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The article presents a method for solving characteristic singular integral equations perturbed with compact operators. The method consists in reducing the solution of these equations to the solution of the systems of singular (unperturbed) equations, which are solved by factoring the coefficients of the obtained systems. The method presented concerns some results of Gohberg and Krupnik and can be used in solving other classes of functional equations with composite operators that fit into the scheme described by Theorem 1.1

Multiple processes in physics and technology are simulated using singular integral equations in exceptional cases, which necessitates development of approximate methods for solving such equations. A computational scheme using the Fourier transform is proposed in this work. [ABSTRACT FROM AUTHOR]

This article attempts to investigate the numerical solution of a weakly singular Fredholm integral equation of the first kind, involving a logarithmic singularity in the reproducing kernel Hilbert space W 2 3 [ 0 , b ] . The approximate solution is represented in the form of series including finite number of terms. Convergence rate of the approximate solution is studied using error analysis. Numerical examples carried out to demonstrate the error analysis estimates. [ABSTRACT FROM AUTHOR]

This paper considers the fluid flow through a porous medium containing intersecting fractures and presents three main analytical findings, namely: (1) mass exchange between fractures and surrounding matrix at the fracture intersection; (2) fluid potential solution (pressure field) within the whole domain under the form of a single singular integral equation; and (3) closed‐form solutions of fluid flow in and around a crack disc under a far field pressure gradient. The crack is represented mathematically by a 2D smooth surface (i.e., zero thickness) within a 3D porous medium, while physically by a constant aperture. The fluid flow within the crack obeys Poisseuille's law, while Darcy's law is used to represent the fluid flow in the surrounding matrix. The general solution of pressure field for the general case of multiple intersecting cracks is firstly derived under a singular integral equation form. The mass exchange between the porous matrix and the crack, as well as the mass conservation at the intersection between cracks are the keys to obtaining this general solution. Then, the general solution is written for the case of a single crack. Rigorous derivation of the latter equation allows obtaining a closed‐form solution of flow through a single crack. Introducing this solution of flow into the general equation gives the pressure field around the crack. The solution derived in this paper for a crack disk with Poisseuille's flow is slightly different from the well‐known Eshelby's solution for the case of flattened inclusion in which the flow obeys Darcy's law. [ABSTRACT FROM AUTHOR]

• The electro-mechanical coupling in a nanoscale Mode III crack, considering surface effects and direct flexoelectricity was first studied. • The corresponding governing equations, along with the associated boundary conditions are derived. • Fourier transform reduces the mixed boundary value problem to a hypersingular integral equation explicitly incorporating surface effects. • The influence of surface and flexoelectric effects on the elastic and electric field are revealed. Flexoelectric materials exhibit mechano-electro coupling between the strain gradients and electric polarization (direct flexoelectricity) and/or between electric field gradients and elastic strains (converse flexoelectricity). As the design of flexoelectric structures is required more and more at decreasing length scales, surface effects become more significant, often resulting in considerable size effects which affect overall deformation of the structure. This further complicates fracture mechanisms in flexoelectric materials particularly at micro- and nano-sized structures. In this paper, we incorporate surface effects with the direct flexoelectricity, to study the electro-mechanical coupling effects of a Mode III crack at the nanoscale structures. We derive the corresponding governing equations together with the associated boundary conditions. Using Williams' eigenfunction expansion and the J-integral's path independence, we have obtained the asymptotic field to analyze the singularity indices at the Mode III crack tip in flexoelectric materials. With the use of Fourier transforms, the corresponding mixed boundary value problem is reduced to a hypersingular integro-differential equation in which surface effects are incorporated explicitly. The resulting hypersingular integral equation is solved numerically using Chebyshev polynomials. The influence of both surface and flexoelectric effects on the displacement field, polarization field, strain gradient field along with the actual physical stress field are revealed and displayed graphically. [ABSTRACT FROM AUTHOR]

This paper investigates the frictional contact problem of a layer indented by a rigid punch within the framework of the couple stress elasticity. It is assumed that the layer is homogeneous, isotropic, and fully bonded to a rigid substrate. The mixed-boundary value problem is converted using Fourier transform into a singular integral equation in which the unknown is the contact pressure between the layer and the punch. The integral equation is further derived for the flat and cylindrical punch case profiles, normalized and then solved numerically using the Gauss–Jacobi integration formula. The obtained results are first validated based on those published for the case of a frictionless contact problem of a half-plane indented by a rigid punch and solved within the context of couple stress theory. An extensive parametric study is then conducted to investigate the effect of several parameters on the contact stresses for the both the flat and cylindrical punch profiles. These parameters include the characteristic material length, the layer height, the friction coefficient, the indentation load, and the shear modulus. [ABSTRACT FROM AUTHOR]

