Propagation of plane elastic waves at a plane interface between two electro-microelastic solid half-spaces

This work is concerned with the wave propagation and their reflection and transmission from a plane interface between two different electro-microelastic solid half-spaces in perfect contact. It is found that there exist five basic waves in an infinite electro-microelastic solid, namely an independent longitudinal micro-rotational wave, two sets of coupled longitudinal waves influenced by the electric effect, and two sets of coupled transverse waves. The existence of the two sets of coupled longitudinal waves is new. In the absence of microstretch and electric effects, these two coupled longitudinal waves reduce to a longitudinal displacement wave of micropolar elasticity. Amplitude and energy ratios of various reflected and transmitted waves are presented when (i) a set of coupled longitudinal wave is made incident and (ii) a set of coupled transverse wave is made incident. Numerical computations have been performed for a particular model and the variations of amplitude and energy ratios are obtained against the angle of incidence. The results obtained are depicted graphically. It has been verified that the sum of energy ratios is equal to unity at the interface and the amplitude ratios of reflected and transmitted waves depend upon the angle of incidence, frequency and elastic properties of the media. Results of some earlier workers have also been reduced from the present formulation.     

二维弹性问题边界元法中边界层效应问题的变换法

基于间接规则化边界积分方程,有效估计奇异边界积分,准确求得边界量,为场变量的计算奠定了基础。在计算场变量时,针对二维弹性力学边界元法中出现的几乎奇异积分,本文采用一类非线性变量替换法,有效地改善了被积函数的震荡特性,从而消除了核积分的几乎奇异性;在不增加计算量的情况下,极大地改进了几乎奇异积分计算的精度,成功地求解了弹性体近边界点上的力学参量,避免了边界层效应。此外,本文引入一种精确几何单元逼近,对于圆弧边界,这样的插值逼近几乎是精确的,提高了计算精度。数值算例表明,本文算法稳定,效率高,并可达到很高的计算精度,即使场点非常靠近边界,如场点到积分单元的距离小到纳米级,仍可避免边界层效应现象。     

The boundary element method applied to elastic contact problems with friction

In the current paper the boundary integral equations (BIE) for elastic contact problems with friction are derived from the incremental virtual work principle. After introducing contact conditions of adhesion and slip into BIE all variants on boundary are made to discretize by quadratic isoparametric boundary element. In the current paper not only an auto-increment loading law is presented but also the iterative calculation laws for open, slip and adhesion condition are given. The results of numerical examples are satisfactory.     

Transverse wave at a plane interface between isotropic elastic and unsaturated porous elastic solid half-spaces

Based on the poroelasticity theory, this article investigates the reflection and transmission characteristics of an incident plane transverse wave at a plane interface between an isotropic elastic half-space and an unsaturated poroelastic solid half-space. For this purpose, the effect of the saturation degree and frequency on the properties of the four bulk waves in unsaturated porous medium, i.e., three longitudinal waves and one transverse wave, are discussed at first. Two general cases of mode conversion are considered: (i) The initial transverse wave is incident from an unsaturated poroelastic half-space to the interface, and (ii) the initial transverse wave is incident from an elastic solid half-space to the interface. The expressions for the partition of energy at the interface during transmission and reflection process of waves are presented in explicit forms. At last, numerical computations are performed for these two cases and the results obtained are depicted, respectively. The variation of the amplitude ratios and energy ratios with the saturation degree and incident angle is illustrated in detail. It is also verified that, at the interface, the sum of energy ratios is approximately equal to unity as expected.     

Experimental study of plane elastic contact problems by the pseudocaustics method

The optical method of pseudocaustics can be used for the experimental solution of plane elasticity, smooth contact problems for finite or infinite media in contact of arbitrary shape. This technique constitutes an alternative to the various numerical and experimental techniques for the approximate solution of plane elasticity contact problems. The success in the application of the method of pseudocaustics to plane elasticity contact problems is due to the possibility inherent in this method of the direct determination of the derivative Φ'(z) of the complex potential Φ(z) of N.I. Muskhelishvili along the boundaries of the media in contact. Then, the Muskhelishvili complex potentials, Φ(z) and Ψ(z), completely characterizing the state of stress and strain in a plane elastic medium, can easily be determined at any point of the media in contact after simple algebraic calculations. Two applications of the proposed method to contact problems of practical interest are also made.     

Some problems in plane elastic dielectrics

Representations of Galerkin type are obtained for the displacement vector, polarization vector and the potential fields in the static plane theory of elastic dielectrics using the method of associated matrices. Fundamental matrix solutions of an infinite elastic dielectric plane subjected to a concentrated body force, electric force and charge density are derived from the singular solutions of harmonic, biharmonic and Helmholtz equations. Using boundary operatorsY, Z, M, the fundamental matrix solutions, and Betti's formulae, a matrix (x, y) is constructed and an integral representation for (u ₁,u ₂,P ₁,P ₂, ) is obtained. Discontinuity theorems are stated for the double layer potential andQ operator of the single layer potential. By means of these theorems, the solutions of interior and exterior boundary value problems are reduced to the solution of a system of five singular integral equations. The index of one of the systems is shown to be zero and it is concluded that Fredholm theorems and its alternatives hold.

