Cauchy-type Integral Equations Research Articles

Analytical solution of two non‐identical edge cracks in an infinite strip under anti‐plane shear wave

AbstractThis article presents an extensive analytical solution addressing the interaction between two non‐identical edge cracks in an infinite orthotropic strip under anti‐plane shear waves. Most studies assume identical cracks or single edge crack in a strip, but this research breaks new ground by considering cracks of different sizes. By incorporating mixed‐type boundary conditions, the study derives dual integral equations. These equations are then transformed into a singular integral equation of Cauchy type with the aid of a trial solution and contour integration technique. The singular integral equation is further converted into a system of integral equations, which are solved numerically utilizing Jacobi polynomials. The obtained solutions are utilized to derive expressions for the stress intensity factor (SIF) and crack opening displacement (COD) at the crack tip using Krenk's interpolation formulae. The derived results are presented graphically and compared against existing solutions for single edge crack and symmetric edge cracks in static scenario.

A contact model for the functionally graded coated elastic structures: Extension of the Hertz theory to the contact of beam structures

The Hertzian displacement assumption is widely used in analyzing the contact problems of non-uniform elastic bodies. It is essential to account for the support conditions of the elastic body as they significantly influence the contact stiffness and distribution of contact pressure. To address this, the deformation of beam structures is integrated into the Hertzian displacement assumption, which leads to the development of an extended contact mechanics model suitable for the elastic bodies of a beam structure which can be non-uniform and with functionally graded coatings. The problem is solved by using a numerical method based on the Gauss-Chebyshev quadrature for the singular integral equation of Cauchy type. In the contact problems of the doubly simply supported (SS-SS) and cantilever beams, the contact pressure and contact stiffness in conjunction with the interactions between the indentation and contact bodies are discussed. An in-depth study on the coupling effects between the structural deformation and functionally graded coatings is presented.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

We have four kinds of solutions for Cauchy type integral equations: by expanding on known functions and using the Maclaurin series, we can convert these four kinds of solutions into linear combinations of some elements that are basis of these solutions. Using these bases gives exact solutions for polynomials and, for some other functions, a high-accuracy approximate solution.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

Analytic solutions of multiple Mode III cracks in a circular bar reinforced by a magneto-electro-elastic layer under torsional loading

In this paper, the Saint-Venant torsion of a circular bar reinforced by a magneto-electro-elastic coating containing multiple cracks is analyzed. Pure Mode III stress intensity factors are studied in this paper. An electric displacement and a magnetic induction are applied to the outer edge of the magneto-electro-elastic coating. The stress field of the circular bar coated by the magneto-electro-elastic layer containing a screw dislocation is first found in terms of the dislocation density using finite Fourier transform. Then, the stress field is reduced to a set of Cauchy-type integral equations in the circular cross-section with the magneto-electro-elastic layer using the distributed dislocation method. The torsional rigidity of the cracked domain and the stress intensity factors (SIFs) are found by a numerical procedure of this type of integral equation. Several numerical examples are presented to check the effect of the magneto-electro-elastic layer and critical parameters on the stress intensity factors and torsional rigidity. In addition, the stress intensity factors can be eliminated using appropriate values of the electric displacement and magnetic induction.

On the receding contact between a graded and a homogeneous layer due to a flat-ended indenter

This paper solves the frictionless receding contact problem between a graded and a homogeneous elastic layer due to a flat-ended rigid indenter. Although its Poisson’s ratio is kept as a constant, the shear modulus in the graded layer is assumed to exponentially vary along the thickness direction. The primary goal of this study is to investigate the functional dependence of both contact pressures and the extent of receding contact on the mechanical and geometric properties. For verification and validation purposes, both theoretical analysis and finite element modelings are conducted. In the analytical formulation, governing equations and boundary conditions of the double contact problem are converted into dual singular integral equations of Cauchy type with the help of Fourier integral transforms. In view of the drastically different singularity behavior of the stationary and receding contact pressures, Gauss–Chebyshev quadratures and collocations of both the first and the second kinds have to be jointly used to transform the dual singular integral equations into an algebraic system. As the resultant algebraic equations are nonlinear with respect to the extent of receding contact, an iterative algorithm based on the method of steepest descent is further developed. The semianalytical results are extensively verified and validated with those obtained from the graded finite element method, whose implementation details are also given for easy reference. Results from both approaches reveal that the property gradation, indenter width, and thickness ratio all play significant roles in the determination of both contact pressures and the receding contact extent. An appropriate combination of these parameters is able to tailor the double contact properties as desired.