In view of the contact problem of functionally graded piezoelectric material (FGPM) coating under different kinds of conducting indenters, effects of the gradient index on the contact mechanical behavior of the FGPM coating were investigated. A model for the multilayer FGPM coating was established. The contact problem of the FGPM coating was transformed into singular integral equations by means of the Fourier integral transform technology and the transfer matrix method. The Gauss-Chebyshev quadrature formula was used to obtain the surface stress distribution and the charge distribution in the FGPM coating-substrate system under a rigid conducting flat indenter and a conducting cylindrical indenter. According to the numerical results, the effects of variations of the FGPM coating parameters on the indentation and electrical potential were analyzed. The distributions of stress and electrical displacement in the FGPM coating were obtained. The results show that, the variations of the FGPM coating parameters have an important influence on the contact behavior of the system. [ABSTRACT FROM AUTHOR]

In the functional space is considered a spectral problem for the Cauchy–Riemann operator with boundary conditions of the Bitsadze–Samarskii type. It is proved under the assumption that when is a non-empty resolvent set of the operator , the considered spectral problem for the Cauchy–Riemann operator is reduced to a singular integral equation with a continuous kernel. [ABSTRACT FROM AUTHOR]

We consider two-dimensional dynamic problems of the theory of elasticity, which can be reduced to singular integral or integrodifferential equations with immobile singularities. These problems include the problems of determination of the stressed state of bodies with edge defects and defects whose cross sections have the shape of broken line and some contact problems. For the solution of the obtained equations, we propose to apply a numerical method that takes into account the actual asymptotics of solutions and is based on the use of special quadrature formulas for singular integrals. [ABSTRACT FROM AUTHOR]

We use complex variable methods and the theory of singular integral equations to study a thin-film-covered mode III crack with dislocation-free zones (DFZs) under uniform remote anti-plane shear stress. The equilibrium condition is formulated in terms of a single singular integral equation constructed in the image plane via the solution of the problem of a single screw dislocation interacting with a completely coated crack and the method of continuously distributed dislocations. The singular integral equation is solved numerically using the Gauss–Chebyshev integration formula to arrive at the dislocation distribution function, the DFZ condition, the total number of dislocations in the plastic zone and the local mode III stress intensity factor at the crack tip. [ABSTRACT FROM AUTHOR]

The severe electric and thermal environments may cause localized deterioration of the contact behavior of thermoelectric devices. The contact responses of the thermoelectricity under the joint effect of the finite size and the friction are analyzed. The law of the Coulomb friction is adopted. The obtained Fredholm kernel functions reveal the influence of the thickness and the friction. The known Jacobi polynomials are employed to discretize the obtained singular integral equation. The effects of the thermoelectric loadings (the total electric current and the total energy flux), friction coefficient, the thermoelectric strip thickness, and the elastic and thermoelectric material constants on the distribution of the normal traction and the surface in-plane stress are demonstrated in detail. The smaller thermal expansion coefficient and shear modulus will contribute to the lower stress concentration at both contact edges. The surface in-plane tensile stress behind the trailing edge can be alleviated as the thermoelectric strip becomes thinner. [ABSTRACT FROM AUTHOR]

The present study establishes a general theory for dynamic frictional contact mechanics between an orthotropic graded half-plane and a moving rigid punch of a flat profile. The rigid punch moves over the orthotropic graded half-plane at a constant subsonic speed. Analytical method is developed for the contact mechanics problem based on two-dimensional elastodynamics. Stresses in the orthotropic graded medium are determined considering plane elasticity. Governing partial differential equations are solved analytically by the use of Galilean and Fourier transformations. The unknown functions appearing in the displacement components are determined considering the surface boundary conditions. The mixed boundary value problem is then reduced to a singular integral equation of the second kind which is solved numerically using an appropriate expansion-collocation technique. The contact stress results of the present analytical study are compared to those generated through a finite element analysis regarding the elastostatic case. Moreover, the contact stress results obtained in the elastodynamic case are also verified using some analytical results available in the literature. Successful agreement is attained in all the comparative analyses. Normal contact stress, lateral contact stress and punch stress intensity factors are calculated for various values of punch speed, coefficient of friction, in-homogeneity constant and orthotropic elastic constants. Results of the present study clearly indicates that the utilization of elastodynamic theory is crucial to get more realizable and accurate estimations of contact stresses especially in high speed sliding conditions. [ABSTRACT FROM AUTHOR]

