International Journal of Engineering Science

International Journal of Engineering Science 63 (2013) 1 9 Contents lists available at SciVerse ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci Numerical study of effective elastic properties of fiber reinforced composites with rhombic cell arrangements and imperfect interface Mathias Würkner, Harald Berger, Ulrich Gabbert Institute of Mechanics, University of Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany article info abstract Article history: Received 19 July 2012 Received in revised form 8 October 2012 Accepted 11 October 2012 Keywords: Composite Fiber Rhombic arrangement Mechanical properties Imperfect contact Finite element method This paper deals with the behavior of unidirectional fiber composites, where on micro scale the reinforcement and the filler are connected by imperfect interface conditions. It is assumed, that the microstructure is periodic and their phases have isotropic and transversely isotropic properties, respectively. The fibers are distributed in such a way, that the microstructure is characterized by a rhomb. By using the concept of a representative volume element (RVE) a finite element model is created, which combines the rhombic distribution of fiber and imperfect contact conditions (spring type) between the phases. Periodic boundary conditions are applied to the RVE, so that effective material properties can be derived. The presented homogenization technique is validated by comparing results with theoretical approach reported in the literature as far as possible. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction There has been done a lot of research in the issue of homogenization and imperfect contact conditions. By homogenization techniques effective material properties for fiber reinforced composites or other complex heterogeneous materials are derivable. Those properties are constants and therefore further investigations of complex mechanical components are possible with a less effort. There are several homogenization methods. For some of them closed-form expressions or formulae are derived. Famous are for example the bounds of Hashin and Shtrikman (1962a, 1962b). Nowadays the concept of the representative volume element (RVE) in connection with a finite element analysis (FE-analysis) gets more and more importance (Kari, Berger, Rodríguez-Ramos, & Gabbert, 2007). A often used assumption for such methods is that the composite consists of two constituents (phases), which are perfectly bonded to each other. For some composites a model with a third constituent (Achenbach & Zhu, 1990; Andrianov, Danishevs kyy, & Kalamkarov, 2008; Christensen & Lo, 1979, 1986) dealing as coating or adhesion is more realistic. Since this separating phase can have a low volume fraction and a small wall thickness, it can be treated as an interface, which fulfills certain conditions (Benveniste & Miloh, 2001; Hashin, 2002). Often used conditions at the interface are that the traction components of the constituents of the composite are proportional to the difference or jump of the displacements. The proportionality factor is given by a constant, which has N/mm 3 as unit of measurement. This imperfect contact (also called spring type ) was studied by for instance Achenbach and Zhu (1989), Andrianov, Bolshakov, Danishevs kyy, and Weichert (2007), Benveniste and Miloh (2001), Hashin (1990), López-Realpozo, Rodríguez-Ramos, Guinovart-Díaz, Bravo-Castillero, and Sabina (2011) and Sevostianov, Rodríguez-Ramos, Guinovart-Díaz, Bravo-Castillero, and Sabina (2012). In some cases effective axial shear properties are presented. Corresponding author. E-mail address: mathias.wuerkner@st.ovgu.de (M. Würkner). 0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2012.10.002

