Singular Integral Equation Approach Research Articles

Moving contact problems involving a rigid punch and a functionally graded coating

Frictional contact mechanics analysis for a rigid moving punch of an arbitrary profile and a functionally graded coating/homogeneous substrate system is carried out. The rigid punch slides over the coating at a constant subsonic speed. Smooth variation of the shear modulus of the graded coating is defined by an exponential function and the variation of the Poisson's ratio is assumed negligible. Coulomb's friction law is adopted. Hence, tangential force is proportional to the normal applied force through the coefficient of friction. An analytical method is developed utilizing the singular integral equation approach. Governing partial differential equations are derived in accordance with the theory of elastodynamics. The mixed boundary value problem is reduced to a singular integral equation of the second kind, which is solved numerically by an expansion-collocation technique. Presented results illustrate the effects of punch speed, coefficient of friction, material inhomogeneity and coating thickness on contact stress distributions and stress intensity factors. Comparisons indicate that the difference between elastodynamic and elastostatic solutions tends to be quite larger especially at higher punch speeds. It is shown that use of the elastodynamic theory provides more realistic results in contact problems involving a moving punch.

Singular integral equation method for 2D fracture analysis of orthotropic solids containing doubly periodic strip-like cracks on rectangular lattice arrays under longitudinal shear loading

The doubly periodic arrays of cracks represent an important mesoscopic model for analysis of the damage and fracture mechanics behaviors of materials. Here, in the framework of a continuously distributed dislocation model and singular integral equation approach, a highly accurate solution is proposed to describe the fracture behavior of orthotropic solids weakened by doubly periodic strip-like cracks on rectangular lattice arrays under a far-field longitudinal shear load. By fully comparing the current numerical results with known analytical and boundary element solutions, the high precision of the proposed solution is verified. Furthermore, the effects of periodic parameters and orthotropic parameter ratio on the stress intensity factor, crack tearing displacement, and effective shear modulus are studied, and an analytically polynomial estimation for the equivalent shear modulus is proposed in a certain range. The interaction distances among the vertical and horizontal periodic cracks are quite different, and their effects vary with the orthotropic parameter ratio. In addition, the dynamic problem is discussed briefly in the case where the material is subjected to harmonic longitudinal shear stress waves. Further work will continue the in-depth study of the dynamics problem of the doubly periodic arrays of cracks.

The effect of orthotropic material gradation on the plane sliding frictional contact mechanics problem

With the advent of new functional and intelligent non-conventional materials, understanding the behavior of these materials in different contact conditions over the conventional materials is one of the most crucial aspects in early design process for coated systems. Therefore, the sliding contact problem for a functionally graded orthotropic coating–substrate system and a rigid cylindrical punch is considered in this article to study the aforementioned aspects. The functionally graded orthotropic coating is modeled to be bonded to an isotropic substrate of finite thickness and is loaded by a sliding rigid cylindrical punch under plane strain conditions. For the material orthotropy, five different real orthotropic materials are utilized and the stiffness coefficients of each principal direction are graded separately. Navier’s equations are converted to ordinary differential equations using the Fourier integral transformation technique. Then, the algebraic equations are solved and the problem is cast into a singular integral equation. A parametric finite element analysis based on augmented contact method is also conducted. The normalized surface stress distributions and the normalized contact boundaries obtained from finite element analysis are validated with the results obtained from singular integral equation approach. The results of this study may be helpful for engineers in design and optimization of the characteristics of non-conventional coatings that are used as thermal or structural barriers and wear-resistant surfaces in engineering applications.

A study on the extent of the contact and stick zones in multiple contacts

In this paper, we analyze a general quasi-static two-dimensional multiple contact problem between two elastically similar half-planes under the constant normal (including applied moments) and oscillatory tangential loading utilizing the classical singular integral equations approach. Boundary conditions at nonsingular edges of discrete contact zones are applied and new side conditions are extracted and named “the consistency conditions” for multiple contacts. These conditions are mandatory for determination of the positions of the nonsingular edges of the contact and stick zones, when the number of them exceeds the number of the discrete contact and stick zones, respectively. Consequently, a mathematical relation between the pressure and shear distribution functions and between the extent of the contact and stick zones is obtained for the mentioned problem that shows all of the contact zones reach the full slip state simultaneously. Moreover, we show that for the weak normal loading, the approximated extent of the contact zones in multiple contacts with nonsingular edges may be estimated conveniently by assuming that the extent of the contact zones is the same as the overlapped extent in the free interpenetration figure.

