3 3D VIEW TOP VIEW FIGURE 2.4 Deformation of a lamina subject to temperature increase. equivalent to that for thermal strains [7,8](neglecting possible pressure dependence of moisture absorption). The constitutive relationship,when it includes mechanical-,thermal-,and moisture-induced strains,takes the following form [1,4] Su S12 0 01 ef] 「εT E2 S2 Sz 0 62 e 立 (2.18) Y12 0 0 S66 T12 0 where superscripts T and M denote temperature-and moisture-induced strains,respectively.Note that shear strains are not induced in the principal material system by a temperature or moisture content change(Figure 2.4). Equation(2.18)is based on the superposition of mechanical-,thermal-,and moisture-induced strains.Inversion of Equation(2.18)gives 5 Q12 (2.19) Consequently,the stress-generating strains are obtained by subtraction of the thermal-and moisture-induced strains from the total strains.The thermal- and moisture-induced strains are often approximated as linear functions of the changes in temperature and moisture concentration, e △T (2.20) 2 △M (2.21) ©2003 by CRC Press LLC
equivalent to that for thermal strains [7,8] (neglecting possible pressure dependence of moisture absorption). The constitutive relationship, when it includes mechanical-, thermal-, and moisture-induced strains, takes the following form [1,4] (2.18) where superscripts T and M denote temperature- and moisture-induced strains, respectively. Note that shear strains are not induced in the principal material system by a temperature or moisture content change (Figure 2.4). Equation (2.18) is based on the superposition of mechanical-, thermal-, and moisture-induced strains. Inversion of Equation (2.18) gives (2.19) Consequently, the stress-generating strains are obtained by subtraction of the thermal- and moisture-induced strains from the total strains. The thermaland moisture-induced strains are often approximated as linear functions of the changes in temperature and moisture concentration, (2.20) (2.21) FIGURE 2.4 Deformation of a lamina subject to temperature increase. 1 2 12 11 12 12 22 66 1 2 12 1 T 2 T 1 M 2 M S S S S S ε ε γ σ σ τ ε ε ε ε = + + 0 0 00 0 0 σ σ τ εεε εεε γ 1 2 12 11 12 11 12 66 111 22 2 12 0 0 0 0 = − − − − Q Q Q Q Q T M T M 1 T 2 T 1 2 T ε ε α α = ∆ 1 M 2 M 1 2 M ε ε β β = ∆ TX001_ch02_Frame Page 16 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC
where AT and AM are the temperature change and moisture concentration change from the reference state. The transformed thermal expansion coefficients (a)are obtained from those in the principal system using Equation(2.13).Note,however, that in the principal material coordinate system,there is no shear deforma- tion induced [4],i.e.,6=B16=0, 0=m201+n202 (2.22a) 0y=n21+m202 (2.22b) y=2mn(a1-02) (2.22c) The moisture expansion coefficients(B,B.,B)are obtained by replacing a with B in Equations(2.22). The transformed constitutive relations for a lamina,when incorporating thermal-and moisture-induced strains,are Ex Ey Oy (2.23) 516 56 Ox Q Q16 Ex-ET-EM Oy Qi2 Qz 36 ey-e时-e (2.24) Q Q36 Y-Ys-Yw 2.2 Micromechanics As schematically illustrated in Figure 2.5,micromechanics aims to describe the moduli and expansion coefficients of the lamina from properties of the fiber and matrix,the microstructure of the composite,and the volume fractions of the constituents.Sometimes,also the small transition region between bulk fiber and bulk matrix,i.e.,interphase,is considered.Much fundamental work has been devoted to the study of the states of strain and stress in the constit- uents,and the formulation of appropriate averaging schemes to allow defini- tion of macroscopic engineering constants.Most micromechanics analyses have focused on unidirectional continuous fiber composites,e.g.[9,10], ©2003 by CRC Press LLC
where ∆T and ∆M are the temperature change and moisture concentration change from the reference state. The transformed thermal expansion coefficients (αx,αy,αxy) are obtained from those in the principal system using Equation (2.13). Note, however, that in the principal material coordinate system, there is no shear deformation induced [4], i.e., α16 = β16 = 0, αx = m2α1 + n2α2 (2.22a) αy = n2α1 + m2α2 (2.22b) αxy = 2mn(α1 – α2) (2.22c) The moisture expansion coefficients (βx,βy,βxy) are obtained by replacing α with β in Equations (2.22). The transformed constitutive relations for a lamina, when incorporating thermal- and moisture-induced strains, are (2.23) (2.24) 2.2 Micromechanics As schematically illustrated in Figure 2.5, micromechanics aims to describe the moduli and expansion