Oscillatory Singularities Research Articles

Application of J-integral to adhesive contact under general plane loading for rolling resistance

In the present work, a mechanical model for two-dimensional non-slipping adhesive contact between dissimilar elastic solids under general loading, namely, normal forces, tangential forces and moments is proposed. The general solutions are obtained analytically with the stresses at the contact edges exhibiting oscillatory singularity, similar to those at a bimaterial interface crack. The well-known J-integral under the current context is analyzed. Its application under the selected integration contour readily gives the relationship between the stress intensity factors and energy release rates at the contact edges. With the results rolling adhesion between two solids with parabolic profiles is considered further. The applied moment can be directly determined by the difference in energy release rates at the trailing and leading edges and hence the rolling resistance even for adhesive contact with cohesive zones. These results provide the foundation for understanding some tribological phenomena associated with adhesion.

Methods and models for fracture mode partitioning: A review

In the problem of a crack lying in the interface between two dissimilar materials, the mathematical expressions for the stress and displacement fields, derived from linear elastic fracture mechanics, feature oscillatory singularities near the crack tip. This makes it impossible to define the mode mixity mathematically. To overcome this issue, several alternative methods for mode partitioning and analytical, semi-analytical, empirical, or combined models performing mode partitioning in linear elastic, mixed-mode fracture problems have been proposed over the last 35 years. This paper attempts the first review of the topic. It begins by recalling the fundamental scientific problem through reference on the fracture mechanics of bimaterial interfaces. Next, it collects over 36 models performing mode partitioning, proposes classifying them into five broad categories (models based on the global method, models based on the local method, models employing crack-tip force measures, models employing a cohesive interface, and special models), and describes and critically reviews them. A comprehensive discussion is then provided, and the paper closes with justification of concluding remarks and prospective directions for future research.

Oscillatory singularity for bending of a partially clamped nanoplate with consideration of surface effect

This article studies a partially clamped flexible ultrathin elastic nanoplate with surface effect. An emphasis is to analyze the singular elastic behavior near the tips of partially clamped boundary. A mixed boundary value problem associated with a partially clamped nanoplate having parabolic deflection is presented and solved. Using the Kirchhoff plate theory incorporating the Gurtin–Murdoch surface elasticity, some basic equations for nanoplates with surface elasticity are established. With the aid of the Fourier transform and the superposition principle, two coupled singular integral equations with Cauchy kernel are derived during bending of the nanoplate and then converted to a singular integral equation with two Cauchy kernels. The exact solution to the singular integral equation with two Cauchy kernels is determined and the oscillatory singular elastic fields near the tips of the clamped boundary are given in elementary functions. The bending moments and the normal stress components in the vicinity of the tips simultaneously have the oscillator singular behavior like r−3/2−iɛ, r being the distance from the tip and ɛ relying on the material properties. Besides, the effective shear force possesses an r−5/2−iɛ singular behavior. The influences of the geometrical parameters and material properties of nanoplates on the oscillatory singular behavior are presented graphically. When surface effect is neglected, the solution is applicable to classical thin plates with partially clamped boundary.

On the Surface Plasmonic Waves Excited by a Dipole Above Anisotropic and Spatially Dispersive Two-Dimensional Surfaces of Infinite Extent in Planarly Layered Media

The surface plasmonic waves excited by a vertical or horizontal oriented Hertzian dipole above anisotropic and spatially dispersive two-dimensional surfaces of infinite extent embedded in planarly layered uniaxial media are investigated using the dyadic Green function approach. The spectral-domain transmission line analogy Green function formulation and iso-frequency-contour equations are derived. The methods to accurately and efficiently evaluate the two-dimensional Fourier integral arising from the spatial-domain Green function computation are also developed. To resolve the numerical inefficiency due to the highly oscillatory integrand and singularities of surface plasmonic waves that possess large wavenumber, the methods of extrapolating the real-axis integration combined with singularity subtraction are proposed and found to be applicable to a wide range of observation distances. As an application example of the proposed formulation, we compute the scattered fields of a vertical dipole above the graphene biased by drift current which exhibits significant spatial dispersion. It is shown that the light-matter interaction of graphene plasmons can be significantly reinforced when placed above uniaxially anisotropic epsilon-near-zero substrates. The proposed formulation may provide the methodology for the computational analysis of two-dimensional materials and surface plasmonic waves.

