Influence of uniaxial and biaxial tension on meso-scale geometry and strain fields in a woven composite

Composite Structures 77 (7) 45 418 www.elsevier.com/locate/compstruct Influence of uniaxial and biaxial tension on meso-scale geometry and strain fields in a woven composite P. Potluri *, V.S. Thammandra Textile Composites Group, Textiles and Paper School of Materials, University of Manchester, Manchester M6 1QD, United Kingdom Abstract Representative volume element (RVE) of a woven composite is often idealised to have identical geometry in the warp and weft directions. While this may be true in the case of consolidating a completely relaxed preform with equal warp and weft densities, majority of woven composites have non-ideal RVE geometry. The present paper investigates the influence of membrane stresses applied during the moulding process on the mechanical properties of the finished composite. Crimp interchange due to uniaxial stress and tow flattening due to biaxial stress have been computed by a fabric compliance model developed by the authors. Compliance model is based on the principle of stationary potential energy by taking into consideration tow bending, compression and extensional energies, and the external work done by the tensile forces. It has been shown that a uniaxial tensile stress applied to the preform results in an increase in tensile modulus and a corresponding reduction in the transverse modulus, as a result of reduced crimp in the loading direction with a corresponding increase in the transverse direction. Tensile strength and stiffness increase in the loading direction, and the mode of failure initiation changes from knee phenomena to tow failure. A biaxial stress applied during the forming stage results in crimp reduction in both axial and transverse directions; this leads to an improved tensile modulus and strength of a composite laminate. RVE of the deformed geometry has been modelled using ABAQUS CAE pre-processor. The elastic moduli were computed using the concept of equivalent strain energy proposed by Zhang and Harding. The resulting strain fields were compared for various cases of uniaxial and biaxial extension. Ó 6 Elsevier Ltd. All rights reserved. Keywords: Woven composites; Fabric tension; Meso-scale; Tow geometry 1. Introduction * Corresponding author. Tel.: +44 161 418; fax: +44 161 955 818. E-mail address: Prasad.Potluri@manchester.ac.uk (P. Potluri). Biaxial woven, non-crimp and braided fabrics are popular reinforcements for moulding composite parts with complex double-curvatures. These fabrics are subjected to in-plane tensile and shear forces, and out-of-plane compression and bending forces during moulding process. While there is a large volume of literature on shear [1 4] and compression [5,6], very little is published on the tensile behaviour. Boisse et al. [7], Sagar et al. [8] presented tensile load-deformation behaviour of plain woven fabrics. Bending is less critical for relatively thin D fabrics bent to modest curvatures, but may become significant for thicker 3D fabrics. The present paper deals with the load-deformation behaviour of woven fabrics subjected to thread-line tensions in uniaxial and biaxial configurations. The main objective here is to study the effect of yarn tensions on meso-scale geometry of textile unit cell, which in turn affects the micro-mechanical behaviour of a composite RVE. 1.1. Woven fabric geometry Woven fabrics are characterised by the interlacement of warp and weft yarns orthogonal to each other, resulting in crimp or waviness in both the directions. The crimp has a significant influence on both the moduli and strength of a composite structure. A multi-filament glass or carbon yarn is normally modelled with an elliptical (Fig. 1a) or a lenticular cross-section. In the case of uniaxial loading, crimp 63-83/$ - see front matter Ó 6 Elsevier Ltd. All rights reserved. doi:1.116/j.compstruct.6.1.5

46 P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 h / Free zone Contact Zone value reduces in the loading direction and increases in the transverse direction. This phenomenon of crimp interchange continues until yarns in the loading direction become straight or the yarns in the transverse direction reach a jammed state (Fig. 1b). In the case of biaxial loading with identical loading in two orthogonal directions, there is a slight reduction in the crimp in both directions due to yarn flattening. However, in the case of unequal biaxial loading, crimp interchange does take place in the direction of higher loading. In this paper, two limit cases, h 1 / Yarn straightening in the loading direction Yarn crimping and jamming in transverse direction Fig. 1. (a) Free and contact zones in a fabric, (b) crimp interchange. d d 1 uniaxial and equi-biaxial (simply referred to as biaxial in the rest of the paper), have been considered. It can be seen from Fig. 1a that the interlacing yarn has two regions, contact zone and a free zone. The cross-sectional shape of the transverse yarns defines the geometry of a contact zone. The main geometric constraint for maintaining the contact between the interlacing yarns is that the sum of the crimp heights is equal to the sum of the yarn thickness. h 1 þ h ¼ b 1 þ b ð1þ where h 1, h are crimp heights and b 1, b are yarn thickness, for warp and weft, respectively. Using suffix 1 to denote the warp yarn or warp direction and suffix for the weft yarn or weft direction, the ordinate of the ellipsoidal yarn path is given by (Fig. a) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ei ðxþ ¼ b ci ða ci a ci a ci x Þ; 6 x 6 x ci ðþ where a ci ¼ ða j þ b i Þ b ci ¼ ðb j þ b i Þ for i ¼ 1ifj¼and i ¼ ifj¼1 The curvature of yarn path at any point along its length is calculated using the following equation q ¼ 1 R ¼ d y dx h i3 1 þ dy dx R 1 θ 1 b θ 1 a Y Y R R O θ 1 X b c1 O (x,y) X θ 1 p 1 / x 1 a c1 Fig.. (a) Elliptical path, (b) tow centre line in contact region, (c) polynomial geometry in free region.

