Numerical Characterization of the In-plane Shear Behaviour of Non-Crimp Fabric Composites

J. Appl. Comput. Mech., 11(2) (2025) 439-450 DOI: 10.22055/jacm.2024.47328.4695 ISSN: 2383-4536 jacm.scu.ac.ir Published online: August 28 2024 Shahid Chamran University of Ahvaz Journal of Applied and Computational Mechanics Research Paper Numerical Characterization of the In-Plane Shear Behaviour of Non-Crimp Fabric Composites L.M. Ferreira 1,2 , E. Graciani 2 , F. París 2 1 Grupo de Elasticidad y Resistencia de Materiales. Escuela Técnica Superior de Ingeniería, Universidad de Sevilla. Camino Descubrimientos, S/N 41092 Sevilla, España, Email: [email protected] 2 Escuela Politécnica Superior, Universidad de Sevilla. C/ Virgen de África, 7, Sevilla, 41011, España, Email: [email protected] (E.G.); [email protected] (F.P.) Received July 02 2024; Revised August 13 2024; Accepted for publication August 13 2024. Corresponding author: L.M. Ferreira ([email protected]) © 2024 Published by Shahid Chamran University of Ahvaz Abstract. Experimental off-axis tensile tests aimed at characterising the in-plane shear behaviour of biaxial non-crimp fabric (NCF) composite laminates obtained from different directions within the same panel (warp and weft) reveal noteworthy variations. The in-plane shear modulus, shear strength, and shear strain at failure are significantly higher in the weft direction compared to the warp direction. To gain insights into the underlying reasons for these discrepancies, a parametric study is performed. This numerical study utilises mesoscopic 3D finite element (FE) models, representing the unit cell of a [+45,-45] 2S NCF laminate. The analysis indicates that neither the presence of stitching yarns nor the out-of-plane fibre crimp induced by the yarns account for the substantial differences observed in the experimental findings. The maximum initial tangent in-plane shear modulus (G xy ), due to out-of-plane fibre crimp is only 2%. However, this difference increases to approximately 6.1% when stitching yarns are also included in the FE model. Moreover, it is found that fibre volume fraction of the tows and the non-linear behaviour negligibly impact the inplane shear performance of the NCF laminate in both directions. In contrast, the study reveals that the primary factor contributing to the differences between the warp and weft directions is related to misalignment between the nominal 45° tows within the NCF composite laminate panels. The misalignment observed in the experimentally tested specimens is approximately 6°, leading to an 8.5% decrease in the G xy of SP laminate (warp direction) compared to ST laminate (weft direction). Keywords: In-plane shear behaviour; Composites; Non-crimp fabric; Finite element analysis; Fibre crimp. 1. Introduction Non-crimp fabric (NCF) composites represent a significant alternative within the realm of high-performance composite applications. The defining feature of NCF composites resides in the arrangement of unidirectional fibre tows, which are not woven but instead positioned side by side and held in place by stitching yarns [1, 2]. The alignment of fibres permits the customisation of mechanical performance, enabling the optimisation of strength and stiffness in multiple directions, thus accommodating specific load-bearing requirements [3]. Moreover, NCF composites offer several advantages over traditional woven fabrics, primarily due to their lower crimp, which leads to enhanced mechanical properties. The alignment of fibres without the interlacing typically seen in woven fabrics reduces stress concentrations and potential weak points within the composite material. This results in better load distribution and higher tensile strength, making NCF composites particularly suitable for applications requiring high performance under complex loading conditions. In the aerospace and automotive industry, for example, the use of NCF composites allows for the creation of lightweight components, contributing to fuel efficiency and overall performance [4-7]. Similarly, in marine applications, the resistance to fatigue and environmental factors. The manufacturing process of NCF composites is another critical aspect that sets them apart. The use of stitching yarns to hold the fibres in place before resin infusion