Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 1, pp. 3-14, Warsaw 2014 THEORETICAL STUDY OF STRESS TRANSFER IN PLATELET REINFORCED COMPOSITES A.M. Fattahi Department of Mechanical Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran e-mail: a.fattahi@iaut.ac.ir. M. Mondali Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran An analytical approach was developed for rectangular platelet reinforced composites which could be used for a 3D elastic stress field distribution subjected to an applied axial load. The ends of the platelet could be bonded to the matrix. Exact displacement solutions were derived for the matrix/platelet fromtheory of elasticity. These displacement solutions were then superposed for achieving analytical expressions for the matrix/platelet 3D stress field components over the entire composite system including the platelet end region, using the adding imaginary fiber technique. The platelet/matrix components could exactly satisfy the equilibrium and compatibility conditions and satisfy the equilibrium requirements and the overall boundary conditions. The obtained analytical results were then validated by FEM and Shear-lag modeling, and some of discrepancies among the shear-lagmodels were resolved.Good agreementswere observed between the analytical and numerical predictions. Key words: analytical modeling, platelet reinforced composites, stress transfer 1. Introduction Over the past few decades, a variety of numerical and analytical models has been developed to investigate different stress transfer problems in composites. These models mainly include 1D models, which are typically based on the shear-lag theory (Agarwal, 1974; Agarwal and Broutman, 1980; Chang and Tarn, 2011; Cox, 1952; Glavinchevski and Piggott, 1973; Haque and Ramasetty, 2005; Hsueh, 1994, 2000; Hsueh et al., 1999; Jiang and Peters, 2008; Jiang et al., 1998; Kim, 1998, 2007, 2008; Kim andKwac, 2009; Kim andNoh, 2004; Kotha et al., 2000; Lusis et al., 1973; Narin, 1997, 2004; Narin and Mendels, 2001; Padawer and Beecher, 1970; Piggott, 1980; Salekin et al., 1992; Taya and Arsenault,1989; Tyson and Davies, 1965; Wu et al., 1997), 3D analytical models based on axisymmetric analyses (Abedian et al., 2007; Jiang et al., 1998, 2004;Wu et al., 1997), the Eshelbymodels based on Eshelby’s equivalent inclusion method (Arsenault andTaya, 1987; Eshelby, 1957; Tanaka et al., 1973; Lee, 2008;Withers et al., 1989) and numerical models (Cannilloa et al., 2003; Lusti et al., 2002; Narin, 2007). Due to the complexity of elasticity fieldequations, analytical closed-formsolutions to fully three-dimensional problems are very difficult to be obtained. Accordingly, many solutions have been developed for reduced problems that are typically composed of axisymmetry or one-dimensionality based on the shear-lag theory for simplifying a particular aspect of the formulation and solution. In mathematical terms, the shear-lag model is the simplest of all models, which is widely used for the stress transfer analysis in unidirectional composites. This model is based on a simple differential equation which relates the fibre axial stress to the interfacial shear stress. It is not applicable at higher volume fractions due to the significant interactions of fibres. Also, the model cannot predict changes of axial stress and strain distributions in the radial direction. As a result, due to these limitations, the shear-lagmodel cannot provide reliable predictions for the 4 A.M. Fattahi, M.Mondali composite properties, and thus, its application for short fibre composites has been limited over time. Most interfacial problems need the detailed distribution of stresses at the interface like the interface friction slip behavior in composites; however, the one-dimensional shear-lag model cannot provide this stress state. Three-dimensional analyticalmodels have been developed based on quite different approaches and aimmostly at the interfacial problems. It has been noted that mostof the three-dimensional analyticalmodelsusually satisfy theequilibriumstates andmostof the boundary and interfacial conditions. This is because the solutions are