR, INSA de Rouen, Avenue de l’Université, 76800 Saint Etienne de Rouvray, France

b

FST Errachidia, BP 509 Boutalamine - Errachidia, Morocco ICD – UTT de Troyes CNRS FRE 2848, 12 rue Marie-Curie, BP 2060 – 10010 Troyes, France

c

article

info

Article history: Received 10 August 2007 Received in revised form 29 August 2008 Accepted 16 September 2008 Keywords: Prepreg woven fabric Augmented Lagrangian Conjugate gradient Membrane and truss finite elements Co-rotational formulation Deep-drawing and laying-up

a b s t r a c t In this paper, we are interested in the simulation of prepreg composite deformation by deep-drawing and laying-up. It uses new bi-component finite elements made of woven material in which the nodal interior loads are deduced from fibre tensile strain energy and not polymerized resin membrane energy. Specific treatment is used to analyze the frictional-contact problem between the deformable prepreg composite and the steel rigid tools. The frictional-contact method is based on the Lagrangian formulation and the preconditioned conjugate gradient method. Some numerical tests are given to investigate the performance of the numerical strategies. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The use of a composite structure to reduce weight in automotive applications requires specific focus on manufacturing and cost constraints for large series. Forming of composite structure reinforced fabrics exploits the relative movements of fibres made possible by the lack of cohesion of the matrix during the process [1–4]. The matrix can be absent (forming of dry fabrics in the first stage of the R.T.M), be not polymerized yet (prepreg draped before polymerization of their thermoset matrix) or made fluid by heating (thermoplastic matrix). For example, in the Resin Transfer Molding (RTM) process, woven fabric is deformed by a deep-drawing operation and liquid resin is then injected at high temperature [4,5]. Finally the composite structure is obtained after cooling. However the fabrication process is not totally controlled and studies are necessary to determine whether manufacturing of a given shape is possible or not, and how it will be carried out. The fabrics used for RTM process are built to allow these large angular variations and to render the shaping process possible. The main defects that lead to the impossibility of obtaining a correct pre-form have been studied experimentally and are the following [5,6]: 1. Local folding due to overlapping of yarns for too large in-plane angular variations; 2. Fracture of fibres due to excessive tensions; 3. Curl of the yarns due to large compression strains. These defects appear as the main limitations for the strain state in the fabric, and they have to be detected from the strain and tensile state of the fabric. The behaviour of the woven fabric during the shaping process is very different from that of a sheet metal. During the sheet-metal forming process, the blank is usually subjected to large membrane extensions and one

∗

Corresponding author. E-mail addresses: [email protected] (A. El Hami), [email protected] (B. Radi), [email protected] (A. Cherouat).

0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.09.016

1338

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

Fig. 1. Finite deformations of prepreg woven fabric.

of the main problems of the operation is the thickness variation due to large membrane strains — one of the major problems being the knowledge of thickness variations that have to agree with the future use of the sheet-metal part [8,9]. By contrast, the extension of fabric yarns is small, the shape change being mainly due to angular variations between weft and warp yarns. These variations can be large (60◦ ) (see Fig. 1). The successful manufacturing part requires a precise control of the numerous processing parameters that govern the forming process. This is often difficult and time consuming to carry out in practice. The ability to define, in advance, the ply shapes and material orientations by anticipation allows engineers to optimize the structural properties of composite structures [6,7]. The numerical tool of composite fabric forming permits the evaluation of the feasibility of a fabric deformation within any large distortion and damage to fibres. It also gives the direction of the reinforcements after forming. This condition gives one idea about the mechanical behaviour of the final composite structure. For the simulation of dry fabrics forming, many codes have been developed on a geometrical approach (fishnet algorithm) [10–14]. These fast and effective methods make possible the calculation of the angles between the yarns in warp and weft directions in order to compare them with limiting values for which forming is not possible any more without wrinkles [15–17]. On the other hand these methods do not take into account the mechanical behavior of the reinforcements and the boundary conditions (blank holder load for example). Both of these two points affect (sometimes much) the obtained shape. An alternative to these geometrical methods is the use of finite element approaches. The domain can be considered as a continuous media, the mechanical behaviour of which models those of the fabric [18–22]. This is not easy especially because many stiffness of the fabric are very weak compared to the tensile rigidity and the direction of the fibres (or yarns) change rapidly. In this paper an alternative is proposed where finite elements, specific to prepreg fabric materials, are constructed. They are composed of woven fibres in tension and shear and the nodal interior loads are obtained from minimization of the total load tensile and shear strain energy. The proposed model is described by a mesostructural approach for finite strains and geometrical nonlinearities [6,18,23]. The not polymerized matrix (resin) has a viscous behavior and the fibres are treated as either unidirectional elastic behavior. The bi-component element for modeling prepreg woven fabric is based on an association of 3D membrane element representative of resin behavior and truss element representative of warp and weft fibre behavior. Numerical simulations were performed using the commercial FE package ABAQUS, based on incremental finite strain theory with explicit displacement boundary conditions. Contact-friction is known to be one of the main factors involved in shaping processes. In this paper, we investigate a frictional-contact mechanics implementation using the Lagrangian formulation and the preconditioned conjugate gradient method in simulating deep-drawing and laying-up forming processes. The effect of some process parameters (fibre orientation, friction coefficient, punch geometry) has been studied in order to improve the process’s working plan. It has been shown that the mechanical approach coupled numerical model is able to ‘‘optimize’’ composite forming process in order to provide input data for the pre-processing of the final composite piece after polymerization and give the mechanical limits of the fabric during the forming process. This might be of great importance to help the process engineers ‘‘virtually’’ fulfil their planning before its physical realisation. The paper is organized as follows. In Section 2, we describe the modeling behaviour of prepreg woven fabric. In Section 3, the finite element modeling is introduced and discussed. The treatment of the frictional contact is done in the Section 4. In Section 5, the numerical simulation of different forming processes is given and it shows the efficiency of our approach. 2. Modeling behavior of prepreg woven fabric The non-sliding inter-fibre of prepreg woven composite fabrics during the forming process can be experimentally observed by the following transformation of aligned straight lines drawn on the fabric (see Fig. 1). This kinematic characteristic is ensured by reinforcement weaving and the not polymerized viscous resin. The assumption is that each cross connexion of straight warp and weft fibre before deformation remains cross connected during the transformation. The basic assumptions for the mechanical forming are that the woven fabric is considered as a continuous 3D surface. The warp and weft yarns are assimilated as a truss which connecting points are hinged and the resin, considered as a membrane, is E f of warp and weft fibres, in the coupled kinematically to the fabric at these connecting points. For each connecting point X

