N. Abolfathi et aL/ Computational Materials Science 43(2008)1193-1206 load cases are analyzed simultaneously (inverse analysis)to pre- with cross angles 0. 45 and 90. For the parallel distributed fibers dict the material characteristics of the unit cell. The methodology the symmetry lines define the periodic microstructure of a straight has verified itself for cases of known solutions, i.e. when the unit fiber composite(Fig. 1). The geometrical parameters of the unit cell is assumed of the same homogenized pure elastic and thus cells are the cross-sectional width w, height h and length L corre- the input data should be expected from the reverse analysis. The lated to the diameters of fibers to maintain fiber/matrix volume illustrative analysis presented in the current study is limited to fractions. elastic materials; however, the methodology can be used for vari- Different unit cell models were developed to study the effect of ous circumstances of composite characterization procedures, such changing cross angles of the fibers, as well as, different fiber vol- as viscoelastic materials [36] with straight or wavy fibers [191. ume fractions of the bidirectional fibers on the overall material property of the composite. In order to verify the accuracy of the 2. Repeating unit cell(RUC)of the bidirectional fibrous modeling procedure, pure unit cells made of the same material type for both fiber and matrix at different cross angles were exam- ined under the six loading conditions to yield the input character- As shown in Fig. 1, bidirectional fibers with a crossing angle of o istics of the materials bers is assumed to remain constant so that a periodic unit cell 2.1. Loading and periodicity constraints can be defined. The periodicity of microstructure determines the geometry of the unit cell. As shown, a parallelplied geometry made Load cases: To determine the compliance and stiffness coeffi- of the matrix and fibers creates the cross-sectional view of the f- cients of the composite, each individual model was analyzed under brous composite As a bidirectional fibrous composite it is assumed six load scenarios. The six types of loadings include three axial and that the microstructure of the composite along the third direction three shear forces(two direct shears, and one shear due to torsion) (perpendicular to plane of cross-section) remains constant. The Referring to Fig. 2, the directions 1, 2, and 3 correspond to the netric shape of the RUC is shown in Fig. 1c. The fibers are all longitudinal, transverse in the plane of fiber, and transverse normal for such a RUC is shown in Fig. 1d. Fig. 2 shows three different RUC Fig 3)were defined as the following. ctively. Six load cases(see ssumed straight and of circular cross sections. a typical FEM mesh to the plane of fiber directions, res b d Periodic Unit Cod Fig. 1.(a)The bidirectional fibers at cross angles of o embedded in matrix, (b)the cross-sectional view of a unit cell. (c)the periodic 3-D unit cell volume, and(d)the FEm discretized of the ruc.load cases are analyzed simultaneously (inverse analysis) to pre￾dict the material characteristics of the unit cell. The methodology has verified itself for cases of known solutions, i.e., when the unit cell is assumed of the same homogenized pure elastic and thus the input data should be expected from the reverse analysis. The illustrative analysis presented in the current study is limited to elastic materials; however, the methodology can be used for vari￾ous circumstances of composite characterization procedures, such as viscoelastic materials [36] with straight or wavy fibers [19]. 2. Repeating unit cell (RUC) of the bidirectional fibrous composite As shown in Fig. 1, bidirectional fibers with a crossing angle of u as embedded in a matrix are shown. The crossing angle of the fi- bers is assumed to remain constant so that a periodic unit cell can be defined. The periodicity of microstructure determines the geometry of the unit cell. As shown, a parallelplied geometry made of the matrix and fibers creates the cross-sectional view of the fi- brous composite. As a bidirectional fibrous composite it is assumed that the microstructure of the composite along the third direction (perpendicular to plane of cross-section) remains constant. The volumetric shape of the RUC is shown in Fig. 1c. The fibers are all assumed straight and of circular cross sections. A typical FEM mesh for such a RUC is shown in Fig. 1d. Fig. 2 shows three different RUC with cross angles 0, 45 and 90. For the parallel distributed fibers, the symmetry lines define the periodic microstructure of a straight fiber composite (Fig. 1). The geometrical parameters of the unit cells are the cross-sectional width w, height h and length L corre￾lated to the diameters of fibers to maintain fiber/matrix volume fractions. Different unit cell models were developed to study the effect of changing cross angles of the fibers, as well as, different fiber vol￾ume fractions of the bidirectional fibers on the overall material property of the composite. In order to verify the accuracy of the modeling procedure, pure unit cells made of the same material type for both fiber and matrix at different cross angles were exam￾ined under the six loading conditions to yield the input character￾istics of the materials. 2.1. Loading and periodicity constraints Load cases: To determine the compliance and stiffness coeffi- cients of the composite, each individual model was analyzed under six load scenarios. The six types of loadings include three axial and three shear forces (two direct shears, and one shear due to torsion). Referring to Fig. 2, the directions 1, 2, and 3 correspond to the longitudinal, transverse in the plane of fiber, and transverse normal to the plane of fiber directions, respectively. Six load cases (see Fig. 3) were defined as the following: Fig. 1. (a) The bidirectional fibers at cross angles of u embedded in matrix, (b) the cross-sectional view of a unit cell, (c) the periodic 3-D unit cell volume, and (d) the FEM discretized of the RUC. N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206 1195