Elastic Moduli of Woven Fabric Composite by Meshless Local Petrov-Galerkin (MLPG) Method

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1 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 Elastic Moduli of Woven Fabric Composite by Meshless Local Petrov-Galerkin (MLPG) Method P.H. Wen 1 and M.H. Aliabadi 2 Abstract: A meshless local Petrov-Galerkin method, for the micro-mechanical material model of woven fabric composite material is presented in this paper. The material models are based on a repeated unit cell approach and two smooth fibre modes. A unit step function is used as the test functions in the local weak-form which leads to local boundary integral equations. The analysed domain is divided into small sub-domains and the radial basis function interpolation without element mesh is adopted. The woven fabric composite elastic moduli evaluated have been shown to be in good agreement with finite element results. Keywords: meshless local Petrov-Galerkin method, woven fabric composites, elastic moduli 1 Introduction Recent advances in Meshless method have demonstrated its advantage over the finite element method in several applications [Atluri, S.N.; Zhu, T.; (1998a; 1998b; 199), Atluti(24), Atluri, Liu, Han(25); Sladek, Sladek, Krivacek and Zhang (25); Sladek, Sladek, Tanaka and Zhang (25); Wen, P.H., Aliabadi, M.H.; Liu, Y.W.(28), Wen and Aliabadi(21)]. A key feature of these methods is that they do not require a structured grid and therefore are better suited to modelling complex geometries or moving boundary problems. Atluri and co-workers presented a family of the meshless local Petrov-Galerkin (MLPG) method, based on the Local weak Petrov-Galerkin formulation for arbitrary partial differential equations [see Atluri et al (1998a and 199b)] with moving least-square (MLS) approximation]. MLPG is reported to provide a rational basis for constructing meshless methods with a greater degree of flexibility. The Finite Difference Method (FDM), within the framework of the Meshless Local Petrov-Galerkin approach, is proposed for 1 School of Engineering and Material Sciences, Queen Mary, University of London, London, UK, E1 4NS 2 Department of Aeronautics, Imperial College, London, UK, SW7 2BY

2 134 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 solving solid mechanics problems [Atluri and Zhu (1999)]. The implementation of a three-dimensional dynamic code, for contact, impact, and penetration mechanics, based on the Meshless Local Petrov-Galerkin approach has been presented [Atluri, Liu; Han(25)]. A meshless computational method based on the local Petrov- Galerkin approach for the analysis of shell structures is presented for three dimensional solid, allowing the use of completely 3-D constitutive models [Han, Liu, Ranjendran, Atluri(26)]. A meshless method based on the local Petrov-Galerkin approach is proposed for the solution of boundary value problems for coupled thermo-electro-mechanical fields and crack analysis in two-dimensional (2-D) and three-dimensional (3-D) axisymmetric piezoelectric solids with continuously varying material properties [Jarak, Soric,Hoster927); Sladek, Saldek, Zhang, Solek (27a)]. The Local Boundary Integral Equation method (LBIE) with moving least square and polynomial radial basis function (RBF) has been developed by Sladek, Saldek, Zhang, Solek, Starek(27b) and Sladek, Sladek and Zhang (24) for boundary value problems in anisotropic non-homogeneous media including functionally graded materials. Both methods (MLPG and LBIE) are meshless, as no domain/boundary meshes are required in these two approaches. Application of the MPLG to elastodynamic and shear deformable shells can be found in [Wen and Aliabadi(28a, 28b, 29) and Sladek, Sladek, Wen, Aliabadi(26), respectively. A comprehensive review of meshless methods can be found in the book by Atluri (24). Woven composite materials are increasingly utilized as primary structural components due to their low weight to strength ratio. In traditional fibre reinforced composites the fibres are aligned parallel to the plane direction providing an excellent in-plane mechanical property, but are poor in the transverse plane mechanical properties. Woven composites on the other hand are developed to provide increase mechanical properties in both the in-plane and transverse directions. The development of the woven composite technology relies heavily on a better understanding of its micromechanical behaviour. To circumvent large computational models when analyzing these complex composite structures many approaches have been developed towards homogenization of macroscopic properties. An early model for the analysis of woven fabric composites can be found in the work by Ishikawa (1981) and Ishikawa and Chou(1983). The finite element analysis and approximated analytical analysis were carried out by Chung and Tamma (1999) and Tanov and Tabiei (21). Wen and Aliabadi (29b) developed an element-free Galerkin method to composite material for the evaluation of functional elastic moduli in the computational volume cell. Other applications of meshless method to composite materials can be found in [ Dang,Thi, D and Sanker (27, 28)].

