Historical development of geometrical modelling of textiles reinforcements for polymer composites: A review

Abstract

Mechanical behaviour of fibre reinforced polymer composites is geared by the geometrical structure of the textile reinforcements. Consistent permeability, predictable impregnation and even distribution of resin are key factors in reaching targeted mechanical performance, whereas these parameters and others are functions of the geometrical composition. Hence numerous analytical geometrical models have been described in literature covering most of the conventional commonly used weave reinforcements. However, no current model can be broadly generalized to address, for instance, manufacturing deformation of fibres cross-section, periodic variability caused by stitching in 2.5D non-crimp stitched fabrics or arbitrary fibre orientation featuring some recently developed three-dimensional textiles reinforcements. Given that the latter structures can potentially provide more favourable mechanical behaviour in terms of bending stiffness, impact resistance and through-thickness properties, it is worthwhile attempting to adapt the available geometrical modelling concepts so as to address versatile structures. This review aims to trace the physical concepts used for modelling the geometrical structures of textile reinforcements at the mesoscale and compare applicability of various analytical and numerical models to types and geometrical dimensionality of woven textile structures.

Keywords

Polymer–matrix composites textile reinforcement geometrical modelling

Geometrical structure of technical textiles

Textile structures are hierarchical in nature [1]. They can be described in terms of dimensional scale involving five length scales, Figure 1, or alternatively, according to their geometrical dimensionality at the mesoscopic level. Geometrical dimensionality is generally relative to the mode of manufacture starting with one-dimensional (1D) fibre assemblies and progressing towards more complex two-dimensional (2D), 2.5D and three-dimensional (3D) textile structures. This paper reviews geometrical modelling of reinforcements in terms of their geometrical dimensionality at the mesoscale; unit cell.

Figure 1.

Dimensional scales of fibrous composites hierarchy.

Yarns, tows and rovings constitute the common, basic 1D textile form. Yarns are made up of a bundle of twisted staple fibres, whereas a tow is a large strand of continuous fibre filaments devoid of definite twist, collected in loose, rope-like form, usually held together by crimp. In yarn production, a roving is an intermediate state between a sliver and yarn, i.e. a drafted sliver, different from that in composites manufacturing, where a roving is a lightly twisted flat tow [2].

Two-dimensional textiles are classified as being of limited thickness; a maximum of 2 yarns will be superimposed at any point of a 2D textile. Two-dimensional weaves are made of two sets of warp and weft yarns interlacing at right angles in one plane. They are produced on a weaving loom where warp yarns go through a lifting mechanism composed of a number of harnesses, and weft yarns are fed at a constant rate through sheds, which are the gaps formed between warp yarns when the harnesses go up or down [3]. The sequence of lifting warp yarns defines the weave pattern [4] expressed in n/m, meaning that the warp end passes over ‘n’ and under ‘m’ number of weft picks, thus the sum (m+n) indicates the number of harnesses needed to create that pattern. Common patterns are the plain weave (1/1) shown in Figure 2(a), twill weaves (1/2, 2/2, 2/3, etc.), Figure 2(b) and satin weaves (usually 5H or 8H), Figure 2(c). Two-dimensional weaves may be triaxial instead of the common biaxial structure, having two sets of warp yarns interlacing with the weft yarns at 60° from each other [3].

Figure 2.

(a) 1/1 Plain weave, (b) 1/2 twill weave and (c) 1/4 satin weave.

Two-dimensional braids are analogous to weaves but they are formed as flat or tubular textile structures in diagonal interlacing. Common patterns are similar to those of weaves and have special names: diamond (1/1), regular (2/2) and hercules (3/3) as well as tri-axial braid [1]. In terms of performance, composites reinforced with 2D braids show good strength under loading in tension but not in compression [5]. Uozumi et al. [6] proposed using a preform deforming process, whereby near-net shape fibre preforms for composite reinforcements are produced in shaped ‘I’, ‘J’, ‘T’ or ‘Z’ profiles from 2D tubular braids and assembled.

Two-dimensional knitted fabrics are either warp knitted, Figure 3(a), or weft knitted, Figure 3(b). One special case of 1D/1.5D of warp knit is the pillar stitch, with the yarn always wraps around the same needle with no lateral connection between the wales. Pillar stitches are only used in connection with other binding elements, in order to increase fabric stability in the longitudinal direction. Two-dimensional knitted fabrics are not ideally suitable for technical applications, because yarn paths are not conducive to high fibre volume fractions in the preforms.

Figure 3.

(a) Warp knit and (b) weft knit.

Two-dimensional non-woven textiles are fibre-to-fabric products requiring no prerequisite yarn spinning and they used for insulation. Holding of the fibrous layers, called ‘batt’, is performed either mechanically by needle felting, chemically with a bonding agent or by thermal bonding [3].