P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 47 The radius of curvature at the end of contact zone can be expressed as 3 R i ¼ a ci ða ci x ci Þþb ci x ci a 4 cib ci where a ci x ci ¼ sin h i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4þ a ci sin h i þ b ci cos h i The horizontal projection of contact width (x ci ), radius of curvature (R i ) and slope of yarn path (h i ) at the end of contact zone are related by Eqs. (3) and (4). The yarn path in the free zone is represented by a fifthorder polynomial. The degree of polynomial depends on the number of geometric boundary conditions. The polynomial has to satisfy the following five conditions, taking the origin as the centre of the yarn length between intersections (Fig. c): dy dx h i at x ¼ p i =; dy dx h i at x ¼ p i =; q ¼ at x ¼ ; q ¼ 1=R i at x ¼ p i =; q ¼ 1=R i at x ¼ p i =; The first two conditions enforce the continuity of slope at the end of contact zone, the third condition enforces the point of inflexion to occur at the centre of yarn path, the last two conditions enforce the continuity of curvature. Since five conditions are available, it is possible to define a fifth-degree polynomial. y pi ðxþ ¼c 1 x þ c x þ c 3 x 3 þ c 4 x 4 þ c 5 x 5 It has been found that the first two conditions do not lead to two independent equations. However, an additional condition of the slope at the inflexion point (h i ) has been imposed to obtain the following condition: @y pi @x ¼ h i at x ¼ ; On substitution of the above conditions and by solving the resulting simultaneous equations, we can get the constants c 1 to c 5 in terms of the variables. c 1i ¼ h i c i ¼ c 3i ¼ ð1þh i Þ3 3p i R i þ 8 3 c 4i ¼ c 5i ¼ 4 5 h i h i p i ð1þh i Þ3 þ 16 p 3 i Ri 5 ðh i h i Þ p 4 i The spacing, crimp heights and the curvilinear lengths of yarns in the warp and weft directions are given by the following expressions. ð3þ ð5þ ð6þ x i ¼ x ci þ p i h i ¼ y ei ðx ci Þþy pi ðp i =Þ l i ¼ R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ci 1 þ dy ei dx þ R rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p i 1 þ dy pi dx; i ¼ 1; dx dx ð7þ 1.. Tensile load-deformation behaviour The yarn path, consisting of two elliptical sections and a polynomial, can be computed using the principle of stationery potential energy. It can be observed from the geometric equations that there are eight variables namely b 1, h 1, h 1, p 1, b, h, h and p. These variables can be found by minimizing the total potential energy function of the unit cell. The total energy of the unit cell under biaxial loads is given by V ¼ F 1 ðx 1 X 1 Þ F ðx X ÞþU e þ U b þ U c ð8þ where x 1, x are deformed and X 1,X are undeformed yarn spacings (hence, dimensions of the unit cell) in the warp and weft directions, respectively. The first two terms represent the potential energy of external loads from the undeformed configuration and the next three terms represent the strain energy stored in the yarns forming the unit cell due to tensile (U e ), bending (U b ) and compression (U c ). For relatively small tensile forces applied during forming, tensile energy may be neglected for an untwisted glass/carbon fibre tow. U b is computed from the following equation using the moment curvature (M j) relationship of the yarn represented by a function f b (j). The bending moment curvature relationship has been obtained using Kawabata bending tester [9] Z U b ¼ Xi¼ li Z ji f bi ðjþdj ds ð9þ i¼1 U c is computed from the following equation using the normal load compression strain (F c d) curve of the yarn represented by a function f c (d). The load compression curve has been measured using Kawabata compression tester [9]. Z U c ¼ Xi¼ di f ci ðdþdd ð1þ i¼1 The energy function given by Eq. (8) should be minimized subject to the following non-linear constraint that governs the geometric constraint of plain weave fabric (Eq. (1)): h 1 + h = b 1 + b. The solution of the deformed state essentially involves minimization of the potential energy function subject to the geometric constraints involved. For this purpose, it is computationally efficient to utilise various algorithms developed in the field of mathematical programming for constrained optimisation. In this work, NAG library routine (nag-nlp-sol) has been employed to solve the constrained minimisation problem [1].

