Free and forced vibration analyses of functionally graded hollow cylinders and spheres are peiformed and analytical benchmark solutions are presented. The material is assumed to be graded in the radial direction according to a simple power law. The Laplace transform method is used, and the inversion into the time domain is peiformed exactly using calculus of residues. The Complex Laplace parameter in the free vibration equation has directly given natural frequencies, and the results are given in tabular form. On the inner surface, various axi.sxmmetric dynamic pressures are applied, and radial displacement and hoop stress are presented in the form of graphs. The e.xponent in the power law, called the inhomogeneity patameter, essentially refers to the degree of inhomogeneity. Increasing the inhomogeneity parameter provides a stress-shielding effect. Closed-form solutions obtained in the present paper are tractable, and they allow for further parametric studies. The inhomogeneity constant is a useful parameter from a design point of view in that it can be tailored for specific applications to control the stress distribution.
A novel approach is employed to a general solution for one-dimensional steady-state thermal and mechanical stresses in a hollow thick cylinder made of a functionally graded material (FGM). The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the cylinder. The material properties, except Poisson's ratio, are assumed to be exponentially-varying through the thickness. Forcing functions applied to the inner boundary are internal pressures which may be in form of steps. These conditions result in governing differential equations with variable coefficients. Analytical solutions to such equations cannot be obtained except for certain simple grading functions and pressures. Numerical approaches must be adopted to solve the problem in hand. The novelty of the present study lies in the fact that the Complementary Functions Method (CFM) is employed in the analysis. The Complementary Functions method (CFM) will be infused into the analysis to convert the problem into an initial-value problem which can be solved accurately. Benchmark solutions available in the literature are used to validate the results and to observe the convergence of the numerical solutions. The solution procedure is well-structured, simple and efficient and it can be readily applied to cylinders. It is also well suited for problems in which mechanical properties are graded.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.