3 Elastic Moduli of Woven Fabric 135 In this paper, the meshless local Petrov-Galerkin method is developed for the woven fabric composites. The meshless method for the problems of micro-mechanical models is used for the evaluation of functional elastic moduli in the computational volume cell. The accuracy of MLPG method is compared with the results given by FEM [Hardy (1971) and Marrey and Sanker (1977)] and the element-free Galerkin method [Wen and Aliabadi (29)]. 2 The meshless local Petrov-Galerkin method Consider a linear elastic body in a three dimensional domain Ω with boundary Ω. For woven fabric composites, Hooke s law can also be written, in matrix form, as σ 11 C 11 C 12 C 13 C 14 C 15 C 16 ε 11 σ 22 C 21 C 22 C 23 C 24 C 25 C 26 ε 22 σ σ = 33 = C 31 C 32 C 33 C 34 C 35 C 36 ε 33 σ 12 C 41 C 42 C 43 C 44 C 45 C 46 = C(x, y)ε (1) ε 12 σ 23 C 51 C 52 C 53 C 54 C 55 C 56 ε 23 σ 31 C 61 C 62 C 63 C 64 C 65 C 66 ε 31 where C i j = C ji denotes the elasticity tensor, σ i j and ε i j are stress and strain tensors. The equations of equilibrium can be written as σ i j, j + f i =, (2) where f i is the body force. The boundary conditions are given as u i = ū i on Γ u t i = σ i j n j = t i on Γ t in which ū i and t i are the prescribed displacements and tractions respectively on the displacement boundary Γ u and on the traction boundary Γ t, and n i is the unit normal outward to the boundary Γ. In the local Petrov-Galerkin approaches, the weak form of differential equation over a local sub-domain Ω s can be written as Ω s (σ i j, j + f i )u i dω = (3) where u i is a test function. By use of divergence theorem, equation (3) can be rewritten in a symmetric weak form as σ i j n j u i dγ (σ i j u i, j f i u i )dω = (4) Ω s Ω s

4 136 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 If there is an intersection between the local boundary and the global boundary, a local If there is an intersection between the local boundary and the global boundary, a symmetric weak form in linear elasticity may be written as local symmetric weak form in linear elasticity may be written as * * * * * σ ijui, σ i j u jdω tiui dγ t i, jdω t i u iui dγ = t Γ + Ω Ω i dγ Γ t i u iui d f u d i dγ Γ= t i u i i (5) i dγ Ω+ f i u s La su st s i dω (5) Ω s L a Γ su Γ st Ω in which, Ω = Γ L Γ is a part of the local boundary s located on the global boundary and s s s, s L in which, Ω s = Γ s L s, Γ s is a part of the local boundary located on the global s is the other part of the local boundary inside the sub-domain Ω s ; Γ su = Γs Γu is the boundary and L s is the other part of the local boundary inside the sub-domain Ω s ; intersection Γ su = between Γ s Γ u is the thelocal intersection boundary between Ω s and the local the boundary global displacement Ω s and theboundary global Γ u ; displacement boundary Γ u ; Γ st = Γ s Γ t is a part of the traction boundary as shown Γst = Γs in Figure Γt is a part 1. of the traction boundary as shown in Figure 1. support domain of η node η Ω Local boundary Ω s sub-domain Ω s Ω Ω L s Γ s =Γ su +Γ st Figure Figure 1. Local 1: Local boundaries boundaries for weak for weak formulation, formulation, the the domain domain Ω s for Ω s RBF for RBF approximation approximation of the trial function, and support area of weight function around node ξ. of the trial function, and support area of weight function around node ξ. The local weak forms (4) and (5) are a starting point to derive local boundary integral equations if appropriate test functions are selected. A unit step functions The local can weak be used forms as(4) theand test (5) functions are a starting u point to derive local boundary integral equations i each sub-domain if appropriate test { functions are selected. A unit step functions can be used as the test functions * u i in each u 1 at x (Ω s Ω s ) i (x) sub-domain =. (6) at x / Ω s 1 at x ( Ωs Ω ) u * ( ) = s i x. (6) at x Ωs Then, the local weak forms (4) and (5) are transformed into simple local boundary integral equations (equilibrium of sub-domain) as

