Unknown Jumps Research Articles

Centennial-Scale Jumps in CO<sub>2</sub> Driven by Earth’s Tilt

Antarctic ice records uncovered seven previously unknown jumps in atmospheric carbon dioxide. These events may have been driven by changes in Earth’s tilt.

Adaptive Containment Control for Heterogeneous MIMO Nonlinear Multiagent Systems With Unknown Direction Actuator Faults

In this article, a novel output-feedback adaptive containment control scheme is proposed for a class of heterogeneous multi-agent systems, where the agents are nonlinear multi-input multi-output (MIMO) systems whose relative degrees are allowed to be different. Unlike existing results, we only require the leading principal minors of agents' control gain matrices (CGMs) to be nonzero and take into account unknown direction actuator faults, which relaxes the restrictions on CGMs and enhances system reliability. The difficulties jointly caused by the unknown CGMs, unknown parameters, and unknown jumps introduced by the actuator faults are successfully overcome by a novel recursive contradiction argument based on some Nussbaum functions and a matrix similarity transformation. Moreover, an event-triggering mechanism is introduced to avoid continuous communication among agents and reduce the communication burden. It is shown that all closed-loop signals are globally uniformly bounded and the containment errors converge to a residual set that can be made arbitrarily small. Simulation results illustrate the effectiveness of the proposed scheme.

On the jump control problem for boundary-value problems with state-dependent impulses

UDC 517.9 We show how an appropriate parametrization technique and special successive approximations can help to control unknown jumps in the case of nonlinear boundary-value problems with state-dependent impulses. The practical application of the technique is shown on a numerical example.

Fracture analysis of a Griffith crack under thermal-mechanical loading using nonlocal theory

In this paper, the nonlocal theory is utilized to deal with a Griffith crack problem under thermal-mechanical loading for the first time to eliminate stress singularity at the crack tip. A partially, thermally insulated Griffith crack subjected to a remote heat flux and mechanical loading is investigated. Fourier transform is employed to convert the mixed, boundary-value problem to two pairs of dual integral equations, which are then solved using the Schmidt method. The unknown jumps of displacement across crack faces are expressed by a series of Jacobi polynomials. The numerical results show that the stress at the crack tip is finite, and the maximum value occurs at a short distance from the crack tip. The results can be employed to build a maximum stress-based criterion instead of the traditional, stress intensity factor- or energy-based fracture criterion.

Adaptive semiparametric estimation for single index models with jumps

The single index model is one of the most popular semiparametric models in applied quantitative sciences. This paper studies a single index model with unknown jumps (SIMJ) that occur in the link function. An adaptive semiparametric estimation procedure is proposed for estimating the index coefficient and link function. The asymptotic normality of the resulting estimators for both the parametric and nonparametric parts can be established under some mild conditions and without specifying the error distribution. We show that the resulting estimators are robust and efficient for different error distributions. In particular, a modified EM algorithm is developed to implement the adaptive semiparametric estimation in practice. Numerical simulations and real data analysis are conducted to illustrate the finite sample performance of the proposed approach.

Hydroelastic waves propagating in an ice-covered channel

The hydroelastic waves in a channel covered by an ice sheet, without or with crack and subject to various edge constraints at channel banks, are investigated based on the linearized velocity potential theory for the fluid domain and the thin-plate elastic theory for the ice sheet. An effective analytical solution procedure is developed through expanding the velocity potential and the fourth derivative of the ice deflection to a series of cosine functions with unknown coefficients. The latter are integrated to obtain the expression for the deflection, which involves four constants. The procedure is then extended to the case with a longitudinal crack in the ice sheet by using the Dirac delta function and its derivatives at the crack in the dynamic equation, with unknown jumps of deflection and slope at the crack. Conditions at the edges and crack are then imposed, from which a system of linear equations for the unknowns is established. From this, the dispersion relation between the wave frequency and wavenumber is found, as well as the natural frequency of the channel. Extensive results are then provided for wave celerity, wave profiles and strain in the ice sheet. In-depth discussions are made on the effects of the edge condition, and the crack.

Напружений стан у пружному тілі з триланковим жорстким включенням при гармонічних коливаннях поздовжнього зсуву

There is a thin absolutely rigid inclusion that in a cross-section represents three segments broken line in an infinite elastic medium (matrix) that is in the conditions of antiplane strain. The inclusion is under the action of harmonic shear force Pe^{iwt} along the axis Oz. Under the conditions of the antiplane strain the only one different from 0 z-component of displacement vector W (x; y) satisfies the Helmholtz equation. The inclusion is fully couple with the matrix. The tangential stresses are discontinuous on the inclusion with unknown jumps. The method of the solution is based on the representation of displacement W (x; y) by discontinuous solutions of the Helmholtz equation. After the satisfaction of the conditions on the inclusion the system of integral equations relatively unknown jumps is obtained. One of the main results is a numerical method for solving the obtained system, which takes into account the singularity of the solution and is based on the use of the special quadrature formulas for singular integrals.

Elemental enriched spaces for the treatment of weak and strong discontinuous fields

This paper presents a finite element that incorporates weak, strong and both weak plus strong discontinuities with linear interpolations of the unknown jumps for the modeling of internal interfaces. The new enriched space is built by subdividing each triangular or tetrahedral element in several standard linear sub-elements. The new degrees of freedom coming from the assembly of the sub-elements can be eliminated by static condensation at the element level, resulting in two main advantages: first, an elemental enrichment instead of a nodal one, which presents an important reduction of the computing time when the internal interface is moving all around the domain and second, an efficient implementation involving minor modifications allowing to reuse existing finite element codes. The equations for the internal interface are constructed by imposing the local equilibrium between the stresses in the bulk of the element and the tractions driving the cohesive law, with the proper equilibrium operators to account for the linear kinematics of the discontinuity. To improve the continuity of the unknowns on both sides of the elements on which a static condensation is done, a contour integral has been added. These contour integrals named inter-elemental forces can be interpreted as a “do nothing” boundary condition (Coppola-Owen and Codina, 2011) published in another context, or as the usage of weighting functions that ensure convergence of the approach as proposed by J.C. Simo (Simo and Rifai, 1990). A series of numerical tests for scalar unknowns as a simple representation of more general numerical simulations are presented to illustrate the performance of the enriched elemental space.

Interaction of a harmonic torsional wave with ring-shaped defects in an elastic body

We have solved the problem of determination of the stressed state in an isotropic elastic body near ring-shaped defects (a crack or a thin rigid inclusion) as a result of the action of a harmonic torsional wave. The method of solution is based on the use of discontinuous solutions of the equation of torsional vibrations and lies in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of angular displacement or tangential stress.

Interaction of plane elastic nonstationary waves with an elastic inclusion under complete adhesion

We solve the problem on the interaction of plane elastic nonstationary waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) which in under conditions of plane strain. It is assumed that the condition of perfect adhesion between the inclusion and the matrix is satisfied. Because of the small thickness of the inclusion we assume that the bending and shear displacements at any inclusion point coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself are found from the corresponding equations of the theory of plates. The statement of the boundary conditions for these equations takes into account the forces and moments acting on the inclusion edges from the matrix. The solution method is based on representing the displacements in the space of Laplace transforms as a discontinuous solution of the Lame’ equations for the plane strain with subsequent determining the transforms of the unknown jumps from integral equations. The passage to the original functions is performed numerically by methods based on replacement of the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors for the inclusion. These formulas are used to study the time dependence of the stress intensity factors and the influence of the inclusion rigidity on their values. We also study the possibility of treating inclusions of high rigidity as absolutely rigid inclusions.

Determination of the stress state in a half-space in the vicinity of cylindrical defects appearing on the surface under torsional vibrations

We solve the problem of the determination of the stress state under torsional vibrations of a half-space with a cylindrical defect (a crack or a thin rigid inclusion) that crosses its surface. The method of solution is based on the use of discontinuous solutions of the equations of torsional vibrations and consists in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of an angular displacement or a tangential stress.

Interaction of plane nonstationary waves with a thin elastic inclusion under smooth contact conditions

We solve the problem of determining the stress state near a thin elastic inclusion in the form of a strip of finite width in an unbounded elastic body (matrix) with plane nonstationary waves propagating through it and with the forces exerted by the ambient medium taken into account. We assume that the matrix is in the plane strain state, and the smooth contact conditions are realized on both sides of the inclusion. The method for solving this problem consists in using the integral Laplace transform with respect to time and in representing the stress and displacement images in terms of the discontinuous solution of Lame equations in the case of plane strain. As a result, the initial problem is reduced to a system of singular integral equations for the transforms of the unknown stress and displacement jumps. To invert the Laplace transform, we use a numerical method based on replacing the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors (SIF) for the inclusion, which are used to study the SIF time-dependence and its influence on the values of the inclusion rigidity. We also studied the possibility of considering the inclusions of higher rigidity as absolutely rigid inclusions.

Resonance blocking of traveling waves by a system of cracks in an elastic layer

Wave processes that occur in an elastic layer when waves traveling in it are diffracted by a system of horizontal cracks are investigated. Integral representations of wave fields are constructed in terms of the convolution of Green’s matrices and unknown jumps of displacements at the cracks. The displacement jumps are determined from the boundary integral equations, which are obtained from the initial boundary-value problem with the boundary conditions at crack faces being satisfied. The spectrum of the integral operator is studied for different variants of mutual crack arrangement and is compared with the spectrum of the corresponding operators for individual cracks; the relationship between the spectrum and the blocking effects is analyzed. The possibility of obtaining an extended frequency band of waveguide blocking in the case of groups of cracks is demonstrated.

Stress intensity near a crack in the composition of a half space and a layer under harmonic loading

We consider a three-dimensional problem for an elastic bimaterial body formed by a half space and a layer and containing a circular crack. The surface of the crack is subjected to the action of stationary torsional forces. The problem is reduced to the solution of a system of boundary integral equations of Helmholtz-potential type for the unknown jumps of displacements on the crack surfaces. The dynamic stress concentration in the vicinity of the crack contour is investigated.

Basic solution of a Mode-I limited-permeable crack in functionally graded piezoelectric/piezomagnetic materials

In this paper, the basic solution of a Mode-I limited-permeable crack in functionally graded piezoelectric/piezomagnetic materials subjected to a stress loading was investigated using the generalized Almansi’s theorem. In the analysis, the electric permittivity and the magnetic permeability of the air inside the crack were considered. The problem was formulated through Fourier transform into two pairs of dual integral equations about the unknown jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses, the singular electric displacements and the singular magnetic flux at crack tips in functionally graded piezoelectric/piezomagnetic materials carry the same forms as those in homogeneous piezoelectric/piezomagnetic materials; however, the magnitudes of the magnetoelectric intensity factors depend on the gradient of functionally graded piezoelectric/piezomagnetic materials. It is also revealed that the effects of the electric and magnetic boundary conditions on the electric displacement and magnetic flux fields near crack tips can not be ignored.

Interaction of plane harmonic waves with inclusions in the elastic space

We solve the problem of interaction of harmonic elastic waves with a thin elastic inclusion in the form of a strip in an unbounded body (matrix) under the conditions of plane deformation. In view of the small thickness of the inclusion, it is assumed that its bending and shear displacements coincide with the displacements of the corresponding points of its median plane. The displacements of the medium plane are found from the corresponding equations of the theory of plates. The method of solution is based on the representation of displacements in the form of singular solutions of the Lame equations with subsequent determination of the unknown jumps from singular integral equations. The indicated integral equations are solved numerically (by the collocation method). The relations for the approximate evaluation of the stress intensity factors at the ends of the inclusion are obtained.

Stresses Formed Near a Crack in Half Space in Contact with Liquid Under Harmonic Loading

We study a three-dimensional bimaterial object formed by an elastic half space with cracks and a liquid. The crack surfaces are subjected to the action of harmonic loads. The problem is reduced to the solution of a system of two-dimensional boundary integral equations of Helmholtz-potential type for the unknown jumps of displacements on the crack surfaces. We analyze the case of a circular crack whose lips are subjected to the action of stationary loads. The inertial effects in the vicinity of the crack contour are investigated.

Temperature Stresses in a Piecewise-Homogeneous Cylindrical Shell Caused by the Presence of a Longitudinal Crack

By using the method of generalized conjugation problems, we propose a numerical scheme for investigation of the redistribution of temperature stresses in a piecewise-homogeneous cylindrical shell caused by a longitudinal crack. This scheme is based on systems of integral equations (some of them are singular) to determine the unknown jumps of integral characteristics of the disturbed temperature field and displacements and their derivatives on the line of a crack and on the interface as well as the derivatives of these integral characteristics with respect to the longitudinal coordinate at the interface.

Comparison of displacement and stress fields upon diffraction of elastic waves at different defects: Cracks and thin rigid intrusions

This paper considers the problem of diffraction of elastic waves at rectilinear defects (cracks or thin rigid intrusions) in an unbounded elastic medium. A discontinuous solution of the Helmholtz equation is used to reduce the problem to a singular integral equation in unknown jumps. There is a detailed analysis and comparison of the stressed state close to the defects and the diffraction field at large distances from the defect for defects of different types.

Interaction of a plane harmonic Rayleigh wave with a thin rigid edge inclusion coupled with an elastic medium

Oscillations of a rigid edge inclusion placed in an elastic half-plane, coupled with an elastic medium and extending onto a surface perpendicular to it, are considered. The oscillations are induced by a plane harmonic Rayleigh surface wave propagating in the elastic medium. To solve this problem, the displacement field in the half-plane is expressed as a superposition of the displacements induced by the propagating Rayleigh wave and two discontinuous solutions of the Lamé equations with jumps at the boundary of the half-plane and at the line along which the inclusion is situated. The unknown jumps are determined from the boundary conditions and the conditions of the interaction of the inclusion with the medium. This reduces the problem to the solution of a system of singular integral equations, with a stationary singularity, for the jumps of the stresses on the line of the inclusion. The system is solved numerically by mechanical quadratures. The parameters of the motion of the inclusion and the stressed state of the medium near it are investigated.