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Ebook Mechanics of composite materials with MATLAB: Part 2

(BQ) Part 2 book "Mechanics of composite materials with MATLAB" has contents: Effective elastic constants of a laminate, failure theories of a lamina, introduction to homogenization of composite materials, introduction to damage mechanics of composite materials. | 9 Effective Elastic Constants of a Laminate 9.1 Basic Equations In this chapter, we introduce the concept of effective elastic constants for the lam¯ inate. These constants are the effective extensional modulus in the x direction Ex , ¯ the effective extensional modulus in the y direction Ey , the effective Poisson’s ratios ¯ ¯ νxy and νyx , and the effective shear modulus in the x-y plane Gxy . ¯ The effective elastic constants are usually defined when considering the inplane loading of symmetric balanced laminates. In the following equations, we consider only symmetric balanced or symmetric cross-ply laminates. We therefore define the following three average laminate stresses [1]: σx = ¯ 1 H σy = ¯ 1 H τxy = ¯ 1 H H/2 −H/2 σx dz (9.1) σy dz (9.2) τxy dz (9.3) H/2 −H/2 H/2 −H/2 where H is the thickness of the laminate. Comparing (9.1), (9.2), and (9.3) with (7.13), we obtain the following relations between the average stresses and the force resultants: 1 Nx H 1 σy = Ny ¯ H 1 τxy = Nxy ¯ H σx = ¯ (9.4) (9.5) (9.6) Solving (9.4), (9.5), and (9.6) for Nx , Ny , and Nxy , and substituting the results into (8.11) and (8.12) for symmetric balanced laminates, we obtain: 170 9 Effective Elastic Constants of a Laminate ⎧ 0 ⎫ ⎡ ⎫ ⎤⎧ 0 a11 H a12 H ⎪ εx ⎪ ⎪ σx ⎪ ⎨ ⎬ ⎨ ¯ ⎬ ⎥ ⎢ ε0 0 ⎦ σy ¯ = ⎣ a12 H a22 H y ⎪ 0 ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ γxy τxy ¯ 0 0 a66 H (9.7) The above 3 × 3 matrix is defined as the laminate compliance matrix for symmetric balanced laminates. Therefore, by analogy with (4.5), we obtain the following effective elastic constants for the laminate: 1 a11 H 1 ¯ Ey = a22 H 1 ¯ Gxy = a66 H a12 νxy = − ¯ a11 a12 νyx = − ¯ a22 ¯ Ex = (9.8a) (9.8b) (9.8c) (9.8d) (9.8e) It is clear from the above equations that νxy and νyx are not independent and ¯ ¯ are related by the following reciprocity relation: νyx ¯ νxy ¯ ¯ = Ey ¯ Ex (9.9) Finally, we note that the expressions of the effective elastic constants of (9.8) can be re-written in terms of the components Aij of the .