Order Of Singularity Research Articles

A note on Pageau et al. (1994) results for the order of stress singularities in multimaterial junctions

In 1994, Pageau, Joseph and Biggers Jr. evaluated “The order of stress singularities for bonded and disbonded three-material junctions”. Their work represented an important finding on multimaterial corners, since their formulation can be used to study the stress state on these junctions. Despite the relevance, the graphs of some results presented in their paper might lead to an improper interpretation, and consequently to a wrong asymptotic stress representation. This is evidenced and discussed in the present work, in which an analytical algorithm was implemented, allowing to reproduce, to complete and to clarify the original results. The physical interpretation of the divergences indicates, among other observations, that the failure mode associated with the highest stress singularity order in a multimaterial junction may change by only varying the Young’s modulus of one of the materials.

Задачи типа Гильберта для уравнения Коши–Римана с сингулярной окружностью в младших коэффициентах

The Cauchy-Riemann system of equations occupies a special place in the class of first-order elliptic systems. Such a system with lowest terms and a right-hand side is called the generalized Cauchy-Riemann system (GCRS), which can be conveniently studied by making a transition from a real space into a complex space. There are several different mathematical theories of equations that generalize the methods of the theory of functions of a complex variable. In this regard, the work of L. Bers, in which the integration operations with respect to a complex variable for a generalized system of Cauchy-Riemann type are generalized, should primarily be mentioned. This approach has received a well-known finalization in the theory of pseudo-analytic functions. In the works of G.N. Polozhiy, the theory of p-analytic functions was developed, which is close in its ideas to the works of L. Bers. Another, more progressive research area, called «generalized analytical functions», was developed by the school of I.N. Vekua and his followers (B.V. Boyarsky and others). Here, the idea of correspondence between the functions of a complex variable and the solutions of the generalized Cauchy–Riemann equation using the functional analysis techniques is developed. The Vekua theory is constructed on the assumption that the coefficients of the function’s lowest terms belong to the space of summable functions with degree р > 2. The coefficients of such systems may admit «weak» singularities limited by the requirement of p-integrability. Thus, the Vekua theory does not cover even equations with coefficients having first-order singularities. However, other problems, the lowest coefficients of which admit first- order singularities or «strong singularities», boil down to GCRS. In this study, Hilbert type problems are solved for the GCRS the lowest coefficients of which admit strong singularity in a circle. &nbsp

Fundamental Solutions for a Class of Multidimensional Elliptic Equations with Several Singular Coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple hypergeometric functions and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, such a formula is proved instead of a previously existing recurrence formula.The order of singularity and other properties of the fundamental solutions that are necessary for solving boundary value problems for degenerate second-order elliptic equations are determined

The strain energy density approach applied to bonded joints

Adhesively bonded joints can be treated as bi-materials characterized by a stress singularity promoted by both geometrical and constitutive discontinuities, very close to the point where the interface between two elastic solids intersects a traction-free edge. Different approaches have been proposed to predict their both static and fatigue strength. Most of them are based on the stress distribution at the interface between the adherent and the adhesive. More recently, energy density-based criteria have been developed to predict the static and fatigue strength of mono-material notched components, like welded joints. The most promising one is the strain energy density approach in which it is assumed that failure occurs when the strain energy density, averaged over a control volume of critical radius Rc surrounding the singularity point, will reach a critical value. Advantages are different. Among the others, the strain energy density doesn’t depend on the singularity order and doesn’t require a fine mesh.This contribution is aimed at applying the strain energy density criterion to bonded joints.

The stress singularity order in a composite wedge of functionally graded materials under antiplane deformation

In this paper, finding the order of singularity in multi-wedge systems containing elements made of functionally gradient material (FGM) with an angular gradient under antiplane deformation is studied. These elements are proposed to be modeled by means of multi-wedge composite, where the shear modulus changes from wedge to wedge according to a certain functional dependence (in

Energy Spectrum of Linear Internal Wave Field in the Vicinity of Continental Slope

The purpose of the research was to investigate two-dimensional modeling of efficiency of mixing, resulting from the reflection of a linear internal wave field (IWF) off a continental slope. Efficiency of deep ocean mixing was associated with the energy balance of the radiating IWF into an interior of the ocean in the vicinity of a sloping bottom topography. Since waves are generated not only at the fundamental frequency but also at all of its harmonics ωn = nω less than buoyancy frequency N and greater than Coriolis frequency f, our analysis includes, in general, an infinite number of discrete internal wave modes n satisfying the dispersion relationship for internal waves. However, since we are interested only in the radiating part of the field, the mode numbers are limited. Due to multiple singularities of order two caused by resonance in the vicinity of critical slope, the energy is visualized in Lδ -norm with δ > 2. Research results include the visualization of the effects of the continental slope and the Earth’s rotation on resulting energy in the vicinity of the slope.

