Evolution of Phase Boundaries by Configurational Forces

An initial boundary value problem modeling the evolution of phase interfaces in materials showing martensitic transformations is studied. The model, which is derived rigorously from a sharp interface model with phase interfaces driven by configurational forces and which generalizes that model, consists of the equations of linear elasticity coupled with a nonlinear partial differential equation of hyperbolic character governing the evolution of the order parameter. It is proved that in one space dimension, global solutions exist for which the order parameter belongs to the space of functions of bounded variation. Other models for interface motion by martensitic transformations and by interface diffusion are suggested.     

Nonlinear vibrations of a beam with time-varying rigidity and mass

Nonlinear Dynamics - We consider asymptotic solutions for nonlinear beams that can be described by a fourth order hyperbolic equation with an integral nonlinearity and some space and time dependent...     

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

We study the nonlinear hyperbolic partial differential equation, (u _t+uu_x)_x=1/2u _x ² . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are no a priori estimates on the second derivatives in any L ^p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.     

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

The cases of large Reynolds number and small expansion ratio for the asym- metric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Nu- merical methods are also designed to confirm the correctness of the present asymptotic solutions.     

Uniform difference scheme for a singularly perturbed linear 2nd order hyperbolic problem with zeroth order reduced equation

In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.     

A phase field model for failure in interconnect lines due to coupled diffusion mechanisms

We describe a diffuse interface, or phase field model for simulating electromigration and stress-induced void evolution and growth in interconnect lines. Microstructural evolution is tracked by defining an order parameter, which takes on distinct uniform values within solid material and voids, and varying rapidly from one to the other over narrow interfacial layers associated with the void surfaces. The order parameter is governed by a form of the Cahn-Hilliard equation. An asymptotic analysis demonstrates that the zero contour of order parameter tracks the motion of a void evolving by coupled surface and lattice diffusion, driven by stress, electron wind and vacancy concentration gradients. Efficient finite element schemes are described to solve the modified Cahn-Hilliard equation, as well as the equations associated with the accompanying mechanical, electrical and bulk diffusion problems. The accuracy and convergence of the numerical scheme is investigated by comparing results to known analytical solutions. The method is applied to solve various problems involving void growth and evolution in representative interconnect geometries.     

The asymptotic behavior of solutions to the Kirchhoff equation with a viscous damping term

We study the asymptotic behavior of solutions to an equation describing non-linear vibration of a string with viscosity. In the case when the string is unstretched (the degenerate case), we determine the decay order of solutions by investigating the dynamics near an infinite-dimensional center manifold. Moreover, we classify the asymptotic behavior of all solutions from a dynamical systems point of view. We also deal with the case where the string is stretched (the nondegenerate case).     

High Frequency Solutions for the Singularly-Perturbed One-Dimensional Nonlinear Schrödinger Equation

This article is devoted to the nonlinear Schrödinger equation when the parameter ε approaches zero. All possible asymptotic behaviors of bounded solutions can be described by means of envelopes, or alternatively by adiabatic profiles. We prove that for every envelope, there exists a family of solutions reaching that asymptotic behavior, in the case of bounded intervals. We use a combination of the Nehari finite dimensional reduction together with degree theory. Our main contribution is to compute the degree of each cluster, which is a key piece of information in order to glue them.     

Stability and Freezing of Nonlinear Waves in First Order Hyperbolic PDEs

It is a well-known problem to derive nonlinear stability of a traveling wave from the spectral stability of a linearization. In this paper we prove such a result for a large class of hyperbolic systems. To cope with the unknown asymptotic phase, the problem is reformulated as a partial differential algebraic equation for which asymptotic stability becomes usual Lyapunov stability. The stability proof is then based on linear estimates from (Rottmann-Matthes, J Dyn Diff Equat 23:365–393, 2011) and a careful analysis of the nonlinear terms. Moreover, we show that the freezing method (Beyn and Thümmler, SIAM J Appl Dyn Syst 3:85–116, 2004; Rowley et al. Nonlinearity 16:1257–1275, 2003) is well-suited for the long time simulation and numerical approximation of the asymptotic behavior. The theory is illustrated by numerical examples, including a hyperbolic version of the Hodgkin–Huxley equations.     

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEMS FOR SECOND ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS ON INFINITE INTERVAL(Ⅱ)

In this paper the existence of solutions of the singularly perturbed boundary valueproblems on infinite interval for the second order nonlinear equation containing a smallparameterε>0 :is examined,whereα_i,βare constants,and i=0,1 .Moreover,asymptoticestimates of the solutions for the above problems are given.     

Compression of an axisymmetric layer on a rigid mandrel in creep

An approximate solution describing the compression of an axisymmetric layer ofmaterial on a rigid mandrel under the equations of the creep theory is constructed. The constitutive equation is introduced so that the equivalent stress tends to a finite value as the equivalent strain rate tends to infinity. Such a constitutive equation leads to a qualitatively different asymptotic behavior of the solution near the mandrel surface, on which the maximum friction law is satisfied, compared with the well-known solution for the creep model based on the power-law relationship between the equivalent stress and the equivalent strain rate. It is shown that the solution existence depends on the value of one of the parameters contained in the constitutive equations. If the solution exists, then the equivalent strain rate tends to infinity as the maximum friction surface is approached, and the qualitative asymptotic behavior of the solution depends on the value of the same parameter. There is a range of variation of this parameter for which the qualitative behavior of the equivalent strain rate near the maximum friction surface coincides with the behavior of the same variable in ideally rigid-plastic solutions.     

