Accepted Manuscript

Compaction behaviour of dense sheared woven preforms: experimental observations and analytical predictions

Dmitry S. Ivanov, Stepan V. Lomov

PII: DOI:

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S1359-835X(14)00128-6

http://dx.doi.org/10.1016/j.compositesa.2014.05.002 JCOMA 3617

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Please cite this article as: Ivanov, D.S., Lomov, S.V., Compaction behaviour of dense sheared woven preforms: experimental observations and analytical predictions, Composites: Part A (2014), doi: http://dx.doi.org/10.1016/ j.compositesa.2014.05.002

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COMPACTION BEHAVIOUR OF DENSE SHEARED WOVEN PREFORMS: EXPERIMENTAL OBSERVATIONS AND ANALYTICAL PREDICTIONS

Dmitry S. Ivanov*1, Stepan V. Lomov2

Advanced Composites Center for Innovation and Science (ACCIS), University of Bristol, 2 Department MTM, KU Leuven, Belgium Corresponding author: dmitry.ivanov@bristol.ac.uk ABSTRACT

Yarn-scale modelling of textile preforms relies on the realistic reconstruction of internal geometry. The paper presents a model of fabric compaction, which shows that fine features of dense woven preforms, conventionally neglected in consolidation analysis, have a significant impact on thickness and consequently on fibre volume fraction and yarn crimp inside the composite. The model relies on single yarn compaction test and geometrical characteristics of preform in compacted and original configurations. The model is validated against compaction experiments on dense sheared single ply and nested carbon twill 2/2 fabrics. This exercise aims at decreasing or eliminating phenomenological parameters being introduced when calibrating geometrically and physically complex numerical models at the yarn scale.

is; C. A

KEYWORDS

A. Fabrics/textiles; C. Analytical modelling; E. Consolidation; E. Preform 1. INTRODUCTION

e of the essential control issues of resin infusion is the behaviour of draped preforms in ompression. In contrast to the RTM process where a rigid upper mould is typically used, a flexible film constraints the impregnated material when vacuum pressure is applied. Pressure imposed by the film is sometimes complemented by added weight. Drape affects the yarn architecture and the fibre volume fraction (FVF), leading to a variation of preform properties when it is laid on a tool. Reconfiguration of yarns results in substantial increase in thickness of preform and yarns, as measured, for instance, by Chang et al [1] for carbon satin fabric by

optical microscopy, by Lomov et al [2] for carbon NCF preforms by a contact device in a picture frame test, by Putluri et al [3] for glass plain weave by examining cross-sectioning of sheared composites, and by Badel et al [4] for glass plain weave by X-ray tomography of deformed fabrics.

Changes in yarn architecture cause evolution of composite compressibility and, as a result, may cause non-homogeneity in component thickness or excessive tooling constraints in rigid mould processes. A strong coupling between the in-plane drape deformations and out-of plane compaction response is clearly demonstrated by models of shear and biaxial deformation of woven fabrics as proposed by Lomov et al [5], Charmetant et al [6], and Nguyen et al [7]. Preforms consolidated at low pressure are also susceptible to dimensional defects and exhibit higher yarn crimp. Thus, understanding preform deformation under compression is essential both for prognosis of the dimensions of manufactured components (particularly relevant for thick parts) and for optimising composite performance.

The major challenges of preform modelling are the intrinsic complexity of textile preforms, high compliance of dry fabrics and yarns which leads to uncertainty in its geometrical characteristics, high sensitivity of material to external loads and handling, non-elastic history-dependent deformations of fibrous media, high scatter of properties, and multiple possible deformation modes at the yarn scale. It appears to be rather challenging to predict the results of even a simple compaction test based on the properties of constituents: fibres and yarns. ft

The prediction of compaction behaviour was attempted with various analytical and finite element

dels. Grishanov et al [8] described interaction of two component yarns based on the tensile, bending, and compressive energy of individual strands. Geometrical and topological considerations allowed Chen and Chou to construct analytical models for single layer [9] and multi-ply woven preforms [10] where compaction response at high pressures is primarily caused by yarn bending. Lee et al [11] implemented numerical visco-elastic model of woven fabric

compaction at the yarn-scale to describe both compression and relaxation of the fabric.

To be predictive, the yarn scale modelling demands well-adjusted yarn properties and calibrated yarn geometry. The yarn properties can be determined either by micro-mechanical models, by direct measurements, or by inverse property identification methods. There is a long history of micro-structural models for non-wovens with randomly oriented fibres through bending and contact-slip mechanisms of fibre interaction, originated from van Wyk [12]. Similar models for unidirectional fibre bundles, where friction and Hertzian contacts prevails over bending [13], are not yet in mature state. Promising numerical simulations of unidirectional fibre bundle at the fibrous scale were demonstrated by Sherburn [14].

inverse property identification has a significant potential as it allows relaxing requirements on the accuracy of geometrical model of preform. Charmetant et al [6] used equibiaxial tensile tests on woven fabric to identify the effective compaction response of yarns and validated the yarn scale finite element model against material response in the in-plane shear test. Nguyen et al [7] used a similar scheme and predicted the compaction response of sheared and nested preforms with a good accuracy.

The direct measurements of yarn properties (compaction, bending, friction, tensile tests) remains the tempting route as it enables the prediction of the compaction response for various preforms based on geometry-independent properties. Lomov and Verpoest [15] suggested an analytical model of compressibility where the fabric geometry is calculated based on the minimisation of energy of interacting yarns. This model, fed by bending and compression properties of single rovings, was validated against the compression tests on glass plain woven fabrics. This model

s then generalised by Lomov et al [16] to analyse the compressibility of nested multi-ply preforms. The downside of the direct measurements is a need in sophisticated calibration of the yarn scale geometry. Potluri and Sagar [17] successfully used energy-based method to predict compressibility of plain weave 2D and 3D woven preforms.