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Zusammenfassung Durch Anwendung der Assoziativ-Matrizen-Methode werden Galerkinische Darstellungen für den Verschiebungsvektor, den Polarisierungsvektor und die potentiellen Prüffelder einer statischen ebenen Theorie des elastischen Dielektrikums entwickelt. Von den singulären Lösungen der harmonischen, biharmonischen und Helmholtz-Gleichungen werden grundlegende Matrizenlösungen für das unendliche, ebene und elastische Dielektrikum, das durch konzentrierte Raumkräfte, elektrische Kräfte und Ladungsdichte beansprucht wird, abgeleitet. Unter Verwendung der Randwert-OperatorenY, Z, M, der grundlegenden Matrizen-Lösungen und Betti's Formel, wird eine Spezial-Matrize (x, y) konstruiert und eine Integraldarstellung für (u ₁,u ₂,P ₁,P ₂, ) erhalten. Unstetigkeitssätze werden für das Doppelschicht-Potential und denQ-Operator eines einschichtigen Potentials angeführt. Durch Anwendung dieser Sätze werden die Lösungen der inneren und äusseren Grenzwertprobleme zur Lösung eines Systems von fünf singulären Integralgleichungen reduziert. Der Index eines der Systeme wird als Null bewiesen und es wird die Schlussfolgerung gezogen, dass der Fredholmsche Satz und seine Alternativen fur diese Theorie anwendbar sind.

     

弹性力学平面问题的无奇异边界积分方程

将弹性力学平面问题归化成无奇异边界积分方程,避免了传统的边界元法中的柯西主值（CPV）积分和Hadamard-Finite-Parts(HFP)积分的计算,建立完整的数值求解体系。     

The perturbation-iterative method applied to the problems of the large deflection of the elastic circular thin plates

In this paper, we present a perturbation-iterative method for solving certain boundary value problems encountered in the nonlinear theory of elastic circular thin plates. At the same time, with this method, we strictly prove the convergence of the solutions for the large deflection equations of circular plates subjected to certain distributed loads.     

BOUNDARY INTEGRAL FORMULA OF ELASTIC PROBLEMS IN CIRCLE PLANE

IntroductionConcerningtheelasticplaneprobleminaunitcircle ,ZhengShenzhouandZhengXueliangdevelopedaboundaryintegralformulaofthestressfunction[1]:Φ(r,θ) =-( 1 -r2 ) 24π ∫2π0ν( φ)1 -2rcos(θ-φ) r2 dφ　　 12π∫2π011 -2rcos(θ-ω) r2 dω∫2π0μ( φ)1 -cos(ω-φ) dφ　　 1 -r22π∫2π0μ( φ)1 -2rcos(θ -φ) r2 dφ　　 ( 0 ≤r <1 ) ,( 1 )whereμ(θ) =Φ(r,θ) ｜r=1,ν(θ) = Φ n r=1= Φ r r=1.Intheformula ( 1 )theseconditemisastrongsingularintegral,itshouldbeunderstoodasanintegra…     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

Dynamic meshless method applied to nonlocal crack problems

The dynamic meshless methods for local and nonlocal field theories are formulated in this paper. Application to two crack problems is presented. The meshless method of local theory gives solution that is in good agreement with the classical analytical crack tip solution, while the nonlocal theory yields a solution without stress singularity at the crack tip. The numerical results also show the embedded nonlocal nature of meshless methods.     

Domain decomposition methods applied to solve frictionless-contact problems for multilayer elastic bodies

A parallel Dirichlet–Dirichlet domain-decomposition algorithm for solving frictionless-contact problems for elastic bodies made of composite materials is proposed and justified. Numerical results that demonstrate the effectiveness of the approach and its software implementation are presented     

Interfacial crack between elastic half-spaces under harmonic loading

A system of boundary integral equations that allows evaluating the displacement and stress fields for an interfacial crack under harmonic loading is presented. Expressions for the integral kernels are obtained. A numerical solution for a penny-shaped crack between steel and aluminum half-spaces under normally incident compression-rarefaction wave is given __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 47–56, July 2007.     

The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces

The matrix sign function is shown to provide a simple and direct method to derive some fundamental results in the theory of surface waves in anisotropic materials. It is used to establish a shortcut to the basic formulas of the Barnett–Lothe integral formalism and to obtain an explicit solution of the algebraic matrix Riccati equation for the surface impedance. The matrix sign function allows the Barnett–Lothe formalism to be readily generalized for the problem of finding the surface wave speed in a periodically inhomogeneous half-space with material properties that are independent of depth. No partial wave solutions need to be found; the surface wave dispersion equation is formulated instead in terms of blocks of the matrix sign function of i$i$ times the Stroh matrix.     

二维非局部线弹性平面问题的辛分析

将二维非局部线弹性理论引入到Hamilton体系下,基于变分原理推导得出了二维线弹性理论的对偶方程和相应的边界条件.在分析验证对偶方程的准确性的基础上,该套方法被应用于二维弹性平面波问题的求解.将精细积分与扩展的W-W算法相结合在Hamilton体系下建立了求解平面Rayleigh波的数值算法.从推导到计算的保辛性确保了辛体系非局部理论与算法的准确性.通过对不同算例的数值计算,分析和对比了非局部理论方法与传统局部理论方法的差别,并进一步指出了该套算法的适用性和优势所在.     

Complementary energy variational approach for plane elastic problems with singularities

Presented is the numerical analysis of plane elastic problems involving stress concentrations and/or singularities using a physically meaningful complementary energy variational approach. The continuum body is modeled by a non-conventional truss structure. Stress distributions in laminated composite bodies and orthotropic sheets with a through crack are obtained. The present results are compared with the analytical solutions for different numerical methods.     

The Galerkin method as applied to problems in viscoelasticity

This paper establishes the convergence of the continuous-time Galerkin technique as applied to quasi-static, linear viscoelasticity.