generally based on the stress equilibrium equations; but the compatibility conditions are only partly or approximately satisfied because of different degrees of applied approximations and simplifications. Therefore, since the excessive complicatedmathematical derivations are involved, very limited efforts could be observed for the exact solution of such problems in the literature. One of the most precise and relatively simple three-dimensional analytical solutions was done by Jiang et al. (2004). In thismodel, two sets of exact displacement solutions for thematrix, i.e. the far-field solution and the transient solution, were derived.Afterwards, the theory of elasticity and superposition of the simplified analytical expressions were used to find all the stress components in the matrix and fibre. It is worthmentioning that the fibre end region could be also included using the imaginary fibre technique. Jiang’s analytical model was modified and developed by Abedian et al. (2007). The latter was greatly improved and could predict capability of the composite behavior. In brief, for Jiang’s and Abedian’s models, the following assumptions weremade: a perfect bond existed at the fibre-matrix interface and both fibre andmatrix were isotropic. Most authors have made emphasis on platelets as reinforcement instead of fibers because of their two-dimensional stiffening effects (Chou and Green, 1993; Piggott, 1980; Salekin et al., 1992) andcost factor.Thetwo-dimensional stress transfermodel ofplatelet reinforced composites was presented by Tyson and Davis (1965), which was derived following the same analysis as applied in the classical shear-lagmodel byCox (1952). Later,Hsueh (1994, 1999, 2000) proposed amore rigorous two-dimensional stress transfermodel for platelet reinforcementwhile assuming that the shear stress in the matrix decreased linearly from the interface between the platelet and matrix to the edges. Also, the effects of matrix bonding at the ends, Young’s modulus, and the aspect ratio of the platelet were investigated on stress transfer. A simple second order model and amore complicated fourth ordermodel were developed for simulating stress transfer in overlapped platelets by Kotha et al. (2000). Some other models based on shear-lag theory were presented by Narin (1999, 2001, 2004). The present study tried to derive such an analytical 3D modeling of platelet reinforced composites which were presented according to Jiang’s and Abedian’s methods and boundary conditions.While considering the typical complexity of stress field distribution in platelet rein- forced composites, the analysis aimed at a platelet reinforced composite subjected to an applied axial load. Two sets of matrix/platelet displacement solutions, the far-field solution and the transient solution were precisely derived based on the theory of elasticity and were superposed to achieve simplified analytical expressions for the matrix/platelet stress field components and the platelet axial stress field components in the entire composite system,which included the pla- telet end region, using the technique of adding the imaginary fibre. The finite element numerical calculations were then conducted to examine the validity of this analytical model. At last, some concluding remarkswere presented on the present analytical model. Both bonded and debonded platelet end cases were introduced in this research. 2. Analyses 2.1. Composite model In thecurrent study, a rectangular unit cell (as shown inFig. 1)wasused formodelingplatelet reinforced composites. The platelet with the width of 2a and length of 2l was embedded in the Theoretical study of stress transfer in platelet reinforced composites 5 center of a matrix with the width of 2b and length of 2l′. The area fraction and aspect ratio of the platelet were defined as follows f = a/b and s= l/a, respectively. The axial stress (σ0) was considered to be uniformly applied to the end faces of the unit cell. A Cartesian coordinate system (x,y) was also applied with its origin located in the centre of the unit cell, as given in Fig. 1. For simplicity, a perfect bondwas assumed at the platelet/matrix