Nearly all the yarn scale models presented in literature describe relatively sparse fabrics without

a pronounced lateral interaction of yarns. As have been shown in [18], a different approach may

intact in

be required for the dense and draped preforms. In dense preforms lateral jamming of parallel yarns leads to a pronounced evolution of the compressibility. The current paper discusses the compaction mechanisms and significance of geometrical factors resulting from lateral contact in the material response to applied load. A model, predicting thickness as function of internal geometry and processing pressure for dense sheared nested preforms is suggested. Following the established approach for the analysis of fabric deformations [e.g. 5, 8, 17, 19] it utilises the energy method. The major feature of the suggested concept is that the lateral yarn interaction is explicitly included into the energy balance in a pragmatic way. The model departs from a schematic representation of lateral jamming leading to the estimation of fabric compaction limit (thickness at high pressure) and the actual fibre length at this state. The parameters in a function approximating the yarn midline at various compaction levels are then adjusted to keep the fibre length constant throughout the compaction process. This allows a simple derivation of the preform compaction curve obtained from a direct test on yarns extracted from fabric, as well as the geometrical characteristics of the preform. The results are validated against the pressure-thickness curves obtained from compaction of sheared preforms. The primary goal of the model is to prioritise the importance of various geometrical and mechanical factors when creating multi-parametric, topologically complex, highly non-linear structural models.

2. EXPERIMENTS ON DRY PREFORMS

For clarity of arguments the paper discusses one fabric type: carbon 2/2 twill woven fabric with al density of 370 g/m2, and ends/picks count of 2.36/cm (the number of yarns per cm). This is a standard aerospace grade fabric. Epoxy binder is applied on its surface to enhance dimensional stability. Yarns have fibre count of 12K, linear density 810 tex, the initial thickness of yarns varies in the range of 0.16-0.22 mm and the width is approximately 5-7 mm. Note that the yarns are wider than the average distance between yarn centre lines (4.2 mm). The experimental program described below includes fabric characterisation in its pre-sheared and deformed

configurations, the compaction tests for the sheared single-ply and nested preforms as well as the tests on individual yarns extracted from these fabrics. Thickness of unsheared and sheared preform

The initial configuration of the unsheared and sheared fabric prior to compaction is studied laser and optical methods. Prior to the measurement the fabric is fixed in a picture frar commonly used for shear resistance measurements ("KUL picture frame" described in [20, 21]). The picture frame set-up presents a metal frame with hinges in the frame corners. The cross-like sample of textile is gripped at the edges. The frame is mounted on the tensile machine (Instron 4467, 1 kN load cell) so that the pulling movement of the machine causes the frame to be closed, the frame sides rotate on hinges, and, eventually, textile is subjected to a state, which is close to pure shear. To characterize the changes of the fabric internal configuration imposed by the gripping tension, the fabric was speckle-painted to eliminate reflection of the shiny carbon yarns, and images from two cameras controlled and processed by LIMESS digital image correlation system were used to measure the surface profile of the fabric. Thickness laser scan measurements however were conducted on dry non-painted preforms.

When fixed in the picture frame, the fabric is stretched to ensure that the textile lies in one plane. The tension is realized by wavy surface of the grips, which are brought together by the screws. The mechanisms of pretension and its influence on the results of "KUL picture frame" test are described by Lomov et al [21, 22]. In the current study gripping is used to reduce the scatter of thickness measurements by eliminating random movements of compliant preform. The evolution

he surface as the result of gripping is shown on Figure 1. The 3D surface is calculated using commercially available digital image correlation software Vic3D. It can be seen that the geometry of the fabric evolves with the gripping. The longitudinal tensile deformations in the yarn direction are measured along the yarn midlines to characterize the stretching of the fabric. 25 locations in warp yarns between the closest cross-overs with the weft yarns are investigated.

The measurements give the following average and standard deviation of stretching deformations in the direction of fibres after gripping: 0.10 ± 0.09 %.

After gripping the preform thickness is scanned using laser sensors. Laser sensors Acuity AR700-4 are set on both the fabric surfaces. The sensor measurements are based on the optical triangulation method: they measure distance by projecting a beam of laser light that creates a spot on a target surface with a fixed dimension. Reflected light from the surface is viewed from an angle by a detector array. The target's distance is calculated from the image pixel data using the sensor's microprocessor. Laser spot size is 70 |im and resolution, defined as smallest increment of change in distance that a sensor can detect, is 5.1 |m. The sensors point at the same location and are calibrated for thickness measurements. The lasers can be moved both vertically

and horizontally. The vertical movement is realized by a carriage with a constant pre-defined speed, the horizontal movement is implemented by a manually driven spindle.

To measure thickness evolution as function of shear angle, the test is paused at a pre-defined shear angle, then textile thickness is scanned in the vertical direction through the length of the sample along three randomly chosen vertical lines - Figure 2a. The average value of the obtained profile is taken as the "true" thickness of the textile. The load cycle is repeated two-three times to check if hysteresis affects the thickness of sheared fabric. The samples were sheared to 50°, then unloaded and sheared again with the same procedure of thickness measurements. The results are shown on Figure 2b.

results The thi ^oft

ickness of all tested samples follows a similar trend up to 30-40° when the global buckling of the fabric disrupts the measurements. A substantial change in thickness values is seen for the twill: 0.4-0.6 mm for 0° and 0.5-0.7 mm for 30°-shear. The experimental curves on Figure 2b are compared with a constant fibre volume fraction fit, which shows thickness evolution of a fabric preserving a constant volume while being sheared. Comparing the fit with the experimental

curves of the same initial thickness (samples 2 and 3), it can be suggested that there is a certain increase of initial fibre volume fraction as the shear angle increases. In contrast to the force required to shear the fabric [23], the thickness evolution is not much affected by repeating the loading cycle. Even though a substantial variation of the test results can be seen (sample 2,3 and samples 1,4 on Figure 2b), thickness measurements at the second cycle nearly repeat the curves of the first one. This indicates that the experimental scatter between different samples is attributed to intrinsic material variability and, possibly, to clamping pressure

variation rather than to a thickness measurement error.

To test the compaction response of sheared preforms the cruciform samples were manually sheared in a square metal frame to a predefined angle and then compacted in a 5 kN load cell machine by a 7 cm diameter circular plate on a rotating pivot - Figure 3. The same test configuration was used in Vanclooster et al [24] for studies of compressibility of sheared steel fibre knitted fabrics and in Pazmino et al [25] - for 3D woven glass fabrics. The compaction tests were conducted at room temperature. At these conditions the binder is not activated which allows exploring the preform behaviour in isolation from the process of chemical binding and thus, focusing on the role of preform architecture in the compaction process. It is believed that the basic features of the compaction process and geometry reconfiguration remain similar even at elevated temperatures.

After shearing the compaction pressure was increased to about 1.2 MPa. The machine

mpliance is measured prior to the test and the machine displacement at a given load is subtracted from the measured displacement in the post-processing of the results. Each sample was cycled three times in loading/unloading. It has been found that after the second cycle the material behaviour does not evolve substantially. Compression curves are recorded for 0°, 10°, 20° and 30° fabric shearing. Single fabric layers and the double layer stack are tested. Three

samples for every material configuration have been used to verify the test repeatability. A

moderate scatter of the compaction curves has been observed - for a given pressure the thickness generally varies within a band of 0.01-0.03 mm. The variation at larger shear angle is typically higher than at lower angles.

The compaction data are shown in the next paragraph where they are compared to an analyti

model of preform deformation - Figure 4, Table 2. All the pressure-thickness curves follow a

and alm

characteristic asymptotic pattern: rapid hardening from the pressure 0.05.. .0.1 MPa and almost no change in thickness at pressures higher than 0.4 MPa. It is found that the in-plane s hear deformations substantially affect the compressibility of the fabric. Even at high pressure levels (1 MPa) there is 20% difference between the thickness of non-sheared preform and the one sheared to 30°. The difference is higher at lower pressure levels.

As well known, nesting in two ply preforms increases fabric compressibility. Average thickness of ply in nested two ply preform is about 5-10% lower compared to the single ply preforms. Geometry of compacted preforms

In the process of the compaction and

spacing is reduced since the distance between the yarn central lines is proportional to the cosine of the shear angle. Yarn may spread due to the compaction. Hence, at some level of deformation yarn width may exceed yarn spacing. This causes peculiarities of fibre path: jamming of parallel yarns, pronounced variation in width along yarn, and side crimp of yarns. These deformations are typical for dense fabric. Figure 5 shows the compacted geometry of the twill fabric. At the locations of cross-overs, where yarns of one direction dive under the yarns of another direction,

fibre bundles are laterally constrained and their width is decreased at the expense of local thickness increase. As a result, the fibres at cross-overs are packed stronger than elsewhere. The projections of neighbouring yarn on the fabric plane overlap. The overlap regions increase the local amount of fibres per unit area. This limits the fabric compressibility and, hence, this structural feature is important for the correct yarn-scale analysis. The increase of thickness of

sheared preform, Figure 2b, is an indication of this process.

3. EXPERIMENTS ON DRY YARNS

The behaviour of yarns extracted from twill fabric was explored in a flat compaction test on glass table with a camera monitoring transverse expansion through the glass. A 1 kN load cell was mounted above a 50 mm diameter metal punch, which allowed 1 MPa pressure to be exceeded. Parallelism of the punch and the supporting glass was verified in an experiment with a pressure sensor placed between the indenter and the compaction base. The homogeneity of pressure distribution over the compacted surface has confirmed a satisfactory level of indenter alignment. Machine compliance was measured and eliminated in the result post-processing. Yarn widening was measured using digital image correlation (Vic 2D software), which follows the evolution of a grey-scale speckle pattern naturally created by the epoxy binder. The strain averaged over the width of the yarn is taken as the measure of the expansion. The transverse strain field on the surface of the yarn was fairly uniform in all the tests. Three loading-unloading cycles were conducted to explore the irreversible deformations of the yarns.

The transverse expansion of a yarn hardly exceeds 1% of initial width. It has to be noted that the yarn in preform may be subjected to constraints which are not necessarily reproduced by the conditions of the compaction experiment on extracted yarns: there is no lateral constraints in the yarn compaction test and tool-yarn friction is likely to be higher compared to the inter-yarn friction. Both the factors may lead to a different yarn spreading in the compaction of single yarns and yarns in fabric. The lack of spreading may also be owed to low twist of the yarns, which a weak coupling between spreading and compaction deformations of yarns. It is resting to note that a spreading limit is also observed in prepregs [26], where high shear deformation at the yarn edges result in a state of flow locking. Barnes and Cogswell [27] suggested that the fibre misalignment and non-uniformity of deformation under no-slip condition results in twist of fibres at the yarn edge, which prevents the transverse shear flow.

The yarn compaction curve shows asymptotic hardening with a limit compaction state

ami \ a i i

leads to inte

(—0.13 mm) achieved at ~0.4 MPa - Figure 6a. In consequent loading the limit deformations are

increased by ~5 % from the first to the second load cycle, and ~2 % from the second to the third cycle.

to describe ween the lo^

ibe the

ytical

To use this data in further analysis, the yarn behaviour needs to be approximated by an analytical function. The curve response is nearly linear at two loading stages: in the very beginning of compaction and upon approaching the limit deformations. Hence, it is rational to des yarn stiffness evolution E by a bilinear function with a smooth transition between the low

E1y and high E2y stiffness at intermediate deformations. A hyperbolic function is suitable for this purpose:

Ey = 2 E - E2y)- 2 (e2y + Eiy )tanhi

£ = ln

îi ' v'0,

measure

is the logarithmic strain chosen as a measure of compaction deformation, et is the

transition strain and S is a curve shape factor. Integrating this function over strain gives an appropriate pressure-strain dependency:

Py =1 (Ely - E21 £-15 (ei + Eli )ln

i ( £^

cosh\ —-

cosh\ £ S

where Py is the nominal pressure normalised by nominal width and punch diameter. The width

is measured on the surface of compacted fabric in the locations between the cross-overs where it spreads the most: w = 6.3 mm. To define strain, an initial yarn thickness needs to be set. In practice it is uncertain and cannot be precisely measured due to yarn crimp, high compliance of fibre bundles and non-uniformity of the yarn surface. By choosing initial yarn thickness as t0 = 0.31 mm (relatively arbitrary choice based on the first reliable values of thickness detected in the yarn compaction testing) the following parameters are derived from the approximation of average experimental stress-strain curves: E1y = 42.6 kPa, E2 y = 429.1 GPa, S = 0.15, et = 1.67

I configu

;tion to —

(with 95% of confidence interval). It should be noted that all these parameters are not related to the actual properties of fibres but used here to provide a suitable fit for the function. The resultant approximation is shown on Figure 6a, and can be judged as good for a practically interesting range P > 0.05 MPa Intra-yarn fibre volume fraction can be estimated using width/thickness measurements and the presumed shape of the yarn. Figure 6b shows the evolution of the fibre volume fraction as the function of transverse strain. The characteristic value of 70% at the first cycle is achieved at the maximum compaction under assumption that the yarns shape remains elliptical at all deformations. Repeated cycles increase the fibre volume fraction to —75%.

4. KEY EXPERIMENTAL OBSERVATIONS The results demonstrate that:

- there is a high variability in the initial material configuration of the sheared fabrics - Figure 2a. At high pressure levels the scatter is significantly smaller - Figure 4.

- the compaction limit, i.e. the state where preform thickness becomes nearly invariant to pressure increments, explicitly depends on shear angle.

- the primary mechanism for through-thickness yarn deformations, compacted in isolation from the fabric, is yarn densification. Only minor transverse deformations of yarns have been observed. The limited spreading in the test allows establishing the packing relation between applied pressure and fibre-volume fraction evolution.

orm compressibility depends on the internal geometry of fabric. The limit thickness of individual yarn (—0.13 mm) is lower than the half of fabric limit thickness - 0.15 mm at 0° shear and 0.18 mm at 30°. This can only be the case if the yarn width exceeds the inter-yarn distance and the yarn overlaps with neighbouring yarns of the same direction - Figure 7a,b. In other words, the weft yarns are squeezed between two parallel warp yarns and vice versa.

- average thickness of a ply in a two-ply nested pack of twill is about 5-8 % smaller than in the single ply preform. As shown by Lomov et al [16], in n-ply preform the effect is larger in

- prefo

proportion to (n-1)/n, which increases the effect to 10-15% for plain weave fabrics. Nesting is more pronounced for the sheared fabrics due to the elevated roughness of fabric surface.

5. MECHANISMS OF FABRIC COMPRESSIBILITY The difference in compressibility of sheared preforms points at the role of preform geometry in compaction mechanisms. The role of fine geometrical features becomes particularly explicit when considering compaction at high pressure levels. At this deformation stage no significant yarn reconfiguration occurs and hence, no other factors but direct yarn compaction co ntribute to the mechanical response of fabric. A realistic model has to be able to describe yarns reconfiguration which leads to the observed difference in thickness of sheared and non-sheared preforms.

The current paragraph describes an analytical model which explains the observed phenomena. The original motivation is to validate a basic assumption about deformation mechanisms in order to set robust requirements for more elaborated numerical models. The presented model is based on geometry of compacted material and compaction response of single yarns. Starting with the notion of yarn compaction limit and geometry of fully compacted fabric, the model then takes into account yarn bending to explain the compaction at low and moderate pressures. The measurements of pressure-fibre volume fraction relation in combination with the measurements of yarn width in compacted fabric give the basic information required for constructing the model. In making an estimation of the final fabric thickness accounting for the yarn overlap, simple assumptions can be utilised: (1)

Following the notion by Harwood et al [28], it is stated that the compressed yarn has a compaction limit. It is determined at pressures exceeding 1 MPa and it is assumed that the thickness change above this pressure level is negligible. From experiments on single yarns

during the first cycle: tcy = 0.13 mm.

(2) In contrast to the yarn width at the cross-overs, the yarn width measured on the surface of compacted single ply preform does not change during shear, compaction or nesting: w = 6.3 mm.

(3) The inter-yarn distance (defined as distance between yarn centrelines) for the sheared configuration is estimated to be p cos( y), where p = 4.2 mm is the yarn spacing in orthogonal fabric and y is the shear angle. The yarn spacing is assumed to be invariant to the degree of compaction.

(4) The thickness of fabric at the compaction limit tcp is defined by the thickness in the centre

of overlap zone, which can be estimated based on the presumed section shape. This assumption is valid only for dense fabrics, where overlap zone is fully filled by fibrous material. Obviously, for the sparse fabrics (a minor overlap or a gap between the yarns) the densest location is at the intersection between warp/weft midlines. For the sake of clarity the further discussion is limited to the fabrics where the overlap is the densest region in the material. However, all the derivations below can be easily generalised for sparse fabrics as well.

ness in the c

As a result the estimation of thickness at high pressure can be condensed to the following expression:

thickness

tc =XtCy

where tc is the thickness of preform at the compaction limit, tC - is the yarn compaction limit and x - is the thickness excess factor for dense fabrics, which characterises the thickness of ply

in comparison to the thickness of a single yarn. The thickness excess factor for single fabric

eform can be determined from the analysis of overlap cross section. The thickest location is half-way between the yarn edges. It is constituted from one whole yarn (in-plane of Figure 7b) and edges of two yarns perpendicular to it. Yarn cross section can be approximated by an elliptical shape, which gives the following estimate of the excess factor:

X = 1 + 2

p cos( y )

It has to be emphasized that (4) is valid only when the excess factor exceeds 2, i.e. 2

w > wdense = —;= p cos(Y). For smaller or no overlap case, the shift in the location of the densest a/3

region has to be considered. For w < wdense, the excess factor is defined by the location of war weft intersection where x = 2.