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

1339

Fig. 2. Mechanical deformation of prepreg woven fabric.

E m (see Fig. 2). At the connecting point (I, J, K, reference configuration C0 , is associated a material position space of a resin X E f = XE m = XE and the kinematic motion of these points is described by L) we have X  f Ex = Exf (XE f , t ) Exm = Exm (XE m , t )

fibres resin.

(1)

The continuous movement of these connecting points is ensured by the weaving and the friction between yarns. The current space position of (i, j, k, l) points after deformation is obtained by

   dExf = Ff XE , t dXE f   dExm = Fm XE , t dXE m

fibres (2) resin

where Ff and Fm are the deformation gradient tensor of fibre and resin respectively. The crucial role of the deformation gradient tensor is further disclosed in terms of its decomposition into stretch and E , t ) with rotation components. The relationship of the no sliding inter-fibre can write at each connecting point as E x = E x( X Exf = Exm = Ex. For these points we have Ff = Fm = F but for all other point Ff 6= Fm and Fm = F. The gradient of transformation of the pre-impregnated woven fabric F and the pseudo gradient of transformation of the fibre Ff are defined by the function:

 Ff = λfL nfi ⊗ Nif ∂x F = ∂X

fibres (3)

resin

f

f

f

where λL is the longitudinal elongation of each fibre and Ni and ni are the fibre orientations in the initial C0 and the current f

E L (θ0 ) direction in the initial configuration, remains, Ct configurations. The tangent vector of the median line of fibre along N f nL

after deformation, the tangent vector of the median line along E 1

f

EL (F, θ0 ) = n

λ

f L

f

E L (θ0 ) . Ff N

F , θ0 direction in the current configuration f




(4)

Using the above assumption, the mechanical deformation of pre-impregnated woven fabric depends on the deformation of fibres and the deformation of resin. It is possible to decompose the deformation gradient tensor in terms of the rigid rotation f tensor followed by a stretch (F = RU). By the spectral theorem, the right stretch tensors (λi for the fibres and Um for the resin) is defined in the reference configuration C0 as

 

λfi =

q

Um =

f

f

Ni (Ff )T Ff Ni

p

T

F F

fibres resin.

(5)

The shaping problem imposes the use of incremental formulation in finite deformations. In finite strain analysis, a careful distinction has to be made between the coordinate systems that can be chosen to be describing the behaviour of the body. To preserve the objectivity requirement, the rotational objective rates are used to calculate the derivatives of any tensorial variables. This consists to rephrase the rate elastic strain model in the deformed configuration rotated by the orthogonal rotation tensor Q itself solution of the Q T Q˙ = W with W is the spin rate of the rotated frame [24,25]. This rotated description keeps unchanged the basic structure of the constitutive equations as formulated in small strain hypothesis. If Q = R the rigid body rotation tensor or the Green–Naghdi rate is recovered. The problem of the integration of strain rate tensors is a

1340

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

central one in large deformations. Hence, along this paper the constitutive of prepreg fabric equations will be formulated in the rotated frame. The rate strain tensor of woven fabric are obtain by:

   Df =   Dm =

λ˙ fi λfi 1

!



f

f

ni ⊗ ni



fibres (6) T

˙ −1 + F−T F˙ FF



resin.