Displacement discontinuity method for interfacial cracks in one-dimensional hexagonal quasi-crystal coating under thermal-mechanical loading

This study derived Green’s functions of the concentrated interfacial displacement and temperature discontinuities. Green’s functions and the displacement and temperature discontinuity method were used to analyze interfacial cracks in a one-dimensional (1D) hexagonal quasi-crystal (QC) coating under thermal-mechanical loading. The boundary integral-differential equations were established for an interfacial crack, the fundamental solutions were obtained for uniformly distributed temperature, and the displacement discontinuities were applied over a line element. The near-crack-tip oscillatory singularity of stresses was eliminated by introducing the Gaussian distribution function to approximate the delta function. After this treatment, the expressions of the stress and heat flux intensity factors without oscillatory singularity, and the energy release rate (ERR) at the crack tips were obtained. The displacement and temperature discontinuity boundary element method for numerical simulation is proposed, and the influence of relevant factors is briefly discussed.

Stable generalized finite element methods (SGFEM) for interfacial crack problems in bi-materials

The poor conditioning in generalized or extended finite element methods (GFEM/XFEM) for crack problems in homogeneous materials has been studied extensively. However, few research efforts have been devoted to resolving the conditioning difficulty of GFEM/XFEM for bi-material crack problems. The conditioning of these methods for a bi-material crack problem is poorer than that for a homogeneous material crack problem; this is because more complex enrichments, including Heaviside functions, distance functions, and singular functions characterizing radial and oscillatory singularities, are involved in the former. This study addresses the conditioning difficulty for bi-material crack problems by proposing a stable GFEM (SGFEM), which (a) reaches optimal convergence O(h), (b) has a scaled condition number O(h−2) having the same order as that of the standard FEM, and (c) achieves convergence and conditioning without deterioration as the interface lines approach the boundaries of the elements. The proposed SGFEM is based on two stability techniques, namely changing the partition of unity functions and conducting local principal component analysis of multi-fold enrichments at one particular node. Numerical experiments suggest that in comparison with the conventional GFEM/XFEM for bi-material crack problems in the literature, the proposed SGFEM achieves all features (a)–(c). In addition, the effect of the ratio of material coefficients on conditioning is investigated, revealing that the conditioning of the proposed SGFEM is insensitive to the material coefficients.

Log Periodic Power Analysis of Critical Crashes: Evidence from the Portuguese Stock Market

The study of critical phenomena that originated in the natural sciences has been extended to the financial economics’ field, giving researchers new approaches to risk management, forecasting, the study of bubbles and crashes, and many kinds of problems involving complex systems with self-organized criticality (SOC). This study uses the theory of self-similar oscillatory time singularities to analyze stock market crashes. We test the Log Periodic Power Law/Model (LPPM) to analyze the Portuguese stock market, in its crises in 1998, 2007, and 2015. Parameter values are in line with those observed in other markets. This is particularly interesting since if the model performs robustly for Portugal, which is a small market with liquidity issues and the index is only composed of 20 stocks, we provide consistent evidence in favor of the proposed LPPM methodology. The LPPM methodology proposed here would have allowed us to avoid big loses in the 1998 Portuguese crash, and would have permitted us to sell at points near the peak in the 2007 crash. In the case of the 2015 crisis, we would have obtained a good indication of the moment where the lowest data point was going to be achieved.