The fabric consists of as fully-loaded dense overlap locations and under-compressed zones, which constitute the major fraction of fabric surface. In a nested multi-ply preform, a coincident in-plane location of fully-compacted zones in interacting plies is improbable. Hence, it is logical to suggest that the total excess factor for nested preform x can be derived from a simple superposition of under-loaded and fully-compacted areas. Normalised by the number of plies, the excess factor is:

+1 (5)

The values of the excess factors and the overlap area fraction for sheared single ply of the studied twill fabric and nested fabrics are summarised in Table 1. It can be seen that for the given geometrical parameters the overlap area is in the range of one quarter - one third of fabric area.

Compared to a sparse fabric, where the preform thickness is double of yarn thickness (x = 2),

the excess factor for the dense preform are 47-62% higher. Nesting decreases the excess to 2432%.

Despite crudeness of the utilised geometrical assumptions, the model provides sensible results. It shows not only the trend but also predicts the compaction limit with satisfactory accuracy for the range of considered shear angles - Table 2 (compaction limit section).

To cover the entire pressure range, the analytical compaction model has to be generalised. For pressure levels typical for flexible moulding processes the compaction response is determined not only by the preform geometry but also by the mechanical factors such as yarn bending, inter and intra-ply friction. In the next paragraph the concurrent yarn bending and compaction are

discussed in greater details to explore the relevance of this factor in the compaction response. The exact derivations depend on the twill interlacing pattern of the fabric being used as an example, but they can be generalised for any 2D weave.

6. ANALYTICAL MODEL OF PREFORM COMPACTION

Utilising classical beam theory, the balance of external work and deformation energies for a single yarn in the fabric can be written in the following form:

D 4d 1 4d

— {№+4d i -(<: - t. >. = 0

where the first term is the normalised potential energy of bending, the second one is the specific energy of yarn compaction, and the third one is the work of external forces, u is the through-thickness displacement of the yarn midline, t.0, t. are the initial and the current amplitudes of the

yarn mid-line, ty are the local yarn thickness, P. W applied pressure, Py,£y are the through-

thickness stress and strain due to compaction and D is the yarn bending rigidity. For simplicity the bending stiffness is approximated as the sum of bending rigidities of all fibres D = :.25nEfNfrf (Ef,Nf,rf stand for fibre stiffness, number of fibres in yarn, and fibre

radius) in the assumption that inter-yarn friction is small and twisting effects are negligible in bending response, i.e. all the fibres in bending act separately. This approximation must give the lowest estimate of the bending contribution to fabric response. In [29] it was found to work well for typical carbon tows, including the ones discussed in this paper. A quadratic empirical endency of D over linear density of glass rovings can be found in [21]

Instead of a conventional solution routine, where the displacement of a yarn u is found in a boundary value problem, a simpler approach is envisaged. Yarn mid-lines are approximated by an analytical function. The coefficients of this approximation are calculated for a given amplitude t. based on the assumption of fibre incompressibility and, consequently, the vertical displacement u( t.) can be calculated for any point along the fibre length. Hence, the first two

for typic dep

terms of (6) can be explicitly integrated and the applied pressure Pa is then determined for a

rns. It

given yarn mid-line amplitude ta.

Yarn movement in an interlaced fabric is limited due to the constraints imposed by other yarns. It is fair to assume that the only difference between yarn paths in the deformed and initial configurations are (a) the vertical (through-thickness) displacement of yarn mid-line and (b) a certain shape adaption required for preserving the constant length of inextensible fibre. It is convenient to approximate yarn mid-line in both the configurations by a simple sigmoid function (Figure 8): straight intervals at the preform face and smooth transition between the front and back faces at the cross-overs. For the span of one unit cell in the twill fabric this approximation can be written in the following form:

y(t) = 2

( ( tanh

Z(ta),

- tanh

: amplitui

yarn thicknesses, 4p - the unit cell size of twill, p and 3p are the cross-over coordinates corresponding to the twill pattern, Z - is a coefficient responsible for the shape of the mid-line path, which evolves as the function of fabric thickness. The span of the yarn path transition from top to bottom fabric surface is estimated to be 4Z - Figure 8 ( y( p + 2Z ) = 0.96 y( 2 p) ).

The invariance of fibre length to compaction deformations gives a condition to derive Z . For small crimp the length of the curve can be defined as:

de and de and

f x-3p ^

VZ(tah J

(0,4 p )

where ta = tf - ty is the mid-line amplitude and calculated as the difference between fabric and

small cr

L=IfW=!(>+2 ( î Î >=4 p+0 î J2 dx

For yarns in twill fabric that gives:

L S 4 p + -a-2Z

tanh\^-\ -1 tanh3 f — Z J 3 I Z

If the transition region is smaller than the distance between the yarns (4Z < 2p) then equation (9) can be simplified to

3(L - 4 p)

Once the shape parameter is defined, the displacement can be calculated as u = y(t.) - y(t.).

Differentiating it analytically and substituting it to (6) gives the contribution of the bending energy to the deformation energy of the fabric.

At this stage two parameters remain undetermined: (a) the initial thickness of preform, (b) the total length of yarn in the unit cell L. The laser scan measurements are used to determine sensible values for the initial thickness of the preform. In order to minimise the effect of fabric pretension in the grips, the thickness of non-sheared fabric is chosen as the upper limit of experimental values t0 a=0 = 0.55 mm. The initial thickness of sheared fabric is calculated under

assumption that the fibre volume fraction remains constant in process of shearing. The following approximation is used then (shown as dot line on Figure 2b):

t°: = t°: a=: / cos(Y) (11)

L is a critical parameter. Minor variation in fibre length may have a significant impact on the bending response of a yarn. However, it is difficult to measure fibre length directly. In this model, L is determined analytically for the non-sheared single-ply fabric. The obtained value is used for all the other sheared and nested configurations of the preform. At the state of the

ximum compaction preform thickness and fibre geometry are known. To make it consistent with the compaction limit model, the transition span has to be equal to the dimensions of the overlap and hence 4Z = w - p . Substituting this approximation to (10) results in an estimate for the fibre length:

3(w - p)

4tc2 (x-1)2 + 4d = y,(X 1 + 4d

3(w - p)

For the considered preform fibre L appears to be longer than its projection to a fraction of per cent (0.14%) only, which is in correspondence with the crimp value in the studied fabric, estimated by direct measurement as 0.2%.

