CT. Herakovich/ Mechanics Research Communications 41(2012)1-20 Effective plane strain bulk modulus MOCT Mori-Tanaka 172. 1/-km+(-pm)/3]+(1-v)(km+4m/3) Effective axial shear modulus. 12f(1+V)+pm(1-v) 68.9 It is evident from eqs. (4)and(5 that the first two terms cor- respond to a rule of mixtures. The last term is typically very small for most composites in use today. Thus, the rule of mixtures (i.e. Voigt upper bound) is a very good predictor for the effective axial 0.25005000.750 modulus and effective axial Poisson s ratio. this cannot be said for he other two properties. Fig. 6. Axial modulus predictions for carbon/epoxy Chamis and Sendeckyj(1968)presented an extensive critique of the theories known at the time for predicting the thermoelastic properties of fibrous composites. The theories reviewed were clas- 4.1. Micromechanics model comparisons sified as: netting analysis, mechanics of materials, self-consistent model, variational, exact, statistical, discrete element, semi empir- Figs 6-9 show comparisons of micromechanics predictions for the effective properties E1, E2, V12, and Gu of unidirectional ca ical methods, and theories accounting for microstructure. They bonepoxy(Lissenden and Herakovich, 1992)as a function of the included comparisons of predictions by different theories for uni- directional glass-epoxy, boron-epoxy and graphite-epoxy. fiber volume fraction. The methods compared include: Voigt, Reuss, Hashin(1972) gave an extensive theoretical treatment of concentric cylinder assemblage, self-consistent, method of cells hermoelastic properties, thermal and electrical conduction, and Several important features are evident from these comparisons. electrostatics and magnetostatics behavior. For the effective axial modulus, E:(Fig. essentially all models give Achenbach(1974) and Achenbach(1975)considered wave the same prediction, with the lower bound reuse model being the The composite with microstructure is distinguished from a com- provides excellent predictions for the effective axial modulus posite that is modeled as a homogeneous, anisotropic continuum Schapery(1967)has shown that the results for linear elastic using effective properties. The point is made that for dynamic materials can be extended to linear viscoelastic materials in a sim- response such as wave propagation, the characteristic lengths of ple and accurate manne the deformation be small and the effective modulus theory ay not suffice. The proposed theory showed good comparison 5. Lamination theory with ultrasonic data for fibrous composites and finite element pre dictions. Lectures on this subject were given at the International Possibly the most fundamental result for the application of Centre for Mechanical Sciences(CISM)in Udine, Italy, in July 1973 fibrous composites in structural and devices is Classical lamination for composite materials and provided an in-depth analysis of the The theory considers an assemblage of layers bonded together Method of Cells for thermo-elastic, viscoelastic, nonlinear behav- to form a laminate. The individual layers are taken to be homoge ior of resin matrix composites, initial yield surfaces and inelastic neous with properties that can range from isotropic to anisotropic. behavior of metal matrix composites, and composites with imper- Typically, the layers are unidirectional fibrous composites with the fect bonding. The method of cells consists of a periodic square array of rectangular subcells, one representing the fiber and three 2.50 172 similar subcells representing the matrix. This model provides a -I MOC-Tl computationally efficient method for predicting inelastic response 丰 The effects of different types of fiber orthotropy on the effective roperties of composites were considered by Knott and Herakovich 拿εCN (1991a). Nemat-Nasser and Hori (1993)presented a treatise on the mechanics of solids with microdefects such as cavities cracks, and inclusions, including elastic composites 68 Discrete element methods such as the finite element method have been used to predict effective properties of unidirectional to 0500 3.4 be that of Foye who studied effective elastic properties, inelastic response, and stress distributions in unidirectional boron/epoxy. Finite element studies can be valuable when the fiber distribution 0.250 0.500 100 as shown for the ceramic fiber in a titanium matrix less so for random fiber distributions such as the arbon/epoxy of Fig 5a. Fig. 7. Transverse modulus predictions for carbon/epoxy
6 C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 Effective plane strain bulk modulus: K∗ 23 = km + m 3 + Vf 1/[kf − km + (f − m)/3] + (1 − Vf )/(km + 4m/3) (6) Effective axial shear modulus: ∗ 12 m = f (1 + Vf ) + m(1 − Vf ) f (1 − Vf ) + m(1 + Vf ) (7) It is evident from Eqs. (4) and (5) that the first two terms correspond to a rule of mixtures. The last term is typically very small for most composites in use today. Thus, the rule of mixtures (i.e. Voigt upper bound) is a very good predictor for the effective axial modulus and effective axial Poisson’s ratio. This cannot be said for the other two properties. Chamis and Sendeckyj (1968) presented an extensive critique of the theories known at the time for predicting the thermoelastic properties of fibrous composites. The theories reviewed were classified as: netting analysis, mechanics of materials, self-consistent model, variational, exact, statistical, discrete element, semi empirical methods, and theories accounting for microstructure. They included comparisons of predictions by different theories for unidirectional glass-epoxy, boron-epoxy and graphite-epoxy. Hashin (1972) gave an extensive theoretical treatment of micromechanics. He considered effective elastic, viscoelastic and thermoelastic properties, thermal and electrical conduction, and electrostatics and magnetostatics behavior. Achenbach (1974) and Achenbach (1975) considered wave propagation in fiber-reinforced composites with microstructure. The composite with microstructure is distinguished from a composite that is modeled as a homogeneous, anisotropic continuum using effective properties. The point is made that for dynamic response such as wave propagation, the characteristic lengths of the deformations may be small and the effective modulus theory may not suffice. The proposed theory showed good comparison with ultrasonic data for fibrous composites and finite element predictions. Lectures on this subject were given at the International Centre for Mechanical Sciences (CISM) in Udine, Italy, in July 1973 with publication of the (expanded) monograph in 1975. Aboudi (1991) presented micromechanical analysis methods for composite materials and provided an in-depth analysis of the Method of Cells for thermo-elastic, viscoelastic, nonlinear behavior of resin matrix composites, initial yield surfaces and inelastic behavior of metal matrix composites, and composites with imperfect bonding. The method of cells consists of a periodic square array of rectangular subcells, one representing the fiber and three similar subcells representing the matrix. This model provides a computationally efficient method for predicting inelastic response of composites. The effects of different types of fiber orthotropy on the effective properties of composites were considered by Knott and Herakovich (1991a). Nemat-Nasser and Hori (1993) presented a treatise on the mechanics of solids with microdefects such as cavities, cracks, and inclusions, including elastic composites. Discrete element methods such as the finite element method have been used to predict effective properties of unidirectional composites. The earliest work using finite elements appears to be that of Foye who studied effective elastic properties, inelastic response, and stress distributions in unidirectional boron/epoxy. Finite element studies can be valuable when the fiber distribution is very regular as shown for the ceramic fiber in a titanium matrix of Fig. 5b, but less so for random fiber distributions such as the carbon/epoxy of Fig. 5a. Fig. 6. Axial modulus predictions for carbon/epoxy. 4.1. Micromechanics model comparisons Figs. 6–9 show comparisons of micromechanics predictions for the effective properties E1, E2, 12, and G12 of unidirectional carbon/epoxy (Lissenden and Herakovich, 1992) as a function of the fiber volume fraction. The methods compared include:Voigt, Reuss, concentric cylinder assemblage, self-consistent, method of cells, Mori-Tanaka and strength of materials. Several important features are evident from these comparisons. For the effective axial modulus, E∗ 1 (Fig. 6) essentially all models give the same prediction, with the lower bound Reuse model being the exception. Thus, a simple rule of mixtures (the Voigt upper bound), provides excellent predictions for the effective axial modulus. Schapery (1967) has shown that the results for linear elastic materials can be extended to linear viscoelastic materials in a simple and accurate manner. 5. Lamination theory Possibly the most fundamental result for the application of fibrous composites in structural and devices is Classical Lamination Theory. The theory follows the original works of Pister and Dong (1959), Reissner and Stavsky (1961) and Dong et al. (1962) The theory considers an assemblage of layers bonded together to form a laminate. The individual layers are taken to be homogeneous with properties that can range from isotropic to anisotropic. Typically, the layers are unidirectional fibrous composites with the Fig. 7. Transverse modulus predictions for carbon/epoxy
CT. Herakovich/Mechanics Research Communications 41(2012)1-20 4.0 MOC-Tl L St Matls Fig 10. Composite laminate. 0,25非 0750 Poisson ratio: Ey Fig 8. Shear modulus predictions for carbon/epoxy Vxy 8x (11) fibers in the kth layer oriented at an angle ek from a global x-axis s depicted in Fig. 10. ix (12) Analysis results in the fundamental equation relating the inplane forces N and moments ( M acting on the laminate to Coefficient of mutual influence the midplane strains E) and curvatures( x) through coefficients [A] [B]and ( D] that are functions of the material properties, layers nxy xei ai1 thickness and stacking sequence of the layers N A B The coefficient of mutual influence(13)quantifies the shear ( 8) strain associated with normal strain; it is non-zero when the lam- inate compliance t The effective engineering properties of symmetric laminates can Specific examples of the range of engineering properties thatcan be predicted from Eq ( 8)through a series of thought experimen be affected through the choice of material and stacking sequence here the laminate is subjected to a series of specified loadings. are presented in Figs 11-13. These figures show the variation in With the laminate compliance defined: axial modulus, Poisson ratio and shear modulus for T300/5208 car- bon/epoxy. la]=2HIAJ-I (9) These three figures show that the effective engineering prop- for the engineering properties of the laminate. Examples are. ons erties of angle-ply laminates are higher than those of the Axial modulus: inates can exhibit values greater than 1.0, and the shear modulus of angle-ply laminates is largest at 45. Ex Another most interesting result for laminated composites (Fig. 14)is the fact that the through-the-thickness P 20.0 0.400 0.200 MOC-TI 0.100 .Mori-Tanaka 含 St matls 0.0 0.00 0.0 0.250 0.750 20.00 8000 Fig. 11. Axial modulus -unidirectional and angle-ply laminates
C.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 7 Fig. 8. Shear modulus predictions for carbon/epoxy. fibers in the kth layer oriented at an angle k from a global x-axis as depicted in Fig. 10. Analysis results in the fundamental equation relating the inplane forces {N} and moments {M} acting on the laminate to the midplane strains {ε◦} and curvatures { } through coefficients [A], [B] and [D] that are functions of the material properties, layers thickness and stacking sequence of the layers. N M = A B B D ε◦ (8) The effective engineering properties of symmetric laminates can be predicted from Eq. (8) through a series of thought experiments where the laminate is subjected to a series of specified loadings. With the laminate compliance defined: [a∗] ≡ 2H[A] −1 (9) The results of these thought experiments provide expressions for the engineering properties of the laminate. Examples are: Axial modulus: Ex = ¯ x ε◦ x = 1 a∗ 11 (10) Fig. 9. Poisson’s ratio predictions for carbon/epoxy. Fig. 10. Composite laminate. Poisson ratio: xy = −ε◦ y ε◦ x = −a∗ 12 a∗ 11 (11) Shear modulus: Gxy = ¯xy ◦ xy = 1 a∗ 66 (12) Coefficient of mutual influence: xy,x = ◦ xy ε◦ x = a∗ 16 a∗ 11 (13) The coefficient of mutual influence (13) quantifies the shear strain associated with normal strain; it is non-zero when the laminate compliance term a∗ 16 is non-zero. Specific examples ofthe range of engineering properties that can be affected through the choice of material and stacking sequence are presented in Figs. 11–13. These figures show the variation in axial modulus, Poisson ratio and shear modulus for T300/5208 carbon/epoxy. These three figures show that the effective engineering properties of angle-ply laminates are higher than those of the corresponding laminae. Further, Poisson’s ratio of angle-play laminates can exhibit values greater than 1.0, and the shear modulus of angle-ply laminates is largest at 45◦. Another most interesting result for laminated composites (Fig. 14) is the fact that the through-the-thickness Poisson’s ratio Fig. 11. Axial modulus – unidirectional and angle-ply laminates.