Faber Series Expansion Research Articles

Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear

We investigate the anti-plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. The problem is first reduced to a non-homogeneous Riemann–Hilbert problem, the solution of which can be obtained by evaluating the associated Cauchy integral. A set of linear algebraic equations is obtained from the compatibility condition imposed on the resulting analytic function defined in the inhomogeneity and its Faber series expansion. Each of the unknown coefficients in the corresponding analytic functions can then be uniquely determined by solving the linear algebraic equations, which are written concisely in matrix form. The resulting analytical solution is then used to quantify the displacement jump across the debonded section of the interface as well as the traction distribution along the bonded section of the interface. In addition, our solution allows us to obtain mode-III stress intensity factors at the two crack tips. The solution to the anti-plane problem of a partially debonded elliptical inhomogeneity containing a confocal crack is also derived using a similar method.

An arbitrary piezoelectric inclusion with weakly and highly conducting imperfect interface

In the current study, we rigorously analyze an arbitrarily shaped piezoelectric inclusion surrounded by an infinite isotropic piezoelectric matrix subject to antiplane shear and in plane electric field loadings. The inclusion and matrix are separated by a homogeneously imperfect interface that characterizes a spring type interaction between the elastic and electric interfacial boundary conditions. Furthermore, the boundary conditions for a mechanically compliant, weakly conducting and mechanically compliant, highly conducting interface are incorporated into the analysis. Using complex variable techniques the potential function inside the inclusion is formulated as a Faber Series expansion and a system of linear algebraic equations for a closed form solution is developed for the corresponding Faber coefficients under a finite number of terms. Under this approach, expressions for both the elastic and electric fields are developed for the inclusion and matrix. The results are presented in exact form for an elliptic inclusion and numerically simulated for a finite number of terms for purposes of verification. Additionally, the cases of a square and star inclusion geometry are analyzed and results are presented numerically. The results clearly demonstrate that not only is the stress distribution inside the inclusion interface non-uniform, but that the magnitude of the peak stresses are highly dependent on the inclusion shape and imperfect interface condition.

Interaction of a Screw Dislocation With an Arbitrary Shaped Elastic Inhomogeneity

Abstract In this paper, the interaction between a screw dislocation and an arbitrary shaped elastic inhomogeneity with different material properties than the surrounding matrix is investigated. The exact solution to this problem is derived by means of complex variable methods and Faber series expansion. Specifically, the conformal mapping function maps the matrix region surrounding the inhomogeneity onto the outside of a unit circle in the image plane, while the analytic function defined in the elastic inhomogeneity is expressed in terms of a Faber series expansion. Once the series form solution is obtained, the stress fields due to the screw dislocation can be obtained. Also the image force on the screw dislocation due to its interaction with the elastic inhomogeneity is derived. Three examples of a screw dislocation interacting with (1) an equilateral triangular inhomogeneity, (2) a square inhomogeneity, and (3) a five-pointed star-shaped inhomogeneity are presented to illustrate how the stiffness of the triangular, square or five-pointed star-shaped inhomogeneity can influence the mobility of the screw dislocation.

Efficient approximation of the exponential operator for discrete 2D advection–diffusion problems

AbstractIn this paper we compare Krylov subspace methods with Faber series expansion for approximating the matrix exponential operator on large, sparse, non‐symmetric matrices. We consider in particular the case of Chebyshev series, corresponding to an initial estimate of the spectrum of the matrix by a suitable ellipse. Experimental results upon matrices with large size, arising from space discretization of 2D advection–diffusion problems, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques. Copyright © 2002 John Wiley & Sons, Ltd.

Strength prediction of composite laminates with multiple elliptical holes

Based on the classical laminated plate theory, a finite composite plate with multiple elliptical holes is treated as an anisotropic multiple connected plate. Using the complex potential method in the plane theory of elasticity of an anisotropic body, a series solution to the title problem is obtained by means of the Faber series expansion, the conformal mapping and the least squares boundary collocation techniques. Laminate strength is predicted by using the concept of characteristic curve and the Yamada–Sun failure criterion. The effects of the layups, the hole sizes, the ellipticity of the holes, the loading conditions, the relative distance between holes, the total number of holes and their locations on the strength of laminates are studied in detail. Some useful conclusions are drawn.