coefficients of the lamina from properties of the fiber and matrix, the microstructure of the composite, and the volume fractions of the constituents. Sometimes, also the small transition region between bulk fiber and bulk matrix, i.e., interphase, is considered. Much fundamental work has been devoted to the study of the states of strain and stress in the constituents, and the formulation of appropriate averaging schemes to allow definition of macroscopic engineering constants. Most micromechanics analyses have focused on unidirectional continuous fiber composites, e.g. [9,10], x y xy x y xy x T y T xy T x M y M xy M = SSS SSS SSS ε ε γ σ σ τ ε ε γ ε ε γ + + 11 12 16 12 22 26 16 26 66 x y xy 11 12 16 12 22 26 16 26 66 x xT x M y yT y M xy xy T xy M = QQQ QQQ QQQ σ σ τ εεε εεε γγγ − − − − − − TX001_ch02_Frame Page 17 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC
Matrix Properties Fiber Properties Micromechanics Lamina Properties FIGURE 2.5 Role of micromechanics. although properties of composites with woven fabric reinforcements can also be predicted with reasonable accuracy,see Reference [11]. The objective of this section is not to review the various micromechanics developments.The interested reader can find ample information in the above-referenced review articles.In this section,we will limit the presenta- tion to some commonly used estimates of the stiffness constants,E,E2,Viz, va,and Gu2,and thermal expansion coefficients a and a required for describing the small strain response of a unidirectional lamina under mechanical and thermal loads(see Section 2.1).Such estimates may be useful for comparison to experimentally measured quantities. 2.2.1 Stiffness Properties of Unidirectional Composites Although most matrices are isotropic,many fibers such as carbon and Kevlar (E.I.du Pont de Nemours and Company,Wilmington,DE,)have directional properties because of molecular or crystal plane orientation effects [4].As a result,the axial stiffness of such fibers is much greater than the transverse stiffness.The thermal expansion coefficients along and transverse to the fiber axis also are quite different [4].It is common to assume cylindrical orthotropy for fibers with axisymmetric microstructure.The stiffness constants required for plane stress analysis of a composite with such fibers are EL,Er,Vr,and GLr,where L and T denote the longitudinal and transverse directions of a fiber.The corresponding thermal expansion coefficients are o and ar. The mechanics of materials approach reviewed in Reference [10]yields E=ELiVi+EmVm (2.25a) EnEm E,FEV。+EnV (2.25b) V12 VLTIVi Vm Vm (2.25c） ©2003 by CRC Press LLC
although properties of composites with woven fabric reinforcements can also be predicted with reasonable accuracy, see Reference [11]. The objective of this section is not to review the various micromechanics developments. The interested reader can find ample information in the above-referenced review articles. In this section, we will limit the presentation to some commonly used estimates of the stiffness constants, E1, E2, ν12, ν21, and G12, and thermal expansion coefficients α1 and α2 required for describing the small strain response of a unidirectional lamina under mechanical and thermal loads (see Section 2.1). Such estimates may be useful for comparison to experimentally measured quantities. 2.2.1 Stiffness Properties of Unidirectional Composites Although most matrices are isotropic, many fibers such as carbon and Kevlar (E.I. du Pont de Nemours and Company, Wilmington, DE, ) have directional properties because of molecular or crystal plane orientation effects [4]. As a result, the axial stiffness of such fibers is much greater than the transverse stiffness. The thermal expansion coefficients along and transverse to the fiber axis also are quite different [4]. It is common to assume cylindrical orthotropy for fibers with axisymmetric microstructure. The stiffness constants required for plane stress analysis of a composite with such fibers are EL, ET, νLT, and GLT, where L and T denote the longitudinal and transverse directions of a fiber. The corresponding thermal expansion coefficients are αL and αT. The mechanics of materials approach reviewed in Reference [10] yields E1 = ELfVf + EmVm (2.25a) (2.25b) ν12 = νLTfVf + νmVm (2.25c) FIGURE 2.5 Role of micromechanics. E E E EV EV Tf m Tf m m f 2 = + TX001_ch02_Frame Page 18 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC
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 corresponding 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