一类带位移的非线性奇异积分方程解的存在性

一类带位移的非线性奇异积分方程,在广义Holdar空间当参数|λ|充分小时解存在。

Cracked elastic substrate strip with functionally graded coating under thermal-mechanical loading

This paper investigates the functionally graded coating bonded to an elastic strip with a crack under thermal- mechanical loading. Considering some new boundary conditions, it is assumed that the temperature drop across the crack surface is the result of the thermal conductivity index which controls heat conduction through the crack region. By the Fourier transforms, the thermal-elastic mixed boundary value problems are reduced to a system of singular integral equations which can be approximately solved by applying the Chebyshev polynomials. The numerical computation methods for the temperature, the displacement field and the thermal stress intensity factors （TSIFs） are presented. The normal temperature distributions （NTD） with different parameters along the crack surface are analyzed by numerical examples. The influence of the crack position and the thermal-elastic non- homogeneous parameters on the TSIFs of modes I and 11 at the crack tip is presented. Results show that the variation of the thickness of the graded coating has a significant effect on the temperature jump across the crack surfaces when keeping the thickness of the substrate constant, and the thickness of functionally graded material （FGM） coating has a significant effect on the crack in the substrate. The results can be expected to be used for the purpose of gaining better understanding of the thermal-mechanical behavior of graded coatings.

Wave Interaction with Dual Circular Porous Plates

In this paper we have investigated the reflection and the transmission of a system of two symmetric circular-arc-shaped thin porous plates submerged in deep water within the context of linear theory. The hypersingular integral equation technique has been used to analyze the problem mathematically. The integral equations are formulated by applying Green＇s integral theorem to the fundamental potential function and the scattered potential function into a suitable fluid region, and then using the boundary condition on the porous plate surface. These are solved approximately using an expansion-cure-collocation method where the behaviour of the potential functions at the tips of the plates have been used. This method ultimately produces a very good numerical approximation for the reflection and the transmission coefficients and hydrodynamic force components. The numerical results are depicted graphically against the wave number for a variety of layouts of the arc. Some results are compared with known results for similar configurations of dual rigid plate systems available in the literature with good agreement.

Analysis of unsteady supercavitating flow around a wedge

Supercavitating flow around a slender symmetric wedge moving at variable velocity in static fluid has been studied. Singular integral equation for the flow has been founded through distributing the sources and sinks on the symmetrical axis. The supereavity length at each moment is determined by solving the singular integral equation with finite difference method. The supercavity shape at each moment is obtained by solving the partial differential equation with variable coefficient. For the case that the wedge takes the impulse and uniformly variable motion, numerical results of time history of the supercavity length and shape are presented. The calculated results indicate that the shape and the length of the supercavity vary in a similar way to the case that the wedge takes variable motion, and there is a time lag in unsteady supercavitating flow induced by the variation of wedge velocity.

A Quadrature Approach for N-Collinear Crack Problem in an Orthotropic Strip

This study presents the determination of the stress intensity factors （SIFs） at the edges of the cracks in an elastic strip weakened by N-collinear cracks. The problem of an orthotropic elastic strip is reduced to a system of Cauchy type singular integral equations. The system of singular integral equations is approached by a Quadrature technique. Under two different loading conditions, the results are obtained for the different cases of crack numbers. The resistance of the strip is examined by considering the orthotropic properties of the strip material. Finally, the crack interactions are clarified during the analysis.

CAUCHY PROBLEM FOR A NONLINEAR SINGULAR INTEGRAL－DIFFERENTIAL EVOLUTION SYSTEM

We study the Cauchy problem for a nonlinear evolution system with singularintegral differential terms.By means of some a priori estimates of the solution and theLeray-Schauder's fixed point theorem, we prove the existence and the uniqueness theoremsof the generalized global solution of the mentioned problem.

Integral Representation of the Manifold Solution for New Class of the Volterra Type Integral Equation with a Boundary Singularity in the Case, When Kernel Contain Logarithmic Singularity and its Power

In this work we suggestion new methods investigation the model Volterra type integral equation with logarithmic singularity, kernel which consisting from composition polynomial function with logarithmic singularity and function with singular point. The problem investigation this type integral equation at n = 2m reduce to problem investigate the Volterra type integral equation （1） for n = 2 the theory for which was constructed in [2]. In this work, we investigation integral equation （1） at = 2m ＋ 1 In this case, we investigate integral equation （1） reduction it＇s to m integral equation type [2] φ（x）＋∫xa[p1＋p2 ln（x-a/t-a）]φ（t）/t-a dt=f（x）and one the following integral equation [1] ω（x）＋p3∫xω（t）/ a t-adt=g（x）.

Two coplanar cracks in a functionally graded piezoelectric material strip under mechanical and transient thermal loadings

In this paper,the fracture problem of a functionally graded piezoelectric material strip(FGPM strip) containing two coplanar cracks perpendicular to its boundaries is considered.The problem is solved for an FGPM strip that is suddenly heated from the bottom surface under static mechanical loading.The top surface is maintained at the initial temperature.The crack faces are supposed to be completely insulated.Material properties are assumed to be exponentially dependent on the distance from the bottom surface.By using the Laplace and Fourier transforms,the thermoelectromechanical fracture problem is reduced to a set of singular integral equations,which are solved numerically.The stress intensity factors for the cases of the two embedded cracks,two edge cracks,and an embedded crack and an edge crack are computed and presented as a function of the normalized time,the nonhomogeneous and geometric parameters.

Influence of residual surface stress on the fracture of nanoscale piezoelectric materials with conducting cracks

In this paper,we analyze the stress and electric field intensity factors affected by residual surface stress for conducting cracks in piezoelectric nanomaterials.The problem is reduced to a system of non-linear singular integral equations,whose solution is determined by iteration technique.Numerical results indicate that the residual surface stress can significantly alter the crack tip fields at nanometer length scales.Due to the residual surface stress,281he electric field can produce stress around crack tip.This suggests a strong electromechanical coupling crack tip field for nanoscale piezoelectric materials.Such a finding is considerably different from the classical fracture mechanics results.A transit electric field to stress load ratio is identified,for which influences of residual surface stresses vanish.The research is useful for the applications of nanoscale piezoelectric devices.

Fracture analysis of a plane crack problem under chemo-mechanical loading

Fracture analysis of a plane crack problem under chemo-mechanical loading is presented based on a linear chemo-elasticity model.The flux conductivity is introduced to characterize the influence of the crack defect on the diffusion process.Using Fourier transform and the dislocation density functions,the crack problem is reduced to a set of singular integral equations,which are solved numerically by the Lobatto-Chebyshev method.Parametric studies are conducted to reveal the effects of flux conductivity,geometric configuration,chemical and mechanical loads on the crack tip field.The numerical results show that the stress singularity at the crack tip is usually a mixture of mode Ⅰ and mode Ⅱ types.

Stress fields caused by a dislocation in an anisotropic 3-layer system

We investigated the stress fields caused by a dislocation in an anisotropic 3-layer system. Based on the image method, the original 3-layer system is firstly decomposed into three infinite homogenous systems. The image dislocation densities used as unknowns are then strategically distributed in order to satisfy the boundary conditions. The resulting governing equations are singular Cauchy integral ones. Removing the singular terms yields non-linear Fredhom integral equations of the second kind. The obtained stress fields satisfy the boundary conditions, i.e., the traction free condition on the free surface and continuous conditions across the interfaces. Also, a comparison with previous results is made and good agreement is achieved. Numerical investigations show that under the plain strain condition, layer thickness and dislocation position play stronger roles in the stress fields than crystallographic orientation, and these effects more significantly affect the stress fields caused by an edge dislocation than by a screw dislocation.

A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

In this work,we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels.Particularly,we consider the case when the underlying solutions are sufficiently smooth.In this case,the proposed method leads to a fully discrete linear system.We show that the fully discrete integral operator is stable in both infinite and weighted square norms.Furthermore,we establish that the approximate solution arrives at an optimal convergence order under the two norms.Finally,we give some numerical examples,which confirm the theoretical prediction of the exponential rate of convergence.

A CRACK PROBLEM WITH A BROKEN LINE INTERFACE

The equilibrium problem for the infinite elastic plane consisting of two different media is considered, in which the interface is a broken line, there is a straight crack touching the vertex of the broken line with some symmetry and the same uniform pressures are applied to both of its sides. The problem is reduced to a uniquely solvable singular integral equation on the interface and the crack. The order of singularity at the vertex is considered, which may be determined by a transcendental equation.

Effect of electromagnetic boundary condition on dynamo actions

In this paper, based on the mean field dynamo theory, the influence of the electromagnetic boundary condition on the dynamo actions driven by the small scale turbulent flows in a cylindrical vessel is investigated by the integral equation approach. The numerical results show that the increase of the electrical conductivity or magnetic permeability of the walls of the cylindrical vessel can reduce the critical magnetic Reynolds number. Furthermore, the critical magnetic Reynolds number is more sensi- tive to the varying electrical conductivity of the end wall or magnetic permeability of the side wall. For the anisotropic dynamo which is the mean field model of the Karlsruhe experiment, when the relative electrical conductivity of the side wall or the rel- ative magnetic permeability of the end wall is less than some critical value, the m=l （m is the azimuthal wave number） mag- netic mode is the dominant mode, otherwise the m=0 mode predominates the excited magnetic field. Therefore, by changing the material of the walls of the cylindrical vessel, one can select the magnetic mode excited by the anisotropic dynamo.