Textiles that are relatively thick and feature a higher number of superimposed layers made of multiple planes of 2D structures with possibly some light yarns interlaced in the third dimension to bind the structure, but devoid of significant vertical interlacing from different layers are sometimes labelled as 2.5D textiles. Some non-wovens, non-crimp stitched fabrics (NCS), velvets and interlock weaves can be considered as 2.5D. They generally behave much like 2D textiles with the exception of greater bending rigidity due to their thickness. Carbon reinforcements made of NCSs, Figure (4), combine unidirectional crimp-free yarn layers laminated at different angles and assembled by warp knitting; the layered structure being designed such that the polymer–matrix composite part to be produced can withstand the expected stresses in a specific required direction [3].

Figure 4.

Non-crimp stitched fabric (NCS).

Interlock weaves are another form of 2.5D textiles. In 2.5D interlock weaves, warp weavers penetrate through planes of warp and weft yarns at an angle of inclination to form the integral structure. If the warp weavers bind two successive layers, the structure is called ‘layer to layer angle interlock’, whereas if they pass through more than two layers, it is called ‘layer-layer angle interlock’ [5]. If they pass through the entire thickness, it is called ‘through-thickness angle interlock’ (Figure 5) [7].

Figure 5.

Through-thickness angle interlock. Modified from Lomov et al. [1] and Cox and Flanagan [7].

The 3D orthogonal interlock weave, Figure 6, is different from the 2.5D (or 3D in some literature) through-thickness angle interlock weave, Figure 5, where the former has warp and weft yarns laid straight while vertical yarns pass through the entire thickness orthogonal to the direction of both [7].

Figure 6.

Three-dimensional orthogonal interlock weave. Modified from Lomov et al. [1] and Cox and Flanagan [7].

Generally, 3D textiles are flexible textile products, either planar with a substantial thickness compared to the other two planar dimensions, or produced in complex non-planar shapes. Three-dimensional reinforcements in the composites industry may optionally replace multilayer laminated 2D reinforcements for providing better through-thickness and inter-laminar properties, higher impact resistance [4] and avoiding the weak resin-rich interfaces commonly present in laminated 2D reinforcements [8].

Examples of recent developments in 3D weaving structures include: 3WEAVE® developed by Mohamed et al. [9] and introduced by 3TEX Inc. [10] and Zplex™, also developed by 3TEX Inc. [10]; featuring a 3D conformable sandwich preform combining integral skins and slender foam cores woven together by a 3D weaving loom with z-column fibres. Other examples are the spacer fabrics, Figure 7, which are composed of two parallel layers of woven or warp knitted fabrics with z-directional yarns inserted between both layers to bond the structure and create its thickness. They provide good dimensional stability and high stiffness to weight ratios in the composite parts [11]. Particularly for composites production using resin vacuum infusion, Mack and Smith [12] recommended 3D spacer fabrics as reinforcing interlaminar infusion mediums to enhance resin distribution and facilitate its penetration. Another major application of developed forms of 3D warp knitted spacer fabrics is textile reinforced concrete [13].

Figure 7.

Spacer fabric manufactured in single weaving step.

As opposed to 2D braids, 3D braided composites combine shear stiffness and strength, reduce or eliminate delamination, and provide high impact resistance, high conformability and high torsional rigidity. Carvelli et al. [14] studied the mechanical properties of 3D braided carbon/epoxy composites and proved that they had superior mechanical properties over other 2D or 3D preform composite reinforcements.

One last form of 3D textiles are the crimp-free 3D orthogonal non-woven preforms, which refer to structures of orthogonally non-interlaced yarns that can be produced on 3D weaving machines. The process involves the insertion of pure matrix to fill the areas between fibre bundles, to form composites in a further processing step [4,7].

General geometrical models for dry fabrics

Geometrical modelling of textile structures has been the subject of investigations starting from the early decades of the 20th century. The introduction of the following early basic models paved the way to the mechanical analysis of fabrics – plain weaves in particular – mainly aimed at the production of apparel, the major consumer of fabrics at that time.

It started with the early attempt introduced in classical Peirce [15] model for monofilament plain weave fabrics defined at the mesoscale. A wavy yarn was divided into intervals of crimp with length P2 (Figure 8). The yarn path was defined by combining segments of straight line running along the geometric centre of a warp yarn, crossing over a circle representing the cross-section of a weft yarn. Peirce [15] assumed that the yarns were incompressible and that their path followed a circular arc at yarn crossovers, achieving perfect contact between warp and weft yarns with the cross-sections being constant throughout the fabric. However, for multifilament yarns, the assumption of a constant circular yarn cross-section becomes unrealistic. Moreover, Peirce's yarns were assumed to be perfectly flexible, hence bending rigidity was not considered.

Figure 8.

Peirce's model. Modified from Peirce [15].

Later, Peirce improved his first model by considering an elliptical cross-section to account for yarn flattening. The accuracy of the model was improved, but its application is still limited to simple plain weaves [1].

Another approach towards the same objective was pursued by studying the problem of the elastica, Latin for a ‘thin strip of elastic material’. A classical theory involving a family of curves was formerly proposed by Bernoulli and Euler. Love [16] modelled yarn paths by considering yarn crimp in isolation, found an elastic line satisfying boundary conditions and minimized the bending strain energy. Elliptical integrals were used and a spline approximation was applied to get an average bending stiffness which is a function of yarn curvature. Comparison of these early models is shown in Figure 9.

Figure 9.

Early models describing the textile structure. After Lomov et al. [1].

Starting from Peirce's model, Kemp [17] introduced an extension of the theory to non-circular yarns, named the ‘racetrack model’ to account for compression loads leading to the flattening of yarns (Figure 10). The author assumed the constant yarn cross-section to be rectangular with semi-circular ends, and derived mathematical equations calculating a value of e, a proposed flattening factor. Hearle and Shanahan [18,19] criticized this model, as the assumption of highly curved semi-circular ends falsely induces an increase in bending energy which is not representative of flattened yarns that actually store less bending strain energy due to their reduced curvature. The authors [18] favoured the lenticular cross-section as an alternative to the rectangular one on applying this model, as depicted in Figure 12.

Figure 10.

Kemp's racetrack cross-section geometry. Modified from Kemp [17].

Figure 12.

Hearle's modification of Kemp's model. After Hearle JWS and Shanahan [19].

Mechanics of dry fabrics

Yarn tensile properties and deformations were studied through two mechanical approaches: the force vectors equilibrium approach adopted by Kawabata et al. [20 –22] and the more versatile minimum energy method introduced by Hearle and Shanahan [18,19]. However, the rise of technical textiles demanded more thorough geometrical analysis.

Kawabata et al. [20 –22] modelled the geometry of plain weave fabrics by considering warp and weft axes as straight lines with small radii, acute bends at the crossovers, and solved for yarn deformations in cases of biaxial [20], uniaxial [21] and shear [22] deformations imposed on the fabrics. In each case, modelled in Figure 11(a) to (c), tensile forces in both warp and weft directions were studied, and theoretical elongations obtained by applying force equilibrium. Both laterally compressible and incompressible yarns were studied, assuming elastic axial behaviour and no bending stiffness.

Figure 11.

Kawabata's model for (a) biaxial, (b) uniaxial and (c) shear deformations. After Kawabata et al. [20 –22].

Hearle and Shanahan [18,19] regarded fabrics as load bearing mechanisms that contain strain energy. The internal strain energy may be accounted for if yarn bending rigidity is considered. By applying the principles of minimum energy equilibrium, linear and angular deformations in the yarns are obtained as functions of applied forces by differentiating any set of algebraic equations defining the geometry; Peirce's [15] and Kemp's [17] models provided examples of basic geometrical equations, with Kemp's cross-section modified to a lenticular shape (Figure 12). The method of minimum strain energy inspired other research teams who worked later on textile geometrical modelling.

To avoid complex computations, Leaf and Kandil [23] simplified Peirce's elliptic integrals, introducing the saw tooth model of the plain woven fabric in which warp and weft yarns in a unit cell were assumed to be straight piecewise and forming pointed teeth at the crossovers (Figure 13). The authors applied the strain energy method on the deformed yarns subjected to axial loadings and bending moments at their ends.

Figure 13.

Saw-tooth model. After Leaf and Kandil [23].

Fabric geometrical models for composites

In the 1980s, industry experienced the development of more widespread applications of polymer-reinforced composite materials where different textile reinforcements are used for providing strength and stiffness. The matrix prevents the motion between the fibres; hence the mechanical behaviour of dry reinforcements (without resin) and that of composites made of reinforcements and matrix are quite different. Since then, the analytical geometric modelling of textile reinforcements of composites was the focus of numerous studies giving a body of valuable research work directed towards studying the mechanical properties of the composites and predicting their structural behaviour. In the last decade, software packages aiming at modelling generalized geometrical structures were introduced. TexGen [24] and WiseTex [25] are two main mesoscale models.

One-dimensional analytical geometrical models of textile composites

Ishikawa and Chou [26] developed three 1D analytical geometrical models among the first that were designed especially for textile composites. The first is the mosaic model, Figure 14(a), which assumes that textile composites behave as asymmetric assemblies of cross-ply laminates with fibre undulation entirely neglected. The second, named the fibre undulation model, Figure 14(b), accounts for fibre undulation and continuity in the warp direction and is applicable to plain weave composites. It expresses the yarn path through sinusoidal functions. The third one, the bridging model, Figure 14(c), accounts for the yarn undulation in the loading weft direction. This model, applicable to satin weave composites, distinguishes interlaced yarn regions from straight yarn regions that act as bridges to transfer loads. These models study the knee phenomenon and in-plane stiffness coefficients of composites based on the classical laminate theory (CLT) introduced by Tsai and Hahn [27].

Figure 14.

(a) (i) Woven fabric, (ii) impregnated in resin, (iii) idealization of the composite; (b) fibre undulation model and (c) idealization of the bridging model. After Ishikawa and Chou [26].

Using a finite element method, Zhang and Harding [28] studied the elastic constants of plain weave composites based on the principle of minimization of the strain energy, but again yarn undulation was modelled in one direction only. The above models are categorized as 1D because they consider geometrical parameters in the loading direction only, which is a major limitation to accuracy.

Two-dimensional analytical geometrical models of textile composites

Yang et al. [29] developed the fibre inclination model, which simply describes yarns as straight segments having angles of inclination to a global reference. Another simplified model, the diagonal brick model introduced by Pastore et al. [30], follows a mechanical method to predict the geometry of the reinforcements by defining yarn path and twist based on the tensile forces acting on it, in relation to its stiffness. The model evolved into a software code, BRAID, which is limited to braided composites. These two models are not perceived as sufficiently accurate because the geometry was idealized considerably by assuming a constant yarn cross-section to facilitate computations.

Naik and Shembekar introduced a series of 2D analytical models for analysing the elastic moduli of plain weave lamina [31] and plain weave laminated composites [32], and also for optimizing laminate design [33]. The models incorporate the undulation in the warp and weft directions, strand continuity and local yarn angle. A unit cell of plain weave fabric was sectioned across both the warp and weft directions, Figure 15, and then the sinusoidal shape functions developed by Ishikawa and Chou [26] were used for defining the yarn geometry. The in-plane elastic properties of the composite were predicted by using the CLT theory [27], applying the isostrain assumption to the sections dominated by warp yarns and the isostress assumption to the sections dominated by weft yarns. These models are limited to plain weaves.

Figure 15.

(a) Unit cell of plain weave lamina and (b) a section of a woven fabric laminate. After Naik and Shembekar [31,32].

Three-dimensional analytical geometrical models of textile composites

Codes based on mathematical description of unit cell.

Researchers at NASA Langley Research Center [7] conducted thorough development work on analytical geometrical models of composites, ultimately aimed at predicting their mechanical properties primarily for aircraft applications. A review of codes introduced, dated from 1997, is presented hereunder.

In a further of development of the aforementioned 1D fibre undulation and bridging models [26], Raju and Wang [34] introduced a modified CLT for the 3D mosaic model. Three-dimensional geometrical descriptions of the repeating unit cell of plain weave, 5H and 8H satin weaves were developed assuming yarns with a rectangular cross-section and undulating in the gap between orthogonal tows. The codes PW, SAT5 and SAT8 were formulated for predicting the thermoelastic behaviour of the composites, but results of the coefficients of thermal expansion were not reliable.

Pochiraju [7] developed the CCM-TEX code applicable to 3D braids and interlock weaves to compute 3D stiffness and estimates strength. Cox and Dadkhah [35] developed the WEAVE code which models the geometry of 3D interlock weaves by studying the variation of waviness angle. Cox et al. [36] formulated the code BINMOD through developing the binary model which addresses the non-linear elastic behaviour of 3D textile reinforced composites. Marrey and Sankar [7] developed the µTex-10 and µTex-20. The generalized geometry of reinforcements was modelled as a combination of yarn paths, represented by defining cross-section parameters and locating their centres. The codes predict stiffness properties and thermal expansion, but they are time intensive.

Whitcomb et al. [37] developed the SAWC code, assuming that yarns follow sinusoidal paths. The finite element method was then used for predicting a stiffness matrix, but this code is limited in application to plain weave reinforcements.

Naik [38] developed the TEXCAD code, the most generalized model that addresses different reinforcements: plain weaves, 5H and 8H satin weaves, different forms of braids, Figure 16, laminates and interlock weaves. With a few geometrical parameters, yarn paths are defined as a combination of straight and sinusoidal segments, neglecting the effect of nesting. The yarn structure is modelled discretely. Applying the isostrain assumption throughout the unit cell, stiffness of the reinforcements of the composites is obtained along with the 3D stiffness constants, thermal expansion coefficients, initial failure and progressive failure strength.

Figure 16.

Unit cell geometry of 2 × 2 2D triaxial braided composite. After Naik [38].

Figure 17.

Unit cell parameter. After Sheng and Hoa [39].

Figure 18.

Wrapping segments and rectilinear segments in yarn crimp. After Sheng and Hoa [39].

Cox and Flanagan [7] compared the aforementioned codes in terms of capabilities, unit cell representation and methodology. Tables 1 and 2 summarize their findings.

Table 1.

Summary of NASA codes in terms of textile forms and geometry of unit cell. Modified from Cox and Flanagan [7].

Code	Capabilities	Summary of method
PW SAT 5 SAT 8	Thermal expansion - Plate stiffness	A, B and D matrices computed for a single layer Stack of layers not treated (subsequent laminate analysis required)
CCM-TEX	3D stiffness	3D stiffness matrix computed for whole thickness
○Tex-10 ○Tex-20	3D stiffness - Thermal expansion - Plate stiffness	Full thickness FEM calculation Traction free surfaces included in boundary conditions A, B and D matrices computed for full thickness
SAW C	3D stiffness	3D stiffness matrix computed for single layer Macro elements can be used to make stacked laminate
TEXCAD	3D stiffness - Thermal expansion - Plate stiffness	3D stiffness matrices computed for a single layer A, B and D matrices calculated by standard laminate theory Plane stress assumed
WEAVE	3D stiffness	Plate treated as homogeneous, orthotropic body
BINMOD	3D stiffness	Boundary conditions, including traction-free surfaces defined by user

Table 2.

Summary of NASA codes in terms of capabilities and methodology. Modified from Cox and Flanagan [7].

