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Using the method of the boundary integral equation, a set of singular integral equations of the heat transfer problems and the thermo-elastic problems of a crack embedded in a two-dimensional finite body is derived, and then its numerical method is proposed by the numerical method of the singular integral equations combined with boundary element method. Moreover, the singular nature of temperature gradient field near the crack front is proved by the main-part analysis method of the singular integral equation, and the singular temperature gradients are exactly obtained. Finally, several typical examples are calculated.It is known that most agricultural products and foods are processed and transported under certain temperature conditions, and the structural components also work under a thermal environment. Temperature induced stresses usually lead to damage of flawed solids. Thus, the investigation of the crack problems in thermally stressed planar solids is very important.As early as in 1946, Motz [1] analyzed the Laplace equation in a square plate with a discontinuous slit, and found that there is a l/veT singularity of gradient near the crack tip. In the literature, the crack problems in a finite thermally stressed solid have been studied by the ffmite element method [~-] and the boundary element alternating technique [3] . Besides, there are two kinds of integral equation methods developed from the boundary integral equation method in fracture mechanics, i.e. singular integral equation method and hypersingular integral equation methodC4-6]. The general advantages of these approaches are that the crack problems can be conveniently treated through numerical integration over the crack surface and get high precision solutions. Although these two approaches have been widely used in fracture mechanics, there are few works on crack problems using singular integral equation method or hypersingular integral

Atime domain numericalmethod is proposed to study the dynamic interaction of multiple planar cracks in a piezoelectric solid. Arbitrary oriented planar cracks, which are acted upon by a transient load, in an infinite piezoelectric space is considered. By using the reciprocity theorem and the technique of integral by part, a time domain singular integral equations are obtained. Convolution quadrature method are applied to discrete the time convolution and Gauss Chebyshev method are applied to approximate the singular spatial integral. Numerical examples are carried out to examine the accuracy of the proposed method. And numerical results show the dynamic interaction of the cracks under impact loading.

Stress intensity factors for a three dimensional rectangular interfacial crack were considered using the body force method. In the numerical calculations, unknown body force densities were approximated by the products of the fundamental densities and power series; here the fundamental densities are chosen to express singular stress fields due to an interface crack exactly. The calculation shows that the numerical results are satisfied. The stress intensity factors for a rectangular interface crack were indicated accurately with the varying aspect ratio, and bimaterial parameter.