On blow up for the energy super critical defocusing nonlinear Schr\xf6dinger equations

We consider the energy supercritical defocusing nonlinear Schrödinger equation i∂tu+Δu-u|u|p-1=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} i\\partial _tu+\\Delta u-u|u|^{p-1}=0 \\end{aligned}$$\\end{document}in dimension dge 5. In a suitable range of energy supercritical parameters (d, p), we prove the existence of {mathcal {C}}^infty well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of {mathcal {C}}^infty spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

The elastodynamic bimaterial interface under mode I and mode II loading

This article provides the mathematical solutions to the elastodynamic fields of a semi-infinite interface lying along two dissimilar media subjected to sudden loading. The article offers the solution to the cases when: (1) the interface slips, i.e., it cannot transfer shear stress from one material to the other, which represents a Hertzian contact; (2) the interface is welded, i.e., all stress components are transferred, in which case it acts as a crack. We obtain the full, explicit analytic solutions to the fields of the interface along the slipping boundary via the Wiener–Hopf and Cagniard-de Hoop techniques. We show that such interface does not entail an oscillatory singularity at the crack tip owing to the fact that shear forces are not transferred across the interface. The welded interface crack leads to a matricial Wiener–Hopf problem that is not reducible to any form that would allow an immediate analytic factorisation of the resulting scattering kernel matrix. The factorisation in this case is achieved via successive Abrahams approximations of the scattering kernel itself, rather than via a Williams expansion of the elastic displacement field. This leads to a quickly convergent solution that retains the asymptotic character in the near and in the far field. Explicit proof of the nature of the oscillatory singularity at the crack tip is provided by studying the scattering matrix, which in the near field is shown to reduce to a Daniele-Khrapkov form amenable to analytic factorisation. The solutions presented in this article are explicit, and will prove eminently useful in the modelling of fast fibre debonding in composite materials, and in the study of the scattering of seismic waves by cracks and faults in layered media.

Entanglement entropy at critical points of classical evolution in oscillatory and exotic singularity multiverse models

Using the 3rd quantization formalism we study the quantum entanglement of universes created in pairs within the framework of standard homogeneous and isotropic cosmology. In particular, we investigate entanglement quantities (entropy, temperature) around maxima, minima and inflection points of the classical evolution. The novelty from previous works is that we show how the entanglement changes in an extended minisuperspace parameterised by the scale factor and additionally, by the massless scalar field. We study the entanglement quantities for the universes which classically exhibit Big-Bang and other than Big-Bang (exotic) singularities such as Big-Brake, Big-Freeze, Big-Separation, and Little-Rip. While taking into account the scalar field, we find that the entanglement entropy is finite at the Big-Bang singularity and diverges at maxima or minima of expansion. As for the exotic singularity models we find that the entanglement entropy or the temperature in all the critical points and singularities is either finite or infinite, but it never vanishes. This shows that each of the universes of a pair is entangled to a degree parametrized by the entanglement quantities which measure the quantumness of the system. Apart from the von Neumann entanglement entropy, we also check the behaviour of the the Tsallis and the Renyi entanglement entropies, and find that they behave similarly as the meters of the quantumness. Finally, we find that the best-fit relation between the entanglement entropy and the Hubble parameter (which classically marks special points of the universe evolution) is of the logarithmic shape, and not polynomial, as one could initially expect.

Analysis of interface cracks in one-dimensional hexagonal quasi-crystal coating under in-plane loads

The displacement discontinuity (DD) method is proposed to analyze interface cracks in one-dimensional hexagonal quasi-crystal (QC) coating under in-plane loads. According to the general solutions and Fourier transform, fundamental solutions are derived for unit-point DDs at interface, with explicit expressions of displacements and stresses. The DD hyper-singular boundary integral–differential equations in terms of DDs are established across interface cracks. The fundamental solutions for uniformly distributed DDs are obtained over a constant line element. To eliminate the near-crack-tip oscillatory singularity, the delta function in fundamental solutions is approximated by the Gaussian distribution function. The expressions of stress intensity factors without oscillatory singularity and energy release rate are presented in terms of DDs. Finally, the DD boundary element method with checking accuracy is proposed for numerical simulation, and the influence of coating thickness, material combination, crack length and crack distance on the fracture behavior is comprehensively studied.