Thermoelasticity if isotropic solids containing non-deformable thread-like inclusions

The paper derives integral equations of heat conduction and thermoelasticity of isotropic solids with non-deformable perfectly thermally conducting thread-like inclusions. It is observed that, in spite of the order of singularity, the integral equations obtained are hypersingular due to the symmetry of the kernels. Non-integral terms of these equations are derived. A boundary element method scheme for numerical solution of formulated problems is proposed. A numerical example is provided.

A new error upper bound formula for Gaussian integration in boundary integral equations

This paper proposes a new error upper bound formula for the Gaussian integration of the near-singular integral using the Boundary Element Method. First, this study found through numerical tests that the maximum relative error of the Gaussian integration has a downward concave shape but an approximately linear relationship with the relative distance, which is defined as the ratio of the distance from the source point to the element over the element length in a semi-logarithmic plot. Thus, the error upper bound can be defined as a line that closely approaches the computed error data points from the upper side. This line can be obtained by connecting two specified data points that are located outside, but very close to, the considered error range. Further research indicates that one parameter of the fitted line has a linear relationship with the number of Gaussian integration points and singularity orders and the other parameter can be treated as a constant, which together make the proposed Gaussian integration error upper bound formula widely applicable. Compared to the Lachat and Watson criterion, the proposed formula requires fewer integration points when the source point is very close to the element and thus serves to improve computational efficiency. The proposed formula also avoids calculation failure that can occur when using the Davies and Bu criterion. The numerical example results show that the proposed error upper bound formula can evaluate the integration accuracy well and improve computational efficiency when using an adaptive Gaussian integration method.

Hydrodynamic attraction of bacteria to gas and liquid interfaces.

Near an interface, the distribution of swimming microorganisms such as bacteria is distinguished from inert colloidal particles because of the interfacial hydrodynamics induced by swimming. In this work, we use nontumbling flagellated bacteria, Escherichia coli, to study cell distribution near gas and liquid interfaces and compare it to the case of a solid wall. For low-viscosity ratios such as gas interfaces, we observe a stronger cell accumulation compared to that near liquid and solid surfaces. This contradicts known theoretical predictions. Therefore, we develop a model based on Brownian dynamics, including hydrodynamic effects and short-range physiochemical interactions between bacteria and interfaces. This model explains our experimental findings and can predict cell distribution near clean and surfactant-contaminated interfaces. By considering higher order singularities, this study helps explain bacteria orientation, trajectories, and cell density.

Two dimensional displacement and stress fields for tri-material V-notches and sharp inclusions in anisotropic plates

The singular behavior for multi-material V-notch notched plates is studied in this paper, considering both isotropic and anisotropic materials. Firstly, by introducing the asymptotic expansions of displacements near the apex of the sharp notch, the elastic governing equations are transformed into eigen ordinary differential equations. The stress singularity orders and eigen angular functions are then obtained solving the established equations by using the interpolating matrix method. Secondly, using in combination the results from proper finite element analyses and the asymptotic analytical solution derived, an over-deterministic method is employed to calculate the amplitudes of the coefficients in the asymptotic expansions of displacements and stresses near the V-notch tip. The efficiency of the proposed method is assessed analyzing several notched specimens made of different materials. The accuracy of the notch stress intensity factors and the displacement and stress fields are evaluated by comparison with results available in the literature, as well as the results obtained from a number of finite element analyses.

Estimates for The First Eigenvalue of the Sturm–Liouville Problem with Potentials in Weighted Spaces

We consider the Sturm–Liouville problem on the interval [0, 1] with the Dirichlet boundary conditions and a weighted integral condition on the potential function, which allows the potential to have singularities of different orders at the end-points. For some values of the parameters of the weight functions, estimates are obtained for the first eigenvalue of this problem, and a method is proposed for finding precise bounds for this eigenvalue in some cases.

