Pair Of Singular Integral Equations Research Articles

Interaction among offset parallel cracks in an orthotropic plane under thermo‐mechanical loading

AbstractThe article deals with the problem of three thermally insulated cracks embedded in a fibrous composite material subjected to thermo‐mechanical loading. Orthotropy is ensured by the unidirectional reinforcement of fiber. The considered infinite orthotropic plane is divided mathematically into three parts. The present mathematical model comprises of a central crack parallel to two symmetric collinear cracks at an offset distance to it. All the cracks under consideration are thermally insulated and aligned in the fiber direction. Using the Fourier integral transformed technique, the governing equations are reduced into a pair of singular integral equations. The Chebyshev polynomials are used to solve the integral equations of first kind with Cauchy kernel functions. Numerical values of the mode I stress intensity factors (SIFs) at the tips of the cracks are found for different lengths and distance between those. The effect of cracks' interaction on the SIFs is depicted graphically for the fiber‐reinforced composite (FRC) material constituted by carbon‐based fiber—Graphite and matrix—Epoxy.

Receding Contact Problem of Multi-Layered Elastic Structures Involving Functionally Graded Materials

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

On the Receding Contact Between a Homogeneous Elastic Layer and a Half-plane Substrate Coated with Functionally Graded Materials

Mechanical contact threatens the integrity of engineering structures. In particular, receding contact is one of the primary causes that are responsible for the delamination of multilayered elastic structures. This paper aims to analyze the receding contact between a homogeneous elastic layer and a half-plane reinforced by a functionally graded coating. The multilayered structure is indented by a rigid stamp of convex profile. Governing equations and mixed boundary conditions of the double contact problem are converted into a pair of singular integral equations by Fourier integral transforms. The dual integral equations are numerically solved by Gauss–Chebyshev quadrature for the contact pressure and contact length at both interfaces of contact. Taking a circular punch as a case study, the developed algorithm is first validated against classical models available in the literature. Extensive parametric studies are subsequently performed to illustrate the effects of indentation load, geometry and material properties of individual components. Numerical results suggest the possibility of optimizing multilayered structures by the introduction of functionally graded materials (FGMs) as coatings or transitional layers.

Effects of surface tension and Steigmann–Ogden surface elasticity on Hertzian contact properties

Surface tension, membrane stiffness and flexural rigidity all have been found to play significant roles in the determination of mechanical properties of nanosized materials and structures. In this work, Steigmann–Ogden theory of surface elasticity is applied for solving the contact properties between a rigid sphere and an elastic half-space. By integrating the surface Green’s function of the corresponding nonclassical Boussinesq solution, the mixed boundary-value problem is transformed into a pair of singular integral equations, which are subsequently reduced to a nonlinear algebraic system of equations by Gauss–Chebyshev quadrature. Extensive numerical studies are performed to investigate the coupling effects of surface tension, membrane stiffness and flexural rigidity on contact radius, contact pressure, contact stiffness and displacement and stress distributions in the half-space. For microsized rigid spherical indenters, the inclusion of Steigmann–Ogden surface theory results in appreciably smaller contact radius, lower contact pressure, displacements and stresses, thus leading to higher contact stiffness. This observation suggests the increasing importance of the half-space boundary in load-carrying capability as the strength of surface material properties increases or the indenter size decreases. Moreover, least-squares based regression analysis proves that, in the presence of surface effects, the distribution of contact pressure obeys an elliptical curve less than a half. In addition, the sharp corners that are conventionally found in classical contact stress and subsidence displacement are replaced by smooth transitions across the circular contact perimeter.

Two interfacial collinear Griffith cracks in thermo-elastic composite media

The objective of the article is to find the stress intensity factors and crack energy for a pair of collinear Griffith cracks situated at the interface of the two orthotropic materials under steady-state temperature field. The problem is reduced to a pair of singular integral equations, which are solved using Jacobi?s polynomials. Numerical computations are carried out for two different pairs of orthotropic materials for different particular cases, which are depicted through figures. The effect of material constants and temperature coefficients on the behavior of physical quantities viz., stress intensity factors and crack energy of the interfacial cracks is the key feature of the present article.

Double contact analysis of multilayered elastic structures involving functionally graded materials

This paper analyzes the frictionless double contact problem of a two-layer laminate pressed against a homogeneous half-plane substrate by a rigid punch. The laminate is composed of a homogeneous elastic strip and a functionally graded layer, perfectly bonded along their interface. The mechanical properties of the graded layer are modeled by an exponentially varying shear modulus and constant Poisson’s ratio. Both the governing equations and the boundary conditions of the double contact problem are converted into a pair of singular integral equations by Fourier integral transforms, which are numerically integrated by Chebyshev–Gauss quadrature. The contact pressure and the contact size at both the advancing and the receding contact interface are eventually obtained by an iterative algorithm, developed from the method of steepest descent. Extensive parametric studies suggest that it is possible to control contact stress and contact size by introducing functionally graded materials into multilayered elastic structures.

