Elasticity Solutions For Anisotropic Functionally Graded Plane Beams

Functionally graded materials(FGMs) are a kind of special non-homogenous materials,whose properties varying continuously along certain directions.There are no internal interfaces in FGMs.Compared with traditional composites,FGMs have a great deal of superiorities.FGMs will be widely applied in structure engineering and intelligent structures.Elasticity solutions are studied in this work for anisotropic functionally graded elastic,piezoelectric and magneto-electro-elastic finite-length beams with different kinds of conditions at their two ends,and subjected to polynomial and arbitrary loads on their upper and lower surfaces.Firstly,based on elementary equations of anisotropic elastic materials,piezoelectric materials and magneto-electro-elastic materials,the partial differential equations are derided for the stress functions,electric displacement function and magnetic induction function of the plane problems.The coefficients of elastic compliance,piezoelectricity, dielectric impermeability,piezo-magnetism,magneto-electricity,and magnetic permeability are functions of the thickness coordinate.The material coefficients are assumed to be arbitrary continuous functions of the thickness coordinate so as to obtain the analytical solutions of the beams subjected to polynomial loads.When the beams are subjected to arbitrary loads,these coefficients are assumed to be exponential or power functions for obtaining analytical solutions,and to be any arbitrary functions for seeking semi-analytical solutions.Secondly,for beams subjected to polynomial loads,the stress function,electric displacement function and magnetic induction function are assumed in forms of polynomials in the longitudinal coordinate,with undetermined functions of the thickness coordinate.For beams subjected to arbitrary loads,the stress function,electric displacement function and magnetic induction function are assumed to consist of two parts,respectively.One is a product of a trigonometric function of the longitudinal coordinate and an undetermined function of the thickness coordinate,and the other a linear polynomial of longitudinal coordinate with unknown coefficients depending on thickness coordinate.Thirdly,the expressions of stress function,electric displacement function and magnetic induction function with undetermined constants are acquired by virtue of the compatibility equations of strain,electric field and magnetic field.The analytical expressions of stresses,electric displacements,magnetic inductions,axial force, bending moment,shear force,average electric displacement,average magnetic induction, displacements,electric potential and magnetic potential are then deduced,with integral constants determinable from the boundary conditions.Finally,the expressions are educed of the stresses,electric displacements,magnetic inductions,displacements,electric potential and magnetic potential.Thus the elasticity solutions are obtained for the anisotropic functionally graded beams.Numerical examples of these elasticity solutions are presented.Comparisons are made among numerical examples of other literatures,by finite element method and by this work.Similarities and differences are found and discussed.