Three-Dimensional Piezoelectric Boundary Element Method

The boundary element method is applied to problems of three-dimensional linear piezoelectricity. The continuum equations for conservation of linear momentum and charge are combined into one governing equation for piezoelectricity. A single boundary integral equation is developed from this combined e eld equation and Green’ s solution for a piezoelectric medium. Green’ s function and its derivatives are derived using the radon transform, and the resulting solution is represented by a line integral that is evaluated numerically using standard Gaussian quadrature. The boundary integral equation is discretized using eight-node isoparametric quadratic elements, resulting in a matrixsystem ofequations. Thesolution of theboundary problem forpiezoelectricmaterialsconsistsof elastic displacements, tractions, electric potentials, and normal charge e ux densities. The complete e eld solutions can be obtained once all boundary values have been determined. The accuracy of this linear piezoelectric boundary element method is illustrated with two numerical examples. The e rst involves a unit cube of material with an applied mechanical load. Thesecond exampleconsistsof a spherical holein an ine nitepiezoelectricbody loaded by a unit traction on itsboundary. Comparisonsare made to the analytical solution for the cube and an axisymmetric e nite element solution for the spherical hole. The boundary element method is shown to be an accurate solution procedure for general three-dimensional linear piezoelectric material problems. I. Introduction T HEbrothersPierreandJacquesCuriediscoveredpiezoelectricity in 1880 during their investigation of crystals. A coupling occurs between the electrical and mechanical e elds in these materials, such that application of an external mechanical loading induces achangeinpolarizationproportionaltoit.Conversely,applicationof an electric e eld produces a proportional material deformation. This phenomenon was discovered in ceramics in the 1940s, and the subsequent manufactured piezoceramic materialshave higher coupling properties than their naturally occurring predecessors. This strong piezoelectric effect has made these materials an important part of the emerging technologies of smart materials and structures. Smart structures are those that incorporate actuators and sensors that are integrated into the structure and have structural functionality. The sensors and actuators are used to ine uence the structure’ s states