48 P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 Fig. 3 shows the yarn path when the fabric is in a relaxed state. It may be noted that for computing relaxed yarn path, bending energy is the only significant term in Eq. (8). Fig. 4 shows the yarn path under biaxial tension of Ordinate of centre line y (mm).4.35.3.5..15.1.5.5 1 1.5.5 Distance along warp x (mm) Fig. 3. Geometry of centre line of yarn path under relaxed state. (a 1 = a = 1.76 mm, b 1 = b =.36 mm, p 1 = p =.116 mm; h 1 = h =.; h 1 = h = /76). Ordinate of centre line y (mm) Ordinate of centre line y (mm)..15.1.5-1.5 -.75 -.5.5.75 1.5 -.5 -.1 -.15 -. Distance along warp x (mm) Undeformed Biaxially Extended Fig. 4. Yarn centre line under (equi-)biaxial deformation..4.3..1-1.5 -.75 -.5.5.75 1.5 -.1 -. -.3 -.4 Distance along warp x (mm) warp-deformed undeformed weft-deformed Fig. 5. Yarn geometry due to uniaxial loading. N per yarn applied in both warp and weft directions. There is a slight reduction in crimp in both directions due to yarn flattening. Fig. 5 shows the yarn path under uniaxial loading of N/yarn. It can be seen that the yarn (warp) in the loading direction is nearly straight while the transverse yarn (weft) is highly crimped.. Micro-mechanical analysis of woven textile composites Various analytical techniques have been developed to predict the elastic properties of representative volume elements (RVEs) of textile composites. Bogdanovich and Pastore [11] presented a concise review of all these methods. Among these techniques are methods of averaging mechanical properties of the constituent materials, property predictions based upon the detailed geometric descriptions of the reinforcement, and finite element methods treating matrix and fibre as discrete components. Many models of the mechanical response of textile composites have their roots in the analysis of curved or wavy fibres within single laminates. Tarnopol skii et al. [1] developed early models to average the response of laminates containing curved filaments. Again Tarnopol skii et al. [13] were perhaps the first authors to present analytical techniques intended specifically for textile reinforced composites. They approached the problem as a generalisation of unidirectional composites using the so-called modified matrix method. The concept was to reduce the three-dimensional extent of the problem by combining the fibres in a particular direction with the matrix material to create an effective medium in the sense of unidirectional micromechanics. Out of various models based on stiffness averaging, the models put forward by Ishikawa and Chou [14] were more rational as they take into account fibre continuity in the thickness direction and the fibre undulation, which are distinctive features of textile composites. The basic principle in their models is to treat a system of two orthogonal sets of fibres in a woven composite as a laminate with two laminae placed in /9 direction. They proposed three different models, i.e., mosaic model, crimp (fibre undulation) model and bridging model. The mosaic model idealises the fabric composite as an assemblage of asymmetric cross-ply laminates and the elastic properties of the composite were found by stiffness (parallel model) or compliance (series model) averaging which give upper and lower bounds for the stiffness constants of the composite. The key simplification is the omission of the fibre continuity and undulation (crimp) that exists in an actual fabric. The crimp model incorporates the fibre continuity and undulation in one direction by describing the geometry of reinforcement using trigonometric functions. The bridging model employs both stiffness and compliance averaging in two orthogonal directions and is suitable to model satin weave composites. Naik and Ganesh [15] extended the models of Ishikawa and Chou by considering the fibre undulations in two orthogonal directions using more detailed trigonometric functions. Models based on a finite element approach offer to examine the detailed stress

P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 49 field throughout the RVE, potentially providing the information necessary for failure analysis and damage propagation studies. Three-dimensional models employing conventional finite elements have been presented by many researchers but notable work was done by Whitcomb [16] and Dasgupta et al. [17]. Typically the yarns are described as lenticular in cross-section and the yarn path is described through some trigonometric relationship. The yarn was treated as a transversely isotropic material and the matrix was treated as an isotropic material. The stress field under in-plane loads was created by imposing appropriate displacements consistent with the boundary conditions. The constraint forces generated at the faces of RVE were related to the external forces and average normal strains and average Poisson s ratios were determined. The effective Young s modulus in a particular direction was then calculated using the energy balance equation by equating the work done by the resultant force in that direction to the strain energy stored in RVE due to the displacement in the corresponding direction. The analysis was used to study several different variations of plain weaves that illustrate the effect of tow waviness on composite modulii, Poisson s ratios and internal strain distributions. A more detailed geometric description of plain weave fabric geometry for creating FEA models was attempted by Kuhn and Charlambides [18] by taking into account the asymmetry of the cross-section and variation of the cross-sectional