Boundary Integral Equation Formulations for Piezoelectric Solids Containing Thin Shapes

A weakly-singular form of the piezoelectric boundary integral equation (BIE) formulation is introduced in this paper, which eliminates the calculation of any singular integrals in the piezoelectric BIE. The crucial question of whether or not the piezoelectric BIE will degenerate when applied to crack and thin shell-like problems is discussed. It can be shown analytically that the conventional BIE for piezoelectricity does degenerate for crack problems, but does not degenerate for thin piezoelectric shells. The latter has significant implications in applications of the piezoelectric BIE to analyze thin piezoelectric films used widely as sensors and actuators. Implementation of the boundary element method (BEM) for the developed piezoelectric BIE in analyzing thin piezoelectric solids, including treatment of the nearly-singular integrals existing in such cases, are discussed and two numerical examples are presented. Introduction Piezoelectric materials can be used as sensors and actuators in smart materials or microelectro-mechanical systems (MEMS), because they have many desirable properties (see, e.g., Ref. [1]). An electric current is induced in a piezoelectric solid when it is applied with a mechanical load. This property can be used to design sensors to monitor the deformation of a structure. Conversely, a mechanical deformation is produced when it is applied with a voltage. This property can be used to design actuators to control vibrations or noises of a structure. Simulations of piezoelectric solids, on the other hand, are very difficulty, because of the anisotropy in piezoelectric materials, coupling of elastic and electric fields, and thinness of the piezoelectric devices (the thickness of sensors/actuators is in the range of a few μm to a few hundred μm). Accurate 3-D modeling and analysis should be employed for such analysis. In 3-D analysis, the BIE/BEM approach has been demonstrated to be a viable alternative to the finite element method (FEM) for many problems, due to its features of surface-only discretization and high accuracy in stress and fracture analyses. Another advantage of the BIE/BEM, which was recognized only in recent years, is its high accuracy and efficiency in handling thin shell-like structures, layered structures (multicoatings or thin films), thin voids or open cracks [2-8]. It has been demonstrated that the BIE/BEM can handle the various thin-body problems very effectively, regardless of the thinness of the structures or voids, or non-uniform thickness, as long as the nearlysingular integrals are computed accurately [3, 4, 9, 10]. Much fewer boundary elements can be used to solve these problems for which the number of required finite elements is at least two-orders larger to achieve the same accuracy in stress analysis [3-6]. Considering the fact that the piezoelectric sensors and actuators are often made in thin shapes (films or patches), the BIE/BEM with thin body capabilities has the potential to provide a very efficient and accurate tool in the analysis of such piezoelectric materials. In this paper, the recent development in the BIE formulation for piezoelectric materials and the BEM implementation is summarized. The weakly-singular form of the piezoelectric BIE is introduced, which can eliminate the calculation of any singular integrals in the BEM. The crucial question of whether or not the piezoelectric BIE will degenerate when applied to crack and thin shell-like structures is discussed. It can be shown that the conventional BIE for piezoelectricity does degenerate for crack problems, but does not degenerate for shell-like structures, in the limit as the two opposing surfaces approaching each other. The latter has significant implications in the applications of the piezoelectric BIE to piezoelectric films used widely as sensors and actuators in smart materials and MEMS. How to deal with the nearly-singular integrals in the piezoelectric BIE when they are applied to thin shapes (thin voids, open cracks, or thin films and coatings) are discussed. Numerical tests to show the degeneracy of the piezoelectric BIE for crack problems, and the non-degeneracy and accuracy in analyzing thin piezoelectric solids using the developed piezoelectric BEM, are also presented. The Boundary Integral Equations for Piezoelectricity Consider a piezoelectric solid occupying domain V with boundary S. The basic equations governing the elastic and electric fields in a linear piezoelectric material can be written as follows (see, e.g., Refs. [11]) (index notation is used here). Equilibrium Equations: , 0 , = + i j ij f σ (1) , 0 , = − q D i i (2) where ij σ is the stress tensor, i f the body force vector per unit volume, i D the electric displacement vector and q the intrinsic electric charge per unit volume. Constitutive Equations: , k kij kl ijkl ij E e s C − = σ (converse effect) (3) , k ik kl ikl i E s e D ε + = (direct effect) (4) where kl s is the strain tensor, k E the electric field, ijkl C the elastic modulus tensor measured in a constant electric field, ijk e the piezoelectric tensor and ij ε the dielectric tensor measured at constant strains. Strain and Electric Fields: , ) ( 2 1 , , i j j i ij u u s + = (5)