Search

Sandwich structures with ceramic honeycomb cores are extensively employed in thermal protection systems owing to their exceptional ability to resist high temperatures. This work aims at exploring the effect of cracking on the transient thermal process of the sandwich panel subject to impulsive and cyclic thermal loadings. Both the conventional and re-entrant hexagonal alumina honeycombs are considered for the core material. By the integral transform method, combined with singular integral equations, the transient temperatures of the whole sandwich panel are determined from the semi-analytical solution. The straightforward temperature difference of the crack face's midpoints is exploited to characterize the heat intensification near the crack. Parametric investigations are carried out for the internal cell angle, the relative density, crack length, crack position, and thickness of face sheets, which provides a better understanding of the honeycomb materials working in thermal protection systems. [ABSTRACT FROM AUTHOR]

In this article, the propagation and scattering of Rayleigh wave from an interfacial crack in a bilayer structure, a viscoelastic waveguide layer attached to a piezoelectric substrate, have been examined. The Principle of Superposition is used to formulate the problem. The Fourier transform method and stiffness matrix formulation have been used to arrive at singular integral equations. These singular integral equations are then reduced to a system of linear algebraic equations by applying the Gauss-Jacobi quadrature technique. A parametric study has been examined for phase velocity, attenuation, and dynamic stress intensity factor of mode-I and mode-II at both crack-tip. [ABSTRACT FROM AUTHOR]

Copyright of Chinese Journal of Applied Mechanics is the property of Chinese Journal of Applied Mechanics Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper investigates the bending fracture problem of a micro/nanoscale cantilever thin plate with surface energy, where the clamped boundary is partially debonded along the thickness direction. Some fundamental mechanical equations for the bending problem of micro/nanoscale plates are given by the Kirchhoff theory of thin plates, incorporating the Gurtin-Murdoch surface elasticity theory. For two typical cases of constant bending moment and uniform shear force in the debonded segment, the associated problems are reduced to two mixed boundary value problems. By solving the resulting mixed boundary value problems using the Fourier integral transform, a new type of singular integral equation with two Cauchy kernels is obtained for each case, and the exact solutions in terms of the fundamental functions are determined using the Poincare-Bertrand formula. Asymptotic elastic fields near the debonded tips including the bending moment, effective shear force, and bulk stress components exhibit the oscillatory singularity. The dependence relations among the singular fields, the material constants, and the plate's thickness are analyzed for partially debonded cantilever micro-plates. If surface energy is neglected, these results reduce the bending fracture of a macroscale partially debonded cantilever plate, which has not been previously reported. [ABSTRACT FROM AUTHOR]

This paper solves the dynamic contact problem when a rigid flat punch indents into an exponentially graded (FG) viscoelastic coated homogeneous half-plane. A harmonic vertical force is applied to the FG coating, and the solution is obtained for the stress and displacement for both the FG viscoelastic coating and the half-plane using the Helmholtz functions and the Fourier integral transform technique. By applying specific boundary conditions, the contact mechanics problem is converted into a singular integral equation of the first kind. This equation is then solved numerically using the Gauss-Chebyshev integration formulas. The analysis provides detailed insights into how various parameters—such as external excitation frequency, loss factor ratio, Young's modulus ratio, density ratio, Poisson's ratio, indentation load, and punch length—affect the dynamic contact stress and dynamic in-plane stress. [ABSTRACT FROM AUTHOR]

We study the unique solvability of a nonlocal boundary value problem with conditions of the Bitsadze–Samarskii and Frankl type for a mixed type equation with singular coefficients. [ABSTRACT FROM AUTHOR]

The H-polarized eigenmodes of dielectric waveguide scattering by a finite system of graphene strips are considered in the THz frequency range. The waveguide has one perfectly electric conducting wall. The solution is based on the singular integral equations with Nystrom discretization. Such an approach is efficient and has guaranteed convergence. Dependences of the scattering characteristics on the frequency and the radiation patterns are presented. We study the properties of periodic graphene strip grating with the same values of the chemical potential on every strip and with different values of the chemical potential on every strip. In the first case, the radiation patterns with relatively high side-lobe level are obtained. In the second case, the side-lobe level is reduced to −20 dB. [ABSTRACT FROM AUTHOR]

For the equation considered in some unbounded mixed domain, uniqueness and existence theorems are proved for a solution to the problem with the missing shift condition on the boundary characteristics and an analogue of the Frankl-type condition on the interval of degeneracy of the equation. [ABSTRACT FROM AUTHOR]