Mixed-mode fracture analysis for two dissimilar half-planes with multiple interface moving cracks

In this paper, the distributed dislocation technique (DDT) is used to calculate the stress intensity factors (SIFs) at the tip of several moving cracks which are located at the interface between two dissimilar non-homogeneous half-planes. In this study, in-plane loading is considered and it is assumed that the properties of the non-homogeneous material change exponentially. First, by employing Fourier and Galilean transforms, the problem of Volterra type climb and glide edge dislocation is solved at the interface of two dissimilar materials, and then the DDT is used to obtain the singular integral equations of Cauchy type. The resulting singular equations are numerically discretized and solved to obtain dislocation density on the crack surfaces to determine the SIFs. Finally, the effects of change in parameters such as gradient non-homogeneous constant, Poisson ratio, crack growth rate and interaction between cracks on SIFs are shown graphically.

On the Stress Transfer and Mode Mixity for a Film of Finite Length Bonded to an Elastic Substrate

The configuration of film-substrate structure is widely involved in many aspects of technologies. To ensure high reliability and sound performance, it is of importance to understand the stress distribution and the adhesion at the interface. In the present work, within mind the film of finite width, a mechanical model has been proposed to analyze the traction at the interface between the film and an elastic substrate that may be subjected to either uniaxial stretching or bending. For incompressible substrates, the problem may be reduced to the solution of singular integral equations of Cauchy type and the tractions at the film’s edges have been revealed to exhibit the square-root singularity. Mode mixity has been calculated. By comparing the strain in the film when the system is subjected to uniaxial stretching with that from experiments on the adhesion between graphene and substrates, an agreement has been found and the implications of the present model in the measurement of adhesion energy have been discussed.

On inverse problem with phase retrieval for an inclined plane in the parabolic approximation

The inverse problem of amplitude reconstruction on an inclined plane based on the values of amplitude or its module as recorded on semi-infinite plane orthogonal to the beam propagation direction is considered within the framework of 2D parabolic equation. It is demonstrated that this inverse problem, in case when the complex image plane amplitude is known, can be reduced to a singular Cauchy-type integral equation. The existence of its solutions requires that certain conditions be met but if a solution exists it is necessary unique. The obtained integral equation is then approximated piece-wisely and the resulting linear algebraic system is solved numerically while applying necessary regularization procedures to enhance the stability of its solutions. Finally, an iterative method of phase retrieval is developed and a set of numerical experiments is performed.

Neumann inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation

This paper is the first attempt to give a rigorous mathematical study of Neumann initial boundary value problems for the multidimensional dispersive evolution equations considering as example famous nonlinear Schrodinger equation. We consider the inhomogeneous initial-boundary value problem for the nonlinear Schrodinger equation, formulated on upper right-quarter plane with initial data $$u({\mathbf {x}},t)\left| _{t=0}\right. =u_{0}({\mathbf {x}})$$ and Neumann boundary data $$u_{x_{1}}\left| _{\partial _{1}{\mathbf {D}}}\right. =h_{1}(x_{2},t),u_{x_{2}}\left| _{\partial _{2}{\mathbf {D}}}\right. =h_{2}(x_{1},t)$$ given in a suitable weighted Lebesgue spaces. We are interested in the study of the influence of the Neumann boundary data on the asymptotic behavior of solutions for large time. We show that problem admits global solutions whose long-time behavior essentially depends on the boundary data. To get a nonlinear theory for the multidimensional model. we propose general method based on Riemann–Hilbert approach and theory Cauchy type integral equations. The advantage of this method is that it can also be applied to non-integrable equations with general inhomogeneous boundary data.

A receding contact analysis for an elastic layer reinforced with a functionally graded coating and pressed against a half-plane

We aim to analyze the receding contact mechanics of an elastic layer reinforced by a functionally graded coating, pressed against a half-plane substrate as an integral assembly. The graded reinforcement is modeled by an inhomogeneous medium whose shear modulus is allowed to vary exponentially. A smooth receding contact is arisen due to the normal tractions applied over a finite segment of the reinforcement surface. The primary goal of this study is the determination of both the receding contact pressure and the extent of receding contact along the assembly and substrate interface. Using both an analytical and a finite element method solves the problem. In the former approach, Fourier integral transforms help convert the problem into a singular integral equation of Cauchy type, which is numerically integrated with Gauss-Chebyshev quadrature. In the numerical approach, multiple homogeneous layers approximate the graded coating. The stresses at those nodes lying on sublayer interfaces are averaged over their surrounding elements for guaranteeing the continuity in stress field. Very good agreements are found between the results obtained from the two methods. Extensive parametric case studies reveal that material properties, loading configuration, and geometric parameters all play important roles in the determination of both the contact pressure distribution and the length of contact. These receding contact properties can therefore be optimized as desired by appropriately varying the influential parameters. Both the solution methodologies and the numerical results presented in this work can provide some useful guidelines on the better design of multilayered functionally graded structures.

Multidimensional inhomogeneous mixed initial-boundary value problem for the nonlinear Schrödinger equation

We consider the inhomogeneous mixed initial-boundary value problem for the nonlinear multidimensional Schrödinger equation, formulated on upper right-quarter plane. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time. Also we are interested in the study of the influence of the mixta boundary data on the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on the studying a linear theory and then using the fixed point argument. To get a linear theory for the multidimensional model we propose general method based on Laplace approach and theory Cauchy type integral equations. To get smooth solutions in L∞ we modify a method based on the factorization for the free Schrödinger evolution group. The advantage of our approach is that it can also be applied to non-integrable equations and arbitrary boundary conditions. This approach is new and it is not standard. We believe that the results of this paper could be applicable to study a wide class of dissipative multidimensional nonlinear equations by the use of techniques of nonlinear analysis.

Inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation

We consider the inhomogeneous Dirichlet initial-boundary value problem for the nonlinear multidimensional Schrödinger equation, formulated on an upper right-quarter plane. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time. Also we are interested in the study of the influence of the Dirichlet boundary data on the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on studying a linear theory and then using the fixed point argument. To get a linear theory for the multidimensional model, we proposed a general method based on the Riemann–Hilbert approach and theory Cauchy type integral equations. To get smooth solutions in L∞, we modify a method based on the factorization for the free Schrödinger evolution group.

Neumann problem for the capillary wave equation

We study the initial–boundary value problem IBV problem for the cubic capillary wave equation iut+|u|2u=|∂x|32u,t>0,x>0;u(x,0)=u0(x),x>0,ux(0,t)=h(t),t>0,where |∂x|32u=12π∫0∞sign (x−y)|x−y|uyy(y)dy.We prove the global in time existence of solutions of IBV problem for nonlinear Fractional Schrödinger equation with inhomogeneous Neumann boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on the study of a linear theory and then using the fixed point argument. To get a linear theory we propose general method based on Riemann–Hilbert approach and theory Cauchy type integral equations. To get smooth solutions in L∞ we modify a method based on the factorization for the free Fractional Schrödinger evolution group.

Surface/interface effect on the scattering of Love waves by a nano-size surface-breaking crack within an ultra-thin layer bonded to an elastic half-space

Using the mathematical framework of surface/interface elasticity theory, an analytical solution is presented for the antiplane scattering of Love waves by a surface-breaking crack normal to the free surface of an ultra-thin layer bonded to a semi-infinite medium. Fourier transform technique is used to formulate the corresponding mixed-boundary value problem as a singular integral equation of Cauchy type. Within the context of the present theory, the scattered stress field, the Mode III dynamic stress intensity factor (DSIF) at the crack-tip, and the reflection coefficient at large distances from the crack face are computed and plotted for different wave numbers and surface/interface parameters. The dispersion relation is also derived within this framework. Contrary to the predictions based on classical theory, surface elasticity theory predicts that Love waves can propagate:(a) in an elastic half-space without the over-layer, and (b) in the layer-substrate system when the shear wave velocity of the layer is greater than that of the substrate. In the given numerical examples, the inevitable role of surfaces and interfaces in the presence of nano-structure is well approved and it is shown that the surface/interface effect is particularly remarkable for short wavelengths which are comparable to the surface/interface characteristic length.

Stress state of two collinear stamps over the surface of orthotropic materials

An exact contact analysis of two collinear stamps over the surface of orthotropic materials is performed. Eigenvalue analyses related to the governing equations are conducted, and real fundamental solutions are obtained for each eigenvalue distribution. Using Fourier sine or cosine transforms, a singular integral equation of Cauchy type is obtained. In particular, the exact solution of the reduced singular integral equation is obtained in the case of two collinear flat or cylindrical stamps. Explicit expressions for various stresses are obtained in terms of elementary functions. Numerical results are presented, and interesting observations are found. For two collinear flat stamps, detailed calculations are provided to show influences of distance between two flat stamps on the contact stress and in-plane stress. For a cylindrical stamp, influences of distance between two cylindrical stamps, mechanical loading P, and the cylindrical stamp radius R on the surface normal stress and surface in-plane stress are detailed.

Exact solutions of the 2‐D frictional sliding contact problem of electrically insulated triangular and cylindrical punches on piezoelectric materials

AbstractThe present paper performs an exact electro‐mechanical contact analysis for piezoelectric materials under a frictional sliding triangular or cylindrical punch. The Coulomb frictional law is used inside the contact region. Appropriate fundamental solutions that can lead to real solutions of electro‐mechanical quantities are derived for real and complex eigenvalues. A second kind singular integral equation of Cauchy type is obtained for the stated problem, whose exact solution can be found by using the excellent properties of Jacobi Polynomials. Closed‐form expressions of stress components and electrical displacement are obtained, which provide benchmark solutions for examining the frictional contact behaviors of piezoelectric materials. Numeric results are presented. The influences of the friction coefficient on the contact region, stress components and electric displacement are detailed.

The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

A method for finding the numerical solution of a weakly singular Fredholm integral equation of the second kind is presented. The Taylor series is used to remove singularity and Legendre polynomials are used as a basis. Furthermore, the Legendre function of the second kind is used to remove singularity in the Cauchy type integral equation. The integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the Cauchy principal value sense without using numerical quadratures. Four examples are given to show the accuracy of the method.

A Note on the Numerical Solution of Singular Integral Equations of Cauchy Type

This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebyshev polynomials of the first kind. It is shown that the numerical solution of characteristic singular integral equation is identical with the exact solution, when the force function is a cubic function. Moreover, it also shown that this numerical method gives exact solution for other singular integral equations with degenerate kernels.

Stress intensity factor for an edge crack in two bonded dissimilar materials under convective cooling

The transient thermal stress crack problem for two bonded dissimilar materials subjected to a convective cooling on the surface containing an edge crack perpendicular to the interface is considered. The problem is solved using the principle of superposition and the uncoupled quasi-static thermoelasticity. The crack problem is formulated by applying the transient thermal stresses obtained from the uncracked medium with opposite sign on the crack surfaces to be the only external loads. Fourier integral transform is used to solve the perturbation problem resulting in a singular integral equation of Cauchy type in which the derivative of the crack surface displacement is the unknown function. The numerical results of the stress intensity factors are calculated for both the edge crack and the crack terminating at the interface using two different composite materials and illustrated as a function of time, crack length, coefficient of heat transfer, and the thickness ratio.