Rolling contact problem for an orthotropic medium

In this paper, the two-dimensional tractive rolling contact problem between a rigid cylinder and an orthotropic half plane is considered. A pair of singular integral equations is derived for the contact tractions using the Fourier transform technique. The governing integral equations are decoupled using the Goodman approximation and then solved numerically. Finally, the distribution of surface tractions is achieved in the contact region. This study emphasizes the significant effect of the orthotropic material parameters and the coefficient of friction on the contact tractions as well as the rolling contact characteristics for an orthotropic medium.

Interaction between Interfacial Collinear Griffith Cracks in Composite Media under Thermal Loading

AbstractThis article deals with the interactions between a central crack and a pair of outer cracks situated at the interface of orthotropic elastic half planes under thermo-mechanical loading. The mixed boundary value problem has been reduced to a pair of singular integral equations which has been solved numerically using Jacobi polynomial method. The interaction effects have been obtained in terms of stress magnification factors depending on the crack spacing and crack length. The phenomena of crack shielding and crack amplification have been depicted through graphs for different particular cases.

Multiple crack propagation along the interface of a non-homogeneous composite subjected to anti-plane shear loading

The present work concerns with the multiple crack propagation along the interface of two bonded dissimilar strips. The crack faces are subjected to anti-plane shear traction. Galilean transformation is employed to reduce the problem to a quasi-static one. Then, using Fourier transforms and asymptotic analysis, the quasi-static problem is reduced to a pair of singular integral equations. That are solved numerically, using Gauss-Chebyshev integration formulae. The values of the dynamic stress intensity factors are obtained and compared with the previous similar works. Further, a parametric study is introduced to investigate the effect of crack growth rate, geometric and elastic characteristics of the composite on the values of dynamic stress intensity factors.

Interfacial crack problem in layered soft ferromagnetic materials in uniform magnetic field

The magnetoelastic plane strain problem of an interfacial Griffith crack between two dissimilar soft ferromagnetic elastic materials subjected to a uniform magnetostatic field is considered within the framework of linear magnetoelasticity. By making use of the Fourier integral transform technique, the mixed boundary problem is then reduced to a pair of singular integral equations of the second kind. Solutions of the singular integral equations are obtained numerically by means of a Jacobi polynomial expansion method. Effects of the magnetic field, the combinations of the magnetic properties of materials and the geometric parameters on the magnetoelastic stress intensity factors in the vicinity of crack tip are shown graphically.

Cylindrical cavity with a circumferential edge crack subjected to a uniform shearing stress

This paper contains an analysis of the stress distribution in an elastic medium having cylindrical cavity with a circumferential edge crack when it is deformed by the application of uniform shearing stress. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a pair of singular integral equations. This pair of singular integral equations is solved numerically, and the stress intensity factor due to the effect of the crack size is calculated. Also the crack opening displacements are displayed in graphical forms.

Moving interfacial Griffith crack between bonded dissimilar media

The plane strain problem of determining strain energy release rate, crack energy, and crack‐opening displacement (COD) for a moving Griffith crack at the interface of two dissimilar orthotropic half‐planes is considered. The problem is reduced to a pair of singular integral equations of second kind which have finally been solved by using Jacobi polynomials. Graphical plots of the strain energy release rate, crack energy, and crack‐opening displacement for the problem in different particular cases are presented.

Interface crack problems in graded orthotropic media: Analytical and computational approaches

Interface crack problems in graded orthotropic media are considered using analytical and computational techniques. In the analytical formulation an interface crack between a graded orthotropic coating and a homogeneous orthotropic substrate is considered. The principal axes of orthotropy are assumed to be parallel and perpendicular to the crack plane. Mechanical properties of the medium are assumed to be continuous with discontinuous derivatives at the interface. The problem is formulated in terms of the averaged constants of plane orthotropic elasticity and reduced to a pair of singular integral equations which are solved numerically to compute the mixed mode stress intensity factors and the energy release rate. In the second part of the study, enriched finite elements are formulated and implemented for graded orthotropic materials. Comparisons of the finite element and analytical results show that enriched finite element technique is capable of producing highly accurate results for crack problems in graded orthotropic media. Finally, periodic interface cracking and the four point bending test for graded orthotropic solids are modeled using enriched finite elements and the results are briefly discussed.