allows for better control over fibre orientation and placement, leading to more consistent mechanical properties throughout the material. This process also facilitates the production of large, complex shapes. Despite their advantages, the manufacturing process of NCF composites introduces certain challenges, such as the potential for in-plane and out-of-plane fibre crimp and the creation of resin-rich areas, all of which can impact the overall performance of the composite. The fibre crimp of the tows is influenced by several factors, including the tension in the stitching yarns, and the nesting of the tows during layup [8-13]. Additionally, resin pockets form between the tows during manufacturing, further complicating the internal structure [14-16]. Collectively, these mesoscale geometric features contribute to a complex three-dimensional (3D) internal architecture that significantly impacts the mechanical performance of NCF composites. The goal of this work is to address the challenges posed by this complex internal architecture in accurately modelling NCF composites using Finite Element (FE) methods. Some earlier representative studies employed a 2D modelling approach [11, 17-21]. However, the mesoscopic architecture of these materials is not fully captured by a 2D FE model [22], as it exhibits considerable 440 L.M. Ferreira et al., Vol. 11, No. 2, 2025 Journal of Applied and Computational Mechanics, Vol. 11, No. 2, (2025), 439-450 irregularity along all material directions, leading to inaccuracies in stress and strain predictions. Consequently, the literature has seen the development of several 3D FE models designed to replicate the intricate internal structure of NCF laminates [14, 23-27]. The modelling strategy adopted in this study builds upon the approach introduced by [24] and further developed in subsequent works [25, 26, 28], which focused on predicting the compressive response of NCF composites through numerical analysis. This approach involves modelling the tows using a geometrically straight 3D FE mesh, with fibre crimp accounted for by rotating the elements’ coordinate system to align with the actual fibre direction. This method has been noted for its simplicity in modelling the tow geometry and its parametrization [22]. However, it also highlights the ongoing tension between model fidelity and computational efficiency, a recurring theme in the literature [24, 29-31]. This study builds on the experimental characterisation of NCF composites that was carried out in [32, 33], where differences in the in-plane shear behaviour of [+45,-45] 2S laminates were observed during off-axis tensile tests in different loading directions (warp and weft). To understand the underlying causes of these experimentally observed differences, a parametric study was performed using a mesoscopic scale 3D FE model of a [+45,-45] 2S NCF composite laminate. The FE model specifically considered out-of-plane fibre crimp induced by non-structural stitching yarns, a potential factor influencing the mechanical response of the laminates [34, 35]. Parameters analysed included the effect of fibre crimp due to the stitching, variations in fibre volume fraction, the non-linear behaviour of both tows and resin, and the orientation of the tows. The study systematically explored how these factors contribute to the differences in mechanical performance between laminates cut in the warp and weft directions. The findings were then compared and correlated with the experimental results reported in [32, 33] to deepen the understanding of the structural behaviour of NCF composites and to identify critical factors influencing their in-plane shear response. In this way, the novelty of this study lies in its extended parametric analysis using a mesoscopic-scale 3D finite element model to explore the impact of several different parameters, on the in-plane shear behaviour of NCF composites, providing new insights into the causes of experimentally observed directional differences in mechanical performance. Additionally, this work is motivated by the need to bridge the gap between the current modelling approaches and the complex reality of NCF composites, aiming to develop accurate and computationally efficient FE models. In this context, our approach aligns with recent