5 Elastic Moduli of Woven Fabric 137 Then, the local weak forms (4) and (5) are transformed into simple local boundary integral equations (equilibrium of sub-domain) as t i dγ = Ω s Ω s f i dω (7) and t i dγ = L a +Γ su Γ st t i dγ Ω s f i dω (8) By the use of shape function, we then haveu = Φû, ε = Bû, where Φ = {ϕ 1,ϕ 2,...,ϕ n } is shape function discussed in section 3. Matrix B is defined by ϕ i x ϕ i y ϕ B i = i z ϕ i ϕ i y y, (9) ϕ i ϕ i z ϕ i z y ϕ i x û i (ξ ) denotes the nodal value at point ξ k = {x k,y k,z k }, k = 1,2,...,n, and n is the total number of nodes in the sub-domain as shown in Figure 1. Therefore the stress tensor can be written as σ = C(x)Bû (1) and traction tensor on the boundary Ω s is t = n σ = N(x)C(x)Bû (11) where the matrix N(x) is related to the normal vector n of Ω s. 3 Shape function and approximation schemes Consider a sub-domain Ω s shown in Figure 1, which is the neighbourhood of a point η (= {x,y,z}) and is considered as the domain of definition of the RBF approximation for the trail function at η and also called as support domain to an arbitrary point η. To interpolate the distribution of function u in the sub-domain

6 138 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 Ω s over a number of randomly distributed nodes ξ [= {ξ 1,ξ 2,...,ξ n }, ξ i = (x i,y i,z i ), i=1,2,...,n], the approximation of function u at the point η can be expressed by u(η) = n k=1 R k (η,ξ )a k = R T (η,ξ )a(η) (12) where R T (η,ξ ) = {R 1 (η,ξ ),R 2 (η,ξ ),...,R n (η,ξ )} is the set of radial basis functions centred around the point η, {a k } n k=1 are the unknown coefficients to be determined. The radial basis function selected multi-quadrics [Hardy (1971), Marrey and Sanker(1977)] as R k (η,ξ ) = c 2 + α 1 (x x k ) 2 + α 2 (y y k ) 2 + α 3 (z z k ) 2 (13) where c is a free parameter (is chosen to unit in this paper) and α i are scale factors. From the interpolation equation (12) for the compact support RBFs a linear system for the unknowns coefficients a is obtained as R a = u (14) where u T = {u 1,u 2,...,u n } (15) are the nodal values, u i = u(ξ i ) and R 1 (ξ 1 ) R 2 (ξ 1 )... R n (ξ 1 ) R 1 (ξ 2 ) R 2 (ξ 2 )... R n (ξ 2 ) R (ξ ) = (16) R 1 (ξ n ) R 2 (ξ n )... R n (ξ n ) It is evident that the interpolation of field variable is satisfied exactly at each node. As the RBFs are positive definite, the matrix R is assured to be invertible. Therefore, we can obtain the vector of unknowns from Eq. (14) a = R 1 (ξ )u(ξ ). (17) So that the approximation u(y) can be represented, at domain point y, as u(η) = R T (η)r 1 (ξ )u(ξ ) = Φ(η,ξ )u(ξ ) = n k=1 φ k u k (18)

