Set Of Singular Integral Equations Research Articles

Random Dynamic Responses of Two Parallel Interfacial Cracks Between A Functionally Graded Material Strip And Two Dissimilar Elastic Strips

Abstract An analytical approach is presented in this article for the random dynamic study of two parallel interfacial cracks in a functionally graded material (FGM) strip that is bonded between two distinct elastic strips. One of the parallel cracks is placed at the interface of the elastic strip I and the FGM strip, and another is at the interface of the FGM strip and elastic strip II. A stationary stochastic process of time is used to model the dynamic loadings that are applied to the crack faces. To find the solution, the FGM strip is splited into a number of sub-strips, and using an average method, the material properties of each sub-strip are reduced to random variables. A fundamental problem is formulated to find the solution. The boundary conditions are reduced to a set of singular integral equations employing the Fourier sine, Fourier cosine, and Laplace transforms, which are solved by using the Collocation method. Further, the analytical expressions of dynamic stress intensity factors (DSIFs) about the crack tips in the time domain are obtained with the help of the Improved Dubner and Abate's method. Finally, the Monte Carlo method is used to obtain the mathematical expectation and standard deviation of DSIFs. The outcomes of the present study are also verified. The unique aspect of this study is the pictorial illustration of mathematical expectation and standard deviation as functions of the number of sub-strips, functionally graded parameter, thickness of the strips, and length of parallel interfacial cracks.

Theoretical analysis of the interaction between blasting stress wave and linear interface crack under high in-situ stress in deep rock mass

This study theoretically investigates the scattering characteristics of blasting stress waves, treaded as elastic waves, at a linear interface crack in a deep rock mass with an even compressive in-situ stress field. First, the basic equations of the deep rock mass theoretically treaded as a homogeneous isotropic medium with an initial stress field are derived based on the linear elastic, finite deformation, plane strain, isotropic, and small deformation hypotheses, and incremental stress theory. The linear interface crack is theoretically treaded as two displacement-free boundaries, of which the interpolation displacements are described by introducing two dislocation density functions. Next, the mathematical model of the blasting stress wave scattering at the linear interface crack is established through a set of singular integral equations of the first type for these two dislocation density functions with Cauchy kernels. Finally, the dislocating and opening displacements of the linear interface crack and the displacement and effective stress fields of the scattered stress wave are numerically calculated. The principal stress field of the scattered stress wave is compared with the experimental data to qualitatively verify the rationality and correctness of the mathematical model. Theoretically, it is found that even compressive in-situ stress with an amplitude of tens of megapascals significantly affects the displacement and effective stress fields of the scattered stress wave near the linear interface crack. In conclusion, considering even compressive in-situ stress is greatly significant when studying the wave and scattering behaviors of deep rock masses. This mathematical model provides a theoretical guidance for the study of stress wave energy transformation and crack propagation in the process of rock blasting and mineral mining.

Thermal shock fracture in a cracked strip: Incorporating convective heat transfer between lateral surfaces and ambient environment

This paper presents a comprehensive analysis of transient temperature and thermal stress around a thermal insulated crack in a finite thermoelastic strip subjected to thermal shock based on the dual-phase-lag (DPL) heat conduction. A detailed investigation of the effect of convective heat exchange between the lateral surfaces of the strip and the ambient environment is conducted. By employing integral transform technique, the thermoelastic crack problem is reduced to a set of singular integral equations, and solutions for the transient temperature distribution and dynamic stress intensity factors (SIFs) at the crack tips are then obtained. Numerical results reveal a considerable temperature gradient near the heating surface when both the high convective heat transfer coefficient and thin strip are involved. Consequently, the SIFs in such scenarios where cracks are located close to the heating surface are significantly larger compared to those calculated by neglecting the convective heat loss. Furthermore, during the initial period following a thermal shock, the SIFs may exhibit higher magnitudes than the steady-state values. These findings emphasize the importance of accounting for the influence of convective heat exchange with the ambient environment in material design and optimization to enhance fracture resistance under transient thermal loading.