A new DBEM for solving crack problems in arbitrary dissimilar materials

In this paper, a new and efficient Dual Boundary Element Method (DBEM) named Dual Boundary-Interface Integral Equation Method (DBIIEM) is presented to solve crack problems in objects composed of arbitrary number of different materials. The conventional DBEM has been widely used in crack analysis and proved to be one of the most effective BEM for solving crack problems, however it cannot be directly employed to solve more complicated multi-medium crack problems. The presented DBIIEM can remedy this deficiency by dually collocating the displacement boundary-interface integral equation on one of the multi-medium crack surface and traction boundary-interface integral equation on the other. The newly derived displacement and traction boundary-interface integral equations are naturally fit for solving multi-medium elasticity problems, and independent each other, therefore they can be employed together in multi-medium crack analysis. The established DBIIEM inherits the advantage of single -region formulation of DBEM for solving more complicated crack problems. Complex stress intensity factors which characterize the oscillatory singularity of crack tips located at the interface crack are evaluated. Several numerical examples are given to verify correctness of the presented method.

Near-tip singular fields for an interface anticrack between dissimilar anisotropic elastic materials

We study the near-tip singular fields for an interface anticrack between dissimilar anisotropic elastic materials using the sextic Stroh formalism. The stress and strain fields exhibit an oscillatory singularity and a square root singularity at the anticrack tip. A complex and a real strain/rotation intensity factors are introduced to scale, respectively, the two singularities. An explicit expression for the negative anticrack contraction force is obtained in terms of the proposed intensity factors. We illustrate our results using an example in which the bimaterial is orthotropic. In this typical case, a complex stress intensity factor can also be defined to replace the complex strain/rotation intensity factor.

Asymptotic Field Near the Tip of a Debonded Anticrack in an Anisotropic Elastic Material

SummaryWe use the sextic Stroh formalism to study the asymptotic elastic field near the tip of a debonded anticrack in a generally anisotropic elastic material under generalised plane strain deformations. The stresses near the tip of the debonded anticrack exhibit the oscillatory singularities $r^{-3/4\pm i\varepsilon }$ and $r^{-1/4\pm i\varepsilon }$ (where $\varepsilon $ is the oscillatory index) as well as the real power-type singularities $r^{-3/4}$ and $r^{-1/4}$. Two complex-valued stress intensity factors and two real-valued stress intensity factors are introduced to respectively scale the two oscillatory and two real power-type singularities. The corresponding three-dimensional analytic vector function is derived explicitly, and the material force on the debonded anticrack is obtained. Our solution is illustrated using an example involving orthotropic materials.

Fundamental solutions and analysis of the interface crack for two-dimensional decagonal quasicrystal bimaterial via the displacement discontinuity method

The extended displacement discontinuity method is developed to analyze an interface line crack in a two-dimensional decagonal quasicrystal bimaterial. According to the general solutions and Fourier transform, fundamental solutions for phonon and phason displacement discontinuities and phonon and phason stresses are obtained. The delta function in the phonon and phason stresses is replaced by the Gaussian distribution function to eliminate the oscillatory singularity near the crack front. The expressions of phonon and phason stress intensity factors are given in terms of phonon and phason displacement discontinuities. The finite element analysis software COMSOL is used to validate the developed method.

Boundary element analysis of the stress intensity factors of plane interface cracks between dissimilarly adjoined anisotropic materials

For assuring structural integrity, the study of interface cracks has remained to be an important topic in engineering analysis. It is well known to have stresses with oscillatory singularity in the vicinity of crack tips between dissimilarly adjoined anisotropic materials. The present works targets plane analysis of the stress intensity factors (SIF) of interface cracks by the boundary element method (BEM). For modeling multiply adjoined domains, the conventional sub-region technique in the BEM is applied. For calculating the SIF of interface cracks, the path-independent H-integral is evaluated by the solutions of stresses and displacements at internal points along the integration path in each irrespective domain. For resolving the problem of near singularity when the internal Gauss points are close to the interfaces or crack surfaces, all boundary integrals are regularized to calculate the displacements and stresses at the internal points. From a few examples tested, the approach is verified to be very efficient for accurate assessment of the SIFs of interface cracks between dissimilarly adjoined anisotropic materials.