geometry from contact region to free region. They developed continuous mathematical functions to describe the yarn s surface. Glaessgen et al. [19] were the first to construct geometry of a preform relating it to the mechanical properties of yarns and used the resultant geometry for finite element modelling of composite RVE. They were able to achieve a maximum fibre volume fraction of 4% compared to 5 3% achieved by all other models which used idealised descriptive geometric functions to represent the yarn path. Zhang and Harding [] used 3D finite element analysis to find the strain energy stored in the whole RVE under uniform uniaxial extension and obtained the effective elastic constant using the principle of equivalent strain energy. The yarn geometry was idealised as straight inclined segments and the undulation was also considered in only one direction. Nevertheless, the concept of strain energy equivalence gave a reasonable prediction of the elastic properties of a woven fabric composite..1. Creation of tow geometry and representative volume element (RVE) The resin impregnated fibre bundle (roving) in a composite is commonly referred as tow. The analytical/fem models that predict the effective modulii of woven composites require a geometric description of the tow path and hence the fibre volume. The usual practice in formulating all these models has been to explicitly describe the fibre path using some trignometrical functions without making any reference to the mechanical properties of the fibre bundle. The rovings used in textile composites have a finite bending rigidity and have to obey certain geometric constraints governed by the fabric structure. Hence the explicit geometric description of textile reinforcement may not represent the actual orientation or actual volume contained in the composite. This may be particularly significant in unbalanced weaves where both the warp and weft spacings are different or warp and weft yarns of different linear densities are used. The yarn path in the initial state of the fabric has been obtained by minimizing the bending energy of the yarn for the given spacings and thickness of the yarn (Fig. 3). The fabric under consideration here is the plain-woven glass fabric woven from 6 Tex roving. Fabric specifications are presented in Table 1. The composite specimen was formed using an unsaturated polyester resin matrix. Fig. 6 shows the 3D FEA model constructed using the ABAQUS CAE pre-processor. The yarn cross-section was swept along the yarn path shown in Fig. 6b to produce Table 1 Fabric specifications Specification Warp Weft Linear density (tex) 6 6 Sett 4.74 ends/cm 4.74 picks/cm Yarn thickness (mm).36.36 Yarn width (mm) 1.76 1.76 Crimp (%) 1. 1.6 Fig. 6. (a) RVE model generated using ABAQUS CAE pre-processor, (b) Tow path divided into regions and assigned with local coordinate system for material orientation. (a =.116 mm, b t =.36 mm, b m =.1 mm).

41 P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 the yarn shape. Cutting the volume occupied by the yarns in the solid matrix volume generated the matrix pockets. Copies of the individual parts i.e. yarn and matrix were then generated and assembled to produce the unit cell of a plain weave composite representative volume element (RVE)... Specification of tow properties The elastic properties of resin-impregnated tow were determined using the equations of micro-mechanics [1]. The tow is considered as a transversely isotropic material and hence it requires the specification of six elastic constants, i.e. E 11, E, m 1, m 3, G 1,G 3 with reference to an orthogonal material coordinate system (Fig. 6b). There are different approaches to the prediction of these constants, some based on an elementary mechanics approach i.e. stiffness averaging (rule of mixtures), compliance averaging, and others based on advanced approaches i.e. strain energy approach, elasticity solutions and charts developed from FEA. While stiffness averaging gives sufficiently accurate values for E 11 and m 11, some refinement is necessary to predict E and G 1.G 3 is taken as equal to G 1 and m 3 is found by considering the isotropic behaviour in plane 3 []. E 11 and m 1 can be found by the well-known rule of mixtures given by E 11 ¼ E f V f þ E m V m ð11þ m 1 ¼ m f1 V f þ m m V m ð1þ E can be found by the equation based on the method of averaging sub regions due to Chamis [1] 3 E ¼ 1 ffiffiffiffiffi pffiffiffiffiffi p 4 V f V f þ E p m 1 ffiffiffiffiffi 5 ð13þ V f 1 Em E f 3 G 1 ¼ 1 ffiffiffiffiffi pffiffiffiffiffi p 4 V f V f þ G p m 1 ffiffiffiffiffi 5 ð14þ V f 1 G m Gf The glass fibre is assumed to be isotropic and hence E f is taken equal to E f. The properties of the fibre and matrix material are shown in Table. The volume fraction was found using the following equation V f ¼ q y ð15þ q f A y where q y is the linear density of yarn, i.e., weight of yarn per unit length, q f is the fibre density and A y is the area of cross-section of yarn. Table Properties of E-glass, polyester Property E-glass (N/mm ) Polyester (N/mm ) E 7,5 5 G 9,713 95 m..35.3. Specification of displacement field and solution of model The stress strain state corresponding to a uniform displacement (strain) along a material principal axis is predicted here. The effective modulus of elasticity of the composite is then determined using the concept of equivalent strain energy following Zhang and Harding []. For Fig. 7. (a) Finite element mesh of RVE. (b) Deformed state of RVE (.5% uniaxial strain). Stress (MPa) 6 5 4 3 1..4.6.8 Strain Fig. 8. Experimental stress strain curve of composite specimen.

P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 411 Fig. 9. Schematic diagram describing yarn flattening in the case of biaxial extension. (a) Initial state; (b) after applying biaxial tension. example, applying a uniform strain e 11 along the warp direction U c ¼ 1 Z E c11 e 11 dv ¼ 1 v E c11e 11 V ð16þ where E c11 is the required homogeneous equivalent modulus of the composite and V is the volume of the unit cell (RVE). The strain energy of RVE (U c ) is equal to the sum of the strain energy stored in the warp yarn, weft yarn and the matrix pocket. Therefore, we have U c ¼ U warp þ U weft þ U matrix and E c11 ¼ ðu warp þ U weft þ U matrix Þ e 11 V ð17þ The software allows output of the strain energy stored in each component and the volume of each component. Fig. 7(a) and (b) show the FEM mesh of the RVE and the stress field corresponding to a uniaxial strain of.5% in the X-direction. The nodes on the plane x = are constrained in the X-direction (1) while a displacement of.158 mm which corresponds to.5% strain was applied to all the nodes lying in the plane x = a thus specifying a uniform displacement field along one principal direction. The solution of the model yields a stress state corresponding to the prescribed displacement as shown in Fig. 7(b). The strain energy stored in the whole model (RVE) for this displacement was found to be.495 N-mm and the volume of RVE was found to be 4.138 mm 3. Substitution of these values in Eq. (17) gives the effective E 11 value of the composite to be equal to 9495 N/mm. Fig. 8 shows the experimental stress strain curve of the composite specimen and the average slope of the curve in the strain range.5% is taken as the modulus of the specimen. The modulus decreases beyond a strain of.35%, probably due to the visco-elastic nature of the polyester matrix. However, the initial modulus of 91 N/mm is however very close to the predicted modulus of 9495 N/ mm. Table 3 Properties of E-glass/epoxy tow after biaxial extension of preform Elastic constant Unloaded state (V f =.49) E 11 37,31 43,5 E = E 33 8389 1,5 G 1 = G 13 3133 394 G 3 3133 394 m 1 = m 13.8.7 m 3.33.33 Biaxially loaded state (V f =.58) Fig. 1. (a) FEM mesh of RVE (a =.136; b m =.1; b t =.3; after biaxial extension). (b) Deformed state of RVE corresponding to.5% uniaxial strain.

41 P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 3. Analysis of stress field corresponding to biaxial fabric strain The objective of this analysis is to study the changes in magnitude of the stresses due to the change in geometry of the preform. The fabric deformed state corresponding to a load of N each in the warp and weft directions is obtained using the energy model described in Section 1., and Fig. 4 shows the geometry of the yarn path. The application of equal biaxial loads in the case of a balanced plainwoven fabric results in equal loss of crimp and hence equal increase in spacing in both the directions. The reduction of yarn thickness due to the inter-yarn compressive force leads to a further loss of crimp and hence the yarns would Table 4 Comparison of stress and strain magnitudes in RVEs of composite a (biaxial extension of preform) Component Unloaded Biaxially loaded (tow V f =.49) Biaxially loaded (tow V f =.58) S 11 34. 39.7 413.3 S 4.5.5 8.79 S 33 11.5 77.79 16.4 e 11 7.38e 3 8.66e 3 9.37e 3 e 6.567e 4 8.487e 5.9e 4 e 33 1.85e 8.75e 3 9.496e 3 S 11 (matrix) 49.51 4.93 8.6 e 11 (matrix) 1.16e 6.67e 3 6.88e 3 Effective in-plane modulus E 11 = E 33 11,794 1,1 13,973 a The stress and modulii are reported in MPa (N/mm ). Fig. 11. Longitudinal strain field (e 11 ) in tow parallel to the direction of imposed displacement: (a) composite with unloaded preform; (b) composite with biaxially loaded preform.

P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 413 tend to become flatter compared to the case where there was no yarn compression. Fig. 9 shows the schematic diagram describing the effect of yarn flattening during the biaxial fabric strain applied during processing. It has been computed from the energy model that there is an increase in the spacing of the yarns from.116 mm to.136 mm and a decrease in the yarn thickness from.36 mm to.3 mm. Hence the cross-section area decreases from.491 mm to.413 mm and hence there is an increase in the volume fraction from.49 to.58 (+18.36%). Since the tow properties directly depend upon the volume fraction, the increase needs to be accounted for and Table 3 shows the tow properties corresponding to both the unloaded and biaxially strained state. The unloaded as well as the biaxially strained states refer to the geometry of a dry fabric. Fig. 1 shows the FEM mesh corresponding to the composite made with biaxially strained fabric. The RVE of this composite is subjected to a uniaxial strain of.5%. The mesh size is kept the same as in the case of RVE corresponding to the unloaded state (Fig. 7) so as to have an objective comparison of the stress field. Table 4 shows the comparison of the stress components of the composite produced with an unloaded and a biaxially strained fabric, both corresponding to a uniaxial strain of.5%. The results of the composite with a biaxially strained fabric are shown separately for original and modified tow properties. The stress components S 11, S and S 33 correspond to the local coordinate system shown in Fig. 6. S 11 refers to the stress in the tow along the fibre direction, S 33 refers to the stress perpendicular to the fibre direction and S refers to the stress in the out of plane direction. It can be seen that there is an increase of stress magnitude S 11 in the biaxially strained state, which is accompanied by a decrease in stress magnitude S 33. This is mainly due to the reduction of crimp, i.e., fibre waviness, which occurs when the fabric is biaxially strained. As the tow waviness decreases, the stress in the tow increases for the same average strain but the transfer of stress to the yarn in the transverse direction would become less and hence the maximum transverse stress and strain would decrease with the decrease of tow waviness. This prediction also corroborates the findings of Whitcomb [16]. Figs. 11 13 show the strain field in different components of composites made with unloaded and biaxially strained fabrics. A careful examination (Fig. 11) shows that the maximum longitudinal stress in tows parallel to the direction of the imposed displacement occurs at the middle of the edge where the cross-sectional area is least (the region marked in figure). The increase of strain (e 11 ) (in Fig. 11b) is due to the change in tow orientation after the biaxial fabric strain and also due to the reduction in thickness at this region. It is observed that the maximum transverse strain (e 33 ) occurs in tows perpendicular to the direction of the imposed displacement. Fig. 1 shows the transverse strain magnitude in the tows perpendicular to the direction of the imposed displacement. It can be seen that there is a reduction of transverse strain by as much as 3% in the case of a composite with a biaxially strained Fig. 1. Transverse strain field (e 33 ) in tow perpendicular to the direction of imposed displacement: (a) composite with unloaded preform; (b) composite with biaxially loaded preform.

414 P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 fabric. This is mainly due to the reduction in inclination of the tow path at the centre. The higher the inclination of the longitudinal tows (in the direction of imposed displacement), the higher the transfer of stress to the perpendicular tows. It should be noted that transverse stress and transverse strain are critical in tows, which have low break strains in the transverse direction compared to the longitudinal direction (Fig. 14). It was well reported that the breaking strain is first reached at this point and progressive failure of the tow starts. This is known as the knee phenomenon [3]. For the biaxially strained case with modified tow properties that take into account the increase in the volume fraction, the longitudinal stress is increased significantly due to the higher longitudinal modulus of the tow and there is decrease of transverse stress when compared to composite with unloaded preform. Fig. 1 shows the stress field in the matrix. The reasons as given for transverse stress can also be attributed to the reduction of stress in the matrix corresponding to the biaxially loaded preform. In the unstrained preform, the higher inclination of the tow-path creates narrow matrix pockets and leads to stress concentration along the ridgeline. The effect of equal biaxial fabric Failure of Transverse yarn Fig. 14. Schematic diagram describing the knee phenomenon in fabric composites. lower weft upper warp Fig. 15. Effect of increase of crimp of weft yarn on warp yarn s crosssection. Fig. 13. Strain field in matrix of composite with: (a) unloaded preform; (b) biaxially loaded preform.

P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 415 strain leads to a decrease of stress in the matrix material. The effective in-plane modulus has been computed following Zhang s strain energy equivalence approach. It can be seen that the effect of the change in crimp after biaxial deformation does not seem to significantly affect the effective modulus of the composite as it increases by only.7%, but the increase of the modulus is quite significant (+18.48%) if the modified tow properties are used in the computation. 4. Analysis of stress field corresponding to uniaxial fabric strain The effect of uniaxial extension on the dry fabric preform is to increase the fabric strain, i.e., the yarn spacing in the direction of loading while simultaneously decreasing the yarn spacing in the other direction. While the yarn path in the loaded direction would become straight, the waviness (crimp) of the yarn path in the other direction increases. This particular phenomenon is called crimp exchange and has already been explained in Section 1.1. As there would be significant change in the fibre orientation after uniaxial strain compared to biaxial strain, it would be interesting to study the change of the effective modulii of composite. The glass fabric used in the present study is subjected to a load of N per yarn in the warp direction. The change of yarn path due to this uniaxial fabric strain is computed using the energy model as shown in Fig. 5. It can be seen that the yarn path in the warp direction is nearly straight while that in the weft direction shows higher crimp. The part of warp cross section that is in contact with the weft yarn tends to become more rounded as it follows the weft yarn path. On the other hand, the lower half of the cross-section that is not in contact tends to retain its original shape. Hence, the cross-section now becomes unsymmetrical in the contact region and gradually changes to a symmetrical one at the middle and is again reversed at Table 5 Properties of E-glass/epoxy tow after uniaxial extension of preform Elastic constant Warp tow (V f =.64) Weft tow (V f =.51) E 11 47,66 38,69 E = E 33 1,43 883 G 1 = G 13 4673 39 G 3 4673 39 m 1 = m 13.6.8 m 3.33.34 Table 6 Effective modulus of RVE for different cases a Preform E (MPa) Unloaded 14,559 Uniaxially loaded (weft) 3 Uniaxially loaded (warp),75 Biaxially loaded 17,44 a Matrix pockets removed. Fig. 16. (a) RVE of a uniaxially strained fabric; (b) RVE applied with.5% strain along the transverse tows (without resin pockets); (c) cross-section along weft tow, showing the reduction in warp tow cross-section area due to simplification.