In this study, the plane continuous and discontinuous contact problems between a graded piezoelectric layer and a homogeneous piezoelectric half plane are considered. The layer is pressed by a rigid conducting flat punch with an electrical load. The problem is converted using Fourier integral transform into a singular integral equation. This equation is then solved numerically by applying the Gauss-Chebychev integration formula. The primary objective of this article is to investigate the effect of varying several parameters including the contact length, the mechanical and electrical loads, and the graded layer non-homogeneity parameter on the contact stresses and on the normal electric displacement. [ABSTRACT FROM AUTHOR]

A nonaxisymmetric contact problem on a circular interfacial inclusion of arbitrary shape under mixed boundary conditions with a piecewise homogeneous transversely isotropic space is considered. One edge of the inclusion is in smooth contact (tangential stresses are zero), while the other is in full contact with the medium. The problem is reduced to a system of five two-dimensional singular integral equations. For the inclusion, which has an axisymmetric shape under arbitrary loads, an exact solution of the system is obtained and its asymptotic properties are determined. On the basis of the latter, an efficient numerical-analytical method for solving the system of two-dimensional singular integral equations is proposed. The stress distribution in the vicinity of a coin-shaped inclusion is obtained and its features are revealed. [ABSTRACT FROM AUTHOR]

The axisymmetric contact problem on steady vibrations of a circular or ring punch on an elastic half-space without taking into account friction forces is considered. Convenient and efficient formulas for the displacements of the elastic foundation are obtained. These formulas were used in the formulation of integral equations for the unknown contact stress, which allowed to apply the method of orthogonal polynomials to the solution. The dependence of the contact stress amplitudes on the mass and vibration frequency of the punch is studied. [ABSTRACT FROM AUTHOR]

Piezoelectric semiconductor materials possess a unique combination of piezoelectric and semiconductor effects, exhibiting multifaceted coupling properties such as electromechanical, acoustic, photoelectric, photovoltaic, thermal, and thermoelectric capabilities. This study delves into the anti-plane mechanical model of an interface crack between a strip of piezoelectric semiconductor material and an elastic material. By introducing two boundary conditions, the mixed boundary value problem is reformulated into a set of singular integral equations with a Cauchy kernel. The details of carrier concentration, current density, and electric displacement near the crack are provided in a numerical analysis. The findings reveal that the distribution of the current density, carrier concentration, and electric displacement is intricately influenced by the doping concentration of the piezoelectric semiconductor. Moreover, the presence of mechanical and electric loads can either expedite or decelerate the growth of the crack, highlighting the pivotal role of external stimuli in influencing material behavior. [ABSTRACT FROM AUTHOR]

This study investigates the contact mechanics of a thin film laying on an elastic substrate within the context of couple-stress elasticity. It aims to introduce the effects of material internal length scale, which has proven an effective way of modeling structures at micro- to nano-scales, allowing to capture their size-dependent behavior. Specifically, stress analysis for a thin film bonded to a couple-stress elastic half-space is considered under plane strain loading conditions by assuming that both shear stress and couple tractions are exchanged between the thin film and the substrate. The problem is converted to a singular integral equation, which is solved by expanding the shear stress tractions as a Chebyshev series. The results show that the introduction of couple tractions decreases the shear stress tractions and the axial load in the thin film. When the characteristic length is sufficiently small, but still finite, the results for classical elastic behavior are approached. [ABSTRACT FROM AUTHOR]

This paper theoretically studies the axisymmetric frictionless indentation of a transversely isotropic piezoelectric semiconductor (PSC) half-space subject to a rigid flat-ended cylindrical indenter. The contact area and other surface of the PSC half-space are assumed to be electrically insulating. By the Hankel integral transformation, the problem is reduced to the Fredholm integral equation of the second kind. This equation is solved numerically to obtain the indentation behaviors of the PSC half-space, mainly including the indentation force-depth relation and the electric potential-depth relation. The results show that the effect of the semiconductor property on the indentation responses is limited within a certain range of variation of the steady carrier concentration. The dependence of indentation behavior on material properties is also analyzed by two different kinds of PSCs. Finite element simulations are conducted to verify the results calculated by the integral equation technique, and good agreement is demonstrated. [ABSTRACT FROM AUTHOR]

Two different direct methods are proposed to solve Cauchy singular integral equations on the real line. The aforementioned methods differ in order to be able to prove their convergence which depends on the smoothness of the known term function in the integral equation. [ABSTRACT FROM AUTHOR]

The objective of this paper is to explore the transient behaviors of fiber reinforced composites with two embedded collinear cracks subject to a sudden thermal shock. The orthotropic composites with both the vertically oriented and horizontally aligned fibers are examined to present a systematic comparison. By means of the integral transform methodology and singular integral equations, the thermal and elastic governing equations are solved numerically to determine the transient thermal process and stress intensifications around the crack tips. Parametric investigations are carried out for fiber volume fractions and crack distances. The findings would be beneficial to a more thorough knowledge of fibrous composites’ fracture characteristics under thermal shocks, which helps material design and structural optimization of the composites with fiber reinforcements.