2 M. Würkner et al. / International Journal of Engineering Science 63 (2013) 1 9 In this paper a numerical technique is developed, which consider a unidirectional fiber reinforced composite, where the fibers are arranged in a rhombic periodic pattern and the interface between fiber and matrix is treated as imperfect i.e., the contrast or jump of the displacements in the interface is proportional to the corresponding component of the tension in the interface in terms of a parameter given by the constant of a spring. Such rhombic structures in case of perfect bond were studied by Jiang, Xu, Cheung, and Lo (2004), Golovchan and Nikityuk (1981), Rodríguez-Ramos et al. (2011), in which derived effective axial shear coefficients are presented. Hexagonal (60 ) and square arrangements (90 ) are special cases of a rhombic structure. So by the possibility of modeling arbitrary rhombic pattern a wider spectrum of composite structures can be observed. This paper is treated as an extension of a previous work of Würkner, Berger, and Gabbert (2011), where only perfect contact for rhombic cells is concerned. Results within this paper are compared with others reported in the literature for verification of the present model. 2. Numerical procedure and imperfect modeling This section contains fundamentals of a homogenization procedure based on the considered elastic theory and the development of an appropriate representative volume element (RVE) in order to create a finite element model, which consists of two different material phases (matrix, fiber) and an imperfect interface. 2.1. Homogenization method In this subsection a two phase model is considered, which consist of a fiber and a matrix phase, where all fibers have the same radius and they are unidirectional orientated. The interface between them (fiber and matrix phase) is assumed to be perfect. The fibers have a circular cross section with equal diameter. In addition with respect to the transversal plane they are arranged in a rhombic pattern (Fig. 1). All phases have isotropic material properties. It is assumed, that the constitutive law in each phase of the composite is given by the Hooke s law, which is written in Einstein summation convention r ij ¼ C ijkl e kl ; i; j; k; l ¼ 1; 2; 3; ð1þ where r ij, e kl and C ijkl are the coefficients of the stress tensor, the linear strain tensor and the stiffness tensor, respectively. They are related to a previously chosen Cartesian coordinate system. In Fig. 1 the chosen RVE can be seen. It is related to the coordinates x 1, x 2 and x 3. It is assumed, that the coefficients of Eq. (1) fulfill the symmetry conditions of the linear elasticity (Torquato, 2002). In case of the stiffness tensor the coefficients are treated as to be constant in each phase. By homogenization techniques it is sufficient to consider a periodic micro structure denoted as V rve, which forms the RVE shown in Fig. 1. Due to linear elasticity in each phase the following equations of equilibrium @ @x j r ij ðxþ ¼0; hold. At the interface C between the phases the continuity of stresses and displacements is assumed (Aboudi, 1991) ( u f i ¼ u m i ; x 2 C: r f ij nf j ¼ r m ij nm j ; x 2 C: ð2þ ð3þ The n k j ; ðk ¼ m; f Þ denote the components of the unit outward normal vector with respect to fiber (superscript f) and matrix (superscript m). On the boundary of the RVE, denoted by @V rve, the following periodicity condition is assumed u i ðxþ e 0 ij x j @V rve periodic: ð4þ The quantities e 0 ij are the given constant coefficients of the macroscopic strain tensor, and u is the displacement field in the microstructure. The condition in formulation Eq. (4) ensures both the continuity of extensions of displacements on adjoining cells and derivation of six independent load cases on the microstructure. In order to avoid rigid body motions one point of the RVE has to be fixed. The choice of this point is independent with respect to the effective (homogenized) stiffness Fig. 1. Rhombic cell arrangement and the rectangular RVE.