interface and isotropic constituents for the composite. Due to the symmetric geometry in the x-y plane and boundary conditions, only one quarter of the unit cell was considered in the analysis. In this work, the solutionswereperformed in two separate areas: in theplatelet region (0¬ y¬ l) and theplatelet end region (l¬ y¬ l′). They were undertaken using the imaginary fibre technique (Abedian et al., 2007; Jiang et al., 2004), which assumed that the platelet end region was composed of an imaginary platelet surrounded with a rectangular matrix while the properties of the imaginary platelet were considered the same as those of the matrix. Fig. 1. Schematic illustrations of the unit cell 2.2. General solutions of matrix and platelet displacements The equilibrium equations for plane stresses (Timoshenko andGoodier, 1951) are as follows ∂σx ∂x + ∂τxy ∂y =0 ∂σy ∂y + ∂τxy ∂x =0 (2.1) which lead to the equations 2 1+ν ∂2u ∂x2 + 1−ν 1+ν ∂2u ∂y2 + ∂2w ∂x∂y =0 1−ν 1+ν ∂2w ∂x2 + 2 1+ν ∂2w ∂y2 + ∂2u ∂x∂y =0 (2.2) where u is the displacement in the x-direction, and w is the displacement in the y-direction, E sYoung’smodulus and ν is Poisson’s ratio. By simple algebraicmanipulation of the operators in Eqs. (2.1) and (2.2), the governing equations of w and u can be derived. The equation with respect to u could be found in the following manner ∂4u ∂x4 +2 ∂4u ∂x2∂y2 + ∂4u ∂y4 =0 (2.3) Themethod of separation variables verifies the last equation and results in different answers. For the case of this reserach, the Laplace equation could be obtained according to the condi- tions (see Appendix A). Thus, the following general solution can be found for the x-direction displacement um(T)(x,y)= (A1 sinnx+A2cosnx)(A3 sinhny+A4coshny) (2.4) 6 A.M. Fattahi, M.Mondali where Ai (i = 1− 4) is an integral constant and n is a constant related to the eigenvalue. By substituting Eq. (2.4) into Eq. (2.1) or Eq. (2.2), the general solution for the y-direction displacement (w) can be derived as follows wm(T)(x,y)= (−A1cosnx+A2 sinnx)(A3coshny+A4 sinhny) (2.5) Another set of the displacement general solutions for zero-eigenvalue were derived as demon- strated below (see Appendix A) um(0)(x,y)=B1+B2x w m(0)(x,y) =B3+B4y (2.6) where Bj (j = 1, . . . ,4) is the integral constant. The above displacement solutions could be superimposed to write the total value of displacement components in the matrix as follows um =um(0)+um(T) wm =wm(0)+wm(T) (2.7) The general solutions for the x- and y-directions displacements in the platelet could be found in Eqs. (2.8)-(2.10). It should be noted that, due to the continuity condition, A3and A4 were considered the same as those for the matrix expressions. But new boundary conditions were required for successful application in these equations, as presented in the next section. The details of derivation of the coefficients by the proposed boundary conditions will be also given in this investigation up(T)(x,y)= (A5 sinnx+A6cosnx)(A3 sinhny+A4coshny) wp(T)(x,y) = (−A5cosnx+A6 sinnx)(A3coshny+A4 sinhny) (2.8) and up(0)(x,y)=B5+B6x w p(0)(x,y)=B7+B8y (2.9) and up =up(0)+up(T) wp =wp(0)+wp(T) (2.10) Thus, the general solutions for the stress and strain components which correspond to the above two sets of displacement solutions were obtained using Hook’s law (see Appendix B). Therefore, the stress field describedby displacementEqs. (2.6) and (2.9) was identical to the one inacompositewith infinitely longplateletswhichwere subjected toa far-field loadcorresponding to the uniform portion of the total stress. However, the stress field described by Eqs. (2.4), (2.5), and (2.8) corresponded to the non-uniform portion of the total stress field since all the components of the stress field depended on both x- and y-directions. 