(e0i ⊗ e0i )

2 The frame associated with the Green–Naghdi derivative is defined, at the material point considered, by the rigid body f f rotation (Rf for the fibres and R for the resin) of the triplet orthogonal material directions (Ni ⊗ Ni for the fibres and e0i ⊗ e0i for the resin), which, after a small time increment, remain orthogonal to the first order. Since this triplet is the same for any observer, the frame and the associated derivative are objective. The stretching tensors Eq. (6) are written in the rigid body rotation frames using Eqs. (4) and (5) as:

!  f    f f D¯ fR = λ˙ i N ⊗ N i i λfi   ¯ mR m m−1 ˙ U D =U (e0i ⊗ e0i )

fibres

(7)

resin.

  ¯ fR are obtained by the unidirectional = λ˙ fL /λfL and the transversal components D¯ fR T , D3  −νLT f f f ¯ fR ¯ fR ¯ fR behaviour of fibre as D νLT is the Poisson’s ratio of fibre. Using the Green–Naghdi T = D3 = −νLT DL ⇒ λT = λ3 = λL fR The longitudinal component is DL

¯

fR

fR

¯ L , and the stress rate tensor of the objective tensor stress, the stress rate σ¯ L , depending on the stretching deformation D mR

¯ mR , depending on the tensor deformation rate D¯ membrane resin σ    f  σ˙¯ fR = E f λf λ˙ L (τ ) L L L λfL (τ )  mR  ˙ mR m σ¯ = C (τ ) : D¯

, can be written at each time

fibres

(8)

resin. f

  f

The constitutive law of fibres is nonlinear and is written in terms of longitudinal modulus of stretching EL λL . Later is f

function of principal elongation of fibre λL , elastic modulus of fibre E¯ f and undulation factor εsh . The viscoelastic behaviour law of resin is formulated in the time domain by the hereditary integral and using the relaxation time τk and the fourth order k

relaxation tensor, which are material parameters Cijm [26]. Approximating the creep functions by a Prony series we have:

     f f f f −λ˙ /λ εsh  ¯  EL λL = Ef 1 − e L L k X k m m∞  Cijm e−t /τk  Cij (t ) = Cij +

fibres (9) resin

1

in which

∞ Cijm ,

k Cijm

and τk are the experimental resin parameters.

3. Finite element modeling Each material point moving as in a continuum, ensured by the non-sliding of fibres due to fabric weaving and resin behaviour, means that a nodal approximation for the displacement can be used. The deformation of woven fabric is described ˙ ) = Pint − Pext (Pint is the internal virtual power with membrane assumptions. The global equilibrium is obtained by Π (u and Pext the external power):

Π (u˙ ) =

Z V resin

σ¯ mR : δ D¯ mR dV +

XZ fibres

Lfibre

fR

fR

¯ L ds − S f σ¯ L δ D

Z

fs .˙udS − Su

Z

fc .˙udS

(10)

Sc

where fs is the externally applied surface forces, fc is the contact load; Su is the surface of the deformed body and Sc is the surface contact. The spatial discretization of fabric is established in the current configuration using isoparametric elements to interpolate the geometry of the element on the connecting points (nodes): nodes

x=

X k=1

nodes

N k (ξ , η) X k +

X

N k (ξ , η) uk

(11)

k=1

where uk and X k are the nodal displacements and positions of each connecting points k and N k (ξ , η) are the standard nodal interpolation functions.

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

1341

Eqs. (7) and (11), enable us to express the rate strain tensors as function of velocity and interpolation functions:

(n

fR

¯L D

o

  = B¯ fibres u˙  mR   ¯ = B¯ resin u˙ D

fibres

(12)

resin

where (B¯ fibres , B¯ resin ) are respectively, the strain–displacement matrix in the current configuration of the warp or the weft fibres and the resin. The elementary stiffness matrix of fabric is function of the strain–displacement matrix of fibres and resin can be written as:

 e Z X  T  e   f f  e  S0 EL B¯ efibres B¯ fibres dl Kfibres = f L0 fibres Z  e T  m   e   e  ¯  hm K = C B¯ resin dS 0 Bresin resin 

fibres (13) resin

S0m

f

in which hm 0 is the initial thickness of the membrane resin and S0 the initial effective cross section of fibre. The effective cross f

f f

section of the truss elements S0 is calculated using the fibre volume fraction Vf of the prepreg woven fabric V fabric = Vf S0 L0 . The initial thickness of the membrane resin is calculated using the resin volume fraction Vm of the prepreg woven fabric m V fabric = Vm V resin = (1 − Vf )V resin = (1 − Vf )hm 0 S0 . The interior load vector is also contain the a contribution of the warp and the weft fibre behaviours and the resin int int behaviour Fint = Ffibres + Fresin

 e Z  int e X  T fR  f   F = S0 B¯ efibres σ¯ L dl  fibres f Zfibres L0  int e  e T mR   ¯  hm σ¯ dS 0 Bresin  Fresin =

fibres (14) resin

S0m

and the external vector force is obtained by

{Fext }e =

Z

N T fse dS − Su

Z

N T fce dS .

(15)

Sc

The nonlinear equilibrium equation (10) on the current configuration (kinematic nonlinearities, material nonlinearities and contact with friction evolution) is linearized by an iterative Newton method. The discretized equations containing the global ˙ ) = 0. At each co-ordinates of the grid node as unknowns are derived using the stationary total potential energy δ Π (u iteration i, for the increment under consideration, the equilibrium equation after discretization is expressed as:

i

n

e e e Aelt Kfabric + Kce ∆uin = Aelt Fext − Fint




i o

(16)

e e e where Kfabric = Kfibres + Kresin is the tangent stiffness elementary matrix of prepreg woven fabric, containing a contribution   of the warp and the weft fibre tensile behaviours and the resin membrane behaviour and Kce the contact stiffness matrix. According to the different modes of deformation occurring in the pre-impregnated fabric during the shaping process, bi-component finite elements are developed to characterize the mechanical behaviour of thin composite structures. The new finite elements contain two orthogonal truss finite elements representing the warp and weft fibre behavior and thin membrane finite element representing the resin behaviour (see Fig. 3). M3D3 and M3D4 membrane finite elements and T3D2 truss finite elements of Abaqus software library are used to simulation the prepreg woven fabric behaviour [26]. These finite elements are complementary in the finite element discretization (isoparametric and use 3 DOF per node) and use the same mechanical formulation in finite deformations (the Green–Naghdi approach). The longitudinal nonlinear constitutive equation of fibre (Eq. (9)) have been implemented in the Abaqus using VUMAT subroutine. The governing equilibrium equations are solved as a dynamic problem using explicit integration. This approach has proven to be, in particular, suitable to highly nonlinear geometric and material problems, particularly where a large amount of contact between different structural parts occurs. The analyses are carried out using the FEA computer code Abaqus/Explicit (ABAQUS, 2005). The detailed formulations are described elsewhere [26,27].

4. Treatment of the frictional contact During the shaping process, boundary conditions relative to the contact and friction are changing. From a geometrical point of view, thus suppose that the connecting nodes move in function of the tools movement. On the contact surface
Ec , Et1 , Et2 is defined on each node, where between the deformable fabric and the tools, one orthonormal local reference n

Ec is the outside normal to the surface and Et1 , Et2 represents the tangent plane to the surface. We use the following n


1342

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

Fig. 3. Bi-component prepreg finite element.

decomposition into  the normal and tangential components of the displacements uE and the contact force vector to the membrane resin σ¯ mR :



E.Enc and uEt = uE − un .E un = u n  σn = σ¯ mR .Enc and {σt } = σ¯ mR − σn . nEc .


(17)

The unilateral contact conditions, written on the parts Sc of the boundary, can then been written by: un ≤ 0

σn ≤ 0

un .σn = 0.

(18)

The case (un = 0, σn ≤ 0) characterizes the contact zone between the fabric and the tool and the case (un < 0, σn = 0) corresponds to the separate part. As a friction condition for the tangential direction, we take Coulomb’s friction law with constant coefficient kσt k ≤ νf |σn |. Following Duvaut [28], this law can be written:

kσt k < νf |σn | ⇒ u.t = 0E kσt k = νf |σn | ⇒ ∃α ≥ 0/u.t = −ασt



sliding sticking

(19)

where νf represents the friction coefficient between the resin and the rigid tool. Keeping these conditions into consideration, the weak form (Eq. (10)) of our problem can be written as follows with δ W (η, u) the total tensile and shear energy of fabric:

δ Π (η, u) = δ W (η, u) −

Z

fs .ηdS −

Su

Z

(rn δ un + Ert .δEut )dS = 0

(20)

Sc

which must hold for all η E satisfying ηE .Enc ≤ 0 on Sc . So an additional variable λ (Lagrange multiplier) is introduced over Sc relative to the non-penetrability condition (un ≤ 0). This variable must satisfy over Sc the following condition: λ ≥ 0. So, the following variational equations, corresponding to the Lagrange multipliers formulation, are:

 Z   Π ( u , v) + λvn dΓ = 0  Sc Z    µvn dΓ = 0

(21)

Sc

which must hold for all admissible v and µ where µ ≥ 0 on Sc . In our approach, we treat the contact and the friction phenomena separately. In the framework of the finite element approximation, the obtained discrete system is:



[Kfabric ] {u} + C t {λ} = {F } [C ] {u} = 0

 

(22)

where (u, λ) represents the saddle point, [C ] represents the interaction matrix compatible with the non-penetrability condition and {F } is the residual vector. If the matrix [Kfabric ] (Eq. (13)) is invertible and we eliminate the unknown {u} in the equation, the Lagrange multiplier {λ} satisfies the following equation: [C1 ] {λ} = {b}

(23)

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

1343

where



[C1 ] = [C ]t [Kfabric ]−1 [C ] {b} = [C ]t [Kfabric ]−1 {F } .

