MODELLING 3D FABRICS AND 3D-REINFORCED COMPOSITES ... · 1ST WORLD CONFERENCE ON 3D FABRICS MANCHESTER 9-11.04.2008 S.V. LOMOV ET AL MODELLING 3D FABRICS AND 3D-REINFORCED COMPOSITES:

1ST WORLD CONFERENCE ON 3D FABRICS MANCHESTER 9-11.04.2008

S.V. LOMOV ET AL MODELLING 3D FABRICS AND 3D-REINFORCED COMPOSITES: CHALLENGES AND SOLUTIONS 1

MODELLING 3D FABRICS AND 3D-REINFORCED COMPOSITES: CHALLENGES AND SOLUTIONS S.V. Lomov, D.S. Ivanov, G. Perie, I. Verpoest Department MTM, K.U. Leuven – Kasteelpark Arenberg, 44 B-3001 Leuven Belgium [email protected] Abstract The vast research area of modelling of 3D woven fabrics can be subdivided in several domains, each of them having its own challenges and difficulties, - some are already solved and implemented in software, some still wait for their solution. The paper discusses some of these problems and reviews the available solutions.

1 INTRODUCTION A clear “road map” is established for modeling a textile material at meso structural level – the level of a unit cell (representative volume element, repeat) of the material structure [1-4]. It consists of the following steps: − Establish mathematical coding of the topology of the structure, which would enable to

answer questions like: are this and this yarn in contact in a particular position in the structure? what is the relative positions (e.g., up-down) of the contacting yarns?

− Build a geometrical model of the fabric which translates the topology into actual placement of the yarns in space and calculates dimensions of the yarns. Such a model should account for the interaction between the yarns in the relaxed state of the fabric, which is defined by the forces due to bending and compression of the yarns.

− Calculate the deformation resistance of the fabric (in tension, shear and compression), based on the geometrical model and mechanical behaviour of the yarns in tension, bending and compression

− The deformation modeling can be done using finite element (FE) method. Then the yarn volumes should be meshed correctly, and contacts between the yarns have to be represented in the model

− Finally, if the fabric serves as reinforcement to composite, then mechanical properties of composites can be calculated based on the model of internal geometry of the fabric.

This “road map” is perfectly applicable to 3D fabrics, and the present paper presents the overview of its steps.

2 CODING OF THE STRUCTURE OF 3D WEAVES 2.1 Matrix coding of warp-interlaced 3D weave

Consider a warp-interlaced 3D weave (rather “multi-layered” weave) – Figure 1. The topological coding of a multi-layered weave is based on the warp yarns paths. The i-th warp path is coded by a sequence of intersection levels wij – denoting either the index number of the weft layer situated above the warp yarn in its intersection with the jk -th weft row, or 0, if the warp yarn lies on the face of the fabric. Let a fabric have L weft layers. Then warp in intersections with the weft can occupy L+1 levels, level 0 corresponding to the face of the fabric, level L – to the back of the fabric. Each warp can be now coded as a sequence of level codes, and the entire weave – as a matrix, as shown in Figure 1. The matrix coding of a one-layer weave also represents a checkerboard pattern, if level 0 were identified with a black square, level 1 – with a white square. In the pattern shown in Figure 1a all the warps are situated side by side. It is very often in composite reinforcements that warps also are layered, as shown in Figure 1b. Paths of the



warps in this case also can be coded as a sequence of level codes. To represent their layered positioning, a notion of warp zones is introduced. A warp zone is a set of warp yarns layered one over another. The yarns going through the thickness of a fabric are called Z-yarns. Multilayered weaves for composite reinforcements are classified as orthogonal (Z yarns go along columns of weft), through-the-thickness angle interlock (Z yarns go across columns of weft, connecting face and back of the fabric) and angle interlock (Z yarns connect separate layers of the fabric). These types of weaves are illustrated in Figure 1b,c.