3. Solution for platelet 3.1. Far-field solution The surface conditions for the far-field solution in the platelet region for the platelet bonded end case could be presented by τm(0)xy (b,y)= 0 w m(0)(x,0)= 0 up(0)(0,y) = 0 (3.1) The above conditions and ε m(0) yy = ε0 can be used for deriving the following constants B3 =B5 =0 B4 = ε0 (3.2) Theoretical study of stress transfer in platelet reinforced composites 7 where ε0 is the far-field strain. UsingEq. (B.7), other constants could be found as demonstrated below B8 = ε0 (3.3) Using um(0)(a,y) =up(0)(a,y), the B1 coefficient can be obtained as follows B1 =(B6−B2)a (3.4) Equation (B.8) and the equation of the equilibrium, i.e. Eq. (3.5)1, lead to obtaining B2 and B6 as in Eqs. (3.5)2,3 σ0 = fσ p(0) yy +(1−f)σ m(0) yy B2 = p21σ0+p22ε0 B6 = p61σ0+p62ε0 (3.5) Also, p21, p22, p61 and p62 were found in Matlab software. Due to being so lengthy, they were not cited in this paper. Then, the stresses in the imaginary platelet region (l ¬ y ¬ l′) were calculated. Similarly, the far-field solution and all the corresponding expressions of the general solution for the stress and strain components of this region could be obtained directly from those of the platelet region by considering Ep =Em and νp = νm. Using the same derivation procedures as the ones in the platelet region, the integral constants in this region were obtained B̃1 = B̃5 =0 B̃4 = B̃8 = ε̃0 B̃3 = B̃7 =(ε0− ε̃0)l B̃2 = B̃6 = (1−ν2m)σ0−Emε̃0 νmEm (3.6) To calculate the far-field strains ε0 and ε̃0, the following two boundary conditions could be used um(0)(b,y)= ũm(0)(b,y) l l′ [fEp+(1−f)Em]ε0+ ( 1− l l′ ) Emε̃0 =σ0 (3.7) Here, Matlab software was used to find ε0 and ε̃0 strains. Because of being lengthy, they were not cited in this paper. Equation (3.7)2 can be obtained from the boundary conditions and some simplifications which lead to having l and l′ inside the strain equations, equilibrium equations l l′ σ0+ ( 1− l l′ ) σ0 =σ0 σ0 = fσ p(0) yy +(1−f)σ m(0) yy σ0 = fσ̃ p(0) yy +(1−f)σ̃ m(0) yy σ m(0) yy =Emε0 σ p(0) yy =Epε0 σ̃ m(0) yy = σ̃ p(0) yy =Emε̃0 (3.8) Finally, by substituting ε0 and ε̃0 and the above-obtained constants in the stress equations, the expressions for the far-field solutions in the twomentioned regions can be foundas a function of σ0. 3.2. Transient solution In this part, the transient solution in the platelet region (0¬ y¬ l) will be first determined. The surface boundary conditions for the transient solution can be presented by wm(T)(x,0)= 0 um(T)(b,y)= 0 (3.9) It is implied by these conditions that the transient solution did not alter the shape of the unit cell. Substituting the above conditions in the stress, the strain anddisplacement expressions resulted in A3 =0 A2 =−A1 tannb (3.10) 8 A.M. Fattahi, M.Mondali To determine two other coefficients, the following boundary conditions were used up(T)(0,y)= 0 σm(T)xx (a,y)=σ p(T) xx (a,y) (3.11) Then, A5 and A6 were derived A6 =0 A5 = Em Ep 1+νp 1+νm (A1−A2 tanna) (3.12) To find the unknown n, axial force equilibrium Eq. (3.13) was used. Matlab software was used for finding the optimum n fσp(T)yy +(1−f)σ m(T) yy =0 (3.13) Afterwards, the transient solution was specified in the platelet end region (l¬ y¬ l′). The transient solution and all the corresponding expressions for the general solution of the stress and strain components in this region could be directly obtained from those of the platelet region by considering Ep = Em and νp = νm. Using the same derivation procedure and corresponding boundaryconditions as theones in theplatelet region, the correspondingconstantswere achieved as follows τ̃m(T)xy (b,y)= 0 w̃ m(T)(a,y) = w̃p(T)(a,y) σ̃m(T)xx (a,y)= σ̃ p(T) xx (a,y) (3.14) Also, the following constants were obtained Ã6 =0 Ã5 = Ã1− Ã2 tan ña Ã2 =−Ã1 tan ñb (3.15) To find the unknown ñ, equilibrium Eq. (3.16) was used, similar to the real platelet field fσ̃ p(T) yy +(1−f)σ̃ m(T) yy =0 (3.16) Using the following surface condition, Eq. (3.17)1, Ã3 coefficient was also derived τ̃mxy(x,l ′)= τ̃pxy(x,l ′)= 0 Ã3 =−Ã4 sinh ñl′ cosh ñl′ (3.17) Finally, A4 and Ã4 coefficients were found using the following boundary conditions σpyy(x,l)= σ̃ p yy(x,l) τ m xy(a,y) = τ̃ m xy(a,y) (3.18) 4. Analytical and numerical predictions Tomonitor the validity of the current analytical model, a comparison was made with the FEM calculations by ANSYS software. The schematic FEM model is also demonstrated in Fig. 1. The numerical calculations were performed on one quarter of the unit cell due to the existing symmetric