Interaction of a harmonic torsional wave with ring-shaped defects in an elastic body

We have solved the problem of determination of the stressed state in an isotropic elastic body near ring-shaped defects (a crack or a thin rigid inclusion) as a result of the action of a harmonic torsional wave. The method of solution is based on the use of discontinuous solutions of the equation of torsional vibrations and lies in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of angular displacement or tangential stress.     

Boundary element method for simulating the coupled motion of a fluid and a three-dimensional body

A boundary element method for potential flow problem coupled with the dynamics of rigid body was developed to determine numerically the resultant force and moment of force acting on an arbitrarily three-dimensional solid body and its motion in a current of an infinite fluid. An accurate integration method for singular integrands occurring in the boundary integral equations, a computational method for the tangential gradient of a velocity potential on a surface, and a method to properly treat the singularities appearing in the system of the dynamic equations of a rigid body, were proposed to complete the numerical solution of the problem. Several numerical examples were given to show the validity of the method.     

A new approach to Cagniard's problem

Cagniard problem refers to the class of linear reflection and transmission problem for pulsed line and point sources, which have solution methods leading to exact algebraic representations of the wave fields. All previous methods have relied heavily on integral or differential transforms. We present in this paper a new and direct approach to the problem which involves only the wave equation and its associated characteristic equation. We illustrate the new method by applying it to the problem of the reflection and transmission of acoustic waves radiating from a line source in the vicinity of a plane boundary separating two uniform acoustic media.     

Solution of problems of oblique penetration of axisymmetric projectiles into soft soil based on local interaction models

A comparative analysis of the solutions of the three-dimensional problem of the oblique penetration of a rigid body into soft soil is carried out arsing interaction models based on one-dimensional solutions of the problem of the spherical cavity expansion. Both the well-known self-similar analytical solutions for an incompressible medium as well as the generalized solution for a compressible elastoplastic medium with separation of the shock wave which arises are considered. Use of the incompressible medium hypothesis, disregarding flow separation, in estimating the maximum values of the resistive forces leads to large errors. Taking account of compressibility enables the resistive forces to be refined appreciably and enables a satisfactory estimate of the deviation of the trajectories of bodies from the initial direction of motion to the obtained. In the proposed method of solving oblique penetration problems, a three-dimensional problem is reduced, on the basis of the plane sections hypothesis and disregarding peripheral mass and momentum flows, to the combined solution of a number of axisymmetric problems for each meridional section. It is shown that, with well-known local interaction models, this approach enables the reliability of the calculation of both the force and the kinematic characteristics of the penetration process to be increased considerably due to the fact that the dynamics of the free surface and cavitation effects of the covitating flow are taken into account.     

非齐次弹性力学问题双互易边界元方法研究北大核心CSCD

基于弹性力学边界元方法理论,将边界元法与双互易法结合,采用指数型基函数对非齐次项进行插值得到双互易边界积分方程.将边界积分方程离散为代数方程组,利用已知边界条件和方程特解求解方程组,得出域内位移和边界面力.指数型基函数的形状参数是由插值点最近距离的最小值决定,采用这种形状参数变化方案,分析径向基函数(RBF)插值精度以及插值稳定性.再次将指数型基函数应用到双互易边界元法中,分析双互易边界元方法下计算精度及稳定性,验证了指数型插值函数作为双互易边界元方法的径向基函数解决弹性力学域内体力项问题的有效性.     

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.     

Hertzian contact problem: Numerical reduction and volumetric modification

A technique for the analytical formulation and numerical implementation of an elastic contact model for rigid bodies in the framework of the Hertzian contact problem is described. The normal elastic force and the semiaxes of the contact area are computed so that the problem is sequentially reduced to a scalar transcendental equation depending on complete elliptic integrals of the first and second kinds. Based on the classical solution to the Hertzian contact problem, an invariant volumetric force function is proposed that depends on the geometric characteristics of interpenetration of two undeformed bodies. The normal forces computed using the force function agree with results obtained previously for non-Hertzian contact of elastic bodies. As an example, a ball bearing is used to compare the contact dynamics of elastic bodies simulated in the classical Hertzian model and its volumetric modification.     

