# On the influence of fabric layer shifts on the strain distributions in a multi-layer woven composite

Aurélien Doitrand, Christian Fagiano, Francois-Henri Leroy, Anne Mavel, Martin Hirsekorn

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## To cite this version:

Aurélien Doitrand, Christian Fagiano, Francois-Henri Leroy, Anne Mavel, Martin Hirsekorn. On the influence of fabric layer shifts on the strain distributions in a multi-layer woven composite. Composite Structures, 2016, 145, pp.15-25. 10.1016/j.compstruct.2016.02.054. hal-01929284

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# On the influence of fabric layer shifts on the strain distributions in a multi-layer woven composite

Aurélien Doitrand<sup>a,*</sup>, Christian Fagiano<sup>a</sup>, François-Henri Leroy<sup>a</sup>, Anne Mavel<sup>a</sup>, Martin Hirsekorn<sup>a</sup>

<sup>a</sup> ONERA - The French Aerospace Lab - F-92322 Châtillon - France

## Abstract

The influence of the relative shift between fabric layers on the local strain distributions at the mesoscopic scale of a four-layer plain weave glass fiber/epoxy matrix composite is studied through Finite Element (FE) modeling. The surface strain fields of several representative unit cells, consisting of compacted and nested plain weave layups with different layer shifts and the matrix complement, are compared to strain fields measured experimentally by digital image correlation. The layer shifts only have a small impact on the calculated homogenized macroscopic mechanical properties. However, the local strain fields are influenced significantly. Good quantitative agreement is obtained between the measured surface strain distributions and the numerical results if the layer shifts of the tested specimen are used in the FE model. The most frequently used models without layer shifts or with maximum nesting do not provide satisfactory surface strain distributions.

**Keywords:** Textile composites, Finite element analysis, Multiscale modeling

## 1. Introduction

Composite materials containing textile reinforcements are increasingly used for aero- nautical and automotive lightweight applications. One of the advantages of woven com-

<footnote>*Corresponding author, tel.: +33(0)146734658</footnote>

<footnote>Email adress: aurelien.doitrand@onera.fr (Aurélien Doitrand)</footnote>
posites is the high flexibility of the fiber reinforcement, which can be directly shaped
to the final form of a part, resulting in a reduction of assembly operations and of the
number of weak points in a structure.

The reinforcement architecture of a woven composite has a strong influence on dam-
age onset and propagation [1, 2]. For instance, different damage kinetics are observed
for 2D [2, 3] and 3D woven composites [4]. Therefore, an experimental characterization
of the damage mechanisms and of their influence on the material behavior is needed
each time a different architecture is considered. The number of tests may be reduced
using predictive models that take into account the reinforcement architecture, which
is defined at the so-called meso-scale [5]. At this scale, the composite reinforcement is
represented by two sets of interlaced yarns (warp and weft yarns) that are modeled as
homogeneous materials embedded in the matrix.

At the meso-scale, the fiber reinforcement of woven composites is approximatively
periodic, even if small variations are induced by (i) the weaving process, (ii) the handling
of the fabric and (iii) the resin injection during composite manufacturing. Olave et
al. [6] showed that the mechanical properties of the composites are not significantly
influenced by these variations. Moreover, repeating damage patterns were observed
along the edge of a woven composite specimen [7]. Therefore, perfect periodicity is
often assumed to model woven composites, using a pattern representative of the whole
material (called a representative unit cell, RUC), in order to reduce computational
costs.

The shifts between the layers in a multi-layer woven composite may be controlled
during the manufacturing process [1, 8]. However, this is a difficult operation and, in
practice, random layer shifts are observed in most woven composites [6, 7, 9, 10]. These
layer shifts are often neglected in meso-scale models. In several published papers, only
one layer is modeled, assuming 3D periodic boundary conditions [11, 12, 13, 14, 15, 16].
Some authors [17, 18, 19] modeled all the fabric layers of multi-layer composites, but using in-phase layer stacks, thus not taking into account the layer shifts.

The influence of layer shifts on the mechanical behavior of woven composites was studied by several authors. Breiling et al. [20] showed that a variation of up to 32% on the ultimate strength of a carbon epoxy five-harness satin was obtained by varying the stacking configuration. Le Page et al. [21], solving a 2D problem, showed that the energy release rate of a cracked textile composite is strongly dependent on the stacking pattern. Ivanov et al. [22] found that the strain distributions were influenced by the layer shifts by comparing an in-phase and a shifted layup of an idealized geometry of a twill. The same authors showed in another paper [23] that damage onset was also strongly influenced. Daggumati et al. [24] studied the sensitivity of the surface strain distributions to different simplified stacking sequences (in phase, out-of phase and step stacking) of a 5-harness satin. A step stacking gave a better qualitative agreement compared to experimental data than in phase and out-of-phase stacks. However, the real layer shifts of the tested specimen, the fabric compaction and the nesting between the layers were not taken into account in the model.

In most models, the layer shifts are simplified for numerical convenience, thus not representing experimentally observed layups. Moreover, during the production process of a textile composite, the reinforcing fabric is usually compacted in order to increase the fiber volume fraction. As a consequence, the fabric layers nest into each other. The aim of this article is to evaluate numerically the influence of the layer shifts on the surface strain distributions in the case of a compacted and nested four-layer plain weave glass fiber/epoxy matrix composite. The results are compared to surface strain distributions measured experimentally by digital image correlation (DIC) on the specimen surfaces. The experimental tests are presented in section 2. Then the generation and meshing of five different RUCs is described in section 3. These RUCs contain compacted reinforce-
ments with (i) no shifts between the layers, (ii) shifts resulting in maximum nesting and (iii) the layer shifts determined experimentally on three specimens. Section 3 also includes the comparison between a geometry resulting from the numerical simulation of the compaction and a geometry extracted from a micro-computed tomography (μ-CT) observation of a specimen. In section 4, the homogenized properties and the surface strain distributions obtained numerically with the different RUCs are compared with the experimental observations.

## 2. Experimental analysis

The composite under investigation consists of four layers of a plain weave reinforcement of E-glass fibers embedded in Araldite LY564 epoxy resin. The thread count of the unbalanced plain fabric is 2.2 warps.cm⁻¹ x 2 wefts.cm⁻¹. The fabric mass per unit area is 504±40 g.m⁻² and the mass density of the fibers is 2.54 g.cm⁻³. The four layers composing the dry fabric were placed into a steel mold and compacted by tightening the screws that keep the mold closed. Due to the relative shift between adjacent layers, nesting occurs during compaction. After compaction and matrix injection, the mean fiber volume fraction in the material is 47%. Three rectangular specimens were tested under tensile loading in order to evaluate the in-plane elastic properties of the material and to measure the surface strain distributions using DIC. The orientation of the fiber reinforcement in the load direction of each specimen is respectively 0° (warp direction), 90° (weft direction) and 45°.

### 2.1. Reinforcement architecture

First, the specimen edges were observed using an optical microscope in order to determine the layer shifts of each specimen. These shifts are given in Table 1 for the three lower layers with respect to the top layer. The layer shifts that are modeled in most published works, i.e., in-phase stacking resulting in no nesting and the shifts
resulting in maximum nesting, are also given. These five stacks are used in Section 3
for the generation of the Finite Element (FE) meshes of the composite RUCs.

Microscope observations on the edge of the specimens only provide a surface de-
scription of the reinforcement architecture. Therefore, the 0° specimen was scanned by
means of an X-ray µ-CT, in order to obtain a full 3D description of the reinforcement
architecture of one of the specimens (Figure 1a). A 3D geometrical representation of
the reinforcement was generated by manually selecting each yarn outline on 20 2D to-
mography slices in warp and weft directions (Figure 1b). The resulting geometry is
used in section 3 to validate the geometries obtained from the numerical modeling of
the fabric compaction.