Code	Textile forms	Geometry of unit cell
PW SAT 5 SAT 8	Plain weave	2D or 3D unit cells for single layer
	5 harness satin	Yarn paths defined as straight and sinusoidal segments
	8 harness satin	Yarn cross-sections rectangular
CCM-TEX	3D weave	Unit cell for full thickness
	2-step braids	Yarn paths defined by simple analytical functions
	4-step braids	Geometry computed from idealization of textile process
○Tex-10	General	Yarn paths defined as piecewise linear, lying in planes
○Tex-20	User defined	Yarn cross-sections defined as polygons
SAW C	Plain weave	Yarn paths defined as sinusoidal
SAW C	(FE code general)	Complex geometries formed by stacking units
TEXCAD	2D weave	Yarn paths defined as straight and sinusoidal segments
	2D braids
	User defined
WEAVE	3D weaves	Quasi-laminar 3D interlock weaves reduced to laminates
BINMOD	3D weaves	Non-periodic representative volume element

Forward to a next generation of models, Sheng and Hoa [39] developed a 3D micro-mechanical model to predict 3D stiffness of composites by applying a variational potential energy method and a variational complementary energy method. Implementation was based on 3D geometrical analysis of general woven composites where, instead of using simplified sinusoidal functions to describe the yarn paths, the model identifies the paths by defining planar orientation angles θ_i and crimp function f(x_i) (Figure 17). The crimp function consists of two mathematical functions: an elliptical function for wrapping yarn segments and a rectilinear function for rectilinear yarn segments (Figure 18). Geometrical calculations involve the parameters shown in Figures 17 and 18, in addition to manufacturer's specifications regarding number of fibres in the yarns, fibre diameter and packing fraction. This model was implemented in code named TEXPROP_WFCM written in MATLAB53.

In an altogether simpler approach and basically applicable on dry fabrics used as reinforcements for composites, Hivet and Boisse [40] introduced the consistent 3D geometrical model for 2D dry non-deformed fabrics, calling it consistent as it guarantees no penetration between yarns. The authors proved experimentally that the section varies along the trajectory, and this was accounted for by using control sections at control points, assuming straight trajectories in the contact-free zones based on small bending stiffness. This approach gives way to considering the relation between the contact points between yarns and their varying cross-section.

The novelty lies in that the model can be identified using as few as three parameters per direction for a balanced plain weave fabric and up to seven parameters per direction for an unbalanced twill woven fabric. Input parameters measured are seen in Figure 19. Control points M_αi are defined by calculating the coordinates of each point in 3D, where α is the weave direction, i.e. α=1 for weft and α=2 for warp.

Figure 19.

Parameterization of the transverse cut (direction α=1) of a twill 3/2. After Hivet and Boisse [40].

This model was used for predicting the permeability of fibrous reinforcement during resin transfer moulding composites processing [41]. X-ray tomography was used for analysing the internal textile geometry. However, a limitation to this model is that it is not applicable to 2.5D and 3D fabrics, knitted or braided fabrics. The authors later presented an extension to the same approach [42] that applies to 2.5D weaves, interlock fabrics in particular, but due to the diversity of fabric structures in 2.5D this recent approach was totally different; unit cells were built by selecting from a library of predefined parameterized sections and conics.

Three-dimensional analytical models based on decomposition of the unit cell.

Hewitt et al. [43] developed computer-generated models of general fabric unit cells for 2D woven reinforcements. This involved compiling a library of 32 subcells which represent various fibre undulations, with the capability of assembling them according to the weave pattern being investigated. The model was introduced to aid the design of dry woven fabrics and provide a geometric description towards modelling the properties of textile reinforced composites. The library of cells describes general yarn interlacing patterns but does not give sufficient information for highly accurate geometric modelling. Vandeurzen et al. [44] developed the combi-cell model (CCM) using TEXCOMP code written in EXCEL®, which generally describes unit cells for any 2D woven reinforcements for macro–micro analysis. At the macro level, the pattern is split into two layers across the thickness, with each layer containing a number of squares composing the pattern altogether. Each of these squares is further split into four rectangular macro-cells (Figure 20). A library of 108 macro-cells was designed for expressing any weave pattern when properly assembled. Further micro-partitioning takes place, either along 1D by dividing each macro-cell into four micro-cells, or along 2D by dividing the macro-cell into 50 micro-cells. Then this micro-partitioned geometrical structure is analysed mathematically using the code developed by the authors and assuming idealized geometry with no nesting or irregularities; a perfect lenticular shape is selected for describing the yarn cross-section. This model was successfully applied in Vandeurzen et al. [45] to predict in-plane stiffness of 2D reinforced composites, but the geometrical partitioning adopted is labour intensive and some cases of yarn interaction or very high crimp cannot be modelled. Besides, assuming a perfect idealized geometry throughout the structure is unrealistic.

Figure 20.

Macro-partition of the plain-weave fabric unit cell. After Vandeurzen et al. [44].

The same research discussed drawbacks of the CCM [44] under computing out-of-plane stresses in hybrid textile composites, mainly associated with applying the isostrain assumption throughout the composite and assuming a homogeneous fabric. To overcome this, the authors developed the TEXCOMP-CEM (complementary energy model) [46]. The model implements multilevel geometrical analysis by decomposing the unit cell over five levels of hierarchy and a multistep homogenization procedure through computing stress concentration factors at each step. These stress concentration factors are tensors that relate the average stress in each decomposed level to the uniform stress on the boundary of the unit cell. This approach is parallel to the Finite Element Method (FEM) and the TEXCAD [38] code used for computing stiffness properties.

In the porous matrix model, Kuhn and Charalambides [47] treated the composite unit cell as four separate unidirectional layers of non-uniform thickness; these four layers being labelled as the lower matrix layer, fill tow layer, warp tow layer and upper matrix layer (Figure 21). The model aimed at producing piecewise continuous shape functions representing the top and bottom surfaces of each of the four layers by employing unit-cell dimensional parameters, Figure 22, in describing yarn profiles. These surface functions were then implemented in the modified CLT [27] to compute stiffness; for further validation, results were compared with those obtained by applying a 3D finite element method. The same procedure was followed with ceramic matrix composite characterized by the presence of large-scale voids but this case is not of interest here. The developed surface functions represented a plain weave composite, but they can be employed to model other structures as well. The model is accurate and easy to apply but requires further development to include other complex reinforcements.

Figure 21.

Plots of surface functions describing the top and bottom surfaces of each of the layers in the porous matrix model. After Kuhn and Charalambides [47].

Figure 22.

Plain weave fabric repeating unit-cell cross-sectional geometry and related parameters. After Kuhn and Charalambides [47].

Moving further to more complex reinforcements and inspired by the method of non-uniform layer separation introduced by Kuhn and Charalambides [47], Rao et al. [48] modelled unit cells of 4H, 5H and 8H satin weave composites while introducing a new middle matrix layer. The five layers were the warp and fill tow layers, and the upper, lower and middle matrix layers. The same modelling procedure was followed, first by describing the geometry of the unit cell through a different approach where it was subdivided into binary subcells using the method introduced by Hewitt et al. [43]. The authors found a library of six binary subcells to be sufficient, when correctly assembled, for constructing the repeat unit cell of 4H, 5H or 8H satin woven structures. Two examples of 4HS and 8HS woven structures are illustrated in Figure 23, and an isometric view of the 5H woven structure appears in Figure 24. Then, the authors [48] adapted the mathematical surface functions developed in the porous matrix model [47] to describe the spatial variation of the satin weave micro-cells layered in the five non-uniform layers. The resulting equations were employed for 3D finite element analysis in predicting micro-damage and composite fracture as well as the thermal behaviour of satin and plain woven fabric polymer and ceramic matrix composites.

Figure 23.

Top views of the RUC of the (a) 4HS woven System, (b) the 8HS woven system and (c) top view of each binary subcell. After Rao et al. [48].

Figure 24.

(a) Top view of the RUC of the 5HS woven system and (b) an isometric view of the corresponding 3D finite element mesh of the 5HS weave pattern. After Rao et al. [48].

Summaries of 3D analytical geometrical models, applicable textile structures and output mechanical properties are presented in Tables 3 and 4.

Table 3.

Structure forms applicable to 3D geometrical models.

Unit cell model	Author	Code	Textile structure
Mathematical description	Sheng and Hoa³⁹	TEXPROP_WFCM	General
Mathematical description	Hivet and Boisse⁴⁰		2D dry fabrics
Decomposition	Hewitt et al.⁴³		2D woven structures
	Vandeurzen et al.⁴⁵	TEXCOMP	2D woven structures
	Vandeurzen et al.⁴⁶	TEXCOMP-CEM	2D woven structures
	Kuhn and Charalambides⁴⁷	Porous matrix model	Plain weave composite
	Rao et al.⁴⁸		4H, 5H and 8H satin weave composites

Table 4.

Input data and successful applications of 3D geometrical models.

Unit cell model	Author	Inputs	Applications
Mathematical description	Sheng and Hoa⁹	Planar orientation angles θ_i and crimp function f(x_i)	3D stillnesses
Mathematical description	Hiveta and Boisse⁴⁰	Three to 7 parameters	Permeability during RTM
Decomposition	Hewitt et al.⁴³	Library of 32 subcells	Design of woven fabrics
	Vandeurzen et al.⁴⁵	Library of 108 macro-cells	In-plane stiffness
	Vandeurzen et al.⁴⁶	Decomposing the unit cell over five levels of hierarchy	Stiffness properties
	Kuhn and Charalambides⁴⁷	Four separate unidirectional layers of non-uniform thicknes	Stiffness
	Rao et al.⁴⁸	Five separate unidirectional layers of non-uniform thicknes library of six binary subcells	Micro-damage- composite fracture- thermal behaviour

Software packages

TechText CAD.

TechText CAD software [49] marketed by TexEng Software Ltd. [50] can be used for defining the internal geometry by minimizing the sum of extension, bending and flattening yarn energies based on the method of Hearle and Shanahan [18]. Potential energy is calculated through the product of forces or moments and displacements or rotations, while bending energy is the product of bending moment and curvature. The biaxial deformation is determined by defining the axial and transverse displacements. Yarn paths are defined by interpolation of B-splines, while yarn cross-section is defined by the radial lengths of a series of angles around the yarn circumference. This tool is limited to studying the uniaxial and biaxial behaviour of the plain weave fabrics. Its application on studying mechanical properties of the textile composites is hence undeveloped.

TexGen.

TexGen was originally developed by Robitaille et al. [24,51,52] and later Sherburn [53] modified the code introducing TexGen version2 in 2003 and version3 in 2007. The aim was to implement an algorithm for a generic geometric model at the level of the unit cell that can be applicable to a large array of 2D and 3D textile structures. Yarns are treated as solid volumes and the cross-section is approximated to be the smallest region that encompasses all of the fibres within the yarn. The yarn path is modelled by specifying its centreline, defined by a position in 3D space as a function of the distance along the yarn. The resulting geometric definitions were supplemented with draping models for studying the mechanical properties of reinforcements. The mathematical model in Robitaille et al. [51] is applicable to woven and braided textiles, whereas in Robitaille et al. [52] a similar set of geometric equations defined non-crimp preforms assembled by warp knitting and multilayer preforms assembled by stitching.

Sherburn [53] modified the software by defining discrete points at the crossovers to describe the yarn path, for which it was enough to specify one or two points known as master nodes. Different cross-sections can be assigned to a selected sector; the ellipse, power ellipse and lenticular sections can be used. An interpolation function, either smooth or polar, is imbedded in TexGen, describing the way in which the cross-section varies along the length of the yarn.

The TexGen software [53] was utilized by Ruijter et al. [54] and others. In the work done by Ruijter et al. [54], TexGen was implemented to mesh the geometry of plain weave glass reinforced composite in order to plot the V_f variation in relation to the damage distribution, focusing on nesting and other geometrical parameters influencing the fibre distribution.

WiseTex.

WiseTex is assumed to be the only multiscale numerical model integrating the micro–meso–macro characteristic hierarchy of textile reinforcements. The WiseTex software package was introduced by Verpoest and Lomov [25]. It is a comprehensive multiscale tool that integrates a family of fabrics geometrical models along with other predictive models. Different universities and research centres in Japan, France, Spain, Italy, Czech Republic and Belgium contributed to the development of this software that is applicable to 2D or 3D woven structures.

WiseTex is based on an older model, CETKA-KUL (CETKA means ‘net’ in Russian) developed by Lomov et al. [55], which serves as a preprocessor for mechanical models at the meso-level. Based on using the principle of minimum energy, and given the weave geometrical and mechanical parameters, the software defines all bend intervals and consequently computes the spatial displacement of fibres. The data generated are further implemented in the micro-level fibre distribution mode or the meso-level yarn path mode.

For further predicting the mechanical properties, some analytical models were integrated into the WiseTex software package, allowing virtual design of textile patterns and their simulation in a composite structure. Moreover, some processing or performance properties of the textiles or composites may be predicted. Among the integrated models are the TEXCOMP model [44 -46], lattice Boltzmann solver [56] and LamTex model [57] predicting the stiffness of textile composites, permeability of textile reinforcements and internal geometry of textile laminates, respectively.

The focus of reviewing these models is to study concepts employed for geometrical modelling of textiles composites at the mesoscopic scale. The reader is referred to Vassiliadis et al. [58] for a review of other numerical models investigating the three basic modelling scales; micro–meso–macro.

Digital fabric mechanics analyser (DFMA).

Wang and Sun [59] used the ‘multichain digital element analysis’ approach, in which yarns are represented at the micro-scale as a chain of rod elements connected by frictionless pins. The contact between yarns is simulated by inserting contact elements. Digital fibres are then assembled into a woven topology and follow a relaxation process to predict the geometry; hence defining fabric geometry using topology and fibre properties. The output of simulation results is organized as reinforcement surface mesh (STL format) or point cloud data.

Overview of approaches in modelling cross-section and yarn path

To analyse a fabric geometrical structure, both the yarn cross-section and path are to be defined. The elliptical shape, first introduced by Peirce [15], is one of the simplest approximations with given width w and height h, and was considered by Sheng and Hoa [39] in representing their geometrical 3D model.

The lenticular cross-section is another assumption defined by the intersection of two circles of radii r₁ and r₂ each offset vertically from the section centre by distances o₁ and o₂, respectively [54]. This shape was used by Hearle and Shanahan [19] as a modification to the rectangular cross-section introduced by Kemp [17] on verifying the principle of minimum energy [18]. Both research teams of Vandeurzen et al. [46] and Hivet and Boisse [40] also selected a lenticular shape to describe the cross-section of the yarn based on their microscopic observations, where they worked on 2D woven fabrics. They assumed that the flattening of yarns that resulted from their interlacing during processing leads to lenticular cross-section. This is not generally true when dealing with other fabric structures; NCS fabrics, where the yarns experience no undulations and thus the cross-section keeps closer to its original shape.