In this study, the plane thermoelastic frictional contact problem between a magneto-electro-elastic (MEE) layer and a rigid cylindrical or flat punch is discussed. Punches are assumed to be perfect electromagnetic and thermal insulators; they slide over the layer with a small constant speed, and due to the friction, heat flux occurs. The general stress, displacement, and electromagnetic expressions for the contact problem are derived using the theory of thermoelasticity. Applying the boundary conditions, the contact problem is reduced to a Cauchy-type singular integral equation, which is then solved numerically using the Gauss-Jacobi integration formula. [ABSTRACT FROM AUTHOR]

in this work, we present an advanced interpolation method via the Vandermonde matrix for solving weakly singular Volterra integral equations of the second kind. The optimal rules for the node distributions of the two kernel variables were created to guarantee that the kernel's singularity was isolated. The unknown function is interpolated using three matrices: one of which is the monomial matrix, based on the Vandermonde matrix and Chebyshev nodes; the second is the known square Vandermyde matrix, and the third is the unknown coefficient matrix. The singular kernel is interpolated twice and transformed into a double-interpolated non-singular function through five matrices, two of which are monomials. A linear algebraic system can be obtained without using the collocation points by inserting the interpolated unknown function on the left and right sides of the integral equation. The solution of the obtained system yields the unknown coefficients matrix and thereby finds the interpolated solution. The obtained results from solving six examples are faster to converge to the exact ones using the lowest degree of interpolants and are better than those achieved by the other indicated method, which confirms the novelty and efficiency of the presented method. [ABSTRACT FROM AUTHOR]

Using the method of continuously distributed dislocations, we study the distribution of a pileup of screw dislocations against an interface between two elastic half-planes. We incorporate surface elasticity on the bimaterial interface using the continuum-based surface/interface model of Gurtin and Murdoch. The equilibrium condition is formulated in terms of a singular integral equation. The singular integral equation is solved numerically using the Gauss–Chebyshev integration formula to arrive at the dislocation distribution function and the number of dislocations in the pileup. [ABSTRACT FROM AUTHOR]

The method of continuously distributed dislocations is used to study the distribution of screw dislocations in a linear array piled up near the interface of a two-phase isotropic elastic thin film with equal thickness in each phase. The resulting singular integral equation is solved numerically using the Gauss–Chebyshev integration formula to arrive at the dislocation distribution function and the number of dislocations in the pileup. [ABSTRACT FROM AUTHOR]

The fracture mechanics of cortical bone has received much attention in biomedical engineering. It is a fundamental question how the material constants and the geometric parameters of the cortical bone affect the fracture behavior of the cortical bone. In this work, the plane problem for cortical bone with a microcrack located in the interstitial tissue under tensile loading was considered. Using the solution for the continuously distributed edge dislocations as Green's functions, the problem was formulated as singular integral equations with Cauchy kernels. The numerical results suggest that a soft osteon promotes microcrack propagation, while a stiff osteon repels it, but the interaction effect between the microcrack and the osteon is limited near the osteon. This study not only sheds light on the fracture mechanics behavior of cortical bone but also offers inspiration for the design of bioinspired materials in biomedical engineering. [ABSTRACT FROM AUTHOR]

• The effects of finite size and punch shape on the contact behavior of the graded thermoelectric strip are explored. • The analysis encompasses flat, triangular, and cylindrical rigid punches, each requiring unique collocation techniques. • The contact stresses can be precisely managed by fine-tuning multiple electric-thermo-elastic parameters. • For the triangle punch, the stress concentration is effectively mitigated by reducing the normalized electrical conductivity. • In the case of the rigid cylindrical punch, increasing the layer thickness is shown to effectively diminish contact stress. The performance enhancement of functionally graded thermoelectric (FGTE) devices is significantly influenced by contact studies of the FGTE materials. It is unclear how the finite thickness and the punch geometry influence the FGTE materials' contact behaviors. This paper investigates the frictionless contact problem between three types of rigid punches (flat, triangular, and cylindrical) and the FGTE strip with finite thickness. The electric-thermo-elastic parameters of the FGTE strip vary in the thickness direction according to an exponential function. Based on the Fourier integral transform and the transformation matrix method, the problem is transformed into the numerical solution of three sets of singular integral equations. The presence of singular features on either side of the punch demands the adoption of specific collocation strategies. The distribution of the normal current density, the normal energy flux, and the normal contact stress is obtained by adjusting multiple electric-thermo-elastic parameters. The contact stresses in the case of punches with varying shapes can be effectively controlled by modulating the coefficient of thermal expansion and the strip thickness, whereas the effect of the electrical conductivity, the shear modulus, and the thermoelectric load on these stresses depends on whether they are increased or decreased. [ABSTRACT FROM AUTHOR]

Auxetic honeycomb materials with negative Poisson's ratio received rapidly growing attention in recent years owing to their exceptional thermomechanical properties in constructing sandwich composites. The objective of this article is to investigate the transient thermal behavior and fracture risk of a honeycomb layer with a central crack subject to a sudden thermal shock by theoretical modeling. Two different material configurations along both orthogonal directions, as well as the conventional and auxetic hexagonal alumina honeycomb cells, are examined to illustrate their effects on the thermal stress intensity factors. To solve the thermoelastic governing equations subject to complex boundary conditions, the methodology of integral transform with singular integral equation is employed. The numerical results demonstrate that the auxetic honeycombs can reduce the thermal stress intensity factors significantly compared to their conventional counterparts. The ratio of stress intensity factors in auxetic to non-auxetic honeycombs under the same absolute value of the internal cell angle θ decreases monotonically with increasing | θ |. In addition, the effects of the relative density, crack length, and crack position are investigated. Our findings would provide a more comprehensive understanding of the honeycomb's fracture behaviors and contribute to the material design of the sandwich composites with honeycomb cores. [ABSTRACT FROM AUTHOR]

Strongly singular integral equations of order four have applications in fracture mechanics, and Gegenbauer polynomials have never been used to solve these equations. This motivated us to develop a Gegenbauer polynomial-based Galerkin method to solve a singular integral equation of order four. We first prove the problem's well-posedness. Then, we show the theoretical convergence of the numerical scheme and derive the rate of convergence and the error estimates. We validate the theoretical error estimates numerically in test examples. We implement the proposed method to a crack problem and compare it with existing results in the literature. • A residual-based Galerkin's method using Gegenbauer polynomials has been developed. • The well-posedness of a strong singular integral equation has been shown using the operator theory. • The error bounds, convergence as well as the rate of convergence has been theoretically derived. • Comparison between our proposed method and the existing collocation method has been shown numerically. [ABSTRACT FROM AUTHOR]

• The functionally graded layer with a central crack is considered. • The Guyer-Krumhansl model is employed to analyze the unclassical thermoelastic responses. • The effects of thermal lagging time and nonlocal length on the transient thermoelastic responses are examined. Recent experimental evidence proves the Guyer-Krumhansl (G-K) model's capability of predicting the nonclassical transient heat transfer process in heterogeneous composites. The objective of this article is to explore the dynamic thermal and fracture behaviors of a cracked orthotropic functionally graded material (FGM) layer subject to abrupt thermal shocks via the G-K equation. The thermal and elastic governing equations determined by the G-K model are transformed into the singular integral equations, which are calculated numerically to assess the non-Fourier transient temperatures and the resultant dynamic thermal stress intensity factors. Parametric comparisons are carried out for the effects of the non-Fourier thermal lagging time and thermal nonlocal length on the transient thermoelastic responses. The findings would benefit the applications of unconventional heat conduction theories and shed light on the scientific understanding of FGM's fracture behaviors. [ABSTRACT FROM AUTHOR]

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. • Development of a Hertz contact model with the effect of boundary supports. • The "local/global" method for local elastic and global structural deformations. • Incorporation of the gradient effect due to surface coatings. [ABSTRACT FROM AUTHOR]

We analyze a nonlinear integral equation for calculating free‐surface divergence that was proposed by Szeri (2017, https://doi.org/10.1002/2016JC012312). When given the temperature and heat flux at a free surface, the surface divergence can be calculated through a nonlinear singular Volterra‐type integral equation. The two given functions in the integral equation satisfy auxiliary conditions through a higher dimensional partial differential equation. We prove the existence and uniqueness of the solution of the integral equation. We also prove the local linear convergence of the corresponding Picard iteration method for solving the integral equation when the surface heat flux is a real‐analytic function of time. The rate of convergence is derived explicitly, which depends on the function of surface heat flux. Numerical examples are provided to validate the convergence performance. [ABSTRACT FROM AUTHOR]

