The State-space Solution On The Thermal-mechanical-electrical Coupling Problem Of Layered Piezoelectric Structure

The State-space technique is an effective method to solve problems of the layeredstructure, because of the state variable matrix selected, which can be simply associated withthe boundary conditions both on the upper and under. With the popularity of computer, theimprovement of speed and reduce programming difficulty, the state variable method is appliedmore and more in numerical analysis of layered structure. In previous research by applyingthe method of state variables in the process of the layered piezoelectric, Theelectro-mechanical coupling effect is often considered by researchers, however theenvironment temperature change seldom considered. In fact, the work environment ofPiezoelectric smart structure is often in a certain temperature，and, at the same time the workproject will inevitably produce heat, in order to guarantee the analysis process can reflectmore true behavior of the piezoelectric structure, thermal-mechanical-electric field couplingeffect should be considered.Considering the factors on the basis of electro-mechanical coupling coupling equationsof piezoelectric, the state variable equation under cartesian coordinate system is establishedby using energy theory derived transversely isotropic piezoelectric layer Thermal-mechanical-electric coupling of the constitutive equations, the very coefficient equations aretransformed into a linear constant coefficient homogeneous state equation of state by theLaplace transform, then we can get the analytical solution to pass transversely isotropicpiezoelectric single model matrix and the initial state variable matrix in Laplace transformspace. Analytical expressions are given under the condition of value of four different possibleby using the Cayley-Hamilton theorem.On the basis of state equation of thermal-mechanical-electric coupling transverseisotropic single piezoelectric layer, assuming the complete continuity of the state variablesbetween the layers, analyzes the problem of solving the finite thickness of layeredpiezoelectric system and semi-infinite piezoelectric system. This paper presents the solvingprocess detailed of the transition matrix method and the layer element analytic method, whichused in layered piezoelectric layer system thermal-mechanical-electric coupling problem:the transfer matrix method calculates the transfer matrix of each layer at first, and thenpresents the relation between the stress, electric displacement; Layer element analyticalmethod is the construction of the relationship between the matrix composed of stress,potential and temperature composition matrix and displacement, electric potential and thethermal field by solving the single layer element stiffness matrix, finally gets the connectionof expression by total stiffness matrix of laminated piezoelectric system, after the process ofoverlaying the layer’s stiffness matrix of dislocation. In addition, this article also discussed the domain surface load of the expressions of theinitial boundary conditions in Laplace transform. It is convenient for applicating the thesolution of the layered piezoelectric system of this paper in engineering. In the end, throughthree layers of piezoelectric structure examples verify the feasibility of theory and analysesthe calculation of this paper are correct.