Analytical and numerical solutions for heat transfer and effective thermal conductivity of cracked media

This paper presents analytical solutions to heat transfer problems around a crack and derive an adaptive model for effective thermal conductivity of cracked materials based on singular integral equation approach. Potential solution of heat diffusion through two-dimensional cracked media, where crack filled by air behaves as insulator to heat flow, is obtained in a singular integral equation form. It is demonstrated that the temperature field can be described as a function of temperature and rate of heat flow on the boundary and the temperature jump across the cracks. Numerical resolution of this boundary integral equation allows determining heat conduction and effective thermal conductivity of cracked media. Moreover, writing this boundary integral equation for an infinite medium embedding a single crack under a far-field condition allows deriving the closed-form solution of temperature discontinuity on the crack and particularly the closed-form solution of temperature field around the crack. These formulas are then used to establish analytical effective medium estimates. Finally, the comparison between the developed numerical and analytical solutions allows developing an adaptive model for effective thermal conductivity of cracked media. This model takes into account both the interaction between cracks and the percolation threshold.

Beam-Forming Capabilities of Waveguide Feeds Assisted by Corrugated Flanges

The physical mechanisms behind the formation of shaped-beam radiation patterns by flat horns (i.e., open ended waveguide feeds with finite-size corrugated flanges) are studied in a rigorous manner using the singular integral equation (SIE) approach. The problem is considered for TM polarization in the two-dimensional (2-D) formulation. First, dispersion curves of periodic grating are presented and used to analyze the modes supported by the waveguide flanges. This includes the surface wave, stopband, and leaky-wave regimes. The modes corresponding to the poles on the bottom “nonphysical” Riemann sheets are also considered. Finally, various types of radiation patterns produced by flat horns with gratings operating in different regimes are illustrated with an emphasis on generating sectoral beams being of interest for many practical applications. The results are validated via comparison with full-wave three-dimensional (3-D) analysis.

On the plastic zone size of solids containing doubly periodic rectangular-shaped arrays of cracks under longitudinal shear

Multiple cracks interaction plays an important role in fracture behavior of materials. A number of studies have been devoted to analytical and numerical analyses of the doubly periodic arrays of cracks. A very natural and highly accurate solution procedure is proposed to describe the interaction effect among the doubly periodic rectangular-shaped arrays of cracks. The proposed solution is implemented in the framework of continuously distributed dislocation model and singular integral equation approach. The accuracy of this solution is proved through a comparison of results from the present simulation and known closed form solutions. Further, the interaction effects among the periodic cracks on the plastic zone size and crack tip opening displacement are studied. It is found that the interaction distance among the vertical and horizontal periodic cracks is quite different.

Two-dimensional symmetric double contacts of elastically similar materials

The two-dimensional contact problem for an elastic body indenting an elastically similar half plane resulting in double contacts is important for various applications. In this paper, a generic quasi-static two-dimensional symmetric double contact problem with nonsingular end points between two elastically similar half planes, under the constant normal and oscillatory tangential loading, is analyzed. The classical singular integral equations approach is utilized to extract the pressure and shear functions in the contact zones; subsequently boundary conditions at end points are applied and a new side condition is derived and titled “the consistency condition” for symmetric double contacts. This condition is necessary for determining the extent of the contact and stick zones. Next, this analytical approach is applied to the symmetric indentation of a flat surface by two rigidly interconnected wedge-shaped punches.

On the contact mechanics of a rolling cylinder on a graded coating. Part 1: Analytical formulation

In this paper, the fully coupled rolling contact problem of a graded coating/substrate system under the action of a rigid cylinder is investigated. Using the singular integral equation approach, the governing equations of the rolling contact problem are constructed for all possible stick/slip regimes. Applying the Gauss–Chebyshev numerical integration method, the governing equations are converted to systems of algebraic equations. A new numerical algorithm is proposed to solve these systems of equations. Both the coupled and the uncoupled solutions to the problem are found through an implemented iterative procedure. In Part I of this paper, the analytical formulation of the rolling contact problem and the discretization of the governing equations are introduced for all assumed stick/slip regimes. A detailed discussion of the proposed numerical algorithm, the iteration procedure and the numerical results, obtained using the analytical formulation, are given in Part II.

On the rolling contact problem of two elastic solids with graded coatings

In this study, the rolling contact problem of two elastically similar cylinders coated with functionally graded materials is investigated. The singular integral equations approach is utilized to extract the governing equations of the problem. The governing equations are discretized into systems of algebraic equations by virtue of the Gauss–Chebyshev integration method. Comprehensive parametric study is performed to examine the effects of mechanical properties, coating thickness and external loads on the surface stress components, stick/slip transition, creep ratio and power loss. This study aimed to emphasize the capability of graded materials in improving the rolling contact fatigue of tribological components, decreasing the wear and increasing the efficiency of power transmitting systems.

Determination of higher order coefficients and zones of dominance using a singular integral equation approach

A method to determine higher order coefficients from the solution of a singular integral equation is presented. The coefficients are defined by σ rr ( r , 0 ) = ∑ n = 0 ∞ k n ( 2 r ) n - 1 2 + T n ( 2 r ) n , which gives the radial stress at a distance, r, in front of the crack tip. In this asymptotic series the stress intensity factor, k 0, is the first coefficient, and the T-stress, T 0, is the second coefficient. For the example of an edge crack in a half space, converged values of the first 12 mode I coefficients ( k n and T n , n = 0, … , 5) have been determined, and for an edge crack in a finite width strip, the first six coefficients are presented. Coefficients for an internal crack in a half space are also presented. Results for an edge crack in a finite width strip are used to quantify the size of the k-dominant zone, the kT-dominant zone and the zones associated with three and four terms, taking into account the entire region around the crack tip.

