Temperature and Displacement Discontinuity Boundary Element Method for Analysis of Cracks in Three-Dimensional Isotropic Thermoelastic Media

Abstract

For the analysis of cracks in three-dimensional isotropic thermoelastic media, a temperature and displacement discontinuity boundary element method is developed. The Green functions for unit-point temperature and displacement discontinuities are derived, and the temperature and displacement discontinuity boundary integral equations are obtained for an arbitrarily shaped planar crack. Our boundary element method is based on the Green functions for a triangular element. As an application, elliptical cracks are analyzed to validate the developed method. The influence of various thermal boundary conditions is studied. Keywords: boundary element method, boundary integral equation method, displacement and temperature discontinuity, Green function, isotropic thermoelastic medium, planar crack, stress intensity factor, thermal boundary condition, triangular element. INTRODUCTIONMuch research has been conducted in the analysis of cracks in thermoelastic materials using analytical or numerical methods [1][2][3][4]. Because solving complicated practical problems analytically is difficult, numerical methods are needed. Among the various numerical methods, the boundary element method is very convenient and efficient in such analyses, and many studies have adopted this method to analyze crack problems in thermoelastic media [2][3][4]. Furthermore, the subsequent displacement discontinuity boundary integral equation and boundary element method are more efficient in studying crack problems, as they grasp the basic characteristic of crack problems; specifically, fields are discontinuous across crack faces. The displacement discontinuity method was first proposed by Crouch [5] to study two-dimensional crack problems, and was then extended to three-dimensional elastic media, piezoelectric media, and magnetoelectroelastic media [6][7][8][9], where the electric and magnetic potential discontinuities were introduced across crack faces. Based on previous work, this paper develops the temperature and displacement discontinuity boundary integral equation and boundary element method for which the temperature distribution across the crack faces is assumed discontinuous. The Green functions for unit-point temperature and displacement discontinuities are derived, and the temperature and displacement discontinuity boundary integral equations are obtained for an arbitrarily shaped planar crack. The singular fields ahead of the crack front are discussed, and the stress intensity factors are obtained. For numerical simulations, the Green functions for a triangular element are obtained, and an elliptical crack is analyzed to validate the correctness of the analytical solution and the proposed numerical method. The influence of the temperature and different thermal conditions are also discussed.

Effects of thermal and electric boundary conditions on fracture of three-dimensional thermopiezoelectric media

“…Next, the singular behavior of the near-crack border fields is studied, and the ESIFs are derived using the method of analysis of Zhao et al (2016) for thermoelastic material. To begin, select an arbitrary point P on the edge of the crack and establish a local Cartesian coordinate system oxyz (Figure 3) with the x -axis perpendicular to the crack border and directed toward the interior of the crack, the y -axis tangential to the crack edge, and the z -axis normal to the crack face.…”

“…Consider a neighborhood Σ of point P , which is part of the penny-shaped area of infinitesimal radius ε on the crack S (Figure 3). In neighborhood Σ, the EDDs are assumed to be (Zhao et al, 1997b, 2016)…”

“…By virtue of the approach taken by Zhao et al (2016) and the finite-part integrals (Ioakimidis, 1982), the singular indexes are obtained…”

“…If the integration area S ( e ) contains the field point ( x i , y i ), the integrals are divergent. In this case, the concept of Hadamard finite-part is used (Ioakimidis, 1982; Zhang et al 2014; Zhao et al 2016). Finally, closed-form expressions are obtained and proved to hold for both regular and divergent cases…”

“…However, thermal effects were not considered in these literatures. Recently, the displacement and temperature discontinuity boundary integral equation and BEM were developed for cracks in thermoelastic materials (Zhao et al, 2016).…”