7 Elastic Moduli of Woven Fabric 139 were the nodal shape function are defined by Φ(η,ξ ) = R T (η)r 1 (ξ ). (19) It is worth noting that the shape function depends uniquely on the distribution of scattered nodes within the support domain and has the Kronecker Delta property. As the inverse matrix of coefficient R 1 (ξ ) is function only of distributed node x in the support domain, it is much simpler to evaluate the partial derivatives of shape function. In order to guarantee unique solution of the interpolation problem, a polynomial term is added to the interpolation (18), giving u(η) = n k=1 R k (η,ξ )a k + along with the constraints t P j (η)b j = R (η,ξ )a + P(η)b (2) j=1 t P k (ξ j )a j =, 1 k t (21) j=1 where {P k } t k=1 is a basis for P m 1, the set of d-variate polynomials of degree m 1, and ([ ]) m + d 1 t = (22) d is the dimension of P m 1. A set of linear equations can be written, in the matrix form, as R a + P T b = u, Pa = (23) where matrix P 1 (ξ 1 ) P 2 (ξ 1 )... P t (ξ 1 ) P 1 (ξ 2 ) P 2 (ξ 2 )... P t (ξ 2 ) P(ξ ) = (24) P 1 (ξ n ) P 2 (ξ n )... P t (ξ n ) Solving these equations in Eq.(23) gives b = ( [ P T R 1 P) 1 P T R 1 u, a = R 1 I P ( ] P T R 1 P) 1 P T R 1 u (25)

8 14 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 where I denotes the diagonal unit matrix. Substituting the coefficients a and b from Eq. (25) into Eq. (2), we can obtain the approximation of the field function in terms of the nodal values u(η) = n k=1 φ k (η,ξ )u(ξ k ). (26) It is clear that the coefficient a and b are functions of nodal positions x with nodal values u. Furthermore, by substituting the RBF approximations (7) and (8) for the unknown fields into the local boundary domain integral equation (7), we obtain the discretized equations (LIE) Ω s and L s +Γ su n k=1 N(x)C(x)B k (x)û k dγ = n k=1 N(x)C(x)B k (x)û k dγ = t(x)dγ Γ st Ω s F(x)dΩ (27) Ω s F(x)dΩ (28) from equation (8). It is worth to point out that the essential boundary conditions can be imposed directly with the use of the RBF approximation. Although the essential boundary conditions are satisfied on the prescribed displacement boundary, these conditions are still not satisfied for all points on the boundary Γ su except the nodal points. It is useful to point out for multiscale problem that the integral equation (7) can be written as t i dγ = f i dω (29) Ω I s+ Ω II s Ω I s+ω II s The local integral boundary with interface is shown in Figure 2. It is apparent that the continuity of the displacement is satisfied everywhere in the local domain including the interface. But there are jumps for tractions on the interface in each local integral domain. However, if the size of the local domain is small enough, the influence of these discontinuities of the tractions can be ignored. Therefore, in the numerical implementation, the interface does not appear in the local integral equation (29). In the numerical procedure, we do not need to distribute node on the interface and consider the continuity of tractions. This simplification works very well for multiscale problems. But the interface (surface) is used in the numerical procedure to define the boundaries for each different material.

9 distribute node on the interface and consider the continuity of tractions. This simplification works very well for multiscale problems. But the interface (surface) is used in the numerical procedure to define the boundaries for each different material. Elastic Moduli of Woven Fabric 141 Integration point Support domain I Ω s I Ω s II Ω s II Ω s Interface Figure 2: Interfacial problem. Figure 2. Interfacial problem. 4 The volume cell models 4. The volume cell models The micromechanical model is based on the method of cells, which consists in dividing the representative volume cell (RVC) in different blocks or cells, each of which can be further divided into sub-cells. These sub-cells are an idealization of actual RVC s sub-volumes, i.e. part of a unidirectional (UD) yarn or a pure matrix region as shown in Figure 3. The entire woven fabric lamina can be constructed by using the RVC shown in Figure 3(b) as a building block. Assuming that the fiber yarns in both direction (the fill and warp) have the same structure and properties, that part of the woven RVC can be constructed using four-cell model as shown in Figure 3(c). Therefore, only a quarter of the composite s RVE needs to be considered to derive the homogenized stiffness properties of the whole woven composite s RVC. The homogenized stiffness properties of the woven composite s RVC are then used to predict the stress-strain behavior of the plain-woven composite at the lamina level. 4.1 The four-cell model To obtain analytical solution of mechanical property of composite material and keep the formulation simple, the cross section of the fill and warp yarns are assumed rectangular and their undulating form is approximated with only horizontal and inclined at angle θ sections as shown in Figure 4, where θ denotes the average undulation angle. In this approach investigated by Tanov and Tabiei (21), RVC

