Vol. 9, No. 3, 2014

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ISSN: 1559-3959
Multi-region Trefftz collocation grains (MTCGs) for modeling piezoelectric composite and porous materials in direct and inverse problems

Peter L. Bishay, Abdullah Alotaibi and Satya N. Atluri

Vol. 9 (2014), No. 3, 287–312
Abstract

A simple and efficient method for modeling piezoelectric composite and porous materials to solve direct and inverse 2D problems is presented in this paper. The method is based on discretizing the problem domain into arbitrary polygonal-shaped regions that resemble the physical shapes of grains in piezoelectric polycrystalline materials, and utilizing the Trefftz solution functions derived from the Lekhnitskii formulation for piezoelectric materials, or for elastic dielectric materials, to express the mechanical and electrical fields in the interior of each grain or region. A simple collocation method is used to enforce the continuity of the inter-region primary and secondary fields, as well as the essential and natural boundary conditions. Each region may contain a void, an elastic dielectric inclusion, or a piezoelectric inclusion. The void/inclusion interface conditions are enforced using the collocation method, or using the special solution set which is available only for the case of voids (traction-free, charge-free boundary conditions). The potential functions are written in terms of Laurent series which can describe interior or exterior domains, while the negative exponents are used only in the latter case. Because Lekhnitskii’s solution for piezoelectric materials breaks down if there is no coupling between mechanical and electrical variables, the paper presents this solution in a general form that can be used for coupled (piezoelectric) as well as uncoupled (elastic dielectric) materials. Hence, the matrix or the inclusion can be piezoelectric or elastic dielectric to allow modeling of different types of piezoelectric composites. The present method can be used for determining the meso/macro physical properties of these materials as well as for studying the mechanics of damage initiation at the micro level in such materials. The inverse formulation can be used for determining the primary and secondary fields over some unreachable boundaries in piezoelectric composites and devices; this enables direct numerical simulation (DNS) and health monitoring of such composites and devices. Several examples are presented to show the efficiency of the method in modeling different piezoelectric composite and porous materials in different direct and inverse problems.

Keywords
piezoelectric, composites, porous, Trefftz, Lekhnitskii, Voronoi cells, void, inclusion, collocation, inverse problems
Milestones
Received: 7 December 2013
Revised: 10 April 2014
Accepted: 8 May 2014
Published: 4 October 2014
Authors
Peter L. Bishay
Center for Aerospace Research and Education (CARE)
The Henry Samueli School of Engineering
University of California, Irvine
4200 Engineering Gateway
Irvine, CA 92697
United States
The Hal and Inge Marcus School of Engineering
Saint Martin’s University
5000 Abbey Way SE, OM 329
Lacey, WA 98503-7500
United States
Abdullah Alotaibi
Department of Mathematics
Faculty of Sciences
King Abdulaziz University
P.O. Box 80203, Jeddah 21589
Saudi Arabia
Satya N. Atluri
Center for Aerospace Research and Education (CARE)
The Henry Samueli School of Engineering
University of California, Irvine
4200 Engineering Gateway
Irvine, CA 92697
United States
Faculty of Engineering
King Abdulaziz University
Jeddah 22254
Saudi Arabia