Convergence of the Cahn-Hilliard equation to the Hele-Shaw model

We prove that level surfaces of solutions to the Cahn-Hilliard equation tend to solutions of the Hele-Shaw problem under the assumption that classical solutions of the latter exist. The method is based on a new matched asymptotic expansion for solutions, a spectral analysis for linearizd operators, and an estimate for the difference between the true solutions and certain approximate ones.     

Mode i and mixed mode fields with incomplete crack tip plasticity

Plane strain slip line fields, in which plasticity does not fully surround the crack tip have been developed for mode I and mixed mode I\II cracks under contained yielding. Analytical solutions have been assembled using slip line theory for the plastic sectors and semi-infinite wedge solutions for the elastic sectors. These solutions are compared with finite element solutions based on modified boundary layer formulations. The analytical solutions agree well with numerical solutions, and form a family of fields with incomplete plasticity around the crack tip.     

非均质流固耦合介质轴对称动力问题时域解

将地基视为流固两相介质并考虑其非均质成层特性，推导了多层地基动力问题时域解．文中首先建立了一组解耦的两相介质动力控制方程；而后利用Ｌａｐｌａｃｅ－Ｈａｎｋｅｌ变换推导了单层地基象空间初参数解答；再利用初参数法及传递矩阵技术导出了任意多层地基瞬态解的一般解析算式．本文获得的解答可方便地退化为现有理想弹性介质的解答     

Dissipative and Entropy Solutions to Non-Isotropic Degenerate Parabolic Balance Laws

We propose a new notion of weak solutions (dissipative solutions) for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. As an application we prove the strong convergence of a general relaxation-type approximation for such equations.     

Water Redistribution in Irrigated Soil Profiles: Invariant Solutions of the Governing Equation

Exact solutions for a class of nonlinear partial differential equations modelling soil water infiltration and redistribution in irrigation systems are studied. These solutions are invariant under two-parameter symmetry groups obtained by the group classification of the governing equation. A general procedure for constructing invariant solutions is presented in a way convenient for investigating numerous new exact solutions.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR GENERALIZED DRINFELD-SOKOLOV EQUATIONS

Ansatz method and the theory of dynamical systems are used to study the traveling wave solutions for the generalized Drinfeld-Sokolov equations. Under two groups of the parametric conditions, more solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions are obtained. Exact explicit parametric representations of these travelling wave solutions are given.     

A computational study of local stress intensity factor solutions for kinked cracks near spot welds in lap-shear specimens

In this paper, the local stress intensity factor solutions for kinked cracks near spot welds in lap-shear specimens are investigated by finite element analyses. Based on the experimental observations of kinked crack growth mechanisms in lap-shear specimens under cyclic loading conditions, three-dimensional and two-dimensional plane-strain finite element models are established to investigate the local stress intensity factor solutions for kinked cracks emanating from the main crack. Semi-elliptical cracks with various kink depths are assumed in the three-dimensional finite element analysis. The local stress intensity factor solutions at the critical locations or at the maximum depths of the kinked cracks are obtained. The computational local stress intensity factor solutions at the critical locations of the kinked cracks of finite depths are expressed in terms of those for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. The three-dimensional finite element computational results show that the critical local mode I stress intensity factor solution increases and then decreases as the kink depth increases. When the kink depth approaches to 0, the critical local mode I stress intensity factor solution appears to approach to that for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. The two-dimensional plane-strain computational results indicate that the critical local mode I stress intensity factor solution increases monotonically and increases substantially more than that based on the three-dimensional computational results as the kink depth increases. The local stress intensity factor solutions of the kinked cracks of finite depths are also presented in terms of those for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. Finally, the implications of the local stress intensity factor solutions for kinked cracks on fatigue life prediction are discussed.     

