Lamina Off-Axis Tensile Response The off-axis tension test of unidirectional composites has received consider- able attention by the composites community."Off-axis"here refers to the material axes(1-2)being rotated through an angle 0 with respect to the specimen axis and direction of loading(Figure 9.1).The off-axis specimen is typically 230 mm long and between 12.7 and 25.4 mm wide.A thickness of eight plies is common(0.127 mm ply thickness). The off-axis tension test is rarely used to determine basic ply properties. Most commonly,the purpose of this test is to verify material properties determined in tension,compression,and shear,as discussed in Chapters 5-7, using the transformed constitutive relations discussed in Chapter 2.Testing of specimens at off-axis angles between 10 and 20 produces significant shear in the principal material system.Consequently,the 10 off-axis test has been proposed as a simple way to conduct a shear test [1].The test has been used FIGURE 9.1 Geometry of the off-axis tensile coupon. ©2003 by CRC Press LLC
9 Lamina Off-Axis Tensile Response The off-axis tension test of unidirectional composites has received considerable attention by the composites community. “Off-axis” here refers to the material axes (1-2) being rotated through an angle θ with respect to the specimen axis and direction of loading (Figure 9.1). The off-axis specimen is typically 230 mm long and between 12.7 and 25.4 mm wide. A thickness of eight plies is common (0.127 mm ply thickness). The off-axis tension test is rarely used to determine basic ply properties. Most commonly, the purpose of this test is to verify material properties determined in tension, compression, and shear, as discussed in Chapters 5–7, using the transformed constitutive relations discussed in Chapter 2. Testing of specimens at off-axis angles between 10 and 20° produces significant shear in the principal material system. Consequently, the 10° off-axis test has been proposed as a simple way to conduct a shear test [1]. The test has been used FIGURE 9.1 Geometry of the off-axis tensile coupon. w LT LT LG L I x y 2 θ TX001_ch09_Frame Page 131 Saturday, September 21, 2002 5:01 AM © 2003 by CRC Press LLC
also to verify biaxial strength criteria because,as will be discussed,uniaxial loading will lead to a combined state of stress in the principal material system. 9.1 Deformation and Stress in an Unconstrained Specimen Because of the off-axis configuration of the specimen,the in-plane response is characterized by a fully populated compliance matrix,as shown in Equation (2.16) Ex 52 516 Ox Ey S12 y (9.1) 56 526 where the x-y system is defined in Figure 9.1,and expressions for the trans- formed compliance elements Su are given in Appendix A. For an ideal,uniformly stressed off-axis tensile coupon,the only stress acting is ox,(oy=txy=0),and Equations (9.1)give the state of strain in the specimen, Ex Ey S (9.2) S.c] Consequently,the off-axis coupon subjected to a uniform uniaxial state of stress thus exhibits shear strain()in addition to the axial and transverse strains(Ex and )(Figure 9.2). A set of material properties may be evaluated based on measurement of axial stress (and axial,transverse,and shear strains (x yYy).It is customary to determine the axial Young's modulus(E)and Poisson's ratio (Vxy)of the off-axis specimen (9.3) Ex Vxy= y (9.4) Ex ©2003 by CRC Press LLC
also to verify biaxial strength criteria because, as will be discussed, uniaxial loading will lead to a combined state of stress in the principal material system. 9.1 Deformation and Stress in an Unconstrained Specimen Because of the off-axis configuration of the specimen, the in-plane response is characterized by a fully populated compliance matrix, as shown in Equation (2.16) (9.1) where the x-y system is defined in Figure 9.1, and expressions for the transformed compliance elements Sij are given in Appendix A. For an ideal, uniformly stressed off-axis tensile coupon, the only stress acting is σx, (σy = τxy = 0), and Equations (9.1) give the state of strain in the specimen, (9.2) Consequently, the off-axis coupon subjected to a uniform uniaxial state of stress thus exhibits shear strain (γxy) in addition to the axial and transverse strains (εx and εy) (Figure 9.2). A set of material properties may be evaluated based on measurement of axial stress (σx) and axial, transverse, and shear strains (εx, εy, γxy). It is customary to determine the axial Young’s modulus (Ex) and Poisson’s ratio (νxy) of the off-axis specimen (9.3) (9.4) x y xy x y xy SSS SSS SSS ε ε γ σ σ τ = 11 12 16 12 22 26 16 26 66 x y xy x S S S ε ε γ σ = 11 12 16 Ex x x = σ ε ν ε ε xy y x = − TX001_ch09_Frame Page 132 Saturday, September 21, 2002 5:01 AM © 2003 by CRC Press LLC
In addition,a ratio (n),which quantifies coupling between shear and axial strains,is defined according to (9.5) Ex The off-axis tension test may also be used to determine the in-plane shear modulus,G2,in the principal material coordinate system.This property is, according to Equation(2.9),defined by (9.6) Y12 Consequently,determination of G2 requires determination of shear stress and strain in the principal material coordinate system.Equations(2.12)and (2.14)yield T12=-7110x (9.7) where m=cos 0 and n=sin 0.The shear strain is obtained from Equations(2.14) Y2 2mn(y -E)+(m2-n2)Yy (9.8) where the strain(e)is directly measured,and the transverse strain ()and shear strain()are determined as subsequently explained. The properties EVyny and Gi2 can be evaluated from test data using procedures detailed later in this chapter.The mechanical properties so deter- mined can be compared to theoretical values calculated from the compliance relations,Equations (9.2),and the definitions in Equations (9.3-9.5), 1 (9.9a) Vxy=- 52 (9.9b) (9.9c) If the principal (basic)material properties(E,E2,Vi2,and G12)are known from previous tests (Chapters 5-7),it is possible to calculate the off-axis properties E,Vsy and nxy using Equations(A.1)(Appendix A),and compare those to the experimentally determined values.G12 may be compared to the modulus measured in the off-axis test(Equation (9.6)). ©2003 by CRC Press LLC
In addition, a ratio (ηxy), which quantifies coupling between shear and axial strains, is defined according to (9.5) The off-axis tension test may also be used to determine the in-plane shear modulus, G12, in the principal material coordinate system. This property is, according to Equation (2.9), defined by (9.6) Consequently, determination of G12 requires determination of shear stress and strain in the principal material coordinate system. Equations (2.12) and (2.14) yield τ12 = –mnσx (9.7) where m = cos θ and n = sin θ. The shear strain is obtained from Equations (2.14) γ12 = 2mn(εy – εx) + (m2 – n2)γxy (9.8) where the strain (εx) is directly measured, and the transverse strain (εy) and shear strain (γxy) are determined as subsequently explained. The properties Ex, νxy, ηxy, and G12 can be evaluated from test data using procedures detailed later in this chapter. The mechanical properties so determined can be compared to theoretical values calculated from the compliance relations, Equations (9.2), and the definitions in Equations (9.3–9.5), (9.9a) (9.9b) (9.9c) If the principal (basic) material properties (E1, E2, ν12, and G12) are known from previous tests (Chapters 5–7), it is possible to calculate the off-axis properties Ex, νxy, and ηxy using Equations (A.1) (Appendix A), and compare those to the experimentally determined values. G12 may be compared to the modulus measured in the off-axis test (Equation (9.6)). η γ ε xy xy x = G12 12 12 = τ γ E S x = 1 11 νxy S S = − 12 11 ηxy S S = 16 11 TX001_ch09_Frame Page 133 Saturday, September 21, 2002 5:01 AM © 2003 by CRC Press LLC