Now all the parameters are defined and the energy balance (6) can be numerically integrated to find the fabric thickness as the function of applied pressure Pa (ta). The modelling results are compared to the experimental data in Figure 4. It can be seen that above atmospheric pressure the model describes reality quite well. A certain disagreement between the model and experiment is seen at the initial stage of compaction, which is due to the uncertain initial thickness of yarn and fabric. The predictions at atmospheric pressures and above are more reliable. Table 2 shows that the model predictions at 0.1 MPa are well in line with the experimental observations. The largest deviation from the experimental data is seen for single ply preform sheared to 30°, yet the maximum deviation hardly exceeds 0.1 mm. The reason for this deviation is under-prediction of compaction limit thickness, which indicates a larger overlap at high shear angles than predicted by the simple schematic model.

The contribution of bending mechanisms in compaction response of the fabric depends on the deformation stage. Figure 9 shows the fraction of bending energy in the total internal deformation energy. At the initial deformation stage the bending factor is substantial; however, it

e. Figur i energy. At t

rapidly drops to 10-20%. Upon approaching the compaction limit the bending factor becomes ible.

negligible.

7. DISCUSSION

The experimental results give a ground for estimating the scale of error which can be introduced in thickness estimation when neglecting various geometrical and mechanical factors, such as fabric shear, nesting, and lateral yarn interaction. Table 3 presents estimates for these errors based on the experimental data discussed above. It can be seen that from thickness perspective, the most important factors are draping and overlap resulting from the lateral interaction of yarns.

bric topol

dsted ya

Apart from these factors, the analytical model suggests that at moderate pressure levels 10-20% error can occur if bending rigidity of a yarn is not taken into account. It has to be emphasized that these estimates were obtained for dense stable (possibly pre-consolidated) preform containing epoxy binder and made of slightly twisted yarns. The stability of the studied preforms allows exploring draping deformations and lateral yarn interaction in isolation from the other factors. The role of the same factors could have been dramatically different for another fabric. For instance, nesting in a sparse triaxial braided fabric can affect up to 50% of average ply thickness [30], and fibre spreading of non-twisted yarns can be significant particularly for heavy tows.

8. CONCLUSIONS The presented analytical model gives an insight into fabric topology and illustrates the importance of modelling some geometrical features when setting up a finite element model of compaction. The input of the analytical model is limited to the geometrical characteristics of the preform and the mechanical response of single yarns. No phenomenological parameters are used in these estimations. It is shown that, the lateral yarn interaction, generally neglected in almost all known yarn-scale models of composites, has a significant impact on compressibility of the considered fabric. The errors in thickness estimation may lead to wrong predictions of fibre crimp, excessive intra-yarn fibre volume fraction and, as a result, damage and strength properties [31].

A tool creating appropriate FE models of the dense sheared overlapped fabrics is demonstrated

32]. It allows creating the models of the fabrics with overlaps and to avoid a general problem of approximate models - interpenetrations of yarn volumes. Figure 10 demonstrates a cross-section of the fabric model with this feature - it can be seen that neighbouring warp yarns extend beyond the size of inter-yarn spacing creating a densely overlapped region, which impacts the preform compressibility.

ACKNOWLEDGEMENTS

The work of S.V. Lomov has been funded by European FP7 Infucomp project "Simulation Based Solutions for Industrial Manufacture of Large Infusion Composite Parts", the work of D.S. Ivanov has been funded by the EPSRC Centre for Innovative Manufacturing in Composites in the frame of feasibility study "Dimensionally stable textile preforms for use in liquid resin

n of to 11 1

infusion manufacture" (EP/1033513/1). REFERENCES

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3 Potluri P., Perez Ciurezu D.A., Ramgulam R.B. Measurement of meso-scale shear deformations for modelling textile composites, Composites: Part A 37 (2006) 303-314

4 Badel P, Vidal-Sallé E, Maire E, Boisse P. Simulation and tomography analysis of textile composite reinforcement deformation at the mesoscopic scale. Compos Sci Technol (2008) 68(12):2433-40.

5 Lomov SV, Verpoest I. Model of shear of woven fabric and parametric description of shear resistance of glass woven reinforcements, Compos Sci Technol 66 (2006) 919-933.

6 Charmetant A., Vidal-Sallé E., Boisse P. Hyperelastic modelling for mesoscopic analyses of composite reinforcements, Compos Sci Technol 71 (2011) 1623-1631

7 Nguyen Q.T., Vidal-Sallé E., Boisse P., Park C.H., Saouab A., Bréard J., Hivet G. Mesoscopic scale analyses of textile composite reinforcement compaction, Composites: Part B 44 (2013) 231-241.

8 Grishanov S. A., Lomov S. V., Harwood R. J., Cassidy T., Farrer C., (1997) The Simulation of the Geometry of Two-component Yarns. Part I: The Mechanics of Strand Compression: Simulating Yarn Cross-section Shape, Journal of The Textile Institute, 88:2, 118-131.

9 Chen B., Chou T-W. Compaction of woven-fabric preforms in liquid composite molding processes: single-layer deformation, Compos Sci Technol 59: 1519-1526, 1999.

10 Chen B., Chou T-W. Compaction of woven-fabric preforms: nesting and multi-layer deformation, ipos Sci Technol 60: 2223-2231, 2000.

11 Lee S.H., Han J. H., Kim S.Y., Youn J.R. Song Y.S. Compression and Relaxation Behavior of Dry Fiber Preforms for Resin Transfer Molding, Journal of Composite Materials 44:1801, 2010

12 van Wyk CM. Note on the compressibility of wool. J Text Inst, 37: 285-92, 1946.