trends in the literature that seek to enhance model accuracy without prohibitive increases in computational cost [31, 36, 37]. The document is structured as follows: Section 2 showcases the experimental findings, emphasising the disparities observed in laminates cut in two different directions. Section 3 details the developed numerical models, covering the geometric parameters, material properties, boundary conditions, and element types. Section 4 presents the results from the parametric study, examining the influence of factors such as fibre crimp, non-structural stitching, fibre volume fraction of the tows, nonlinear material behaviour of the tows, and tows misalignment. Subsequently, Section 5 juxtaposes the numerical results with the experimental evidence. Finally, the study’s main findings are presented in Section 6. 2. Experimental Evidence To characterise the in-plane shear properties of NCF composites, experimental tensile off-axis tests were carried out according to the EN6031 standard [38] on [+45,-45] 2S NCF laminates as part of the “Failure, performance and processing prediction for enhanced design with non-crimp fabric composites project (FALCOM)” [39]. The layout of the 8-ply standard tested specimens is shown in Fig. 1(a), featuring a length of 250 mm, a width of 25 mm and a thickness of 3 mm, as detailed in Table 1. As it is possible to observe, longitudinal strain was measured using an extensometer attached to the specimen, while transverse strain was measured with a strain gauge glued to the specimen. The specimens were cut in two distinct orientations: one aligned with the stitching yarns, denoted as “SP” (warp direction), and the other perpendicular to the stitching yarns, referred to as “ST” (weft direction). A representation of SP and ST specimens is depicted in Fig. 1(b), providing an illustration of the differences between the two orientations. (a) (b) Fig. 1. (a) Layout of the standard tested specimen, (b) differences between SP and ST specimens. Table 1. Specifications of the tested specimen. Layup Length [mm] Width [mm] Thickness [mm] Fibre volume fraction 𝑉 𝑓 𝑙 [+45,-45]2S 250 25 3 60% Numerical Characterization of the In-Plane Shear Behaviour of Non-Crimp Fabric Composites 441 Journal of Applied and Computational Mechanics, Vol. 11, No. 2, (2025), 439-450 Table 2. Experimental results for Gxy and Sxy with standard deviations, obtained for different NCF laminate configurations [31, 34]. Layup Specimen Gxy [GPa] Sxy [MPa] [+45,-45]2S SP 4.95±0.29 62.27±1.25 ST 5.43±0.29 80.54±1.83 This study utilised NCF laminates manufactured from biaxial Tenax ® HTS carbon fibre 5632 12k fabric, with a weight of 534 g/m 2 . Resin film infusion (RFI) using the HexFlow ® RTM6 resin system was employed [33, 40]. Nonstructural Sinterama Zerbion ® 50 dtex polyester stitching yarns were added to maintain the fabric’s structural integrity. The NCF laminates exhibited a fibre volume fraction of approximately 𝑉 𝑓 𝑙 = 60%. Given that the specimens were extracted from a common panel, similar results were expected for both directions. However, the ST specimens demonstrated superior shear modulus, shear strength, and shear strain at failure compared to their SP counterparts. Characteristic shear stress-stain curves obtained for the off-axis tensile tests in both orientations are presented in Fig. 2 along with the upper and lower limits of the experimental data. Results pertaining to initial tangent in-plane shear modulus G xy and in-plane shear strength S xy are outlined in Table 2. Notice that the x-axis and y-axis correspond to the transverse and longitudinal directions of the specimens, respectively, as shown in Fig. 1. The longitudinal and transverse strains were measured in the specimens using strain gauges positioned at 0° (longitudinal) and 90° (transverse) directions. In this way, 𝜎 𝑥𝑦 was calculated using Eq. (1): 𝜎 𝑥𝑦 = 𝑃 2 𝑤𝑡 (1) where, P represents the applied tensile load, and w and t denote the width and thickness of the specimen, respectively. S xy was calculated considering the highest tensile load P max sustained by the specimens during testing. Regarding G xy , it was determined by applying a standard linear fitting procedure to the stress-strain curve within the range of 0.1% to 0.4% shear strain. It is possible to observe that in the case of G xy , the differences between specimens ST and SP can reach around 9%. In the case of S xy , the differences are more pronounced, reaching approximately 25%. Noteworthy, this mechanical behaviour of the biaxial NCF specimens, when cut in two different orientations within the same panel, was also reported by González et al. in [41]. 