CT. Herakovich/Mechanics Research Communications 41(2012)1-20 x axI(±)l nterface 12.0 21,6 16.2 Generic Cross- Section I08 Generic Quarter-Section 0.0150 45.060.075.0 Fig. 16. Finite width coupon under axial load. Fig 15. CTE-unidirectional and angle-ply laminates. 6. 2. Moisture effec or the axial and transverse thermal coefficients of expansion for The analysis of moisture effects in organic matrix composites is unidirectional, fiber-reinforced composites were presented. Hashin analogous to that for thermal effects at both the micromechanics (1979)extended Schapery's elastic results for composites with and laminate levels. Much of this work is detailed in three volumes transversely isotropic phases. The final forms of the predictions for edited by Springer(1981), Springer(1984), and Springer(1988a) the axial and transverse thermo-elastic coefficients of expansion Volume 3, Chapter 1(Springer, 1988b) provides a broad review of presented by daniel and Ishai, 1994)are the effects of temperature and moisture on organic matrix com posites. In general, moisture effects are not nearly as significant as Era V+EmamVm (Ea) (14) thermal effects. 7. Interlaminar stresses for the coefficient in the fiber direction and The first publication concerned with interlaminar stresses in laminated composites ap be that of Hayashi(1967)who 02=a2f +1 investigated interlaminar shear stresses in an idealized lam consisting of orthotropic layers separated by isotropic shear (U12 V+v12m Vm) Ear)1 (15) ers. Other important early works include those by Bogy(1968 who investigated the singular behavior of stresses at the inte section of a boundary and bonded dissimilar isotropic materials he coefficient in the transverse direction. In the above, f andand the first three-dimensional(numerical)analysis of inter- fer to fiber and matrix, respectively, V is volume fraction, E laminar stresses in laminated Is modulus, a is coefficient of thermal expansion and v is Pois-(1970) son's ratio,(Ea)1=Ea Vf+ Emam Vm and En is the rule of mixture Pipes and Pagano provided the first complete analysis of the mposite modulus in the fiber direction. problem of an axially loaded, laminated coupon with free edges Additional works on thermal effects in composites include the(Fig. 16). They formulated a reduced system of elasticity equa view article by Tauchert(1986)and that by Herakovich and tions governing the laminate behavior by assuming independence Aboudi(1999). of the stress and strain state on the axial coordinate and then The first presentation of the thermal-elastic formulation for solved the system of equations using the finite difference method posite laminates was by Tsai(1968). An early textbook presen- Their results showed the existence of all three interlaminar stress tation of the formulation is that by Calcote(1969). Amostimportant components in the boundary layer regions along the free edges result of the formulation is an expression for the effective coeffi- of finite width laminated coupons under inplane tensile load ent of thermal expansion (@) for a symmetric N-layered laminate, ing. They presented results for a variety of fiber orientations and laminate stacking sequences and showed that the width of the boundary layer is approximately equal to the thickness of the lam- l2=A∑广at nate, that the interlaminar normal stress oz and the interlaminar (16) shear stress tz can exhibit singular behavior as the free edge is approached, at the sign and magnitude of the interlami nar stresses are functions of the laminate configuration including Ashton et al. 1969 presented results for the varia- material type, fiber orientations, layer thicknesses and stacking tion of thermal strains as a function of fber orientation sequence. The free edge problem has been studied on a continuing basis shows that rather large, negative coefficients of ther- ever since the original work in the late 1960s. The finite dif- are possible for a typical carbon/epoxy ference solution of Pipes and Pagano was followed quickly by a naterial(T300/5208 in Fig. 15)over a range of fiber orientations three-dimensional finite element solution by Rybicki( 1971). Later. for angel-ply laminates it was recognized that the tensile coupon problem also could beC.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 9 Fig. 15. CTE – unidirectional and angle-ply laminates. for the axial and transverse thermal coefficients of expansion for unidirectional,fiber-reinforcedcomposites werepresented.Hashin (1979) extended Schapery’s elastic results for composites with transversely isotropic phases. The final forms of the predictions for the axial and transverse thermo-elastic coefficients of expansion (as presented by Daniel and Ishai, 1994) are: ˛1 = Ef ˛f Vf + Em˛mVm Ef Vf + EmVm = (E˛)1 E1 (14) for the coefficient in the fiber direction, and ˛2 = ˛2f Vf  1 + 12f ˛1f ˛2f  + ˛2mVm  1 + 12m ˛1m ˛2m  − (12f Vf + 12mVm) (E˛)1 E1 (15) for the coefficient in the transverse direction. In the above, f and m refer to fiber and matrix, respectively, V is volume fraction, E is modulus, ˛ is coefficient of thermal expansion and  is Pois￾son’s ratio, (E˛)1 = Ef˛fVf + Em˛mVm and E1 is the rule of mixture composite modulus in the fiber direction. Additional works on thermal effects in composites include the review article by Tauchert (1986) and that by Herakovich and Aboudi (1999). The first presentation of the thermal-elastic formulation for composite laminates was by Tsai (1968). An early textbook presen￾tationofthe formulationis that by Calcote (1969).Amostimportant result of the formulation is an expression for the effective coeffi- cient of thermal expansion {˛¯ } for a symmetric N-layered laminate, namely: {˛¯ } = [A] −1 N k=1 [Q¯ ] k {˛} ktk (16) Ashton et al., 1969 presented results for the varia￾tion of thermal strains as a function of fiber orientation for unidirectional and angle play laminates. Fig. 15 shows that rather large, negative coefficients of ther￾mal expansion are possible for a typical carbon/epoxy material (T300/5208 in Fig. 15) over a range of fiber orientations for angel-ply laminates. Fig. 16. Finite width coupon under axial load. 6.2. Moisture effects The analysis of moisture effects in organic matrix composites is analogous to that for thermal effects at both the micromechanics and laminate levels. Much of this work is detailed in three volumes edited by Springer (1981), Springer (1984), and Springer (1988a). Volume 3, Chapter 1 (Springer, 1988b) provides a broad review of the effects of temperature and moisture on organic matrix com￾posites. In general, moisture effects are not nearly as significant as thermal effects. 7. Interlaminar stresses The first publication concerned with interlaminar stresses in laminated composites appears to be that of Hayashi (1967) who investigated interlaminar shear stresses in an idealized laminate consisting of orthotropic layers separated by isotropic shear lay￾ers. Other important early works include those by Bogy (1968) who investigated the singular behavior of stresses at the inter￾section of a boundary and bonded dissimilar isotropic materials, and the first three-dimensional (numerical) analysis of inter￾laminar stresses in laminated composites by Pipes and Pagano (1970). Pipes and Pagano provided the first complete analysis of the problem of an axially loaded, laminated coupon with free edges (Fig. 16). They formulated a reduced system of elasticity equa￾tions governing the laminate behavior by assuming independence of the stress and strain state on the axial coordinate and then solved the system of equations using the finite difference method. Their results showed the existence of all three interlaminar stress components in the boundary layer regions along the free edges of finite width laminated coupons under inplane tensile load￾ing. They presented results for a variety of fiber orientations and laminate stacking sequences and showed that the width of the boundary layer is approximately equal to the thickness of the lam￾inate, that the interlaminar normal stress z and the interlaminar shear stress zx can exhibit singular behavior as the free edge is approached, and that the sign and magnitude of the interlami￾nar stresses are functions of the laminate configuration including material type, fiber orientations, layer thicknesses and stacking sequence. The free edge problem has been studied on a continuing basis ever since the original work in the late 1960s. The finite dif￾ference solution of Pipes and Pagano was followed quickly by a three-dimensional finite element solution by Rybicki (1971). Later, it was recognized that the tensile coupon problem also could be