On the steady propagation of a semi-infinite crack

We consider the rectilinear propagation of a semi-infinite crack with constant velocity in a crystal structure. We obtain the solutions of homogeneous boundary-value problems for the corresponding difference-differential operators in spaces of one and two dimensions. We give a justification of the computational aspect of the problem. Bibliography: 8 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 127–153.     

On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.     

On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.     

On the asymptotic behaviour of the empirical random field of the brownian motion

Let ξ_t, t ? 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ??∫^t₀?(ξ_s) ds is investigated, where ? belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm.     

An asymptotic approach in problems of crack identification

An asymptotic approach to solving problems of the identification of a rectilinear crack of small relative size is presented. The solution of the direct problem is reduced to solving a boundary integral equation. Using the proposed approach, its kernel is investigated, and the main part of the asymptotic form is singled out. The inverse problem of determining the crack parameters from prescribed information on the amplitudes of the displacement on the boundary of a layer is solved. Transcendental equations are obtained, from which the characteristics of a crack are determined in stages. Numerical results of the solution of the inverse problem are presented.     

Stochastic modeling of fatigue crack propagation

This paper presents a stochastic model of fatigue-induced crack propagation in metallic materials. The crack growth rate predicted by the model is guaranteed to be non-negative. The model structure is built upon the underlying principle of Karhunen–Loève expansion and does not require solutions of stochastic differential equations in either Wiener integral or Itô integral setting. As such this crack propagation model can be readily adapted to damage monitoring and remaining life prediction of stressed structures. The model results have been verified by comparison with experimental data of time-dependent fatigue crack statistics for 2024-T3 and 7075-T6 aluminum alloys.     

On asymptotic properties of the empirical correlation matrix of a homogeneous vector-valued Gaussian field

We investigate properties of the empirical correlation matrix of a centered stationary Gaussian vector field in various function spaces. We prove that, under the condition of integrability of the square of the spectral density of the field, the normalization effect takes place for a correlogram and integral functional of it.     

关于有限域中一个方程及其渐近公式

利用解析方法研究了有限域中一个方程的性质，并给出了其解数的渐近公式．     

Optimal lower bounds on asymptotic support propagation rates for the thin-film equation

We derive lower bounds on asymptotic support propagation rates for strong solutions of the Cauchy problem for the thin-film equation. The bounds coincide up to a constant factor with the previously known upper bounds and thus are sharp. Our results hold in case of at most three spatial dimensions and n∈(1,2.92)$n\in (1,2.92)$. The result is established using weighted backward entropy inequalities with singular weight functions to yield a differential inequality; combined with some entropy production estimates, the optimal rate of propagation is obtained. To the best of our knowledge, these are the first lower bounds on asymptotic support propagation rates for higher-order nonnegativity-preserving parabolic equations.     

On asymptotic distribution on the a-adic integers

We show that the values of a polynomial with a-adic coefficients at integer and rational prime arguments are asymptotically distributed on the a-adic integers and that the integer parts of certain sequences known to be uniformly distributed modulo one, are uniformly distributed on the a-adic integers.     

3D Ductile crack propagation with the XFEM

Ductile fracture is of high interest for industrial Manufacturing. Here we attempt to model this material degradation with a combined approach of non-local continuum damage mechanics and discrete fracture mechanics using the eXtended Finite Element Method. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a numerical scheme for curved crack propagation based on configurational forces and maximum dissipation

A numerical scheme is presented to predict crack trajectories in two dimensional components. First a relation between the curvature in mixed–mode crack propagation and the corresponding configurational forces is derived, based on the principle of maximum dissipation. With the help of this, a numerical scheme is presented which is based on a predictor–corrector method using the configurational forces acting on the crack together with their derivatives along real and test paths. With the help of this scheme it is possible to take bigger than usual propagation steps, represented by splines. Essential for this approach is the correct numerical determination of the configurational forces acting on the crack tip. The methods used by other authors are shortly reviewed and an approach valid for arbitrary non–homogenous and non–linear materials with mixed–mode cracks is presented. Numerical examples show, that the method is a able to predict the crack paths in components with holes, stiffeners etc. with good accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An asymptotic approach to the problem of the distant propagation of waves

An asymptotic approach is used in this paper to examine solutions of the problem of distant propagation of Love waves in a medium consisting of an inhomogeneous layer lying on a homogeneous half-space. The dispersion properties of these waves and their attenuation are estimated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 125–135, 1986.     

On a model of electromagnetic field propagation in ferroelectric media

The Maxwell system in an anisotropic, inhomogeneous medium with non-linear memory effect produced by a Maxwell type system for the polarization is investigated under low regularity assumptions on data and domain. The particular form of memory in the system is motivated by a model for electromagnetic wave propagation in ferromagnetic materials suggested by Greenberg, MacCamy and Coffman [J.M. Greenberg, R.C. MacCamy, C.V. Coffman, On the long-time behavior of ferroelectric systems, Phys. D 134 (1999) 362-383]. To avoid unnecessary regularity requirements the problem is approached as a system of space-time operator equation in the framework of extrapolation spaces (Sobolev lattices), a theoretical framework developed in [R. Picard, Evolution equations as space-time operator equations, Math. Anal. Appl. 173 (2) (1993) 436-458; R. Picard, Evolution equations as operator equations in lattices of Hilbert spaces, Glasnik Mat. 35 (2000) 111-136]. A solution theory for a large class of ferromagnetic materials confined to an arbitrary open set (with suitably generalized boundary conditions) is obtained.     

On the asymptotic linking number

We prove a theorem formulated by V. I. Arnold concerning a relation between the asymptotic linking number and the Hopf invariant of divergence-free vector fields. Using a modified definition for the system of short paths, we prove their existence in the general case.

     

Homogenization and flame propagation in periodic excitable media: The asymptotic speed of propagation

We study the effect of homogenization on flame propagation in periodic excitable media when the width of the flame is much smaller than the characteristic size of the heterogeneities. © 2005 Wiley Periodicals, Inc.