Thermo-elastic singularities of V-notches in orthotropic materials are studied. By introducing series asymptotic expansions of physics fields near notch tip, stress and flux equilibrium equations are transformed into characteristic ordinary differential equations with respect to singularity orders. Singularity orders and corresponding characteristic angular functions can be derived synchronously as interpolating matrix method is introduced to solve characteristice quations. The method evaluates stress and flux singularity orders at the same time. Numerical results show that it has high accuracy and strong adaptability.

With asymptotic assumption for physical field near notch tip, characteristic differential equations for electroelastic singularities of wedges that contain bounded piezo/piezo materials are built from three-dimensional equilibrium equations and Maxell equations. Mechanical and electric boundary conditions are expressed by combination of singularity orders and characteristic angle functions. Thus, evaluation of singularity orders is transformed into solving ordinary differential equations (ODEs) under designated boundary conditions. Interpolating matrix method is introduced to solve derivative ODEs. More electroelastic singularity orders and associated eigenfunctions in wedges that comprise two bounded transverse isotropic piezoelectrics materials are obtained. It shows that the method is efficient and has high accuracy compared with existent solutions.

Boundary element method (BEM) is used to study propagation of crack under loading. First, all leading unknown coefficients in Williams series expansion and complete stress field for notched structures are calculated with BEM and former eigenanalysis for notch tip region. Then, with consideration of non-singular stress term and maximum circumferential stress criterion of brittle fracture, crack initiation extended direction from crack tip in a semicircular bending specimen is determined by BEM. Strategy for BEM tracking crack propagation is given. Numerical examples show that the method is correct and effective in simulating propagation of plane crack.

Based on asymptotic extension of displacement field at a composite notch tip, equilibrium equation for a notch subjected to anti-plane loading is transformed into a characteristic differential equation respects to notch singularity orders. A transformation is applied to convert the equation into a set of characteristic linear ordinary differential equations. Interpolate matrix method is introduced to solve the equations for getting notch singularity orders. A single material notch, a bi-material notch and a notch terminated at bimaterial interface are studied successively. Examples indicate that the method provides all stress singularity orders synchronously. Though singular stress state is not shown with regard to non-singular orders, non-singular stress orders are indispensable parameters as evaluating complete stress field at notch tip region.

An algorithm is developed to calculate stresses at the interior points near boundary in boundary element method of thermoelasticity. A series of transformations is manipulated on conventional derivative boundary integral equations(BIE). It leads to a new natural BIE in thermoelasticity problem named thermal stress natural boundary integral equation(NBIE). Hyper-singularity and strong-singularity in conventional BIE are converted into strong-singularity in NBIE. Nearly strong singular integrals are evaluated in NBIE by the regularization algorithm. Thermal stresses at points near boundary are calculated by NBIE. Numerical examples illustrate efficiency of the method.