Mode I stress intensity factor solutions for spot welds in lap-shear specimens

The analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens are investigated based on the classical Kirchhoff plate theory for linear elastic materials. First, closed-form solutions for an infinite plate containing a rigid inclusion under counter bending conditions are derived. The development of the closed-form solutions is then used as a guide to develop approximate closed-form solutions for a finite square plate containing a rigid inclusion under counter bending conditions. Based on the J integral, the closed-form solutions are used to develop the analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens of large and finite sizes. The analytical solutions of the mode I stress intensity factor based on the solutions for infinite and finite square plates with an inclusion are compared with the results of the three-dimensional finite element computations of lap-shear specimens with various ratios of the specimen half width to the nugget radius. The results indicate that the mode I stress intensity factor solution based on the finite square plate model with an inclusion agrees well with the computational results for lap-shear specimens for the ratio of the half specimen width to the nugget radius between 4 and 15. Finally, a set of the closed-form stress intensity factor solutions for lap-shear specimens at the critical locations are proposed for future applications.     

Singular solutions and Green's method in micropolar theory of elasticity

The matrices of fundamental solutions are constructed for a concentrated force as well as a concentrated couple varying harmonically in time and acting in an unbounded micropolar elastic continuum. These solutions are then used to obtain solutions for some other loading singularities. Integral representations, for the displacement and the rotation vectors are obtained by making use of the basic singular solutions. The formal solutions to two fundamental boundary value problems are expressed in terms of integrals which include given surface and body data and Green's functions.     

Further improved F-expansion and new exact solutions for nonlinear evolution equations

The improved F-expansion method with a computerized symbolic computation is used to construct the exact traveling wave solutions of four nonlinear evolution equations in physics. As a result, many exact traveling wave solutions are obtained which include new soliton-like solutions, trigonometric function solutions, and rational solutions. The method is straightforward and concise, and it holds promise for many applications.     

Note on the Heat Transfer of Flows over a Stretching Wall in Porous Media: Exact Solutions

In this note, heat transfer over a stretching sheet with mass transfer in a porous medium is revisited. Analytical solutions are presented for two cases including a prescribed power-law wall temperature case and a prescribed power-law wall heat flux case. The solutions are expressed by the Kummer’s function. Closed-form solutions are found and presented for some special parameters. The solutions might offer more insights of the heat transfer characteristics compared with the numerical solutions.     

Large amplitude non-linear oscillations of a general conservative system

This paper presents new, approximate analytical solutions to large-amplitude oscillations of a general, inclusive of odd and non-odd non-linearity, conservative single-degree-of-freedom system. Based on the original general non-linear oscillating system, two new systems with odd non-linearity are to be addressed. Building on the approximate analytical solutions of odd non-linear systems developed by the authors earlier, we construct the new approximate analytical solutions to the original general non-linear system by combinatory piecing of the approximate solutions corresponding to, respectively, the two new systems introduced. These approximate solutions are valid for small as well as large amplitudes of oscillation for which the perturbation method either provides inaccurate solutions or is inapplicable. Two examples with excellent approximate analytical solutions are presented to illustrate the great accuracy and simplicity of the new formulation.     

New solitons and periodic wave solutions for nonlinear physical models

In this paper, an extended tanh method with computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks, and plane periodic solutions. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.     

A note on longitudinal oscillations of a generalized Burgers fluid in cylindrical domains

The starting solutions for the oscillating motion of a generalized Burgers fluid due to longitudinal oscillations of an infinite circular cylinder, as well as those corresponding to an oscillating pressure gradient, are established as Fourier–Bessel series in terms of some suitable eigenfunctions. These solutions, presented as sum of steady-state and transient solutions, describe the motion of the fluid for some time after its initiation. After that time, when the transients disappear, the motion of the fluid is described by the steady-state solutions which are periodic in time and independent of the initial conditions. These solutions are also presented in simpler but equivalent forms in terms of modified Bessel functions of first and second kind. In both forms, the steady-state solutions can be specialized to give the similar solutions for Burgers, Oldroyd-B, Maxwell, second grade and Newtonian fluids performing the same motions. Finally, the required time to reach the steady-state for cosine and sine oscillations of the boundary is obtained by graphical illustrations.     