Dynamic crack propagation between two bonded orthotropic plates

The problem of crack propagation along the interface of two bonded dissimilar orthotropic plates is considered. Using Galilean transformation, the problem is reduced to a quasistatic one. Then, using Fourier transforms and asymptotic analysis, the problem is reduced to a pair of singular integral equations with Cauchy‐type singularity. These equations are solved using Gauss‐Chebyshev quadrature formulae. The dynamic stress intensity factors are obtained in closed form expressions. Furthermore, a parametric study is introduced to investigate the effect of crack growth rate and geometric and elastic characteristics of the plates on values of dynamic stress intensity factors.

A long circular cylinder with a circumferential edge crack subjected to a uniform shearing stress

This paper contains an analysis of the stress distribution in a long circular cylinder of isotropic elastic material with a circumferential edge crack when it is deformed by the application of a uniform shearing stress. The crack with its center on the axis of the cylinder lies on the plane perpendicular to that axis, and the cylindrical surface is stress-free. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a pair of singular integral equations. This pair of singular integral equations is solved numerically, and the stress intensity factor due to the effect of the crack size is tabulated.

Dynamic Motion of Two Elastic Half-Spaces in Relative Sliding Without Slipping

Two isotropic linear elastic half-spaces of different material properties are pressed together by a uniform pressure and subjected to a constant shearing stress, both of which are applied far away from the interface. The shear stress is arbitrarily less than is required to produce slipping according to Coulomb’s friction law. Nonetheless, it is found here that the two bodies can slide with respect to each other due to the presence off a separation wave pulse in which all of the interface sticks, except for the finite-width separation-pulse region. In this type of pulse, the separation zone has a vanishing slope at its leading edge and an infinite slope at its trailing edge. Nonetheless, the order of the singularity at the trailing edge is small enough so as not to produce an energy sink. The problem is reduced to the solution of a pair of singular integral equations of the second kind which are solved numerically using a variation of the well-known method of Erdogan et al. (1973). Results are given for various material combinations and for a range of the remote shear-to-normal-stress ratio.

Frictional heating of a half-space with cracks. I. Single or periodic system of subsurface cracks

The paper deals with the two-dimensional problem of frictional heating for an elastic half-space containing one or a system of periodic subsurface cracks. The generation of frictional heating in the region of contact is estimated by the use of sliding speed, frictional coefficient and contact pressure. The problem is reduced to a pair of singular integral equations which are solved numerically. The influence of the location of cracks on the stress intensity factor is investigated.

The axisymmetric crack problem in a non-homogeneous interfacial region between homogeneous half-spaces

In this paper, the axisymmetric crack problem in a non-homogeneous interfacial region between two homogeneous half-spaces is considered. It is assumed that the shear modulus varies continuously between that of the two half-spaces; and the shear modulus for the interface region is approximated by μ = μ0 emz. By using Hankel transform technique the problem is reduced to a pair of singular integral equations. The solutions of the problem are obtained for different material combinations and loading conditions; and modes I and II stress intensity factors, and the direction of a probable crack growth are calculated.

Mutual Interference of Two Surface Cracks in a Semi-Infinite Body Due to Rolling Contact with Frictional Heating by a Rigid Roller

This paper deals with the two-dimensional thermoelastic contact problem of a rolling rigid cylinder of specified shape, which induces effects of friction and heat generation in the contact region, moving with constant velocity in an elastic half-space containing two surface cracks located close to each other. In the present temperature analysis, the speed of the moving heat source is assumed to be much greater than the ratio of the thermal diffusivity to the contact length. The problem is solved using complex-variable techniques and is reduced to a pair of singular integral equations which are solved numerically. Numerical results of stress intensity factors are obtained for the case of two parallel cracks. The variance in interference effects on the stress intensity factors with distance between two cracks, and the effects of the frictional coefficient, the sliding/rolling ratio and the distribution of heat generation on the results are considered.

Surface and internal cracks in a residually stressed plate

The problem of a surface or an internal crack in a plate which contains residual stresses is examined. The line spring model, which reduces a three-dimensional elasticity problem into a two-dimensional problem in plate theory, is used to model the crack. The Reissner plate theory, which takes into account transverse shear deformations, is used to model the plate. The formulation is based on Fourier Transforms which lead to a pair of singular integral equations that are solved numerically. The line spring method requires the plane strain solution to both the edge and internally cracked strip with crack surface loads representative of tension, bending, and the given residual stress distribution. For general use, plane strain solutions are presented for polynomial loading through the thickness up to the fifth order. Comparisons are made between the results given by the line spring model for the Reissner plate theory and the finite element method.