CT. Herakovich/ Mechanics Research Communications 41(2012)1-20 0600 120 0.450 00 0.800 0.150 H儿L 0.150 0.300 0015003000450060.007500 30.04560.0 75,0 Fig. 12. Poisson s ratio- unidirectional and angle-ply laminates Fig. 14. Through-the-thickness poissons ratio. Vxz is negative over a significant range of fber orientations for some Composites often are the material of choice where thermal stresses ngle-ply laminates( Herakovich, 1984). xpansion are important. The coefficient of thermal Another interesting feature of laminates is that, depending expansion in the fiber direction of unidirectional composites is n the stacking sequence of the layers, they can exhibit cou- often near zero and can be slightly negative. This has huge pling between inplane and bending effects. Laminates that are consequences when designing laminates for low,or matching, coef- unsymmetric about the laminate midplane have a non-zero [b] ficients of thermal expansion. Thermal stresses can be extremely matrix resulting in coupling between inplane and out-of-plane important for the application of fibrous composite materials as responses( see Eq(8) Unsymmetric laminates exhibit curvature essentially all composite materials are fabricated at an elevate hen subjected to pure inplane loading. Likewise, unsymmetric temperature. The constituent phases become bonded at anelevated laminates exhibit inplane strains when subjected to pure bending temperature resulting in residual thermal stresses in the composite moments. More on unsymmetric laminates is provided in a later after it has cooled to room temperature section Fundamental problems at the micromechanics level are predic tion of the residual stresses al 6. Environmental effects unidirectional composites. At the laminate level, it is necessary to predict the residual stresses and the laminate effective coefficient 6. 1. Thermal effects of thermal expansion(CTE). This latter property is very important as it is one of the unique aspects of laminated composite materi- ronmental effects often play a critical role in the choice als: composite laminates can exhibit ChE values over a wide range erial for many applications in devices and structures. including zero, positive and negative. The earliest papers dealing with thermal effects in anisotropic materials appear to be those by Ambartsumyan (1952)who consid- ered thermal stresses in anisotropic, laminated plates, and hayashi (1956)who considered thermal stresses in orthotropic plates. The earliest works at the micromechanics level appears to be that of Van Fo Fy(1965) who considered thermal effects in com- posites consisting of periodic arrays of continuous, circular glass fibers. He used stress analysis to determine exact thermal coef- ficients for specific phase geometries. Levin(1967) presented an approach for determining the effective coefficients of thermal expansion for two phase composites with isotropic phases. The 300 work used an extension Hill,s approach and included bounds on the expansion coefficients of transversely isotropic, undirect 200 fiber-reinforced composites. Rosen(1968)investigated th expansion coefficients for composite materials. Much of this is incorporated in the later paper by Rosen and Hashin (1970)on 100 expansion coefficients. Schapery(1968)derived upper and lower bounds as well as specific approximations for thermal expansion coefficients of lin- ear elastic and viscoelastic composite materials. He extended the 000 6000 00 previous work of Levin and Van Fo Fy for an arbitrary number e of constituents and phase geometries, for isotropic phases. The approach provided upper and lower bounds using the principles Fig 13. Shear modulu of complementary and potential energy. Approximate expressions8 C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 Fig. 12. Poisson’s ratio – unidirectional and angle-ply laminates. xz is negative over a significant range of fiber orientations for some angle-ply laminates (Herakovich, 1984). Another interesting feature of laminates is that, depending on the stacking sequence of the layers, they can exhibit cou￾pling between inplane and bending effects. Laminates that are unsymmetric about the laminate midplane have a non-zero [B] matrix resulting in coupling between inplane and out-of-plane responses (see Eq. (8)). Unsymmetric laminates exhibit curvature when subjected to pure inplane loading. Likewise, unsymmetric laminates exhibit inplane strains when subjected to pure bending moments. More on unsymmetric laminates is provided in a later section. 6. Environmental effects 6.1. Thermal effects Environmental effects often play a critical role in the choice of material for many applications in devices and structures. Fig. 13. Shear modulus – unidirectional and angle-ply laminates. Fig. 14. Through-the-thickness Poisson’s ratio. Composites often are the material of choice where thermal stresses or thermal expansion are important. The coefficient of thermal expansion in the fiber direction of unidirectional composites is often near zero and can be slightly negative. This has huge consequences when designing laminates for low, or matching, coef- ficients of thermal expansion. Thermal stresses can be extremely important for the application of fibrous composite materials as essentially all composite materials are fabricated at an elevated temperature. The constituent phases become bonded at an elevated temperature resulting in residualthermal stresses in the composite after it has cooled to room temperature. Fundamental problems at the micromechanics level are predic￾tion of the residual stresses and the effective thermal properties of unidirectional composites. At the laminate level, it is necessary to predict the residual stresses and the laminate effective coefficient of thermal expansion (CTE). This latter property is very important as it is one of the unique aspects of laminated composite materi￾als: composite laminates can exhibit CTE values over a wide range including zero, positive and negative. The earliest papers dealing with thermal effects in anisotropic materials appear to be those by Ambartsumyan (1952) who consid￾ered thermal stresses in anisotropic, laminated plates, and Hayashi (1956) who considered thermal stresses in orthotropic plates. The earliest works at the micromechanics level appears to be that of Van Fo Fy (1965) who considered thermal effects in com￾posites consisting of periodic arrays of continuous, circular glass fibers. He used stress analysis to determine exact thermal coef- ficients for specific phase geometries. Levin (1967) presented an approach for determining the effective coefficients of thermal expansion for two phase composites with isotropic phases. The work used an extension Hill’s approach and included bounds on the expansion coefficients of transversely isotropic, unidirectional, fiber-reinforced composites. Rosen (1968) investigated thermal expansion coefficients for composite materials. Much of this work is incorporated in the later paper by Rosen and Hashin (1970) on expansion coefficients. Schapery (1968) derived upper and lower bounds as well as specific approximations for thermal expansion coefficients of lin￾ear elastic and viscoelastic composite materials. He extended the previous work of Levin and Van Fo Fy for an arbitrary number of constituents and phase geometries, for isotropic phases. The approach provided upper and lower bounds using the principles of complementary and potential energy. Approximate expressions