(24)

Since [C1 ] is a symmetric and positive definite matrix, we can use the preconditioned conjugate gradient method to solve the linear system equation (23). This method only requires the application of the matrix [C ] in the descent p. This application is performed in two steps: 1. Solving the linear system: [Kfabric ] wk = [C ] p.

(25)

2. Computing the matrix product: [C ]t wk = [C ]t [Kfabric ]−1 [C ] p.

(26)

So, after some algebraic manipulations of {λ} corresponding the reaction at the contact zone, one can compute the displacement {u} by solving: [Kfabric ] {u} = {F } − [C ] {λ} .

(27)

The tangential contact stress {σt }is computed as follows:

{σt } = {Fint } − λEnc .

(28)

Ec , Et1 , Et2 . The treatment where {Fint } corresponds to the internal effort (see Eq. (14)) written in the local reference frame n of the friction is done according to Coulomb’s law:


(i) if kσEt k ≤ νf λ, the computational tangential constraint is accepted and this corresponds to sticking zone and we know that u.t is equal to zero.

(ii) if kσEt k ≥ νf λ, we impose that σEt = νf λut . / u.t . The friction model is implemented in ABAQUS/EXPLICIT using user subroutine UFRIC. The interaction between the composite fabric and the tools is formulated using finite sliding approach, which allows for the possibility of separation between the two surfaces during sliding. For more details, one can see [32–34]. Remarks: 1. The frictional-contact model used in this paper is very simple but in many numerical tests had showing its efficiency (in the metal forming process per example [29,30]). In our study, the adhesion is not taking into account [31]. 2. The use of the elimination technique combined with the preconditioned conjugate gradient method to solve the condensed system (23) let the introduction of the Lagrangian multiplier attractive because one of the disadvantage of the Lagrangian formulation is to solve enlarge the dimension of the system but with our technique, we reduce the dimension of the system to be solve. 3. Others techniques can be used to solve this frictional-contact problem [34–38] but these techniques use more parameters (penalizations and Lagrange multipliers) thus need more numerical simulations to find the good parameters. Our proposed technique need only one Lagrange multiplier and the use of the preconditioned conjugate gradient let our approach robust. 5. Numerical simulation of forming process To establish the validity of the finite element model and the computational formulation procedure (prepreg behaviour, finite element discretization and friction model), some numerical simulations have been carried out. The constitutive behaviour of prepreg woven fabric is validated from the tensile shear test (see [18] for more information). A shaping simulation of composite fabrics manufactured by hemispherical tool is presented in order to study the fibre orientations, the distortion map, the fabric shrinkage and the ply shape after the forming process. The effect of the friction coefficient between the prepreg and the rigid tools is study numerically. Numerical examples show the agreement of the deformed shapes and angular distortion with the experimental ones. 5.1. Laying-up of composite fabric on a hemispherical surface The first experimental process of 3D laying-up fabric on a rigid mold is proposed order to determine the ability of Serge 2 × 2 carbon fabric to be formed into a particular size and geometry. An initially square non-polymerized woven fabric made of carbon fibre (350 × 350 mm) is draped on a rigid hemispherical surface (r = 120 mm). Eight loaded springs are used to maintain the fabric during the shaping. The hemispherical mold (punch) displacement is about 300 mm (see Fig. 4). The fabric is modeled with 1600 membrane elements (three nodes M3D3 or four nodes M3D4 of ABAQUS element library) representing the resin deformation and 3200 linear truss elements (T3D2) representative of warp and weft fibre

1344

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

Fig. 4. Laying up tools on hemispherical surface.

Table 1 Mechanical characteristics of prepreg composite carbon fabric.

εsh

Ef (MPa) 230 000 Em (MPa) 45 Time (s) mk Shear C12

0.005

ν

0.45 0.01 0.02332

0.1 0.023332

1 0.083509

10 0.11723

100 0.14423

1000 0.178

Fig. 5. Experimental draping of a hemispherical piece obtained with (0◦ /90◦ ) and (−45◦ /45◦ ) fibre orientations.