4

1 2 3

1 2 3 4 layer 1

layer 2

level 0

level 1

level 2

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1210012110122101warp 1

warp 2

warp 3

warp 4

a

1 2-1

2-2

2-3

3 4-1

4-2

4-3

0 4 1 1 2 2 3 3 4 0 1 1 2 2 3 3

1

2

3

4

warp zones b

c Figure 1 Matrix coding of a multilayered weave: (a) building the matrix; (b) warp in zones; (c)

angle interlock The matrix weave coding for 3D fabrics was proposed in [5, 6], and is implemented in software WiseTex with GUI for definition and editing the weave (the images in this paper are produced using this software). Note that this approach differs from the approach used in [7-9], who aim on technological issues as, for example, the shedding lifting plan for a loom. In [10] the specific matrix coding is applied to produce 3D images of a fabric and their approach is closer to the one described here. 2.2 Analysing weave topology

Consider a warp yarn between two intersections with weft. Which weft yarns it is interacting with in these intersections? What is its position vis-à-vis these yarns? An answer to these questions is evident for one-layer weave, but for multi-layered weaves it needs analysing the weave code. Consider a warp yarn first, e.g., the first warp yarn in Figure 1a, up. Its level codes are {wi}= {0,2,4,2}.The yarn can be subdivided into crimp intervals, which constitute a part of the yarn between two intersections. At the first crimp interval, the yarn is supported (interacts with at the ends of the interval) with weft yarns in the layers l1

1=1 and l12=2 (the subscript gives the

number of the crimp interval, the superscript identify one of its ends. The yarn is situated above its supporting weft at the left end of the crimp interval (P1

1 =1) and below the supporting weft an the right end (P1

2 =-1). For all the crimp intervals of this yarn



1,1,1,2

1,1,3,4

1,1,4,3

1,1,2,1

21

11

21

14

21

11

21

13

22

12

22

12

21

11

21

11

+=−===

+=−===

−=+===

−=+===

PPll

PPll

PPll

PPll

To calculate these values from the intersection codes {wi}, the following algorithm is employed:

⎩⎨⎧

=−<+

==+==⇒= + LwLw

PPLwllwwi

iiiiiiii ,1

,1),,1min( 2121

1

1,1,1, 211

211 −=+=+==⇒< ++ iiiiiiii PPwlwlww

1,1,,1 211

211 +=−==+=⇒> ++ iiiiiiii PPwlwlww

To construct the same description for a weft yarn, the intersection codes and parameters of crimp intervals of warp are used. Consider a weft yarn i at layer l. First, looking up the lists of crimp interval parameters, find the first warp which has in its lists li

1=l or li2=l (i.e., supported

by the weft yarn i at layer l). This would be the left end of the first crimp interval on the weft yarn. The support warp number is thus found, and the position sign of the weft would be inverse to the position sign of the warp. Then find the next warp yarn supported by the weft (i,l). This would be the right end of the first crimp interval on the weft yarn, and the left end of the second crimp interval. Continue till all the crimp intervals would be defined. Note that in muli-layered structures the number of crimp intervals on a weft yarn can be smaller then the total number of the warp zones. For example, the first weft yarn in the first layer of the weave Figure 1a, up (eight warp zones) has only five crimp intervals, as it does not intersect with three of the Z-yarns. 2.3 Complex weft placement, complex yarn paths

The matrix coding is further developed to represent weaves which do not have necessarily the same number/placement of the weft yarns in the weft rows/layers. The simple solution to represent such weaves is to skip weft yarns in certain positions. This is done by introducing Boolean values WElj, l=1…L, j=NWe, where NWe is the number of weft rows, which are true if the weft yarn is present in the position and false if not. When processing the weave topology, the weft yarns with WElj = false are considered as not present.

Figure 2 Complex placement of weft yarns in the layers

The coding based on the intersection levels can be further developed for more complex yarn paths. First, a modification of the matrix coding is needed to take into account the eventual "missing" wefts. To describe cases where a warp yarn goes through a space where a weft yarn



have been removed, negative values for the matrix coding have been introduced. Consider a missing weft on the first weft layer number L: if a warp goes though this empty space, the corresponding value of the matrix coding will be equal to -L. In this case the matrix coding value does not correspond directly to the supporting weft layer but indicates the position of the warp yarn in the weft network. These negative values are easily handled in the existing code by using absolute values. Then a modified algorithm of definition of supporting wefts is used to decide which of the above or below weft yarn is considered as supportive. On the weave example on Figure 2, in each weft layer one weft yarn on two has been removed to obtain this shifted weft layers configuration. 2.4 Challenges

With the coding mathematics well developed and GUI tools available, the main challenge is to connect the coding of 3D weaves representing their topology with the coding, which controls production of the fabric on the loom. Another important challenge is development of coding for weave architectures, not yet covered, for example, weaves with different numbers of weft yarns in different layers, weft-interlaced weaves etc.

