Papers by Keyword: Singular Integral Equation

Abstract: This paper presents an analytical model for analyzing two bonded dissimilar piezoelectric materials weakened by multiple embedded cracks. The medium is subjected to anti-plane mechanical and in-plane electrical loading. We consider the interface imperfect to address potential electro-mechanical damage at the interface, with the electro-mechanical imperfection represented by a linear spring model. First, the solution for a dynamic electro-elastic dislocation in the piezoelectric layer is obtained using the integral transforms technique. Subsequently, the dislocation solutions are utilized to transform the problems into a set of singular integral equations featuring Cauchy kernels, which are then solved numerically in the Laplace transform domain. The numerical Laplace inversion technique calculates the dynamic stress intensity factors (DSIFs). Several examples are analyzed to derive DSIFs for varying crack lengths and the spring constants reflecting imperfect electro-mechanical bonding and the material properties of the piezoelectric layers.

Abstract: In this study, the contact problem for a graded elastic half-plane in frictional contact with a rigid stamp is considered. The plane contact problem is assumed to be linear elastic and the Poisson's ratio is assumed to be constant. Analytical formulation of the study includes Fourier transforms of the governing equations and boundary conditions. The resulting integral equation is solved numerically. Contact pressure, in-plane stress and the stress intensity factor at the sharp edges of the contact are evaluated and demonstrated for various stamp profiles. The results are compared with a closed form solution for homogeneous isotropic half-plane indented by rigid stamps. The effects of the nonhomogeneity parameter, coefficient of friction and stamp profiles on the contact and in-plane stresses are analyzed in detail.

Abstract: In this paper, overlaid cement concrete pavement with crack is considered based on fracture mechanics. Cement concrete pavement is reduced to an elastic plate on Winkler foundation. Fourier integral transform, residue theorem and Lobatto-Chebyshev integration formula are used to obtain the analytical solutions of this problem. To analyze the effect of overlay for the old pavement, stress intensity factors ofItype crack and II type crack are calculated respectively for various overlay thicknesses, elastic moduli of the overlay. The stress intensity factors of the crack tips in the pavement with 10cm thickness overlay are reduced heavily. However, stress intensity factors are less affected by elastic modulus of overlay material.

Abstract: In this paper, the frictional contact problem of a homogeneous orthotropic material in contact with a wedge-shaped punch is considered. Materials can behave anisotropically depending on the nature of the processing techniques; hence it is necessary to develop an efficient method to solve the contact problems for orthotropic materials. The aim of this work is to develop a solution method for the contact mechanics problems arising from a rigid wedge-shaped punch sliding over a homogeneous orthotropic half-plane. In the formulation of the plane contact problem, it is assumed that the principal axes of orthotropy are parallel and perpendicular to the contact. Four independent engineering constants , , , are replaced by a stiffness parameter, , a stiffness ratio, a shear parameter, , and an effective Poisson’s ratio, . The corresponding mixed boundary problem is reduced to a singular integral equation using Fourier transform and solved analytically. In the parametric analysis, the effects of the material orthotropy parameters and the coefficient of friction on the contact stress distributions are investigated.

Abstract: In this paper, the Fourier integral transform-singular integral equation method is presented for the Mode I crack problem of the functionally graded orthotropic coating-substrate structure. The elastic property of the material is assumed vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the boundaries of the strip. Numerical examples are presented to illustrate the effects of the crack length, the material nonhomogeneity and the thickness of coating on the stress intensity factors.

Abstract: We discussed a kind of singular integral equation with Hilbert kernel on open arcs lying in a period strip. By using the method of complex functions, we obtained the extended Plemelj Formula with Hilbert kernel, and based on this, we obtained the general solutions and the solvable conditions for this kind of characteristic singular integral equation with Hilbert kernel on open arcs.

Abstract: We considered the regularization method for a kind of complete singular integral equation with Hilbert kernel on open arcs lying in a period strip. And based on this, we obtained the solvable Noether theorem for this kind of complete singular integral equations.

Abstract: We considered a kind of singular integral equation with Hilbert kernel on closed contours. By using the method of complex functions, we obtain the extended Plemelj Formula with Hilbert kernel, and based on this, we obtained the related conditions of solvability and the general solution for the characteristic singular integral equation with Hilbert kernel on closed contours.

Abstract: The elastostatic antiplane problem of a Dugdale crack at the interface of two different materials is considered. Using integral transform, the problem is reduced to a single integral equation. The integral equation is solved numerically. The evolution of the crack for different values of the physical and geometrical parameters of the problem is studied. A comparison between the results obtained with the Griffith and Dugdale models is presented.

Abstract: In this paper, the scattering problems of SH waves on periodic cracks in an infinite of piezoelectric/piezomagnic composite materials bonded to an infinite of homogeneous piezoelectric materials is investigated, the Fourier transform techniques are used to reduce the problem to the solution of Hilbert singular integral equation, the latter is solved by Lobotto-Chebyshev and Gauss integral equation, at last, numerical results showed the effect of the frequency of wave, sizes and so on upon the normalized stress intensity factor.