《纺织复合材料》课程参考文献（Experimental Characterization of Advanced Composite Materials, Third Edition）02 Analysis of Composite Materials

2 Analysis of Composite Materials 2.1 Constitutive Relations Laminated composites are typically constructed from orthotropic plies (laminae)containing unidirectional fibers or woven fabric.Generally,in a macroscopic sense,the lamina is assumed to behave as a homogeneous orthotropic material.The constitutive relation for a linear elastic orthotropic material in the material coordinate system (Figure 2.1)is [1-6] Su S12 S13 0 0 07 01 E2 S S23 0 0 0 02 E3 9 S23 S3 0 0 0 63 0 0 Sa 0 (2.1) 0 0 个9 Y13 0 0 0 0 Ss5 0 0 0 0 0 0 2 where the stress components(o,t)are defined in Figure 2.1 and the S are elements of the compliance matrix.The engineering strain components(Y) are defined as implied in Figure 2.2. In a thin lamina,a state of plane stress is commonly assumed by setting 03=t23=t13=0 (2.2) For Equation(2.1)this assumption leads to E3=S1301+S2302 (2.3a) Y2s=h3=0 (2.3b) Thus,for plane stress the through-the-thickness strain E3 is not an independent quantity and does not need to be included in the constitutive relationship. Equation (2.1)becomes ©2003 by CRC Press LLC

2 Analysis of Composite Materials 2.1 Constitutive Relations Laminated composites are typically constructed from orthotropic plies (laminae) containing unidirectional fibers or woven fabric. Generally, in a macroscopic sense, the lamina is assumed to behave as a homogeneous orthotropic material. The constitutive relation for a linear elastic orthotropic material in the material coordinate system (Figure 2.1) is [1–6] (2.1) where the stress components (σi, τij) are defined in Figure 2.1 and the Sij are elements of the compliance matrix. The engineering strain components (εi , γij) are defined as implied in Figure 2.2. In a thin lamina, a state of plane stress is commonly assumed by setting σ3 = τ23 = τ13 = 0 (2.2) For Equation (2.1) this assumption leads to ε3 = S13σ1 + S23σ2 (2.3a) γ23 = γ13 = 0 (2.3b) Thus, for plane stress the through-the-thickness strain ε3 is not an independent quantity and does not need to be included in the constitutive relationship. Equation (2.1) becomes ε ε ε γ γ γ σ σ σ τ τ τ 1 2 3 23 13 12 11 12 13 12 22 23 13 23 33 44 55 66 1 2 3 23 13 12 000 000 000 000 00 0000 0 00000                     =                       SSS SSS SSS S S S                   TX001_ch02_Frame Page 11 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC

◆3 8..8888。 T23 02 FIGURE 2.1 Definitions of principal material directions for an orthotropic lamina and stress components. E1 Su S12 0 E2 S12 0 62 (2.4 Y12 0 0 The compliance elements Si may be related to the engineering constants (E1,E2,G12V12V2 S11=1/E1,S2=-V12/E1=-V2:/E2 (2.5a) S2=1/E2,S6=1/G12 (2.5b) The engineering constants are average properties of the composite ply.The quantities E and vi2 are the Young's modulus and Poisson's ratio,respectively, corresponding to stress o1(Figure 2.2a) E1=o1/e1 (2.6a) V12=-e2/e (2.6b) E2 and v2 correspond to stress 02(Figure 2.2b) E2=o2/e2 (2.7a V21=-e1/e2 (2.7b) For a unidirectional composite E2 is much less than E,and va is much less than v12.For a balanced fabric composite E=E2 and vi2=V2.The Poisson's ratios vz and vz are not independent ©2003 by CRC Press LLC

(2.4) The compliance elements Sij may be related to the engineering constants (E1, E2, G12, ν12, ν21), S11 = 1/E1, S12 = –ν12/E1 = –ν21/E2 (2.5a) S22 = 1/E2, S66 = 1/G12 (2.5b) The engineering constants are average properties of the composite ply. The quantities E1 and ν12 are the Young’s modulus and Poisson’s ratio, respectively, corresponding to stress σ1 (Figure 2.2a) E1 = σ1/ε1 (2.6a) ν12 = –ε2/ε1 (2.6b) E2 and ν21 correspond to stress σ2 (Figure 2.2b) E2 = σ2/ε2 (2.7a) ν21 = –ε1/ε2 (2.7b) For a unidirectional composite E2 is much less than E1, and ν21 is much less than ν12. For a balanced fabric composite E1 ≈ E2 and ν12 ≈ ν21. The Poisson’s ratios ν12 and ν21 are not independent FIGURE 2.1 Definitions of principal material directions for an orthotropic lamina and stress components. 1 2 12 11 12 12 22 66 S S S S S ε ε γ σ σ τ             =                         0 0 0 0 1 2 12 TX001_ch02_Frame Page 12 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC

3 *2 3D VIEW TOP VIEW () ↑3 2 2 61 3D VIEW 6) TOP VIEW 13 12 2 2 T12 F12 -12 T12 3D VIEW ( TOP VIEW FIGURE 2.2 Illustration of deformations of an orthotropic material due to (a)stress o,(b)stress o2,and(c) stress t2. V21=V12E2/E (2.8) The in-plane shear modulus,G2,is defined as(Figure 2.2c) G12=t12/M12 (2.9) It is often convenient to express stresses as functions of strains.This is accomplished by inversion of Equation(2.4) 1 Qi Q12 0 E1 Q12 Q22 0 (2.10) 0 0 Q66 Y12 ©2003 by CRC Press LLC

ν21 = ν12E2/E1 (2.8) The in-plane shear modulus, G12, is defined as (Figure 2.2c) G12 = τ12/γ12 (2.9) It is often convenient to express stresses as functions of strains. This is accomplished by inversion of Equation (2.4) (2.10) FIGURE 2.2 Illustration of deformations of an orthotropic material due to (a) stress σ1, (b) stress σ2, and (c) stress τ12. 1 2 12 11 12 12 22 66 1 2 12 = Q Q Q Q Q σ σ τ ε ε γ                                   0 0 0 0 TX001_ch02_Frame Page 13 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC

where the reduced stiffnesses,Qcan be expressed in terms of the engineering constants Q1=E1/(1-V12V2) (2.11a) Q12=V12E2/(1-V12V2i)=V2E1/(1-V12V2) (2.11b) Q22=E2/(1-v12V2) (2.11c) Q66=G12 2.11d) 2.1.1 Transformation of Stresses and Strains For a lamina whose principal material axes(1,2)are oriented at an angle,0, with respect to the x,y coordinate system(Figure 2.3),the stresses and strains can be transformed.It may be shown [1-6]that both the stresses and strains transform according to 01 Ox 02 =[T] y (2.12) Ty】 and E2 (2.13) Y2/2 Y/2 z,3 2 8o888 FIGURE 2.3 Positive (counterclockwise)rotation of principal material axes(1,2)from arbitrary x,y axes. ©2003 by CRC Press LLC

where the reduced stiffnesses, Qij, can be expressed in terms of the engineering constants Q11 = E1/(1 – ν12ν21) (2.11a) Q12 = ν12E2/(1 – ν12ν21) = ν21E1/(1 – ν12ν21) (2.11b) Q22 = E2/(1 – ν12ν21) (2.11c) Q66 = G12 (2.11d) 2.1.1 Transformation of Stresses and Strains For a lamina whose principal material axes (1,2) are oriented at an angle, θ, with respect to the x,y coordinate system (Figure 2.3), the stresses and strains can be transformed. It may be shown [1–6] that both the stresses and strains transform according to (2.12) and (2.13) FIGURE 2.3 Positive (counterclockwise) rotation of principal material axes (1,2) from arbitrary x,y axes. 1 2 12 x y xy T σ σ τ σ σ τ           = [ ]             ε ε γ ε ε γ 1 2 12 / / 2 2           = [ ]           T x y xy TX001_ch02_Frame Page 14 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC

where the transformation matrix is [1-6] 3 n2 2mn [T]= n2 m2 -2mn (2.14) -mn mn m2-n2 and m cos 0 (2.15a) n=sin 0 (2.15b) From Equations (2.12)and (2.13)it is possible to establish the lamina strain-stress relations in the (x,y)coordinate system [1-6] Ex 516 6 Ey 69 (2.16) 56 Sao The S terms are the transformed compliances defined in Appendix A. Similarly,the lamina stress-strain relations become x Q16 y Qu2 y (2.17)） txy] Q16 Q where the overbars denote transformed reduced stiffness elements,defined in Appendix A. 2.1.2 Hygrothermal Strains If fibrous composite materials are processed at elevated temperatures,ther- mal strains are introduced during cooling to room temperature,leading to residual stresses and dimensional changes.Figure 2.4 illustrates dimensional changes of a composite subjected to a temperature increase of AT from the reference temperature T.Furthermore,polymer matrices are commonly hygroscopic,and absorbing moisture leads to swelling of the material.The analysis of moisture expansion strains in composites is mathematically ©2003 by CRC Press LLC

where the transformation matrix is [1–6] (2.14) and m = cos θ (2.15a) n = sin θ (2.15b) From Equations (2.12) and (2.13) it is possible to establish the lamina strain–stress relations in the (x,y) coordinate system [1–6] (2.16) The Sij terms are the transformed compliances defined in Appendix A. Similarly, the lamina stress–strain relations become (2.17) where the overbars denote transformed reduced stiffness elements, defined in Appendix A. 2.1.2 Hygrothermal Strains If fibrous composite materials are processed at elevated temperatures, ther￾mal strains are introduced during cooling to room temperature, leading to residual stresses and dimensional changes. Figure 2.4 illustrates dimensional changes of a composite subjected to a temperature increase of ∆T from the reference temperature T. Furthermore, polymer matrices are commonly hygroscopic, and absorbing moisture leads to swelling of the material. The analysis of moisture expansion strains in composites is mathematically T = m n mn n m mn mn mn m n [ ] − − −             2 2 2 2 2 2 2 2 x y xy y xy S S S SSS SSS ε ε γ σ σ τ             =                             11 12 16 12 22 26 16 26 66 x x y xy x y xy Q Q Q QQQ QQQ σ σ τ ε ε γ             =                           11 12 16 12 22 26 16 26 66 TX001_ch02_Frame Page 15 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC

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 thermal￾and 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 deforma￾tion 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 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], 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 presenta￾tion 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 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

FIGURE 2.6 Laminate coordinate system. 2.3 Laminated Plate Theory Structures fabricated from composite materials rarely utilize a single com- posite lamina because this unit is thin and anisotropic.To achieve a thicker cross section and more balanced properties,plies of prepreg or fiber mats are stacked in specified directions.Such a structure is called a laminate (Figure 2.6).Most analyses of laminated structures are limited to flat panels (see,e.g.,References [1,2]).Extension to curved laminated shell structures may be found in References [5,14,15]. In this section,attention will be limited to a flat laminated plate under in-plane and bending loads.The classical theory of such plates is based on the assumption that a line originally straight and perpendicular to the middle surface remains straight and normal to the middle surface,and that the length of the line remains unchanged during deformation of the plate [1-6]. These assumptions lead to the vanishing of the out-of-plane shear and exten- sional strains: Yz =Yyz =E2=0 (2.29) where the laminate coordinate system (x,y,z)is indicated in Figure 2.6. Consequently,the laminate strains are reduced tox,y,and Yy.The assump- tion that the cross sections undergo only stretching and rotation leads to the following strain distribution [1-6]: Ex Kx Ey e号 +Z Ky (2.30) Yy】 Kxy whereandare the midplane strains and curvatures, respectively,and z is the distance from the midplane. ©2003 by CRC Press LLC

2.3 Laminated Plate Theory Structures fabricated from composite materials rarely utilize a single com￾posite lamina because this unit is thin and anisotropic. To achieve a thicker cross section and more balanced properties, plies of prepreg or fiber mats are stacked in specified directions. Such a structure is called a laminate (Figure 2.6). Most analyses of laminated structures are limited to flat panels (see, e.g., References [1,2]). Extension to curved laminated shell structures may be found in References [5,14,15]. In this section, attention will be limited to a flat laminated plate under in-plane and bending loads. The classical theory of such plates is based on the assumption that a line originally straight and perpendicular to the middle surface remains straight and normal to the middle surface, and that the length of the line remains unchanged during deformation of the plate [1–6]. These assumptions lead to the vanishing of the out-of-plane shear and exten￾sional strains: γxz = γyz = εz = 0 (2.29) where the laminate coordinate system (x,y,z) is indicated in Figure 2.6. Consequently, the laminate strains are reduced to εx, εy, and γxy. The assump￾tion that the cross sections undergo only stretching and rotation leads to the following strain distribution [1–6]: (2.30) where are the midplane strains and curvatures, respectively, and z is the distance from the midplane. FIGURE 2.6 Laminate coordinate system. x y xy x 0 y 0 xy 0 x y xy = +z ε ε γ ε ε γ κ κ κ                                       x 0 y 0 xy 0 [ ] ε ε, , γ and [ ] κκκ x y xy , , TX001_ch02_Frame Page 20 Saturday, September 21, 2002 4:48 AM © 2003 by CRC Press LLC