10 RVC. The homogenized stiffness properties of the woven composite s RVC are then used to predict the stress-strain behavior of the plain-woven composite at the lamina level. 142 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 a) e) b) c) d) 删除 Figure 3: Woven fabric composite and micromechanical model: typical plain woven composites; representative volume cell andone quarter cell. contains four sub-cells, i.e. sub-cell ff, fm, mf and mm. Sub-cell fm consists of yarn and matrix portions as illustrated in Figure 4. From the geometry of the four-cell model, the height unit of RVC can be written as H = 4(1 Vy ) tan θ (3) where Vy is overall yarn volume fraction, θ is the undulation angle. The width and depth of RVC is unit. Analytical solutions for the homogenization of the stiffness properties in Ref. by [Tanov and Tabiei (21)] were derived based on the parallelseries assumptions such as for some of the strain components the adjacent cells are assumed to work in parallel and the corresponding strains are equal, while for the

11 shown in Figure 4, where θ denotes the average undulation angle. In this approach inv by Tanov and Tabiei (21), RVC contains four sub-cells, i.e. sub-cell ff, fm, mm. Sub-cell fm consists of yarn and matrix portions as illustrated in Figure 4. Elastic Moduli of Woven Fabric 143 z fm mm ff mf θ H/4 H/2 fill x y H/4 θ warp V y 1-V y 1-V y V y Figure 4: Geometry of the RVC for the four-cell model [Tanov and Tabiei(21)]. Figure 4. Geometry of the RVC for the four-cell model [Tanov and Tabiei(21)]. rest of the strains, the cells are assumed to work in series, i.e. the corresponding stress components are equal and the strains are averaged to get the whole resultant strains. A similar parallel-series approach can be applied to the whole RVC and thus the sub-cell strains can be determined. From the geometry of the four-cell model, the height unit of RVC can be written as H = 4(1 Vy ) tanθ 4.2 The smooth fabric models Two 3D woven fabric composite unit cells were constructed to determine its elastic properties as shown in Figure 5(a) and (b). In this case, the fibers in yarn volume are unidirectional and smooth distributed along the fill/warp directions. We assume that the variation of bottom fiber in the fill direction on section y = is where V y is overall yarn volume fraction, θ is the undulation angle. The width and RVC is unit. Analytical solutions for the homogenization of the stiffness propertie were derived based on the parallel-series assumptions such as for some of t z 1 (x) = z H 4 + H πx cos 4 2 components the adjacent cells are assumed to work in parallel and the corresponding s (31a) where z denotes the location of a fibre at x = y =. For the bottom fibre, z = H/2 and the fibre location in the xoz becomes z 1 (x) = H 4 (1 + cos πx 2 ) (31b) Smooth fiber model I.

12 fibre location in the xoz becomes H πx z1( x) = (1 + cos ) Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 (31b) z z matrix H/4 H/2 fill x z z 1 H/4 warp 1-λ z 2 x y 1-λ λ λ λ 1-λ Figure 5: RVC with smooth fibres and geometry of yarn volume for model I. Figure 5. The RVC geometry with smooth of smooth fibres fiber and model geometry I is illustrated of yarn volume in Figure for model 5. Suppose I. the configuration for the yarn volume in the warp direction to be 删 Smooth z fiber model I. 2 (x) = αx 2 (32) The coefficient α is determined by considering the location of joint for fibres of fill and warp α = H 4λ 2 (1 + cos πλ 2 ) (33) where λ denotes the coordinate of joint shown in Figure 5. Considering the portion of fabric yarn volume in the RVC, we have following integral V y = 2 λ [ ] H πx (1 + cos 4 2 ) αx2 dx (34) Then we have following equation to evaluate the location of joint 4λ = 6V y + 3(λ 2 π sin πλ 2 ) λ(1 cos πλ 2 ) (35) The root of λ can be easily determined by the alterative method. Therefore, the top

