A simplified finite element model for draping of woven material

S.B. Sharma, M.P.F. Sutcliffe*

Department of Engineering, Cambridge University, Cambridge CB2 1PZ, UK

Received 29 August 2003; revised 16 February 2004; accepted 24 February 2004

Abstract

A model for draping of woven fabric is described which is intermediate to the two main approaches currently used, kinematic modelling

and comprehensive finite element modelling. The approach used is a simple unit-cell finite element model, with the fabric modelled by a

network of simple truss elements, connected across the ‘diagonals’ by soft elements to mimic the shear stiffness of the material. Bias

extension tests on dry fabric are used both to calibrate and validate the model. The model is then used to simulate draping of an ellipsoid. The

capability of the model is illustrated by construction of a simple process optimisation map for this geometry.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: A. Fabrics/textiles; Carbon fibre; C. Finite element analysis; E. Forming; Draping

1. Introduction

Woven fabrics have been increasingly used in composite

material applications because of their attractive forming

characteristics and mechanical properties. To a first

approximation the material undergoes pure shear during

draping, with the tows rotating freely at the crossover

points—the pin-jointed net (PJN) approximation [1–4].

Advantages of this approach (termed ‘kinematic’ model-

ling) are simplicity and speed of calculation, making it

suitable for commercialisation [5]. However, it is likely to

be increasingly inaccurate for components which do not

have uniformly curved surfaces, and ignores fabric proper-

ties which are known empirically to play an important role.

For example, while it has been shown to model the broad-

brush deformation for a helmet component, it is not able to

model tow slippage or mimic details around the ‘ear’

features on this component [6,7]. It is expected that in-plane

membrane forces and normal tool contact forces developed

during draping will influence the changes in the tow

architecture and hence change the forming response and

final material properties [8]. Biaxial tests have been reported

by Boisse et al. [9] and Sharma et al. [10] to account for

the effects of membrane stresses on fabric deformation.

To model these effects it is necessary to use a more

sophisticated approach.

An alternative approach uses a comprehensive finite

element (FE) method to model the material behaviour

during draping. For example ESI PAM/FORM [11] uses

several layers of 2D elements to represent the fabric. This

program requires several input parameters such as the blank

holder pressure, fabric shear stiffness, tool velocity, ply/tool

and ply/ply friction coefficients. It has been successfully

used to simulate the interaction of the woven ply with

the tool as it isdeformed [12]. Although the method overcomes

most of the problems associated with the kinematic model, it

requires significant computer power and an experienced

engineer to drive the software. Examples of research focusing

on this approach are listed in Refs. [9,11–15].

The aim of the present work is to provide a drape model

intermediate to the above two approaches. An earlier proof-

of-concept study by the authors described how a kinematic

drape simulation could be adapted to include membrane

stresses [16]. However, an alternative approach, described

in this paper, has been pursued because of the greater

flexibility and ease of exploitation that it allowed. This

‘intermediate drape model’ is based on standard FE

algorithms, thus exploiting their ability to model complex

boundary conditions as well as their user-friendly input and

output interfaces. The model uses a novel unit-cell

implementation of the fabric, with material properties

derived from standard tests. The bias extension test is

1359-835X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compositesa.2004.02.013

Composites: Part A 35 (2004) 637–643

www.elsevier.com/locate/compositesa

* Corresponding author. Tel.: þ44-1223-332996; fax: þ44-1223-

332662.

E-mail address: mpfs@eng.cam.ac.uk (M.P.F. Sutcliffe).

used both to extract appropriate material shear properties

from a ‘gauge section’ where the PJN approximation is

good, and additional material parameters from consider-

ation of regions near the grips where the PJN model is less

accurate. The model is implemented in an FE code and

applied to draping of an ellipsoid.

2. Unit cell model

Fig. 1 shows the undeformed square unit cell of side

length D used to represent the material. The four outer

elements model the tows while the diagonal element endows

the cell with shear stiffness. The four tow elements are

connected at the corners via pin joints, to allow the trellising

action observed in practice. All elements are modelled using

two-noded 3D truss elements of area A: The strain 1 in both

types of element follows the Green–Lagrange definition

of strain used by the MARC.MSC FE implementation

adopted [17,18]

1 ¼u

D0

þ1

2

u

D0

� �2

ð1Þ

Here u and D0 are the corresponding extension and initial

length of an element. With this strain formulation and

assuming small extensions of the tow elements, the strains

are equal and opposite for the two diagonals of the unit cell,

so that the same material response can be used for the

diagonal element, irrespective or whether it is in com-

pression or tension.