boundary conditions. Eight nodded PLANE82 solid elements were applied for the purpose of meshing. Loading in ANSYSwas via pressure applied to the element boundary line. All the simulations were performed under the plane stress condition. The boundary conditions required fixing both the x-displacement at x=0 and the y-displacement at y=0.The loading was applied to y= l′. Only typical results were presented for b/a=1.75, l/a=5 and l/l′ =0.5 (platelet volume fraction was almost 0.285) in order to decrease the number of figures. The material properties were selected as Em = 63GPa, Ep = 402GPa, νm = 0.22 and νp = 0.23 (Cannilloa et al., 2003). The applied stress was taken as σ0 =200MPa. Theoretical study of stress transfer in platelet reinforced composites 9 Fig. 2. Comparison and validation of the present model with Hsueh’s model and numerical results Figure 2 demonstrates the comparison and validation of the present model with the Hsueh model and its numerical results. Compared with Hsueh’s results, the current model and FEM values of the stress components for the case ofperfectbondwere foundtobe inagoodagreement. To demonstrate the capabilities of the present model, displacements, strains, stresses in the matrix and platelet were given in the x- and y-directions. It is worth considering that only typical results were presented in order to reduce the number of figures. The analytical and numerical curves of the normalized matrix in displacements in the x- and y-directions on the outer surface of the unit cell and the platelet/matrix interface versus the normalized axial position are demonstrated in Fig. 3. As can be observed, considerable agreements were obtained between the analytical and numerical predictions. Figure 4 depicts the analytical and numerical curves of the shear stress and x-direction stresses on the outer surface of the unit cell (x= b) and the platelet-matrix interface versus the normalized axial position y/a. Fig. 3. Analytical and numerical curves of the matrix displacements vs. the normalized axial position Fig. 4. Analytical and numerical curves of the matrix stress vs. the normalized axial position 10 A.M. Fattahi, M.Mondali The given comparisons confirmed considerable strong prediction ability of the analytical solution for the elastic field distribution in such a complicated system. There are some small inconsistencies in singular behavior on the platelet end plane between the analytical and nume- rical solutions, which can be probably attributed to the limited prediction ability of the finite element numerical method for the stress singularity. These calculations also demonstrated that the same consistent predictions could be achieved for all the strain components and over a great range of material properties and geometries, which were not included here. 5. Conclusions An analytical model was developed for the analysis of the 3D elastic stress field in a platelet reinforced composite subjected to an axial load. The present stress field solution involved two regions, i.e. thematrix region surrounding the platelet and the platelet region. The derived ana- lytical expressions for the matrix and platelet stress fields precisely satisfied both equilibrium and compatibility conditions of the theory of elasticity. Furthermore, since the interface con- tinuity conditions and axial force equilibrium conditions were taken into account, the overall equilibriumwas ensured rigorously within the platelet, matrix and between the platelet and the matrix. Apart from the investigations based on the shear-lag theory, the stress transfer from the matrix to theplateletwasobtained throughthe interface continuity conditionsandtheaxial force equilibrium conditions. Furthermore, this model of the platelet reinforced composite presented all stress, strain and displacement components in thematrix and platelet; in contrast, shear-lag model only could predict the y-direction average stress in the platelet and the interface shear stress. Because the number of boundary conditions, equilibrium and compatibility conditions was more than the number of unknown coefficients (Aj,Bj, n, ñ), optimization