By Hankel integral transform, the problem is reduced to a set of dual integral equations which are transformed into a set of singular integral equations.

The exact analytic solution of a Mode Ⅲ Griffith crack in the material was obtained by using the Fourier transform and dual integral equations theory, and so the displacement and stress fields, the stress intensity factor and strain energy release rate were determined.

by using the Fourier integral transform and the boundary conditions, the problem is reduced to a dual integral equations. The dynamic stress intensity factors at the crack tip are obtained by the using Copson methods and the numerical integral technique. As an example, the eects of the parameter and the frequency of SH wave on norm dynamic stress intensity factors are discussed.

：This paper studies the propagation of crack problem in dynamic fracture mechanics, and obtains the analytical expressions of the stress and displacement fields and dynamic stress intensity factor by using Fourier analysis and theory of dual integral equation.

The mixed boundary value problem is reduced to a dual integral equation by means of nonlocal linear elasticity theory and integral transform method The stress field and displacement field for an infinite strip of FGM are solved near the tip of a crack by using Schmidt’s method.

The mixed boundary value problem is reduced to a dual integral equation by means of nonlocal linear elasticity theory and Fourier integral transform method. The stress field and displacement field for an infinite strip of FGM are solved near the tip of a crack by using Schmidt's method.

By using the Fourier transform,the problem can be solved with the help of a pair of dual integral equations that the unknown variable is the jump of the displacement across the crack surfaces.

Fourier transforms are used to reduce the problem to a pair of dual integral equations, which are then expressed in terms of a Fredholm integral equation of the second kind. The dynamic stress intensity factor is determined.

Based on the Biot’s dynamic equations, the vertical vibration of a rigid strip foundation resting on partially saturated soil subgrade which is composed of a dry elastic layer and a saturated substratum is studied. The analysis relied on the use of Fourier integral transform techniques and a pair of dual integral equations governing the vertical vibration of the rigid foundation is listed under the consideration of mixed boundary value condition and the continuity condition.

Based on the theory of elastic wave propagation in saturated soil subgrade established by the author of this paper, the axisymmetric vertical vibration of a rigid circular foundation resting on partially saturated soil subgrade which is composed of a dry elastic layer and a saturated substratum is studied. The analysis relied on the use of integral transform techniques and a pair of dual integral equations governing the vertical vibration of the rigid foundation is listed under the consideration of mixed boundary-value condition.

At first, the governing differential equations are solved by Fourier transform, then, under consideration of the mixed boundary value condition, a pair of dual integral equations about the vertical vibration are listed which are converted to linear algebra equations by the Jacobi orthogonal polynomial and solved by numerical procedures. Consequently, the dynamic compliance coefficient Cv versus the dimensionless frequency is derived, and the program is compiled.

This study gives the computation of some typical dynamic stress intensity factors of mode I, II and III for two or three dimensional cracks under transient loading, making use of integral transforms and dual integral epuations.

By using the Copson method, one of these equations is transformed into a Fredholm integral equation of the second kind to obtain the solution. Furthermore, the intensity factors, the total energy release rate and the mechanical strain energy release rate are obtained and compared with the classical results without considering the electric field gradient effects.

We get the displacement, stress, electric potential and electric displacement around the crack in an infinite plane of functionally graded piezoelectric materials, a half plane of functionally graded piezoelectric materials and a strip of functionally graded piezoelectric materials by Fourier Transforms, respectively.

By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable was the displacement on the crack surfaces.

To solve the dual integral equations, the displacement on the crack surfaces was expanded in a series of Jacobi polynomials.

Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables were the jumps of the displacements across the crack surfaces.

To solve the dual integral equations, the jumps of the displacements across the crack surfaces were expanded in a series of Jacobi polynomials.

This problem is reduced by means of Fourier transforms to the standard set of dual integral equations with two variables.

Thus, the analytical solution for the temperature with respect to the spatial variables reduces to a dual integral equation.

The transient response of a central crack in an orthotropic strip under the in-plane shear impact loading is studied by using the dual integral equation method proposed by Copson and Sih.

The results of this paper are very close to those given by the two-dimensional dual integral equation method.

The present problem can be solved by using the Fourier transform and the technique of dual integral equation, in which the unknown variable is the jump of displacement across the crack surfaces, not the dislocation density function.

A recently developed integral equation method has been used to solve the corresponding two dimensional simultaneous dual integral equation involving the displacement discontinuity across the crack faces that arises in such an interaction problem.

Thus, the analytical solution for the temperature with respect to the spatial variables reduces to a dual integral equation.

The transient response of a central crack in an orthotropic strip under the in-plane shear impact loading is studied by using the dual integral equation method proposed by Copson and Sih.

The results of this paper are very close to those given by the two-dimensional dual integral equation method.

The present problem can be solved by using the Fourier transform and the technique of dual integral equation, in which the unknown variable is the jump of displacement across the crack surfaces, not the dislocation density function.

A recently developed integral equation method has been used to solve the corresponding two dimensional simultaneous dual integral equation involving the displacement discontinuity across the crack faces that arises in such an interaction problem.

By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable was the displacement on the crack surfaces.

The analysis relied on the use of integral transform techniques and a pair of dual integral equations governing the vertical vibration of the rigid foundation is listed under the consideration of mixed boundary-value condition.

By using the Fourier transform, the problem can be solved with a pair of dual integral equations in which the unknown variable was the jump of the displacements across the crack surfaces.

Then, under the contact conditions, the problem leads to a pair of dual integral equations which describe the mixed boundary-value problem.

By using the Fourier transform, the problem can be solved with a pair of dual integral equations in which the unknown variable is the jumps of the displacements across the crack surfaces.

In this paper, the three-dimensional elastic solid with internal rectangular crack is considered. Let the crack surfaces be subjected to equal and opposite normal tractions p0. This problem is reduced, by means of Fourier transforms, to the standard set of dual integral equations with two variables. Then the fomulas of analytic solution of the displacements on the crack surfaces and of the stress-intensity factors of crack border are obtained.

In this paper, the problem of the plane circular crack in a homogenous and isotropic infinte elastic body under circumferential load at infinity parallel to the plane of the crack is studied within the linearized couple-stress theory by Mindlin and Tiersten. The problem of the crack is reduced to a simultaneus system of dual integral equation by means of the method of integral transforms.This system, in turn,is reduced to a one-dimensional integral equation of Fredholm's second kind that is amenable to a numerical...

In this paper, the problem of the plane circular crack in a homogenous and isotropic infinte elastic body under circumferential load at infinity parallel to the plane of the crack is studied within the linearized couple-stress theory by Mindlin and Tiersten. The problem of the crack is reduced to a simultaneus system of dual integral equation by means of the method of integral transforms.This system, in turn,is reduced to a one-dimensional integral equation of Fredholm's second kind that is amenable to a numerical treatment. Stress singularities around the edge of the crack are considered, and the stress intensity factor is calculated. The results obtained show that the order of the stress sigularities is the same as that of the classical stress singularities, but owing to the effect of the couple-stress, the magnitude of the stress factor is lower than the classical theory's and changes with parameters of the material, 1 and η.