N Bolat Other than the periodic constraints enforced on the node-pairs on composites as a means of comparison with the results from the pposite surfaces of the unit cell, translational and rotational con- developed algorithm Assuming 123 to be the principal(orthotro- straints to prevent the singularity of rigid body modes are also pic) directions and xyz be general anisotropic directions in a lami- enforced nate. The stress and strains components in these two coordinate Rigid body constraints: The central node at the center of body is systems are correlated by a geometrical transformation matrix fixed in all directions. To prevent translations and rotations the [T] as center-nodes on faces 1 and 2 are fixed in directions 2 and 3 and finally a node on one of the edges of face 1 is fixed in direction 2 Exx Eyy Ezz xy 7x y) or 3 depending on the edge to prevent rigid rotation about the =[T1E11 222 233 712 713 723] length of the model 3. The constitutive micromechanical material approach 833712713723 (11) where [T], the transformation matrix is written as nFig2, three rucs at fiber cross angles of0°,45°and90°are shown. If the fibers are arranged at zero crossing angles, i.e. parallel mi ni Pi 2m,n 2p, m, 2n,P, fibers, the homogenized properties of the composite should follow m3 n2 p2 2m2n2 2p, m2 2n2p2 those of unidirectional fibrous composites. If the fibers are ar- m=I m? n, p2 2m3, ranged at 90 cross angles, i.e., perpendicular to each others, due mim2 nn2 p1P2 1P2+n2P, p, m2+p2m1 min2+m2n1 creation of three perpendicular planes of material symmetry. m,m, nan, P3P, naP,+, P,m,+p, m, m,n+m, n,I the composite material is orthotropic. At any other cross angle m2m3 nzn3 pp3 nip,+n2p3 p3m2 +p2m m2n3+m3n2 the composite has only one plane of symmetry and therefore the (12) clinic isotropic with the material chosen. If the material is aniso- m ni pi(i=1, 3)are the cosine directions of the coordinate system tropic, then the compliance matrix is fully populated. Thus the xyz with respect to 123. The elastic material constitutive relations constitutive relations in an anisotropic coordinate system 1-2-3 cal an be utilized to correlate the elastic constants in different direc- can be written in the following form: tions or coordinate systems. Therefore, the following relations be tween the material compliance or stiffness in principal directions S111 S1122 S1133 S1112 S1113 S11231(a1 and in generalized anisotropic directions can be concluded S3311 S3322 S3333 S3312 S3313 S3323 S1211S1 (9)Syox Syyy Sya Syyay Sym Syyyz S2311S232S23S2312S233S2323 Sxyax Sxyyy Sxyzz Sxyay Sxya sxyz Sxxx Sxy Sxzn Sxzxy Sxzxz Sxz where Sijk represents are elements of the 6 x 6 matrix of compli- Syax Syzyy Syzz Syzxy Syzxz Sy ance coefficients for a completely anisotropic material. For an S1l Fuzz S anisotropic material one has 21 independent coefficients. In the haracterization procedure, the rUC of the material will S221S222S223000 ected to six independent loading scenarios. The outcome six scenarios of loading yield six sets of stresses and strains, and therefore, thirty six equations needed to find the 36 0000S130 mponents of the compliance or stiffness matrix in general. The 00000S2323 mpliance and stiffness components and their relations in terms of engineering constants can be found in most composite text- For the bidirectional angle-ply laminate under consideration in this books 38]. paper, the equivalent compliance/stiffness of the laminate can be For each load scenario, the stress-strain information obtained considered a combination of the stiffness/compliance of two lami- from the analysis are volume averaged over the volume of the nas(shown in Fig. 1), that is Cleg=vi[Ch+ vlc (10) where [c [S] and V(i= 1, 2)are the stiffness, compliance matri- where 'V is the volume of the ruc these correlated ces and the volume fractions of the laminas 1 and 2, respectively provide the base for determination of t I indepen- The equivalent compliance/stiffness can then be measured elements of compliance coefficients ing the as- employing Eqs.