Plane-strain crack-tip stress field for anisotropic perfectly-plastic materials

Plane-strain crack-tip stress solutions for anisotropic perfectly-plastic materials are presented. These solutions are obtained using the plane-strain slip-line theory developed by Rice (1973). The plastic anisosotropy is described by the Hill quadratic yield condition. The crack-tip stress solutions under symmetric (Mode I) and anti-symmetric (Mode II) conditions agree well with the low-hardening solutions for the corresponding power-law hardening materials. The crack-tip stress solutions under mixed Mode I and II conditions are also presented. All the solutions indicate that the general features of the slip-line field near a crack tip in orthotropic plastic materials with the elliptical yield contours in the Mohr plane are the same as those associated with isotropic plastic materials. However, the angular variations of the crack-tip stress fields for the materials with large plastic orthotropy differ substantially from those for isotropic plastic materials. Modifications due to polygonal yield contours are outlined and implications of solutions to the fracture analysis of ductile composite materials containing macroscopic flaws are discussed.     

A new method to the (2+1)-dimensional modified KP equation

By means of the auxiliary ordinary differential equation method, we have obtained many solitary wave solutions, periodic wave solutions and variable separation solutions for the (2+1)-dimensional KP equation. Using a mixed method, many exact solutions have been obtained.     

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

This paper presents two exact explicit solutions for the three dimensional dual-phase lag （DLP） heat conduction equation, during the derivation of which the method of trial and error and the authors＇ previous experiences are utilized. To the authors＇ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.     

A new branch of solutions of boundary-layer flows over a permeable stretching plate

The steady-state boundary-layer flows over a permeable stretching sheet are investigated by an analytic method for strongly non-linear problems, namely the homotopy analysis method (HAM). Two branches of solutions are obtained. One of them agrees well with the known numerical solutions. The other is new and has never been reported in general cases. The entrainment velocity of the new branch of solutions is always smaller than that of the known ones. For permeable stretching sheet with sufficiently large suction of mass flux, the difference between the shear stresses and velocity profiles of two branches of solutions is obvious: the shear stress of the new branch of solutions is considerably larger than that of the known ones. However, for impermeable sheet and permeable sheet with injection or small suction of mass flux, the shear stress and the velocity profile of two branches of solutions are rather close: in some cases the difference is so small that the new branch of solutions might be neglected even by numerical techniques. This reveals the reason why the new branch of solutions has not been reported. This work also illustrates that, for some non-linear problems having multiple solutions, analytic techniques are sometimes more effective than numerical methods.     

The analyticity of solutions of the Stefan problem

The Stefan problem of a semi-infinite body with arbitrarily prescribed initial and boundary conditions is studied. One of the objectives of the paper is to investigate the analyticity of the solutions. For this purpose, the prescribed initial and boundary conditions are considered to be series of fractional powers of their arguments. It is found that the exact solutions of the problem for various forms of the initial and boundary conditions can be established in series of parabolic cylinder functions and time t. Existence and convergence of the series solutions are studied and proved. The present solutions include the known exact solutions as special cases. On the basis of the present solutions, the question of the analyticity of solutions of the Stefan problem, raised by Rubinstein in his book, can be answered. Conditions for analyticity of the solutions with various initial and boundary conditions are fully discussed.     

Some Flows in Shape Optimization

Geometric flows related to shape optimization problems of the Bernoulli type are investigated. The evolution law is the sum of a curvature term and a nonlocal term of Hele–Shaw type. We introduce generalized set solutions, the definition of which is widely inspired by viscosity solutions. The main result is an inclusion preservation principle for generalized solutions. As a consequence, we obtain existence, uniqueness and stability of solutions. Asymptotic behavior for the flow is discussed:we prove that the solutions converge to a generalized Bernoulli exterior free-boundary problem.