13 Elastic Moduli of Woven Fabric 145 and bottom surfaces of yarn volume in warp direction can be written as z bottom (x,y) = H 4 z top (x,y) = H 2 + H 4 = H 4 (1 cos πy 2 ) + αx2 (1 cos πy 2 ) (2 + cos πx 2 and the slop of fibres is dz dy = Hπ πy sin 8 2 cos πy 2 ) [ H 2 H ] πx (1 + cos 4 2 ) (36) (37) The derivative above will be used to determine the elasticity energy stored in yarn volume. Similarly to the yarn volume in fill direction, the top and bottom surfaces of yarn volume in warp direction can be written as z top (x,y) = H αy 2 H 4 z bottom (x,y) = H πx (2 + cos 4 2 and the derivative of the fibre is dz dx = Hπ πx sin 8 2 Smooth fibre model II. (1 cos πx 2 ) cos πy 2 ) (38) The geometry of smooth fiber model II is illustrated in Figure 6. The configuration for yarn volume in the warp direction is assumed to be a half of elliptic and defined as z 2 (x) = H 4 (1 ± 1 x2 λ 2 ) (4) (39) where λ is the principle axis of elliptic which can be determined by the portion of yarn volume as λ = 4V y π (41) Model II is more suitable for a small portion of yarn volume composites. For instance, if the yarn volume portion V y =.5385, then λ =.8162 and.6856 for the smooth fibre model I and model II respectively.

14 146 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 z z Matrix H/4 H/2 fill x z z 1 H/4 war z 2 1-λ x y 1-λ λ λ λ 1-λ Figure 6: RVC with smooth fibres and geometry of yarn volume for model II. Figure 6. RVC with smooth fibres and geometry of yarn volume for model II. 5 Results and discussion In this section, the application of mesh free method to the woven fabric composites is demonstrated. To deal with smooth fibre mode, we need to analyse transforming constitutive matrix by rotating the local coordinate system at angle θ respecting to the axis y. The fill constitutive matrix can be written as C 11 C 12 C 12 C 12 C 22 C 23 C = C 12 C 23 C 22 C 44. (42) C 55 C 44 So the constitutive matrix in RVC global coordinate system by rotating the local system at angle θ becomes [Tanov and Tabiei(21)] C 11 C 12 C 12 C 16 C 12 C 22 C 23 C 26 C (θ) = C 12 C 23 C 22 C 36 C 44 C 45 C 45 C 55 C 16 C 26 C 36 C 66 (43)

15 Elastic Moduli of Woven Fabric 147 where C 11 = C 11 cos 4 θ +C 33 sin 4 θ + 2(C C 66 )sin 2 θ cos 2 θ C 12 = C 12 cos 2 θ +C 23 sin 2 θ C 13 = C 13 (cos 4 θ + sin 4 θ) + (C 11 +C 33 4C 66 )sin 2 θ cos 2 θ C 16 = [(C 13 C 11 )cos 2 θ + (C 33 C 13 )sin 2 θ + 2C 66 (cos 2 θ sin 2 θ)]sin 2 θ cos 2 θ C 22 = C 22 (44) C 23 = C 12 sin 2 θ +C 23 cos 2 θ C 26 = (C 23 C 12 )sinθ cosθ C 33 = C 11 sin 4 θ +C 33 cos 4 θ + 2(C C 66 )sin 2 θ cos 2 θ C 36 = [(C 13 C 11 )sin 2 θ + (C 33 C 13 )cos 2 θ 2C 66 (cos 2 θ sin 2 θ)]sin 2 θ cos 2 θ C 44 = C 44 cos 2 θ +C 55 sin 2 θ C 45 = (C 55 C 44 )sinθ cosθ C 55 = C 44 sin 4 θ +C 55 cos 4 θ C 66 = (C 11 +C 33 2C 13 )sin 2 θ cos 2 θ +C 66 (cos 2 θ sin 2 θ) 2 where θ is the angle of fibres and defined as from Eq. (39) θ = tan 1 ( Hπ 8 sin πx 2 ). (45) Similarly to the yarn in warp volume, the constitutive matrix is as follows C 11 C 12 C 12 C 16 C 12 C 22 C 23 C 26 C(θ) = C 12 C 23 C 22 C 36 C 44 C 45. (46) C 16 C 26 C 36 C 55 C 45 C 66 where θ is the angle of fibres and defined as from Eq. (37) θ = tan 1 ( Hπ 8 sin πy 2 ). (47) A 3D woven fabric composite unit cell was constructed to determine its elastic properties by the finite element method by Chung and Tamma (1999). The RVC