(12)-(14)to be compared with the solutions of constitutive relations response analysis under different loading scenarios will be carried principal directions, and are orthotropic in their respective prind out by finite elements. The volume average data and inverse pal directions. One lamina remains unidirectional in the global characterization analysis will be interfaced with the finite ele- 123 directions(shown in Fig. 1). and the second one is only ment package orthotropic along the principal directions oriented at a cross an gle of o with respect to direction 1. The cosine directions become 3.1. Determination of angle-ply elastic constants from lamination mpler for the transformation, ie, m,=COSp, m 2=sine, m3=0 1=-sin, n2=COS, n3=0: p1=0. P2=0, P3=1. In the modeling procedure one laminate is unidirectional and therefore the co As a means of characterization of fibrous composites, lamina- pliance and stiffness remain constant in 123 directions. The sec- tion theory can be used with a degree of mation. Here this ond laminate has a principal directions oriented along the fiber approximated theory is tailored for the solution of the angle-ply directionsOther than the periodic constraints enforced on the node-pairs on opposite surfaces of the unit cell, translational and rotational con￾straints to prevent the singularity of rigid body modes are also enforced. Rigid body constraints: The central node at the center of body is fixed in all directions. To prevent translations and rotations the center-nodes on faces 1 and 2 are fixed in directions 2 and 3 and finally a node on one of the edges of face 1 is fixed in direction 2 or 3 depending on the edge to prevent rigid rotation about the length of the model. 3. The constitutive micromechanical material approach In Fig. 2, three RUCs at fiber cross angles of 0, 45 and 90 are shown. If the fibers are arranged at zero crossing angles, i.e. parallel fibers, the homogenized properties of the composite should follow those of unidirectional fibrous composites. If the fibers are ar￾ranged at 90 cross angles, i.e., perpendicular to each others, due to creation of three perpendicular planes of material symmetry, the composite material is orthotropic. At any other cross angle, the composite has only one plane of symmetry and therefore, the homogenized properties of the composite are considered mono￾clinic isotropic with the material chosen. If the material is aniso￾tropic, then the compliance matrix is fully populated. Thus the constitutive relations in an anisotropic coordinate system 1–2–3 can be written in the following form: e11 e22 e33 e12 e13 e23 8 >>>>>>>>< >>>>>>>>: 9 >>>>>>>>= >>>>>>>>; ¼ S1111 S1122 S1133 S1112 S1113 S1123 S2211 S2222 S2233 S2212 S2213 S2223 S3311 S3322 S3333 S3312 S3313 S3323 S1211 S1222 S1233 S1212 S1213 S1223 S1311 S1322 S1333 S1312 S1313 S1323 S2311 S2322 S2333 S2312 S2313 S2323 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 r11 r22 r33 s12 s13 s23 8 >>>>>>>>< >>>>>>>>: 9 >>>>>>>>= >>>>>>>>; ð9Þ where Sijkl represents are elements of the 6  6 matrix of compli￾ance coefficients for a completely anisotropic material. For an anisotropic material one has 21 independent coefficients. In the material characterization procedure, the RUC of the material will be subjected to six independent loading scenarios. The outcome of these six scenarios of loading yield six sets of stresses and strains, and therefore, thirty six equations needed to find the 36 components of the compliance or stiffness matrix in general. The compliance and stiffness components and their relations in terms of engineering constants can be found in most composite text￾books [38]. For each load scenario, the stress–strain information obtained from the analysis are volume averaged over the volume of the RUC, i.e., rij ¼ 1 V Z v rij dv; eij ¼ 1 V Z v eij dv ð10Þ where ‘V’ is the volume of the RUC. These averaged correlated data provide the base for determination of the 36 (21 indepen￾dent) elements of compliance coefficients Sijkl or Cijkl using the as￾sumed constitutive relations eij ¼ Sijklrkl or rkl ¼ Cijkleij. The response analysis under different loading scenarios will be carried out by finite elements. The volume average data and inverse characterization analysis will be interfaced with the finite ele￾ment package. 3.1. Determination of angle-ply elastic constants from lamination theory As a means of characterization of fibrous composites, lamina￾tion theory can be used with a degree of approximation. Here this approximated theory is tailored for the solution of the angle-ply composites as a means of comparison with the results from the developed algorithm. Assuming 123 to be the principal (orthotro￾pic) directions and xyz be general anisotropic directions in a lami￾nate. The stress and strains components in these two coordinate systems are correlated by a geometrical transformation matrix [T] as, f exx eyy ezz cxy cxz cyz gT ¼ ½T 1 f e11 e22 e33 c12 c13 c23 gT f exx eyy ezz cxy cxz cyz gT ¼ ½T 1 f e11 e22 e33 c12 c13 c23 gT ð11Þ where [T], the transformation matrix is written as ½T ¼ m2 1 n2 1 p2 1 2m1n1 2p1m1 2n1p1 m2 2 n2 2 p2 2 2m2n2 2p2m2 2n2p2 m2 1 n2 1 p2 1 2m3n3 2p3m3 2n3p3 m1m2 n1n2 p1p2 n1p2 þn2p1 p1m2 þp2m1 m1n2 þm2n1 m3m1 n3n1 p3p1 n3p1 þn1p3 p3m1 þp1m3 m3n1 þm1n3 m2m3 n2n3 p2p3 n3p2 þn2p3 p3m2 þp2m3 m2n3 þm3n2 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ð12Þ mi, ni, pi (i = 1, 3) are the cosine directions of the coordinate system xyz with respect to 123. The elastic material constitutive relations can be utilized to correlate the elastic constants in different direc￾tions or coordinate systems. Therefore, the following relations be￾tween the material compliance or stiffness in principal directions and in generalized anisotropic directions can be concluded. Sxxxx Sxxyy Sxxzz Sxxxy Sxxxz Sxxyz Syyxx Syyyy Syyzz Syyxy Syyxz Syyyz Szzxx Szzyy Szzzz Szzxy Szzxz Szzyz Sxyxx Sxyyy Sxyzz Sxyxy Sxyxz Sxyyz Sxzxx Sxzyy Sxzzz Sxzxy Sxzxz Sxzyz Syzx Syzyy Syzzz Syzxy Syzxz Syzyz 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ¼ ½T 1 S1111 S1122 S1133 000 S2211 S2222 S2233 000 S3311 S3322 S3333 000 000 S1212 0 0 0000 S1313 0 00000 S2323 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ½T ð13Þ For the bidirectional angle-ply laminate under consideration in this paper, the equivalent compliance/stiffness of the laminate can be considered a combination of the stiffness/compliance of two lami￾nas (shown in Fig. 1), that is, ½C eq ¼ V1½C 1 þ V2½C 2 ½S eq ¼ ½C 1 eq ; ½S 1 ¼ ½C 1 1 ; ½S 2 ¼ ½C 1 2 ð14Þ where [C]i, [S]i and Vi (i = 1,2) are the stiffness, compliance matri￾ces and the volume fractions of the laminas 1 and 2, respectively. The equivalent compliance/stiffness can then be measured employing Eqs. (12)–(14) to be compared with the solutions of the micromechanics formulations. Note that both laminas have principal directions, and are orthotropic in their respective princi￾pal directions. One lamina remains unidirectional in the global 123 directions (shown in Fig. 1), and the second one is only orthotropic along the principal directions oriented at a cross an￾gle of u with respect to direction 1. The cosine directions become simpler for the transformation, i.e., m1 = cosu, m2 = sinu, m3 = 0; n1 = sinu, n2 = cosu, n3 = 0; p1 = 0, p2 = 0, p3 = 1. In the modeling procedure one laminate is unidirectional and therefore the com￾pliance and stiffness remain constant in 123 directions. The sec￾ond laminate has a principal directions oriented along the fiber directions. 1198 N. Abolfathi et al. / Computational Materials Science 43 (2008) 1193–1206