## 2.2. *Surface strain distributions*

The differences between the mechanical behavior of the stiff fibers and the compli-
ant matrix result in heterogeneous strain distributions with strong gradients, especially
around yarn crimp regions. Classical electrical resistance strain gages can thus not
provide an adequate spatial resolution. Therefore, the local surface strains in woven
composites are often measured using DIC. For example, Anzelotti *et al.* [25] and Nico-
letto *et al.* [26] highlighted the weave-specific surface strain features of a specimen
containing a single layer of twill-weave. Daggumati *et al.* [10] performed a local strain
analysis on a 5-harness satin composite and matched the strain pattern with the rein-
forcement architecture and the locations of the resin rich areas. Koohbor *et al.* [27]
showed that the in-plane surface strains have a strong influence on the failure mech-
anisms in several glass fiber and epoxy matrix specimens with different reinforcement
orientations.

In this study, the surface strain distributions on the 3 specimens have been measured
using 3D stereo-DIC. The displacements measured using 3D DIC are directly computed
in the plane of the specimen, which is an advantage compared to 2D DIC where the
specimen must be perfectly aligned with the camera. In order to capture the surface
strain distributions during the tensile tests, the specimen surfaces have been painted
with a white spray followed by a black speckle pattern. Then, several images have been
captured during the loading process. The local strains have been computed through
the correlation between these images and an initial image of the unloaded specimen
using the 'Limess-VIC3D' software [28]. The correlation step size was 3 pixels and
the correlation subset size was 17 pixels. The strains obtained by derivation of the
displacement fields are averaged over a 15 pixels window using a Gaussian filter, as
explained in [28].

**3. Finite element modeling of a Representative Unit Cell**

3.1. *Geometry generation*

The dry reinforcement is usually compacted before adding the matrix in order to
increase the fiber volume fraction and hence the mechanical properties of the compos-
ite. The compaction process leads to complex yarn shapes including local variations
of both the section and the fiber volume fraction [6, 29]. If the fabric layer position
is not precisely controled during the manufacturing process, random shifts and nesting
between the layers are obtained [30, 31]. Since the reinforcement architecture has a
strong influence on the mechanical behavior of the composites, e.g., on damage on-
set and propagation, it is essential to use a geometrical model of the reinforcement
architecture that is as close as possible to that of the real specimen.

There are different methods to generate geometrical models of fabric reinforcements.
First, the reinforcement architecture of a woven composite can be determined by post-
treating µ-CT data [32]. The main advantage of this method is that the geometry
exactly represents the real reinforcement, taking into account all the local variations
of the architecture. However, since the material is not perfectly periodic, the choice
of a RUC and of the associated boundary conditions for FE analysis is not obvious. Geometrical models of fabric reinforcements can also be generated using automated tools that cover a large variety of weaving patterns [33, 34, 35, 36]. These models often neglect the deformation due to preforming, resulting in idealized geometries of the fabric and in much larger resin rich areas than observed experimentally. Therefore, in order to preserve the overall fiber volume fraction in the composite, the fiber volume fraction in the yarn must be set to unrealistically high values [22]. Geometries closer to the real composite can be obtained by modeling the preforming step of the dry fabric [7, 37, 38, 39]. The main objective of the compaction simulation is to obtain a realistic geometry of the reinforcement architecture including (i) a correct yarn volume fraction, (ii) a qualitative reproduction of the nesting between layers and (iii) an appropriate deformation of the yarn cross sections due to the neighboring yarns in contact. In this work, the simplified method described in [7], is used, which includes (i) a simplified mechanical behavior of the yarns, (ii) idealized yarn shapes in the non-compacted reinforcement and (iii) periodic boundary conditions on a single RUC, thus not accounting for the geometrical variability of the reinforcement architecture observed experimentally. This method is used to generate five geometries with different shifts between the fabric layers: (i) no shifts, (ii) shifts resulting in maximum nesting and (iii) the layer shifts measured on the three tested specimens (0°, 90° and 45°). The layer shifts of each RUC are summarized in table 1. Despite the simplified compaction modeling, the layer nestings in the generated RUCs agree well with optical microscope observations on the edges of the three specimens. The microscope images are shown together with the geometries in Figure 2.

