Evaluation of mixed mode stress intensity factors for interface cracks using EFGM

This paper presents the implementation of element free Galerkin method for the stress analysis of structures having cracks at the interface of two dissimilar materials. The material discontinuity at the interface has been modeled using a jump function with a jump parameter that governs its strength. The jump function enriches the approximation by the addition of special shape function that contains discontinuities in the derivative. The trial and test functions of the weak form are constructed using moving least-square interpolants in each material domain. An intrinsic enrichment criterion with enriched basis has been used to model the crack tip stress fields. The mixed mode (complex) stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The numerical results are obtained for edge and center cracks lying at the bi-material interface, and are found to be in good agreement with the reference solutions for the interfacial crack problems.     

A conductive crack and a remote electrode at the interface between two piezoelectric materials

An interaction of a tunnel conductive crack and a distant strip electrode situated at the interface between two piezoelectric semi-infinite spaces is studied. The bimaterial is subject by an in-plane electrical field parallel to the interface and by an anti-plane mechanical loading. Using the presentations of electromechanical quantities at the interface via sectionally-analytic functions the problem is reduced to a combined Dirichlet-Riemann boundary value problem. Solution of this problem is found in an analytical form excepting some one-dimensional integrals calculations. Closed form expressions for the stress, the electric field and their intensity factors, as well as for the crack faces displacement jump are derived. On the base of these presentations the energy release rate is also found. The obtained solution is compared with simple particular case of a single crack without electrode and the excellent agreement is found out. An auxiliary plane problem for open and closed cracks between two isotropic materials is also considered. The mathematical model of this problem is identical to the above one, therefore, the obtained solution is used for this model. It is compared with finite element solution of a similar problem and good agreement is found out.     

Single-frequency oscillations of non-linear systems with distributed parameters

A weakly non-linear oscillatory system with distributed parameters is investigated An asymptotic method of constructing a solution, which describes the oscillatory motions of the single-mode (single-frequency) approximation, which is usually implemented in practical problems, is described and justified. Constructive sufficient conditions are formulated and the closeness of the approximate single-frequency solution to the exact solution in an asymptotically long time interval is proved. Possible extensions of the structure of the perturbing functions are considered and the case of the finite-mode approximation is investigated. Solutions of specific problems, which are of practical interest, are constructed to illustrate the effectiveness of the single-frequency approximation method.     

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong‐form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second‐order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well‐known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher‐order derivatives using the approximation of lower‐order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second‐ and higher‐orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031–1053, 2015     

The problem of an inclusion in a three-dimensional elastic wedge

Fundamental solutions for a three-dimensional wedge are used to investigate problems of a thin, rigid, elliptic inclusion in a wedge. A regular asymptotic form is employed which has previously been used in contact problems for a wedge [1] and in problems of a crack in a wedge [2] in the case of an elliptic shape of the contact region or crack. The method is effective in the case of an inclusion which is sufficiently distant from an edge of the wedge when the known exact solution for the space [3] can be taken as the zeroth approximation. A numerical analysis and comparison of different characteristics of wedge problems is carried out.     

Static and extending interface cracks with contact zones in anisotropic as well as in piezoelectric bimaterials

An interface crack with an electrically permeable and mechanically frictionless contact zone in a piezoelectric bimaterial under the action of a remote mixed mode mechanical loading as well as thermal and electrical fields is considered in the first part of this paper. By use of the matrix‐vector representations of thermal, mechanical and electrical fields via sectionally‐holomorphic functions the problems of linear relationships are formulated and solved exactly both for an electrically permeable and an electrically impermeable interface crack. For these cases the transcendental equations and clear analytical formulas are derived for the determination of the contact zone lengths and the associated fracture mechanical parameters. A plane strain problem for a crack with a frictionless contact zone at the leading crack tip extending stationary along an interface of two semi‐infinite anisotropic spaces with a subsonic speed under the action of various loading is considered in the second part of this paper. By introducing of a moving coordinate system connected with the crack tip and by using the formal similarity of static and propagating crack problems the combined Dirichlet‐Riemann boundary value problem is formulated and solved exactly for this case as well and a transcendental equation is obtained for the determination of the real contact zone length. It is found that the increase of the crack speed leads to an increase of the real contact zone length and the correspondent stress intensity factors which increase significantly for a quasi‐Rayleigh wave speed.     

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.     