Asymptotic analysis of turbulent boundary layer flow on a flat plate at high Reynolds numbers

The equations of the turbulent boundary layer contain a small parameter — the reciprocal of the Reynolds number, which makes it possible to carry out an asymptotic analysis of the solutions with respect to that small parameter. Such analyses have been the subject of a number of studies [1–5]. In [2, 5] for closing the momentum equation algebraic Prandtl and turbulent viscosity models were used. In [1, 3, 4] the structure of the boundary layer was analyzed in general form without formulating specific closing hypothesis but under additional assumptions concerning the nature of the asymptotic behavior of the limiting solutions in the various regions.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 106–117, May-June, 1993.     

A dynamical study of the evolution of pressure waves propagating through a semi-infinite region of homogeneous gas combustion subject to a time-harmonic signal at the boundary

In this paper, we study the evolution of pressure waves propagating in a region of gas combustion subject to a time-harmonic signal at the boundary. The problem is modeled by a non-linear, hyperbolic partial differential equation. Steady-state behavior is investigated using the perturbation method to ensure that enough time has passed for transient effects to have dissipated. The zeroth-order and first two approximations are obtained. Furthermore, the behavior of the following quantities is investigated, with particular attention paid to the low and high-frequency limits: the location of the peak of the first-order approximation, dispersion relations and phase speeds. Additionally, a maximum value of the perturbation parameter is determined ensuring boundedness of the solution. Next, approximate solutions are obtained in the low and high frequency limits and a comparison is made with the corresponding perturbation solution. Finally, the solution obtained from the perturbation method is compared with the long-time solution obtained by a non-standard finite-difference scheme.     

Transient spherical flow of non-newtonian power-law fluids in porous media

The transient spherical flow behavior of a slightly compressible non-Newtonian, power-law fluids in porous media is studied. A nonlinear partial differential equation of parabolic type is derived. The diffusivity equation for spherical flow is a special case of the new equation. We obtain analytical, asymptotic and approximate solutions by using the methods of Laplace transform and weighted mass conservation. The structures of asymptotic and approximate solutions are similar, which enriches the theory of one-dimensional flow of non-Newtonian fluids through porous media.     

Non-linear vibrating systems in resonance governed by hyperbolic differential equations

The asymptotic solutions of second order hyperbolic differential equations with weak non-linearities in the case of internal and external resonance are found. The method used is an extension of the Krylov-Bogoliubov-Mitropolskii method. An application is made to the longitudinal vibrations of a rod in which non-linear elastic behaviour and linear viscoelastic damping occur.     

Limiting Solutions of Nonlocal Dispersal Problem in Inhomogeneous Media

This paper is concerned with the nonlocal dispersal problem in inhomogeneous media. Our goal is to show the limiting behavior of perturbation equation with parameters. By analyzing the asymptotic behavior of solutions when the parameter is small, we find that convection appears in inhomogeneous media. Moreover, if the effect of inhomogeneous media changes, then we prove a convergence result that convection disappears in nonlocal dispersal problems.

     

Plasticity,internal structure and phase field model

The model presented here is based on the assumption that the plastic phase is due to a variation of the strain generating a dislocation migration, which in turn implies a different material response. That is, the transition from elastic to plastic behavior is related with two different internal or mesoscopic structures. So, within the Landau theory on phase transitions, and via the notion of order parameter, we suggest a model for a hardening plasticity by a second order phase transition, able to describe the elastic–plastic transformation. The differential problem related with this new variable will be represented by the Ginzburg–Landau equation. The model is supplemented by a differential constitutive equation among the strain, the stress, and the order parameter. By this system, we are able to obtain the classical behavior of hardening plastic phase diagrams.     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

Diffusive limits of nonlinear hyperbolic systems with variable coefficients

We consider the initial-boundary value problem for a 2-speed system of first-order nonhomogeneous semilinear hyperbolic equations whose leading terms have a small positive parameter. Using energy estimates and a compactness lemma, we show that the diffusion limit of the sum of the solutions of the hyperbolic system, as the parameter tends to zero, verifies the nonlinear parabolic equation of the p-Laplacian type.     

The asymptotic solving equations of thick ring shell and its solution under moment Mo

In this paper, from the fundamental equations of three dimensional elastic mechanics, we have found a sequence of asymptotic solving equations of thick ring shell (or body) applied arbitrary loads by the perturbation method based upon a geometric small parameter a=r_o/R_o, which may be divided into two independent equation groups which are similar to the equation groups for plane strain and torsional problems. Using these equations, we have also found first order and second order approximate solutions of thick ring shell applied moment M_o.