## 3.2. *Comparison with µ-CT observations*

The 3 RUC geometries corresponding to the tested specimens have been generated using the layer shifts measured on the edges of the specimens. It can be seen from Figure
2 that the reinforcement geometry obtained from the compaction simulation using the layer shifts of the 0° specimen qualitatively agrees well with the μ-CT observation. For a quantitative comparison, we evaluated (i) the yarn (warp and weft) wavelength, (ii) the spacing between the yarn middle lines of each layer, (iii) the yarn volume fraction and (iv) the yarn mean cross section area. The averages of each of these parameters taken separately in warp and weft direction are summarized in Table 2. The differences between the modeled geometry and the μ-CT observation are small (< 3.5%). Therefore, we conclude that the simplified compaction modeling yields geometries that are representative of the geometric features of a compacted and nested fiber reinforcement.

### 3.3. *Mesh generation of the RUCs*

Meso-scale FE analysis of woven composites requires the generation of a mesh of the RUC containing the deformed reinforcement and the matrix complement. A method that is widely used, due to its simplicity, is voxel meshing [11, 12, 40, 41, 42, 43, 44]. This method allows for a correct prediction of the elastic properties of undamaged materials, provided that a converged mesh is used. However, several limitations arise when predicting damage onset [7] since the yarn surfaces are not correctly represented by the element faces. The generation of consistent FE meshes with element faces that follow accurately the yarn surfaces is complex [38], especially in the zones where yarns are in contact. Therefore, a matrix layer is often inserted between the yarns [5, 15, 18, 45], which considerably simplifies the meshing procedure since each yarn can be meshed separately. If the layer is thin, very small elements are needed to avoid bad mesh quality, and if it is thick, a lower yarn volume fraction is obtained in the RUC. Therefore, the method of Grail [38], which allows the generation of consistent meshes of 2D woven composite RUCs, is used in this work to generate the FE meshes of the geometries obtained by compaction simulation. The meshes that are conformal at the contact zones between the yarns and between matrix and yarns, are composed of
quadratic tetrahedral elements to ensure an accurate modeling of the stress and strain gradients.

