3.2 STIFFNESS MATRICES OF THIN LAMINATES 79 Inversion of matrices [A]and [D]yields the compliance matrices [a]and [d] (see Eqs.3.29 and 3.30).The nonzero elements are 1 2(1+v)1 a11三E a12=-va11 a66= Eh Gh (3.43) 12 d41二Ef 24(1+v)12 d12=-vd1 d66= Eh> Gh Ouasi-isotropic laminate.A laminate is quasi-isotropic when there are at least three fiber directions; the orientation (fiber angle of each ply is =i/I,where i is an integer (i =1,2,...,1),and I is the total number of fiber orientations (I>3); the number of plies in each fiber direction is the same;and each ply is made of the same material and has the same thickness. For example,inπ/4 laminates(page6)there are fibers in the0°，45°，-45°， and 90 directions,and I =4. For each ply,the elements of the [O]matrix are obtained by substituting= i180/I into the expressions in Table 3.1 (page 70).Then,by substituting these elements into the expression for the [A]matrix (see Eq.3.20),we obtain the following nonzero elements of the [A]matrix: 3 A:=(01+Q如)+h02+0s A22=A1 (3.44) -5h(:+Q)+0-h0w 3 46=1-A2 The stiffness matrix [A]may be written as 41 A2 0 1 [A= A22 0 =hR (3.45) 0 0 A66 where R and viso are parameters defined as 1 R=8(0+Qa)+4m+i0s (3.46) o=8R(++62-40 1 and h is the thickness of the laminate.It is stated here without proof that Eqs.(3.45) and (3.46)are valid2 for all values of I as long as I >3. 2 S.W.Tsaiand H.T.Hahn.Introduction to Composite Materials.Technomic,Lancaster.Pennsylvania, 1980.p.145.3.2 STIFFNESS MATRICES OF THIN LAMINATES 79 Inversion of matrices [A] and [D] yields the compliance matrices [a] and [d] (see Eqs. 3.29 and 3.30). The nonzero elements are a11 = 1 Eh a12 = −νa11 a66 = 2 (1 + ν) Eh = 1 Gh d11 = 12 Eh3 d12 = −νd11 d66 = 24 (1 + ν) Eh3 = 12 Gh3 . (3.43) Quasi-isotropic laminate. A laminate is quasi-isotropic when
there are at least three fiber directions;
the orientation (fiber angle ) of each ply is  = iπ/I, where i is an integer (i = 1, 2,... , I), and I is the total number of fiber orientations (I ≥ 3);
the number of plies in each fiber direction is the same; and
each ply is made of the same material and has the same thickness. For example, in π/4 laminates (page 65) there are fibers in the 0◦, 45◦, −45◦, and 90◦ directions, and I = 4. For each ply, the elements of the [Q] matrix are obtained by substituting  = i180◦/I into the expressions in Table 3.1 (page 70). Then, by substituting these elements into the expression for the [A] matrix (see Eq. 3.20), we obtain the following nonzero elements of the [A] matrix: A11 = 3 8 h (Q11 + Q22) + 1 4 hQ12 + 1 2 hQ66 A22 = A11 A12 = 1 8 h (Q11 + Q22) + 3 4 hQ12 − 1 2 hQ66 A66 = A11 − A12 2 . (3.44) The stiffness matrix [A] may be written as [A] =    A11 A12 0 A12 A22 0 0 0 A66    = hR    1 νiso νiso 1 1−νiso 2    , (3.45) where R and νiso are parameters defined as R = 3 8 (Q11 + Q22) + 1 4 Q12 + 1 2 Q66 νiso = 1 8R(Q11 + Q22 + 6Q12 − 4Q66), (3.46) and h is the thickness of the laminate. It is stated here without proof that Eqs. (3.45) and (3.46) are valid2 for all values of I as long as I ≥ 3. 2 S. W. Tsai and H. T. Hahn,Introduction to Composite Materials. Technomic, Lancaster, Pennsylvania, 1980, p. 145