Root Singularity Research Articles

Scattering in Quantum Dots via Noncommutative Rational Functions

In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via Nll M channels, the density rho of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio phi := N/Mle 1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit phi rightarrow 0, we recover the formula for the density rho that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any phi <1 but in the borderline case phi =1 an anomalous lambda ^{-2/3} singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.

Creation of discontinuities in circle maps.

Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and ‘threshold’ systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.

A new system of singular integral equations for a curvilinear crack in bonded materials

The modified complex potentials (MCPs) functions are used to develop a new system of singular integral equations (SIEs) for a curvilinear crack in the upper part of bonded materials subjected to shear mode stress with the help of continuity conditions for resultant force and displacement functions. The unknown dislocation distribution function is mapped into a square root singularity function by using curved length coordinate method and the traction along the crack as the right hand term. The Gaussian quadrature rules were used to obtain the numerical solution for a new system of SIEs in order to compute the nondimensional stress intensity factors (SIFs) for these problems. Our results agree with those of the previous works. The findings have revealed that the nondimensional SIFs depend on the elastic constant ratio, crack geometries and the position of the cracks.

Unconventional singularities and energy balance in frictional rupture

A widespread framework for understanding frictional rupture, such as earthquakes along geological faults, invokes an analogy to ordinary cracks. A distinct feature of ordinary cracks is that their near edge fields are characterized by a square root singularity, which is intimately related to the existence of strict dissipation-related lengthscale separation and edge-localized energy balance. Yet, the interrelations between the singularity order, lengthscale separation and edge-localized energy balance in frictional rupture are not fully understood, even in physical situations in which the conventional square root singularity remains approximately valid. Here we develop a macroscopic theory that shows that the generic rate-dependent nature of friction leads to deviations from the conventional singularity, and that even if this deviation is small, significant non-edge-localized rupture-related dissipation emerges. The physical origin of the latter, which is predicted to vanish identically in the crack analogy, is the breakdown of scale separation that leads an accumulated spatially-extended dissipation, involving macroscopic scales. The non-edge-localized rupture-related dissipation is also predicted to be position dependent. The theoretical predictions are quantitatively supported by available numerical results, and their possible implications for earthquake physics are discussed.

A crack on a cycloid rough bimaterial interface

We conduct an analytical study of a finite mode III Griffith or Zener-Stroh crack lying on a cycloid rough interface between an elastic body and a rigid substrate. Closed-form expressions of the analytic function characterizing the stress and displacement fields are derived via the solution of a Riemann-Hilbert problem with discontinuous coefficients. Stress distributions are obtained in both Cartesian and curvilinear coordinate systems. In general, the stresses exhibit a square root singularity at the two interface crack tips and at all cusp tips. Cusps located on the interface crack surface behave as cracks while those outside the interface crack act as anticracks or rigid line inclusions. The stress intensity factors scaling the square root singularity in the stresses are extracted from the full-field solutions. When one crack tip is lodged only at a cusp tip, the stronger r−3/4 and the weaker r−1/4 power-type singularities at this crack-cusp tip are observed and the stress intensity factors scaling these power-type singularities are derived.

An elastic half-plane with a fixed cycloid wavy boundary

We present a rigorous analysis of the plane problem of an elastic half-plane with a fixed cycloid wavy boundary under biaxial tension using Muskhelishvili’s complex variable method. A closed-form expression for the pair of analytic functions characterizing the stresses and displacements in the elastic body is derived by means of analytic continuation. Furthermore, elementary expressions for the interfacial and hoop stresses along the cycloid interface and stress distributions along the right of an interface valley are obtained. For a cusped cycloid interface, the stresses exhibit the square root singularity at the cusp tips. The mode I stress intensity factor at a cusp tip is expediently extracted from the resulting analytic functions. The interfacial normal stress and hoop stress are constant along a cusped cycloid interface except at the isolated cusp tips.

ПРО ВЗАЄМОДІЮ ДВОХ ТРІЩИН НА МЕЖІ ПОДІЛУ МАТЕРІАЛІВ

Plane deformation problem of the interaction of two collinear cracks between isotropic heterogeneous half-spaces loaded at infinity by normal and tangential stresses is considered. To find the solution, the representation of stresses and displacements through a function that is analytical in the entire complex plane except of the crack regions is used. With the help of these representations, the problem is reduced to the Riemann-Hilbert problem, for the specified piecewise analytical function with jumps along the segments of cracks. An analytical representation of the solution with unknown coefficients, which is determined from the conditions at infinity and the conditions of unambiguous displacements when traversing the contours of cracks, is written. The implementation of the latter leads to the need of calculatson the integrals from the functions with oscillating root singularities. Based on the obtained solution, analytical expressions are written for stresses at different segments outside the crack and jumps of displacements on the cracks. The behavior of stresses and derivatives of the displacements jumps at different parts of the material interface are established. Complex coefficients of stress intensity and energy release rate near the vertices of the right crack are found. The dependences of these factors on the mechanical characteristics of materials, size and mutual location of cracks, as well as on the intensities of the remote normal and tangential stresses are analyzed. The results are presented in the form of tables and graphs. In particular, an increase of the energy release rate for the crack tip when another crack approaches is shown. Graphs of changes in the jump of displacements for different ratios of modulus of elasticity of the upper and lower materials and different ratios of intensities of external normal and tangential stresses are also shown. It was found, in particular, that with a significant difference in the modulus of elasticity and with a significant shear stress field, the interpenetration of materials occurring because of the oscillating singularity near the crack vertices becomes noticeable.

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.

Pre-fracture zone modeling for an interface crack in an isotropic bimaterial

An interface crack in an infinite bimaterial space under remote combining loading is considered. The complex potentials approach is applied and an exact analytical solution of this problem is presented. To remove the oscillating singularities which occur in this solution a model based on the introduction of the shear stress pre-fracture zones at the crack tips is suggested. In the framework of this assumption the nonhomogeneous combined Dirichlet-Riemann boundary value problem with the conditions at infinity is formulated and its analytical solution is presented. The length of this zone is found from the condition of restriction of the shear stress at the end point of the zone. In this case the shear stress becomes finite in the right hand side of the pre-fracture zone while the normal stress has a square root singularity at the crack tip. The energy release rates (ERR) at the crack tip and also along the pre-fracture zone are found and their total values are compared the ERR of the classical model. For the case of a similar problem, but for finite sized body the finite element method is applied. The crack length is assumed to be much smaller then characteristic body size. The finite element net with two levels of concentration is constructed. The first level provides the uniform concentration from the boundaries of the body to the crack and the second level assumes the similar concentration at the singular points of the pre-fracture zone. The different values of the shear stress in the pre-fracture zone are considered and the local ERR at the singular points as well as the global energy release rate is found. It is shown that for different values of the mentioned shear stress the global energy release rate remains almost invariable and is in a a good agreement with the analytical solution.

Interaction between a screw dislocation and a completely coated semi‐infinite anticrack

AbstractWe use complex variable methods to derive analytic solutions to the problem associated with the interaction between a screw dislocation and a completely coated semi‐infinite anticrack. The two sides of the anticrack or rigid line inclusion are perfectly bonded to the surrounding coating, which is also perfectly bonded to its neighboring matrix. The screw dislocation can be located anywhere in the composite structure. We obtain explicit expressions for the resultant stress intensity factor characterizing the square root singularity in the stress distribution at the anticrack tip, the resultant anticrack contraction force, which is always negative, and the image force acting on the screw dislocation.

Investigation of the large deformations of composite plane with interface crack loaded by uniform pressure

The analytical solution to a nonlinear problem of elasticity for the composite plane formed by connection of two half-planes from different materials with an interface crack is obtained. The elastic properties of half-planes are modeled by semi-linear harmonic material. External loading is uniform pressure, which is orthogonal to the deformed surfaces of a crack coasts, the stresses at infinity are absent. For the solution to a nonlinear plane problem methods of the theory of the complex variable functions are used. An existence of critical values of pressure at the coasts of a crack which excess conducts to loss of stability of a material and to large post-critical displacements, deformations and stresses in a vicinity of a crack is established. The critical pressures have the order of the shear modules of the materials and are really possible for low-modules rubber-like materials (elastomers). For the rigid materials with the large module of shear, in particular metals, the critical pressures really are not achieved. The formulas for disclosing of the crack coasts depending on value of pressure and parameters of materials are obtained, the plots of displacements of the crack surfaces for different values of pressure are presented. On the base of a common solution the asymptotic expansions are constructed for nominal (Piola) and Cauchy stresses in vicinities of a crack ends. The nominal stresses have the root singularities, the Cauchy stresses have no singularities at the tips of crack.

Elastic field near the tip of an anticrack in a decagonal quasicrystalline material

We investigate the elastic field near the tip of an anticrack in a homogeneous decagonal quasicrystalline material subject to plane strain deformations. The phonon and phason stresses exhibit a square root singularity at the anticrack tip. Two real-valued phonon stress intensity factors and two real-valued phason stress intensity factors are introduced to scale four separate modes of deformation. We obtain four analytic functions which completely characterize the induced phonon and phason stresses as well as the displacement field. In particular, we derive a concise yet elegant representation of the anticrack contraction force.

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.

Uniform stress state inside a non-elliptical inhomogeneity near an irregularly shaped hole in antiplane shear

Analytic continuation and conformal mapping techniques are applied to establish that the state of stress inside a non-elliptical elastic inhomogeneity can remain uniform despite the presence of a nearby irregularly shaped hole when the surrounding matrix is subjected to uniform remote antiplane shear stresses. The hole boundary is assumed to be either traction-free or subjected to antiplane line forces. Detailed numerical results are presented to demonstrate the resulting analytical solutions. Our results indicate that in maintaining a uniform stress distribution inside the inhomogeneity, it is permissible for the stresses in the matrix to exhibit either a square root singularity at sharp corners of a hole boundary or a high level of stress concentration at rounded corners of a hole.

Quadratic Vector Equations On Complex Upper Half-Plane

We consider the nonlinear equation $-\frac{1}{m}=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb{H} $, where $S$ is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in $ \mathbb{H}$ is unique and its $z$-dependence is conveniently described as the Stieltjes transforms of a family of measures $v$ on $\mathbb{R}$. In [AEK17a] we qualitatively identified the possible singular behaviors of $v$: under suitable conditions on $S$ we showed that in the density of $v$ only algebraic singularities of degree two or three may occur. In this paper we give a comprehensive analysis of these singularities with uniform quantitative controls. We also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the companion paper [AEK16b], we present a complete stability analysis of the equation for any $z\in \mathbb{H}$, including the vicinity of the singularities.

Mean field theory of jamming of nonspherical particles

Recent computer simulations have uncovered the striking difference between the jamming transition of spherical and non-spherical particles. While systems of spherical particles are isostatic at the jamming point, systems of nonspherical particles are not: the contact number and shear modulus of the former exhibit a square root singularity near jamming, while those of the latter are linearly proportional to the distance from jamming. Furthermore, while our theoretical understanding of jamming of spherical particles is well developed, the same is not true for nonspherical particles. To understand jamming of non-spherical particles, in the previous work Brito (2018 Proc. Natl Acad. Sci. 115 11736), we extended the perceptron model, whose SAT/UNSAT transition belongs to the same universality class of jamming of spherical particles, to include additional variables accounting for the rotational degrees of freedom of nonspherical particles. In this paper, we give more detailed investigations of the full scaling behavior of the model near the jamming transition point in both convex and non-convex phases.

A Crack between Orthotropic Materials with a Shear Yield Zone at the Crack Tip

A plane problem for a crack between two anisotropic semi-infinite spaces under remote tensile-shear loading is considered. In the framework of the assumption that the crack faces are free of stresses an exact analytical solution of the problem is given on basis of the complex potentials approach. This solution possesses oscillating square root singularities in stresses and in the derivatives of the displacement jumps at the crack tips. To remove these singularities a new model founded on the introduction of the shear yield zones at the crack tips is suggested. This model is appropriate for the cases where interface adhesive layer is softer than the surrounding matrixes. Under this assumption the problem is reduced to the nonhomogeneous combined Dirichlet-Riemann boundary value problem with the conditions at infinity. An exact analytical solution of this problem is presented for the case of a single yield zone. The length of this zone is found from the finiteness of the shear stress at the end point of the zone. Due to such simulation the shear stress becomes finite at any point and the normal stress possesses only square root singularity at the crack tip. Therefore, the conventional stress intensity factor of the normal stress at the crack tip is used. The numerical illustration of the obtained solution is given.

Formation of Weak Singularities on the Surface of a Dielectric Fluid in a Tangential Electric Field

The process of interaction between nonlinear waves on a free surface of a nonconducting fluid in a strong tangential electric field is simulated numerically (effects of the force of gravity and capillarity are neglected). It is shown that singular points are formed at the fluid boundary in a finite time; at these points, the boundary curvature significantly increases and undergoes a discontinuity. The amplitude and slope angles of the boundary remain small. The singular behavior of the system is demonstrated by spectral functions of the fluid surface - they acquire a power dependence. Near the singularity, the boundary curvature demonstrates a self-similar behavior typical for weak root singularities.

Кругове теплоактивне міжфазне включення в кусково-однорідному трансверсального-ізотропному просторі

An exact solution of the stationary thermoelasticity problem about interfacial circular absolutely rigid inclusion, which is under conditions of complete adhesion and under conditions of smooth contact with transversely homogeneous spaces, is constructed. The task with the help of the constructed discontinuous solution, by the method of singular integral relations, is reduced to a system of singular integral equations (SIE). An exact solution has been built for the specified systems of two-dimensional singular integral equations. As a result, dependences jumps of stresses and displacement on temperature, equivalent load, main moments and thermomechanical characteristics of transversally isotropic materials. The influence of the type of contact interaction on the behavior of the solutions is established. In particular, it has been shown that the stresses in the neighborhood of the inclusion with a smooth contact have a root singularity, and with complete coupling, the root singularity, which is amplified by oscillation. The behavior of the generalized intensity coefficient (GCIN) was studied for the combination of various transversely isotropic materials at different power and temperature loads.

Формирование слабых особенностей на поверхности диэлектрической жидкости в тангенциальном электрическом поле

AbstractThe process of interaction between nonlinear waves on a free surface of a nonconducting fluid in a strong tangential electric field is simulated numerically (effects of the force of gravity and capillarity are neglected). It is shown that singular points are formed at the fluid boundary in a finite time; at these points, the boundary curvature significantly increases and undergoes a discontinuity. The amplitude and slope angles of the boundary remain small. The singular behavior of the system is demonstrated by spectral functions of the fluid surface—they acquire a power dependence. Near the singularity, the boundary curvature demonstrates a self-similar behavior typical for weak root singularities.