3 BUILDING A GEOMETRICAL MODEL OF 3D FABRIC The topology of interlacing of yarns, set by the weave pattern, defines the waviness (or crimp) of the yarns inside the fabric. The approach described below was first formulated in [11, 12], and further developed in [1-4].

x

z

p

h z(x)

Q

Q

d2

d1

∆Z

A

B

a

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

A

F

h/p b

Figure 3 Elementary crimp interval: (a) scheme; (b) characteristic functions 3.1 Yarn crimp

A waved shape of a yarn inside a woven fabric can be subdivided in intervals of crimp (between two intersections A and B, Figure 3). Assume that we are considering a warp yarn (the case of weft is treated in the same way). Let p and h be distances between the points A and B in the direction of the yarn x and in vertical direction z. Distance p is defined by the yarn spacing. Distance h is called a crimp height. Consider it as given (returning to its calculation later), and consider a problem of finding the shape of the middle line if the yarn in the crimp interval,

0)(;2/)(;0)0(;2/)0(:)( =′−==′= pzhpzzhzxz (3.1)

Also consider as given the dimensions of the cross-section of the warp and weft yarn (distinguished by superscripts “Wa” and “We”) d1 and d2, the former being the dimension in z direction, the latter – in orthogonal direction. These dimensions are affected by forces of the yarn interaction Q, but for a moment we consider them as “frozen”.



Our aim is to calculate forces of the interaction of the intersecting yarns. To do this, we consider an elastica model: find an elastic line satisfying boundary conditions (3.1) and minimising the bending energy

( ) ( )( )( )∫ →

′+

′′=

p

dxz

zBW0

2/52

2

min12

1 κ (3.2)

where B(κ) is the bending rigidity of the yarn, depending on the local curvature. This approach is simplified as it does not take into account interactions with weft yarns. Further simplifying (3.2), assume that the bending rigidity can be taken out of the integral as a certain average value, characteristic for the given range of curvatures. Eq. (3.2) will be then

( )( )( )∫ →

′+

′′p

dxz

z

02/52

2

min1

(3.3)

This is well known problem of elastica. Its solution with boundary conditions (3.1) can be written using elliptical integrals. Calculations are made easier with an approximation of the exact solution by

( ) ( )pxxxxx

phAxx

hz

=⎟⎠⎞

⎜⎝⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+−= ,

211164

21 2223 (3.4)

where function A(h/p) is shown in Fig.2.24b. The value A=3.5 provides a good approximation in the range 0<h/p<1. The first term of this formula, underlined in (3.4), is a solution for a linearised problem (3.3) (neglecting z´ in the denominator). The underlined expression is a cubic spline. It provides good approximation of the yarn line (the maximum value of the second term is about A/100). Consider the shape of the yarn defined by the solution (3.4). It is parameterised with a non-dimensional parameter h/p. Hence all the properties, associated with the bent centreline of the yarn depend solely of this parameter. This allows introduction of a characteristic function F (Figure 3b) of the elementary crimp interval, which defines these properties: the bending energy of the yarn W, transversal forces at the ends of the interval Q and average curvature κ:

( ) ( )( )( )

( )⎟⎟⎠

⎞⎜⎜⎝

⎛=

′+

′′= ∫ p

hFp

Bdxz

zBWp κκ0

2/52

2

121

(3.5)

( )⎟⎟⎠

⎞⎜⎜⎝

⎛==

phF

phB

hWQ κ22

(3.6)

( )( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

′+

′′= ∫ p

hFp

dxz

zp

p 1

1

1

02/52

2

κ (3.7)

If a spline approximation of (3.4) and linear approximation were used, then

( ) ( ) ( ) 3

2

0

2 621

phBdxzBW

p

κκ =′′= ∫ (3.8)

and F≈6(h/p)2, which is shown in Figure 3b as a dotted line. The difference with the exact F(h/p) is quite small. 3.2 From the weave coding to internal geometry of the fabric

Having the coding of the structure, next step will be creating a full description of internal geometry of multi-layred woven fabric.



Throughout the section, subscript i, i=1..NWa designates a warp yarn, subscripts j=1..NWe, l=1..L designate a weft yarn, NWa is number of warps in the fabric, NWe is the number of wefts in each layer of the fabric and L is the number of weft layers. The following input data are given: 1. Fabric weave, given by a matrix of warp levels 2. Compression and bending behaviour of warp and weft yarns (there can be any number of

different types of yarns in both warp and weft) 3. Spacing of warp and weft yarns (which can be non-uniform) 4. Shift between the weft layers in the warp direction. This is defined by the weft insertion

and battening process. Two typical cases are zero shift (weft yarns are one above another) and shift of 50% of the weft spacing (weft yarns of the upper layer in between yarns of the lower layer).

