Stability analysis of inhomogeneous, fibrous composite plates

Abstract Materials with deliberately designed forms of anisotropy and inhomogeneity, obtained by manipulating microstructural composition, can provide excellent solutions to existing design constraints and create new design opportunities. This new class of problems, concerning optimization of the internal structures of elastic bodies, is the subject matter of the present investigation. The paper focuses attention on deliberately designed inhomogeneity by controlling spatial fiber distribution in a lamina, for improving uniaxial and shear buckling behavior of rectangular, unidirectional and cross-ply laminates under a variety of boundary conditions. In the literature, optimization of the orientation of fibers (through thickness) in fibrous composite laminates with respect to the buckling load is discussed extensively on the assumption of uniform spatial fiber distribution in the plane of the plate. Design involving non-uniform fiber distribution, which has received much less attention, appears to be an attractive option, at least from a theoretical point of view. Also, the motivation comes from reinforced concrete structures where the non-uniform spacing of reinforcing bars is quite common practice. Non-uniform fiber distribution leads to the problem of inhomogeneous, orthotropic plate buckling which is solved in two steps. Firstly, the prebuckling stress field is derived because the assumption of uniform, uniaxial stress, common in homogeneous plates, is theoretically no longer valid. Finally, out-of-plane buckling is analysed incorporating the prebuckling field derived earlier. Within the framework of the Ritz method, a stress function formulation for plane-stress (stretching) and a displacement formulation for buckling analysis, are employed. An important feature of the analysis is using the classical analogy between in-plane stress function and out-of-plane buckling displacement formulations which not only provides a unified analytical treatment but also reduces the problem size significantly. For the analysis, a computerized Rayleigh-Ritz method, in conjunction with Gram-Schmidt orthogonal polynomials as coordinate functions, is developed, which is capable of modeling a variety of boundary conditions, viz. simple, clamped, free and their combinations. Uniaxial and shear buckling coefficients of unidirectional and cross-ply laminates are computed for various cases of sinusoidal fiber distribution. It is found that, for given constant fiber volume, a higher fiber concentration at the middle of the plate would generally increase the buckling load by as much as 25% over a uniform distribution of the same amount of fibers. The paper highlights the unusual tailoring capabilities offered by advanced composite materials.