# 4. Comparison between numerical and experimental results

## 4.1. *Comparison of macroscopic homogenized properties*

The macroscopic elastic properties of the studied material are calculated by means of meso-macro homogenization on the five meshed RUCs previously presented. The matrix is supposed to be linear elastic. The mechanical properties of the matrix given by the manufacturer are: Young's modulus $E_m = 3.2$ GPa and Poisson's ratio $\nu_m = 0.35$. The yarn behavior is obtained by micro-meso homogenization, as in Ref. [15], using $E_f = 73.6$ GPa [46] and $\nu_f = 0.3$ for the fibers. The fiber volume fraction in the yarns is chosen such that the overall fiber volume fraction in the RUC is similar to the fiber volume fraction in the composite, i.e., 47%. Since the layer shifts are different for the five studied RUCs, the compaction process leads to different layer nestings with more or less deformed yarns. Therefore, yarn volume fractions are not the same for each RUC. As a consequence, the fiber volume fraction in the yarns has been adapted for each RUC to reach an overall fiber volume fraction of 47%. Moreover, the ratio between warp and weft volume fractions after the compaction simulation is not exactly equal to the ratio between the respective thread counts in the fabric. Therefore, different fiber volume fractions are used for warp and weft yarns such that the ratio between the fiber volumes in warp and weft yarns corresponds to that of the real material. The transverse isotropic elastic yarn properties calculated for each RUC are summarized in Table 3. The local orientation of the yarn material is calculated separately at each integration point through orthogonal projection of its position on the central line of the yarn. The transverse isotropy axis of the yarn material, i.e., the fiber direction, is determined as the tangent to the central line at the projected point. Periodic boundary conditions are
applied in the fabric plane directions of the RUC, whereas the top and bottom surfaces are left free in order to correctly represent the boundary conditions of the composite specimens in a tensile test. A uniaxial stress state in warp (respectively weft) direction (parallel to the Y-axis (respectively X-axis)) is applied to the RUC in order to model a tensile test on the 0° (respectively 90°) specimen. It is given by $\underline{\sigma}$ in equation 1. In the 45° specimen, the warp and weft directions are rotated by 45° with respect to the loading direction. Therefore, the stress tensor $\underline{\sigma}_{\underline{R}}$ is applied to the RUC with the directions of periodicity (directions of the warp and weft yarns) parallel to the X and Y axes. It is obtained from $\underline{\sigma}$ by a rotation of 45° around the Z-axis:

$$ \underline{\sigma}_{\underline{R}} = P \cdot \underline{\sigma} \cdot P^T = \begin{pmatrix} \frac{\sigma}{2} & -\frac{\sigma}{2} & 0 \\ -\frac{\sigma}{2} & \frac{\sigma}{2} & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \text{with } P = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad \text{and} \quad \underline{\sigma} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & \sigma & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad (1) $$

The measured elastic properties and those obtained through homogenization on the 5 RUCs are compared in Table 4. The influence of the layer shifts on the homogenized elastic properties is small.

## 4.2. Comparison of surface strain distributions

The strain distributions measured on the three studied specimens and the surface strain distributions obtained numerically are shown in Figure 3 for the 0° specimen, in Figure 4 for the 90° specimen and in Figure 5 for the 45° specimen. $\epsilon_{yy}$ is the longitudinal and $\epsilon_{xx}$ the transverse strain component under tensile load in the y-direction. The numerically calculated strain distributions are averaged using a Gaussian filter over the same window size as in the post treatment of the DIC data (see Section 2). During the tests of the 45° and the 90° specimens, an extensometer was fixed to the specimen. Therefore the measured strain distributions are separated in three parts since the
speckle was hidden by the extensometer in two areas.

The surface strain fields obtained from the RUCs without shifts and with maximum nesting of the reinforcement differ significantly from the strain patterns observed experimentally. These types of RUC, which are mostly used in published articles about meso-scale modeling of woven composites, do not provide satisfactory surface strain distributions. Fewer differences are observed between the results obtained from the three RUCs with layer shifts determined experimentally on the specimen edges. The best qualitative agreement between the experimental and numerically calculated strain distributions is obtained if the layer shifts corresponding to the considered specimen are used. A quantitative comparison of the strain fields measured experimentally on the three specimens and those computed with the corresponding RUC with the same layer shifts is shown in Figures 6-8. The longitudinal (Figure 6-8a) and transverse (Figure 6-8b) strain evolutions are calculated along several lines on the specimen surface (Figure 6-8c and d), two consecutive lines being separated by one RUC length. The variations between the experimental strains along different lines can be explained by small variations of the reinforcement architecture induced during the manufacturing process. Since the RUC is periodic, the numerically calculated strain evolutions are repeated periodically for the comparison with the experimental data (Figure 6-8c and d). The strains on the surface of the 45° specimen are shown based on the coordinates of the experimental setup (tension parallel to the Y-axis), i.e., the strain fields obtained from the FE calculations are rotated back by 45° (the opposite of the rotation described in Eq. 1). Since the fabric is slightly unbalanced (the RUC is not exactly a square in the X-Y plane), the directions of periodicity are not exactly parallel to the X and Y axes in the experimental setup. The strain profiles shown in Figure 8 are taken along these directions of periodicity. The measured strains agree well with the calculated strains along the corresponding line on the respective RUC.
5. Conclusion

The three RUCs generated by means of the simplified numerical simulation of the fabric compaction described in [6], using the layer shifts observed on the specimen edges, provide a satisfactory model of the composite specimens with realistic fiber volume fractions and yarn shapes, including nesting between the fabric layers. This has been shown by comparison with 2D optical microscope images of the specimen edges and in 3D by comparison of the whole RUC of the 0° specimen with a μ-CT scan.

The differences between the homogenized elastic properties calculated using five
different RUCs with different relative layer shifts, including those of the three tested
specimens and the two extreme cases without layer shifts and with shifts maximizing
the nesting between the layers, are relatively small. The calculated elastic properties
are in good agreement with the experimental values. Therefore, for the prediction of the
homogenized elastic properties, arbitrary layer shifts may be used, including simplified
models without shifts, as in [18, 19, 24].

A significant influence of the layer shifts on the surface strain distributions has been
observed. In particular, the idealized cases of in-phase stacking and maximum nesting
yield strain fields that differ considerably from those observed experimentally by DIC.
If random layer shifts are used, the differences decrease significantly. The RUC with
the layer shifts determined on the edges of the corresponding specimen yields the best
qualitative (location of the extrema) and quantitative (absolute values) agreement of
the surface strain fields with the experimental observations.

Since damage onset in the composite depends on local stress or strain concentrations,
the RUCs with the experimentally determined layer shifts seem appropriate to model
damage onset and propagation at the mesoscopic scale. Simplified layups that do not
take into account the shifts and the nesting between the fabric layers may lead to
erroneous predictions of damage onset locations and propagation.
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## Figure captions

**Fig. 1** - Geometry of the reinforcement of the 0° specimen (one RUC) obtained (a) from the μ-CT scan and (b) from compaction modeling.

**Fig. 2** - Comparison between microscope observations (top) of the edge of the (a) 90°, (b) 0° and (c) 45° specimens and the edge of the corresponding RUC (bottom) in weft (left) and warp (right) direction. (d) RUC with maximum nesting and (e) RUC without nesting.

**Fig. 3** - Longitudinal ($\epsilon_{yy}$, left) and transverse ($\epsilon_{xx}$, right) strain distributions under tensile loading in the y-direction obtained using (a) the RUC determined from the 0° specimen, (b) the RUC determined from the 90° specimen, (c) the RUC without nesting, (d) the RUC determined from the 45° specimen, (e) the RUC with maximum nesting, and (f) observed by DIC on the 0° specimen.

**Fig. 4** - Longitudinal ($\epsilon_{yy}$, left) and transverse ($\epsilon_{xx}$, right) strain distributions under tensile loading in the y-direction obtained using (a) the RUC determined from the 0° specimen, (b) the RUC determined from the 90° specimen, (c) the RUC without nesting, (d) the RUC determined from the 45° specimen, (e) the RUC with maximum nesting, and (f) observed by DIC on the 90° specimen.

**Fig. 5** - Longitudinal ($\epsilon_{yy}$, left) and transverse ($\epsilon_{xx}$, right) strain distributions under tensile loading in the y-direction obtained using (a) the RUC determined from the 0° specimen, (b) the RUC determined from the 90° specimen, (c) the RUC without nesting, (d) the RUC determined from the 45° specimen, (e) the RUC with maximum nesting, and (f) observed by DIC on the 45° specimen.

**Fig. 6** - Longitudinal ($\epsilon_{yy}$, left) and transverse ($\epsilon_{xx}$, right) strain distributions under tensile loading in the y-direction (a) observed experimentally for the 0° specimen and (b) calculated numerically using the corresponding RUC. Quantitative comparison
along several lines of (c) longitudinal and (d) transverse experimental (lines drawn in (a)) and numerical (line drawn in (b)) strain profiles.

