Asymptotic behaviour of solution for fourth order wave equation with dispersive and dissipative terms

This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.     

The influence of a magnetic field on the mechanical behavior of a fluid interface

This work focuses on a theoretical investigation of the shape and equilibrium height of a magnetic liquid–liquid interface formed between two vertical flat plates in response to vertical magnetic fields. The formulation is based on an extension of the so called Young–Laplace equation for an incompressible magnetic fluid forming a two-dimensional free interface. A first order dependence of the fluid susceptibility with respect to the magnetic field is considered. The formulation results in a hydrodynamic-magnetic coupled problem governed by a nonlinear second order differential equation that describes the liquid–liquid meniscus shape. According to this formulation, five relevant physical parameters are revealed in this fluid static problem. The standard gravitational Bond number, the contact angle and three new parameters related to magnetic effects in the present study: the magnetic Bond number, the magnetic susceptibility and its derivative with respect to the field. The nonlinear governing equation is integrated numerically using a fourth order Runge-Kutta method with a Newton–Raphson scheme, in order to accelerate the convergence of the solution. The influence of the relevant parameters on the rise and shape of the liquid–liquid interface is examined. The interface shape response in the presence of a magnetic field varying with characteristic wavenumbers is also explored. The numerical results are compared with asymptotic predictions also derived here for small values of the magnetic Bond number and constant susceptibility. A very good agreement is observed. In addition, all the parameters are varied in order to understand how the scales influence the meniscus shape. Finally, we discuss how to control the shape of the meniscus by applying a magnetic field.     

Buckling of a twisted and compressed rod

We consider the problem of determining the stability boundary for an elastic rod under thrust and torsion. The constitutive equations of the rod are such that both shear of the cross-section and compressibility of the rod axis are considered. The stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations. The type of bifurcation is determined for parameter values. It is shown that the bifurcating branch is the branch with minimal energy. Finally, by using the first integral, the solution for one specific dependent variable is expressed in terms of elliptic integrals. The solution pertaining to the complete set of equilibrium equations is obtained by numerical integration.     

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.     

Formulation of strain gradient plasticity with interface energy in a consistent thermodynamic framework

In this work, the strain gradient formulation is used within the context of the thermodynamic principle, internal state variables, and thermodynamic and dissipation potentials. These in turn provide balance of momentum, boundary conditions, yield condition and flow rule, and free energy and dissipative energies. This new formulation contributes to the following important related issues: (i) the effects of interface energy that are incorporated into the formulation to address various boundary conditions, strengthening and formation of the boundary layers, (ii) nonlocal temperature effects that are crucial, for instance, for simulating the behavior of high speed machining for metallic materials using the strain gradient plasticity models, (iii) a new form of the nonlocal flow rule, (iv) physical bases of the length scale parameter and its identification using nano-indentation experiments and (v) a wide range of applications of the theory. Applications to thin films on thick substrates for various loading conditions and torsion of thin wires are investigated here along with the appropriate length scale effect on the behavior of these structures. Numerical issues of the theory are discussed and results are obtained using Matlab and Mathematica for the nonlinear ordinary differential equations (NODE) which constitute the boundary value problem.This study reveals that the micro-stress term has an important effect on the development of the boundary layers and hardening of the material at both hard and soft interface boundary conditions in thin films. The interface boundary conditions are described by the interfacial length scale and interfacial strength parameters. These parameters are important to control the size effect and hardening of the material. For more complex geometries the generalized form of the boundary value problem using the nonlocal finite element formulation is required to address the problems involved.     

Asymptotic form of the free surface of a uniformly rotating liquid for large bond numbers

In [1–5] boundary-layer methods were used to solve problems concerned with the equilibrium and motion of a liquid with surface tension in a strong gravitational field (for large Bond numbers Bo). In the present paper we apply these methods to problems involving the equilibrium shape of a uniformly rotating liquid, contained in a cylindrical container of arbitrary cross section or in a container which is a surface of revolution about the z axis. Both of these problems reduce to the asymptotic integration of an equation with a small parameter involving a quasilinear elliptic operator with a nonlinear boundary condition. In the second case, owing to radial symmetry, the equation for the problem goes over into an ordinary equation; however, the wetted boundary is not known beforehand. This boundary, together with the equilibrium shape, is also determined asymptotically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–12, November–December, 1973.The authors thank L. A. Slobozhanin for his help in the preparation of this paper.     

SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM FOR A KIND OF VOLTERRA TYPE FUNCTIONAL DIFFERENTIAL EQUATION

IntroductionThereweresomeresultsofstudyingonboundaryvalueproblemsforfunctionaldifferentialequation[1～6 ]byemployingthetoplolgicaldegreetheoryandsomefixedpointprinciplesinrecentyears.Buttheworktostudyboundaryvalueproblemsfordelaydifferentialequationwithsmallparameterbymeansofthetheoryofsingularperturbationrarelyappeared[7～11].Thereasonforitisthattheworktoconstructtheuppersolutionandlowersolutionforthecaseofdifferentialequationwithdelayisdifficult.Theauthorhasstudiedakindofboundaryvalueproblem…     

功能梯度变曲率曲梁的几何非线性模型及其数值解

基于弹性曲梁平面问题的精确几何非线性理论,建立了功能梯度变曲率曲梁在机械和热载荷共同作用下的无量纲控制方程和边界条件,其中基本未知量均被表示为变形前的轴线坐标的函数。以椭圆弧曲梁为例,采用打靶法求解非线性常微分方程的两点边值问题,获得了两端固定功能梯度椭圆弧曲梁在横向非均匀升温下的热弯曲变形数值解,分析了材料梯度指数、温度参数、结构几何参数等对曲梁受力及变形的影响。     

Singular perturbation of general boundary value problem for higher order quasilinear elliptic equation involving many small parameters

In this paper applying M. I. Visik’s and L. R. Lyuster-nik’s^[1] asymptotic method and principle of fixed point of functional analysis, we study the singular perturbation of general boundary value problem for higher order quasilinear elliptic equation in the case of boundary perturbation combined with equation perturbation. We prove the existence and uniqueness of solution for perturbed problem. We give its asymptotic approximation and estimation of related remainder term.     

ON THE MATHEMATICAL PROBLEMS OF COMPOSITE MATERIALS WITH A DOUBLY PERIODIC SET OF CRACKS

In this paper,the mathematical problem of the second fundamental problem of composite materials with a doubly periodic set of arbitrary shape cracks are investigated,and the interface are arbitrary smooth closed contours.At first,we establish mathematical models by using Muskhelisvili complex variable methods,change the primitive problems into searching complex stress functions which satisfy four boundary value problems and construct forms of the solution,then,under some general restrictions it is reduced to normal type singular integral equation,the unique solvability is proved mathematically     

The optimal point of the gradient of finite element solution

We consider the first boundary value problem of the second order elliptic equation and serendipity rectangular elements. Papers [2,3,9] proved that the gradients of finite element solution possess superconvergence at Gaussianpoint. In this paper, we extend the results in papers [2,3,9] in the sense that the coefficients of the elliptic equations are discontinuous on a curve S which lies in the bounded domain .     

On the determination of missing boundary data for solids with nonlinear material behaviors,using displacement fields measured on a part of their boundaries

The paper is devoted to the derivation of a numerical method for expanding available mechanical fields (stress vector and displacements) on a part of the boundary of a solid into its interior and up to unreachable parts of its boundary (with possibly internal surfaces). This expansion enables various identification or inverse problems to be solved in mechanics. The method is based on the solution of a nonlinear elliptic Cauchy problem because the mechanical behavior of the solid is considered as nonlinear (hyperelastic or elastoplastic medium). Advantage is taken of the assumption of convexity of the potentials used for modeling the constitutive equation, encompassing previous work by the authors for linear elastic solids, in order to derive an appropriate error functional. Two illustrations are given in order to evaluate the overall efficiency of the proposed method within the framework of small strains and isothermal transformation.     

The meshless analog equation method: I. Solution of elliptic partial differential equations

A new purely meshless method for solving elliptic partial differential equations (PDEs) is presented. The method is based on the principle of the analog equation of Katsikadelis, hence its name meshless analog equation method (MAEM), which converts the original equation into a simple solvable substitute one of the same order under a fictitious source. The fictitious source is represented by multiquadric radial basis functions (MQ-RBFs). The integration of the analog equation yields new RBFs, which are used to approximate the sought solution. Then inserting the approximate solution into the PDE and boundary conditions (BCs) and collocating at the mesh-free nodal points results in a system of linear equations, which permit the evaluation of the expansion coefficients of the RBFs series. The method exhibits key advantages over other RBF collocation methods as it is highly accurate and the coefficient matrix of the resulting linear equations is always invertible. The accuracy is achieved using optimal values for the shape parameters and the centers of the multiquadrics as well as of the integration constants of the analog equation, which are obtained by minimizing the functional that produces the PDE. Without restricting its generality, the method is illustrated by applying it to the general second order 2D and 3D elliptic PDEs. The studied examples demonstrate the efficiency and high accuracy of the developed method.     