3. Numerical Model A mesoscopic scale 3D FE model was developed for the representative unit cell (RUC) of a [+45,-45] s NCF laminate using ANSYS FE code [42]. The RUC comprises of four laminas stacked with corresponding +45° and -45° orientations, and each lamina features two half-rectangular cross-section tows with resin rich areas between them, as shown in Fig. 3. With reference to the local coordinate system, the RUC’s plane corresponds to the 12-plane, with directions 1 and 2 corresponding to the fibre direction and in-plane direction normal to the fibre. Direction 3 represents the through-thickness direction of the laminate. Non-structural stitching yarns, along with the out-of-plane fibre crimp they induce across the length of the tows, were also incorporated into the numerical model. Since a straight 3D FE mesh was employed, the out-of-plane fibre crimp was modelled following the approach outlined in [24] and which has proven successful in previous studies [25, 26, 28]. Each colour used in the RUC depicted in Fig. 3 represents a distinct fibre orientation angle, denoted as α. In this study, a linear variation in fibre crimp angle is assumed. Consequently, the theoretical and the approximate rotations are determined based on the y-coordinate of the point. Figure 3(c) illustrates the variation of the fibre orientation angle α concerning the maximum crimp angle β for each column of elements along the length of the RUC (in the y-direction). It’s important to note that the rotation of the element’s axis is executed about the x-axis. The modelling approach used enables the definition of both constant and variable outof-plane fibre crimp angles, as well as distinct crimp orientations for each lamina (either curved upwards or downwards). However, for the purpose of maximising the impact of fibre crimp, all laminas were uniformly curved for the same side, characterised by a consistent through-thickness angle β. As mentioned before, there are two primary factors contributing to fibre crimp: the presence of resin pockets, which may lead to the nesting of tows, and the existence of stitching yarns with a determinate stitching tension. It’s worth noting that previous studies predominantly focused on out-of-plane fibre crimp resulting from the presence of resin pockets [24–26, 28]. However, in this study, this fibre crimp is overlooked, and the focus is on examining the “stitching yarns” fibre crimp. Fig. 2. Typical shear stress-strain curves obtained from the off-axis tensile tests [31, 34]. 442 L.M. Ferreira et al., Vol. 11, No. 2, 2025 Journal of Applied and Computational Mechanics, Vol. 11, No. 2, (2025), 439-450 Fig. 3. 3D FE model for the RUC of a [+45,-45]s NCF laminate: (a) Shape of the tows, (b) complete RUC, including stitching and out-of-plane fibre crimp, (c) geometric parameters of the complete RUC illustrated through detailed drawing views. The geometry and mechanical properties of the constituents of the laminas were parametrically defined, enabling the study of the effect of dimensions and mechanical properties on the in-plane shear behaviour of the NCF laminate. Figure 3(c) depicts the geometrical parameters employed, with a representing the length of the RUC, t denoting the thickness of each lamina, and g indicating the width of the gap between two adjacent tows of each lamina. In this study, the numerical values of the geometric parameters were determined for a 𝑉 𝑓 𝑙 = 60%, and they were estimated based on the internal geometry and fibre content of the materials analysed in [33]. Consequently, the specific parameter values were obtained: a = 3.67 mm, t = 0.24 mm and g = 0.26 mm. 3.1. Material Properties The tows were modelled as a homogeneous transversely isotropic elastic material with bilinear shear behaviour. This nonlinearity manifests in the longitudinal and transverse through-thickness planes, specifically the 12-plane and 13-plane, where the response of the tows