Analyticity of a nonlinear operator associated to the conformal representation of a doubly connected domain in schauder spaces

We consider a suitably normalized Riemann map gζ of the plane annulus with , to the plane doubly connected domain enclosed by the pair of Jordan curves , and we present a nonlinear singular integral equation approach to prove that the nonlinear operator which takes the pair of functions to the triple of functions is real analytic in Schauder spaces.

Analyticity of a Nonlinear Operator Associated to the Conformal Representation in Schauder Spaces. An Integral Equation Approach

We consider the Riemann map gζ,w of the complex unit disk to the plane domain 𝕀[ζ] enclosed by the Jordan curve ζ and normalized by the conditions gζ,w(0) = w, g′ζ,w(0) > 0, where w is a point of 𝕀[ζ], and we present a nonlinear singular integral equation approach to prove that the nonlinear operator which takes the pair (ζ, w) to the map g(–1)ζ,w ○ ζ is real analytic in Schauder spaces.

On the singular integral equations approach to the interface crack problem for piezoelectric materials

A plane strain problem for an interface crack with poling axis orthogonal to the crack plane is considered. The contact zone model with an artificial contact zone is considered for electrically permeable crack. By means of the method of singular integral equations, the quasi-invariance of the energy release rate with respect to the contact zone length is demonstrated. The appearance of a singularity of real power type instead of an oscillating singularity for insulated crack faces for most combinations of ceramics is shown. In a numerical way, the comparison of stresses and electrical displacements corresponding to the different interface crack models is employed.

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.

A closed crack tip model for interface cracks inthermopiezoelectric materials

The interface crack problem of a bimaterial thermopiezoelectric solid was treated byapplying the extended version of Strohs formalism and singular integral equation approach. Theinterface crack considered is subjected to combined thermal, mechanical and electric loads.Under the applied loading, the interface crack is assumed to be partially opened. Formulation ofthe problem results in a set of singular integral equations which are solved numerically. Thestudy shows that the contact zone is extremely small in comparison with the crack length. Basedon the formulation, some physically meaningful quantities of interest such as stress intensityfactors and size of contact zone for a particular material group are analyzed.

Cruciform Rigid Line Problem in an Infinite Plate

The objective of this paper is to provide the singular integral equation approach for the problem. Secondly, numerical results are present to show the interaction between rigid branches. In addition, some particular features in the rigid line problem are compared with those in the crack problem

A SINGULAR INTEGRAL EQUATION SOLUTION FOR THE LINEAR ELASTIC CRACK OPENING DISPLACEMENT OF AN ARBITRARILY SHAPED PLANE CRACK: PART I FINITE‐PART INTEGRAL SOLUTIONS

Abstract— The aim of the paper is to compute the local crack face displacements of a linear elastic body containing an arbitrarily shaped plane crack. From the crack face displacements the local stress intensity factors can be derived.The boundary value problem for a plane crack of arbitrary shape, embedded in a linear elastic medium, has been treated by several authors by the singular integral equation (SIE) approach. Their computations lead to a set of hyper‐singular integral equations for the Cartesian components of the unknown crack face displacements. To solve these equations the authors present a discretization procedure based on six‐node triangular finite elements. A total set of 24 finite‐part integrals defined over a triangular area can be developed. These 2D‐finite‐part integrals can be split into both a 1D‐regular and a 1D‐finite‐part‐integral by means of the polar coordinates so that they can be solved in closed form. Finally, the investigation of the SIEs is reduced to a discrete set of linear algebraic equations for the unknown nodal point values. The necessary steps will be demonstrated in detail. The derived closed‐form solutions will be offered in the text and in the appendices.

Singular Integral Equation Approach to Push-In Tests in Fiber-Reinforced Composites

Singular Integral Equation Approach to Push-In Tests in Fiber-Reinforced Composites

Strong interactions of morphologically complex cracks

Previous works on crack morphology have focused on such cracks as a kinked crack, a branched crack, and an inclined array of identical branched cracks. In this paper, the strong interactions between two cracks in two-dimensional solids under remote tension are investigated. Three morphological types are considered: kinks, branches and zigzags. The method of analysis follows the singular integral equation approach in which the deviations from the main cracks are modeled by distributions of dislocations. Investigations are made on the dependence of the stress intensity factors on the asymmetry of the crack configuration, the crack separation, and the shape of the cracks. The results show that (i) strong interactions can have significant effects on the mode mixity of the stress intensity factors, (ii) a small asymmetry of the crack configuration can cause significant changes to the stress intensity factors, and (iii) zigzag cracks with rectangular steps reduce the stress intensity factors more efficiently than those with triangular or trapezoidal steps.