transformation. The contact between the rigid surface (tools) and a deformable body (composite fabric) is modeled with rigid surface contact elements. The hemispherical rigid tool is modeled by 1600 triangular and quadrangular rigid element (T3 R3D3 and Q4 R3D4). The fabric holding system is modeled by 24 spring elements (SPINGA) with a nonlinear stiffness. In this case, the behaviour of the resin is considered as isotropic elastic or viscoelastic and the fibre as elastic linear (see the mechanical parameters of prepreg fabric in Table 1). The friction between the carbon fabric and the steel punch is simulated by different values of the friction parameter (νf ). Accordingly, the minimum value νf = 0 represents the perfectly sliding contact between the fabric and the punch, while the maximum value νf = 0.5 represents a ‘‘stick’’ contact. As for the first example, we consider the two fibre orientations: (0◦ /90◦ ) and (−45◦ /45◦ ) compared with punch directions. The friction between the punch and the fabric is equal to 0.01. Fig. 5 shows the experimental draping with respect to these orientations. Notice that the woven fabric material is highly anisotropic and the initial directions of the fibre influence the final resulting shape. Fig. 6 illustrates the numerical final shapes (for each orientation) obtained by mechanical draping for 300 mm punch displacement. As we can see, the numerical shapes sound like the experimental shapes. Fig. 7, showing the variation of fibre shear angles along the diagonal and the median axes respectively, confirm these shape likeness. The angular fibre distortion exceeds 35◦ along the diagonal axis for (0◦ /90◦ ) fibre orientations and along the diagonal axis for (−45◦ /45◦ ) fibre orientations. Notice that in the case where the behaviour of the resin is viscoelastic, the resulting numerical angular fibre distortion approaches better the experimental results. A lower loading velocity will generate lower viscous forces at the intra-ply shearing. This will minimize the difficulty for a ply to conform to a given shape and should lead to less wrinkling. As mentioned above the contact-steel–prepreg effect is simulated by different values of the friction coefficient νf . Accordingly, νf = 0.0 represents the slide condition, while νf = 0.5 represents a stick contact, νf = 0.1 and νf = 0.2 represent moderate or medium friction condition. The maximum angular distortion between fibres are summarized in

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

1345

Fig. 6. Mechanical draping of a hemispherical piece obtained with (0◦ /90◦ ) and (−45◦ /45◦ ) fibre orientations.

Fig. 7. Angular distortions along the diagonal for (0◦ /90◦ ) fibre orientation and along the median for (−45◦ /45◦ ) fibre orientations. Table 2 Maximum angular distortions for different friction coefficients νf . Fabric orientation

Experimental

Slide contact νf = 0.0

νf = 0.1

Moderate contact

νf = 0.2

Medium contact

νf = 0.5

Stick contact

0◦ /90◦ −45◦ /45◦

31◦ 28◦

29.12◦ 30.57◦

29.85◦ 29.72◦

27.7◦ 25.22◦

25.247◦ 23.501◦

Table 3 Mechanical characteristics of composite Serge 4 × 3 glass fabric. Ef (MPa) 75 000 Em (MPa) 30

εsh 0.012

ν

0.41

Table 2 for the used fabric orientation (0◦ /90◦ ) and (−45◦ /45◦ ). It is clear that the chosen values of νf have a relatively great effect on the angular distortion between the warp and the weft fibre. 5.2. Deep-drawing of composite fabric by hemispherical tools A set of shaping experiments with a hemispherical tool set has been carried out in order to validate the proposed computational procedure. An initially square blank (300 × 300 mm) made of a Serge 4 × 3 glass fibre are shaped into a hemispherical rigid tools (Fig. 8). The behaviour of the resin is supposed elastic linear (see Table 3). The behaviour of the dry woven fabric (serge 4 × 3) is elastic linear with density of fabric = 2.2 yarns/cm and initial thickness of fabric h0 = 0.57 mm. The effects of fibre undulations and resin viscosity are negligible for the fabric used in this test. A pressure equal to 3.25 MPa is applied on the blank-holder and the friction between the fabric and the tool is set to νf = 0.1. For (0◦ /90◦ ) or (−45◦ /+45◦ ) fabric, Fig. 9 shows the contour of final shapes after deep-drawing. For the usual glass fibre fabrics used in the R.T.M. process the maximum value of these angular variations are close to 60◦ . We can notice that the final shape contour of the fabric obtained with (0◦ /90◦ ) orientation is very different then (−45◦ /45◦ ) orientation and reveals the strong anisotropy effects of woven fabric material on the final composite shape and the fibre distortions.

1346

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

Fig. 8. Finite element meshing of tools of the deep-drawing process.

Fig. 9. Experimental shapes of (−45◦ /45◦ ) and (0◦ /90◦ ) fabric.

Fig. 10. Deformed mesh of (−45◦ /45◦ ) and (0◦ /90◦ ) fabrics by the deep-drawing process.

Results of simulation of the deep-drawing are shown in Fig. 10 for (0◦ /90◦ ) fabric deformation and for (−45◦ /+45◦ ) fabric deformation for 100 mm displacement of the punch. We can notice that the final shape obtained with (0◦ /90◦ ) fabric is very different from (−45◦ /45◦ ) fabric and reveals strong anisotropy effects of woven fabric material. We note that these shear angles values are very large along the diagonal line for (0◦ /90◦ ) fabric and along the median line for (−45◦ /+45◦ ) fabric (Fig. 11). As we can see, the numerical shapes sound like the experimental shapes. Fig. 12, showing the variation of fibre shear angles along the diagonal and median axes of the final shape. The angular fibre distortion exceeds 45◦ . The comparison between the experimental and numerical results shows that the proposed numerical model takes the highly anisotropic material of fabric and the intrinsic behaviour of composite fabric into account. The numerical simulation also predicts the thickening due to the necking in zones with strong membrane stretching the strains and stresses of the resin and the tension forces in the fibres. As mentioned above the ‘‘lubricant’’ of friction effect is simulated by different values of the friction coefficient. Accordingly, νf = 0.01 represents the ‘‘slide’’ condition, while νf = 0.4 represents a ‘‘stick’’ condition. Fig. 13 shows the prepreg shrinkage for (0◦ /90◦ ) and (−45◦ /45◦ ) fibre orientation have a big influence on the fibre distortions between the warp and the weft fibre. It is clear that the chosen values of νf have a relatively big influence on the blank shrinkage and the final shape of the composite product after resin polymerization. Table 4 summarizes the numerical results of the shrinkage phenomena for different values of the friction coefficient.