Because it is difficult to measure the thickness of fabric

material accurately, estimates of the applied stress are not

straightforward. Instead we follow textile practice and

characterise the applied loads using forces per unit length, or

‘line forces’. This can be separated into components NL and

NS acting along the tows and in the shear direction,

respectively [10].

The tow elements are given an appropriate constant

stiffness EL to allow for decrimping or tow slip during

deformation, by fitting the simulation to the measured bias

extension deformation response, as described in Section 3.

Conversion between the element stiffness EL and the

corresponding effective modulus E0L of the fabric along

the tow direction is straightforward, requiring comparison

of forces over the unit cell. The effective modulus E0L is

defined by

E0L ¼

NL

1L

ð2Þ

where 1L is the strain along the tow direction. Since each

tow element carries a load equal to the product of the

applied load NL and the side length D of the unit cell,

Hooke’s law for the element gives

EL ¼NLD

A1L

ð3Þ

so that the tow element stiffness EL corresponding to a

fabric effective modulus E0L is given by

EL ¼E0

LD

Að4Þ

The effect of changes in geometry (e.g. large defor-

mations) on the tow elastic modulus are not included, nor is

the small contribution from the shear element to stretching

along the tow directions.

The diagonal element is given a non-linear stress strain

response sS ¼ f ð1SÞ; which is implemented here using a

specified stress–strain curve within the elastic–plastic

option of MARC. This curve is derived from the material

response Nx ¼ f1ð1xÞ; measured along a bias direction (see

Section 3), where Nx is the nominal line load applied in the

bias direction (i.e. applied force divided by original

specimen width) and 1x is the corresponding strain in the

bias direction. Again the element and corresponding

material responses can be related by considering forces on

the unit cell. As the unit cell is pin-jointed, the only

resistance to elongation along the bias direction is provided

by the shear element. Equating the external force applied to

the unit cell of initial widthffiffi2

pD with the force in the shear

element gives

Nx

ffiffi2

pD ¼ Ass ð5Þ

By using a nominal line load formulation, changes in unit

cell width do not need to be considered in the above

expression. Hence, the element response sS ¼ f ð1SÞ is

related to the material response Nx ¼ f1ð1xÞ via a simple

scale factor

ss ¼

ffiffi2

pD

Af1ð1sÞ ð6Þ

The ‘bracing’ nature of this element effectively gives the

material a non-linear shear stiffness. The exact form of

material response Nx ¼ f1ð1xÞ used for the shear elements in

the simulation is given in Fig. 4, derived from the material

response of the ‘gauge section’ of a bias extension test. In

practice the area of each element was set to unity,Fig. 1. Unit cell model of material behaviour.

S.B. Sharma, M.P.F. Sutcliffe / Composites: Part A 35 (2004) 637–643638

and the appropriate stiffness derived for the corresponding

mesh density and material properties using Eqs. (3) and (6).

3. Measurement of material parameters for a dry fabric

Fabric tests are described in this section which are used

to estimate values for the tow and shear springs of the unit

cell model. Material property data for the simulation are

based on measurements on Tenax HTA 6k carbon fibre dry

fabric, with 5% binder, in a five-harness satin weave,

manufactured by Hexcel Composites.

Picture frame and bias extension tests have been widely

used to characterise the shear response of dry fabric,

prepregs and thermoplastic fabrics, and results from the two

types of test have been compared recently by Sharma et al.

[10] and Harrison et al. [19]. In a picture frame test the

fabric is clamped in a square frame and deforms uniformly

as the frame is deformed into a rhombus shape. This test

provides a direct strain measurement but the stress

calculation is less reliable due to unknown tension in the

fabric. Although the picture frame test would provide

suitable shear data [19], here we use instead the bias

extension test. In this test the fabric specimen is elongated

along the bias directions (i.e. ^458 to the warp and weft

directions), as shown in the sketch of Fig. 2. The specimen

has initial length L ¼ 200 and width W ¼ 75 mm and the

grips are moved apart at a constant relative velocity

V ¼ 60 mm/min. Further details of the bias extension tests

used in the present work are given in Ref. [10]. The bias

extension test was chosen for the work because the axial tow

stresses are well characterised, allowing extraction of a

suitable axial tow stiffness, and indeed are probably more

representative of those present during draping. The non-

uniformities in shear arising in this test were treated using

suitable image analysis, and in fact provided an opportunity

to extract suitable material properties from different

regions of the specimen. Tests on the material at 60 and

100 mm/min produced similar results, indicating that an

effect of binder on the shear response was small. Although

some influence of binder cannot be ruled out, it is expected

to be relatively small, certainly in comparison with the effect

of matrix in prepreg material. The fabric showed some

asymmetry in shear (see for example the deformed speci-

men, Fig. 6). However, the effect appeared to be relatively

minor and has not been considered in detail.