methods were used for finding the best answers. A. Solution for matrix displacements ∂2u ∂x2 + ∂2u ∂y2 =0 (A.1) Using themethod of separation of variables for writting u(x,y) as u(x,y) =Y (y)X(x) (A.2) substitution of u(x,y) into Eq. (A.1) leads to ∂2X ∂x2 +nX =0 ∂2Y ∂x2 −nY =0 (A.3) Then, the following can be given X =A1 sinnx+A2cosnx Y =A3 sinhny+A4coshny (A.4) So um(T)(x,y)= (A1 sinnx+A2cosnx)(A3 sinhny+A4coshny) (A.5) For the far-field case n=0, then Xm(0) =B1+B2x Y m(0) =C1+C2y (A.6) Theoretical study of stress transfer in platelet reinforced composites 11 So um(0)(x,y)= (B1+B2x)(C1+C2y) (A.7) where B1, B2, C1 and C2 are constants. The boundary conditions can be used with some simplifications, which results in um(0)(x,y)=B1+B2x (A.8) Substitution of elements of Eqs. (A.5), (A.8) into Eq. (2.1) leads to wm(T)(x,y) = (−A1cosnx+A2 sinnx)(A3coshny+A4 sinhny) wm(0)(x,y)=B3+B4y (A.9) B. Boundary conditions um(b,y)= ũm(b,y) τmxy(b,y)= τ̃ m xy(b,y)= 0 wm(x,0)=wp(x,0)= 0 τmxy(x,0)= τ p xy(x,0)= 0 w̃m(x,l′)= w̃p(x,l′) τ̃mxy(x,l ′)= τ̃pxy(x,l ′)= 0 (B.1) The interface continuity conditions on the platelet-matrix interface (x= a) εm(a,y) = εp(a,y) σmxx(a,y)=σ p xx(a,y) ε̃m(a,y) = ε̃p(a,y) σ̃mxx(a,y)= σ̃ p xx(a,y) (B.2) The continuity conditions on the platelet end surface (x= l) σmyy(x,l)= σ̃ m yy(x,l) σ p yy(x,l)= σ̃ p yy(x,l) τmyy(x,l)= τ̃ m yy(x,l) w m(x,l)= w̃m(x,l) (B.3) C. General solutions for the stress and strain components According to the theory of elasticity, the elastic constituent relations could bewritten as follows τxy = τyx = E 2(1−ν) γxy σxx = E 1−ν2 (εxx+νεyy) σyy = E 1−ν2 (νεxx+εyy) εxx = ∂u ∂x εyy = ∂w ∂y γxy = γyx = ∂u ∂y + ∂w ∂x (C.1) where σ, ε, τ and γ are thenormal stress and strain and the shear stress and strain, respectively. The following equations present the far-field solution of thematrix. Note that, for the end fiber region; i.e. l¬ y¬ l′, all the coefficients should have the tilde σ m(0) xx (x,y)= Em 1−ν2m (B2+νmB4) σ m(0) yy (x,y)= Em 1−ν2m (νmB2+B4) τ m(0) xy (x,y)= 0 ε m(0) xx (x,y)=B2 ε m(0) yy (x,y) =B4 γ m(0) xy (x,y)= γ m(0) yx =0 (C.2) 12 A.M. Fattahi, M.Mondali The far-field solution for the fiber and imaginary fiber regions will be shown as follows. It is worth considering that, for the imaginary fiber, Ep and νp should be converted to Em and νm, and all the coefficients should have the tilde σ p(0) xx (x,y) = Ep 1−ν2p (B6+νpB8) σ p(0) y (x,y)= Ep 1−ν2p (B8+νpB6) τ p(0) xy (x,y)= 0 ε p(0) xx (x,y)=B6 ε p(0) yy (x,y)=B8 γ p(0) xy (x,y)= γ p(0) yx (x,y)= 0 (C.3) The transient solution of thematrix is given below.For the endplatelet region; i.e. l¬ y¬ l′, all the coefficients should have the tilde σm(T)xx (x,y)= nEm 1+νm (A1cosnx−A2 sinnx)(A3 sinhny+A4coshny) σm(T)yy (x,y)= nEm 1+νm (−A1cosnx+A2 sinnx)(A3 sinhny+A4coshny) τm(T)xy (x,y)= nEm 1+νm (A1 sinnx+A2cosnx)(A3coshny+A4 sinhny) εm(T)xx (x,y)=n(A1cosnx−A2 sinnx)(A3 sinhny+A4coshny) εm(T)yy (x,y)=n(−A1cosnx+A2 sinnx)(A3 sinhny+A4coshny) γm(T)xy (x,y)= 2n(A1 sinnx+A2cosnx)(A3coshny+A4 sinhny) (C.4) For the transient solution of the platelet and imaginary platelet, one may reach the following equations. For the imaginary platelet, the Ep and νp should be converted to Em and νm, and all the coefficients should have the tilde σp(T)xx (x,y)= nEp 1+νp (A5cosnx−A6 sinnx)(A3 sinhny+A4coshny) σp(T)yy (x,y)= nEp 1+νp (−A5cosnx+A6 sinnx)(A3 sinhny+A4coshny) τp(T)xy (x,y)= nEp 1+νp (A5 sinnx+A6cosnx)(A3coshny+A4 sinhny) εp(T)xx (x,y)=n(A5cosnx−A6 sinnx)(A3 sinhny+A4coshny) εp(T)yy (x,y)=n(−A5cosnx+A6 sinnx)(A3 sinhny+A4coshny) γp(T)xy (x,y)= 2n(A5 sinnx+A6cosnx)(A3coshny+A4 sinhny) (C.5) References 1. AbedianA.,MondaliM.,PahlavanpourM., 2007,Basicmodifications in 3Dmicromechanical modeling of short fibre composites with bonded and debonded fibre end,Computational Materials Science, 40, 421-433 2. Agarwal B.D., Broutman L.J., 1980, Analysis and Performance of Fibre Composites, John Wiley and Sons, NewYork 3. Agarwal B.D., Lifsitz J.M., Broutman L.J., 1974, Elastic plastic finite element analysis of short fibre composites,Fibre Science and Technology, 7, 45-62 4. 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