13 Wielhorski, Y., D. Durville, and L. Marcin, Finite element simulations of 3D interlock fabrics. 13th International Conference on Textile Composites (TexComp-13), Leuven, 2013: electronic edition, s.p.

14 Sherburn M. Geometric and Mechanical Modelling of Textiles, PhD Thesis, The University of Nottingham, July 2007.

15 Lomov, S.V., Verpoest I., Compression of woven reinforcements: a mathematical model. Journal of Reinforced Plastics and Composites, 19(16): p. 1329-1350, 2000.

10 U Com,

IhiKAI-

16 Lomov S.V., Verpoest I., Peeters T., Roose D., Zako M., Nesting in textile laminates: Geometrical modelling of the laminate. Compos Sci Technol, 2002. 63(7): p. 993-1007.

17 Potluri P., Sagar T.V. Compaction modelling of textile preforms for composite structures, Composite Structures 86 (2008) 177-185.

18 Ivanov, D.S., Van Gestel C., Lomov S.V., Verpoest I., Local compressibility of draped woven fabrics, in 15th European Conference on Composite Materials (ECCM-15), 2012: Venice.

19 Hearle J.W.S., Shanahan W.J. An energy method for calculations in fabric mechanics, Part I: Principles of the method. J Textile Institute: 69(4):81-91, 1978.

mechan

;a, P. Har

20 J. Cao, R. Akkerman, P. Boisse, J. Chen, H.S. Cheng, E.F. de Graaf, J.L. Gorczyca, P. Harrison, G. Hivet, J. Launay, W. Lee, L. Liu, S.V. Lomov, A. Long, E. de Luycker, F. Morestin, J. Padvoiskis, X.Q. Peng, J. Sherwood, Tz. Stoilova, X.M. Tao, I. Verpoest, A. Willems, J. Wiggers, T.X. Yu, B. Zhu, Characterization of mechanical behavior of woven fabrics: Experimental methods and benchmark results, Composites: Part A 39 (2008) 1037-1053.

21 Lomov S.V., Stoilova T., Verpoest I. Shear of woven fabrics: theoretical model, numerical experiments and full field strain measurements, in Proceedings of the 7th Esaform conference on material forming. Trondheim, Norway; 2004, p. 345-48.

22 Lomov S.V., Verpoest I., Model of shear of woven fabric and parametric description of shear resistance of glass woven reinforcements. Compos Sci Technol, 2006. 66: 919-933.

23 Ivanov D.S., Van Gestel C., Lomov S.V., Verpoest I. In-situ measurements of fabric thickness evolution during draping, Proceedings of Esaform-2011, Belfast, UK, 2011.

24 Vanclooster K., Barburski M., Lomov S.V., Verpoest I., Deridder F., Lanckmans F. Experimental Characterisation of Steel Fibre Knitted Fabrics Deformability, Experimental Techniques (2012), Society for Experimental Mechanics

25 Carvelli, V., J. Pazmino, S.V. Lomov, and I. Verpoest, Deformability of a non-crimp 3D orthogonal weave E-glass composite reinforcement. Compos Sci Technol, 2012. 73: 9-18

26 Ivanov D., Li Y., Ward C., Potter K. Transitional behaviour of prepregs in Automated fibre deposition processes, Proceedings of International Conference on Composite Materials -19, Montreal, Canada, 2013

27 Barnes J.A., Cogswell F.N. Transverse flow processes in continuous fibre-reinforced thermoplastic composites, Composites 20 (1989) 1: 38-42

28 Harwood, R.H., S.A. Grishanov, S.V. Lomov, and T. Cassidy, Modelling of two-component yarns. Part I.: The compressibility of yarns. Journal of the Textile Institute, 88(1): 373-384 ,1997

29 Lomov S.V., Verpoest I., Barburski M., Laperre J., Carbon composites based on multiaxial multiply stitched preforms. Part 2: KES-F characterisation of the deformability of the preforms at low loads. Composites part A, 2003. 34(4): 359-370.

vanov D.S., Lomov S.V, Baudry F., Xie H., Van Den Broucke B., Verpoest I, Failure analysis of triaxial braided composite, Compos Sci Technol, 69(9), 1372-1380, 2009.

31 Ivanov D.S., Lomov S.V. Modeling of 2D and 3D woven composites, in "Polymer composites in the aerospace industry", Woodhead Publishing Limited 2012, (in print).

32 Lomov S.V., Verpoest I., Cichosz J., Hahn Ch., Ivanov D.S., Verleye B. Meso-level textile composites simulations: open data exchange and scripting, Journal of Composite Materials, 48(5): 621-637, 2014.

1. FIGURE CAPTIONS Figure Error! Main Document Only.. Z-coordinate [mm] of twill surface (approximately 50 by 50 mm) in the picture frame: before and after gripping. The scales for the in-plane and Z-coordinates are the same.

n-situ thick

Figure Error! Main Document Only.. (a) The scheme of laser scanning for in-situ thickness measurement: profile scanning, (b) the obtained average thickness as function of the shear angle. One point on the thickness diagram is average thickness over all three scan lines. Solid blue lines indicate thickness evolution in the first load cycle, dot red lines correspond to the second cycle.

Black dash line shows a constant fibre volume fraction fit: tf a=0 / cos( a ), where

100 a=0 = 0.55 mm is the maximum thickness of non-sheared preform over all samples, a is the

shear angle.

Figure Error! Main Document Only.. The compression of preform: (a) shearing in an aluminium frame prior to compaction (for illustration purpose only - the clamped fabric is not the one discussed in this paper); (b) a scheme of the compaction test.

Figure Error! Main Document Only.. Average ply thickness in single and double ply preforms. Experimental results (three tests per configuration) are compared to the analytical model based on assumptions that compression is caused by the compaction resistance of yarns and yarn bending.