Thermal Fatigue Evaluation Model of a Microelectronic Chip in Terms of Interfacial Singularity

Abstract Thermal fatigue failure of microelectronic chip often initiates from the interface between solder and substrate, and the service life of the chip is largely dependent on the singular stress–strain at this interface. To provide a reasonable life evaluation method, three thermal fatigue evaluation models, including strain-based and stress–strain based, have been established in terms of the interfacial singular fields. Thermal fatigue lives of different chips under different thermal cycles are obtained by thermal fatigue tests, and the stress and strain intensity factors and singular orders at the solder/substrate interface are computed at the same conditions, to determine the material constants in the established models. The thermal fatigue lives predicted are in acceptable agreement with the experimental results. What is more, the application of these thermal fatigue models demonstrates a fact that the thermal fatigue of the microelectronic chips can be evaluated uniformly no matter what the shapes, dimensions of the chip, and the thermomechanical properties of the solders are, as long as the relevant stress–strain intensity factors and singular orders are obtained.

Multicritical Fermi Surface Topological Transitions.

A wide variety of complex phases in quantum materials are driven by electron-electron interactions, which are enhanced through density of states peaks. A well-known example occurs at van Hove singularities where the Fermi surface undergoes a topological transition. Here we show that higher order singularities, where multiple disconnected leaves of Fermi surface touch all at once, naturally occur at points of high symmetry in the Brillouin zone. Such multicritical singularities can lead to stronger divergences in the density of states than canonical van Hove singularities, and critically boost the formation of complex quantum phases via interactions. As a concrete example of the power of these Fermi surface topological transitions, we demonstrate how they can be used in the analysis of experimental data on Sr_{3}Ru_{2}O_{7}. Understanding the related mechanisms opens up new avenues in material design of complex quantum phases.

Abnormal and Singular Solutions in the Target Guarding Problem with Dynamics

The topic of this paper is a two-player zero-sum differential game known as the target guarding problem. After a brief review of Isaacs’ original problem and solution, a problem with second-order dynamics and acceleration control is considered. It is shown that there are four solution classes satisfying the necessary conditions. The four classes are (i) abnormal and non-singular, (ii) normal and non-singular, (iii) normal and pursuer singular, (iv) normal and evader singular. The normal and totally singular case is ruled out. Closed-form solutions are provided for cases ii–iv. The order of singularity in all cases is infinite. Thus, the problem exhibits many interesting properties: normality, abnormality, non-singularity, infinite-order singularity, and non-uniqueness. A practical example of each class is provided.

Premature failures in standard test specimens with composite materials induced by stress singularities in adhesive joints

This work presents three examples of standard test configurations, involving composite materials, where the presence of stress singularities induced by adhesive joints causes premature failures leading to underestimated strength values. Slight modifications of the local geometry, removing or reducing, in these critical points, the order of the stress singularities, have shown to give higher experimental failure loads, with almost double failure loads in the most striking cases.The three examples analyzed in the present work are: a) the tensile and shear strength determination of a bimaterial interface, b) the off-axis tension test, for the intralaminar shear strength of unidirectional long fibre composite materials and c) the compression test of thick composite laminates. In two of the three cases, the premature failure occurs at the corner where the tab (which is necessary for the grip jaw faces of the testing machine) is bonded to the composite laminate, while in the third one the failure occurs at the bimaterial interface of the two materials, all of them involving an adhesively bonded joint and corner configurations with stress singularities.With the use of a semi-analytical tool, developed by the authors, to calculate the order of stress singularities, slight and very local geometrical modifications have been successfully carried out to eliminate the stress singularity configuration. After the modifications, higher failure loads have been obtained in the tests carried out.The major outcome of the present work is the role that the singular stresses play in the premature failure of standard tests involving adhesive bonding. A correlation between the removal of the singularity stress configuration in critical parts of the samples and a higher experimental failure load, in all studied cases, has been clearly shown.

