172 9 Effective Elastic Constants of a Laminate laminates. 名 H-thickness of laminate a inv(A); y=1/(日*a(3,3))； Example 9.1 Show that the effective elastic constants for the laminate can be written in terms of the components A;of the [A]matrix as follows: 瓦=AAA22-A经 A22H (9.10a) A11AA22-A2 ,= AuH (9.10b) 0y= A12 (9.10c A22 Dus A12 (9.10d) A11 Gv=Ace H (9.10e) Solution Starting with (8.11)and(8.12)as follows: N: A11 A12 A12 A22 (9.11) 0 A66 take the inverse of(9.11)to obtain: a11 a12 112 a22 (9.12) 0 0 66 N where A22 a11= (9.13a) A11A22-A72 Au 022= A11A22-A2 (9.13b) A12 a12=A11A2-A (9.13c 1 a66=A66 (9.13d)172 9 Effective Elastic Constants of a Laminate % laminates. % H - thickness of laminate a = inv(A); y = 1/(H*a(3,3)); Example 9.1 Show that the effective elastic constants for the laminate can be written in terms of the components Aij of the [A] matrix as follows: E¯x = A11AA22 − A2 12 A22H (9.10a) E¯y = A11AA22 − A2 12 A11H (9.10b) ν¯xy = A12 A22 (9.10c) ν¯yx = A12 A11 (9.10d) G¯xy = A66 H (9.10e) Solution Starting with (8.11) and (8.12) as follows: ⎧ ⎪⎨ ⎪⎩ Nx Ny Nxy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ A11 A12 0 A12 A22 0 0 0 A66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ ε0 x ε0 y γ0 xy ⎫ ⎪⎬ ⎪⎭ (9.11) take the inverse of (9.11) to obtain: ⎧ ⎪⎨ ⎪⎩ ε0 x ε0 y γ0 xy ⎫ ⎪⎬ ⎪⎭ = ⎡ ⎢ ⎣ a11 a12 0 a12 a22 0 0 0 a66 ⎤ ⎥ ⎦ ⎧ ⎪⎨ ⎪⎩ Nx Ny Nxy ⎫ ⎪⎬ ⎪⎭ (9.12) where a11 = A22 A11A22 − A2 12 (9.13a) a22 = A11 A11A22 − A2 12 (9.13b) a12 = A12 A11A22 − A2 12 (9.13c) a66 = 1 A66 (9.13d)