M. Würkner et al. / International Journal of Engineering Science 63 (2013) 1 9 3 coefficients. So any point of the RVE can be chosen. The stiffness coefficients, which represent the material behavior on macro level, are calculated from hr ij i¼ ijkl he kli; where the quantities hr ij i¼ 1 Z r ij ðxþdx; jv rve j V rve he kl i¼ 1 Z e kl ðxþdx; jv rve j V rve ð5þ ð6þ ð7þ are the corresponding stress and strain components on macro scale. Using the divergence theorem one can show, that he kl i¼e 0 kl ; ð8þ holds. By setting e 0 ij, three pure normal strain states and three pure shear states are derived with respect to the macro scale. In case of pure normal states the right hand side of Eq. (5) is only dependent on one strain component he kl i, since the others are zero. For the shear case the same dependence is achieved with regard of the symmetry of the strain tensor. Therefore it is possible to calculate all effective coefficients by ¼ hr iji ijkl he ; k ¼ l; kl i ¼ hr ð9þ iji ijkl 2he ; k l: kl i 2.2. Imperfect interface model In the previous subsection the interface between the phases are treated as perfect. This means continuity in displacements and stresses. The perfect interface contact is now replaced by imperfect interface conditions (Andrianov, Bolshakov, Danishevs kyy, & Weichert, 2007; Hashin, 1990, 2002), which have the form r m rr ¼ rf rr ¼ K rjju r jj r m rh ¼ rf rh ¼ K hjju h jj: r m rz ¼ rf rz ¼ K zjju z jj: ð10þ The values r i rr ; ri rh and ri rz are the surface traction components with respect to a cylindrical coordinate system, which are related to either the fiber (i = f) or the matrix phase (i = m). The quantities K i, i = r, h, z are spring type parameters, which have the dimension of stresses divided by length. The double bar defines differences of displacements between matrix and fiber phase jju i jj ¼ u m i u f i ; i ¼ r; h; z: ð11þ From Hashin (2002) the parameters K i, i = r, h, z can be defined, since the imperfect contact formulation stems from considering a three phase problem with thin interphase thickness. If the interphase material behaves isotropic, they have the form E i 1 m i K r ¼ K h ¼ Gi t K z ¼ Gi t ; tð1 2m i Þð1 þ m i Þ where E i, m i and t are the Young s modulus, Poisson s ratio and the radial directional thickness of the interphase, respectively. It will be assumed, that here also holds, that with the boundary conditions in Eq. (4) the quantities e 0 ij describe the macroscopic strains. Due to the imperfect contact the averaged strains Eq. (7) are changed to he kl i¼ 1 Z e kl dv þ 1 Z e kl dv þ 1 Z jju k jjn l þjju l jjn k ds; ð13þ jv rve j V m jv rve j V f 2jV rve j C where V m, V f and C are the matrix phase, fiber phase and the interface between the matrix and fibers, respectively (Caporale, Luciano, & Sacco, 2006; Nemat-Nasser & Hori, 1999). The values n k and n l are the k-th and l-th component of the outer normal vector, respectively, where the normal vector is related to the fiber phase. ð12þ

4 M. Würkner et al. / International Journal of Engineering Science 63 (2013) 1 9 2.3. Numerical consideration This subsection contains basic information about the implementation in ANSYS. The model of the micro cell (see Fig. 2) consists of three-dimensional linear SOLID45 elements. These elements are characterized by eight nodes. The cell is rectangular shaped with the geometry characterizing parameters l 1 ¼ 2 cos a 2 ; l 2 ¼ 2 sin a 2 ; ð14þ where l 1 and l 2 are the width (x 1 -direction) and the height (x 2 -direction), respectively. In the depth (x 3 -direction) the model has one finite element discretization. To achieve also double symmetry in the mesh first a quarter of the mesh is built. Then it is copied to x 1 - and x 2 -direction symmetrically. For this geometric model the maximum fiber volume fraction is limited and dependent of the angle a in order to avoid geometric overlapping. This is taken into account by the inequality r f < min l 2 2 ; 0:5 : ð15þ The parameter r f is the radius of the fiber. For the imperfect contact behavior spring elements are used between matrix and fiber. Here every nodal pair fiber and matrix phase have on the surface coincident node discretization and location is connected by three one dimensional COMBIN14 elements. So the imperfect contact relations Eq. (10), which set stresses proportional to displacement differences, changes to the numerical relations F r ¼ K r jju rjj; F h ¼ K h jju hjj; F z ¼ K z jju zjj; ð16þ where F j, j = r, h, z, the force components, are proportional to the displacement differences. The parameters K i ; i ¼ r; h; z represent spring constants, which are related to the spring type parameters by K i ¼ K i A n ; i ¼ r; h; z: ð17þ Here A n is the area of contact surface related to node n, which is a node of the coincident nodal pair. In the case that due to finite element discretization the fiber geometry is represented by a non-oblique prism with an equilateral polygon as base area the corresponding area A n related to node n is given by (Caporale et al., 2006) A n ¼ r f e sin p : ð18þ n p Here e and n p are the depth (x 3 -direction) of the RVE and the number of nodes of the polygon. This formula holds only for a fiber, which is completely embedded in the RVE. Due to finite element discretization the integrals in Eqs. (6) and (7) for calculation averaged stresses and strains are replaced by sums over all element contributions. Furthermore the surface integral in Eq. (13) is replaced by summation over nodal displacement differences. Fig. 2. Meshed RVE for 0.4 fiber volume fraction and 75 fiber arrangement.

M. Würkner et al. / International Journal of Engineering Science 63 (2013) 1 9 5 6.6 6 5.4 4.8 k eff transversal /Gm 4.2 3.6 3 3 phase model,fem spring model,fem 3 phase model (CCA),Hashin2002 0 Fig. 3. Transversal bulk modulus for 0.4 fiber volume fraction and g = 0.001. 2.2 2 G eff axial /Gm 1.6 1 0.8 3 phase model,fem spring model,fem 3 phase model (CCA),Hashin2002 0.4 Fig. 4. Axial shear modulus for 0.4 fiber volume fraction and g = 0.001. 3. Comparisons and results Firstly, a comparison with results reported by Hashin (2002) is presented in order to verify the correctness of the implemented algorithm. Effective material constants are listed and plotted in order to compare the mechanical behavior of composites under hexagonal fiber arrangements. Afterward selected macroscopic stiffness coefficients according to several rhombic fiber arrangements are presented in dependence of imperfect contact behavior. A RVE is considered, where the rhombic structure is characterized by an angle of 60. The volume fraction of the fiber is fixed to a value of 0.4. The material properties for the matrix and fiber phase are characterized by the ratio of the shear moduli (Hashin, 2002) G f G m ¼ 10: ð19þ The quantities are the shear modulus of the fiber (index f ) and the matrix (index m ) phase, respectively. In addition to the shear ratio Eq. (19) the Poisson s ratios of the matrix and fiber phase are 0.35 and 0.2, respectively. Since the imperfect contact condition is derived from a three phase problem, where the interphase has isotropic material behavior, the Poisson s

6 M. Würkner et al. / International Journal of Engineering Science 63 (2013) 1 9 6.6 6 5.4 4.8 k eff transversal /Gm 4.2 3.6 3 3 phase model,fem spring model,fem 3 phase model (CCA),Hashin2002 0 log 10 Fig. 5. Transversal bulk modulus for 0.4 fiber volume fraction and g = 0.01. 2.2 2 G eff axial /Gm 1.6 1 0.8 3 phase model,fem spring model,fem 3 phase model (CCA),Hashin2002 0.4 Fig. 6. Axial shear modulus for 0.4 fiber volume fraction and g = 0.01. ratio is defined by 0.3. The shear modulus is also predefined, but not fixed, since we want to vary it. The interphase thickness is given by the dimensionless parameter g ¼ t r f ; ð20þ which takes values of 0.001 and 0.01. In Figs. 3 and 4 the normalized transverse bulk modulus and the normalized axial shear modulus are plotted. The graphs contain results obtained by FEM procedures and formulae taken from Hashin (2002) related to a three phase CCA-model. The FEM procedures are based on a three phase composite model and a two phase composite model with imperfect interface conditions (spring model). The results for the transverse bulk modulus as well as the axial shear modulus show good agreement between all three methods for values log < 2. A very small value represents almost no interactions between fiber and matrix, which is equivalent to a model with voids. For values higher than two only the three phase models are in good agreement. Here the influence of the third phase for the averaging process gets higher and higher, since interphase stiffness is increased. The two phase imperfect model converges to the value of a two phase model with perfect contact.

M. Würkner et al. / International Journal of Engineering Science 63 (2013) 1 9 7 Table 1 Material phase properties. Epoxy Graphite E [GPa] m E a (GPa) m a E t (GPa) m t G a (GPa) G t (GPa) 3.45 0.35 345 0.2 9.66 0.3 2.07 3.72 5.6 4.9 4.2 iiii /Gm 3.5 2.8 2.1 1111 /Gm 2222 /Gm Fig. 7. Transversal coefficients for 0.4 fiber volume fraction and a = 45. 4.9 4.2 iiii /Gm 3.5 2.8 2.1 1111 /Gm 2222 /Gm Fig. 8. Transversal coefficients for 0.4 fiber volume fraction and a = 75. In Figs. 5 and 6 the results for the effective transversal bulk modulus and shear modulus are plotted, where g = 0.01. This means for the three phase model the thickness of the interphase is increased by the factor 10. It can be recognized from the graphs, that the bound for normalized interphase shear value, where all model problems show good agreements in the effective values, is decreased significantly. Across the bound as in the results before the values of the imperfect model converges to the value of the perfect two phase model and the remaining models show almost coincidence. In the following a more realistic composite model with unidirectional fibers is considered. Here the fibers have transversely isotropic material properties and the matrix is isotropic. The considered materials are Epoxy and Graphite, stated as in Hashin (1979). For detailed information see Table 1.

8 M. Würkner et al. / International Journal of Engineering Science 63 (2013) 1 9 3 2.7 2.1 iijj /Gm 1.5 0.9 0.3 0 1122 /Gm,45 2211 /Gm,45 1122 /Gm,75 2211 /Gm,75 Fig. 9. Transversal coefficients for 0.4 fiber volume fraction. 1.6 1212 /Gm 1 0.8 0.4 0.2 G eff 1212 /Gm,45 G eff 1212 /Gm,75 Fig. 10. Transversal shear coefficients for 0.4 fiber volume fraction. In Figs. 7 10 transversal effective coefficients for rhombic fiber arrangements are plotted. The angle, which characterizes the distribution, is 45 and 75, respectively. The fiber volume fraction is 0.4. The interface contact parameters are characterized by Eq. (12), where the thickness parameter t is given by the dimensionless parameter g = 0.001. All values for the effective stiffness coefficients are referred to the coordinate system x 1, x 2 and x 3. Figs. 7 and 8 show the principal diagonal coefficients 1111 and Ceff 2222. Small differences of the values can be observed with respect to high imperfect contact parameters, which represent perfect interface contact. For 75 this difference is much smaller than for 45. A more obvious anisotropy can be observed (Würkner et al., 2011) in the case of 75 with higher volume fraction of the fiber phase. With respect to low contact parameters the effective material behavior is as in previous figures (model with voids). For this range the coefficients show a more significant anisotropy. In Fig. 9 the off-diagonal coefficients 1122 and Ceff 2211 for a =45 and a =75 are plotted. For each angle these effective components show coincidence, which confirms the correctness of the used procedure. The transversal shear coefficient for 45 and 75 can be observed in Fig. 10. It can be seen, that in case of very low interphase moduli the coefficient for 75 is much lower than for 45.

M. Würkner et al. / International Journal of Engineering Science 63 (2013) 1 9 9 4. Conclusion In this paper homogenized coefficients for unidirectional fiber reinforced composites with isotropic and transversal isotropic constituents are derived from a numerical method, where the micro structure of the composite is characterized by a rhombic fiber distribution and imperfect interface between fiber and matrix. For the realization the development of an appropriate model for a FE-analysis in ANSYS is described. In order to verify the present method, effective coefficients are compared to values derived from a CCA-model (Hashin (2002)). Also a FE-analysis considering a three-phase-geometry is studied as another comparison. There is a good agreement with the compared models in the case of g ¼ 0:001 and low interphase properties. Especially the CCA-model and the FE-analysis with three constituents show in all figures of comparison almost coincidence. 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