On stress singularity near the tip of a crack with surface stresses

In the framework of the simplified linear Gurtin–Murdoch surface elasticity we discuss a singularity of stresses and displacements in the vicinity of a mode III crack. We show that inhomogeneity in surface elastic properties may significantly affect the solution and to change the order of singularity. We also demonstrate that implicitly or explicitly assumed symmetry of the problem may also lead to changes in solutions. Considering various loading and symmetry conditions we show that the stresses may have logarithmic or square root singularity or be bounded in the vicinity of a crack tip.

Determination of stress intensity factors of V-notch structures by characteristic analysis coupled with isogeometric boundary element method

To describe the physical fields of the V-notch structure accurately, the structure is first divided into two parts, the notch tip sector containing singular stress field and the non-singular remained part. A twofold procedure is then carried out to obtain the stress intensity factors: (I) the asymptotic expansions are introduced to describe physical fields of the singular region and transform the elastic governing equation to the characteristic ordinary differential equations. The interpolating matrix method is then implemented to provide stress singular orders and related characteristic angular functions of this singular area. (II) The boundary integral equation for the left part is then discretized by the non-uniform rational B-splines (NURBS) elements and is solved under the isogeometric framework. The amplitudes of the asymptotic expansions in process (I) are calculated by coupling procedures (I) and (II) through utilizing the interfacial continuity conditions. The stress intensity factors are finally obtained and compared with the referenced results for both symmetric and inclined V-notch structures. Benefiting from the isogeometric analysis, satisfied results can be provided by discretizing the artificial arc boundary between the singular and non-singular parts with fewer NURBS elements. This method can also be integrated with the preprocessing of computer aided design (CAD) conveniently.

Singular functions for heterogeneous composites with cracks and notches; the use of equilibrated singular basis functions

In this paper a novel method is presented to solve problems with weak singularities in two-dimensional heterogeneous media using equilibrated singular basis functions. This especially includes crack problems in composites of functionally graded material types. The method is mainly presented in a boundary formulation, although it may be found quite useful in other mesh-based or mesh-less approaches as the eXtended Finite Element Method (XFEM). The present paper considers harmonic and elasticity problems. The most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem. To this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach. After developing the formulation in a mapped polar co-ordination, some primary basis functions generated from Chebyshev polynomials and trigonometric functions, along with corresponding weight functions are employed. The numerical examples, either selected from the well-known literature or solved by well-established techniques, will demonstrate the capability of the method in problems related to composite materials.

Fracture analysis of cracks terminating at the interface of elastic-piezoelectric bimaterials

In this paper, the fracture behaviors of a piezoelectric-elastic bimaterial with cracks terminating at the interface are investigated by a symplectic approach. In the Hamiltonian system, the Hamiltonian forms of governing equations is derived by the Hamiltonian variational principle and a total unknown vector consisted of generalized displacements and stresses. The interface fracture problem is reduced into a symplectic eigenproblem which can be directly solved by the method of separation of variables. Thus, the total unknown vector is expanded in terms of symplectic eigenfunctions. The unknown coefficients of the symplectic series can be determined from the continuity conditions at the interface and outer boundary conditions. Consequently, exact solutions for the singular electro-elastic fields and explicit expression of electric/elastic intensity factors are obtained simultaneously. Results indicate that the electro-elastic singularities and intensity factors only depend on the first few terms of symplectic eigenfunctions with non-zero eigenvalues. Numerical examples are presented to show the effects of key influencing factors on the singularity orders and intensity factors of such interface cracks. Some new results are given also.

Cloaking of irregularly shaped bodies using coordinate transformation

The inception of transformation optics has opened avenues for designing of a plethora of applications related to the propagation of electromagnetic waves in anisotropic media. In this paper, an algorithm is proposed using a coordinate transformation for the purpose of designing a cloak for a body having an arbitrary convex geometry. For the purpose of verification of the algorithm, a ray tracing process is carried out for an ellipsoid as well as an irregularly shaped body, both having axial symmetry. The ray tracing is carried out in a two dimensional plane since the transmitted ray vector would lie in the plane of incidence containing the incident ray vector and the normal vector to the cloak at the point of incidence. The approach not only reduces the numerical and analytical complexity compared to the conventional approach, but also reduces the order of singularity of the equation of Hamiltonian, improving the ability of the algorithm to cloak bodies with complex geometries which cannot be done using the conventional method.