The contact problem with the bulk application of intermolecular interaction forces (a refined formulation)

A refined formulation of the contact problem when there are intermolecular interaction forces between the contacting bodies is considered. Unlike the traditional formulation, it is assumed that these forces are applied to points within the body, rather than to the surface of the deformable body as a contact pressure, and that the body surface is load-free. Solutions of the contact problems for a thin elastic layer attached to an absolutely rigid substrate and for an elastic half-space are analysed. The refined and traditional formulations of the problem when there is intermolecular interaction are compared. ©2013     

新的三维力学GELD正演和反演算法

在本文中 ,我们提出了新的整体积分和局部微分GILD的力学正演和反演方法 .我们建立了弹性和塑性力学的体积分微分方程 .我们证明了这个体积分方程和伽辽金虚功原理等价 .新的GILD方法是基于这个体积分微分方程 .GL方法是进一步的发展 ,GL是一种整体场和局部场相互作用的全新方法 .在这个方法中 ,仅仅需要解 3× 3或者 6 × 6的局部小矩阵 .特别是 ,用GL方法求解无限域的偏微分方程时 ,不需要任何人工边界 ,不需要任何吸收边界条件和不需要任何边界积分方程 .新的三维力学GILD正演和反演算法已被应用研究奈米材料的力学性质的模拟计算 .我们获得非常好的奈米材料的力学变形的超拉力的力学性质 .我们提出了新的奈米地球物理新概念和发现了GILD数值量子     

Axisymmetric torsional fretting contact between a spherical punch and an FGPM coating

The axisymmetric torsional fretting contact between a rigid conducting spherical punch and a functionally graded piezoelectric material (FGPM) coating is studied in this paper. The exponential model is used to simulate the inhomogeneous electro-mechanical properties of the FGPMs coating. The conducting spherical punch with a constant surface electric potential is considered in the contact. A normal force and a cyclic torque are applied to the two contact bodies. The applied torque produces an outer annular slip area and an inner stick area. The torsion angle is produced within the inner stick area as a rigid body. With the aid of the Hankel integral transform technique, we can reduce the contact problem to the singular integral equations of the Cauchy type. Then the unknown electro-mechanical fields and stick/slip area can be obtained numerically. The effect of the gradient index on the surface electro-mechanical fields is discussed at loading and unloading phases. The Mises stress and principal stress at the contact surface are also discussed to predict the possible location of fretting damage and failure.     

Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid body in a nonconservative field

We present complete results devoted to the study of the equations of motion of a dynamically symmetric four-dimensional rigid body in a nonconservative force field. The form of the body is taken from the dynamics of real two- or three-dimensional rigid bodies interacting with a resisting medium according to the streamline flow around laws under which a non-conservative pair of forces acts on the body and forces the body center of mass to move rectilinearly and uniformly.     

推广的KdV方程特征问题解的存在唯一性

推广的KdV方程u_t+αuu_x+μu_x3+εu_x5=0[1]是典型的可积方程.它先后在研究冷等离子体中磁声波的传播[2],传输线中孤立波[3]和分层流体中界面孤立波[4]时导出.本文对推广的KdV方程的特征问题,在Riemann函数的基础上,设计一恰当结构,并由此化待征问题为一与之等价的积分微分方程.而该积分微分方程对应的映射E是列自身的映射[5],依不动点原理,积分微分方程有唯一的正则解,即推广的KdV方程的特征问题有唯一解,且由积分微分方程序列所得的迭代解于Ω上一致收敛.     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.     