Sherburn [53] first chose the elliptical cross-section then modified his assumption and used the lenticular cross-section to simplify the procedure of taking measurements from several sections and finally averaging the parameters. The power ellipse cross-section might be chosen as well; that is a flexible function which describes the cross-section in terms of a power n resulting in a more rectangular shape as n approaches zero and a lenticular one as n > 1 (Figure 25). On the other hand, Lomov et al. assumed three possible yarn cross-section shapes in their CETKA-KUL model [55]; either elliptical for twisted yarns, lenticular for low- or no-twist yarns or rectangular for highly packed 3D fabrics, an assumption that remains closer to realistic observations.

Figure 25.

Power elliptical cross-sections. Modified from Sherburn [53].

Due to different deformation levels in the yarn along its length, the cross-section is normally subject to variations. It is thus more accurate to define the cross-section as a function of distance along the yarn. In their studies on modelling 2D woven fabrics as reinforcements for composites, Kuhn and Charalambides [47] assumed, as later Hivet and Boisse [40] did, that the section varies along the trajectory taking into account the reorganization of the fibres near the contact zone (Figure 26). However, most of the researchers who developed 1D and 2D geometrical models assumed an idealized constant cross-section of yarns, an assumption used for simplifying the computations, but lacks accuracy and results in misleading estimations of fibre volume fraction V_f.

Figure 26.

Shapes of cross-section along the tow. After Kuhn and Charalambides [47].

Implementation of geometrical models in finite element tools

Drawbacks faced with finite element mesh generation of mesoscale geometrical models in case of compacted plain weave textile reinforcements were highlighted by Grail et al. [60], who presented an algorithm defining the smooth contact zones at yarn surfaces obtained from available geometrical models [25,40,53]; hence eliminating voids and interpenetration associated with the interfaces at these surfaces. Another approach is the spatial modelling of 3D textiles introduced by Stig and Hallström [61] where the authors employed TexGen [53] for generating periodic geometry of the repeated volume element (RVE) of 3D textile reinforcements, then exported the model for meshing. Smitheman et al. [62] also used TexGen [53] to provide the geometric model of RVE which is then divided into voxels and homogenized. Finite Element Analysis (FEA) tool was used to obtain thermomechanical properties of the bulk composite.

Drach et al. [63] employed the DFMA [59] for providing point clouds of 3D fabric reinforcements used to develop FEA mesh and then run numerical simulations. The artefacts of the geometrical model were accounted for by implementing a code written in Matlab and by using FEA geometrical preprocessors. In its latest release, WiseTex version 3.0, 2012, Lomov et al. [64] presented meso-level textile processing via Matlab scripting framework that utilizes WiseTex [25] for providing the geometrical model. The python script integrates input data from image analysis tools, textile deformation data and manufacturing specifications of textile reinforcements. Yarns cross-sections are modified and interpenetrations are removed for further export to FEA Abaqus tool for simulating 3D stress analysis.

For reviewing adaptability of the available geometrical models to implementation into FEA methods, it is recommended to refer to NASA report [65] for comparing surface-to-surface contact analysis and other challenges that preclude the feasibility of such models. In a related context, Hallal et al. [66] evaluated the available analytical models from a different perspective by classifying them in terms of micro–macro homogenization methods, i.e. by applying CLT, isostress and/or isostrain assumptions or inclusion method. The authors concluded that the CLT can be reliable in predicting stiffness properties in case of laminates or simple 2D geometrical structures where subdivision of the RVE is straightforward, whereas the isostrain assumption, while can be more generic, but lacks accuracy. By comparing analytical results versus experimental data, the authors [66] recommended inclusion methods or methods of cells for predicting stiffness properties.

Conclusion

Generally, analytical models of textile reinforcements depend on available descriptions of the internal geometry of the textile obtained empirically, which means that the fabric has to be produced and then measured [25], and so the analytical approaches result in models that describe existing textile structures but do not predict the generalized geometry. They are also limited to representing one specific fabric structure and aim at predicting one specific mechanical property of the composite. Another major drawback is that their use is typically time consuming, for they usually require collecting a large amount of data and performing lengthy calculations.

On the other hand, software packages provide a versatile tool for modelling any type of weave structure more efficiently with an easy implementation in the FEA methods. However, challenges are envisaged with simulating deformed scenarios.

Geometrical analytical models reviewed in this work vary in their complexity and modelling strategies. They cover most of the commonly used weave reinforcements but no model can be generalized to include all 2D, 2.5D or 3D weave structures in any deformed state. Hence, defining one convenient and reliable model in terms of accuracy is a complicated task, especially that most of the analytical models are restricted in their applicability to certain textile structures and/or output information. It is aimed at presenting a comprehensive comparison of their accuracy in a separate paper.

To the authors' knowledge, no model can be generalized to include 2.5D stitched non-woven structures that are characterized by variability in in-plane fibre configurations and probable occurrence of defects as a result of stitching. Moreover, as compared to realistic observations, most of these models do not account for variability in the internal yarn and fibre configurations, i.e. misalignments, defects, nesting and varying yarn sections. These drawbacks were precluded in the work done in the literature [60 -64]. The so mentioned limitations offer an open channel for future investigation.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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