Pair Of Dual Integral Equations Research Articles

Hypersingular integral equations over a disc: Convergence of a spectral method and connection with Tranter’s method

Two-dimensional hypersingular equations over a disc are considered. A spectral method is developed, using Fourier series in the azimuthal direction and orthogonal polynomials in the radial direction. The method is proved to be convergent. Then, Tranter’s method is discussed. This method was devised in the 1950s to solve certain pairs of dual integral equations. It is shown that this method is also convergent because it leads to the same algebraic system as the spectral method.

Basic solution to four three-dimensional rectangular limited-permeable cracks in transversely isotropic magneto-electro-elastic material

The solution to four three-dimensional rectangular cracks in magneto-electro-elastic material is proposed by using the generalized Almansi’s theorem and the Schmidt method under limited-permeable boundary conditions. The problem is formulated through Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. For solving the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the effects of the electric permittivity, the magnetic permeability of the air inside the crack, the geometric shape of the rectangular crack and the distance among four rectangular cracks on the stress intensity factors, the electric displacement intensity factors and the magnetic flux intensity factors in magneto-electro-elastic material are presented and the magneto-electro-elastic coupling effects are also discussed.

The basic solution of a 3-d rectangular permeable crack in a piezoelectric/piezomagnetic composite material

The solution of a 3-D rectangular permeable crack in a piezoelectric/piezomagnetic composite material was investigated by using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relations between the electric filed, the magnetic flux field and the stress field near the crack edges were obtained and the effects of the shape of the rectangular crack on the stress, the electric displacement and magnetic flux intensity factors in a piezoelectric/piezomagnetic composite material were analyzed.

Intervallic Coiflets for Numerical Calculation of Dynamic Stress Intensity Factor

There are lots of practical problems which are related to the solution of Fredholm integral equations of the second kind. The present work proposes intervallic Coiflets for solving the equations. Illustrative problem involving dynamic stress and electric fields of a cracked piezoelectric excited by anti-plane shear wave is addressed. Permeable boundary condition has been used to obtain a pair of dual integral equations of the symmetric and antisymmetric parts which can be reduced to the solutions of two Fredholm integral equations of the second kind. The dynamic stress intensity factor is expressed in terms of the right-end values of two unknown functions in Fredholm integral equations. The two unknown functions are solved by intervallic Coiflets which have less the endpoints error. And intervallic Coiflets have low calculation cost and high accuracy due to the wavelet expansion coefficients are exactly obtained without calculating the wavelet integrations. The calculation results agree well with the existing method, which show the high accuracy of the estimation and demonstrate validity and applicability of the method.

Non-local theory solution to two collinear limited-permeable mode-I cracks in a piezoelectric/piezomagnetic material plane

The non-local theory solution to two collinear limited-permeable mode-I cracks in a piezoelectric/piezomagnetic medium was investigated by using the generalized Almansi’s theorem and the Schmidt method in the present paper. The problem was formulated through Fourier transformation into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. For solving the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length, the distance between the two collinear cracks, the lattice parameter, the electric permittivity and the magnetic permeability of the air inside the crack on the stress fields, the electric displacement fields and the magnetic flux fields near the crack tips in a piezoelectric/piezomagnetic medium. Different from the classical solutions, the present solution exhibits no stress, electric displacement and magnetic flux singularities at the crack tips in a piezoelectric/piezomagnetic medium.

Exact Solution for an Anti-Plane Interface Crack between Two Dissimilar Magneto-Electro-Elastic Half-Spaces

This paper investigated the fracture behaviour of a piezo-electro-magneto-elastic medium subjected to electro-magneto-mechanical loads. The bimaterial medium contains a crack which lies at interface and is parallel to their poling direction. Fourier transform technique is used to reduce the problem to three pairs of dual integral equations. These equations are solved exactly. The semipermeable crack-face magneto-electric boundary conditions are utilized. Field intensity factors of stress, electric displacement, magnetic induction, cracks displacement, electric and magnetic potentials, and the energy release rate are determined. The electric displacement and magnetic induction of crack interior are discussed. Obtained results indicate that the stress field and electric and magnetic fields near the crack tips exhibit square-root singularity.

Basic solution of four 3‐D rectangular limited‐permeable cracks in piezoelectric materials

AbstractThe solution of four 3‐D rectangular limited‐permeable cracks in piezoelectric materials were given by using the generalized Almansi's theorem and the Schmidt method. At the same time, the electric permittivity of the air inside the rectangular crack was considered. The problem was formulated through Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the effects of the electric permittivity of the air inside the rectangular crack, the shape of the rectangular crack and the distance among four rectangular cracks on the stress and electric displacement intensity factors in piezoelectric materials were analyzed.

Fracture of a finite piezoelectric layer with a penny-shaped crack

This paper studies a penny-shaped crack in a finite thickness piezoelectric material layer. The piezoelectric medium is subjected to a thermal flux on its top and bottom surfaces. Both thermally insulated crack and heated crack are considered. Numerical solution for the finite layer thickness is obtained through the solution of a pair of dual integral equations. The result reduces to the closed form solution when the thickness of the piezoelectric layer becomes infinite. Exact expressions for the stress and electric displacement at the crack border are given as a function of the stress intensity factor, which is determined by the applied thermal flux. This paper is useful for the reliability design of piezoelectric materials in thermal environments.

Added mass of a pair of discs

Irrotational flow of inviscid incompressible fluid around a pair of coaxial discs of zero thickness, moving along their common axis, is considered. The added mass of an accelerating disc is modified by the presence of a second, coaxial disc. It is reduced if both discs accelerate equally in the same direction along their common axis and increased if their accelerations are equal in magnitude but in opposite directions. Other accelerations can be treated as linear combinations of these two cases. Analytic predictions are obtained when the disc separation 2z0 is large compared with the disc radius a and also when the discs are close (z0≪a). Results for intermediate separations are obtained as numerical solutions of a pair of dual integral equations.

Moving anti-plane shear crack in a piezoelectric layer bonded to dissimilar elastic infinite spaces

The dynamic behavior of a moving Griffith crack under anti-plane shear in a piezoelectric layer bounded by two elastic infinite spaces is considered. The crack is vertical to the interfaces of the piezoelectric layer. By using Fourier transforms, the problem is reduced into two pairs of dual integral equations with cosine functions as kernels. The pairs of dual integral equations are reduced into two pairs of Fredholm integral equations of second kind. Solving the Fredholm integral equations analytically and numerically, the numerical results for stress intensity factors and electric intensity displacement factors at the edged of the crack and energy release rate are obtained.

Diffraction of antiplane shear waves by a finite crack in a piezoelectric material

AbstractThe problem of diffraction of antiplane shear waves by a crack of finite length under the permeable electric boundary conditions is investigated analytically. Using Fourier transforms the mixed boundary value problem is reduced to two pairs of dual integral equations. These dual integral equations are further reduced to a pair of Fredholm integral equations of the second kind. The iterative solutions of the Fredholm integral equations have been obtained for small values of the wave number. And analytical expressions for the dynamic stress intensity factor and electric displacement intensity factor are obtained. Finally the numerical results for dynamic stress intensity factor are displayed graphically.

Basic solutions of a 3-D rectangular limited-permeable crack or two 3-D rectangular limited-permeable cracks in piezoelectric materials

The solutions of a 3-D rectangular limited-permeable crack or two 3-D rectangular limited-permeable cracks in piezoelectric materials were given by using the generalized Almansi’s theorem and the Schmidt method. At the same time, the electric permittivity of the air inside the rectangular crack was considered. The problem was formulated through Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the effects of the electric permittivity of the air inside the rectangular crack,the shape of the rectangular crack and the distance between two rectangular cracks on the stress and electric displacement intensity factors in piezoelectric materials were analyzed.

Basic solutions of two parallel limited-permeable mode-I cracks in piezoelectric/piezomagnetic composite materials

The basic solutions of two parallel mode-I limited-permeable cracks in piezoelectric/piezomagnetic composite ma- terials were derived by using the generalized Almansi's theorem in this paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials to obtain the relations among the electric displacement intensity factors, the magnetic flux intensity factors and the stress intensity factors at the crack tips. From the solutions of the present paper, it was obtained that the effects of the electric and the magnetic boundary conditions on the electric displacement fields and the magnetic flux fields near the crack tips are large. The paper presents the interactions of two or four parallel mode-I cracks and the crack shielding effect in piezoelectric/piezomagnetic composite materials.

Non‐local theory solution of a limited‐permeable mode‐I crack in a piezoelectric/piezomagnetic material plane

AbstractThe non‐local theory solution of a limited‐permeable mode‐I crack in a piezoelectric/piezomagnetic media was given by using the generalized Almansi's theorem and the Schmidt method in the present paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the lattice parameter, the crack length, the electric permittivity, and the magnetic permeability of the air inside the crack on the stress field, the electric displacement field, and the magnetic field near the crack tip. Different from the classical solution that the present solution exhibits no stress, electric displacement, and magnetic flux singularities at the crack tips in a piezoelectric/piezomagnetic media.

Dynamic fracture of a rectangular limited-permeable crack in magneto-electro-elastic media under a time-harmonic elastic P-wave

The dynamic behavior of a rectangular limited-permeable crack deeply embedded in transversely isotropic magneto-electro-elastic media was investigated under an incident harmonic stress wave using the generalized Almansi’s theorem and the Schmidt method. With the help of the Fourier transform, this problem was formulated into three pairs of dual integral equations with the jumps in displacement across the crack surfaces as the unknown variables. By directly expanding the jumps in displacement across the crack surfaces into a series of Jacobi polynomials, the solution to the dual integral equations was derived. Finally, the dynamic response of a rectangular crack under a harmonic wave was analyzed, and the effects of the electric permittivity and the magnetic permeability of air inside the crack, the geometric shape of the rectangular crack and the characteristics of the harmonic wave on the stress, the electric displacement and the magnetic flux intensity factors in magneto-electro-elastic media were concluded.

A 3-D rectangular permeable crack or two 3-D rectangular permeable cracks in a piezoelectric material

The solutions of a 3-D rectangular permeable crack and two 3-D rectangular permeable cracks in a piezoelectric material were investigated by using the generalized Almansi’s theorem and the Schmidt method. The problem was formulated through Fourier transform into three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the effects of the shape of the rectangular crack and the distance between two rectangular cracks on the stress and electric displacement intensity factors in a piezoelectric material were analyzed. These results can be used for the strength and the coupling effect evaluation of cracked piezoelectric materials.

Non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane

The non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane was investigated using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the jumps in displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of crack length, the distance between two collinear cracks and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement or magnetic flux singularities are present at the crack tips in a magnetoelectroelastic composite material plane. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.

Dynamic stress intensity factors for two parallel interface cracks between a nonhomogeneous bonding layer and two dissimilar elastic half-planes subject to an impact load

Some composite materials are constructed of two dissimilar half-planes bonded by a nonhomogeneous elastic layer. In the present study, a crack is situated at the interface between the upper half-plane and the bonding layer of such a material, and another crack is located at the interface between the lower half-plane and the bonding layer. The material properties of the bonding layer vary continuously from those of the lower half-plane to those of the upper half-plane. Incoming shock stress waves impinge upon the two interface cracks normal to their surfaces. Fourier transformations were used to reduce the boundary conditions for the cracks to two pairs of dual integral equations in the Laplace domain. To solve these equations, the differences in the crack surface displacements were expanded in a series of functions that are zero-valued outside the cracks. The unknown coefficients in the series were solved using the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors were defined in the Laplace domain and were inverted numerically to physical space. Dynamic stress intensity factors were calculated numerically for selected crack configurations.

Non-local theory solution of a mode-I crack in a piezoelectric/piezomagnetic composite material plane

The non-local theory solution of a mode-I permeable crack in a piezoelectric/piezomagnetic composite material plane was given by using the generalized Almansi’s theorem and the Schmidt method in this paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement and magnetic flux singularities are present at the crack tips in piezoelectric/piezomagnetic composite materials. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.

Dynamic stress intensity factors for two parallel cracks in an infinite orthotropic plate subject to an impact load

Stresses are solved for two parallel cracks in an infinite orthotropic plate during passage of incoming shock stress waves normal to their surfaces. Fourier transformations were used to reduce the boundary conditions with respect to the cracks to two pairs of dual integral equations in the Laplace domain. To solve these equations, the differences in the crack surface displacements were expanded to a series of functions that are zero outside the cracks. The unknown coefficients in the series were solved using the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors were defined in the Laplace domain and were inverted numerically to physical space. Dynamic stress intensity factors were calculated numerically for selected crack configurations.