Stress analysis of finite composite laminate with multiple loaded holes

Based on the classical laminated plate theory, a finite composite plate weakened by multiple elliptical holes is treated as an anisotropic multiple connected plate. Using the complex potential method in the plane theory of elasticity of an anisotropic body, an analytical study concerned with the stress distributions around multiple loaded holes in finite composite laminated plates subjected to arbitrary loads was performed. The analysis makes use of the Faber series expansion, conformal mapping and the least squares boundary collocation techniques. The effects of plate and hole sizes, layups, the relative distance between holes, the total number of holes and their locations on the stress distribution are studied in detail. Some conclusions are drawn.

Stress analysis of finite composite laminates with elliptical inclusion

Using the complex potential method in the plane theory of elasticity of an anisotropic body, we determined the stress distribution in the finite laminates containing a soft/hard inclusion with the help of Faber series expansion and the least squares boundary collocation technique. The effect of lay-ups, laminate’s size, load conditions and the stiffness of inclusion on the stress distribution are studied in detail. Some useful conclusions are drawn.

Stress concentration of finite composite laminates weakened by multiple elliptical holes

Based on the classical plate theory, a finite composite laminated plate weakened by multiple elliptical holes is treated as an anisotropic multiple-connected plate. Using the complex potential method in the plane theory of elasticity of an anisotropic body, a series solution to the title problem is obtained with the help of the Faber series expansion, the conformal mapping and the least squares boundary collocation techniques. The effects of the layups, the ellipticity of holes, the relative distance between holes, total number of holes and their locations on the stress concentration are studied in detail. Some useful conclusions are drawn.

Thermoelasticity analysis of finite composite laminates weakened by multiple elliptical holes

A finite composite laminate weakened by multiple elliptical holes of arbitrary distribution, arbitrary orientation and arbitrary dimension, is treated as an anisotropic, finite multiple connected thin plate. Using the complex potential method in plane theory of heat conduction and elasticity of an anisotropic body, the analytical solution of a finite composite laminated plate subjected to arbitrary mechanical and thermal loads with multiple elliptical holes is obtained by means of the Faber series expansion, mapping and the least square boundary collocation technique. The effects of some parameters on the thermostress distribution are studied in detail. Some conclusions are drawn.

Load distribution of multi-fastener laminated composite joints

In this paper, a general and effective method is developed for load distribution of multi-fastener laminated composite joints by using the Faber series expansion (Cutis, 1971, Am. Math. Monthly78, 577–596; Kosmodamianskii and Chernick, 1981, Appl. Mech. 17(6), 94–100) and a complex potential method (Lekhnitskii, 1957, Anisotropie Plates, Gordon and Breach Science Publishers, New York; Muskhelishvili, 1966, Some Basic Problems of the Mathematical Theory of Elasticity (5th Edn), Nauka, Moscow) in the plane theory of elasticity. As an application, the effects of friction and fitting tolerance on load distribution of a four-fastener joint are studied. The method presented here can be generalized to solve other problems in this field.

Stresses in Orthotropic Laminate with Two Elastic Pins Having Different Fitting Tolerances

Using Faber series expansion and the complex potential method in the plane theory of elasticity of an anisotropic body, stresses in an orthotropic laminate with two elastic pins having snug and interference fitting tolerances are accounted for. It is shown from the numerical results obtained that a little interference fitting tolerance in a mechanically fastened joint may probably be acceptable for a laminated composite structure, and that the case of two elastic pins is at an intermediate stage between two extreme cases: that of two absolutely rigid pins and that of free hole edges or of two absolutely soft pins.

Stress concentration of a laminate weakened by multiple holes

A laminate weakened by multiple equal elliptical holes in series is treated as an anisotropic, infinite, multiply connected and thin plate. Using Faber series expansion and a complex potential method in the plane theory of elasticity of an anisotropic body, the general step to deduce the stress concentration in the laminate subjected to arbitrary in-plane loads at infinity is obtained. As a numerical example, the stress concentration of a particular laminate weakened by two, three or four equal circular holes in series is calculated. Moreover, the effects of the loading type, the number of holes and the relative distance between two neighbouring holes on the stress concentration of the laminate are discussed.