to shear loads is influenced by the non-linear behaviour of the resin [11,17]. To incorporate this non-linearity into the numerical model, a bilinear 𝜎 12 /𝛾 12 relationship was introduced. The stress-strain curve by Ditcher [43] was used as a reference. The transition from the initial slope to the second slope occurs at 𝜎 12 ∗ = 𝜎 13 ∗ = 60 MPa and 𝛾 12 ∗ = 1.4%. Accordingly, the following values were assigned for the initial slope 𝐺 12 𝑖 = 𝐺 13 𝑖 = 4.03 GPa and second slope 𝐺 12 𝑖𝑖 = 𝐺 13 𝑖𝑖 = 0.75 GPa for the shear moduli. Conversely, the resin-rich areas were treated as a homogeneous isotropic material. The elastic constants of the resin-rich areas (𝐸 𝑟 , 𝜈 𝑟 and 𝐺 𝑟 ) and of the stitching yarns (𝐸 𝑠 , 𝜈 𝑠 and 𝐺 𝑠 ) were obtained from [44, 45]. The mechanical properties employed in the parametric are summarised in Table 3, along with the corresponding source references. 3.2. Boundary Conditions and Element Types The applied boundary conditions on the faces of the RUC aim to replicate the tensile test in both the SP and ST directions. For the case in which the load is applied in the SP direction, a symmetry condition was imposed on one of the faces parallel to the yzplane (𝑢 𝑥 = 0, and 𝜎 𝑥𝑦 = 𝜎 𝑥𝑧 = 0), and a pure longitudinal tensile strain was applied along the x-axis on the opposite face (𝜀 𝑥 = 3%, and 𝜎 𝑥𝑦 = 𝜎 𝑥𝑧 = 0). When the load is applied in the ST direction, a symmetry condition was assumed on one of the faces parallel to the xz-plane (𝑢 𝑦 = 0, and 𝜎 𝑥𝑦 = 𝜎 𝑥𝑧 = 0), and a pure longitudinal tensile strain (along the y-axis) was applied on the opposite face (𝜀 𝑦 = 3%, and 𝜎 𝑥𝑦 = 𝜎 𝑥𝑧 = 0). In both cases coupling boundary conditions were applied in the remaining faces and point supports were used to avoid rigid body motion. In this way, the in-plane shear strain 𝛾 𝑥𝑦 for the SP and ST laminates was calculated as the difference between the longitudinal strain and the transverse strain. Table 3. Mechanical properties assigned to the fibre tows, resin-rich areas and stitching yarns. Mechanical property Units Value Fibre tows [24, 26] 𝐸 11 𝑡 GPa 167.6 𝐸 22 𝑡 = 𝐸 33 𝑡 GPa 11.44 𝜈 12 𝑡 = 𝜈 13 𝑡 - 0.3 𝜈 23 𝑡 - 0.42 𝜎 12 ∗ = 𝜎 13 ∗ MPa 60 𝐺 12 𝑖 = 𝐺 13 𝑖 GPa 4.03 𝐺 12 𝑖𝑖 = 𝐺 13 𝑖𝑖 GPa 0.75 𝐺 23 𝑡 GPa 4.03 Resin-rich areas [44, 45] 𝐸 𝑟 GPa 3.5 𝜈 𝑟 - 0.42 𝐺 𝑟 GPa 1.23 Stitching yarns [44, 45] 𝐸 𝑠 GPa 61 𝜈 𝑠 - 0.29 𝐺 𝑠 GPa 2.9 Numerical Characterization of the In-Plane Shear Behaviour of Non-Crimp Fabric Composites 443 Journal of Applied and Computational Mechanics, Vol. 11, No. 2, (2025), 439-450 The tows and the resin rich areas were modelled utilising linear solid elements (SOLID45) with eight nodes and three degrees of freedom at each node (translations in the directions 1, 2 and 3). The non-structural stitching was implemented using 3D spar elements (LINK180). This element works as a uniaxial tension-compression element, providing three degrees of freedom at each node: translation in the nodal x, y, and z directions. This approach was employed in various studies using stitched composites [4649]. The configuration of each NCF lamina, denoted as [+45,-45], requires the duplication of stitching yarns in the middle of the laminate thickness. As a result, the cross-sectional area of the spar elements between two consecutive -45° laminas is twice that of the remaining elements. It is noteworthy that various stitching patterns were tested, but no significant changes in the results were observed. As indication, the complete numerical model is composed of 65,600 elements and 68,901 nodes. 4. Results of the Parametric Study A parametric study was conducted to examine the impact of various parameters on the in-plane shear behaviour of a [+45, -45] n NCF laminate. Parameters under analysis encompassed out-of-plane fibre crimp, non-structural stitching, the non-linear behaviour of the tows, lamina fibre volume fraction, and tows misalignment. The goal was to identify potential factors contributing to the observed differences in in-plane shear performance between the SP and ST laminates, as indicated in the experimental evidence. For this purpose, the numerically predicted values of the in-plane shear stress 𝜎 𝑥𝑦 are plotted against the in-plane shear strain 𝛾 𝑥𝑦 . Considering the geometric parameters defined in the numerical model depicted in Fig. 3, 𝜎 𝑥𝑦 was computed using Eq. (2). Notice that the difference between Eqs. (1) and (2) arises because the total thickness in the numerical model is defined as 4t, as shown in Fig. 3(c), whereas in the experimental samples, it corresponds to the total laminate thickness. 