Interface crack between magnetoelectroelastic and orthotropic half-spaces under in-plane loading

In this paper, an interface crack between magnetoelectroelastic and orthotropic half-spaces has been studied in detail. By using integral transform techniques the present mixed boundary value problem was reduced to the solution of singular integral equations, which can be further reduced to solving a Riemann-Hilbert problem with closed form solution. The crack-tip singularities of the interface crack have been investigated for possible combinations of the magnetoelectroelastic and orthotropic materials, a criterion based on the coefficient of the Riemann-Hilbert problem is introduced to study the possible singularity behavior of the interface crack, and it is shown that there can be either oscillatory or non-oscillatory singularity for the interface crack depending on the particular combination of the bi-materials. A closed form solution for stresses, electric fields, magnetic fields, electric displacement and magnetic induction in the cracked biomaterials is given, and of particular interests, the analytical expression of the stresses, electric displacements and magnetic inductions along the interface has been obtained.

Non‐oscillatory and non‐singular asymptotic solutions to stress fields at interface cracks

AbstractThis work concerns the complex oscillatory singularities revealed in Williams's asymptotic solutions to stress fields around arbitrary interface cracks, which are the foundation of phenomenological interface fracture mechanics. First, we highlight the fatal discrepancy between the asymptotic stress fields for cracks in a homogeneous material obtained by assigning an identical material on both regions embracing an interface crack, and the solutions directly derived from cracks in a single material. Next, following a brief introduction to Williams's formulation process, we adopt the method of repeatedly eliminating variables instead of solving the determinant equation for the coefficient matrix to reformulate the asymptotic analysis of stress fields at arbitrary interface cracks. The resultant stresses get rid of oscillatory character. Further, under two specific loading conditions, namely, remotely uniaxial tension or shear, non‐oscillatory and non‐singular asymptotic solutions to stress fields around interface cracks are obtained.

Analysis of a three-dimensional arbitrarily shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystal bi-material. Part 2: Numerical method

The extended displacement discontinuity boundary integral equation and boundary element method are extended to analyze a three-dimensional (3D) arbitrarily shaped interface crack in a one-dimensional hexagonal, quasicrystal bi-material with both electric and thermal effects under combined phonon-phason-electric-thermal loadings. Based on the analogy with the analysis method for three-dimensional, transversely isotropic magnetoelectrothermoelastic bi-materials (Dang et al., in press) the numerical method is proposed for one-dimensional quasicrystal bi-materials. Firstly, the fundamental solutions for uniformly distributed, extended displacement discontinuities applied over a constant triangular element are obtained by integrating the fundamental solutions for unit-point extended displacement discontinuities given in Part 1 over the triangular area (Zhao et al., 2017). Secondly, in order to eliminate the oscillatory singularity near the crack front, the Delta function in the integral–differential equations is approximated by the Gaussian distribution function, and the Heaviside step function is replaced by the Error function accordingly. Thirdly, the extended stress intensity factors without oscillatory singularities and the energy release rate are all obtained in terms of the extended displacement discontinuities. At last, the extended displacement discontinuity boundary element method is proposed to validate the analytical solution. In the numerical simulation, the multi-physical behavior of an elliptical, interface crack is numerically simulated. The correctness of the proposed numerical method, the influence of the applied, combined phonon-phason-electric-thermal loadings, the material-mismatch, and the ellipticity ratio are all studied.

Analysis of an interface crack of arbitrary shape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bimaterial—part 2: Numerical method

ABSTRACTThe extended displacement discontinuity (EDD) boundary integral equation and boundary-element method are extended and developed to analyze an arbitrarily shaped, planar interface crack in a three-dimensional, transversely isotropic, magnetoelectrothermoelastic bimaterial under combined, thermoelectromagnetomechanical loadings. The fundamental solutions for uniformly distributed EDDs applied over a constant triangular element are obtained through integrating the fundamental solutions for the unit-point EDDs given by Part 1 over the triangular area. To eliminate the oscillatory singularity near the crack front, the Dirac delta function in the integral–differential equations is approximated by the Gaussian distribution function, and accordingly, the Heaviside step function is replaced by the Error function. The extended stress intensity factors without oscillatory singularities, the energy release rate, and the local J-integral in terms of intensity factors are all obtained. To validate the solution, the EDD boundary-element method is proposed. As an application, an elliptical interface crack is numerically simulated. The influences of the applied combined loadings and material-mismatch as well as the ellipticity ratio on the multiphysical response are studied.