An analytical model for thermoelastic crack problems in a strip incorporating convective heat transfer between lateral surfaces and ambient environment

Most existing thermoelastic crack models commonly overlook the convective heat exchange occurring on the lateral surfaces of the structures being studied. Consequently, there is often a concern regarding the accurate estimation of temperature fields and the resulting thermal stresses. To address this concern, this paper presents a two-dimensional (2D) thermoelastic model for a cracked strip that incorporates convective heat transfer between the lateral surfaces and the ambient environment. The crack problem is reduced to a set of singular integral equations with Cauchy kernels which are solved numerically. Analytical solutions for temperature distribution, stress intensity factors (SIFs) and energy release rate at crack tips are obtained. The influence of the ratios of convective heat transfer coefficient to thermal conductivity and thickness to height of the strip on the key fracture parameters is investigated through the numerical results. It has been discovered that classical thermoelastic crack models often exhibit a significant underestimation of the SIFs and energy release rates for cracks located near the hot boundary of the strip. The analytical model proposed in this paper has significant potential to advance our understanding of thermoelastic fracture mechanics.

Terahertz-range plasmon and whispering gallery mode resonances in the plane wave scattering from thin microsize dielectric disk with graphene covers

We analyse the scattering and absorption of a terahertz-range plane wave by a thin circular dielectric disk with two graphene covers. Assuming that the thickness of the whole composite disk is much smaller than the free-space wavelength, we reduce the complexity of the problem with the aid of the generalized boundary conditions. This yields a set of singular integral equations for the effective electric and magnetic currents, induced on an equivalent zero-thickness disk. The adopted advanced numerical solution technique is a version of the method of analytical preconditioning, which uses weighted polynomials as expansion functions in the discretization of the integral equations. Then, the resulting matrix equation has Fredholm second-kind property that enables us to control the computational error by the matrix size and reduce it to the desired level. This accurate analysis reveals resonances on several types of natural modes, best understood via visualization of in-resonance near-fields. They are the plasmon-mode resonances of the graphene disk, perturbed by the presence of the dielectric filler, and the dielectric disk modes, perturbed by the graphene covers. Additionally, quasi-full disk transparency is observed if the transverse resonance condition is fulfilled. Special attention is paid to the tunability of the found effects with the aid of graphene's chemical potential.

The effect of specimen thickness on the dislocation-free zone model of fracture

We use the method of continuously distributed dislocations to study the dislocation-free zone (DFZ) model of fracture in an infinite plate of finite thickness under externally applied uniform anti-plane shear stress. We formulate the equilibrium condition in terms of a set of two coupled singular integral equations using the representation of the total interaction force between two screw dislocations in a thin plate obtained by Eshelby and Stroh (1951). This set of singular integral equations is then solved numerically using the Gauss-Chebyshev integration formula resulting in the dislocation distribution functions in the crack and in the two plastic zones, the total number of screw dislocations in the plastic zone, the DFZ condition for a crack in a thin plate and the local mode III stress intensity factor at the crack tip.

Schmidt method to study the disturbance of steady-state heat flows by an arbitrary oriented crack in bonded functionally graded strips

This article aims to investigate the impact of a partially insulated crack on vertical and horizontal components of heat flow. The crack is arbitrary oriented and its center is located at the interface of bonded finitely thick functionally graded strips. Employing the integral transformation, the thermoelastic equations are reduced to a set of singular integral equations. The unknown variables of these equations are the jump in temperature and displacements across the crack surfaces. Representing the unknowns as a series of Jacobi polynomials, the solutions of the singular integral equations are obtained by employing the Schmidt method which further helps to determine the analytical forms of crack tip heat flux intensity factors (HFIFs) and thermal stress intensity factors (TSIFs). The results of the present study are also validated for a particular case.To quantify the strength of heat flow components, the crack tip HFIFs and TSIFs as a function of crack angle, crack insulation parameter, strips’ width, and non-homogeneity parameters’ ratio are demonstrated pictorially.

Dynamic mode III stress intensity factors of multiple axisymmetric interfacial cracks in an FGM coated orthotropic layer

In the present study, mode III dynamic stress intensity factors (DSIFs) for the multiple axisymmetric interfacial cracks in an orthotropic layer with FGM orthotropic coating under transient torsional loading are formulated. It is assumed that the mass density and the shear modulus of the FGM orthotropic coating vary exponentially and power-law form along the thickness of the layer. At first, the solution for Somigliana-type dynamic rotational ring dislocation in the layer and its coating is obtained by using the Laplace and Hankel transforms. Then, the dislocation solution is used to derive a set of singular integral equations for a system of coaxial axisymmetric interfacial cracks, including penny-shaped and annular cracks. The integral equations are of Cauchy singular type, which are solved by Erdogan’s collocation method for dislocation density on the surface of interfacial crack and the results are used to determine DSIFs for axisymmetric interfacial cracks. Finally, several examples are provided to study the effects of the non-homogeneity constant, orthotropy parameter and thickness of FGM coating as well as the interaction of multiple interfacial cracks on the DSIFs.

Transient behavior of an orthotropic layer with imperfect FGM coating containing multiple interfacial and embedded cracks under anti-plane shear impact load

This paper investigates the transient response of an orthotropic strip with functionally graded (FG) coating weakened by multiple embedded and interfacial cracks. The first part of this paper is concerned with the dynamic response of embedded cracks in the orthotropic substrate with imperfect FG coating. In the second part, the solution of interfacial crack is obtained analytically to extract the dynamic stress intensity factors (DSIFs). In the third part, the proposed method is developed to examine the dynamic fracture behavior of coating–substrate system with an orthotropic FGM coating bonded to an orthotropic substrate through a layer of a bond-coat. In all parts, Laplace and Fourier transforms are used to reduce the problems to a set of singular integral equations with Cauchy kernels which are solved numerically in the Laplace transform domain. The dynamic stress intensity factors are obtained by the numerical Laplace inversion technique. Several examples are solved to obtain DSIFs with various crack lengths and the spring constant of imperfect bonding and material properties of the layers.

Surface contact behavior of functionally graded thermoelectric materials indented by a conducting punch

The contact problem for thermoelectric materials with functionally graded properties is considered. The material properties, such as the electric conductivity, the thermal conductivity, the shear modulus, and the thermal expansion coefficient, vary in an exponential function. Using the Fourier transform technique, the electro-thermoelastic problems are transformed into three sets of singular integral equations which are solved numerically in terms of the unknown normal electric current density, the normal energy flux, and the contact pressure. Meanwhile, the complex homogeneous solutions of the displacement fields caused by the gradient parameters are simplified with the help of Euler’s formula. After addressing the non-linearity excited by thermoelectric effects, the particular solutions of the displacement fields can be assessed. The effects of various combinations of material gradient parameters and thermoelectric loads on the contact behaviors of thermoelectric materials are presented. The results give a deep insight into the contact damage mechanism of functionally graded thermoelectric materials (FGTEMs).

Water Wave Scattering by a Bottom-Standing Thick Rectangular Barrier in the Presence of an Ice Cover

The problem of scattering of a normally incident surface wave train by an obstacle in the form of a thick rectangular vertical bottom-standing barrier submerged in finite-depth water with a free surface is extended here with an ice cover modelled as a thin uniform elastic plate. The problem is reduced to a set of singular integral equations of the first kind. The Galerkin approximation in terms of simple polynomials multiplied by appropriate weight functions whose form is dictated by the behavior of the fluid velocity near the barrier edges is used for solving the integral equations. The reflection and transmission coefficients are obtained numerically, and they are depicted graphically against the wavenumber for various ice cover parameters. It is observed that the barrier thickness plays a significant role in modelling of efficient breakwaters.

Identification of crack parameters and stress intensity factors in finite and semi-infinite plates solving inverse problems of linear elasticity

A method for the detection of a single or multiple straight cracks in finite and semi-infinite plane structures is presented. This allows both the identification of crack parameters such as length, position, and inclination angles with respect to a reference coordinate system and the calculation of stress intensity factors (SIFs). The method is based on strains measured at different locations on the surface of a structure and the application of the dislocation technique of linear elasticity. Cracks and boundaries are modelled by continuous distributions of dislocation densities. This approach gives a set of singular integral equations with Cauchy kernels, which are solved using Gauss–Chebyshev numerical quadrature, and, in contrast to e.g. a finite element calculation, spares a discretization of the structure. Once knowing the dislocation densities, the strain at an arbitrary point can be calculated. The crack parameters as well as external loads are determined by solving the inverse problem with a genetic and a simulated annealing algorithm. Once knowing loading and crack parameters, the SIFs are subsequently calculated. With the presented approach, the unknown parameters can be determined very accurately, being aware of some restrictions, which are thoroughly investigated.

Stress intensity factors of multiple axisymmetric interface cracks in an isotropic layer with FGM coating under torsional loading

Purpose The purpose of this paper is to provide a theoretical analysis of the fracture behavior of multiple axisymmetric interface cracks between a homogeneous isotropic layer and its functionally graded material (FGM) coating under torsional loading. Design/methodology/approach In this paper, the authors employ the distributed dislocation technique to the stress analysis, an FGM coating-substrate system under torsional loading with multiple axisymmetric cracks consist of annular and penny-shaped cracks. First, with the aid of the Hankel transform, the stress fields in the homogeneous layer and its FGM coating are obtained. The problem is then reduced to a set of singular integral equations with a Cauchy-type singularity. Unknown dislocation density is achieved by numerical solution of these integral equations which are employed to calculate the SIFs. Findings From the numerical results, the following key points were observed: first, for two types of the axisymmetric interface cracks, the SIFs decrease with growing in the values of the non-homogeneity. Second, the SIFs increase with increases in interface crack length. Third, the magnitude of the SIFs decreases with increases in the FGM coating thickness. Fourth, the interaction between cracks is an important factor affecting the SIFs of crack tips. Originality/value New analytical dislocation solution in an FGM coating-substrate system is developed.

Mathematical Modeling of Multi-Element Antenna Arrays with Chiral Metamaterials Substrates Using Singular Integral Equations

In this paper, a physical model of the multi-element antenna arrays (MEAA) has been considered and a self-consistent numerical method for solving the problem of current distribution on the MEAA surface with chiral metamaterials substrate has been proposed. The algorithm of the input admittance matrix elements for chiral layer based on the conductive left and right-handed helices has been developed and elements of the matrix surface impedances for investigated structures have been found. A set of singular integral equations with a Cauchy kernel for calculating the current density on the surface of the MEAA has been derived. A numerical solution of the set is a well-posed in the sense of Hadamard. Calculations of current distribution on the MEAA surface, impedance characteristics of a three-element antenna array and the dependence of the isolation levels between emitters on the chirality parameter have been performed and analysed. It has been shown that the use of chiral substrates can substantially increase the isolation between emitters.

Crack detection and parameter identification in finite and semi‐ infinite plane structures based on remote strain fields and the distributed dislocation technique

AbstractA method for the detection of cracks in finite and semi‐infinite plane structures is presented. This allows both the identification of crack position parameters, such as length, location and angles with respect to a reference coordinate system and the calculation of stress intensity factors (SIFs). The method is based on strains measured at different locations on the surface of a structure and the application of the dislocation technique. Cracks and boundaries are modelled by continuous distributions of dislocation densities. This approach gives a set of singular integral equations with Cauchy kernels, which can be solved using Gauss‐Chebyshev numerical quadrature. Once knowing the dislocation densities, the strain at an arbitrary point can be calculated. The crack parameters as well as external loads are parameters which have to be determined by solving the inverse problem with a genetic algorithm. Once knowing loading and crack parameters, the SIFs can be calculated.

Identification of crack positions and crack loading quantities from strain gauge data by inverse problem solution

A method for the detection of cracks in finite and semi-infinite plane structures is presented. This allows both the identification of crack position parameters such as length, location and inclination angles with respect to a reference coordinate system and the calculation of stress intensity factors (SIFs). The method is based on strains measured at different locations on the surface of a structure and the application of the dislocation technique. Cracks and boundaries are modelled by continuous distributions of dislocation densities. This approach gives a set of singular integral equations with Cauchy kernels, which can be solved using Gauss-Chebyshev numerical quadrature [Erdogan et al. (1973)]. Once knowing the dislocation densities, the strain at an arbitrary point can be calculated. The crack parameters as well as external loads are parameters which are determined by solving the inverse problem with a genetic algorithm. Once knowing loading and crack parameters, the SIFs can be calculated.

Semi-analytical study of microscopic two-dimensional partial slip contact problem within the framework of couple stress elasticity: Cylindrical indenter

In the present work, a size-dependent partial slip contact model of an elastic half-plane indented by a rigid cylindrical punch is established within the framework of couple stress elasticity. The proposed model introduces an internal material length to characterize the size effect of materials and assumes that the friction between two bodies is one of the Coulomb friction type. The Fourier transform in conjunction with the generalized stress function formulation is adopted to obtain the relationship between the coupled normal pressure and tangential traction in the form of a set of singular integral equations. Employing an iterative approach, the coupled system of integral equations is numerically solved and the effects of the couple stress and the friction coefficient upon the contact response are highlighted. The results show significant departure from the predictions of previous contact models based on the classical elasticity or the couple stress elasticity under frictionless condition. Moreover, it is demonstrated that the predicted contact responses of microindentation problems are underestimated if the material is assumed to have a smooth surface or treated as a classical elastic material.

Dynamic interaction of plates in an inhomogeneous transversely isotropic space weakened by a crack

AbstractThe problem of axisymmetric vibration of a flat thin rigid circular plate located inside a vertically exponentially graded, transversely isotropic material of infinite extent is addressed by means of a displacement potential method. The contact condition on one side of the foundation is assumed to be the perfect adhesion with the media but known to be faced by a penny‐shaped crack at the other side as it occurs in anchors. The mixed boundary value problem is formulated with the aid of Hankel integral transforms and is written in the form of a set of singular integral equations. The analytical procedure for the special case of vertical movement of the rigid plate results in a closed form solution. The solution is pursued numerically for the general elastodynamic case. The physical quantities, such as contact stress on the plate and the stress and displacement fields in the non‐homogeneous medium are obtained for different materials.

Torsion analysis of a hollow cylinder with an orthotropic coating weakened by multiple cracks

This paper presents an analytical solution for an infinite hollow cylinder with an orthotropic coating containing multiple cracks under Saint-Venant torsion by means of the distribution dislocation technique. First, the solution of a hollow cylinder with an orthotropic coating weakened by a Volterra-type screw dislocation is achieved with the aid of the finite Fourier sine transform. Next, the problem is reduced to a set of singular integral equations with a Cauchy type kernel in the hollow cylinder by using distribution dislocation technique. The singular integral equations are then solved numerically and the stress intensity factors of the crack tips and torsional rigidity in the whole domain are also obtained. Finally, several examples are presented to show the accuracy and efficiency of the dislocation technique in Saint-Venant torsion problems.

Theoretical analysis of thermal shock resistance of ceramic foam coatings

Ceramic foams have a high resistance to corrosion and wear. They also have a good thermal insulation performance because of their high melting point and low thermal conductivity. The thermal shock resistance of a ceramic foam coating with an edge crack under a sudden temperature variation is investigated. The dynamic thermal stress fields in the ceramic foam coating are obtained. Using the superposition principle, the crack problem of the ceramic foam coating is reduced to the solution of a set of singular integral equations. Propagation of the edge crack is analyzed. Effects of the relative density and thermal properties of the ceramic foam and of crack length on the thermal shock resistance are identified. The results obtained can be useful in designing thermal protective ceramic materials for thermal barrier coatings.