A comprehensive study on the 2D boundary element method for anisotropic thermoelectroelastic solids with cracks and thin inhomogeneities Iaroslav Pasternak a,n , Roman Pasternak a , Heorhiy Sulym b a Lutsk National Technical University, Lvivska Str. 75, 43018 Lutsk, Ukraine b Bialystok Technical University, Wiejska Str. 45C, 15-351 Bialystok, Poland article info Article history: Received 22 May 2012 Accepted 5 November 2012 Keywords: Pyroelectric Boundary integral equation Stress and electric displacement intensity factors Thin inclusion Crack abstract This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displace- ment in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results. & 2012 Published by Elsevier Ltd. 1. Introduction Modern high-tech devices often include parts produced from piezoelectric materials, which are used as sensor, precision positioning tools, transducers, etc. The highest electromechanical coupling is possessed by ferroelectric materials, which in turn are pyroelectrics, i.e. materials, which polarize with a temperature change. The pyroelectric phenomenon is widely used, in particu- lar, in infrared radiation sensors [1]. In addition, these effects are utilized in the design of modern smart materials. Due to the thermoelectric and thermomechanical effects, the presence of cracks and inhomogeneities cause high concentration of stress and electric displacement at their tips. Thus, the pyroelectric phenomenon should be accounted for in the analysis of piezo- electric solids. A wide range of publications in the scientific literature pro- vides a study of thermal stress and electric displacement in thermoelectroelastic materials near inclusions, holes and cracks. In particular, Lu et al. [2] obtained a solution for an elliptic hole in piezoelectric medium under a uniform heat flux. Liu et al. [3] studied thermal stress and electric displacement at an elliptic inclusion or a hole in an infinite pyroelectric medium. Gao and Wang [4] and Gao et al. [5] determined stress and electric displacement intensity factors for periodic cracks in thermoelec- troelastic and thermomagnetoelectroelastic media. Kaloerov and Khoroshev [6,7] obtained Lekhnitskii type complex potentials of thermoelectroelasticity and solved a number of problems for solids with holes and cracks using the expansions of these complex potentials into power series, with unknown coefficients determined from the boundary conditions. Qin [8,9] obtained Green’s function for pyroelectric and thermomagnetoelectro- elastic materials with holes and cracks. Hou [10] derived a 2D fundamental solution for orthotropic pyroelectric plane and half- plane. The majority of papers, which study infinite thermoelec- troelastic media, uses the assumption that the remote uniform steady-state heat flux does not cause stress and electric displace- ment in the unnotched pyroelectric medium. However, this assumption was not sufficiently proved, and in some cases of thermal load it can be invalid. The above-mentioned papers generally use analytical or semi- analytical approaches for studying notched pyroelectric solids, which limits their application mainly to solids of a canonical shape. One can overcome these limitations using numerical methods, in particular, the boundary element method (BEM), which possesses high accuracy due to its semi-analytic nature and requires only boundary mesh. Qin [11,12] and Qin et al. [13,14] were the first, who obtained the BEM for anisotropic Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2012 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.enganabound.2012.11.002 n Corresponding author. Tel.: þ380 97 301 68 19. E-mail addresses: pasternak@ukrpost.ua (I. Pasternak), h.sulym@pb.edu.pl (H. Sulym). Engineering Analysis with Boundary Elements 37 (2013) 419–433