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

1347

Fig. 11. Fibre distortion map of (0◦ /90◦ ) and (−45◦ /45◦ ) fibre orientation.

Fig. 12. Angular distortions along the diagonal of (0◦ /90◦ ) and along the median of (−45◦ , 45◦ ).

Fig. 13. Fabric shrinkage for (0◦ /90◦ ) and (−45◦ /45◦ ) fibre orientation.

6. Conclusion A constitutive equation accounting for the isotropic elastic (or viscoelastic) nonlinear and contact-friction algorithm have been used to improve numerically the behaviour of the prepreg fabric shaping. This improvement aims to avoid the fibre

1348

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

Table 4 Shrinkage of fabric for different friction coefficient νf . Initial fabric orientation

0◦ /90◦

−45◦ /45◦

Shrinkage (mm)

Experimental

Simulation

νf = 0.01

νf = 0.1

νf = 0.2

νf = 0.4

DX DY DD DX DY DD

44 44 10 10 10 42

43.8 43.8 9.6 8.3 8.3 38.3

42.3 42.3 8.7 8.3 8.3 38.3

41.0 41.0 8.64 8.3 8.3 38.3

40.2 40.2 8.33 8.21 8.21 37.38

distortion occurrence in order to obtain an acceptable final composite shape. The treatment of the frictional contact leads to the solution of condensed system. The preconditioned gradient conjugate is applied. With this method, it is not necessary to compute the matrix of the condensed system and the initialization is carried out naturally. The effect of some process parameters has been studied in order to improve the process’s working plan. For the studied composite shaping process one can conclude that the friction parameter, the initial fibre orientation, as well as the tool geometry have a strong influence on the final piece before resin polymerization. It has been shown that the mechanical approach coupled numerical model is able to ‘‘optimize’’ composite forming process parameters in order to provide input data for the pre-processing of the final composite piece after polymerization and give the mechanical limits of the fabric during the forming process. This might be of great importance to help the process engineers ‘‘virtually’’ fulfil their planning before its physical realisation. A work is now in progress which, aims firstly to generalise the present model to include the shear behaviour of prepreg fabric and secondly the study of this composite shaping process by taking some structure’s data as random variables [39]. References [1] E.J. Barbero, Introduction to Composite Materials Design, Taylor and Francis, ISBN: 1-56032-701-4, 1998. [2] J.L. Billoët, Introduction Aux Matériaux Composites à Hautes Performances, Teknea, 1993. [3] P. Potluri, I. Parlak, R. Ramgulam, T.V. Sagar, Analysis of tow deformations in textile preforms subjected to forming forces, Composites Science and Technology 66 (2) (2006) 297–305. [4] R. Umer, S. Bickerton, A. Fernyhough, Modelling the application of wood fibre reinforcements within liquid composite moulding processes, Composites Part A: Applied Science and Manufacturing 39 (4) (2008) 624–639. [5] V. Frishfelds, T.S. Lundström, A. Jakovics, Bubble motion through non-crimp fabrics during composites manufacturing, Composites Part A: Applied Science and Manufacturing 39 (2) (2008) 225–243. [6] S. Belhous, A. Cherouat, J.L. Billoet, Two-component finite element model for the simulation of shaping prepreg woven fabric, in: 2nd Inter. Conf. on Integrated Design and Manufacturing in Mechanical Engineering, vol. 2, Compiègne France, 1998, pp. 523–530. [7] A.C. Long, Process modelling for liquid moulding of braided performs, Composites Part A: Applied Science and Manufacturing 32 (7) (2001) 941–953. [8] J.P. Fan, C.Y. Tang, C.P. Tsui, L.C. Chan, T.C. Lee, 3D finite element simulation of deep drawing with damage development, International Journal of Machine Tools and Manufacture 46 (9) (2006) 1035–1044. [9] R. Padmanabhan, M.C. Oliveira, L.F. Menezes, Deep drawing of aluminium–steel tailor-welded blanks, Materials & Design 29 (1) (2008) 154–160. [10] O.K. Bergsma, J. Huisman, Proceeding of the 2nd In. Conference On Computer Aided design in Composite Material Technology, Springer Verlag, 1988, 323–333. [11] H. Borouchaki, A. Cherouat, J.L. Billoët, GeomDrap new computer aided design and manufacturing for advanced textile composites, version 1, 1999. [12] H. Borouchaki, A. Cherouat, Draping of composite fabrics, Comptes Rendus Mécanique 