If the deformation is modelled using pin-jointed analysis

then the constraints at the grips lead to rigid triangular

regions at the two ends of the specimen. The apex of these

triangles is termed here the ‘stationary point’, as it is

supposed using the pin-jointed analysis to remain stationary

with respect to the grip. The central gauge section of the

specimen undergoes uniform shear, while there is a

triangular ‘half-shear’ region towards the ends of the

specimen. The material response for this gauge section is

used to find an appropriate stress–strain curve for the

diagonal element in the unit-cell model (Section 2). The

deformation during the test of a grid marked on the central

50 £ 50 mm2 gauge section is recorded photographically

and these images are used to measure the shear angle u in

the mid-section. The longitudinal strain again follows the

Green–Lagrange definition [17], given by:

1x ¼ux

‘0

þ1

2

ux

‘0

� �2

ð7Þ

where ux is the gauge section extension measured from the

images and ‘0 is the gauge section length at the start of the

test. The nominal applied line force Nx is given by dividing

the measured force F by the initial specimen width W0

Nx ¼ F=W0 ð8Þ

The force–displacement response for the bias extension

specimen, over the whole length of the specimen, is shown in

Fig. 3, showing significant stiffening (lock-up) at a displace-

ment of around 60 mm. The corresponding nominal line

force–strain response, derived now from measurements only

over the gauge section, is given in Fig. 4. This shows an initial

low stiffness region, followed by a large increase in stiffness

at a strain of about 0.5, corresponding to a shear deformation

angle of about 408. The relationship plotted in Fig. 4 is used,

via the scaling factor of Eq. (6), to give the material

properties for the diagonal elements in the FE model.

Fig. 2. Sketch of the bias extension test.

Fig. 3. Typical force–displacement curve over the whole specimen length

for a bias-extension test.

S.B. Sharma, M.P.F. Sutcliffe / Composites: Part A 35 (2004) 637–643 639

In principle it seems reasonable to estimate the fabric

stiffness along the tow directions from appropriate tests of

specimens loaded along this direction. Although this approach

was tried, the values of tow stiffness derived were found to

give a poor fit to the data. It is believed that there were two

reasons for this; firstly that an appropriate stiffness should be

taken from only the initial decrimping part of the load–

displacement response (since this corresponds to the loads

typically found during draping), while in fact much higher

stiffnesses corresponding to higher loads and extension of the

fibres were measured. Secondly, it is likely that fibre slip may

play an important role in the bias extension tests used to

validate the model. Since the conditions in the bias extension

test are closer to draping conditions, the stiffness extracted

from these tests, as described in Section 4, was preferred.

4. Validation using bias extension test

Detailed analysis of the bias extension tests presented

here [8] has shown that the behaviour of the central gauge

section is well modelled using the PJN assumption, while

there is significant deviation from this model at the ends of

the specimen. While the gauge section provides the shear

response of the material, the overall deformation of the

specimen and the behaviour near the grips provides a way of

extracting the tow stiffness and helping validate the overall

model. To simulate the bias extension test, it is modelled

using the unit cell approach, taking a unit cell with diagonal

lengthffiffi2

pD ¼ 5 mm. The material used both for the

experiments and the simulation is the dry woven CFRP

described above. The FE specimen has the same size as the

real specimen (200 £ 75 mm2).

Fig. 5(a) and (b) plot the variation with end displacement

of the width and shear angle of the central gauge section.

The simulation results are shown for three values of the tow

stiffness, and compared with the PJN analysis. The

simulations results approach the PJN analysis at sufficiently

large values of tow stiffness E0L: Experimental results show

significant deviations from the PJN analysis at large

deformations. These deviations are well modelled using a

tow element stiffness E0L ¼ 15 kN/m. Further analysis

shows that the central gauge section is effectively behaving

in a pin-jointed manner [8], so that the observed deviations

from the PJN analysis are due to the behaviour near the ends

of the specimen (for example, the significant slippage

observed in Fig. 6).