416 P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 the contact region of the next cross over point (Fig. 15). The tow geometry needs to be constructed with a varying cross-sectional area [18]. Due to the limitations of ABA- QUS pre-processor, it has not been possible to create tow geometry with cross section varying along the length. As a result, it is assumed here that the cross-section is symmetrical and some simplifications are made to create the geometry of RVE corresponding to the preform subjected to the uniaxial fabric strain (Fig. 16). While the warp tows are straight with zero crimp, the weft tows have relatively higher crimp. It can be seen that the warp cross-section follows the weft yarn path up to half the thickness and is constructed to be symmetrical with respect to the centre, while retaining the original thickness. The area of the resulting cross-section of the warp yarn is now reduced to.37 mm from the original.491 mm. It may be noted that this area reduction is mainly due to the simplification made to the geometry, and not due to the uniaxial strain. Since the fibre volume is the same, the tow properties have been adjusted to take into account the increase in fibre volume fraction. Table 5 shows the properties for the warp and weft tows. Since the volume of the matrix pockets would be different in an RVE corresponding to the uniaxially loaded preform from that of an unloaded preform, it has been proposed to analyse the effective moduli and stress field in the RVE considering only the tows. The effective moduli in the warp and weft directions are now computed using the superposition method proposed by Whitcomb [16] as there are no matrix pockets in the RVE. The procedure involves applying a uniform strain successively in the longitudinal and transverse directions and computing the restraint forces. The modulus is then computed using the work-energy theorem. Table 6 shows the effective in-plane moduli for different cases. It can be seen, in a composite made with uniaxially strained fabric (in the warp direction), the modulus decreases by as much as 77% in the weft direction and increases by 4% in the warp direction due to the uniaxial fabric strain in warp direction. Table 6 shows the stress and strain magnitudes in RVEs with the application of.5% strain along the weft direction. Fig. 17 show the strain contours in the longitudinal and transverse tows, respectively. A look at Table 7 shows that there is a decrease in longitudinal stress and strain (in the case of RVE corresponding to the uniaxially strained fabric) mainly due to the increase of tow waviness. However there is a decrease of transverse stress and strain with an increase of tow waviness, which is contrary to the expected behaviour. This is mainly due to the simplification made in the construction of the tow Fig. 17. (a) Longitudinal strain field (e 33 ) in tow parallel to the direction of imposed displacement. (b) Transverse strain field (e 33 ) in tow perpendicular to the direction of imposed displacement.

P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 417 Table 7 Comparison of stress and strain magnitudes in RVEs of composite with unloaded and uniaxially loaded preforms Component Unloaded Uniaxially loaded S 11 489. 153. S 14.9 4.51 S 33 18.8 9.4 e 11 1.197e 3.873e 3 e 7.87e 4 1.783e 3 e 33 1.49e.738e 3 e 11 (matrix) 1.16e 6.67e 3 geometry that is dictated by the limited features of the software. It can be seen from Fig. 16c that the warp cross-section is reduced and hence the transverse and longitudinal stiffness is increased owing to the increased volume fraction. Moreover the warp cross-section does not have full contact with the weft tow-path up to the centre and hence the transfer of stress from the longitudinal to the transverse tow is less. Hence more detailed tow architecture is necessary to carefully predict the stress and strain field in the case of an RVE corresponding to a uniaxially strained fabric (future versions of ABAQUS may allow us to vary cross-sectional shape along the tow path). 5. Discussion This paper aimed to study the influence of fabric tensions applied during processing/consolidation on the mechanical properties of a finished composite. An energy-based fabric compliance model has been presented to predict the change in yarn geometry due to uniaxial and bi-axial loads. In the case of an equi-biaxial loading, there is a slight reduction in the yarn thickness and a corresponding reduction in crimp in both warp and weft directions. Equi-biaxial loading also leads to an increase in fibre volume fraction in a tow. In the case of uniaxial loading, yarns in the loading direction become relatively straight while the crimp increases in the transverse yarns due to crimp interchange; this crimp interchange has a significant effect on the composite mechanical properties. FEM based micro-mechanical models for three cases of woven composites have been presented: unstrained, biaxially strained and uniaxially strained fabrics. In the case of composites made with biaxially strained fabrics, there is a slight reduction in crimp, tow thickness, laminate thickness and a corresponding increase in tow volume fraction. Tensile modulus increased by 19% in comparison to a composite made with unstrained fabric. A more significant factor is the reduction in transverse strain in tows. The tows are weak in the transverse direction and are prone to failure due to knee phenomena. Hence, with a biaxial loading during the forming stage, the knee phenomena can be postponed until a much higher load. Uniaxial fabric strain leads to straightening of the tows in the loading direction with a corresponding increase in tow waviness in the transverse direction. This exacerbates the knee effect when the composite laminate is loaded in the transverse direction. Hence, composites made with uniaxially strained fabrics may be suitable for components with a significant loading in the axial direction. Due to the limitations of the ABAQUS pre-processor, a number of simplifications have been made to the RVE of a composite made with uniaxially strained preform. Hence, FE analysis of these composites may be used only for a qualitative argument. In future, a more sophisticated pre-processor will be used to create tows with varying cross-sectional shapes along its length. References [1] Potter KD. The influence of accurate stretch data for reinforcements on the production of complex structural mouldings. Composites 1979:161 7. [] Lomov SV, Stoilova Tz, Verpoest I. Shear of woven fabrics: theoretical model, numerical experiments and full field strain measurements. In; Proceedings of ESAFORM 4, Trondheim, Norway, April 8 3; 4. [3] Nguyen M, Herzberg I, Paton R. The shear properties of carbon fabric. Compos Struct 1999;47:767 79. [4] Harrison P, Clifford MJ, Long AC. Shear characterisation of viscous woven textile composites: a comparison between picture frame and bias extension experiments. Compos Sci Technol 4;64:1453 65. [5] Baoxing Chen, Eric JLang, Tsu-Wei Chou. Experimental and theoretical studies of fabric compaction behaviour in resin transfer molding. Mater Sci Eng A 1;317:188 96. [6] Lomov SV, Verpoest I. Compression of woven reinforcements: a mathematical model. J Reinforced Plast Compos ;19:139 5. [7] Boisse P, Gasser A, Hivet G. Analyses of fabric tensile behaviour: determination of the biaxial tension-strain surfaces and their use in forming simulations. Composites: Part A 1;3:1395 414. [8] Sagar T, Potluri P, Hearle JWS. Mesoscale modelling of interlaced fibre assemblies using energy method. Computat Mater Sci 3;8(1):49 6. [9] KES test system, Kato Tekko Ltd., Kyoto, Japan. [1] AG s Fortran 9 Library (fl9), Release4, Numerical Algorithm Group (NAG), UK;. [11] Bogdanovich AE, Pastore CM. Mechanics of textile and laminated composites. 1st ed. Chapman & Hall; 1996. [1] Tarnopol skii YM, Portnov GG, Zhigun IG. Effect of fiber curvature on modulus of elasticity for unidirectional glass-reinforced plastics in tension. Mekhanika Polymerov 1967;3():43 9. [13] Tarnopol skii YM, Polyakov VA, Zhigun IG, Composite materials reinforced with a system of three straight mutually orthogonal fibres: calculation of elastic characteristics. Mekhanika Polymerov 9(5), 853 6. [14] Ishikawa T, Chou TW. One dimensional micro-mechanical analysis of woven fabric composites. AIAA J 1983;1:1714 1. [15] Naik NK, Ganesh VK. Prediction of on-axes elastic properties of plain weave fabric composites. Compos Sci Technol 199;45:135 5. [16] Whitcomb JD. Three dimensional stress analysis of plain weave composites. In: O Brien TK, editor. ASTM composite materials: fatigue and fracture. STP 111, vol. 3. Philadelphia, PA: American Society for Testing and Materials; 1991. p. 417 38. [17] Dasgupta A, Agarwal RK, Bhandarkar SM. Three dimensional modelling of woven fabric composites for effective thermo-mechanical and thermal properties. Compos Sci Technol 1996;56:9 3. [18] Kuhn JL, Charlambides PG. Modelling of plain weave fabric composite geometry. J Compos Mater 1999;33(3):188 19. [19] Glaessgen EH, Pastore CM, Griffin H, Birgir A. Geometrical and finite element modelling of textile composites. Composites: Part B 1996;7B:43 5.

418 P. Potluri, V.S. Thammandra / Composite Structures 77 (7) 45 418 [] Zhang YC, Harding J. A numerical micromechanics analysis of the mechanical properties of a plain weave composite. Computers Struct 199;36(5):839 44. [1] Gibson RF. Principles of composite material mechanics. 1st ed. McGraw-Hill; 1994. [] Tabiei A, Yi W. comparative study of predictive methods for woven fabric composite elastic properties. Compos Struct ;58(1):149 64. [3] Chou TW, Ishikawa T. Analysis and modelling of two-dimensional fabric composites. In: Chou TW, Ko FK, editors. Textile structural composites. Composite materials, vol. 3. Elsevier; 1989.