Construction of two-phase equilibria in a non-elliptic hyperelastic material

This work focuses on the construction of equilibrated two-phase antiplane shear deformations of a non-elliptic isotropic and incompressible hyperelastic material. It is shown that this material can sustain metastable, two-phase equilibria which are neither piecewise homogeneous nor axisymmetric, but, rather, involve non-planar interfaces which completely segregate inhomogeneously deformed material in distinct elliptic phases. These results are obtained by studying a constrained boundary value problem involving an interface across which the deformation gradient jumps. The boundary value problem is recast as an integral equation and conditions on the interface sufficient to guarantee the existence of a solution to this equation are obtained. The contraints, which enforce the segregation of material in the two elliptic phases, are then studied. Sufficient conditions for their satisfaction are also secured. These involve additional restrictions on the interface across which the deformation gradient jumps — which, with all restrictions satisfied, constitutes a phase boundary. An uncountably infinite number of such phase boundaries are shown to exist. It is demonstrated that, for each of these, there exists a solution — unique up to an additive constant — for the constrained boundary value problem. As an illustration, approximate solutions which correspond to a particular class of phase boundaries are then constructed. Finally, the kinetics and stability of an arbitrary element within this class of phase boundaries are analyzed in the context of a quasistatic motion.     

Flow and heat transfer of an electrically conducting third grade fluid past an infinite plate with partial slip

The effects of partial slip on the steady flow and heat transfer of an electrically conducting, incompressible, third grade fluid past a horizontal plate subject to uniform suction and blowing is investigated. Two distinct heat transfer problems are studied. In the first case, the plate is assumed to be at a higher temperature than the fluid; and in the second case, the plate is assumed to be insulated. The momentum equation is characterized by a highly nonlinear boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. Numerical solutions for the governing nonlinear equations are obtained over the entire range of physical parameters. The effects of slip, magnetic parameter, non-Newtonian fluid characteristics on the velocity and temperature fields are discussed in detail and shown graphically. It is interesting to find that the velocity and the thermal boundary layers decrease with an increase in the slip, and as the slip increases to infinity, the flow behaves as though it were inviscid.     

Analytical modeling of the flexible rim of space antenna reflectors

This paper presents a geometrically nonlinear analytical model of the flexible cylindrical rim of a deployable precision large space antenna reflectors made of shape-memory polymer composites. A nonlinear boundary-value problem for the rim in the deformed (folded) configuration is formulated and exact analytical solutions in elliptic functions and integrals describing the deformation modes of the rim are obtained. Exact analytical solutions based on the geometrically nonlinear model are obtained and can be used to determine preliminary geometric dimensions and optimal shape of the flexible rim along with the estimation of the accumulated energy.     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.     

Forced dissipative Boussinesq equation for solitary waves excited by unstable topography

In the paper, the effects of topographic forcing and dissipation on solitary Rossby waves are studied. Special attention is given to solitary Rossby waves excited by unstable topography. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a forced dissipative Boussinesq equation. By using the modified Jacobi elliptic function expansion method and the pseudo-spectral method, the solutions of homogeneous and inhomogeneous dissipative Boussinesq equation are obtained, respectively. With the help of these solutions, the evolutional character of Rossby waves under the influence of dissipation and unstable topography is discussed.     

On the Robin-Transmission Boundary Value Problems for the Nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes Systems

The purpose of this paper is to study a boundary value problem of Robin-transmission type for the nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes systems in two adjacent bounded Lipschitz domains in \({{\mathbb{R}}^{n} (n\in \{2,3\})}\), with linear transmission conditions on the internal Lipschitz interface and a linear Robin condition on the remaining part of the Lipschitz boundary. We also consider a Robin-transmission problem for the same nonlinear systems subject to nonlinear transmission conditions on the internal Lipschitz interface and a nonlinear Robin condition on the remaining part of the boundary. For each of these problems we exploit layer potential theoretic methods combined with fixed point theorems in order to show existence results in Sobolev spaces, when the given data are suitably small in \({L^2}\)-based Sobolev spaces or in some Besov spaces. For the first mentioned problem, which corresponds to linear Robin and transmission conditions, we also show a uniqueness result. Note that the Brinkman–Forchheimer-extended Darcy equation is a nonlinear equation that describes saturated porous media fluid flows.     

Transonic Shocks for the Full Compressible Euler System in a General Two-Dimensional De Laval Nozzle

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p _e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p _e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Hölder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.