A straight line with slope corresponding to this value of

tow stiffness is plotted on the load–strain response of the

diagonal elements Fig. 4, to allow a direct comparison

between the stiffnesses of the tow and diagonal elements.

This figure shows how, with a value of tow stiffness

E0L ¼ 15 kN/m, the shear stiffness of the material is small

compared with the tow stiffness until lock-up is approached

at a strain of about 0.5. At that point the material exhibits

significant shear resistance and departs from PJN behaviour.

A more critical test of the model can be made by

investigating the variation in shear angle along the length of

the specimen. As indicated in Section 3 above, there is a

region near the grips which is nearly rigid, while the central

region undergoes nearly uniform shear. This is illustrated in

the photo of the specimen after an elongation of 40 mm,

Fig. 6. Although the shear response of the material has been

extracted using the centre of the specimen, nevertheless

Fig. 4. Nominal line force–strain response taken from the gauge section of

a bias extension test. This function is used, via the scaling factor given in

Eq. (6), to describe the mechanical response of the shear element of the unit

cell. The dashed line shows a linear stiffness of 15 kN/m, to allow

comparison with the stiffness found appropriate for the tow elements.

Fig. 5. Comparison between measured and theoretical change in specimen

geometry as a function of overall specimen displacement U during a bias

extension test, (a) gauge section width W, (b) shear angle u in the gauge

section.

S.B. Sharma, M.P.F. Sutcliffe / Composites: Part A 35 (2004) 637–643640

the significant non-uniform deformation which occurs near

the grips is sensitive to the in-plane response of the fabric

and acts as a useful test of the overall material model. Fig. 7

compares measurements and predictions of the variation of

shear angle along the centre line of the specimen after an

extension U ¼ 40 mm. Experimental points are taken from

the photographic image, Fig. 6, tracing individual tows in

the deformed specimen. The scatter in the measurements

reflects the difficulty in estimating the shear deformation

from the photograph. In the centre of the specimen, at the

right hand side of the graph, the shear deformation is nearly

uniform. Near the grip on the left hand side of Fig. 7 there is

very little deformation. Predictions using the PJN model

give the dashed line rising abruptly at the ‘stationary’ point.

Fig. 7 shows that inclusion of a more realistic material

model, with again a tow element stiffness E0L ¼ 15 kN/m,

allows a good prediction of the shear response through the

specimen.

5. Application to draping over an ellipsoid

The intermediate drape model described in this paper is

able to take into account processing parameters (e.g. blank

holder forces), taking advantage of standard FE capabilities,

while being sufficiently rapid (around 350 s per simulation)

to permit the many calculations needed when searching for

optimum forming conditions. Here we illustrate the value of

this approach by considering draping of an ellipsoid, again

using an MSC.MARC implementation.

5.1. Description of drape model

The geometry is sketched in cross-section in Fig. 8a, and

as an isometric view of the deformed geometry in Fig. 8b.

The ellipsoidal punch has depth 200 mm and major and

minor axes of length 250 and 200 mm, respectively. The

sheet of material, which is initially square with side length

600 mm, drapes around the rigid punch as it is driven

upwards, making contact with the punch guide at the edges.

The travel of the punch after first contact with the sheet is

equal to 150 mm. Two geometric parameters associated

with the ply orientation are considered, as illustrated in

Fig. 8c. Rotation of the nominally ^458 ply relative to the

axis of the ellipse is given by the ply orientation angle f;

while some pre-shear in the fabric, perhaps generated by in-

plane forces prior to draping, is modelled by a pre-shear

angle a: The material model for dry woven CFRP described

above is again used for the drape simulation, with 1600 unit

cells of side length D ¼ 15 mm. Two values of the tow axial

stiffness E0L of 15 and 8000 kN/m are used. The sheet is held

in place at its edges by springs, which here are used to model

blank holder forces. These springs are given an effective

stiffness of 200 kN/m. A Coulomb friction coefficient of 0.2

is used at all contacts.

5.2. Simulation results

Fig. 9(a) shows the effect of tow stiffness on the variation

of shear angle along the major axis of the ellipsoid. The arc

length has been normalised by the half-perimeter of

the ellipsoid, so that a value of one corresponds to the

Fig. 6. Bias extension specimen after an extension U ¼ 40 mm, illustrating the grid used for measuring deformations outside the gauge section.