A film coating on a rough surface of an elastic body

A solution of the plane problem of the theory of elasticity for a film–substrate composite is solved by a perturbation method for a substrate with a rough surface. An algorithm for calculating any approximation, which ultimately leads to the solution of the same Fredholm equation of the second kind, is given. Formulae for calculating the right-hand side of this equation, which depends on all the preceding approximations, are derived. An exact solution of the integral equation in the form of Fourier series, whose coefficients are expressed in quadratures, is given in the case of a substrate with a periodically curved surface. The stresses on the flat surface of the film and on the interfacial surface are found in a first approximation as functions of the form of bending of the surface, the mean thickness of the film and the ratio of Young's moduli of the film and the substrate. It is shown, in particular, that the greatest stress concentration on the film surface occures on a protrusion of the softer substrate. ©2013     

An improved boundary element Galerkin method for three-dimensional crack problems

In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskin's Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.Dedicated to Prof. Dr.-Ing. W. L. Wendland on the occasion of his 50th birthday.A part of this work was done while the first author was a guest at the Georgia Institute of Technology and while the second author was partially supported by the NSF grant DMS-8501797.     

Finite element treatment of boundary singularities by augmentation with non-exact singular functions

A two-dimensional Poisson problem which contains both an interface and a reentrant corner is considered. For this problem the singular form of the solution at the reentrant corner is not known explicitly, with the result that a (nonexact) approximation to the singular form has to be calculated. The finite element method is applied to the Poisson problem, with the test and trial function spaces augmented with the nonexact singular functions. An error analysis for the nonexact augmentation is presented.     

The global instability of uniform flows in non-one-dimensional regions

The conditions for the instability of flows or states, which are independent of time and coordinates, in extended non-one-dimensional regions are considered in a linear approximation. An extension of the idea of global instability, previously introduced for the one-dimensional case, is given. A method is proposed for weakly unstable flows, which enables one to investigate under what conditions perturbations, which grow without limit with time, and which do not depend on the specific form of boundary conditions (provided they are not degenerate), exist. The case of a two-dimensional rectangular region is considered in detail.     

Stress intensity factors for the Brazilian disc with a short central crack: Opening versus closing cracks

Closed form expressions are obtained for the stress intensity factors (SIFs) in case of a Brazilian disc with a short central crack, the length of which does not exceed one fifth of the disc radius. The disc is loaded by uniform radial pressure along two finite symmetric arcs of its periphery. The solution is achieved using the method of complex potentials introduced by Kolosov and Muskhelishvili. The advantage of the expressions obtained is that they are valid both for cracks under opening mode as well as for closing cracks. For the first case (opening cracks) the results of the present study are compared with existing approximate solutions and it is concluded that the agreement is excellent as long as the length of the crack remains relatively small compared to the radius of the disc. Regarding the case of a closing mode crack the procedure proposed here (based on a recent alternative approach of the cracked Brazilian disc) leads to a physically acceptable deformed crack shape instead to an unnatural crack with overlapped lips. At the same moment the dependence of the SIFs on the properties of the material is eliminated.     

压电复合材料圆形夹杂中螺型位错和界面裂纹的干涉分析

研究了无穷远纵向剪切和面内电场共同作用下,压电复合材料圆形夹杂中螺型位错与界面裂纹的电弹耦合干涉作用．运用Riemann-Schwarz 对称原理,并结合复变函数奇性主部分析方法,获得了该问题的一般解答．作为典型算例,求出了界面含一条裂纹时基体和夹杂区域复势函数和电弹性场的封闭形式解．应用广义Peach-Koehler公式,导出了位错力的解析表达式．分析了裂纹几何参数和材料的电弹性常数对位错力的影响规律．结果表明,界面裂纹对位错力和位错平衡位置有很强的扰动效应,当界面裂纹长度达到临界值时,可以改变位错力的方向．该结果可以作为格林函数研究圆形夹杂内裂纹和界面裂纹的干涉效应．其公式的退化结果与已有文献完全一致．     

Stokes问题的混合有限元分析

在问题(ST)中,u=(u_1,u_2)~T是流体速度,p是压力. 设X_h及M_h分别为(H_0~1(Ω))~2及L_0~2(Ω)的有限元离散空间。且X_h(H_0~1(Ω))~2,M_h?     