**Fig. 7** - Longitudinal ($\epsilon_{yy}$, left) and transverse ($\epsilon_{xx}$, right) strain distributions under tensile loading in the y-direction (a) observed experimentally for the 90° specimen and (b) calculated numerically using the corresponding RUC. Quantitative comparison along several lines of (c) longitudinal and (d) transverse experimental (lines drawn in (a)) and numerical (line drawn in (b)) strain profiles.

**Fig. 8** - Longitudinal ($\epsilon_{yy}$, left) and transverse ($\epsilon_{xx}$, right) strain distributions under tensile loading in the y-direction (a) observed experimentally for the 45° specimen and (b) calculated numerically using the corresponding RUC. Quantitative comparison along several lines of (c) longitudinal and (d) transverse experimental (lines drawn in (a)) and numerical (line drawn in (b)) strain profiles.
<table><caption>Table 1: Layer shifts in warp and weft directions with respect to the top layer (the layers are numbered from top to bottom).</caption><thead><tr><th>Shifts [mm]</th><th>layer 1 (top layer)</th><th>layer 2</th><th>layer 3</th><th>layer 4</th></tr></thead><tbody><tr><td rowspan="2">0°-specimen</td><td>Warp direction</td><td>-0.61</td><td>-0.59</td><td>0.43</td></tr><tr><td>Weft direction</td><td>-0.30</td><td>3.43</td><td>2.93</td></tr><tr><td rowspan="2">90°-specimen</td><td>Warp direction</td><td>1.32</td><td>2.50</td><td>-3.56</td></tr><tr><td>Weft direction</td><td>1.71</td><td>3.24</td><td>4.09</td></tr><tr><td rowspan="2">45°-specimen</td><td>Warp direction</td><td>-2.57</td><td>0.58</td><td>-2.31</td></tr><tr><td>Weft direction</td><td>-0.63</td><td>0.29</td><td>0.59</td></tr><tr><td rowspan="2">No nesting</td><td>Warp direction</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Weft direction</td><td>0</td><td>0</td><td>0</td></tr><tr><td rowspan="2">Nesting max</td><td>Warp direction</td><td>2.27</td><td>0</td><td>2.27</td></tr><tr><td>Weft direction</td><td>2.50</td><td>0</td><td>2.50</td></tr></tbody></table>

