Z-M. Huang/ Computers and Structures 80(2002)1177-1199 The present author has recently proposed a useful shown in Fig. 2 of Ref [ID. This will be done based on the composite theory, the bridging model [5]. By combining classical lamination theory. Suppose that the laminate the bridging model with the classical lamination theory, consists of a sequence of fibrous laminae, stacking in a general constitutive relationship can be established for different ply-angles. Each lamina can have different con- fibrous composite laminate. In this two-parts of stituent materials, different fiber volume fraction, and papers, a systematic description for the general constit even different fiber preform reinforcement. We take a utive theory is presented. In the first part [I], the lamina global coordinate system(X, y, z) with X and y in the analysis based on the bridging model has been described. laminate plane and z along the thickness direction. Let Any single layer continuous fiber reinforced composite the fiber direction of the kth lamina have an inclined ply- lamina was eventually subdivided into a number of uni- angle Ak with the global X direction, as shown in Fig 3 directional (UD) composites to which the bridging model of Ref [l] for a UD lamina. According to the classical can be applied. The response of the original lamina was lamination theory, only in-plane stress and strain in- obtained by means of an assemblage. The purpose of the crements, i. e idol= doxx, do yy, doxy)and deh lamina analysis under an arbitrary load condition was IdExx, dey, 2dexr, are retained, where G refers to the obtain three fundamental quantities, i.e. the instanta global coordinate system. The out-of plane strain com- neous compliance matrix of the lamina, which will be ponents, dExz, dayz, and dez, are assumed to be zero, immediately seen in this paper to be critical for laminate whereas the out-of-plane stress components can be de- analysis, the internal stress state in the constituent fibers, termined using the lamina three-dimensional constitutive and the internal stresses in the constituent matrix mat relationship if necessary. The averaged stress increments rial. Based on the latter two quantities, the lamina load on the kth lamina can be determined from(refer to eq carrying capacity can be understood from the knowledge(13.3)of Ref [ D of the constituent load carrying abilities. In the second part, i.e. the present paper, the lami nate analysis is elaborated. As the laminate consists of doe laminae, the external load applied on the laminate must [(CO)defT -) dT (1) be shared by each lamina involved. This load share can be determined through the classical lamination theor where However, the instantaneous compliance matrix of the lamina is required, since the lamina is a statically inde- terminate structure in the laminate. For the same reason d-{4++2,4 the laminate response can be determined once the lam- ina behavior has been fully understood. Towards this Zk+zk-I dx yr, idexx +(zk+zk-idK purpose, one additional incorporation is of stiffness discount: whenever a lamina fails. its stiffness contribu tion to the remaining laminate must be reduced. a total reduction is adopted in the present paper, which is ver- fied to be pertinent by some experimental evidence Extensive comparisons between simulations and e periments have been demonstrated in the paper using {B={(B1,(B2),(B3)}2=([T1)(S)-2{(xh xample laminates. These examples cover a large variety of laminates reinforced with different fiber preforms and having different constituent materials, under static or depx, dapy, and depr and dxpx, dnr, and dir are the fatigue load conditions, and with or without thermal laminate in- plane strain and curvature increments. Zk load effect. For all these composites, only the material and Zk-I are the Z coordinates of the top and the bottom parameters of their constituent fibers and matrixes are surfaces of the kth lamina. I is the lamina instanta- sed as input data, which are measured independently. neous compliance matrix given in the ply coordinate The predicted ultimate strengths or the stress-strain system, which can be obtained using the formulae pre curves up to failure of all the composite laminates agree sented in Ref. [] for various fiber preform reinforced reasonably well with their experimental counterparts laminae. Specifically, for a UD lamina, its ply coordi- nate system coincides with its local one. Hence, Sk is simply given by Eq (4)of Ref [1]. If the lamina under 2. Laminate analysis ion is a single layer woven/braided fabric composite, [ SI is defined by equation(41)of Ref. Having obtained the instantaneous compliance (see Section 6 of this paper for additional discussion) tries of various fibrous laminae [l], we can now per- Further, Sk is given by equation(43. 1)of Ref [I] if the form the laminate analysis(referring to the flow chart lamina is a single layer knitted fabric composite. TheThe present author has recently proposed a useful composite theory, the bridging model [5]. By combining the bridging model with the classical lamination theory, a general constitutive relationship can be established for any fibrous composite laminate. In this two-parts of papers, a systematic description for the general constit￾utive theory is presented. In the first part [1], the lamina analysis based on the bridging model has been described. Any single layer continuous fiber reinforced composite lamina was eventually subdivided into a number of uni￾directional (UD) composites to which the bridging model can be applied. The response of the original lamina was obtained by means of an assemblage. The purpose of the lamina analysis under an arbitrary load condition was to obtain three fundamental quantities, i.e. the instanta￾neous compliance matrix of the lamina, which will be immediately seen in this paper to be critical for laminate analysis, the internal stress state in the constituent fibers, and the internal stresses in the constituent matrix mate￾rial. Based on the latter two quantities, the lamina load carrying capacity can be understood from the knowledge of the constituent load carrying abilities. In the second part, i.e. the present paper, the lami￾nate analysis is elaborated. As the laminate consists of laminae, the external load applied on the laminate must be shared by each lamina involved. This load share can be determined through the classical lamination theory. However, the instantaneous compliance matrix of the lamina is required, since the lamina is a statically inde￾terminate structure in the laminate. For the same reason, the laminate response can be determined once the lam￾ina behavior has been fully understood. Towards this purpose, one additional incorporation is of stiffness discount: whenever a lamina fails, its stiffness contribu￾tion to the remaining laminate must be reduced. A total reduction is adopted in the present paper, which is ver￾ified to be pertinent by some experimental evidence. Extensive comparisons between simulations and ex￾periments have been demonstrated in the paper using example laminates. These examples cover a large variety of laminates reinforced with different fiber preforms and having different constituent materials, under static or fatigue load conditions, and with or without thermal load effect. For all these composites, only the material parameters of their constituent fibers and matrixes are used as input data, which are measured independently. The predicted ultimate strengths or the stress–strain curves up to failure of all the composite laminates agree reasonably well with their experimental counterparts. 2. Laminate analysis Having obtained the instantaneous compliance ma￾trixes of various fibrous laminae [1], we can now per￾form the laminate analysis (referring to the flow chart shown in Fig. 2 of Ref. [1]). This will be done based on the classical lamination theory. Suppose that the laminate consists of a sequence of fibrous laminae, stacking in different ply-angles. Each lamina can have different con￾stituent materials, different fiber volume fraction, and even different fiber preform reinforcement. We take a global coordinate system (X, Y, Z) with X and Y in the laminate plane and Z along the thickness direction. Let the fiber direction of the kth lamina have an inclined ply￾angle hk with the global X direction, as shown in Fig. 3 of Ref. [1] for a UD lamina. According to the classical lamination theory, only in-plane stress and strain in￾crements, i.e. fdrg G ¼ fdrXX ; drYY ; drXY gT and fdegG ¼ fdeXX ; deYY ; 2deXY gT , are retained, where G refers to the global coordinate system. The out-of plane strain com￾ponents, deXZ ; deYZ , and deZZ , are assumed to be zero, whereas the out-of-plane stress components can be de￾termined using the lamina three-dimensional constitutive relationship if necessary. The averaged stress increments on the kth lamina can be determined from (refer to Eq. (13.3) of Ref. [1]) fdrgG k ¼ ð½T  cÞk ð½S P k Þ 1 ð½T  T c ÞfdegG k  fbgG k dT ¼ ½ðCG ij Þk fdegG k  fbgG k dT ; ð1Þ where fdegG k ¼ de 0 XX
þ Zk þ Zk1 2 dj0 XX ; de 0 YY þ Zk þ Zk1 2 dj0 YY ; 2de 0 XY þ ðZk þ Zk1Þdj0 XY T ; ð2:1Þ and fbgG k ¼ fðb1Þ G k ;ðb2Þ G k ;ðb3Þ G k g T ¼ ð½T  cÞk ð½S P k Þ 1 fagk : ð2:2Þ de0 XX ; de0 YY , and de0 XY and dj0 XX , dj0 YY , and dj0 XY are the laminate in-plane strain and curvature increments. Zk and Zk1 are the Z coordinates of the top and the bottom surfaces of the kth lamina. ½S P k is the lamina instanta￾neous compliance matrix given in the ply coordinate system, which can be obtained using the formulae pre￾sented in Ref. [1] for various fiber preform reinforced laminae. Specifically, for a UD lamina, its ply coordi￾nate system coincides with its local one. Hence, ½S P k is simply given by Eq. (4) of Ref. [1]. If the lamina under consideration is a single layer woven/braided fabric composite, ½S P k is defined by equation (41) of Ref. [1] (see Section 6 of this paper for additional discussion). Further, ½S P k is given by equation (43.1) of Ref. [1] if the lamina is a single layer knitted fabric composite. The 1178 Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199