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.     

Numerical analysis of multi-crack large-scale plane problems with adaptive cross approximation and hierarchical matrices

The problem of interaction of large number of cracks in a plate is considered by the method of singular integral equations (SIE). The corresponding system of SIE is solved by using Gauss–Chebyshev quadratures, which results in a large system of linear algebraic equations. The solution of the latter employs the adaptive cross approximation (ACA) technique that has not previously been applied for studying multi-crack large-scale plane problems. Therefore, several benchmarks problems with large number of cracks modelling periodical arrangements have been tested to investigate performance of the method; these include arrays of collinear cracks, parallel cracks, and double network of parallel cracks. Comparisons with analytical and numerical periodical solutions available for the mentioned cases reveal high accuracy and fast performance of the method. It is also applied for studying effective characteristics of bodies with up to 20,000 cracks and for accurate modelling of interaction of a macrocrack with thousands of microcracks.     

Fracture problems of rubbers: J-integral estimation based upon η factors and an investigation on the strain energy density distribution as a local criterion

The plane elasticity problem studied is of a circular inclusion having a circular arc-crack along the interface and a crack of arbitrary shape in an infinite matrix of different material subjected to uniform stresses at infinity. The solution of the problem is given using Muskhelishvili's complex variable method with sectionally holomorphic functions. First, the solution to the (auxiliary) problem of a dislocation (or force) applied at a point in the matrix with the circular inclusion partially bonded is derived fully in its general form by solving the appropriate Rieman-Hilbert problem. It is subsequently used as the Green's function for the initial problem by introducing an unknown density function associated with a distribution of dislocations along the crack in the matrix. The initial problem is then reduced to a singular integral equation (SIE) over the crack in the matrix only. The SIE is solved numerically by appropriate quadratures and the stress intensity factors reported for the arc-cut and a straight crack in the matrix for a range of values of the geometrical parameters.     

Electromagnetic scattering from cylindrical objects above a conductive surface using a hybrid finite-element-surface integral equation method

This work presents a novel finite-element solution to the problem of scattering from a finite and an infinite array of cylindrical objects with arbitrary shapes and materials over perfectly conducting ground planes. The formulation is based on using the surface integral equation with Green's function of the first or second kind as a boundary constraint. The solution region is divided into interior regions containing the cylindrical objects and the region exterior to all the objects. The finite-element formulation is applied inside the interior regions to derive a linear system of equations associated with nodal field values. Using two-boundary formulation, the surface integral equation is then applied at the truncation boundary as a boundary constraint to connect nodes on the boundaries to interior nodes. The technique presented here is highly efficient in terms of computing resources, versatile, and accurate in comparison with previously published methods. The near and far fields are generated for a finite and an infinite array of objects. While the surface integral equation in combination with the finite-element method was applied before to the problem of scattering from objects in free space, the application of the method to the important problem of scattering from objects above infinite flat ground planes is presented here for the first time, to our knowledge.     

Numerical solutions for polygonal cracks

An effective numerical scheme capable to deal with polygonal and branching cracks in a plane is proposed. It is suggested to decompose the general singular integral equation, SIE, for curvilinear cracks into a set of SIEs for straight cuts coinciding with straight crack segments. Solutions of SIEs are sought as bounded for all internal ends of the cuts and unbounded for the left end of the left cut and the right end of the right cut. The Gauss-Chebyshev quadrature is applied to each SIE that eventually leads to an over-determined system of linear algebraic equations followed by the application of the least squares method to solve this system. Stress intensity factors are calculated for some crack configurations. This scheme provides satisfactory accuracy although no correct asymptotic behaviour of the solution at internal ends is taken into consideration. The results are verified against the known results for branching cracks.     

Mode-III stress intensity factors for two circular inclusions subject to a remote uniform shear load

ABSTRACT

An analytical solution to the antiplane elasticity problem associated with two circular inclusions interacting with a line crack is provided in this article. A series solution for the stress field is derived in an elegant form by using complex variable theory in conjunction with the alternation method. Based on the superposition method, a singular integral equation (SIE) is established from the traction-free condition along the crack surface. After solving the SIE, the mode-III stress intensity factors (SIFs) can be obtained to quantify the singular behavior of the stress field ahead of the crack tips. Numerical results of the SIFs, when a crack is embedded either in the inclusion or in the matrix, are discussed in detail and displayed in graphic form.     