Figure Error! Main Document Only.. The image of the compacted twill fabric. Yarn contours are highlighted to show the lateral crimp.

Figure Error! Main Document Only.. a) Yarn thickness against applied pressure shown for three samples and approximation by function (2), b) Evolution of fibre volume fraction as

on assu

bending Fig

function of the transverse strain, vcf is the characteristic fibre volume fraction at the end of the first cycle.

Figure Error! Main Document Only.. (a) The in-plane view of sheared twill fabric (producei

ng the

by WiseTex software); (b) The scheme illustrating yarn overlap. The cross-section al

along th weft-yarn midline is shown.

Figure Error! Main Document Only.. The warp yarn mid-lines in the process of the compaction a)->b)->c) (the cross-section along warp yarns midlines is shown). The total length of all three curves is the same. Ellipses show the schematic position of the weft yarns in the twill fabric. ta - is the yarn mid-line amplitude, 4Z - is the span of the transition between the top and

bottom surfaces of the fabric shown for the cases a) and c). The same mechanism for weft yarns compaction is assumed.

Figure Error! Main Document Only.. Contribution of yarn bending to the thickness estimation of woven fabric for non-sheared single-ply preform (Note: the scale of x-axis is reversed to illustrate the load increase).

Figure Error! Main Document Only.. The cross-section of finite-element model of the twill fabric with lateral yarn interaction taken into account.

Figure Error! Main Document Only.. Z-coordinate [mm] of twill surface (approximately 50 by 50 mm) in the picture frame: before and after gripping. The scales for the in-plane and Z-coordinates are the same.

(a) Paths of laser scanning (b)

Figure Error! Main Document Only.. (a) The scheme of laser scanning for in-situ thickness measurement: profile scanning, (b) the obtained average thickness as function of the shear angle. one point on the thickness diagram is average thickness over all three scan lines. solid blue lines indicate thickness evolution in the first load cycle, dot red lines correspond to the second cycle.

Black dash line shows a constant fibre volume fraction fit: t0 a=0 / cos( a), where

shear angle.

= 0.55 mm is the maximu

ss of non-sheared preform over all samples, a is the

Figure Error! Main Document Only.. The compression of preform: (a) shearing in an aluminium frame prior to compaction (for illustration purpose only - the clamped fabric is not the one discussed in this paper); (b) a scheme of the compaction test.

Figure Error! Main Document Only.. Average ply thickness in single and double ply preforms. Experimental results (three tests per configuration) are compared to the analytical model based on assumptions that compression is caused by the compaction resistance of yarns and yarn

bending.

ric. Yarn ct

Figure Error! Main Document Only.. The image of the compacted twill fabric. Yarn contours are highlighted to show the lateral crimp.

0.2 0.4 0.6 0.8

Figure Error! Main Document Only.. a) Yarn thickness against applied pressure shown for three samples and approximation by function (2), b) Evolution of fibre volume fraction as function of the transverse strain, vf is the characteristic fibre volume fraction at the end of the first cycle.

Initial Y f weft ^ weft

direction

/ cos(y)

w/(l cos(y))

(|-)j Ply thickness

Figure Error! Main Document Only.. (a) The in-plane view of sheared twill fabric (produced by WiseTex software); (b) The scheme illustrating yarn overlap. The cross-section along the

arn midline is shown.

Figure Error! Main Document Only.. The warp yarn mid-lines in the process of the compaction a)->b)->c) (the cross-section along warp yarns midlines is shown). The total length of all three curves is the same. Ellipses show the schematic position of the weft yarns in the twill fabric. ta - is the yarn mid-line amplitude, 4Z - is the span of the transition between the top and

bottom surfaces of the fabric shown for the cases a) and c). The same mechanism for weft yarns compaction is assumed.

Figure Error! Main Document Only.. Contribution of yarn bending to the thickness estimation of woven fabric for non-sheared single-ply preform (Note: the scale of x-axis is reversed to illustrate the load increase).

Figure Error! Main Document Only.. The cross-section of finite-element model of the twill fabric with lateral yarn interaction taken into account.

1. TABLE CAPTIONS

Table Error! Main Document Only.. Geometric characteristics of the overlap region for the dense twill fabric

Table Error! Main Document Only.. Thickness of sheared fabric

Table Error! Main Document Only.. Estimations of errors associated with neglecting various factors in consolidation modelling

Table Error! Main Document Only.. Geometric characteristics of the overlap region for the dense twill fabric

Shear angle % X

0° 2.47 2.24

10° 2.49 2.25

20° 2.55 2.28

30° 2.62 k2.32

Table Error! Main Document Only.. Thickness of sheared fabric

Shear angle,° 0 10 20 30

Thickness of one ply preform, pressure 0.l MPa

Experiments, mm 0.33 0.35 0.36 0.42

Estimation, mm 0.34 0.34 0.34 0.35

Thickness of one ply in a nested pack, pressure 0.1 MPa

Experiments, mm 0.32 0.33 0.35 0.38

Estimation, mm 0.34 0.34 0.34 0.35

Compaction limit thickness for one ply

Experiments*, mm 0.30 0.31 0.32 0.36

Estimation, mm 0.32 0.32 0.33 0.34

Compaction limit thickness of one ply in a nested pack

Experiments, mm 0.28 0.29 0.30 0.33

Estimation, mm 0.29 0.29 0.30 0.31

bar and 0.01 mm for 0.1 MPa and <1 MPa (except nested/30°: 0.03-0.05mm correspondingly).

eglecting v

Table Error! Main Document Only.. Estimations of errors associated with neglecting various factors in consolidation modelling

Considered factors Estimation is based on comparison of Error

Yarn spreading Transverse spreading of single yarn ~1 %

Ply nesting Thickness of one and two ply preforms at the same pressure level 5-8%

In-plane shear due to draping The limit compaction thicknesses of single orthogonal and 30°-sheared plies ~20%

Lateral yarn interaction (overlap) the limit compaction thickness of one ply and double compaction thickness of single yarn ~28%