Adapted kussmaul formulation of acoustic scattering problems: an overview

Problems of exterior acoustic scattering may be conveniently formulated by means of boundary integral equations. The problem seeks to find a wave function which gives velocity potential profile, pressure density profile, etc. of the acoustic wave at points in space. At the background of the formulations are two theories viz. (Helmholtz) Potential theory and the Green's representation formula. Potential theory gives rise to the so-called indirect formulation and the Green's representation formula to the direct formulations. Classical boundary integral formulations fail at the eigenfrequencies of the interior domain. That is, if a solution is sought of the exterior problem by first solving a homogeneous boundary integral equation, one is inevitably led to the conclusion that these homogeneous boundary equations have nontrivial solutions at certain wave-numbers which are the eigenvalues of the corresponding interior problem. At lower wave-numbers, these eigenfrequencies are thinly distributed but the higher the wave-number, the denser it becomes. This is a well-known drawback for both time-harmonic acoustics and elastodynamics. This is not a physical difficulty but arises entirely as a result of a deficiency in the integral equation is representation. Why then use It? The use has many advantages notably in that the meshing region is reduced from the infinite domain exterior to the body to its finite surface. This created the need for some robust formulations. A proof of the Kussmaul [1] formulation is presented. The formulation has a hypersingular kernel in the integral operator, which creates a havoc in computation (e.g., ill conditioning). The hyper-singularity can be avoided [2], as a result a new formulation is proposed. This paper presents a broad overview of the Adapted Kussmaul Formulation (AKF).     

Wave Field Decomposition in Anisotropic Fluids: A Spectral Theory Approach

An extension of directional wave field decomposition in acoustics from heterogenous isotropic media to generic heterogenous anisotropic media is established. We make a connection between the Dirichlet-to-Neumann map for a level plane, the solution to an algebraic Riccati operator equation, and a projector defined via a Dunford–Taylor type integral over the resolvent of a nonnormal, noncompact matrix operator with continuous spectrum.In the course of the analysis, the spectrum of the Laplace transformed acoustic system's matrix is analyzed and shown to separate into two nontrivial parts. The existence of a projector is established and using a generalized eigenvector procedure, we find the solution to the associated algebraic Riccati operator equation. The solution generates the decomposition of the wave field and is expressed in terms of the elements of a Dunford–Taylor type integral over the resolvent.     

Solution of problems on the anti-plane deformation of elastic bodies with corner points by the method of integral equations

The problem of the antiplane deformation of an elastic cylinder with a multiconnected finite or infinite section, bounded by a system of closed curves that can have corner points, is examined. Forces or displacements are given on the whole boundary of the body. The problem is reduced to an integral equation whose kernel has strong stationary singularities at the corner points. Results of an investigation of the solvability of this equation and the smoothness of its solution are presented. A procedure for the numerical solution of the integral equation is described. A space with a prismatic hole of rectangular section or a rigid inclusion subjected to a uniform tangential force at infinity is considered as an example. The generalized stress intensity factors are calculated.     

Inverse coefficient problem for a wave equation in a bounded domain

The nonlinear inverse problem for a wave equation is investigated in a three-dimensional bounded domain subject to the Dirichlet boundary condition. Given a family of solutions to the equation defined on a closed surface within the original domain, it is required to reconstruct the coefficient determining the velocity of sound in the medium. The solutions used for this purpose correspond to the acoustic medium perturbations localized in the neighborhood of a certain closed surface. The inverse problem is reduced to a linear integral equation of the first kind, and the uniqueness of the solution to this equation is established. Numerical results are presented.     

The contact problem for a two-layer spherical base

Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.     

Summatory and integral equations in periodic diffraction problems for elastic waves at defects in stratified media

We consider the diffraction problem for an elastic wave on a periodic set of defects located at the interface of stratified media. We reduce the mentioned problem to a pair summatory functional equation with respect to coefficients of the expansion of the desired wave by quasiperiodic waves (the Floquet waves). Using the method of integral identities, we reduce the pair equation to a regular infinite system of linear equations. One can solve this system by the truncation method. We prove that the integral identity is the necessary and sufficient condition for the solvability of the auxiliary overspecified problem for a system of equations in a half-plane in the elasticity theory. We obtain integral equations of the second kind which are equivalent to the initial diffraction problem.