Unconstrained Ends P M Constrained Ends FIGURE 9.3 FIGURE 9.2 Influence of gripped end regions on deformation Off-axis coupon under uniform axial stress. of off-axis specimen [2]. 9.2 Influence of End Constraint As first pointed out by Halpin and Pagano [2],most test machines used in testing laboratories employ rigid grips that constrain the shear deformation illustrated in Figure 9.2.As a result,the specimen assumes a shape schematically illustrated in Figure 9.3 [2].To quantify the influence of gripping on the response of an off-axis tension specimen,Halpin and Pagano [2]performed a stress analysis of a constrained coupon and obtained the following expressions for the shear strain and longitudinal strain at the specimen centerline. Yxy=S1eC2-S66Cow2/4 (9.10a) ex=5C2-51Cow2/4 (9.10b) with 125160 C=3w25,56-5+25,民 (9.11a) C。-(356w2+52) C2=1256 (9.11b) where eo =AL/LG (elongation/gage length)and w and LG are specimen width and gage length,respectively(Figure 9.1). ©2003 by CRC Press LLC
9.2 Influence of End Constraint As first pointed out by Halpin and Pagano [2], most test machines used in testing laboratories employ rigid grips that constrain the shear deformation illustrated in Figure 9.2.As a result, the specimen assumes a shape schematically illustrated in Figure 9.3 [2]. To quantify the influence of gripping on the response of an off-axis tension specimen, Halpin and Pagano [2] performed a stress analysis of a constrained coupon and obtained the following expressions for the shear strain and longitudinal strain at the specimen centerline. (9.10a) (9.10b) with (9.11a) (9.11b) where ε0 = ∆L/LG (elongation/gage length) and w and LG are specimen width and gage length, respectively (Figure 9.1). FIGURE 9.2 Off-axis coupon under uniform axial stress. FIGURE 9.3 Influence of gripped end regions on deformation of off-axis specimen [2]. 2 y x σ x x σ w LG P P M V Constrained Ends Unconstrained Ends 1 γ xy = − S C S Cw 16 2 66 0 2 4 εx = − S C S Cw 11 2 16 0 2 4 C S w SS S SLG 0 16 0 2 11 66 16 2 11 2 12 3 2 = ( ) − + ε C C S 2 Sw SLG 0 16 66 2 11 2 12 = + ( ) 3 TX001_ch09_Frame Page 134 Saturday, September 21, 2002 5:01 AM © 2003 by CRC Press LLC
On the basis of this analysis,it is possible to derive an expression for the apparent axial Young's modulus including end constraint (Ex)a=x/Ex (9.12) where o,and are the stress and strain at the centerline of the constrained off-axis coupon.(E)a may be expressed as Ex (但)。=1 (9.13) in which Ex is the modulus for an unconstrained off-axis specimen.The parameterξis given by 35。 (9.14) Su 356+25.L6/w7 Examination of the above equations reveals that0and(E)E=1/Su when LG/w→o. Similarly,Pindera and Herakovich [3]derived an expression for the apparent Poisson's ratio,(Vxy)a 1- B ). (9.15) 1- 3 2 52 where B is given by [3], (9.16) The apparent shear coupling ratio of the specimen subjected to end con- straint is (nxy)a =Ysy/Ex (9.17) 2003 by CRC Press LLC
On the basis of this analysis, it is possible to derive an expression for the apparent axial Young’s modulus including end constraint (Ex)a = σx/εx (9.12) where σx and εx are the stress and strain at the centerline of the constrained off-axis coupon. (Ex)a may be expressed as (9.13) in which Ex is the modulus for an unconstrained off-axis specimen. The parameter ξ is given by (9.14) Examination of the above equations reveals that ξ → 0 and (Ex)a → when LG/w → ∞. Similarly, Pindera and Herakovich [3] derived an expression for the apparent Poisson’s ratio, (νxy)a (9.15) where β is given by [3], (9.16) The apparent shear coupling ratio of the specimen subjected to end constraint is (ηxy)a = γxy/εx (9.17) E E x a x ( ) = 1− ξ ξ = + ( ) 1 S S 11 3S 2S L w G 16 2 66 11 2 3 E S x = 1 11 ν ν β β xy a xy S S S S ( ) = − − 1 3 2 1 3 2 26 11 16 12 β = + w L S S w L S S G G 2 16 11 2 66 11 1 3 2 TX001_ch09_Frame Page 135 Saturday, September 21, 2002 5:01 AM © 2003 by CRC Press LLC