Potential functions of thermoelastodynamic problems for transversely isotropic functional graded materials

Functionally graded materials are the novel class of advanced composite structures with variable properties in one or more directions. The mechanical properties of Functionally Graded Materials (FGM) such as Poisson’s ratio, Young’s modulus of elasticity, material density and shear modulus of elasticity undergo changes gradually and continuously between two surfaces in a predetermined manner. FGM structures are often made of a combination of ceramics and metals in which the metal component provides strength and fracture resistance while ceramic component provides thermal resistance. Due to desirable properties, FGM materials are used in various fields of engineering such as optics, electronics, space vehicles, shipbuilding, mechanical, biomechanical and other engineering structures subjected to high thermal and residual stresses. Therefore, the analytical study of thermoelastodynamic problems is of great importance for the functionally graded media. The use of potential functions to analyze three-dimensional elastic problems is one of the most effective methods. This method facilitates solving three-dimensional elastic problems by uncoupling the set of governing differential equations or at least simplifying them. In the present study, the displacement potential functions for solving thermoelastodynamic problems in the transversely isotropic media with functionally graded materials are introduced. For this purpose, first, the three-dimensional kinematic and thermodynamic equations for the functionally graded materials in the transversely isotropic materials are written and then using a systematic method to separate the equations, the displacement potential functions to solve thermoelastodynamic problems are obtained, which can be used further to solve problems of beams, plates, shells, and infinite and semi-infinite media. The obtained potential functions include two scalar functions F and χ; the scalar function F satisfies the sixth-order partial differential equation while the χ-scalar function satisfies the second-order partial differential equation. In addition, in the present study, the thermal potential functions for the specific state of the isotropic functional graded media are presented.