Nonlinear thermal stability of temperature-dependent metal matrix composite shallow arches with functionally graded fiber reinforcements

Abstract

The present article aims to study the non-linear equilibrium paths of an Al2O3 reinforced metal matrix composite (MMC) shallow arch in a thermal environment. The structure is made up of applicable industrial materials having temperature-dependent thermomechanical properties. In order to describe the configuration of shallow arch, two different geometrical models are implemented in which the first model has a symmetrical configuration but the second model has an unsymmetrical one. The concept of symmetric functionally graded (FG) material is implemented in the laminated composite arch by changing the volume fraction of fibers in each lamina. The elasticity tensor is created based on the plane-stress assumption and implementation of the rule of mixture. The mathematical modeling is developed using first-order shear deformation beam theory under specific assumptions i.e., geometrical nonlinearity, neutral axis concept, and uniform temperature gradient. Next, an analytical procedure is established to solve the three coupled nonlinear equilibrium equations for both clamped-clamped and simply-simply supporting conditions. The obtained exact solutions are capable of determining the critical temperature, in which the perturbed pitchfork bifurcation occurs, as well as the thermal equilibrium path of the structure. The influences of geometrical parameters (i.e., the initial curvature shape and amplitude, the thickness-to-length ratio, and the lamination sequences) as well as physical elements (i.e., matrix materials, fiber volume fraction distribution patterns, and temperature dependence of materials) on the thermal behavior of the system are examined. Eventually, in order to verify the analytical results presented in this study, a finite difference method-based numerical solution (bvp4c in Matlab software) is utilized to solve simultaneously the three coupled nonlinear governing equations.