<table><caption>Table 2: Comparison between geometrical parameters obtained from a μ-CT scan and from simulation of dry fabric compaction with layer shifts determined on the specimen edges.</caption><thead><tr><th>Property</th><th>Yarn type</th><th>Specimen</th><th>RUC-0°</th><th>Difference</th></tr></thead><tbody><tr><td rowspan="2">Yarn wavelength [mm]</td><td>Warps</td><td>9.17</td><td>9.1</td><td>1.9%</td></tr><tr><td>Wefts</td><td>10.3</td><td>10.01</td><td>2.8%</td></tr><tr><td rowspan="2">Spacing between middle lines [mm]</td><td>Warps</td><td>5.01</td><td>5.02</td><td>0.2%</td></tr><tr><td>Wefts</td><td>4.57</td><td>4.55</td><td>0.44%</td></tr><tr><td rowspan="2">Volume fraction</td><td>Warps</td><td>40.4%</td><td>42.4%</td><td>2.0%</td></tr><tr><td>Wefts</td><td>46.3%</td><td>46.2%</td><td>0.1%</td></tr><tr><td rowspan="2">Mean cross section area [mm²]</td><td>Warps</td><td>0.89</td><td>0.88</td><td>1.1%</td></tr><tr><td>Wefts</td><td>0.86</td><td>0.89</td><td>3.5%</td></tr></tbody></table>
<table><caption>Table 3: Homogenized elastic properties of the warp and weft yarns used in the FE simulations of the five RUCs.</caption><thead><tr><th>RUC</th><th>Yarn type</th><th>Volume fraction (%)</th><th>E<sub>l</sub> [GPa]</th><th>E<sub>t</sub> [GPa]</th><th>ν<sub>lt</sub></th><th>ν<sub>tt</sub></th><th>G<sub>lt</sub> [GPa]</th></tr></thead><tbody><tr><td rowspan="2">RUC-0°</td><td>Warps</td><td>41.1</td><td>41.5</td><td>9.7</td><td>0.32</td><td>0.43</td><td>7.1</td></tr><tr><td>Wefts</td><td>44.9</td><td>41.8</td><td>9.8</td><td>0.32</td><td>0.43</td><td>7.2</td></tr><tr><td rowspan="2">RUC-90°</td><td>Warps</td><td>37.3</td><td>45.5</td><td>11.3</td><td>0.32</td><td>0.41</td><td>8.2</td></tr><tr><td>Wefts</td><td>40.8</td><td>45.7</td><td>11.4</td><td>0.32</td><td>0.41</td><td>8.3</td></tr><tr><td rowspan="2">RUC-45°</td><td>Warps</td><td>37.5</td><td>44.3</td><td>11.1</td><td>0.32</td><td>0.41</td><td>8.2</td></tr><tr><td>Wefts</td><td>40.3</td><td>45.2</td><td>11.6</td><td>0.32</td><td>0.41</td><td>8.5</td></tr><tr><td rowspan="2">No nesting</td><td>Warps</td><td>37.5</td><td>45.2</td><td>11.2</td><td>0.32</td><td>0.41</td><td>8.2</td></tr><tr><td>Wefts</td><td>42.9</td><td>43.6</td><td>10.5</td><td>0.32</td><td>0.42</td><td>7.7</td></tr><tr><td rowspan="2">Nesting max</td><td>Warps</td><td>44.3</td><td>38.7</td><td>8.8</td><td>0.32</td><td>0.44</td><td>6.5</td></tr><tr><td>Wefts</td><td>48.6</td><td>38.8</td><td>8.8</td><td>0.32</td><td>0.44</td><td>6.5</td></tr></tbody></table>

<table><caption>Table 4: Macroscopic mechanical properties measured experimentally and calculated by means of FE periodic homogenization using several RUCs with different layer shifts.</caption><thead><tr><th>Properties</th><th>Experimental</th><th>RUC-0°</th><th>RUC-90°</th><th>RUC-45°</th><th>no nesting</th><th>nesting max</th></tr></thead><tbody><tr><td>E<sub>0°</sub> [GPa]</td><td>20.3 ± 0.3</td><td>21.2</td><td>21.7</td><td>20.4</td><td>21.0</td><td>20.7</td></tr><tr><td>ν<sub>0°</sub></td><td>0.126 ± 0.009</td><td>0.132</td><td>0.129</td><td>0.138</td><td>0.142</td><td>0.133</td></tr><tr><td>E<sub>90°</sub> [GPa]</td><td>22.5 ± 0.5</td><td>21.5</td><td>22.9</td><td>21.5</td><td>22.6</td><td>21.9</td></tr><tr><td>ν<sub>90°</sub></td><td>0.147 ± 0.007</td><td>0.134</td><td>0.136</td><td>0.146</td><td>0.153</td><td>0.141</td></tr><tr><td>E<sub>45°</sub> [GPa]</td><td>11.2 ± 0.7</td><td>10.04</td><td>10.66</td><td>10.43</td><td>10.20</td><td>9.87</td></tr><tr><td>ν<sub>45°</sub></td><td>0.517 ± 0.02</td><td>0.59</td><td>0.59</td><td>0.57</td><td>0.60</td><td>0.60</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>Figure 1:</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2:</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3:</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4:</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 5:</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6:</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 7:</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8:</figcaption></figure>