8 Terminology used in micromechanics Ef, Em - Young’s modulus of fiber and matrixGf, Gm - Shear modulus of fiber and matrixνf, νm - Poisson’s ratio of fiber and matrixVf, Vm - Volume fraction of fiber and matrixWf, Wm – Weight fraction of fiber and matrix
9 Elastic properties: Rule of mixtures approach Parallel model - E1 and ν12 (Constant Strains)E1 = Ef1Vf + EmVmν12 = νf Vf + νmVmPredictions agree well with experimental dataSeries model – E2 and G (Constant Stress)E2 = __ __ Ef2 Em________Ef2 Vm + Em VfG12 = __ __ Ef2 Em________Experimental results predicted less accurately
10 Micromechanics example: Volume fraction changes Knowns : Carbon: E = 34.0 x 106 psiEpoxy : E = 0.60 x 106 psiHow much does the longitudinal modulus change when the fiber volume fraction is changed from 58% to 65%?E1 = Ef1 Vf + Em VmFor Vf = 0.58: E1 = (34.0 x 106 psi)(.58) + (0.60 x 106 psi)(0.42) = 20.0 x 106 psiFor Vf = 0.65: E1 = (34.0 x 106 psi)(.65) + (0.60 x 106 psi)(0.35) = 22.3 x 106 psiThus, raising the fiber volume fraction from 58% to 65% increases E1 by 12%
11 Design and analysis of composite laminates: Laminated Plate Theory (LPT) Used to determine the response of a composite laminate based on properties of a layer (or ply)
12 Laminate Ply Orientation Code Designate each ply by it’s fiber orientation angleList plies in sequence starting from top of laminateAdjacent plies are separated by “/” if their angle is differentDesignate groups of plies with same angle using subscriptsEnclose complete laminate in bracketsUse subscript “S” to denote mid plane symmetry, or “T” to denote total laminateBar on the top of the ply indicates mid-plane
14 Special types of laminates Symmetric laminate – for every ply above the laminate mid plane, there is an identical ply(material and orientation) an equal distance below the mid planeBalanced laminate – for every ply at a +θ orientation, there is another ply at the – θ orientation somewhere in the laminateCross-ply laminate – composed of plies of either 0o or 90o (no other ply orientations)Quasi-isotropic laminate – produced using at least three different ply orientations, all with equal angles between them. Exhibits isotropic extensional stiffness properties (have the same E in all in-plane directions)
15 The response of special laminates Balanced, unsymmetric, laminateTensile loading produces twisting curvatureEx: [+θ/0/- θ]τSymmetric, unbalanced laminateTensile loading produces in-plane shearingUnsymmetric cross-ply laminateTensile loading produces bending curvatureEx: [0/90]τBalanced and symmetric laminateTensile loading produces extensionEx: [+θ/- θ]sQuasi isotropic laminate: [+60/0/- 60]s and [+45/0/+45/90]sTensile loading produces extension loading, independent of angle
39 Equivalent Engineering Constants for the Laminate Equation for the composite laminate is Or [N] = [A] [ε] Matrix equation for a single orthotropic lamina or layer is Or [σ] = [M] [ε]
40 [A] = [M] where h = laminate thickness The properties of the single “equivalent” orthotropic layer can be determined from the following equation:[A] = [M] where h = laminate thicknessor A11 = A12 = A66 = E6A22 = A21 =Solving last equation, we have the following elastic constants for the single “equivalent” orthotropic layer:E1 = (1 – ) E2 = (1 – )E6 = A66 ν1 = ν2 =
41 The equation for the composite laminate is or [ε] = [a] [N] where [a] = [A] -1 The equation for a single orthotropic lamina is or [ε] = [C] [σ]
42 [a] h = [C] or a11h = a12h = a66h = a21h = a22h = E1 = E2 = E66 = The properties of the single “equivalent” orthotropic layer is determined by[a] h = [C]or a11h = a12h = a66h =a21h = a22h =elastic constants for the single “equivalent” orthotropic layer:E1 = E2 = E66 =ν1 = – ν2 = –
43 Special Cases:Cross-Ply Laminates: All layers are either 0° or 90°, which results in A16 = A26 = 0. This is sometimes called specially orthotropic w. r. t. in-plane force and strains. Refer to Figure 1 and Table 1.h = total laminate thickness
44 Figure 1 In-Plane Modulus and Compliance of T300/5208 Cross-Ply Laminates.
45 Balanced Angle-Ply Laminates: There are only two orientations of the laminae; same magnitude but opposite in sign. With an equal number of plies with positive and negative orientations. Refer to Figure 2 and table 2.h = total laminate thickness.
46 Figure 2 In-Plane Modulus and Compliance of T300/5208 Angle-Ply Laminates.
47 [0, 90] un-symmetric cross-ply, A16 = A26 = 0 The balanced angle-ply laminate is another case of “special orthotropy” w.r.t. in-plane force and strains; that is A16 = A26 = 0.The values of Table 2 apply for un-symmetric laminate as well as symmetric laminates. For example, the values for [+30, −30, +30, −30] or [+30, +30, −30, −30] are the same as those for [+30, -30, -30, +30].If the laminate is not balanced, such as [+30, −30, +30], then the terms A16 and A26 are not zero.Also remember that A16 = A26 for all cross-plies, whether they are symmetric or un-symmetric.Example:[0, 90] un-symmetric cross-ply, A16 = A26 = 0[0, 90, 0] symmetric cross-ply, A16 = A26 = 0[+30, −30, −30, +30] symmetric balanced angle-ply, A16 = A26 = 0[+30, −30, +30, −30] un-symmetric balanced angle-ply, A16 = A26 = 0[+30, −30, +30] symmetric un-balanced angle-ply, A16, A26 ≠ 0
48 3) Quasi-Isotropic Laminates: Has ‘m’ ply groups spaced at ply orientations of 180/m degrees.Note: This does not apply for m = 1 and m = 2. The laminate need not be symmetric to be quasi-isotropic. For example, the laminate [0, 45, −45, 90] is quasi-isotropic.Examples:[0/60/−60]s m = /m = 60°[0/90/45/−45]s m = /m = 45°[0/60/−60/60/0/−60]s m=3 180/m = 60°The modulus of the quasi-isotropic laminate has the following properties:A11 = A22A16 = A26 = 0A66 = (A11 – A22) which is equivalent to G = for an isotropic material.