Analysis of the weave matrix allows determining sets of crimp intervals on warp and weft yarns. Subscript k designates the interval number, for warp yarns k=1..NWe, for weft yarns k=1..Kjl, the number of crimp intervals on different weft yarns Kjl may be different. The ranges of the subscripts i,j,k are not given explicitly in the formulae below. For each interval the support yarns at the interval ends and signs of the yarn position relative to them (designated as P) are known from the analysis of the weave. If we consider a crimp interval k on a warp yarn i, then the indices of the weft yarn (and crimp intervals of it) supporting the warp at the ends of the interval are designated as j’,l’,k’ and j’’,l’’,k’’. If we consider a crimp interval k on a weft yarn (j,l), then the indices of the supporting warp yarns and intervals on them are designated by i’,k’ and i’’,k’’. Figure 4a explains the indexing.

warp i

warp crimp interval k

weft j’,l’

weft j’’,l’’

weft crimp interval k’

weft crimp interval k’’a

weft j,l+1; interval k2

weft j,l; interval k1

hjlWe

hjl+1We

warp i z

Zl

Zl+1

∆x

b Figure 4 Crimp intervals for calculation of internal geometry

We assume that all the crimp intervals of a weft yarn (j,l) have the same crimp height hjlWe.

The weft yarns deviate at the ends of the crimp intervals by hjlWe/2 in z direction from the

average planes of the weft layers. These average planes have z co-ordinates Zl, Z0=0. Dimensions of cross-sections can differ along a yarn. These dimensions at the ends of crimp intervals of warp and weft yarns are designated as . We

jlkWe

jlkWa

ikWa

ik dddd ,2,1,2,1 ,,,

The parameters listed above allow building a fill description of the internal geometry. Vertical positions of the centres of weft yarns at the ends of crimp intervals are given by

Wejlk

Wejll

Wejlk PhZz ⋅+=

Vertical positions of the warp yarns at the end of crimp intervals are given then as

Waik

Weklj

WaikWe

kilWaik P

ddzz ⎟

⎟⎠

⎞⎜⎜⎝

⎛+−= ′′′

′′ 2211

)(



Positions in x and y directions are determined by the yarn spacing. With dimensions of the yarns and position of the yarns at the end of crimp intervals and support yarns defined, the models of 1.2.3 are employed to generate fill paths of the yarns in the repeat.

Now calculate the unknowns . Wejlk

Wejlk

Waik

Waikl

Wejl ddddZh ,2,1,2,1 ,,,,,

Yarn dimensions are defined by the laws of the yarn compression:

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛= ′′′′

Waik

ljiWai

Wai

WaikWa

ik

ljiWai

Wai

Waik d

Qdd

dQ

dd2

22022

1101 , ηη (3.9)

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛= ′′

Wejlk

jliWejl

Wejlk

WejlkWe

jlk

jliWejl

Wejl

Wejlk d

Qdd

dQ

dd2

22022

1101 , ηη (3.10)

Here and below subscripts with prime designate corresponding indexes of crimp intervals and yarns, which are the support yarns for the crimp interval under consideration. Qijl in (3.9) and (3.10) are transversal forces of interaction if the warp yarn I and weft yarn (j,l):

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+′′+′′′′′′

+′

+′

+′+′′

′

′′

Wekjl

Wejl

Wejl

Wekjl

Wejl

Wekjl

Wejl

Wejl

Wekjl

Wejl

Waki

Waki

Waki

Waki

Wai

Waki

Waki

Waki

Waki

Wai

ijl

ph

Fhp

Bph

Fhp

B

ph

Fhp

Bph

Fhp

BQ

11

1

1

11

21

21

(3.11)

Relation between crimp heights of the warp crimp intervals and the weft crimp is given by a constraint, which describes the contact of the yarns in the adjacent layers:

( ) ( )Weklj

Weklj

Waik

Waikll

Welj

Welj

Waik ddddZZdhh ′′′′′′′′′+′′′′′′′′′ ++++−=++ 111111 2

121

(3.12)

The coordinates of the weft layer planes Zl are calculated from the condition of tight packing of the woven structure in z direction. Consider two weft yarns – (j,l) and (j,l+1) in two consecutive weft layers (Figure 4b). k1 and k2 are indices of crimp intervals on these yarns which either have warp i as a support or are supports for the warp i at a certain crimp interval k’. z-coordinates of the centres of the cross-sections of the weft yarns are given by

Weklj

Weljlklj

Wejlk

Wejlljlk

PhZz

PhZz

22

21

,1,1,1,1, ++++ +=

+= (3.13)

where P are the position codes of the weft yarns derived from the weave analysis. The distance ∆x between the centres of the weft yarns in the warp direction is defined by spacing of weft and possible shift between the weft layers. Consider a distance ∆z between these centres. As the yarns are packed closely, then this is the distance defined by the condition: the distance between contours is equal to dWa

ik’. The solution of such a problem, designated as ∆ztight depends on the shapes of the yarns, their dimensions and ∆x. The value of ∆ztight defines by (3.13) the distance between the layers. As we have assumed existence of a common middle plane of a weft layer, then

( ))

,,,,,,(max

22

212111

,1,1,

1,1,2,1,1211,1

Wejlk

Wejl

Weklj

Welj

Wakj

Weklj

Weklj

Wejlk

Wejlk

Wejl

Wejltightkjll

PhPh

dddddshapeshapezZZ

+−

∆+=

++

′++++ (3.14)

With the condition Z1=0, (3.14) defines all the weft layers positions. Finally, weft crimp heights hjl

We are independent variables in the minimum problem, expressing the minimum of the bending energy:



( ) ( )min

,,,→⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∑∑Σ

kljWejlk

Wejl

Wejlk

Wejlk

Wejlk

kiWaik

Waik

Waik

Waik

Waik

ph

Fp

Bph

Fp

BW

κκ (3.15)

where hWa are related to hWe with (3.4), and average curvatures are calculated with (3.7). The minimum is reached when

0=∂∂ Σ

Wejlh

W (3.16)

The system of equations, which defines the parameters of the fabric internal geometry, is now complete (Table 1). It is solved by an iteration procedure, doing calculations in the order given by Table 1 and checking the convergence by the convergence of weft crimp height values. Figure 5 displays examples of the calculations.

Table 1 Summary of the mathematical description of internal structure of a 3D woven fabric Unknown variables Number Equations Dimensions of warp and weft yarns

Wejlk

Wejlk

Waik

Waik dddd ,2,1,2,1 ,,, ⎟⎟

⎠

⎞⎜⎜⎝

⎛+⋅ ∑∑

= =

L

l

NWe

jjlKNWeNWa

1 12

(3.9 – 3.11)

Vertical positions of mid-planes of weft layers Zl

L (3.14)

Weft crimp heights Wejlh L*NWe (3.15 – 3.16)

Figure 5 Internal geometry of glass woven 3D fabric: (a) X-ray computer tomography; (b)

simulated 3.3 Challenges

Solution of the minimum energy problem. The problem (3.15-3.16) is ill-defined in the case when there are yarns, loosely supported in the structure. This may lead to bad convergence of the minimization process. An example of such situation is provided by straight weft and warp yarns in Figure 1. In this case there is no interaction between the yarns, hence zero transversal forces. Algorithmic shortcuts can be built in to handle this, as imposing zero crimp on such yarns. Another reason for bad convergence is very low compression stiffness of the yarns for the initial stage of compression (3.9 – 3.10). This gives a low norm to the iterative operator of the minimization process. Approximate assumptions: flat middle surface of the layers; constant crimp height in different crimp intervals of the weft. In more elaborate weave architectures this assumption is too restrictive. Assuming individual vertical adjustment of the weft yarns can relax it, still staying inside the paradigm of the model (but adding to the problems of convergence) Shape of the yarn cross-section. In the current implementation of the model in WiseTex software, the cross-sections of the yarns have symmetrical prescribed shape (elliptical, lenticular or rectangular). This leads to not-so-good definition of the yarn volumes when it comes to building of FE model of the fabric unit cell. The alternative, more accurate



descriptions of the cross-sections have been proposed [13, 14], but so far have not been combined with calculations of the yarn paths based on the balance of crimp.

4 DEFORMATION RESISTANCE OF A 3D FABRIC Here we briefly describe the algorithms for calculation of the resistance of 3D fabric for compression, biaxial tension and shear. The full formulation can be found in [15-17]. 4.1 Compression

Research on compressibility of woven fabrics in composite technological processes is mostly empirical. A theoretical model, developed by the authors and published elsewhere [16, 18], uses the WiseTex model as the starting point and accounts for two physical phenomena associated with the fabric compression: change of the yarns crimp and compression of the individual yarns. The result is pressure-thickness curve for the fabric, from which fibre volume fraction for the given preform compaction is easily deduced. When applied to 3D fabrics, the model also computes change of the yarn paths (Figure 6).

Figure 6 Typical deformed Z-yarns (cross-section and computed) in 3D fabrics

4.2 Bi-axial tension

Consider a woven fabric under bi-axial tension characterised by deformations in warp (x-axis) and weft (y-axis) directions ex = Y'/Y-1, ey = X'/X-1, where X and Y are sizes of the fabric repeat, dashed values correspond to the state after the deformation. Inside the WiseTex model, the internal structure of the fabric is described based on weft crimp heights hj

We and weft and warp cross-section dimensions at the intersections dij

Wa and djiWe (subscripts designate

different yarns in the fabric repeat). These values change after the deformation. Tension of the yarns induces transversal forces, which compress the yarns, changing d's. The same transversal forces change the equilibrium conditions between warp and weft, which leads to a redistribution of crimp and change of crimp heights. When the mentioned values in the deformed configuration are computed, the internal geometry of the deformed fabric is built as explained above. Change of length of the yarns determines their average (in the repeat) deformations, which, through the tension-deformation diagrams of the yarns allow computing tensions of the yarns. When summed up, with yarns inclinations due to the crimp accounted for, the yarn tensions are transformed into loads, caused the fabric deformations. This computational scheme has been proposed for plain weaves in 60s-70s [19-22], but never applied to 3D fabrics. The key problem in the bi-axial modelling is computation of crimp heights and transversal forces in the deformed structure. Assuming that the spacing of the yarns in the fabric is changed proportionally to the change of the repeat size, we compute the x and y positions of intersections of warp and weft in the deformed structure. The configuration of the yarns in the crimp intervals between the intersections is determined by these positions and (unknown) crimp heights. Consider some values of the crimp heights. Then the WiseTex geometrical model determines positions of the ends of crimp intervals (warp/weft intersections) and bent shape of the yarns in the intervals. The transversal forces are computed then using the following formula:

θcos2TQQ bend += (4.1)



where Qbend is the transversal force due to the yarn bending, T is the yarn tension, θ is the angle of inclination of the yarn on the crimp interval. Note that T depend on yarn length after the deformation, which in its turn depend on crimp heights and yarn dimensions. The transversal forces compress the yarns according to an experimental law of the compression: d=d(Q). When the yarn dimensions are computed and "frosen", then crimp heights are determined using the minimum energy condition:

min→+= tensbend WWW (4.2)

where Wbend and Wtens are the bending and tension energy of the yarns. The former is computed summing up bending energies of the yarns in crimp intervals between yarn intersections, the latter is the sum of tension energies of all the yarns, which are computed using their (linear or non-linear) tension diagrams and yarn deformations. The computations described above determine one step in the iteration process: started from current values of the crimp heights we compute yarn lengths, yarn tensions, transversal forces, yarn compressed dimensions and then new values of the crimp heights. These iterations are the same as are employed in the model for the relaxed state with the addition of the terms responsible for the yarn tensions in (4.1) and (4.2). If one side of the fabric is kept free (uni-axial tension, say, along the warp), then the described algorithm has another, the outmost iteration loop, searching for X'<X (negative ey) which would lead to zero loads along weft (y) direction. This allows computing Poisson coefficient for the fabric. Figure 7 illustrates the application of this algorithm to a multilayred fabric [12, 23]. Tension along warp

0

10

20

30

40

50

60

70

80

0 10 20 30 40

Deformation, %

Forc

e pe

r ya

rn, N

warp yarn computed

measured

Figure 7 Tensile curve of multilayered fabric. Warp: Polyester yarns in polyamide lining; weft:

polyester monofilaments 4.3 Shear

When a woven fabric is sheared, the orthogonal directions of warp and weft become skewed. Such a configuration is similar to a braided structure. A model for internal geometry of braids is proposed in [24] and is used to construct the sheared woven fabric internal geometry. The problem then is in computation of the loads associated with a given shear deformation. In formulating the model we again follow the lines sketched by S.Kawabata in 70s [25], which are also followed by more recent publications (e.g.[12]). When a fabric is sheared, the deformation is resisted by friction between yarns, bending and compression of the yarns. Friction forces are estimated in the model using normal forces of the yarn interaction, tension being a pretension normally employed in the shear test. The transversal forces are increased by the internal pressure, developed inside yarns due to their lateral compression in the sheared structure. This is taken into account using the experimental compression diagrams of the yarns. Resistance due to bending is estimated using the difference in bending energies in deformed and undeformed configurations, the latter computed with algorithms for non-orthogonal structures [24].



The model is fully described in [15]. Figure 8 illustrates the calculations for two-layered carbon fabric (measurements performed in the group of Ph. Boisse). Note the instabilities in the calculations, caused by instable behaviour of numerical algorithms, which involve extremely low stiffness of the yarns in compression.

Figure 8 Shear diagram of two-layered carbon fabric [26]: line – measured in biaxial test, points

– calculations 4.4 Challenges

Approximate models. The models, presented above, which use principle of minimum energy, approximate description of the yarn paths and approximate assumptions of the nature of the interaction of the yarns, are very approximate. Quite rough assumptions are made for the contact region between the yarns and the associated friction, for uncoupled bending and compression resistance, for the compression of the yarns etc. Limits of validity. If for 2D fabrics such models are discussed since 1970s, for 3D fabrics there is no real validation available for 3D fabrics (examples above use experimental data on two-layered fabrics). This is mostly because of lack of experimental data for 3D fabrics. Use of the approximate models is quite tempting because they are simple and fast, but the limits of their validity should be carefully analised. Dead end. In a certain sense the models presented above, are a “dead end” for approximate textile mechanics. An attempt to make more complex and elaborate treatment of the interaction of the yarns encounters difficulties, which lead to finite element formulation of the problem. This gives generality to the solution – but throws away easy and mechanically clear formulation and speed of the calculations.

5 TRANSFORMATION OF THE GEOMETRICAL MODEL INTO FINITE ELEMENT (FE) MESH

The authors have recently published a paper [27], which treats in detail meso-modelling of textile composites. Here we will focus on one aspect of the problem, which applies to models of dry fabrics as well as to models of composite unit cells: interpenetration of the yarn volumes. If the geometry for the meso-FE model is acquired by direct measurement of the yarn shapes then there are no defects in mutual placement of the yarn volumes. However, such approach has limited predictive capabilities. General-purpose geometrical models, like the models, discussed here, use several simplifying assumptions. One of these assumptions is a fixed shape (but maybe changing dimensions) of the yarns cross-sections. The shape of the yarn middle line prescribes the positions of the centres of the cross-sections. The model calculates dimensions of the cross-sections, ensuring that the distance between the contacting yarn centre lines is equal to the sum of their dimensions. For quite a wide class of 2D woven fabrics such a treatment is sufficient to create geometrical model, which can be meshed in FE



package. However, the condition of point or line contact does not guarantee that the surfaces of the contacting yarn never penetrate one another, and interpenetration may occur. Interpenetrations in modelling of 3D fabrics can be treated by solving an intermediate FE problem: artificial separation of the contacting parts of the yarn and then compressing them together, using a compressible media as a filler of the space around the yarns. This approach was proposed by M. Zako [28], and is explained in detail in [27]. It is implemented in MeshTex software, integrated with WiseTex geometrical simulations of woven (2D and 3D) fabrics.

Figure 9 (M. Zako) Mesh of yarns in 3D fabric: before (left) and after (right) the correction of

interpenetration Recently D. Durville has applied his Multifil method [29, 30] of simulating multiple contacts inside a fibrous assembly, with individual fibres represented by beam elements, to building up consistent (not interpenetrating) models of woven fabrics, based on WiseTex description of the geometry [31]. Figure 10 illustraes application of this method to a multi-layred fabric.

Figure 10 (D. Durville) Correction of yarn volumes build with WiseTex (left) into non-penetrating

configuration (right) 5.1 Challenges

Meso-scale FE model of woven fabrics are successfully used to simulate fabric deformation in tension, compression and shear for 2D fabrics [13, 32-36]. However, FE modeling to 3D woven fabrics still awaits serious implementation.

6 PREDICTION OF ELASTIC MECHANICAL PROPERTIES OF 3D REINFORCED COMPOSITES

6.1 Method of inclusions

To apply the method of inclusions, implemented in the TexComp software [2, 3, 37-39], the yarns in the unit cell are subdivided into a number of smaller segments, where each yarn segment is geometrically characterised by its total volume fraction, spatial orientation, cross-sectional aspect ratio and local curvature (all these parameters are readily provided by the geometrical model). Next, Eshelby's equivalent inclusion principle is adopted to transform each heterogeneous yarn segment into homogeneity with a fictitious transformation strain distribution. The solution makes use of a short fibre equivalent, which physically reflects the



drop in the axial load carrying capability of a curved yarn with respect to an initially straight yarn. Every yarn segment is hence linked to an equivalent short fibre, possessing an identical cross-sectional shape, volume fraction and orientation as the original segment it is derived from. The length of the equivalent fibre on the other hand is related to the curvature of the original yarn. For textiles with smoothly varying curvature radii, a proportional relationship between the short fibre length and the local yarn curvature radius is the most straightforward choice and sufficiently accurate for the present purpose. The interaction problem between the different reinforcing yarns is solved in the traditional way, by averaging out the image stress sampling over the different phases. If a Mori-Tanaka scheme is used, the stiffness tensor CC of the composite is hence obtained as:

1C m s s sm s m sc c c c

−⎡ ⎤ ⎡= + +⎣ ⎦ ⎣ ⎤⎦C A I AC C , where the subscripts m and s denote the matrix

and a yarn segment respectively, ci is the volume fraction of phase i (i = m,s), and the angle brackets denote a configurational average. As follows from this brief description, the homogenisation procedure does not depend on the configuration of the unit cell, and can be applied to geometries with interpenetrating volumes. Being integrated with a geometrical processor WiseTex, the model allows fast calculation of homogenised properties and integration into multi-level micro-meso-macro calculations [40]. 6.2 FE modelling of elastic behaviour and damage

The predictions of homogenized properties of 3D woven composites using inclusion theory are quite accurate. Figure 11 shows a FE model of a 3D woven glass/epoxy composite, the homogenized properties of which were calculated using FE simulation and the inclusion method (both starting from the same geometry, characterized by micro-CT and cross-sectioning, and implemented in WiseTex model [41]). The composite has fibre volume fraction of 49%, and was tested in tension in three directions: along the warp yarns, along the fill yarns and in bias direction (45° angle with the warp). This material, supplied by 3Tex (USA), was studied together with A. Bogdanovich and D. Mungalov.

Figure 11 Finite element models of a 3D woven composite

Table 2 experimental and computed mechanical properties of 3D woven glass/epoxy composite,

VF = 49% Property and test direction Experiment Method of

inclusions FE

Young module, GPa warp 24.3±1.2 25.1 23.1

fill 25.1±2.3 25.7 24.7 bias 12.9±0.5 13.3 13.5

Poisson coefficient warp 0.14±0.07 0.13 0.13 bias 0.50±0.21 0.54 0.50



Table 2 shows that both FE and method of inclusions predict the elastic properties fairly well. We can conclude that the homogenized properties of 3D woven composites can be easily obtained using approximate method, like the inclusion method. Integration with a versatile geometric modeler allows easy study of the influence of parameters and integration with macro-simulations. However, as it comes to damage, the FE modeling is the must, as approximate methods cannot produce local stresses and strains, hence, the deformation criterion cannot be assessed. we have discussed the damage modeling in textile composites elsewhere [27], identifying serious problems in the existing approaches [42]. 3D fabrics present a serious challenge, as complex shapes of the yarns, modelled in an approximate way, may create spurious stress-strain concentrations, predicting too early initiation of damage. For example, in the case shown in Figure 11, the Fe modelling predicts damage initiation at non-realistic strain of 0.1% (tension in the warp direction), whilst experiment gives damage initiation threshold of 0.43±0.04%. This is caused by assumed elliptical shape of the yarns, which does not correspond to the real shape, which is closer to rectangular.

7 CONCLUSION There exists a serious baggage of modelling approaches for 3D woven fabrics, implemented in software tools, which allows: − Creation and easy varying weave architectures, (almost) without restriction of number of

the yarns, layers, interlacing pattern or other complexity factors of the fabric weave − Creation of geometrical models of internal structure of 3D fabrics, adequately

representing yarn paths (hence crimp factors, hence overall parameters of the fabric, as areal density, tightness, porosity…)

− Calculation (with certain reservations vis-à-vis precision) of mechanical response of the fabric to compression, tension and shear

− Modelling of the geometry of deformed fabric − Translation of the fabric geometry model into finite element model for minute simulation

of local stress-strains during deformation − Calculation of effective (homogenised) properties of textile composites with precision

conforming to requirements of macro-structural analysis of composite part − Building meso-level FE models of unit cell of 3D woven composite and approach the

problem of damage prediction

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MODELLING 3D FABRICS AND 3D-REINFORCED COMPOSITES ... · 1ST WORLD CONFERENCE ON 3D FABRICS MANCHESTER 9-11.04.2008 S.V. LOMOV ET AL MODELLING 3D FABRICS AND 3D-REINFORCED COMPOSITES: