Cauchy-type Singular Integral Equations Research Articles

Multi-method examination of contact mechanics in orthotropic layers under gravity

Analyzing the behavior of intelligent unconventional materials under diverse contact scenarios, in comparison to conventional materials, is a critical step in the initial design of engineering systems. This paper presents the development of analytical and numerical approaches for analyzing contact mechanics in a system comprising an orthotropic layer resting on a rigid foundation. Parametric analyses include the consideration of cylindrical and flat punch profiles. Analytical formulation utilizes integral transform methods to convert planar elasticity equations into a Cauchy-type singular integral equation of the second kind. A detailed description of the solution technique for the integral equation is provided, encompassing both the analytical formulation and the required discretization for obtaining the solution. Subsequently, a finite element approach (FEA) is employed to approximate contact bodies using a collection of finite elements, while contact boundaries are approximated by utilizing a set of polygons. Alternatively, the problem is addressed using the Multilayer Perceptron approach, a form of artificial neural network frequently applied in diverse machine learning applications, including scenarios involving contact problems. Finally, the resolution to the problem is achieved by employing the multilayer perceptron (MLP) method. The study yields determinations of contact stresses under the punch, the critical separation load resulting in the detachment of the orthotropic layer from the rigid foundation, and the corresponding separation distance. This analysis considers a range of dimensionless parameters and explored the behavior of diverse orthotropic materials. Comparisons between the results of the analytical method and computations from FEA and MLP reveal the exceptional accuracy attained by all three approaches. As the pioneer study employing three distinct approaches to examine continuous contact mechanics within an orthotropic layer under the influence of gravity, this research constitutes a valuable point of reference for upcoming scholars in the field.

Moving contact problem of a functionally graded orthotropic coated half plane

AbstractThis paper develops a frictional moving contact model for a functionally graded (FG) orthotropic layer pressed by a rigid cylindrical punch. The FG orthotropic layer is fully bonded to the isotropic half-plane. The punch moves to the left on the layer at a constant subsonic velocity and a shear stress arises in the contact zone according to the Coulomb friction law. General expressions of displacements and stresses are derived with the help of the Fourier transform and Galilean transformation. Using boundary conditions, the moving contact problem is reduced to a Cauchy-type singular integral equation, the unknowns of which are contact stress and contact width. Gauss–Jacobi integration formula is used to solve the obtained singular integral equation. The effect of some parameters and material properties on the contact width, contact stress and in-plane stress are given in graphical forms and detailed numerical interpretations are presented.

Interaction between a crack and a circular liquid inclusion

We study the interaction between a radial matrix crack and a circular compressible liquid inclusion embedded in an elastic matrix under uniform remote in-plane normal stresses. The crack is simulated by a continuous distribution of climbing edge dislocations. A Cauchy-type singular integral equation is derived by imposing the traction-free condition on the crack surfaces. The singular integral equation is solved numerically using the Gauss-Chebyshev integration formula, resulting in the mode I stress intensity factors at the two crack tips and the internal uniform hydrostatic tension within the liquid inclusion. Our detailed numerical results indicate that the crack will propagate toward or away from the liquid inclusion depending on the inclusion-crack distance, the crack length, the compressibility of the liquid inclusion, the Poisson’s ratio of the matrix and the external loading. This fact implies that the liquid inclusion exhibits softening as well as stiffening effects.

A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted Lp space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method’s validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others.

An application of Lobatto–Chebyshev method to a nonhomogeneous plane problem with two cracks

In this manuscript, Lobatto–Chebyshev method, which is an effective collocation method, is applied to a system of singular integral equations, which leads from a nonhomogeneous plane problem. It is assumed that there are two cracks in a nonhomogeneous medium. The problem is formulated in the view of the basics of the elasticity theory and boundary conditions of the problem. By using the method of singular integral equation, the problem is converted to a system of first kind Cauchy type singular integral equations. It is aimed to determine the stress intensity factors (SIFs) of the crack problem. It is seen that Lobatto–Chebyshev quadrature has many advantages in determining SIFs. To verify the validity of the method, the obtained results corresponding to the one crack case are compared with the results in literature.

Non-Fourier thermoelastic interaction of two collinear cracks in a functionally graded layer

Designed with a predetermined composition profile, functionally graded materials receive much attention in serving as the thermal barrier coating to resist severe thermal loading. By means of the dual-phase-lag theory, the non-Fourier effect is considered in establishing the theoretical thermoelastic model of the cracked layer under sudden thermal shocks. The novelty of the present work lies in revealing the interaction effects of two collinear Griffith cracks embedded in a functionally graded layer on the transient temperature and thermal stress fields under harsh conditions. Via employing the Fourier sine and cosine transforms, coupled with the Laplace transform, the dynamic mixed mode crack problem is transformed into a group of Cauchy-type singular integral equations. Numerical calculations are implemented to evaluate the transient temperature and stress intensity factors. The influences of the crack spacing, the nonhomogeneous parameters, and the thermal lags on the thermal and stress concentrations are explored. The results indicate that both mode I and II thermal stress intensity factors of the outer crack tips are noticeably higher than those of the inner crack tips, even up to 71% and 128%, respectively. A larger fracture risk may occur with the closer two adjacent cracks.

Error analysis of a residual-based Galerkin’s method for a system of Cauchy singular integral equations with vanishing endpoint conditions

In this paper, we develop Galerkin’s residual-based numerical scheme for solving a system of Cauchy-type singular integral equations of index minus N using Chebyshev polynomials of the first and second kind, where N is the total number of Cauchy-type singular integral equations in the system. Without theoretical analysis, a numerical scheme is not justified. Therefore, first, we prove the well-posedness of the system of Cauchy-type singular integral equations with the help of the compactness of an operator. Further, we derive a theoretical error bound and the order of convergence. Also, we show that the resulting system of equations obtained by applying the algorithm is well-posed together with an explicit representation of the solution in matrix form. Finally, we give some illustrative examples to validate the theoretical error bounds numerically.

Fracture analysis of multiple curved cracks in an orthotropic FGM coating bonded to an orthotropic substrate layer

In this paper, based on the distribution dislocation technique (DDT), multiple curved cracks placed in an orthotropic FGM layer bonded to an orthotropic layer and also interface cracks are considered. With the use of integration of Fourier transform the problem is reduced to a system of Cauchy-type singular integral equations which are solved numerically to compute the dislocation density on the surfaces of the cracks. The distribution dislocation is a powerful method to calculate accurate solutions to plane crack problems; especially this method is very good for solutions of curved crack patterns. Hence this technique allows considering any number of embedded and edge cracks with arbitrary curved shape and also interface cracks. Many numerical examples are given to validate our results with available analytical, finite element and experimental methods. The primary objective of this paper is to investigate the effects of the interaction of multiple curved cracks, the shape of crack, material orthotropy, nonhomogeneity parameters, the thickness ratios and the crack length on the modes I and II SIFs and energy release rate. Numerical results show that the normalized energy release rate and modes I/II SIFs decreasing with the both of the stiffness ratio of the coating and the nonhomogeneity parameter.

Transient heat transfer analysis of a cracked strip irradiated by ultrafast Gaussian laser beam using dual-phase-lag theory

With the rapid development of ultrafast laser technology, the effect of non-Fourier heat conduction on the transient process has received considerable attention. In the present article, the non-Fourier dual-phase-lag heat conduction theory is adopted to analyze the transient thermal process of a cracked strip subjected to ultrafast laser heating. The originality of this work lies in considerations of Gaussian type distribution of the laser beam and the insulated crack's disturbance to the heat flow. Fourier transform and Laplace transform are utilized to reduce the complex boundary-value problems to the Cauchy-type singular integral equation, which is then solved numerically by the Lobatto-Chebyshev technique. With the aid of the numerical inversion of Laplace transform, the transient temperature characteristics of the cracked strip are displayed graphically to illustrate the effects of the two non-Fourier thermal lags, laser incident position, and the Gaussian type heating source.

Sliding moving contact problem between a rigid cylindrical punch and a functionally graded orthotropic layer bonded to an isotropic homogeneous layer

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.

Induced aseismic slip and the onset of seismicity in displaced faults

Abstract We address aseismic fault slip and the onset of seismicity resulting from depletion-induced or injection-induced stresses in reservoirs with pre-existing vertical or inclined faults. Building on classic results, we discuss semi-analytical modelling techniques for fault slip including dislocation theory, Cauchy-type singular integral equations and the use of Chebyshev polynomials for their solution and an eigenvalue-based stability analysis. We consider slip patch development during depletion for faults with zero, constant static and slip-weakening friction, and our results confirm earlier findings based on numerical simulation, in particular the aseismic growth of two slip patches that may subsequently merge and/or become unstable resulting in nucleation of seismic slip. New findings include improved approximate expressions for the induced seismic moment per unit strike length and a description of the effect of coupling between the slip patches which affects both forward simulation and eigenvalue computation for high values of the ratio of fault throw to reservoir height. Our implementation based on analytical inversion and semi-analytical integration with Chebyshev polynomials is more efficient and more robust than our numerical integration approach. It is not yet well suited for Monte Carlo simulation, which typically requires sub-second simulation times, but with some further development that option seems to be within reach. Moreover, our results offer a possibility for embedded fault modelling in large-scale numerical simulation tools.

Thermoelastic contact problem of a rigid punch sliding on a functionally graded piezoelectric layered half plane with heat generation

In this study, the plane steady state thermoelastic frictional contact problem between a rigid cylindrical punch and a functionally graded piezoelectric material (FGPM) layer bonded to the homogeneous half plane is considered. The rigid punch is assumed to be a perfect electric conductor and thermal insulator and slides over the FGPM layer with a small constant speed and heat flux is generated due to the friction. Applying the Fourier integral transform technique and thermoelasticity, general stress, displacement, and electrical expressions for the thermoelastic piezoelectric contact problem are derived. Using the boundary conditions, the contact problem is reduced to Cauchy-type singular integral equations of the second kind in which the contact stress, the electrical displacement and the contact widths are unknown. The system of singular integral equation is solved numerically using Gauss-Jacobi and Gauss-Chebyshev integration formula. The effect of material inhomogeneity parameter, moving velocity, friction coefficient, external load, electric charge, punch radius on the contact stress, in-plane stress, electric displacement, and generated temperature are discussed in detail.

Dynamic response of a cracked thermopiezoelectric strip under thermoelectric loading using fractional heat conduction

This study is concerned with the dynamic behavior of a piezoelectric material strip with a parallel crack under thermal shock and transient electric loading. The governing thermal and electromechanical equations are exactly reduced to a system of Cauchy-type singular integral equations by applying Laplace and Fourier transforms and the dislocation density functions. Numerical examples are discussed in detail to show the effects of the applied heat conduction model, cracked strip configuration, and electrical load variables on the dynamic thermal results and the stress-electric displacement intensity factors. The results reveal that the overshooting phenomenon would be more evident with the longer relaxation time and higher fractional order, while the wave behavior becomes weaker and stronger, respectively. There are apparent inflection points on the dynamic stress intensity factors (DSIFs) and the dynamic electric displacement intensity factor (DEDF) curves for different configurations of the strip and the coefficients of the heat conduction model. In addition, the stress intensity factors are insensitive to the electric load which has a dominant influence on the electric displacement intensity factor.

Thermoelastic contact problem of a magneto-electro-elastic layer indented by a rigid insulating punch

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.

Transient non-Fourier thermoelastic fracture analysis of a cracked orthotropic functionally graded strip

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.

An effective method for the frictional thermoelastic contact of a cylindrical punch on a piezoelectric layer

In this study, the plane steady state thermoelastic frictional contact problem between a rigid cylindrical punch and a homogenous piezoelectric layer bonded to the rigid base is considered. The punch is assumed to be a perfect electric conductor and thermal insulator and slides over the piezoelectric layer with a small constant speed and heat flux is generated due to the friction. The general stress, displacement, and electrical expressions for the thermoelastic piezoelectric contact problem are derived using the theory of thermoelasticity and Fourier integral transform technique. Applying the boundary conditions, the present problem is reduced to Cauchy-type singular integral equations of the second kind in which the contact stress, the electrical displacement, and the contact width are unknown. The singular integral equations are solved numerically using Gauss–Jacobi and Gauss–Chebyshev integration formula with a developed effective method. The effect of moving velocity, friction coefficient, external load, electric charge, punch radius, and layer height on the contact stress, in-plane stress, electric displacement, and generated temperature are discussed in detail. This work is the first study that investigates a cylindrical punch on a thermoelastic contact of a piezoelectric layer.

Thermoelastic receding contact problem of a layer resting on a half plane with frictional heat generation

The plane steady-state thermoelastic receding double contact problem between a rigid cylindrical punch and a homogenous layer lying on a half plane is investigated in this paper. The frictional rigid insulated punch slides over the layer with a constant velocity and heat flux is generated due to the friction. The speed of the punch is taken so small that the inertia effects may be neglected. The general stress and displacement expressions for the thermoelastic contact problem are derived using the theory of thermoelasticity and Fourier integral transform technique. Applying the boundary conditions, the contact problem is reduced to Cauchy-type singular integral equations of the second kind in which the contact stresses and contact widths are unknown. The singular integral equations are solved numerically using the Gauss-Jacobi integration formula. The effect of thermoelastic parameters, friction coefficient, external load, punch radius on the contact stress, and contact widths are discussed in detail. This is the first study to investigate the thermal effect on the receding contact problem.

Thermoviscoelastic fracture analysis of a cracked orthotropic fiber reinforced composite strip by the dual-phase-lag theory

Fiber-reinforced polymer composites have gained tremendous popularities in various engineering applications due to their superior mechanical performances. This work is devoted to investigating the transient thermomechanical responses of an orthotropic fiber-reinforced strip containing a crack aligned along the fiber direction. The non-Fourier, dual-phase-lag theory is employed to account for the finite time required to complete the process of transporting heat and building local thermal equilibrium in composites, while the Generalized Maxwell viscoelastic model is utilized to characterize the constitutive law of the polymeric matrix and the fibers are treated to be elastic. By using Laplace transform and Fourier transform, the mixed, boundary-value problems are finally reduced to the Cauchy-type singular integral equations, which are solved by the Lobatto-Chebyshev technique numerically. The temperatures and thermal stress intensity factors in the time domain are evaluated through the numerical inversion of Laplace transform. Numerical results are displayed graphically based on graphite fiber reinforced epoxy composite, and detailed comparisons are made to investigate the effects of thermal lags, viscoelastic properties, and fiber volume fractions on the thermoviscoelastic responses.

Solution methods for integral equations - a survey

The theory of integral equations has been an active field of research for many years and is inextricably related with other areas of Mathematics such as complex and mathematical analysis, function theory, integral transforms and functional analysis. Integral Equations arise naturally in applications, in many areas of Mathematics, Engineering, Science and Technology and have been studied extensively both at the theoretical and practical level. It is significant to note that a MathSciNet keyword search on Integral Equations returns more than eleven thousand items. In this paper, we do a brief survey of the existing literature on methods of solving integral equations of Volterra and Fredholm type of the first, second and third kind, Cauchy type singular integral equations and integral equations over an infinite interval. The objective is to classify the selected methods and evaluate their applicability while discussing challenges faced by individual researchers in this field. We also provide a rather extensive bibliography for the reader who would be interested in learning more about various theoretical and computational aspects of Integral Equations.

Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function

This paper aims to present a Clenshaw–Curtis–Filon quadrature to approximate thesolution of various cases of Cauchy-type singular integral equations (CSIEs) of the second kind witha highly oscillatory kernel function. We adduce that the zero case oscillation (k = 0) proposed methodgives more accurate results than the scheme introduced in Dezhbord at el. (2016) and Eshkuvatovat el. (2009) for small values of N. Finally, this paper illustrates some error analyses and numericalresults for CSIEs.