𝜎 𝑥𝑦 = 𝑃 8 𝑎𝑡 (2) 4.1. Effect of the Out-of-Plane Fibre Crimp The modelling approach employed for the out-of-plane fibre crimp enables the specification of either constant or variable outof-plane angles, while also allowing distinct crimp orientations for each lamina, as illustrated schematically in Fig. 4. Nevertheless, it was found that these variations exerted a negligible influence on the numerical predictions. In this context, the same fibre crimp orientation and angle β was assumed for all laminas. To assess the effect of β on results in both SP and ST laminates, various crimp angles (ranging from 0° to 45°) were considered. For clarity, only the results obtained without stitching yarns, and with fibre crimp angles of β = 0°, 15°, and 45° are presented. Although β = 45° may be considered a high crimp value, it was chosen as a reference to clearly discern its effect on results between SP and ST laminates. Figure 5 depicts the in-plane shear stress 𝜎 𝑥𝑦 plotted against the in-plane shear strain 𝛾 𝑥𝑦 . The findings reveal that, unless unrealistic high fibre crimp angles are considered, almost no differences can be appreciated in the in-plane shear performance between SP and ST laminates. This is evident in Fig. 5, where the results obtained with β = 15° closely resemble those with perfectly straight tows, and some differences only emerge between SP and ST for β = 45°. The results indicate that this disparity increases upon reaching the second part of the bilinear shear constitutive equation of the tows. Additionally, it is observed that both orientations demonstrate almost identical initial tangent in-plane shear modulus 𝐺 𝑥𝑦 . Specifically, for β = 45°, the 𝐺 𝑥𝑦 value for ST laminates is approximately 2% higher compared to SP laminates. While the numerical predictions suggest enhanced in-plane shear performance in ST laminates, aligning with experimental observations, it can be concluded that incorporating out-of-plane fibre crimp has a marginal effect on the overall in-plane shear performance of both SP and ST laminates. Noticeably, Yin et al. [14] similarly observed the negligible influence of fibre crimp on the in-plane shear strength of NCF composites. Their analysis employed 3D mesoscale FE models incorporating tow waviness angles ranging from 1° to 3°. 4.2. Effect of the Non-Structural Stitching The analysis in this section aims to assess the effect of non-structural stitching yarns on the observed experimental differences between SP and ST laminates. For this purpose, assuming β = 45°, the in-plane shear stress-strain curves obtained with and without stitching yarns are compared in Fig. 6. In terms of modelling, the incorporation of non-structural stitching yarns is accomplished through the utilisation of 3D spar elements, specifically the LINK180, as detailed in section 3.2. Consequently, the FE model without stitching yarns does not include these elements. It is noteworthy that despite considering larger stitching yarn areas and different stitching patterns, no significant impact was observed on the numerical predictions. The introduction of stitching yarns contributes to a slight increase in the stiffness of the NCF laminate, particularly for ST. Consequently, the initial tangent in-plane shear modulus for ST with stitching yarns is about 6.5% higher than without stitching yarns, while for SP, this difference is approximately 2.2%. Moreover, the presence of stitching yarns contributes to an increase in the differences found in 𝐺 𝑥𝑦 between SP and ST. Specifically, with the inclusion of stitching yarns, the maximum difference observed is about 6.1%, whereas without stitching yarns is 2%. These findings suggest that the incorporation of non-structural stitching in the numerical models, coupled with out-of-plane fibre crimp, intensifies the observed differences in in-plane shear performance between SP and ST laminates. Fig. 4. Schematic representation of the out-of-plane crimp orientations considered for the NCF laminas: (a) constant out-of-plane fibre crimp with identical orientation, (b) constant out-of-plane fibre crimp with distinct orientations, (c) variable out-of-plane fibre crimp with distinct orientations. 