Three-dimensional Quasi-static General Solution And Fundamental Solutions For Transversely Isotropic Pyroelectric Materials

With the development of science and technology, pyroelectric materials used in aerospace, machinery and power electronics engineering more and more widely, it also brings a variety of pyroelectric problem. Study on the general solution and the Fundamental function based on pyroelectric governing equations is the basis of pyroelectric problem. The general solutions are the basis of obtained various analytical solutions including fundamental solutions. Based on the superposition principle, the fundamental solutions for a structure under the concentrated loads can be used to construct the analytical solutions of many practical problems. In addition, the fundamental solutions are also the essential in the boundary element method as well as the study of cracks, defects and inclusions. In the previous studies, the quasi-static general solution and dynamic general solution all contain a function which satisfies a complicated high-order partial differential equation, it is not convenient to use. In additions, little literature on the quasi-static fundamental solution of pyroelectric materials can be found, while the quasi-static state is the common working state of pyroelectric materials and devices. it will happen in the change step during the working period from the startup to shutdown, including the times of startup, shutdown and change power, in which the shock loadings can often cause a high stress and result to failure of pyroelectric materials and devices. Under the above background, a compact quasi-static general solution and the fundamental solutions for the transversely isotropic pyroelectric materials will be studied in this paper.First of all, based on constitutive, equilibrium and quasi-static heat conduction equations for transversely isotropic pyroelectric materials, with using the differential operator theory, Almansi basic theory and some mathematical transformations, a compact three-dimensional quasi-static general solution is derived and expressed by seven functions. Four of them satisfy the harmonic equations and the other three satisfy the heat conduction equations.Secondly, the corresponding functions with undetermined constant are constructed for an infinite transversely isotropic pyroelectric material under a pulse point heat source. Using the correlation properties of the Dirac function, to describe the point heat source change with time. Then introduce the error function, construct a novel function containing an undetermined constant that corresponding to the general solution. The fundamental solution can be obtained by substituting these functions into the obtained general solution. And the constant can be determined by the heat conservation equation. Based on the obtained fundamental solution for the pulse point heat source, the contours in different times for the temperature increment and all components of stress and electric displacement components in the infinite transversely isotropic pyroelectric body are plotted. Meanwhile analyzing the contours, some valuable engineering conclusions are obtained.Then, the corresponding functions with undetermined constant are constructed for an infinite transversely isotropic pyroelectric material under a sudden steady point heat source. Using the correlation properties of the Heaviside Step function, to describe the point heat source change with time.The fundamental solution can be obtained by substituting these functions into the obtained general solution. And the constants can be determined by the Fourier’s law of heat conduction. Based on the obtained fundamental solution for the sudden steady point heat source, the contours in different times for the temperature increment and all components of stress and electric displacement components in the infinite transversely isotropic pyroelectric body are plotted. Meanwhile analyzing the contours, some valuable engineering conclusions are obtained.Finally, the corresponding functions with undetermined constant are constructed for an infinite transversely isotropic pyroelectric material under a harmonic point heat source. The fundamental solution could be obtained by substituting these functions into the obtained general solution, and the constants can be determined by the law of heat conservation and the inverse Laplace transform. All components for coupling field harmonic point heat source are dimensionless, on the basis of the obtained fundamental solution for harmonic point heat source, the contours in different times for the temperature increment and all components of stress and electric displacement components in the infinite transversely isotropic pyroelectric body are plotted. Meanwhile analyzing the contours, some valuable engineering conclusions are obtained.