In this study, moving contact problem for a rigid punch and a functionally graded orthotropic layer bonded to an isotropic homogeneous layer is solved, semi-analytically. The cylindrical punch moves steadily with a constant subsonic velocity over the upper layer and transmits concentrated normal and tangential forces. Using Fourier transform and Galilean transformation, the contact problem is converted to a Cauchy-type singular integral equation of the second kind, in which the contact stress and contact width are the unknowns. The numerical solution of the singular integral equation is obtained by using Gauss–Jacobi integration formulas. Numerical results for the contact stress, in-plane stress, and the contact width are given. [ABSTRACT FROM AUTHOR]

Nano-indentation experiments are an important means of studying the mechanical properties and sur- face morphology of materials. With the decrease of the contact area, the adhesion between the indenter and the contact surface of the specimen cannot be ignored. Therefore, the adhesion effect plays an important role in the contact problem under the action of the indenter. The functional graded piezoelectric material (FGPM) has the advantages of both graded and piezoelectric materials, and can effectively avoid contact damage and failure of coatings. The adhesive contact problem of FGPMs under conducting indenters was studied. With exponentially changing material parameters of the FGPM coating, based on the Maugis adhesive model, the control singular integral equation for the 2D frictionless adhesive contact problem of the FGPM coating under the conducting indenter, was obtained through the Fourier integral transform, and the Erdogan-Gupta numerical method was used to solve the equation. The effects of the adhesive stress, the graded parameter and the charge of the indenter on the electro-mechanical coupling response were obtained. The results provide a theoretical basis for improving the contact behavior of material surfaces with FGPM coatings, and help design piezoelectric structures and devices. [ABSTRACT FROM AUTHOR]

In the present work, adhesion has been incorporated in a model for receding contact between an elastic layer and a half plane under flat and parabolic punches. Based on the JKR model for adhesion and with the help of integral transformation, solution of the problem may be reduced to a system of singular integral equations. A numerical scheme has been proposed and verified. The effect of adhesion on contact size and the stresses has been demonstrated. [ABSTRACT FROM AUTHOR]

The work is devoted to the formulation and study of the solvability for a problem with missing conditions on the characteristic boundary of the domain and an analogue of the Frankl condition on the segment of the degeneracy for a hyperbolic equation. The difference between this problem and known local and nonlocal problems is that, firstly, a hyperbolic equation is taken with arbitrary positive power degeneracy and singular coefficients on the part of the boundary, and secondly, the characteristic boundary of the domain is arbitrarily divided into two pieces and the value of the desired function is set on the first piece, and the second piece is freed from the boundary condition and this missing Goursat condition is replaced by an analogue of the Frankl condition on the degeneracy segment, and the value of an unknown function on another characteristic boundary of the domain is also considered to be known. The conditions for the coefficients of the equation and the data of the formulated problem, ensuring the validity of the uniqueness theorem are found. The theorem of the existence of a solution to the problem is proved by reducing to the problem of solving a non-standard singular integral equation with a non-Fredholm integral operator in the non-characteristic part of the equation, the kernel of which has an isolated first-order singularity. Applying the Carleman regularization method to the received equation, the Wiener-Hopf integral equation is obtained. It is proved that the index of the Wiener-Hopf equation is zero, therefore it is uniquely reduced to the Fredholm integral equation of the second kind, the solvability of which follows from the uniqueness of the problem’s solution.

Background. In microwave technology, of great interest is the creation of amplifiers and generators based on thin-layered longitudinal inhomogeneities in rectangular waveguides, which are an integral part of many functional devices in the microwave and EHF ranges. Aim. Therefore, it is of practical interest to calculate one of these inhomogeneities in a rectangular waveguide. Methods. One of the methods for calculating such structures is the diffraction of an electromagnetic wave by a segment of a semiconductor film with negative differential conductivity on a narrow wall of a rectangular waveguide. Based on the boundary conditions, a singular integral equation for the electric component of the electromagnetic field was obtained. By the method of inversion of the integral equation and by means of expansion in terms of Chebyshev polynomials, the integral equation is given in a system of two algebraic equations. The coefficients of the system are calculated according to the theory of residues. Then an equation was obtained for calculating the transmission coefficient, which was calculated in the Matcad environment. Results. Based on the integral equation, the values for calculating the transmission coefficient for the structure under consideration are obtained. Conclusion. From the graphs obtained for the calculated transmission coefficient, it can be concluded that there are frequency zones where an electromagnetic wave is amplified in the structure. So, on the basis of the structure under consideration, it is possible to create amplifiers and generators of the microwave and EHF ranges.

Two different direct methods are proposed to solve Cauchy singular integral equations on the real line. The aforementioned methods differ in order to be able to prove their convergence which depends on the smoothness of the known term function in the integral equation.

We consider a boundary value problem for a system of integro-differential equations arising when modeling the influence of an electric field on a plasma layer. The paper presents an analytical solution of this problem, which is constructed with the use of theory of the Fourier transform of generalized functions and the Gakhov–Muskhelishvili method for solving singular integral equations and the Riemann boundary value problem. [ABSTRACT FROM AUTHOR]

For a singular integral equation on an interval of the real line, we study the behavior of the error of a delta-delta discretization. We show that the convergence is non-uniform, between order O (h 2) in the interior of the interval and a boundary layer where the consistency error does not tend to zero. [ABSTRACT FROM AUTHOR]

Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γ z e − s z cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f (0) . This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients. [ABSTRACT FROM AUTHOR]

In this paper, the electromechanical response of piezoelectric coating-functionally graded piezoelectric (FGP) interfacial layer–piezoelectric substrate system under the rigid spherical punch is considered. It is assumed that the piezoelectric coating and piezoelectric substrate are connected by the FGP interfacial layer which is simulated by a multi-layer model. The axisymmetric frictionless contact problem of piezoelectric coating–substrate system with FGP interfacial layer is formulated in terms of singular integral equation with the method of transfer matrix and Hankel integral transform technique. The effect of the gradient of material properties on the stress components and electric displacement components in piezoelectric coating–substrate system is gained by solving the equation numerically. The present results could serve as a useful reference for the design of piezoelectric coating–substrate system. [ABSTRACT FROM AUTHOR]

To improve the thermoelectric converting performance in applications such as power generation, reutilization of heat energy, refrigeration, and ultrasensitive sensors in scramjet engines, a thermoelectric film/substrate system is widely designed and applied, whose interfacial behavior dominates the strength and service life of thermoelectric devices. Herein, a theoretical model of a thermoelectric film bonded to a graded substrate is proposed. The interfacial shear stress, the normal stress in the thermoelectric film, and the stress intensity factors affected by various material and geometric parameters are comprehensively studied. It is found that adjusting the inhomogeneity parameter of the graded substrate, thermal conductivity, and current density of the thermoelectric film can reduce the risk of interfacial failure of the thermoelectric film/graded substrate system. Selecting a stiffer and thicker thermoelectric film is advantageous to the reliability of the thermoelectric film/graded substrate system. The results should be of great guiding significance for the present and upcoming applications of thermoelectric materials in various fields. [ABSTRACT FROM AUTHOR]

We use the method of continuously distributed edge dislocations to study the dislocation-free zone between an edge crack tip and a plastic zone in an isotropic elastic half-space with a traction-free surface under applied in-plane shear stress. The equilibrium condition is formulated in terms of a set of two coupled singular integral equations. The set of singular integral equations is solved numerically using the Gauss-Chebyshev integration formula resulting in the dislocation distribution function and the number of edge dislocations in each of the edge crack and plastic zone, the condition on the externally applied shear stress and the mode II stress intensity factor at the edge crack tip. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic contact response of a viscoelastic coated half-plane under a forced rigid punch. This punch, acting on the coating surface, is subjected to a concentrated vertical harmonic force. The coating and half-plane are assumed as viscoelastic materials which can be described by the linear hysteretic damping model. Using the Fourier integral transform technique and the asymptotic method, the dynamic contact problem is reduced to a Cauchy singular integral equation. By using appropriate numerical method, the dynamic contact stress and vertical impedance for the flat punch are calculated. It is found that the dynamic contact behaviors are not only affected by the external excitation frequency but also dependent on the material properties of the coated half-plane. [ABSTRACT FROM AUTHOR]

Важным этапом в становлении теории краевых задач стали предложенные А.М. Нахушевым в 1969 году нелокальные задачи нового типа, впоследствии названные у нас краевыми задачами со смещением, а за рубежом — задачами (проблемами) Нахушева. Они являются обобщением задачи Трикоми, а так же содержат широкий класс корректных самосопряженных задач. Эти задачи сразу вызвали большой интерес многих авторов. За последние годы исследования задач со смещением для уравнений смешанного типа ведутся особенно интенсивно. Но в этих работах краевые условия, как правило, содержат классические операторы, в то время как нелокальным краевым задачам, содержащим операторы более сложной структуры и операторы дробного интегро-дифференцирования. Настоящая статья посвящена исследованию вопроса однозначной разрешимости краевых задач со смещением для уравнения смешанного эллиптико-гиперболического типа. Сформулированы корректные краевые задачи со смещением для уравнения смешанного типа. В данной работе исследованы вопросы однозначной разрешимости задач со смешением для уравнения смешанного эллиптико-гиперболического типа в смешанной области Ω, ограниченной в полуплоскости y>0 гладкой кривой Жордана, а в полуплоскости y

Piezoelectric semiconductor materials possess a unique combination of piezoelectric and semiconductor effects, exhibiting multifaceted coupling properties such as electromechanical, acoustic, photoelectric, photovoltaic, thermal, and thermoelectric capabilities. This study delves into the anti-plane mechanical model of an interface crack between a strip of piezoelectric semiconductor material and an elastic material. By introducing two boundary conditions, the mixed boundary value problem is reformulated into a set of singular integral equations with a Cauchy kernel. The details of carrier concentration, current density, and electric displacement near the crack are provided in a numerical analysis. The findings reveal that the distribution of the current density, carrier concentration, and electric displacement is intricately influenced by the doping concentration of the piezoelectric semiconductor. Moreover, the presence of mechanical and electric loads can either expedite or decelerate the growth of the crack, highlighting the pivotal role of external stimuli in influencing material behavior.

This paper considers the size-dependent plane frictional contact problem of a homogeneous coated half-plane indented by a rigid punch based on the couple stress elasticity. Using the Fourier integral transform technique in addition to the boundary and compatibility conditions, the mixed-boundary value problem is converted into a singular integral equation of the second kind. The integral equation is further derived and normalized for the cylindrical and flat punch case profiles. Applying the Gauss–Jacobi integration formula, the resulting singular integral equation is reduced to a system of algebraic equations. The obtained results are first validated based on those published for the case of a frictionless contact problem of a half-plane indented by rigid cylindrical and flat punches and solved based on the couple stress theory. A detailed parametric study is then performed to investigate the effect of the characteristic material length, the friction coefficient, the layer height, the shear modulus, the indentation load and Poisson's ratio on the contact and in-plane stresses. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

At the moment, there is a scientific and technical problem of developing adequate methods and models of conformal radiating structures with chiral filling based on spiral elements, taking into account the frequency dispersion of macroscopic parameters, allowing for high accuracy of calculations with small computational resources. The paper shows a mathematical model of conformal radiating structures with chiral filling based on spiral elements, taking into account the frequency dispersion of macroscopic parameters. A singular integral equation is obtained with respect to an unknown function of the longitudinal distribution of the current density. The advantage of the developed method is the possibility of correct calculation of the characteristics of conformal radiating structures with filling from chiral metamaterials. The results obtained can be used to create conformal antennas based on chiral metamaterials consisting of spiral elements for mobile radio communication systems. [ABSTRACT FROM AUTHOR]

This seems to be the first exact closed form solution to the problem of a penny-shaped interface crack, subjected to an axisymmetric normal and tangential loading. The crack is located at the boundary between two bonded piezo-electric transversely isotropic half-spaces, made of different materials. We use the combination of Green's functions for 2 different half-spaces and Fourier transform. We derive first the governing equations, which are valid for a crack of arbitrary shape. The usual approach leads to 4 hypersingular integral equations, we arrive at 3 integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications or not noticed at all. In the particular case of axial symmetry, the problem is reduced to just one singular equation, for which exact closed form solution is known. We are not aware of any other publication with which our results can be compared. [ABSTRACT FROM AUTHOR]

In this paper, we prove Noetherian solvability and obtain the index formula for a singular integral equation with a Carleman shift in Besov spaces. We also obtain the existence of the solution of the singular integral equation as well as their explicit representations in Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, the Neumann boundary value problem for the Beltrami operator is explicitly solved in a circular ring domain, solvability conditions for this problem are also given in explicit forms. Moreover, the Neumann problem for second-order operators with the Bitsadze/Laplace operator as the main part as combinations of the Cauchy-Riemann and the Beltrami operators is investigated. [ABSTRACT FROM AUTHOR]

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]

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]