Pair Of Triple Integral Equations Research Articles

Scattering of antiplane shear waves by two collinear coplanar rigid strips in a piezoelectric material

In this paper the dynamic interaction between two collinear rigid strips in a piezoelectric material under antiplane shear waves is investigated when the strips are metallic and grounded. By using the Fourier transforms , the solution of the problem is reduced to the solution of a pair of triple integral equations. The pair of triple integral equations are solved analytically. Analytic expressions for shear stress, mechanical shear intensity factors, electric displacement intensity factors, displacement component and electrical potential function are obtained for small wave numbers and the results for mechanical shear intensity factors. electric displacement intensity factors. electric potential and displacement component are shown graphically.

Two collinear Mode-I cracks in piezoelectric/piezomagnetic materials

In this paper, the behavior of two collinear Mode-I cracks in piezoelectric/piezomagnetic materials subjected to a uniform tension loading was investigated by the generalized Almansi's theorem. Through the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations, in which the unknown variables were the jumps of displacements across the crack surfaces. To solve the triple integral equations, the jumps of displacements 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. The interaction of two collinear cracks was also discussed in the present paper.

Dynamic behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips

The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is jump of displacement across the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter, the circular frequency of the incident waves and the thickness of the strip upon stress, electric displacement and magnetic flux intensity factors of cracks.

Non-Local Theory Solution of Two Mode-I Collinear Cracks in the Piezoelectric Materials

In this paper, the non-local theory is applied to obtain the behavior of two collinear cracks in the piezoelectric materials subjected to a uniform tension loading. The permittivity of air in the crack is considered. By means of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the triple integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants, the electric boundary conditions and the lattice parameter upon the stress and the electric displacement fields near the crack tips. It can be obtained that the effects of the electric boundary conditions upon the electric displacement fields are large. Unlike the classical piezoelasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allowing us to use the maximum stress as a fracture criterion.

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in anisotropic material plane by using the non-local theory

In this paper, the dynamic behavior of two collinear cracks in the anisotropic elasticity material plane subjected to the harmonic anti-plane shear waves is investigated by use of the nonlocal theory. To overcome the mathematical difficulties, a one-dimensional nonlocal kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress field near the crack tips. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near crack tips. The nonlocal elasticity solutions yield a finite hoop stress at the crack tips, thus allowing us to using the maximum stress as a fracture criterion. The magnitude of the finite stress field not only depends on the crack length but also on the frequency of the incident waves and the lattice parameter of the materials.

Non-local theory solution for the anti-plane shear of two collinear permeable cracks in functionally graded piezoelectric materials

In this paper, the non-local theory of elasticity is firstly applied to obtain the behavior of two collinear cracks in functionally graded piezoelectric materials under anti-plane shear loading for permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield finite stresses at the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses at the crack tips depend on the distance between two collinear cracks, the functionally graded parameter and the lattice parameter of the materials, respectively.

The nonlocal theory solution for two collinear cracks in functionally graded materials subjected to the harmonic elastic anti-plane shear waves

In this paper, the scattering of harmonic elastic anti-plane shear waves by two collinear cracks in functionally graded materials is investigated by means of nonlocal theory. The traditional concepts of the non-local theory are extended to solve the fracture problem of functionally graded materials. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress field near the crack tips. To make the analysis tractable, it is assumed that the shear modulus and the material density vary exponentially with coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present at crack tips.

The scattering of the harmonic anti-plane shear stress waves by two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material half-infinite planes dynamic loading

In this paper, the dynamic behaviour of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material half-infinite planes subjected to the harmonic anti-plane shear stress waves is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By using the Fourier transform technique, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the jump of the displacements across the crack surfaces. These equations are solved by using the Schmidt method. The relations among the electric field, the magnetic flux field, and the dynamic stress field near the crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter, the distance between two interface cracks, and the circular frequency of the incident waves upon the stress, the electric displacement, and the magnetic flux intensity factors of cracks.

Two Collinear Interface Cracks between Two Dissimilar Functionally Graded Piezoelectric/Piezomagnetic Material Layers under Anti-Plane Shear Loading

In this paper, the behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material layers subjected to an anti-plane shear loading is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved using the Schmidt method. The normalized stress, the electrical displacement and the magnetic flux intensity factors are determined for different geometric for the permeable electric boundary conditions. The relations among the electric filed, the magnetic flux field and the dynamic stress field near the crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter and the thickness of the strip upon the stress, the electric displacement and the magnetic flux intensity factors of the crack.

Investigation the Dynamic Interaction between Two Collinear Cracks in the Functionally Graded Piezoelectric Materials Subjected to the Harmonic Anti-Plane Shear Stress Waves by Using the Non-Local Theory

In this paper, the non-local theory of elasticity is applied to obtain the dynamic interaction between two collinear cracks in functionally graded piezoelectric materials under the harmonic anti-plane shear stress waves for the permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips. The non-local elastic solutions yield a finite stress at the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses at the crack tips depend on the crack length, the distance between two cracks, the functionally graded parameter, the circular frequency of the incident waves and the lattice parameter of the materials, respectively.

Dynamic behavior of two collinear permeable cracks in a piezoelectric layer bonded to two half spaces

The dynamic behavior of two collinear cracks in a piezoelectric layer bonded to two half spaces under harmonic anti-plane shear waves was investigated by means of Schmidt method. The cracks are vertically to the interfaces of the piezoelectric layer. The boundary conditions of the electrical field were assumed to be the permeable crack surface. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. Numerical examples were presented to show the effect of the geometry of the interacting cracks, the piezoelectric constants of the materials and the frequency of the incident waves upon the stress intensity factors. The results show that the dynamic field will impede or enhance the propagation of the crack in a piezoelectric material at different stages of the frequency of the incident waves. It is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than that for the impermeable crack surface conditions.

Investigation of the interaction of two collinear cracks in anisotropic elasticity materials by means of the nonlocal theory

In this paper, the interaction of two collinear cracks in anisotropic elasticity materials subjected to an anti-plane shear loading is investigated by means of the nonlocal theory. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surface. To solve the triple integral equations, the displacement on the crack surface is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present at the crack tip. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress hypothesis as a fracture criterion. The magnitude of the finite stress field depends on the crack length, the distance between two cracks and the lattice parameter of materials.

Nonlocal theory solution of two collinear cracks in the functionally graded materials

In this paper, the interaction of two collinear cracks in functionally graded materials subjected to a uniform anti-plane shear loading is investigated by means of nonlocal theory. The traditional concepts of the nonlocal theory are extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it is assumed that the shear modulus varies exponentially with the coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near the crack tips. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials. The magnitude of the finite stress field depends on the crack length, the distance between two cracks, the parameter describing the functionally graded materials and the lattice parameter of the materials.

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in the piezoelectric plate by using the non-local theory

In this paper, the dynamic interaction between two collinear cracks in a piezoelectric material plate under anti-plane shear waves is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved using the Schmidt method. This method is more reasonable and more appropriate. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis.

Multi-interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading

The behavior of four parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading is investigated. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved by the Schmidt method. This process is quite different from that papers adopted previously. The normalized stress and electrical displacement intensity factors are determined for different geometric and property parameters for permeable crack surface conditions. Numerical examples are provided to show the effect of the geometry of the interacting cracks, the thickness and the materials constants of the piezoelectric layer upon the stress and electric displacement intensity factors of the cracks. It is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions.

The behavior of permeable multi-cracks in a piezoelectric material

In this paper, the behavior of four parallel symmetric cracks in a piezoelectric material under anti-plane shear loading is studied by the Schmidt method for the permeable crack surface boundary conditions. By use of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations that the unknown variables are the jumps of the displacement across the crack surfaces. These equations are solved by means of the Schmidt method. The results show that the stress and the electric displacement intensity factors of cracks depend on the geometry of the crack. Contrary to the impermeable crack surface condition solution, it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions.

Crack bifurcation predicted for dynamic anti-plane collinear cracks in piezoelectric materials using a non-local theory

A non-local theory of elasticity is applied to obtain the dynamic interaction between two collinear cracks in the piezoelectric materials plane under anti-plane shear waves for the permeable crack surface boundary conditions. Unlike the classical elasticity solution, a lattice parameter enters into the problem that make the stresses and the electric displacements finite at the crack tip. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and electric displacement near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations in which the unknown variable is the jump of the displacement across the crack surface. The solutions are obtained by means of the Schmidt method. Crack bifurcation is predicted using the strain energy density criterion. Minimum values of the strain energy density functions are assumed to coincide with the possible locations of fracture initiation. Bifurcation angles of ±5° and ±175° are found. The result of possible crack bifurcation was not expected before hand.

Investigation of the dynamic behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces by a new method

The dynamic behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces subjected to the harmonic waves is investigated by a new method. The cracks are parallel to the interfaces in the mid-plane of the piezoelectric layer. By using the Fourier transform, the problem can be solved with two pairs of triple integral equations. These equations are solved by using Schmidt's method. This process is quite different from that adopted previously. Numerical examples are provided to show the effect of the geometry of cracks, the frequency of the incident wave, the thickness of the piezoelectric layer and the constants of the materials upon the dynamic stress intensity factor of cracks.

Multi-cracks problem for piezoelectric materials strip subjected to dynamic loading

This paper analyses and models the dynamic interaction among permeable multi-cracks in a piezoelectric strip under anti-plane shear waves by the Schmidt method. The Fourier transform is applied and then two pairs of triple integral equations can be solved using the Schmidt method. The results show that the stress and the electric displacement intensity factors of cracks depend on not only the crack length and the piezoelectric coefficient, but also the thickness of the piezoelectric strip, the distance between multi-cracks and the frequency of incident wave.

Investigation of anti-plane shear behavior of two collinear cracks in the piezoelectric materials by using the non-local theory

Abstract In this paper, the interaction between two collinear cracks in the piezoelectric materials under anti-plane shear loading was investigated by using the non-local theory for impermeable crack-face conditions. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. The solutions are obtained by using the Schmidt method. Numerical examples are provided to show the effect of the geometry of the interacting cracks. Contrary to the previous results, it is found that no stress and electric displacement singularity is present near the crack tip.