16 C' 16 C' 26 C' 36 C' 45 where θ is the angle of fibres and defined as from Eq. (3) 148 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 Hπ πy θ = tan 1 sin. (47) 8 2 consists of epoxy matrix and fibre bundles, which are 65 percent E-glass/epoxy A 3D yarns woven and the fabric properties composite are unit shown cell in was Tableconstructed 1. The yarn to volume determine fraction its elastic V y = properties by the.5385 finite element and the average method undulation by Chung angle and Tamma attanθ = (1999). 1/6. The The thickness RVC consists of the RVC of epoxy matrix is H=.377. The uniform distribution of nodes is shown in Figure 7. The total and number fibre bundles, of nodeswhich is N x N are y N 65 z percent ( =847 E-glass/epoxy in this paper) yarns and the the scale properties factors are shown in Table are 1. selected The yarn by α volume 1 = α 2 = fraction 1; α 3 = V N z /(N x H). The elastic constants of RVC can y = and the average undulation angle at tan = 1/ 6 be written as The thickness C of 11 C the RVC 12 C is 13 H=.377. The uniform distribution of nodes is shown in Figure 7. The total number C 12 C 11 of nodes C 13 is N x N y N z ( =847 in this paper) and the scale factors C = C 12 C 13 C 33 C 44. (48) are selected by α 1 = α 2 = 1; α 3 = N z /( N x H ). The elastic constants of RVC can be written as C 55 C11 C12 C13 C 55 C12 C11 C13 Following Ccalculations should be done to determine seven coefficients in above 12 C13 C33 matrix: C =. (48) C44 Boundary conditions for C 55 ε x = 1;ε 2 = ε 3 = γ xy = γ yz = γ zx = C55 C' 55 C' 66 θ. z N z x y N x N y Figure 7: Uniform distribution of nodes in RVC (5 5 3). Table 1: E-glass/epoxy properties materials E x (GPa) E y (GPa) G xy (GPa) G yz (GPa) ν xy ν yz Epoxy Fiber bundles (65%)

17 Elastic Moduli of Woven Fabric 149 x = : u = ; τ xy = τ zx = ; x = 1 : u = 1; τ xy = τ zx = ; y = : v = ; τ xy = τ yz = ; y = 1 : v = ; τ xy = τ yz = ; (49) z = : w = ; τ yz = τ zx = ; z = H : w = ; τ yz = τ zx =. The total forces acting on three surfaces can be determined by following integral F x = 1 H σ x x= dzdy; F y = 1 H σ y y= dzdx; F z = Therefore, the micromechanical properties 1 1 σ z z= dxdy. (5) C 11 = F x H ; C 12 = F y H ; C 13 = F z. (51) Boundary conditions for ε z = 1;ε 1 = ε 1 = γ xy = γ yz = γ zx = x = : u = ; τ xy = τ zx = ; x = 1 : u = ; τ xy = τ zx = ; y = : v = ; τ xy = τ yz = ; y = 1 : v = ; τ xy = τ yz = ; (52) z = : w = ; τ yz = τ zx = ; z = H : w = H; τ yz = τ zx =. The total forces acting on three surfaces can be determined by following integral F z = 1 1 σ z z= dxdy. Thus, the micromechanical property C 33 = F z. Boundary conditions for γ xy = 1;ε 1 = ε 2 = ε 3 = γ yz = γ zx =, can be achieved proximately by following displacements x = : v = ; σ x = τ zx = ; x = 1 : v = 1/2; σ x = τ zx = ; y = : u = ; σ y = τ yz = ; y = 1 : u = 1/2; σ y = τ yz = ; (53) z = : w = ; τ yz = τ zx = ; z = H : w = ; τ yz = τ zx =. The total shear force can be written as F x = 1 H τ xy y= dzdx and, therefore, the micromechanical property C 44 = F x /H. Following the similar way, we can evaluate C 55. Table 2 shows the FEM results for the RVC properties from upper and lower bound estimates by Chung and Tamma (1999), Four/Single model results given by Tanov

