GLTGm Gu-GuVm+G.V (2.25d) where subscripts f and m represent fiber and matrix,respectively,and the symbol V represents volume fraction.Note that once E,E2,and vi2 are calculated from Equations (2.25a),v2 is obtained from Equation (2.8). Equations(2.25a)and(2.25c)provide good estimates of E and v12.Equations (2.25b)and (2.25d),however,substantially underestimate E2 and G2 [10].More realistic estimates of E2 and G2 are provided in References [10,12]. Simple,yet reasonable estimates of E2 and G12 may also be obtained from the Halpin-Tsai equations [13], P=Pm(1+ExVi) (2.26a) 1-xV X=P:-Pm (2.26b) Pi+ξPm where P is the property of interest(E2 or G2)and P:and Pm are the corre- sponding fiber and matrix properties,respectively.The parameter is called the reinforcement efficiency;(E2)=2 and (Gi2)=1,for circular fibers. 2.2.2 Expansion Coefficients Thermal expansion(and moisture swelling)coefficients can be defined by considering a composite subjected to a uniform increase in temperature (or moisture content)(Figure 2.4). The thermal expansion coefficients,o and o,of a unidirectional composite consisting of cylindrically or transversely orthotropic fibers in an isotropic matrix determined using the mechanics of materials approach [10]are -uEu V+amEV (2.27a) EuVi+Em Vm 2=aTrVi+am Vm (2.27b) Predictions of o using Equation(2.27a)are accurate [10],whereas Equation (2.27b)underestimates the actual value of a2.An expression derived by Hyer and Waas [10]provides a more accurate prediction of o2: a,=anV+aV。+CEUVE义man-guVV (2.28) EuVi+EmVm ©2003 by CRC Press LLC(2.25d) where subscripts f and m represent fiber and matrix, respectively, and the symbol V represents volume fraction. Note that once E1, E2, and ν12 are calculated from Equations (2.25a), ν21 is obtained from Equation (2.8). Equations (2.25a) and (2.25c) provide good estimates of E1 and ν12. Equations (2.25b) and (2.25d), however, substantially underestimate E2 and G12 [10]. More realistic estimates of E2 and G12 are provided in References [10,12]. Simple, yet reasonable estimates of E2 and G12 may also be obtained from the Halpin-Tsai equations [13], (2.26a) (2.26b) where P is the property of interest (E2 or G12) and Pf and Pm are the corre￾sponding fiber and matrix properties, respectively. The parameter ξ is called the reinforcement efficiency; ξ(E2) = 2 and ξ(G12) = 1, for circular fibers. 2.2.2 Expansion Coefficients Thermal expansion (and moisture swelling) coefficients can be defined by considering a composite subjected to a uniform increase in temperature (or moisture content) (Figure 2.4). The thermal expansion coefficients, α1 and α2, of a unidirectional composite consisting of cylindrically or transversely orthotropic fibers in an isotropic matrix determined using the mechanics of materials approach [10] are (2.27a) α2 = αTfVf + αmVm (2.27b) Predictions of α1 using Equation (2.27a) are accurate [10], whereas Equation (2.27b) underestimates the actual value of α2. An expression derived by Hyer and Waas [10] provides a more accurate prediction of α2: (2.28) G G G G V GV LTf m LTf m m f 12 = + P P (1+ V ) 1 V m f f = − ξχ χ χ ξ = f m − f m P P + P P α α α 1 = + + Lf Lf f mmm Lf f m m E V EV EV EV αα α ν ν 2 =+ + α α + + − Tf f m m Lf m m LTf Lf f m m V V m Lf f m E E EV EV V V ( ) ( ) TX001_ch02_Frame Page 19 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC