Thermoelastic fracture modeling is an indispensable field of research due to its relevance in numerous engineering applications. In thermoelastic problems, coupling temperature and displacement fields leads to thermal stress, which can become singular in case of fracture. Special purpose materials, such as functionally graded materials (FGMs), are engineered to counter the effects of high thermal stress. In most real-world applications, numerical methods such as the standard finite element method (FEM) have been employed to analyze thermoelastic fracture for decades. However, it requires special treatments to model the singular stress due to its polynomial base interpolation functions. Plenty of other numerical methods have been proposed to overcome the limitations of the standard finite element method, yet thermoelastic fracture modeling is still an active field of research.

This thesis develops a numerical technique to model thermoelastic fracture using the scaled boundary finite element method (SBFEM). The SBFEM is a semi-analytical method in which only the boundary of the computational domain is discretized, and an analytical solution is constructed in the radial direction. In a simulation based on SBFEM, the domain is divided into arbitrary-sided star-convex polygons that can represent cracks naturally using open polygon elements. Polygon elements also facilitate the meshing and re-meshing of complex domains in the case of discrete crack propagation modeling. Contrary to the polynomial base interpolation functions of the FEM, the semi-analytical discretization of the SBFEM leads to power functions of its radial coordinate as the interpolation functions. One of the key features of the SBFEM solution is the direct representation of singular stress without the need for additional post-processing. With the capabilities to mesh and re-mesh complex geometries and the built-in capacity to capture singular stress, the SBFEM is an attractive choice to model fracture.

The work presented in this thesis first outlines the geometric transformation of a computational domain from Cartesian to the scaled boundary coordinates. Then, the SBFEM modeling of Laplace's equation and linear elasticity problem is discussed. Next, the displacement due to a known temperature field is obtained semi-analytically for elastostatics, assuming uni-directional coupling of the temperature and displacement fields. A temperature field leads to a load vector in the SBFE equation in displacement, which then becomes a non-homogeneous differential equation. The non-homogeneous term's particular solution is expressed as integrals in the radial direction, which are evaluated analytically for temperature changes varying as power functions of radial coordinate. Then, the supplementary scaled boundary shape functions are also obtained from the particular solution of the non-homogenous SBFE equation in displacement but without needing a prior solution of temperature field, thus paving the way for modeling fully coupled temperature and displacement fields. The equations of fully coupled thermoelasticity are then discretized using the SBFE shape functions enriched with supplementary shape functions. 

For the SBFEM modeling of FGMs, the spatial variation of material properties is approximated by polynomial functions. The novel semi-analytical integration of thermoelastic material coefficient matrices for FGMs is presented. Furthermore, the generalized stress intensity factors (SIFs) are evaluated using the direct and efficient SBFE approach to capture singular stress. A localized re-meshing algorithm using polygon elements is implemented to model discrete crack propagation problems. Physical quantities are mapped using the SBFEM shape functions, and a novel mapping procedure is presented for the auxiliary variables corresponding to supplementary shape functions. Finally, the developed SBFEM for coupled thermoelastic fracture is validated through various benchmark examples. Uni-directional and full coupling cases are considered, and examples of thermoelastic fracture in standard and functionally graded materials are addressed. Moreover, the modeling of discrete crack propagation problems in uni-directional and full coupling is verified.