18 15 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 and Tabiei (21) and results by proposed method for different models. Compared with the numerical results given by FEM [Chung and Tamma (1999)], approximated analytical solution [Tanov and Taiebei(21)] and the element-free Galerkin method, a good agreement has been achieved. Due to element mesh is free in this approach, the technique is more convenient and powerful dealing with complicated problems such as variation of fibre rotation in the VRC. For woven composites, we only need to define the smooth surfaces for both yarns in the fill and warp directions. In the second example, a Glass/epoxy woven composite, which has been discussed by Marrey and Sankar (1977) by the FEM, with properties for the yarn E L = 58.61GPa, E T = 14.49GPa, G LT = 5.38, ν LT =.25 and ν TT =.247; the isotropic matrix is of Epoxy with E = 3.45GPa, ν =.37 is analyzed. The yarn volume fraction V y =.26 and the average undulation angle θ = 4.2. The thickness of RVC is H= Table 3 shows all values obtained by using different approaches. It is shown that the agreement by the MLPG method is satisfied by comparing with other approaches. In the last example, we consider a Graphite/epoxy RVC, which consists yarns with properties E L = 137.3GPa, E T = 1.79GPa, G LT = 5.394, ν LT =.26 and ν TT =.46; the isotropic matrix is of Epoxy with E = 4.511GPa, ν =.38. The yarn volume fraction, V y =.58 and the average undulation angle at θ = 1.4. From Eq. (3), the thickness of RVC is H=.411 unit. The numerical results by difference approaches are shown in Table 4. In general, good agreement is obtained with the approximated analytical solutions given by Tanov and Tabiei (21) and element free Galerkin method. In the current approach, we consider the smooth woven fibers in the RVC only and believe that the prediction of micro mechanical properties using the meshless technique is more accurate. 6 Conclusions The micromechanical models with smooth woven fabric composites have been established and studied numerically by using meshless local Petrov-Galerkin method. The distribution of the fibres in the RVC can be arbitrary in each numerical model and the prediction of micromechanical properties can be obtained easily by the proposed mesh free method. Compared with the results given by FEM and elementfree Galerkin method, the accuracy of the MLPG method in this paper is satisfacroy. Similar to the element-free Galerkin method, we can conclude with following observations: (1) The smooth fibres model is more realistic; (2) The meshless local Petrov-Galerkin method is one of the most powerful method to evaluate moduli for micromechanical models; (3) More complicated woven fabric composite models can be investigated by the use of the MLPG method; (4) The MLPG method can be

19 Elastic Moduli of Woven Fabric 151 Table 2: RVC properties by different approaches Properties Upper Bound Lower Bound Single-cell Model Smooth Fibre Smooth Fibre Model I Model II Chung et al Chung et al Tanov et al Model I, Wen Model II, Wen (MLPG) (MLPG) (1999) (1999) (21) et al (29) et al (29) C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C44 (GPa) C55 (GPa) Table 3: RVC properties by different approaches Approach Ex(GPa) Ey(GPa) Gxz, Gyz(GPa) Gxy(GPa) νxy νyz Ref [22] Single-cell model Tanov et al (21) Smooth fibre model I Wen et al (29) Smooth fibre model II Wen et al (29) Model I (MLPG) Model II (MLPG)

20 152 Copyright 21 Tech Science Press CMES, vol.61, no.2, pp , 21 Table 4: RVC properties by different approaches Approach Ex(GPa) Ey(GPa) Gxz, Gyz(GPa) Gxy(GPa) νxy νyz Four-cell model Tanov et al (21) Single-cell model Tanov et al (21) Four-cell model Wen et al (29) Smooth fibre model I Wen et al (29) Smooth fibre model II Wen et al (29) o Model I (MLPG) Model II (MLPG)

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