Analysis of Electromechanical Stress Singularity in Piezoelectrics by Computed Eigensolutions and Hybrid-trefftz Finite Element Models

Abstract

This paper concerns the determination of singular electromechanical stress field in piezoelectrics. A one-dimensional finite element procedure is generalized to compute the eigensolutions of the singular electromechanical field. The generalized procedure is capable of taking differently poled piezoelectrics, cracks and ultra-thin electrodes into account.To determine the strength of the singular electromechanical stress field, the hybrid-Trefftz finite element method is adopted. The independently assumed electromechanical stress modes are extracted from the eigensolutions previously computed from the one-dimensional procedure. Since the eigensolutions satisfy all the balance conditions, the hybridTrefftz models can be constructed by boundary integration. This feature enables the models to be interfaced compatibly with conventional finite element models. To illustrate the efficacy of the present approach, the eigensolutions and/or parameters directly related to the electromechanical stress intensity in crack, interfacial crack and bimorph with embedded electrode are considered. The predictions are in good agreement with the reference solutions reported in the literature or computed by using over 10,000 conventional finite elements.

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

“…In designing hybrid elements for stress/structural analyses [1][2][3][4][5]10,30,31], the two common guidelines on the selection of the stress (domain) modes are that the element should be rank sufficient and invariant. Rank sufficiency is guided by (a) (b) Fig.…”

“…Nevertheless, if the element size is very small compared with its distance from the coordinate origin, numerical error may arise. Therefore, it is a common practice to use a point (x c , y c ) local to the element to define ( x, y) which are in turn used in defining the domain modes [1][2][3][4][5]10,22,23,25,27]. …”

“…additional field variable(s) to the displacement obtained by polynomial-based nodal interpolation [1][2][3][4][5][6][7][8][9][10]. In the hybriddisplacement method, the additional field is a domain displacement u d which leads to equilibrating stress and may also satisfy some homogeneous boundary conditions [2][3][4].…”

Four- and eight-node hybrid-Trefftz quadrilateral finite element models for helmholtz problem

“…In the hybrid finite element method for stress/structural analyses, the displacement-based finite element models are enhanced by introducing stress, strain or another displacement as the additional field variable(s) to the displacement obtained by nodal interpolation [1][2][3][4][5][6][7][8][9][10]. In the case of the hybriddisplacement method, the additional field is a domain displacement which leads to equilibrating stress and may also satisfy some homogeneous boundary conditions [2,3,10]. This category of hybrid elements are also known as hybrid-Trefftz or Trefftz elements linked by the displacement-frame or the boundary displacement [4,[6][7][8][9][10].…”

A hybrid‐Trefftz finite element model for Helmholtz problem

“…Moreover, special elements can be designed for stress concentration problems by using the eigen-solutions, which satisfy not only the stress equilibrium but also the boundary conditions [2,3,5]. These special elements are sometimes termed as hybrid-Trefftz or Trefftz elements linked by the inter-element boundary displacement [7][8][9][10][11]. The underlying reason is that the eigen-solution set is also known as the 2049 the numerical examples that the present hybrid element model is markedly more accurate than the conventional model.…”