331 (6) (2003) 437–442. [13] J. Wang, R. Paton, J.R. Page, The draping of woven fabric preforms and prepregs for production of polymer composite components, Composites Part A: Applied Science and Manufacturing 30 (6) (1999) 757–765. [14] F. Van Der Ween, Algorithms for draping fabrics on doubly curved surfaces, International Journal of Numerical Methods in Engineering 31 (1991) 1415–1426. [15] T. Ishikawa, M. Sushima, Y. Hayashi, T. Wei Chou, Experimental confirmation of the theory of elastic of fabric composites, Journal of Composites Materials 19 (1985) 3443–3458. [16] S. Kawabata, M. Niwa, H. Kawai, The finite-deformation theory of plain-weave fabrics, part I, part II and part III, Journal of Textile Institute 64 (1973) 47–61. [17] C.L. Lai, W.B. Young, Modelling the Fibre Slippage During Preforming of Woven Fabric, ICCM12, Paris, 1999. [18] J.L. Billoët, A. Cherouat, Mechanical and numerical modelling of composite manufacturing processes: Deep-drawing and laying-up of thin preimpregnated woven fabrics, in: Proceeding of the Inter. Conf. on Advances in Materials and Processing Technologies, AMPT’99 and 16th Annual Conf. On the Irish Manufacturing Committee, IMC 16, vol. 1, Dublin Ireland, 1999, pp. 329–346. [19] R. Blanlot, J.L Billoet, Loi de comportement orthotrope évolutive pour la simulation de la mise en forme des tissus composites, J.N.C. 9 (1996) 761–772. [20] J.C. Gelin, A. Cherouat, P. Boisse, H. Sabhi, Manufacture of thin composite structures by the RTM process: Numerical simulation of the shaping operation, Composites Science and Technology 56 (1996) 711–718. [21] M.L. Realef, C. Marcy, S. Backer, A Micromechanical approach to modelling tensile behaviour of woven fabrics, Use of Plastic and Plastic Composites: Material and Mechanics Issues ASME 46 (64) (1993) 285–294. [22] T.C. Lim, S. Ramakrishna, H.M. Shang, Strain field of deep drawn knitted fabric reinforced thermoplastic composite sheets, Journal of Materials Processing Technology 97 (1–3) (2000) 95–99. [23] P. Boisse, M. Borr, K. Buet, A. Cherouat, Finite element simulations of textile composite forming including the biaxial fabric behaviour, Composites Part B: Engineering 28 (4) (1997) 453–464. [24] P. Gilormini, P. Roudier, Abaqus and Finite Strain, Intern Report LMT Cachan n◦ 140 France, 1993. [25] F. Sidoroff, Cours sur les grandes déformations, Rapport GRECO n◦ 51, 1982. [26] Hibbit, Karlsson, Soresen, ABAQUS Theory, User’s Manual, 2005. [27] M.A. Crisfield, Non linear Finite Element Analysis of Solids and Structures, vol. 1, John Wiley & Sons, 1990. [28] G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. [29] B. Radi, O.A. Baba, J.-C. Gelin, Treatment of the frictional contact via a Lagrangian formulation, Mathematical and Computer Modelling 28 (4–8) (1998) 407–412 (special issue).

A. El Hami et al. / Mathematical and Computer Modelling 49 (2009) 1337–1349

1349

[30] O.A. Baba, B. Radi, J.-C. Gelin, An augmented Lagrangian treatment of the metal forming process, Mathematical and Computer Modelling 32 (2000) 1171–1179. [31] A.-S. Bretelle, M. Cocou, Y. Monerie, Unilateral contact with adhesion and friction between two hyperelastic bodies, International Journal of Engineering Science 39 (2001) 2015–2032. [32] M. Raous, Numerical methods for solving unilateral contact problem with friction, Journal de Mécanique Théorique et Appliquée 7 (1988) 111–128. [33] M. Raous, Friction modeling of a bolted junction under internal pressure loading, Computers and Structures 43 (1992) 925–933. [34] J.J. Kalker (Ed.), Three-Dimensional Elastic Bodies in Rolling Contact, Kluwer Academic Press, 1990. [35] J.T. Oden, J.A.C. Martins, Models and computational methods for dynamic friction phenomena, Computer Methods in Applied Mechanics and Engineering 52 (1985) 527–534. [36] P. Wriggers, Panagiotopoulos (Eds.), New Development in Contact Problems, Springer-Verlag, 1999. [37] F. Lebon, Contact problems with friction: Models and simulations, Simulation Modelling Practice and Theory 11 (5–6) (2003) 449–463. [38] A. Cherouat, B. Radi, A. El Hami, The frictional contact of the composite fabric’s shaping, Acta Mechanica (2008) doi:10.1007/s00707-007-0566-1. [39] B. Radi, A. El Hami, Reliability analysis of the metal forming process, Mathematical and Computer Modelling 45 (2007) 431–439.