A Finite Element Approach for the Simulation of Quasi-Brittle Fracture

In the context of a strong discontinuity approach, we propose a finite element formulation with an embedded displacement discontinuity. The basic assumption of the proposed approach is the additive split of the total displacement field in a continuous and a discontinuous part. An arbitrary crack splits the linear triangular finite element into two parts, namely a triangular and a quadrilateral part. The discontinuous part of the displacement field in the quadrilateral portion is approximated using linear shape functions. For these purposes, the quadrilateral portion is divided into two triangular parts which is in this way similar to the approach proposed in [5]. In contrast, the discretisation is different compared to formulations proposed in [1] and [3], where the discontinuous part of the displacement field is approximated using bilinear shape functions. The basic theory of the underlying finite element formulation and a cohesive interface model to simulate brittle fracture are presented. By means of representative numerical examples differences and similarities of the present formulation and the formulations proposed in [1] and [3] are highlighted. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interface cracks in anisotropic composites

The linear model equations of elasticity often give rise to oscillatory solutions in some vicinity of interface crack fronts. In this paper we apply the Wiener–Hopf method which yields the asymptotic behaviour of the elastic fields and, in addition, criteria to prevent oscillatory solutions. The exponents of the asymptotic expansions are found as eigenvalues of the symbol of corresponding boundary pseudodifferential equations. The method works for three‐dimensional anisotropic bodies and we demonstrate it for the example of two anisotropic bodies, one of which is bounded and the other one is its exterior complement. The common boundary is a smooth surface. On one part of this surface, called the interface, the bodies are bonded, while on the complementary part there is a crack. By applying the potential method, the problem is reduced to an equivalent system of Boundary Pseudodifferential Equations (BPE) on the interface with the stress vector as the unknown. The BPEs are defined via Poincaré–Steklov operators. We prove the unique solvability of these BPEs and obtain the full asymptotic expansion of the solution near the crack front. As a special case we consider the interface crack between two different isotropic materials and derive an explicit criterion which prevents oscillatory solutions. Copyright © 1999 John Wiley & Sons, Ltd.     

Two arbitrarily located normal forces and a penny‐shaped crack: a complete solution

The following problem is considered: a penny‐shaped crack is located in the plane z=0 of a transversely isotropic elastic space and interacts with two equal and opposite normal forces, which are located arbitrarily, but symmetrically with respect to the plane of the crack. An exact closed‐form solution is obtained and expressed in terms of elementary functions for the fields of stresses and displacements in the whole space. This kind of problem deemed to be intractable by the methods of contemporary mathematical analysis, and has never been attempted before, even in the case of an isotropic body. Copyright © 1999 John Wiley & Sons, Ltd.     

Explicit Approximations for Strictly Nonlinear Oscillators with Slowly Varying Parameters with Applications to Free-Electron Lasers

The first part of this paper summarizes the mathematical modeling of free-electron lasers (FEL), and the remainder concerns general perturbation methods for solving FEL and other strictly nonlinear oscillatory problems with slowly varying parameters and small perturbations. We review and compare the Kuzmak-Luke method and that of near-identity averaging transformations. In order to implement the calculation of explicit solutions we develop two approximation schemes. The first involves use of finite Fourier series to represent either the leading approximation of the solution or the transformation of the governing equations to a standard form appropriate for the method of averaging. In the second scheme we fit a cubic polynomial to the potential such that the leading approximation is expressible in terms of elliptic functions. The ideas are illustrated with a number of examples, which are also solved numerically to assess the accuracy of the various approximations.     

Approximation of a function of two variables by a product of functions of one variable on a given domain

We are concerned with the problem of uniform approximation of a continuous function of two variables by a product of continuous functions of one variable on some domain D. This problem have been examined so far only on a rectangular domain D = U × V, where U and V are compact sets. An algorithm to give a solution of this problem in the discrete case is available. We put forward an algorithm which in certain cases allows one to construct an approximate solution of the problem on a given domain (not necessarily rectangular). This approximate solution is built in the form of interpolating natural splines, which in turn are constructed by means of discrete approximation. Depending on the degree of the splines, the problem can be solved in classes of functions with appropriate degree of smoothness.     

An asymptotic approach in problems of crack identification

An asymptotic approach to solving problems of the identification of a rectilinear crack of small relative size is presented. The solution of the direct problem is reduced to solving a boundary integral equation. Using the proposed approach, its kernel is investigated, and the main part of the asymptotic form is singled out. The inverse problem of determining the crack parameters from prescribed information on the amplitudes of the displacement on the boundary of a layer is solved. Transcendental equations are obtained, from which the characteristics of a crack are determined in stages. Numerical results of the solution of the inverse problem are presented.