Fig. 7. Comparison between measured and theoretical variation of shear

angle u with position along the centre line of the bias extension specimen,

U ¼ 40 mm.

S.B. Sharma, M.P.F. Sutcliffe / Composites: Part A 35 (2004) 637–643 641

base of the punch. For the lower value of stiffness of

E0L ¼ 15 kN/m, corresponding to the value found appro-

priate from the bias extension tests, the peak shear angle

occurs just outside the contact between the ellipsoid and the

punch, taking a value of around 368. The effect of much

stiffer tow stiffness is to increase the shear in the fabric, as

stretching of the fabric becomes more difficult. Fig. 9(b)

illustrates the effect of ‘pre-shear’ in the fabric, plotting the

change in shear angle along both the major and minor axes

of the ellipsoid. The case of no pre-shear corresponds to a

^458 layup, relative to the major axis of the ellipsoid. The

case with a ‘pre-shear’ angle of 108 represents a uniform

shear deformation of the fabric by 108 prior to draping. This

might be associated with in-plane stretching prior to draping

and is implemented simply by taking an initial ply

orientation of ^408 relative to the major axis. Again the

maximum shear angle occurs just outside the contact

between the punch and the material. Application of 108 of

pre-shear causes the position of maximum shear to switch

from the major axis to the minor axis. Fig. 9(b) suggests that

choosing a pre-shear angle of around 58 will lead to a value

of maximum shear angle below both the no-shear and 108

pre-shear cases.

5.3. Process optimisation

Here we illustrate the value of the proposed FE model, by

exploring the effect of ply orientation angle and initial pre-

shear angle on the maximum shear induced when draping

the ellipsoid. Fig. 10 shows a contour plot of the maximum

shear angle in the component as a function of pre-shear and

ply orientation angles. The maximum shear angle has a

minimum when the bias direction is aligned along the major

axis and with about 48 of pre-shear. Pre-shear has a greater

influence on the maximum shear angle than does ply

orientation angle. Similar process optimisation maps could

equally be produced for other process parameters such as

Fig. 8. Details of ellipsoid drape simulation; (a) arrangement seen in cross-

section, (b) isometric view of mesh, (c) definition of initial ply geometry

and orientation.

Fig. 9. Effect of (a) tow stiffness and (b) pre-shear on the variation along the

major and minor axes in the total shear in the material after draping an

ellipsoid. The distance along the arc is normalised by the arc length from

the centre to the edge of the ellipsoid.

S.B. Sharma, M.P.F. Sutcliffe / Composites: Part A 35 (2004) 637–643642

blank holder forces or material properties (perhaps associ-

ated with changes in the pre-preg or with modifications to

the mould temperature during forming).

6. Conclusions

A simplified FE model is proposed to model draping of

composite material. The unit cell consists of a network of

pin-jointed stiff trusses, with shear stiffness introduced via

diagonal bracing elements. Suitable material properties are

extracted from bias extension tests measurements on a dry

CFRP fabric. Measurements of the deformation in the

central gauge section during a bias extension test are used to

extract a suitable stress–strain curve for the diagonal

elements. The stiffness of the trusses is estimated by fitting

bias extension measurements over the whole specimen

length. Predictions of the deformation of a bias extension

specimen away from the gauge section are in good

agreement with measurements, confirming the ability of

the unit cell model to represent the effect of in-plane forces

on shear deformation and tow slippage. It is expected that

this approach would also be applicable to pre-preg woven

material. In this case the flexibility of the model allows

incorporation of the more sophisticated shear response

needed for such fabrics, for example including rate-sensitive

terms. The material response would need to be extracted

from experiments or micromechanics-based models [20].

The model is used to predict the draping of an ellipsoid and

to illustrate how it might be used for process optimisation.

The combination of flexibility in the FE approach and speed

of computation allows optimisation studies to be undertaken

with only modest computing time.

Acknowledgements

The authors gratefully acknowledge the contributions

from the Universities of Nottingham and Leeds, EPSRC,

BAE SYSTEMS, QinetiQ, ESI Software, Ford Motor

Company Ltd, MSC.Software, Saint-Gobain Vetrotex, BP

Amoco and Hexcel.

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S.B. Sharma, M.P.F. Sutcliffe / Composites: Part A 35 (2004) 637–643 643