In the present paper in space the optimal quadrature formulas with derivatives are constructed for approximate solution of a singular integral equation of the first kind with Cauchy kernel. Approximate solution of the singular integral equation is obtained applying the optimal quadrature formulas. Explicit forms of coefficients for the of optimal quadrature formulas are obtained. Some numerical results are presented. [ABSTRACT FROM AUTHOR]

A kind of Riemann problem of (λ , k) bi-analytic functions are studied by using the theory of boundary value problems for analytic functions. The solutions of Riemann problem of (λ , k) bi-analytic function are obtained. The conclusion may be applied to the quasi-static system of thermoelasticity. [ABSTRACT FROM AUTHOR]

We investigate the Tricomi boundary value problem for a differential-difference leading-lagging equation of mixed type with non-Carleman deviations in all in all arguments of the required function. A reduction is applied to a mixed-type equation without deviations. Symmetric pairwise commutative matrices of the coefficients of the equation are used. The theorems of uniqueness and existence are proved. The problem is unambiguously solvable.

The Ambartsumian equation, along with its generalizations, is one of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. The Ambartsumian equation plays an important role in the study of light scattering in media of infinite optical thickness. Nowadays the analytical solution of this equation is not known; therefore, the development of approximate methods for its solution is urgent. To solve the Ambartsumian equation, several iterative methods are proposed that are used in solving practical problems. Methods of collocations and mechanical quadratures have also been constructed, the substantiation of which has been carried out under rather severe conditions. In the previous work of the authors, a spline-collocation method for solving the Ambartsumian equation with zero-order splines was constructed and substantiated. The accuracy of this method is , where N is a number of collocation nodes. It is of considerable interest to construct an iterative method adapted to the smoothness of the coefficients and kernels of the equation. Light scattering in media of finite optical thickness is described by Ambartsumian equation systems, for an approximate solution of which it is necessary to construct and substantiate effective numerical methods. This study is devoted to the construction of such methods. The construction and substantiation of iterative methods for solving systems of Ambartsumian equations is based on a generalization of the continuous method for solving nonlinear operator equations. The method and its generalization are based on the Lyapunov stability theory and are stable against perturbation of the initial conditions, coefficients, and kernels of the equations being solved. An additional advantage of the continuous method for solving nonlinear operator equations is that its implementation does not require the reversibility of the Gateaux derivative of the nonlinear operator. In this work, spline-collocation methods with first-order splines are constructed for solving the Ambartsumian equation and systems of equations, and their justification is given. Model examples that illustrate the effectiveness of the methods are solved. Equations generalizing the classical Ambartsumian equations are considered. To solve them, the computational schemes of spline-collocation methods are constructed and substantiated. The results obtained can be used to solve a number of astrophysical issues. [ABSTRACT FROM AUTHOR]

Ambartsumian equation and its generalizations are some of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. An analytical solution to this equation is currently unknown, and the development of approximate methods is urgent. To solve the Ambartsumian equation, several iterative methods are proposed that are used in solving practical problems. Methods of collocations and mechanical quadratures have also been constructed and substantiated under rather severe conditions. It is of considerable interest to construct an iterative method adapted to the coefficients and kernels of the equation. This paper is devoted to the construction of such method. The construction of the iterative method is based on a continuous method for solving nonlinear operator equations. The method is based on the Lyapunov stability theory and is stable against perturbation of the initial conditions, coefficients, and kernels of the equations being solved. An additional advantage of the continuous method for solving nonlinear operator equations is that its implementation does not require the reversibility of the Gateaux derivative of the nonlinear operator. An iterative method for solving the Ambartsumian equation is constructed and substantiated. Model examples were solved to illustrate the effectiveness of the method. Equations generalizing the classical Ambartsumian equation are considered. To solve them, computational schemes of collocation and mechanical quadrature methods are constructed, which are implemented by a continuous method for solving nonlinear operator equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• A partially insulated crack in FGM coating/substrate structure is analyzed by the dual-phase-lag theory. • Integral transform coupled with singular integral equations are utilized. • Parametric investigations are conducted to benefit the optimizing of these structures. Serving as thermal barrier coatings in the harsh environment, functionally graded materials are usually subject to the extremely high temperature gradient, under which circumstance the classical Fourier's law breaks down and the non-Fourier effect becomes pronounced. The configuration of an orthotropic functionally graded coating bonded to the homogenous substrate containing a Griffith interface crack is considered in this work. The dual-phase-lag heat conduction theory is adopted to analyze the transient heat conduction and the resulting thermal stress intensity factors of the coating/substrate structure. The governing partial differential equations subjected to the complex thermal/mechanical boundary conditions are solved by the integral transform coupled with singular integral equations. A good agreement is achieved between the transient thermal stress intensity factors in the present work and steady-state values from previous literature. The influence of the thermal lags, nonhomogeneous parameters, and the thicknesses of two layers on the thermomechanical responses are investigated. [ABSTRACT FROM AUTHOR]

We consider the scattering of the H-polarized eigenwaves of a planar dielectric waveguide by a coplanar system of graphene strips in the THz range. The strips are placed along the centreline of the waveguide. Our treatment is based on the singular integral equations with the Nystrom-type discretization algorithm. Dependences of the scattering characteristics, near and far fields, are studied. Frequency scanning radiation patterns are presented. Maximum of the radiated power is observed near the plasmon resonances. The resonant frequency and main lobe level can be controlled by variation of the chemical potential. Applied optimization procedure allows to obtain the radiation pattern with the side-lobe level less than − 20 dB. The presented results can be used in designing of graphene leaky-wave antennas. [ABSTRACT FROM AUTHOR]

In this study, the stress analysis for an orthotropic thin film bonded to an orthotropic elastic substrate is addressed using both the analytical and finite element methods. The analytical method employs the integrodifferential formulation with the aid of membrane assumption. Utilizing the finite element method, the effect of orientation of the material principal directions are studied. The loading scenarios include a temperature gradient imposed on the film and a far-field uniaxial tension on the substrate. The results of current study indicate that the ratio of the film to the substrate stiffness plays a leading role in the film stress distribution. For the mechanical loading applied to the substrate, a soft thin film attached to a relatively stiffer substrate is preferred. The film can tolerate the induced thermal stresses as it is bonded to a softer host structure. The rotation angle of material orthotropy directions significantly affects the stress singularity near the film edges up to a certain extent. [ABSTRACT FROM AUTHOR]

A singular integral equation method is proposed to analyze the two-dimensional (2D) multiple cracks in anisotropic piezoelectric bimaterial. Using the Somigliana formula, a set of singular integral equations for the multiple crack problems are derived, in which the unknown functions are the derivatives of the generalized displacement discontinuities of the crack surfaces. Then, the exact analytical solution of the extended singular stresses and extended stress intensity factors near the crack tip is obtained. Singular integrals of the singular integral equations are computed by the Gauss–Chebyshev collocation method. Finally, numerical solutions of the extended stress intensity factors of some examples are presented and discussed. [ABSTRACT FROM AUTHOR]

Background. Ambartsumian equation and its generalizations are one of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. Ambartsumian equation plays an important role in the study of light scattering in media of infinite optical thickness. Nowadays the analytical solution of this equation is not known; therefore, the development of approximate methods for its solution is urgent. To solve the Ambartsumian equation, several iterative methods are proposed that are used in solving practical problems. Methods of collocations and mechanical quadratures have also been constructed, the substantiation of which has been carried out under rather severe conditions. In the previous work of the authors, a spline-collocation method for solving the Ambartsumian equation with zero-order splines was constructed and substantiated. The accuracy of this method is O(N−1) , where O(N) – number of collocation nodes. It is of considerable interest to construct an iterative method adapted to the smoothness of the coefficients and kernels of the equation. Light scattering in media of finite optical thickness is described by Ambartsumian equation systems, for an approximate solution of which it is necessary to construct and substantiate effective numerical methods. This study is devoted to the construction of such methods. Materials and methods. The construction and substantiation of iterative methods for solving systems of Ambartsumian equations is based on a generalization of the continuous method for solving nonlinear operator equations. The method and its generalization are based on the Lyapunov stability theory and are stable against perturbation of the initial conditions, coefficients, and kernels of the equations being solved. An additional advantage of the continuous method for solving nonlinear operator equations is that its implementation does not require the reversibility of the Gateaux derivative of the nonlinear operator. Results. In this work, spline-collocation methods with first-order splines are constructed for solving the Ambartsumian equation and systems of equations, and their justification is given. Model examples, which illustrate the effectiveness of the methods were solved. Conclusions. Equations generalizing the classical Ambartsumyan equations are considered. To solve them, the computational schemes of spline-collocation methods are constructed and substantiated. The results obtained can be used to solve a number of astrophysical issues.

• The unequal arms branch crack problem subjected to traction in plane elasticity was solved using singular integral equation. • To problem is formulated by making used of a point dislocation at origin and distribution dislocation at branches. • We proved the stability, convergence, order of convergence of method and got the estimate error for approximate solution. • To justify the correctness of results, we compared our results with the existing results, and obtained good agreement. • The COD was obtained without evaluating any integration and it depends on length, angle and the number of branch arms. In this paper, an unequal arms branch crack problem in a plane elasticity is treated. Using distribution dislocation function and complex variable potential method, the problem is formulated into a singular integral equation. The appropriate integration scheme, in which a point dislocation is set at the origin and the distribution dislocation, is applied through all arms of the branch crack to solve the obtained singular integral equations numerically. Stability, convergence, the order of convergence, and the error term of the solution are analyzed. Some numerical examples are examined to describe the behavior of stress intensity factors at the endpoints of each branch crack. [ABSTRACT FROM AUTHOR]

In this work, the fracture problem of an orthotropic functionally graded strip containing an internal crack parallel to its surfaces subjected to thermal shocks is examined. To eliminate the paradox of infinite heat propagation speed and take the microstructural interactions of thermal energy carriers into account, the non-Fourier, dual-phase-lag theory is employed to investigate the transient heat conduction and the associated thermal stresses response. By utilizing Laplace transform and Fourier transform, the thermoelastic problems are finally reduced to the Cauchy-type singular integral equations, which are solved by the Lobatto–Chebyshev technique numerically. The temperature field and thermal stress intensity factors are evaluated by the numerical inversion of Laplace transform to illustrate the effects of two thermal lags and nonhomogeneous parameters. The results show the fracture risks accompanied by the dual-phase-lag heat conduction can be higher than the classical analysis and it would be more conservative to consider non-Fourier effects in designing the orthotropic functionally graded materials. [ABSTRACT FROM AUTHOR]

Oblique surface waves incident on a fixed vertical porous membrane of various geometric configurations is analyzed here. The mixed boundary value problem is modified into easily resolvable problems by using a connection. These problems are reduced to that of solving a couple of integral equations. These integral equations are solved by a one-term or a two-term Galerkin method. The method involves a basis functions consists of simple polynomials multiplied with a suitable weight functions induced by the barrier. Coefficient of reflection and total wave energy are numerically evaluated and analyzed against various wave parameters. Enhanced reflection is found for all the four barrier configurations. [ABSTRACT FROM AUTHOR]

This paper studies a receding contact problem of a functionally graded layer laminate pressed against a functionally graded coated homogeneous half-plane substrate by a rigid flat indenter. The shear modulus of the functionally graded materials with a constant Poisson's ratio is modeled by an exponential function which varies along the thickness direction. Both the governing equations and the boundary conditions of the receding contact problem are converted into a pair of singular integral equations using the Fourier integral transforms, which are numerically integrated by the Chebyshev–Gauss quadrature. The contact pressure and the contact size at both contact interfaces are eventually obtained iteratively, as developed from the steepest descent algorithm. Extensive parametric studies suggest that it is possible to regulate the contact pressure and contact size by constructing the top layer from a soft functionally graded material. [ABSTRACT FROM AUTHOR]

An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann–Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin–Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin–Murdoch model with a non-zero surface tension and for the general Steigmann–Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss–Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported. [ABSTRACT FROM AUTHOR]

In the present paper in the the optimal quadrature formulas with derivatives are constructed for approximate solution of a singular integral equation of the first kind with Cauchy kernel. Approximate solution for the singular integral equation is obtained, applying the optimal quadrature formulas (OQF). Explicit forms of coefficients for the optimal quadrature formulas are obtained. Some numerical results are presented. [ABSTRACT FROM AUTHOR]

In this paper we study a problem with local displacement conditions on a segment of the degeneration line and with displacements on pieces of boundary characteristics in an unbounded domain, the elliptical part of which is the upper half-plane, and the hyperbolic part is the characteristic triangle. The uniqueness of the solution of the problem is proved using the extremum principle. The proof of the existence is based on the theory of singular integral equations, Wiener–Hopf equations, and Fredholm integral equations. [ABSTRACT FROM AUTHOR]

In this study, the author implemented integral transform technic, the Mellin-Barnes transforms method for solving singular integral equations and evaluation of certain integrals involving modified Bessel’s functions, moreover an algorithm in terms of the Fourier transforms to invert the Mellin-Barnes integral transform is presented. The obtained results reveal that the transform method is convenient and effective. Constructive examples are given throughout the paper. [ABSTRACT FROM AUTHOR]

• A model of the continuous contact problem between the thermoelectric layer and the rigid layer is established. • The body force of the thermoelectric layer is considered. • The continuous contact problem of finite thermoelectric layer is considered. • The influence of punch width and thermoelectric layer body force on critical load factor is discussed. In this study, the continuous contact problem of thermoelectric layer resting on a rigid substrate and loaded by a rigid punch is examined. Considering the body force of the thermoelectric layer, the model of stress distribution between thermoelectric layer and rigid layer is established. The general expressions for the stress between the thermoelectric layer and the substrate are given. Using the boundary conditions, the singular integral equation of the stress distribution is obtained and solved by Gauss-Chebyshev integral formula. The numerical results of the stress distribution between the rigid punch and the thermoelectric layer are obtained. It is found that the stress between the rigid punch and the thermoelectric layer exhibits singularity at the edge of the rigid punch. The influence of the parameters of the rigid punch and the thermoelectric layer on the critical load factor between the thermoelectric layer and the substrate is analyzed. The results show that the larger rigid punch width and the body force of the thermoelectric layer make it difficult to separate the thermoelectric layer from the rigid substrate. The results of this paper will provide a reference for studying the contact behavior of thermoelectric materials. [ABSTRACT FROM AUTHOR]

In this study, a frictional moving contact problem between an electrically conducting rigid cylindrical punch and a functionally graded piezoelectric material (FGPM) layer bonded to a piezoelectric homogeneous half plane is considered. The punch moves on the layer in the lateral direction at a subsonic constant velocity V and transmits the normal and the tangential loads. The mechanical and the electrical material properties of the layer are assumed to vary exponentially along the thickness direction. Using Fourier integral transform technique and Galilean transformation, the mixed boundary value problem is reduced to the singular integral equations in which the unknowns are the contact stress, the contact width, and the electric charge distribution. The singular integral equations are solved numerically applying the appropriate Gauss-Jacobi integration formulas. Numerical results for the contact width, the contact stress and the electric charge distribution are given as a solution. This work is the first study that investigates the moving contact problem of a graded piezoelectric materials. [ABSTRACT FROM AUTHOR]

Stresses due to frictional sliding contact between a rigid cylinder and laminated glass fiber composites are calculated in this study. A novel analytical formulation based on Cholesky decomposition, Fourier transforms, and singular integral equation is presented to solve the stress and displacement fields both at the contact patch and sub-surface, which can take into account transition of laminae with different fiber angle. An augmented finite element algorithm having an adaptive mesh refinement algorithm is also developed to verify the results of the analytical formulation. A perfect match between these results reveals the success of the new analytical formulation. Then, the formulation is implemented for various loading types and composite-related parameters to reveal the effects of those on the surface and sub-surface stresses. The results and discussions presented can be beneficial for the structural design of laminated composites under severe contact conditions. [ABSTRACT FROM AUTHOR]

A novel and effective method is proposed to determine the temperature and thermal stresses in the case of a finite Griffith crack lying perpendicular to the surface of an isotropic half-plane under uniform remote heat flux or a surface heat source. The surface of the half-plane and the two faces of the crack are otherwise thermally insulating and traction-free. For the heat conduction problem, the thermally insulating crack is simulated by a continuous distribution of heat dipoles, and the resulting Cauchy singular integral equation (SIE) is solved numerically to arrive at the associated density function. The original thermoelastic problem can be treated equivalently as a mode II Zener-Stroh crack (ZSC) under isothermal conditions. The net dislocation of ZSC is determined by the aforementioned density function and the crack is modeled by a pileup of edge dislocations. The resulting SIE is solved numerically leading to the mode II stress intensity factors at the two crack tips induced by uniform remote heat flux or surface heat source. The net dislocation of an actual Zener-Stroh crack can be designed in such a way that the crack will become neutral to the uniform heat flux or surface heat source. [ABSTRACT FROM AUTHOR]