A meshless solution to two-dimensional convection–diffusion problems

An integral equation domain decomposition method has been implemented in a meshless fashion. The method exploits the advantage of placing the source point always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green's identities and the remaining equations are the derivatives of the first equation in respect to space coordinates. Radial basis function interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). Dual reciprocity method (DRM) has been applied to convert the domain integrals into boundary integrals, though the approach is general and can be applied without the DRM. The accuracy and robustness of the method has been tested on a convection–diffusion problem. The results obtained using the current approach have been compared with previously reported results obtained using the finite element method (FEM), and the DRM multi-domain approach (DRM-MD) showing similar level of accuracy.     

Impurity Effect in Silver-Point Realization

CCT-WG1 has recommended the sum of individual estimates (SIE) method to correct for the influence of impurities on the realization of temperature fixed points when a detailed impurity analysis is available. The method to estimate the uncertainty of the SIE has also been reported. On the other hand, most cells are fabricated from commercial fixed-point metals that often have no detailed impurity analysis, so the SIE calculation is impossible in that case. Due to this circumstance, and with the focus on the silver fixed point, a new fixed-point cell was fabricated in such a way that a portion of the silver ingot used was extractable during the silver casting. This portion was then analyzed by glow discharge mass spectrometry (GDMS), and the result used to calculate the SIE correction and its uncertainty. Temperature measurements during melting and freezing were collected using new and existing silver fixed-point cells under various conditions. These measurements were used to derive the slope of the silver freezing curve, from which the effect of impurities was evaluated by thermal analysis. The difference between the SIE and the thermal analysis method was evaluated to check the inaccuracy of the thermal analysis from the SIE point of view.     

Analytical modeling of imperfect bond between coated fibers and matrix material

The present study is intended to find the stress intensity factors (SIF) and strain energy release rates (SERR) at the tips of an interface crack in a non-homogeneous medium. The boundary-value problem governing a three-phase concentric cylinders model is used to analyze annular interfacial crack problems with Love's strain functions. The complex form of a singular integral equation of second kind is formulated using Bessel's functions in Fourier domain. Stress intensity factors (SIF) and total strain energy release rates (SERR) are calculated using Jacoby polynomials. For validity of the equations of Stress Intensity Factors, the Singular Integral Equation (SIE) of a three concentric cylinders model is reduced to the SIE for a two concentric cylinders model and results are compared with previous results of Erdogan.     

The radial basis integral equation method for solving the Helmholtz equation

A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=⁴Rln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results.     

海底沉积层中高阶界面波对多掩埋物散射分析

根据界面波散射迭代公式,并利用含多个异质体弹性动力学的散射积分方程,引出海底沉积层中界面波高阶模式对掩埋物的散射情况。根据建立的分层海底模型,对沉积层中传播的界面波进行模式分解。利用界面波传播的Green函数构建界面波散射迭代积分,并根据弹性波对多异质体散射理论,将界面波散射积分推广到对多目标的散射情况。由模式分解的结果,求解出了前三阶界面波对多目标的散射情况,并对界面波高阶模式传播及对不同目标的散射情况做了分析。     

Cloaking of metallic sub-wavelength objects by plasmonic metamaterial shell in quasistatic limit

Cloaking of metallic sub-wavelength cubic and cylindrical objects by isotropic, homogeneous plasmonic cover is investigated and numerically analysed. Significant reductions in both monostatic and bistatic radiation cross section (RCS) have been observed. In the case of cubic object, oblique incidence is also considered. Furthermore, cloaking of rectangular bar has also been investigated. The cloak in all these cases is a spherical plasmonic shell with dielectric and magnetic losses. The simulation results reveal that a lossy shell slightly degrades the cloaking performance; nevertheless the reduction in RCS is still sufficient for potential practical applications.     

An Electrically-Driven GaAs Nanowire Surface Plasmon Source

Over the past decade, the properties of plasmonic waveguides have extensively been studied as key elements in important applications that include biosensors, optical communication systems, quantum plasmonics, plasmonic logic, and quantum-cascade lasers. Whereas their guiding properties are by now fairly well-understood, practical implementation in chipscale systems is hampered by the lack of convenient electrical excitation schemes. Recently, a variety of surface plasmon lasers have been realized, but they have not yet been waveguide-coupled. Planar incoherent plasmonic sources have recently been coupled to plasmonic guides but routing of plasmonic signals requires coupling to linear waveguides. Here, we present an experimental demonstration of electrically driven GaAs nanowire light sources integrated with plasmonic nanostrip waveguides with a physical cross-section of 0.08λ(2). The excitation and waveguiding of surface plasmon-polaritons (SPPs) is experimentally demonstrated and analyzed with the help of full-field electromagnetic simulations. Splitting and routing of the electrically generated SPP signals around 90° bends are also shown. The realization of integrated plasmon sources greatly increases the applicability range of plasmonic waveguides and routing elements.     

Improving the accuracy of the magnetic field integral equation with the use of a new impedance matrix element formulation

The electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) are widely used in conjunction with method of moments for electromagnetic scattering analysis of three-dimensional conducting objects with closed surfaces. However, the MFIE suffers from an accuracy problem compared with the EFIE with the use of the Rao?Wilton?Glisson (RWG) basis function. This accuracy problem is more serious for objects with sharp edges or corners. To solve this problem, a new technique to compute the impedance matrix elements (IME) of the MFIE using an RWG basis function is presented here. Details to compute the IME and the advantage of this new formulation are displayed. In addition, the relationship between this new IME formulation and the formulation using the low-order curl-conforming basis function for the MFIE is given. Through the computation of the RCS of several relatively small sharp-edged conducting objects, it is shown that the accuracy of the MFIE can be greatly improved by the use of the new IME formulation.     

Electric and magnetic combined source integral equations for electromagnetic transmission problems

Various combined source integral equation (CSIE) formulations are developed for the analysis of electromagnetic scattering by three-dimensional arbitrarily shaped homogeneous penetrable objects. The considered CSIE formulations include the classical “electric” CSIE proposed by Mautz and Harrington in 1979, and its dual “magnetic” CSIE as well as mixed “electric–magnetic” and “magnetic–electric” CSIE formulations. Novel discretization schemes by utilizing the primary (Rao–Wilton–Glisson, RWG) and the dual (Buffa–Christiansen, BC) functions are applied to convert the integral equations into matrix equations. These schemes avoid the numerical problems that would appear if the CSIE formulations were discretized with the conventional Galerkin method using the RWG functions only.     

The Influence of Spatial Coherence on the Monochromatic Optical Transfer Function

It is widespread practice to measure the optical transfer function (OTF) by scanning the images of ‘simple’ objects. However, the illumination used in such systems is (generally) partially coherent, and not incoherent as is required by theory. The influence of spatial coherence is explicitly examined in order to obtain a general equation for the observed OTF for the case of one-dimensional test objects. The equation is applied to the case of a ‘perfect’ rotationally symmetric lens, using a slit object, and, on comparing the results with the true (incoherent) OTF of such a lens, significant differences have been found.     

A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials

This paper presents a meshless local boundary integral equation method (LBIEM) for dynamic analysis of an anti-plane crack in functionally graded materials (FGMs). Local boundary integral equations (LBIEs) are formulated in the Laplace-transform domain. The static fundamental solution for homogeneous elastic solids is used to derive the local boundary–domain integral equations, which are applied to small sub-domains covering the analyzed domain. For the sub-domains a circular shape is chosen, and their centers, the nodal points, correspond to the collocation points. The local boundary–domain integral equations are solved numerically in the Laplace-transform domain by a meshless method based on the moving least–squares (MLS) scheme. Time-domain solutions are obtained by using the Stehfest's inversion algorithm. Numerical examples are given to show the accuracy of the proposed meshless LBIEM.     

Solution of the exterior and interior Dirichlet problem of potential theory in a multiply connected domain

Through use of a complement to the solution of a heat conduction boundary value problem of Dirichlet type (presented classically in the form of a double layer potential) we obtain by means of simple sources singular integral equations (SIE) for exterior and interior multiply connected domains. Algorithms and a computer program were developed to obtain a numerical solution of the SIE.Translated from Inzhenerno-fizicheskii Zhurnal, Vol. 61, No. 5, pp. 858–862, November, 1991.     

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.     

An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation

The boundary integral equation approach has been shown to suffer a nonunique solution when the geometry is equal to a degenerate scale for a potential problem. In this paper, the degenerate scale problem in boundary element method for the two-dimensional Laplace equation is analytically studied in the continuous system by using degenerate kernels and Fourier series instead of using discrete system using circulants [Engng Anal. Bound. Elem. 25 (2001) 819]. For circular and multiply-connected domain problems, the rank-deficiency problem of the degenerate scale is solved by using the combined Helmholtz exterior integral equation formulation (CHEEF) concept. An additional constraint by collocating a point outside the domain is added to promote the rank of influence matrix. Two examples are shown to demonstrate the numerical instability using the singular integral equation for circular and annular domain problems. The CHEEF concept is successfully applied to overcome the degenerate scale and the error is suppressed in the numerical experiment.