532 R.Talreja and C.V.Singh illustrates matrix cracks observed on the free edges induced due to fatigue loading in composite laminates [37].Although matrix cracking does not cause structural failure by itself,it can result in significant degradation in material stiffness and also can induce more severe forms of damage,such as delamination and fiber breakage [44].Numerous studies of micro- cracking initiation were performed in the 1970s and early 1980s [4,13-15, 29,48,49].It was observed that the strain to initiate microcracking increases as the thickness of 90 plies decreases.Also,these microcracks form almost instantaneously across the width of the specimen. The first attempt to predict the strain to first microcrack used the first ply failure theory [18]where it is assumed that the first crack develops when the strain in the plies reaches the strain to failure in the plies.The predictions were not in agreement with the experimental observations since the first ply failure theory predicts that the strain to initiate micro- cracking will be independent of the ply thickness.The experimental obser- vations on laminates with a 90 layer on the surface [90/0s show that the strain to initiate microcracking is lower for laminates with cracks in surface plies than for laminates with cracks in central plies [52,54]. The simplest way to model transverse matrix cracks in composite laminates is to completely neglect the transverse stiffness of cracked plies, called the ply discount method.This method underestimates the stiffness of cracked laminates,since cracked plies,in reality,can take some loading Another simple way is shear lag analysis,wherein the load transfer between plies is assumed to take place in shear layers between neighboring plies. The normal stress in the external load direction is assumed to be constant over the ply thickness.The thicknesses and stiffness of these shear layers are generally unknown,and the variations in the thickness direction of local ply stresses and strains are also neglected in the shear lag theory.The shear lag theory has limited success for crossply laminates [19,25,39,62]. For crossply laminates,the most successful approach is the variational method.By application of the principle of minimum complementary potential energy,Hashin [21,22]derived estimates for thermomechanical properties and local ply stresses,which were in good agreement with experimental data.Varna and Berglund [65]later made improvements to the Hashin model by use of more accurate trial stress functions.A disadvantage of the variational method is that it is extremely difficult to use for laminate lay- ups other than crossplies.McCartney [43]used Reissner's energy function to derive governing equations similar to Hashin's model.He applied this approach to doubly cracked crossply laminates assuming that the in-plane normal stress dependence on the two in-plane coordinates is given by two independent functions.Gudmundson and coworkers [16,17]consideredillustrates matrix cracks observed on the free edges induced due to fatigue loading in composite laminates [37]. Although matrix cracking does not cause structural failure by itself, it can result in significant degradation in material stiffness and also can induce more severe forms of damage, such as delamination and fiber breakage [44]. Numerous studies of micro￾cracking initiation were performed in the 1970s and early 1980s [4, 13–15, 29, 48, 49]. It was observed that the strain to initiate microcracking increases as the thickness of 90° plies decreases. Also, these microcracks form almost instantaneously across the width of the specimen. The first attempt to predict the strain to first microcrack used the first ply failure theory [18] where it is assumed that the first crack develops when the strain in the plies reaches the strain to failure in the plies. The predictions were not in agreement with the experimental observations since the first ply failure theory predicts that the strain to initiate micro￾cracking will be independent of the ply thickness. The experimental obser￾vations on laminates with a 90° layer on the surface [90n/0m]s show that the strain to initiate microcracking is lower for laminates with cracks in surface plies than for laminates with cracks in central plies [52, 54]. The simplest way to model transverse matrix cracks in composite laminates is to completely neglect the transverse stiffness of cracked plies, called the ply discount method. This method underestimates the stiffness of cracked laminates, since cracked plies, in reality, can take some loading. Another simple way is shear lag analysis, wherein the load transfer between plies is assumed to take place in shear layers between neighboring plies. The normal stress in the external load direction is assumed to be constant over the ply thickness. The thicknesses and stiffness of these shear layers are generally unknown, and the variations in the thickness direction of local ply stresses and strains are also neglected in the shear lag theory. The shear lag theory has limited success for crossply laminates [19, 25, 39, 62]. For crossply laminates, the most successful approach is the variational method. By application of the principle of minimum complementary potential energy, Hashin [21, 22] derived estimates for thermomechanical properties and local ply stresses, which were in good agreement with experimental data. Varna and Berglund [65] later made improvements to the Hashin model by use of more accurate trial stress functions. A disadvantage of the variational method is that it is extremely difficult to use for laminate lay￾ups other than crossplies. McCartney [43] used Reissner’s energy function to derive governing equations similar to Hashin’s model. He applied this approach to doubly cracked crossply laminates assuming that the in-plane normal stress dependence on the two in-plane coordinates is given by two independent functions. Gudmundson and coworkers [16, 17] considered 532 R. Talreja and C.V. Singh