MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis 5.2.5 Summary Composite strength analysis is most commonly performed,by industry,on the macromechanics level given that the analysis of composite materials uses effective lamina properties based on average stress and strain. Ply level stresses are the commonly used approach to laminate strength analysis Lamina stress/strain is influenced by many properties of interest,but is dominated by mechanical load and environmental sensitivity. Stress-strain elastic behavior,in its simplest form,may be described as a function of a composite ma- terials constitutive properties (i.e.,E,G,v,a). ● Several practical failure criteria exist today that:1)depend upon cross-plied laminate coupon data to determine lamina stress/strain allowables and 2)identify the failure mode based on the allowable that is first exceeded by its stress/strain counterpart. 5.3 ANALYSIS OF LAMINATES 5.3.1 Lamina stress-strain relations A laminate is composed of unidirectionally-reinforced laminae oriented in various directions with re- spect to the axes of the laminate.The stress-strain relations developed in the Section 5.2 must be trans- formed into the coordinate system of the laminate to perform the laminate stress-strain analysis.A new system of notation for the lamina elastic properties is based on xi in the fiber direction,xz transverse to the fibers in the plane of the lamina,and x;normal to the plane of the lamina. E1=E1 12=i2 E2=E3=E2 V23=V23 5.3.1(a) G12=GI G23=G2 In addition,the laminae are now treated as effective homogeneous,transversely isotropic materials. 1 -V12 -V12 0 0 E 0 EL e11 -V12 1 -V23 0 0 0 EL E2 E2 011 E22 -V12 -V23 1 022 0 0 0 e33 EL E2 E2 033 2e23 5.3.1(b) 0 0 1 0 0 0 023 2e13 G23 013 2e12 0 0 0 0 0 012 G12 0 0 0 0 0 G12 It has become common practice in the analysis of laminates to utilize engineering shear strains rather than tensor shear strains.Thus the factor of two has been introduced into the stress-strain relationship of Equation 5.3.1(b). The most important state of stress in a lamina is plane stress,where 013=023=033=0 5.3.1(c) 5-21
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-21 5.2.5 Summary • Composite strength analysis is most commonly performed, by industry, on the macromechanics level given that the analysis of composite materials uses effective lamina properties based on average stress and strain. • Ply level stresses are the commonly used approach to laminate strength analysis. • Lamina stress/strain is influenced by many properties of interest, but is dominated by mechanical load and environmental sensitivity. • Stress-strain elastic behavior, in its simplest form, may be described as a function of a composite materials constitutive properties (i.e., E, G, ν, α). • Several practical failure criteria exist today that: 1) depend upon cross-plied laminate coupon data to determine lamina stress/strain allowables and 2) identify the failure mode based on the allowable that is first exceeded by its stress/strain counterpart. 5.3 ANALYSIS OF LAMINATES 5.3.1 Lamina stress-strain relations A laminate is composed of unidirectionally-reinforced laminae oriented in various directions with respect to the axes of the laminate. The stress-strain relations developed in the Section 5.2 must be transformed into the coordinate system of the laminate to perform the laminate stress-strain analysis. A new system of notation for the lamina elastic properties is based on x1 in the fiber direction, x2 transverse to the fibers in the plane of the lamina, and x3 normal to the plane of the lamina. 1 1* 12 12* 2 32* 23 23* 12 1 * 23 2 * E =E = E =E = E = G =G G = G ν ν ν ν 5.3.1(a) In addition, the laminae are now treated as effective homogeneous, transversely isotropic materials. ε ε ε ε ε ε ν ν ν ν ν ν σ σ σ σ σ σ 11 22 33 23 13 12 1 12 1 12 1 12 1 2 23 2 12 1 23 2 2 23 12 12 11 22 33 23 13 12 2 2 2 1 000 1 000 1 000 000 1 0 0 0 0 00 1 0 0 0 0 00 1 R S | | | T | | | U V | | | W | | | = − − − − − − L N M M M M M M M M M M M M M M O Q P P P P P P P P P P P P P P R S | | | T | | | U V | | | W | | | EEE EEE EEE G G G 5.3.1(b) It has become common practice in the analysis of laminates to utilize engineering shear strains rather than tensor shear strains. Thus the factor of two has been introduced into the stress-strain relationship of Equation 5.3.1(b). The most important state of stress in a lamina is plane stress, where σ13 23 33 = σ = σ = 0 5.3.1(c)