444 L.M. Ferreira et al., Vol. 11, No. 2, 2025 Journal of Applied and Computational Mechanics, Vol. 11, No. 2, (2025), 439-450 Fig. 5. In-plane shear stress 𝜎𝑥𝑦 vs. the in-plane shear strain 𝛾𝑥𝑦 for SP and ST with a maximum crimp angle β = 45°, 15° and 0°. Fig. 6. In-plane shear stress 𝜎𝑥𝑦 vs. the in-plane shear strain 𝛾𝑥𝑦 for SP and ST with and without stitching yarns and considering β = 45°. Fig. 7. In-plane shear stress 𝜎𝑥𝑦 vs. the in-plane shear strain 𝛾𝑥𝑦 for SP and ST with 𝑉𝑓 𝑙 = 60%, 65% and 70%, considering β = 45° and stitching yarns. 4.3. Effect of the Fibre Volume Fraction Lamina fibre volume fractions of 𝑉 𝑓 𝑙 = 60%, 65% and 70% were considered to assess their impact on the in-plane shear behaviour of SP and ST laminates. For this purpose, the RUC with a fibre crimp angle β = 45° and incorporating stitching yarns was utilised. In Fig. 7, the in-plane shear stress 𝜎 𝑥𝑦 is plotted versus the in-plane shear strain 𝛾 𝑥𝑦 for the different lamina fibre volume fractions. As could be expected, increasing the lamina fibre volume fraction 𝑉 𝑓 𝑙 enhances the initial tangent inplane shear modulus 𝐺 𝑥𝑦 for both SP and ST laminates. For example, raising 𝑉 𝑓 𝑙 from 60% to 65% results in a 16% increase in 𝐺 𝑥𝑦 . However, altering the fibre volume fraction of the lamina does not account for the experimental discrepancies between the two laminates. In fact, an increase in 𝑉 𝑓 𝑙 reduces the differences found between ST and SP. For example, increasing 𝑉 𝑓 𝑙 to 65% and 70% results in differences of approximately 5.3% and 4.6%, respectively. Numerical Characterization of the In-Plane Shear Behaviour of Non-Crimp Fabric Composites 445 Journal of Applied and Computational Mechanics, Vol. 11, No. 2, (2025), 439-450 Table 4. Parameters used to analyse the effect of the in-plane shear moduli 𝐺12 𝑖𝑖 = 𝐺13 𝑖𝑖 . 𝜎 12 ∗ = 𝜎 13 ∗ (MPa) First Slope 𝐺 12 𝑖 = 𝐺 13 𝑖 (GPa) Second Slope 𝐺 12 𝑖𝑖 = 𝐺 13 𝑖𝑖 (GPa) 60 4.03 0.10 0.75 1 Table 5. Parameters used to analyse the effect of the in-plane shear stresses 𝜎12 ∗= 𝜎13 ∗. 𝜎 12 ∗ = 𝜎 13 ∗ (MPa) First Slope 𝐺 12 𝑖 = 𝐺 13 𝑖 (GPa) Second Slope 𝐺 12 𝑖𝑖 = 𝐺 13 𝑖𝑖 (GPa) 40 4.03 0.75 60 80 4.4. Effect of the Non-Linear Behaviour of the Tows To assess the effect of the non-linear shear behaviour of the tows, three different values for the second slope of the in-plane shear moduli 𝐺 12 𝑖𝑖 = 𝐺 13 𝑖𝑖 were analysed. Additionally, three distinct values of 𝜎 12 ∗ = 𝜎 13 ∗ , corresponding to the in-plane shear stress values at which the bilinear material curve transitions to its second slope. It is important to note that, for all configurations, the initial slope of the in-plane shear moduli was kept constant. The numerical models also accounted for the presence of the stitching yarns and a fibre crimp of β = 45°. The parameters used in this study tare detailed in Tables 4 and 5. The effect of varying the second slope of the in-plane shear moduli on the on the in-plane shear stress-strain curves is illustrated in Fig. 8(a). As previously noted, the initial slope remains constant across all configurations. Moreover, the percentage difference between the curves of ST and SP laminates remains largely unchanged, with a consistent value around 6%. The influence of the stress value at which the curve transitions to the second slope is shown in Fig. 8(b). The results clearly demonstrate that choosing different stress values affect the laminates response, but do not contribute to the observed differences in the in-plane shear performance of SP and ST laminates. The difference between the in-plane shear moduli of ST and SP laminates remains consistent throughout the entire curve, with ST values consistently about 6% higher than those of SP. Overall, the parametric study reveals that altering the non-linear behaviour of the tows has no significant impact on the in-plane shear performance of SP and ST laminates. 4.5. Effect of the Misalignment of the Tows Various phases in the manufacturing process of NCF panels can impact the orientation of the fibre tows, leading to a certain degree of rotation or misalignment concerning the projected orientation. This phenomenon is evident in the C-Scan images of the NCF panels tested in [50], as illustrated in Fig. 9. Upon visual inspection of the panels, a misalignment between the nominal direction of the tows was observed, as shown in Fig. 10. The prevailing misalignment was around 6° and displayed a near-symmetry with respect to the direction of the stitching yarns. This misalignment introduced a deviation from theoretical uniformity in the laminates. The smallest angle between the tows consistently appeared in the loading direction of the ST laminates, resulting in the highest angle in the loading direction of the SP laminates, as represented in Fig. 11. By applying the Classical Laminate Theory (CLT) [51, 52], it becomes possible to assess the impact of misalignment of the tows on 𝐺 𝑥𝑦 . For instance, if the misalignment of 6° is taken into account, corresponding to a SP laminate with a stacking sequence [+42, -42] S and a ST laminate with [+48, -48] S as shown in Fig. 11, 𝐺 𝑥𝑦 increases by 12% for ST and decreases by 7% for SP when compared to perfectly aligned tows, i.e., SP = ST = [+45, -45] S . The results from the CLT underscore the impact of misalignment on the in-plane shear modulus. Nevertheless, it is essential to note that these results do not consider the out-of-plane fibre crimp or the presence of stitching yarns, both of which, as previously demonstrated, can also slightly contribute to the observed differences between the two laminates. (a) (b) Fig. 8. In-plane shear stress   versus the in-plane shear strain   for SP and ST considering β = 45° and stitching yarns: (a) 𝐺12 𝑖𝑖 = 𝐺13 𝑖𝑖 = 0.1 GPa, 0.75 GPa and 1 GPa, (b) 𝜎12 ∗ = 𝜎13 ∗ = 40 MPa, 60 MPa and 80 MPa. 446 L.M. Ferreira et al., Vol. 11, No. 2, 2025 Journal of Applied and Computational Mechanics, Vol. 11, No. 2, (2025), 439-450 Fig. 9. C-Scan images of NCF laminate panels [50]. Fig. 10. Misalignment of tows found in SP and ST laminates. Fig. 11. Schematic representation of a misalignment of 6° between the tows in SP and ST laminates. To predict how the misalignment angle of the tows influences in-plane shear performance, new 3D FE models of the RUC were generated with misalignment angles of 2°, 4°, and 6°. As a result, the FE models were modified to adopt a rectangular shape to represent the angle of misalignment. Therefore, the width a used in Eq. (2) to calculate 𝜎 𝑥𝑦 was adjusted to fit the rectangular shape of the RUC. Figure 12 illustrates the new 3D FE model, incorporating the relevant geometric parameters, with a misalignment of 6°. This configuration corresponds to an SP laminate with a stacking sequence of [+42, -42] S and an ST laminate with [+48, -48] S , as shown in Fig. 11. To facilitate the understanding of the numerically predicted in-plane stress strain curves, Fig. 13 only showcases the curves corresponding to 2° and 6°. The results underscore the impact of tows misalignment on the in-plane shear performance of both ST and SP laminates. For example, when an angle of 2° is considered, the in-plane shear modulus (𝐺 𝑥𝑦 ) of SP experiences an 8.5% decrease relative to ST. This percentage change becomes more pronounced with increasing angles, reaching approximately 11.8% and 15.1% for 4° and 6°, respectively. It is important to highlight that these results were derived under the conditions of an out-ofplane fibre crimp angle of 45° and the inclusion of stitching yarns. Notably, when these factors were not considered, the predictions exhibited less pronounced effects. Numerical Characterization of the In-Plane Shear Behaviour of Non-Crimp Fabric Composites 447 Journal of Applied and Computational Mechanics, Vol. 11, No. 2, (2025), 439-450 Fig. 12. Geometric parameters of the 3D FE model of the RUC generated with a misalignment angle of 6° between the tows, including out-of-plane fibre crimp and stitching yarns. Fig. 13. In-plane shear stress 𝜎𝑥𝑦 versus the in-plane shear strain 𝛾𝑥𝑦 for SP and ST with misalignment angles of 2° and 6°, considering β = 45° and stitching yarns. Fig. 14. Numerical-experimental correlation for the In-plane shear stress 𝜎𝑥𝑦 versus the in-plane shear strain 𝛾𝑥𝑦. Table 6. Parameters used to adjust the numerical predictions to the experimental evidence. 𝜎 12 ∗ = 𝜎 13 ∗ (MPa) First Slope 𝐺 12 𝑖 = 𝐺 13 𝑖 (GPa) Second Slope 𝐺 12 𝑖𝑖 = 𝐺 13 𝑖𝑖 (GPa) 55 5.57 0.45