The dynamic problem of thermoelectroelasticity for functionally graded layer

Abstract

A general formulation for the problem of motion of inhomogeneous thermoelectroelastic body is considered. The problem of heat flow impact on the functionally graded layer made of piezoceramics is considered as an example. One plane of the layer is grounded, while the other has an electric potential induced by a pyroelectric effect. Nondimensionalization of the boundary value problem of thermoelectroelasticity allows us to identify the coupling parameters. After excluding the electric potential from the formulation, the problem of thermoelectroelasticity is reduced to the problem of themoelasticity with modified coefficients. Upon the application of the Laplace transformation, the problem of themoelasticity is reduced to the system of the Fredholm integral equations of the second kind and is solved numerically by the collocation method using the trapezoid quadrature formula. The original solutions are found using the residue theory. A comparison between the results of the numerical and analytical solutions is made, using as an example, a homogeneous layer of barium titanate. The influence of different forms of the thermal impact on the behavior of the induced electric potential is investigated. The dependence of the induced electrical potential on the distribution of inhomogeneities in the class of power and exponential functions is explored. It has been found that the form of the laws governing the distribution of inhomogeniety of heat conduction and heat capacity coefficients has an essential effect on the shape of the induced potential. And vice versa, the form of the laws governing the distribution of inhomogeneity of the elastic modulus, material density and coefficient of thermal stresses do not affect the shape of the induced potential. These findings are attributed to smallness of the thermomechanical coupling parameter for real materials and should be taken into account in designing various technical devices, which make use of functionally-graded materials with specified properties.