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis since it occurs from both in-plane loading and bending at sufficient distance from the laminate edges. The plane stress version of Equation 5.3.1(b)is 1 -V12 0 E 11 12 1 E22 0 E2 022 5.3.1(d) E12 012 0 0 G12】 which may be written as ed =[S]od 5.3.1(e) Here [S],the compliance matrix,relates the stress and strain components in the principal material direc- tions.These are called laminae coordinates and are denoted by the subscript ( Equation 5.3.1(d)relates the in-plane strain components to the three in-plane stress components. For the plane stress state,the three additional strains can be found to be E23=e13=0 e33=-011 3-6221 V23 5.3.1(⑤ E E2 and the complete state of stress and strain is determined. The relations 5.3.1(d)can be inverted to yield (od=[ST ed 5.3.1(g) 0 d=QKEd 5.3.1(h) The matrix [Q]is defined as the inverse of the compliance matrix and is known as the reduced lamina stiffness matrix.Its terms can be shown to be [Q11 Q12 0 [Q]= Q12 Q22 0 5.3.1(0 0 0Q66J Q11= EL Q12=M2E2 哈唱 5.3.10 Q22= E2 哥 Q66=G12 In the notation for the [Q]matrix,each pair of subscripts of the stiffness components is replaced by a sin- gle subscript according to the following scheme. 11→122→233→3 23→431→512→6 5-22
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-22 since it occurs from both in-plane loading and bending at sufficient distance from the laminate edges. The plane stress version of Equation 5.3.1(b) is ε ε ε ν ν σ σ σ 11 22 12 1 12 1 12 1 2 12 11 22 12 1 0 1 0 0 0 1 R S | T | U V | W | = − − L N M M M M M M M O Q P P P P P P P R S | T | U V | W | E E E E G 5.3.1(d) which may be written as {ε A A }= S{ } σ 5.3.1(e) Here [S], the compliance matrix, relates the stress and strain components in the principal material directions. These are called laminae coordinates and are denoted by the subscript A . Equation 5.3.1(d) relates the in-plane strain components to the three in-plane stress components. For the plane stress state, the three additional strains can be found to be 23 13 13 23 33 11 22 1 2 = =0 =- - E E ε ε ν ν εσ σ 5.3.1(f) and the complete state of stress and strain is determined. The relations 5.3.1(d) can be inverted to yield { }= S { } -1 σ ε A A 5.3.1(g) or {σ A A }= Q{ } ε 5.3.1(h) The matrix [Q] is defined as the inverse of the compliance matrix and is known as the reduced lamina stiffness matrix. Its terms can be shown to be [Q] = QQ 0 QQ 0 0 0Q 11 12 12 22 66 L N M M M M O Q P P P P 5.3.1(i) 11 1 12 2 2 1 12 12 2 12 2 2 1 22 2 12 2 2 1 66 12 Q = E 1- E E Q = E 1- E E Q = E 1- E E Q = G ν ν ν ν 5.3.1(j) In the notation for the [Q] matrix, each pair of subscripts of the stiffness components is replaced by a single subscript according to the following scheme. 11 1 22 2 33 3 23 4 31 5 12 6 →→→ → →→
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis The reduced stiffness and compliance matrices 5.3.1(i)and(d)relate stresses and strains in the prin- cipal material directions of the material.To define the material response in directions other than these coordinates,transformation relations for the material stiffnesses are needed. In Figure 5.3.1(a),two sets of coordinate systems are depicted.The 1-2 coordinate system corre- sponds to the principal material directions for a lamina,while the x-y coordinates are arbitrary and related to the 1-2 coordinates through a rotation about the axis out of the plane of the figure.The angle 0 is de- fined as the rotation from the arbitrary x-y system to the 1-2 material system. The transformation of stresses from the 1-2 system to the x-y system follows the rules for transforma- tion of tensor components. Oxx m2 n2 -2mn 11 Oyy n2 m2 2mn 022 5.3.1(k) Oxy mn -mn m2-n2 012 or ox=[elod 5.3.10 where m cose,and n sin.In these relations,the subscript x is used as shorthand for the lami- nate coordinate system. The same transformation matrix [can also be used for the tensor strain components.However, since the engineering shear strains have been utilized,a different transformation matrix is required. Exx m2 n2 -mn 11 Eyy 2 03 mn e22 5.3.1(m) 2 Exy 2mn -2mn m2-n2 2e12 or ex=[yled 5.3.1() 1.2 Principal Material Coordinates x,y Laminate,or Arbitrary Coordinates FIGURE 5.3.1(a)Coordinate systems. 5-23
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-23 The reduced stiffness and compliance matrices 5.3.1(i) and (d) relate stresses and strains in the principal material directions of the material. To define the material response in directions other than these coordinates, transformation relations for the material stiffnesses are needed. In Figure 5.3.1(a), two sets of coordinate systems are depicted. The 1-2 coordinate system corresponds to the principal material directions for a lamina, while the x-y coordinates are arbitrary and related to the 1-2 coordinates through a rotation about the axis out of the plane of the figure. The angle θ is defined as the rotation from the arbitrary x-y system to the 1-2 material system. The transformation of stresses from the 1-2 system to the x-y system follows the rules for transformation of tensor components. = m n -2 mn n m 2 mn mn -mn m - n xx yy xy 2 2 2 2 2 2 11 22 12 σ σ σ σ σ σ R S | | T | | U V | | W | | L N M M M M O Q P P P P R S | T | U V | W | 5.3.1(k) or {σ x}= { } θ σ A 5.3.1(l) where m = cosθ , and n = sinθ . In these relations, the subscript x is used as shorthand for the laminate coordinate system. The same transformation matrix [θ ] can also be used for the tensor strain components. However, since the engineering shear strains have been utilized, a different transformation matrix is required. 2 = m n -mn n m mn 2 mn -2 mn m - n 2 xx yy xy 2 2 2 2 2 2 11 22 12 ε ε ε ε ε ε R S || T | | U V || W | | L N M M M M O Q P P P P R S | T | U V | W | 5.3.1(m) or {ε x}= { } ψ ε A 5.3.1(n) FIGURE 5.3.1(a) Coordinate systems
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis Given the transformations for stress and strain to the arbitrary coordinate system,the relations be- tween stress and strain in the laminate system can be determined. fox=Qfex 5.3.1(o) The reduced stiffness matrix [Q]relates the stress and strain components in the laminate coordinate sys- tem. [阿]-[' 5.3.1(p) The terms within [Q]are defined to be 011=Q11m4+Q2n4+2m2n2(Q12+2Q66) Qbar12=m2n2(Q11+Q22-4Q66)+(m4+n4)Q12 16=[Q11m2.Q22n2-(Q12+2Q66m2-n2]mn Q22=Q11n4+Q22m4+2m2n2(Q12+2Q66) 5.3.1(q) Q26=[Q11n2.Q22m2+(Q12+2Q6(m2-n2]mn 066=(Q11+Q22-2Q12)m2n2+Q66(m2-n22 0Q21=Q12Q61=016Q62=026 where the subscript 6 has been retained in keeping with the discussion following Equation 5.3.1(j). 「Q11Q12Q16 [Q]= Q21 Q22 Q26 5.3.1() Q16Q26Q66J A feature of [Q]matrix which is immediately noticeable is that [Q]is fully-populated.The additional terms which have appeared inand relate shear strains to extensional loading and vice versa. This effect of a shear strain resulting from an extensional strain is depicted in Figure 5.3.1(b).From Equations 5.3.1(q),these terms are zero for e equal to 0 or 90.Physically,this means that the fibers are parallel or perpendicular to the loading direction.For this case,extensional-shear coupling does not occur for an orthotropic material since the loadings are in the principal directions.The procedure used to develop the transformed stiffness matrix can also be used to find a transformed compliance matrix. ed=[S]od 5.3.1(s) ex)=[vISTIOT (ox 5.3.1() ex=[5Had 5.3.1(u Noting that the stress-strain relations are now defined in the laminate coordinate system,lamina stiff- nesses can also be defined in this system.Thus,expanding the last of Equations 5.3.1(s)-(u): 5-24
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-24 Given the transformations for stress and strain to the arbitrary coordinate system, the relations between stress and strain in the laminate system can be determined. {σ ε x x }= Q{ } 5.3.1(o) The reduced stiffness matrix [Q] relates the stress and strain components in the laminate coordinate system. Q= Q -1 θ ψ 5.3.1(p) The terms within [Q] are defined to be 11 11 4 22 4 22 12 66 12 2 2 11 22 66 4 4 12 16 11 2 22 2 12 66 2 2 22 11 4 22 4 22 12 66 26 11 2 22 2 12 66 2 2 66 11 22 12 2 2 2 66 2 2 Q = Q m + Q n + 2m n (Q + 2Q ) Qbar = m n (Q + Q - 4Q )+(m + n )Q Q = [Q m - Q n - (Q + 2Q )(m - n )]mn Q = Q n + Q m + 2m n (Q + 2Q ) Q = [Q n - Q m + (Q + 2Q )(m - n )]mn Q = (Q + Q - 2Q )m n + Q (m - n ) 5.3.1(q) Q21 12 61 16 62 26 = Q Q = Q Q = Q where the subscript 6 has been retained in keeping with the discussion following Equation 5.3.1(j). [Q] = QQQ QQQ QQQ 11 12 16 21 22 26 16 26 66 L N M M M M M O Q P P P P P 5.3.1(r) A feature of [Q] matrix which is immediately noticeable is that [Q] is fully-populated. The additional terms which have appeared in Q ,Q16 and Q26, relate shear strains to extensional loading and vice versa. This effect of a shear strain resulting from an extensional strain is depicted in Figure 5.3.1(b). From Equations 5.3.1(q), these terms are zero for θ equal to 0° or 90°. Physically, this means that the fibers are parallel or perpendicular to the loading direction. For this case, extensional-shear coupling does not occur for an orthotropic material since the loadings are in the principal directions. The procedure used to develop the transformed stiffness matrix can also be used to find a transformed compliance matrix. {ε A A }= S{ } σ 5.3.1(s) { x}= S { } -1 ε ψ θ σ x 5.3.1(t) {ε σ x x }= S{ } 5.3.1(u) Noting that the stress-strain relations are now defined in the laminate coordinate system, lamina stiffnesses can also be defined in this system. Thus, expanding the last of Equations 5.3.1(s) - (u):
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis Exx Su S12 S67 Oxx Eyy 521 S22 S26 Oyy 5.3.1() 2 Exy S16 S26 Oxy The engineering constants for the material can be defined by specifying the conditions for an experiment. For oyy=xy=0,the ratio oxx/Exx is Young's modulus in the x direction.For this same stress state, -yy/xxis Poisson's ratio.In this fashion,the lamina stiffnesses in the coordinate system of Equations 5.3.1(s)-(u)are found to be: 1 1 Ex= Su Ey= 22 5.3.1(w Gxy"5c6 1 Vy=- 21=S12 11S11 一一--Deformed Shape Under Extensional Loading FIGURE 5.3.1(b)Extensional-shear coupling. It is sometimes desirable to obtain elastic constants directly from the reduced stiffnesses, [可byus ing Equations 5.3.1(o).In the general case where the matrix is fully populated,this can be accom- plished by using Equations 5.3.1(w)and the solution for Sii as functions of Q obtained from the inverse relationship of the two matrices.An alternative approach is to evaluate extensional properties for the case of zero shear strain.For single stress states and zero shear strain,the elastic constants in terms of the transformed stiffness matrix terms are: 5-25
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-25 2 = SSS SSS SSS xx yy xy 11 12 16 21 22 26 16 26 66 xx yy xy ε ε ε σ σ σ R S || T | | U V || W | | L N M M M M O Q P P P P R S || T | | U V || W | | 5.3.1(v) The engineering constants for the material can be defined by specifying the conditions for an experiment. For σ yy xy = = σ 0 , the ratio σ ε xx xx / is Young's modulus in the x direction. For this same stress state, −ε ε yy xx / is Poisson's ratio. In this fashion, the lamina stiffnesses in the coordinate system of Equations 5.3.1(s) - (u) are found to be: x 11 y 22 xy 66 xy 21 11 12 11 E = 1 S E = 1 S G = 1 S = -S S = -S S ν 5.3.1(w) FIGURE 5.3.1(b) Extensional-shear coupling. It is sometimes desirable to obtain elastic constants directly from the reduced stiffnesses, Q , by using Equations 5.3.1(o). In the general case where the Qij matrix is fully populated, this can be accomplished by using Equations 5.3.1(w) and the solution for Sij as functions of Qij obtained from the inverse relationship of the two matrices. An alternative approach is to evaluate extensional properties for the case of zero shear strain. For single stress states and zero shear strain, the elastic constants in terms of the transformed stiffness matrix terms are: