Hybrid-Trefftz Equilibrium Model for Crack Problems

Abstract

A formulation based on the approximation of the stress field is used to compute directly the stress intensity factors in crack problems. The boundary displacements are independently approximated. In each finite element, the assumed stresses may model multipoint singularities of variable order. The differential equilibrium equations are locally satisfied as solutions of the governing differential system are used to build the stress approximation basis. The approximation on the boundary displacements is constrained to satisfy locally the kinematic boundary conditions. The remaining fundamental conditions, namely the differential compatibility equations, the constitutive relations and the static boundary conditions are enforced through weighted residual statements. The approximation criteria are so chosen as to ensure that the finite element model is described by a sparse, adaptive and symmetric governing system described by structural matrices with boundary integral expressions. Numerical applications are presented to show that accurate solutions can be obtained using structural discretizations based on coarse meshes of few but highly rich elements, each of which may have different geometries and alternative approximation laws.

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

“…The hybrid-displacement or hybrid-Trefftz method is an established finite element method for obtaining high coarse-mesh accuracy for singular stress problems in elastic materials (see, e.g., [2,4,7,12,18,21]). Recently, Qin [22] also adopted the method to solve the anti-plane crack problem in piezoelectrics.…”

“…The problem becomes more complicated if the singularity is not of the conventional 1/ √ r type. To this end, ad hoc finite element methods relying on the availability of the singular eigensolutions can be adopted for enhancing the solution accuracy (see, e.g., [2,4,7,12,18,21,22]). To the best knowledge of the authors, such methods have not been reported and attempted to solve singular electromechanical stress problems in plane piezoelectricity.…”

A Novel Hybrid Element Analysis for Piezoelectric-parent Material Wedges

“…Sound variational basis and high coarse mesh accuracy of crack-tip and wedge-tip hybrid elements for conventional materials have been given in a lot of literatures (see Chen and Sze 2001;Pian and Sumihara 1984;Lee and Gao 1995;Freitas and Ji 1996;Tong and Pian, 1973 …”

Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach

“…Planar crack problems have been examined by Sabino et al [28] using the Trefftz boundary element method and by Freitas and Ji [29] using an equilibrium element model. In the present paper, we confine our attention to the applications of Trefftz FE method to anti-plane electroelastic problems.…”

“…Up to now, T-elements have been successfully applied to problems of elasticity [2,3], Kirchhoff plates [4, 5], moderately thick Reissner-Mindlin plates [6-8], thick plates [9], general 3-D solid mechanics [10, 11], axisymmetric solid mechanics [12], potential problems [13, 14], shells [15], elastodynamic problems [16-18], transient heat conduction analysis [19], geometrically nonlinear plates [20-23] and materially nonlinear elasticity [24,25]. Further, the concept of special purpose functions has been found to be of great efficiency in dealing with various geometry or load-dependent singularities and local effects (e.g., obtuse or reentrant corners, cracks, circular or elliptic holes, concentrated or patch loads) [2-4, 26, 27].Planar crack problems have been examined by Sabino et al [28] using the Trefftz boundary element method and by Freitas and Ji [29] using an equilibrium element model. In the present paper, we confine our attention to the applications of Trefftz FE method to anti-plane electroelastic problems.…”