Two-dimensional Green’s Functions For Pyroelectric Ceramics

Fundamental solutions or Green’s functions play an important role in both appliedand theoretical studies on the physics of solids. They are foundations of a lot of furtherworks. For example, fundamental solutions can be used to construct many analyticalsolutions of practical problems when boundary conditions are imposed. They areessential in the boundary element method as well as the study of cracks, defects andinclusions.Concerning the Green’s functions and fundamental solutions studies ofpiezoelectric ceramics, many references only research electro–mechanical couplingbehavior in isothermal case. The Green’s functions and fundamental solutions forpiezoelectric ceramics with considering temperature effection (this kind of ceramics isnamed pyroelectric ceramics) are studied rarely. This is mainly because adding thermalcoupling equations breaks the fine properties of the pristine equations, which increasesthe troubles. However, for the smart structures and intelligent structure systemsconstructed with pyroelectric ceramics, their practical work environment universally isnot isothermal. This makes that the thermal coupling effects of pyroelectric ceramicscannot be avoided.Against this background, as the study object of the familiar pyroelectric ceramics,the two-dimensional Green’s functions are presented respectively for semi-infinite,infinite and two-phase pyroelectric ceramics which are loaded by steady point heatsource and point force.Firstly, the steady general solution of pyroelectric ceramics is described. Forvarious kinds of point load in an infinite pyroelectric ceramics, their fundamentalsolutions can be obtained by constructing four new harmonic functions withundetermined constants. The undetermined constants of the harmonic functions can beobtained by using the compact general solution and the corresponding equilibrium andcontinuity conditions.Two-dimensional Green’s functions for various kinds of point load on the surfaceand in the interior of a semi-infinite pyroelectric ceramics can be obtained byintroducing four new suitable harmonic functions with undetermined constants,respectively. The undetermined constants can also be obtained by using the compactgeneral solution and corresponding boundary, equilibrium and continuity conditions.According to the same method, the two-dimensional Green’s functions for variouskinds of point load in two-phase pyroelectric ceramics is presented by introducing eight new harmonic functions with undetermined constants. The undeterminedconstants can also be obtained by using the compact general solution and consideringthe corresponding boundary, continuity and equilibrium conditions.Finally, numerical results of corresponding coupled fields are given graphicallyby contours, including temperature increment, stress, electric displacement induction.Meanwhile the contours are analyzed, and some valuable engineering conclusions areobtained by analyzing contours.