Asymptotics of the elastic solution of a plane problem near the crack tip terminating at a nonideal bimaterial interface (mode I and II)

The local stress-strain state in the vicinity of the crack tip in a composite is studied, taking into account the mechanical and geometric features of the nearest interface. The modeling of Mode I and II problems for a semi-infinite crack terminating normally at a nonideal interface in the bimaterial plane is considered. The constituents, of the composite are assumed to be elastic, homogeneous, and isotropic. The intermediate zone between the constituents is modeled by interfacial conditions in the form: [ _n ]=0, [u]=r _n , where [u] and [ _n ] are jumps of the vectors of displacements and tractions along the interface. The diagonal matrix with nonnegative components and the parameter, 0 are defined by the mechanical and geometric characteristics of the intermediate zone, respectively. Thus, the case =0 corresponds to the usual ideal contact conditions along the interface. Using the method of integral transformations, the corresponding problems are reduced to systems of functional equations, and later to systems of integral equations with fixed point singularities. The solvability of the systems of integral equations is proved and the asymptotics of their solutions is found. Based on these results, the local distributions of the displacements and stresses near the crack tip are obtained. It is shown that the interfacial parameters and greatly influence the stress not only qualitatively (the character of the stress singularity near the crack tip changes), but also quantitatively (number of singular terms in the asymptotics increases). The graphs illustrating these results are presented as the values of the interfacial parameters and , as well as the ratio of the shear moduli ₀/₁ of the constituents.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Polytechnical Institute Poland. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 5, pp. 621–642, September–October, 1998.     

Investigation of the prefracture zone at the tip of a mode I crack reaching a nonsmooth interface of an elastic media

We perform calculation of the initial prefracture zone at the tip of a mode I crack that reaches a nonsmooth interface of two dissimilar elastic media at its corner point by the Wiener–Hopf method. The zone is modeled by a line of normal displacement fracture on a crack continuation. Expressions for the length of the prefracture zone and the potential energy accumulated in it are obtained. Their numerical values are compared with the corresponding values for the prefracture zone in a bonding material on the interface of the media, on the basis of which we draw a conclusion on the possible direction of development of the zone.     

On the Number of Appearances of a Word in a Sequence of I.I.D. Trials

Let X ₁,...,X _n be a sequence of i.i.d. random variables taking values in an alphabet =₁,...,_q,q 2, with probabilities P(X _a=_i)=p _i,a=1,...,n,i=1,...,q. We consider a fixed h-letter word W=w₁...w_h which is produced under the above scheme. We define by R(W) the number of appearances of W as Renewal (which is equal with the maximum number of non-overlapping appearances) and by N(W) the number of total appearances of W (overlapping ones) in the sequence X _{a 1} a1n under the i.i.d. hypothesis. We derive a bound on the total variation distance between the distribution (R(W)) of the r.v. R(W) and that of a Poisson with parameter E(R(W)). We use the Stein-Chen method and related results from Barbour et al. (1992), as well as, combinatorial results from Schbath (1995b) concerning the periodic structure of the word W. Analogous results are obtained for the total variation distance between the distribution of the r.v. N(W) and that of an appropriate Compound Poisson r.v. Related limit theorems are obtained and via numerical computations our bounds are presented in tables.     

Basic solutions of a 3D rectangular limited-permeable crack or two 3D rectangular limited-permeable cracks in the piezoelectric/piezomagnetic composite materials

The solutions of a three-dimensional rectangular limited-permeable crack or two three-dimensional rectangular limited-permeable cracks in the piezoelectric/piezomagnetic composite materials were investigated by using the generalized Almansi’s theorem and the Schmidt method. Finally, the relations among the electric field, the magnetic flux field and the stress field near the crack tips were obtained and the effects of the electric permittivity, the magnetic permeability of the air inside the crack, the shape of the rectangular crack on the stress, the electric displacement and magnetic flux intensity factors in the piezoelectric/piezomagnetic composite materials were analyzed.     

On the Azlarov-Volodin theorem for sums of I.I.D. Random elements in banach spaces

For partial sums Sn of i.i.d. random elements {V_n, n 1) in a real separable Banac space, Azlarov and Volodin proved that if EV₁^p<∞ where 0<p<1, then

S_n/n_1/p→0 almost certainly. In this note a simple proof is given of their result and this proof allows for the deletion of the independence assumption. Some remarks are provided which relate the extended result to other results in the literature or which illustrate its sharpness     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

Divergence Criterion for a Class of Random Series Related to the Partial Sums of I.I.D. Random Variables

Journal of Theoretical Probability - Let $$ \{X, X_{n};~n \ge 1 \}$$ be a sequence of independent and identically distributed Banach space valued random variables. This paper is devoted to...     

Nonlinear evolution of a first mode oblique wave in a compressible boundary layer: I. Heated/cooled walls

The nonlinear stability of an oblique mode propagating in atwo-dimensional compressible boundary layer is considered underthe long wavelength approximation. The growth rate of the waveis assumed to be small so that the ideas of unsteady nonlinearcritical layers can be applied. It is shown that the spatial/temporalevolution of the mode is governed by a pair of coupled unsteadynonlinear equations for the disturbance vorticity and density.Expressions for the linear growth rate show clearly the effectsof wall heating and cooling, and in particular how heating destabilizesthe boundary layer for these long wavelength inviscid modesat O(1) Mach numbers. A generalized expression for the lineargrowth rate is obtained and is shown to compare very well fora range of frequencies and wave angles at moderate Mach numberswith full numerical solutions of the linear stability problem.The numerical solution of the nonlinear unsteady critical layerproblem using a novel method based on Fourier decompositionand Chebyshev collocation is discussed and some results arepresented.     

Boundary layers for the 3D primitive equations in a cube: The Zero-mode

We establish the vanishing viscosity limit of the zero-mode of the linearized Primitive Equations in a cube. Our method is based on the explicit construction and estimates of the boundary layers. This result, together with that in [12, 15], allows us to conclude the vanishing viscosity limit of the linearized Primitive Equations in a cube.     

Dynamics of droplets in an agitated dispersion with multiple breakage. Part I: formulation of the model and physical consistency

In (Proc. Symp. Lectures Appl. Math. 2000; 123–141) a new model for the evolution of a system of droplets dispersed in an agitated liquid was presented, with the inclusion of the so‐called volume scattering effect (a combination of coalescence and breakage). In that paper droplets breakage was considered to be binary, in order to simplify exposition. Here we remove that limitation, investigating the effect of each breakage mode and of scattering with multiple exits. We also allow the breakage kernel, at each mode, to become singular when droplets approach their finite maximum admissible size. Copyright © 2004 John Wiley & Sons, Ltd.     

On a Long-Standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property

This paper studies a conjecture made by De Giorgi in 1978 concerning the one-dimensional character (or symmetry) of bounded, monotone in one direction, solutions of semilinear elliptic equations Δu=F′(u) in all of R ⁿ . We extend to all nonlinearities F∈C ² the symmetry result in dimension n=3 previously established by the second and third authors for a class of nonlinearities F which included the model case F′(u)=u ³?u. The extension of the present paper is based on new energy estimates which follow from a local minimality property of u. In addition, we prove a symmetry result for semilinear equations in the halfspace R ₊ ⁴. Finally, we establish that an asymptotic version of the conjecture of De Giorgi is true when n≤8, namely that the level sets of u are flat at infinity.     

Application of M-PML reflectionless boundary conditions to the numerical simulation of wave propagation in anisotropic media. Part I: Reflectivity

This paper presents a detailed study of the construction of reflectionless boundary conditions for anisotropic elastic problems. A Multiaxial Perfectly Matched Layer (M-PML) approach is considered. With a proper stabilization parameter, the M-PML ensures solution stability for arbitrary anisotropic media. It is proved that this M-PML modification is not perfectly matched, and the reflectivity of the M-PML exceeds that of the standard PML. Moreover, the reflection coefficient linearly depends on the stabilization parameter. A problem of constructing an optimal stabilization parameter is formulated as follows: find a minimum possible parameter that ensures stability. This problem is considered in the second part of this work.     

Free boundary effects on stability of two phase planar fluid motion in annulus. Part I: Migration of the stable mode

We study the linearized stability of a planar dynamical model describing two-phase perfect fluid circulating around a circle with a sufficiently large radius within a central gravitational field. The model is associated with the spatial and temporal structure of the zonally averaged global-scale atmospheric longitudinal circulation around the Earth. Two cases are studied separately; in the first one, the simulations were carried out using the rigid lid approximation at the upper boundary of the outer atmospheric layer. In the second one, the free boundary nonlinear conditions (kinematic and dynamic) were assumed on the outer atmospheric layer. For the both cases, a certain family of steady, explicit solutions which have circular streamlines was considered. The governing equations were linearized at these solutions to find the typical wave numbers of the interfacial wave perturbation to the basic state at which the destabilizing effect of shear, which overcomes the stabilizing effect of stratification, occurs. It is shown that for the both cases, the model always have the same two potentially unstable wave modes while there always exist two wave modes which are stable for any wavelengths. The behavior of the stable and unstable modes were compared for the both cases to investigate the effects of the free boundary on the mixing process at the interface.     

A criterion for straightening of a lipschitz surface in the Lizorkin-Triebel sense. I

We study the conditions when the trace of a Lizorkin-Triebel space on a Lipschitz surface coincides with the trace of this space on a hyperplane. A criterion in terms of a dyadic weighted inequality is found for a wide range of indices.     

Stabilization in a 3D eco-epidemiological model: From the complete extinction of a predator population to their self-healing

In this paper, using the localization method of compact invariant sets, we examine the ultimate dynamics of the 3D prey–predator model containing two subpopulations of susceptible and infected predators. Our attention is focused to finding ultimate sizes of interacting populations, and, in addition, we show the existence of a global attracting set. Then, we derive various global conditions of ultimate extinction of at least one of the predators subpopulations and describe conditions under which all types of internal bounded dynamics are ruled out. In particular, we describe convergence conditions to omega-limit sets located (1) in the intersection of the prey-free plane with the infected predators-free plane and (2) in the infected predators-free plane. Based on the dynamical analysis of the 2D infection-free subsystem, we obtain conditions of global attraction to (i) the prey-only disease-free equilibrium point, (ii) the disease-free prey-predator equilibrium point (self-healing of the predator population), and (iii) the omega-limit set containing an equilibrium point or a periodic orbit. Main theoretical results are illustrated by numerical simulation. Tools and techniques developed in this work can be appropriated in the studies within predictive population ecology of more complex eco-epidemiological models.     

The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow

We show that for a certain family of initial data, there exist non-unique weak solutions to the 3D incompressible Euler equations satisfying the weak energy inequality, whereas the weak limit of every sequence of Leray–Hopf weak solutions for the Navier–Stokes equations, with the same initial data, and as the viscosity tends to zero, is uniquely determined and equals the shear flow solution of the Euler equations corresponding to this initial data. This simple example suggests that, also in more general situations, the vanishing viscosity limit of the Navier–Stokes equations could serve as a uniqueness criterion for weak solutions of the Euler equations.     

Solutions with peaks for a coagulation–fragmentation equation. Part I: stability of the tails

Abstract

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation–fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behavior is stable. In a companion paper, we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.