Fredholm integral equation approach for multiple crack problem in antiplane shear and inplane electric field of piezoelectric materials

In this paper, the basic presentation in antiplane shear and inplane electric field of piezoelectric materials is refreshed. In order that the functions used in the formulation can be distinguished by their usage, four analytic functions, or four complex potentials, are introduced. A multiple crack problem for piezoelectric materials is studied. After taking the traction or the electric displacement on the crack face as unknown functions, one can naturally obtain a Fredholm integral equation for the multiple crack problem. It is found that the Fredholm integral equation approach is effective for solving the multiple crack problem. Finally, numerical examples are given.     

A necessary and sufficient set of boundary integral equations with indirect unknowns for plate bending problem

A boundary integral representation of plane biharmonic function is established rigorously by the method of unanalytical continuation in the present paper. In this representation there are two boundary functions and four constants which bear a one to one correspondence to biharmonic functions. Therefore the set of boundary integral equations with indirect unknowns based on this representation is equivalent to the original differential equation formulation.     

On the regularization method of the first kind of Fredholm integral equation with a complex kernel and its application

The regularized integrodifferential equation for the first kind of Fredholm integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-demensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given. Project supported by the National Natural Science Foundation, of China     

The anisotropic and angularly inhomogeneous elastic wedge under a monomial load distribution

In this study, the generally anisotropic and angularly inhomogeneous wedge under a monomial type of distributed loading of order n of, the radial coordinate r at its external faces is considered. At first, using variable separable relations in the equilibrium equations, the strain–stress relations and the strain compatibility equation, a differential system of equations, is constructed. Decoupling this system, an ordinary differential equation is derived and the stress and displacement fields may be determined. The proposed procedure is also applied to the elastostatic problem of an isotropic and angularly inhomogeneous wedge. The special cases of loading of order n=−1 and n=−2, where the self-similarity approach is not valid, are examined and the stress and displacements fields are derived. Applications are presented for the cases of an angularly inhomogeneous wedge and in the case of a bi-material isotropic wedge.     

Boundary integral equations for 2D elasticity and its application in discrete dislocation simulation in finite body: 2. Numerical implementation

In the previous paper by Yu and Diab (2013), several sets of boundary integral equations are derived for general anisotropic materials and corresponding equations for materials with different classes of symmetry are deduced. The work presented herein implements two sets of boundary element schemes to numerically solve the stress field. The integration on the element that has the singular point of the kernel is bounded and can be evaluated analytically. Four benchmark elastic problems are solved numerically to show the advantage of the two schemes over the conventional boundary element formulation in eliminating the boundary layer effect. The one with the weaker singularity has better convergence and gives more accurate results. The presented formulation also provides a direct approach to solve for stress field in a finite solid body in the presence of dislocations. Combined with discrete dislocations dynamics, boundary value problems with dislocations in finite bodies can be solved. Two examples, bending of a single crystal beam and pure shearing of a polycrystalline solid, are simulated by discrete dislocation dynamics using the scheme that has the weaker singularity. The comparisons with the published results using the well-established superposition technique validate the proposed formulation and show its quick convergence.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.     

Boundary integral equations for 2D elasticity and its application in discrete dislocation dynamics simulation in finite body: 1. General theory

A unified approach, originating from Cauchy integral theorem, is presented to derive boundary integral equations for two dimensional elasticity problems. Several sets of boundary integral equations are derived and their relations are revealed. Explicit expressions for materials with different symmetry planes are listed. Special attention is given to the formulation that is based on the tractions and the tangential derivatives of displacements along solid boundary, since its integral kernels have the weakest singularities. The formulation is further extended to include singular points, such as dislocations and line forces, in a finite body, so that the singular stress field can be directly obtained from solving the integral equations on the external boundary, without involving the linear superposition technique that was often used in the literature. Its application in simulating discrete dislocation motion in a finite solid body is discussed.     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.     

Solutions for a system of nonlinear random integral and differential equations under weak topology

I.IntroductionTheexistenceandcomparisonresultsofsolutionsfornonlinearVolterraintegralequationsinstrongtopologyofBanachspaceshavebeenobtainedbyVaughnI"'"],LakshmikanthamIl3]andLakshmikantham-Leela114l.TheexistenceresultsofweaksolutionsfortheCauchyproblemof…     

Passages to the limit in the dynamic problem for a crack at the interface between elastic media

The paper analyzes numerically the passages to the limit in the dynamic problem for a penny-shaped crack at the interface between dissimilar linear elastic, homogeneous, isotropic materials as either the frequency of harmonic load or the difference between the properties of the materials decreases. It is shown that as the frequency decreases, the solution of the dynamic problem tends to that of the static problem, and as the physical and mechanical properties of the materials become less different, the original problem goes into the dynamic problem for a crack in a homogeneous body __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 26–34, July 2008.     

Lubrication theory for micropolar fluids and its application to a journal bearing with finite length

In this paper, the field equation of micropolar fluid with general lubrication theory assumptions is simplified into two systems of coupled ordinary differential equation. The analytical solutions of velocity and microrotation velocity are obtained. Micropolar fluid lubrication Reynolds equation is deduced. By means of numerical method, the characteristics of a finitely long journal bearing under various dynamic parameters, geometrical parameters and micropolar parameters are shown in curve form. These characteristics are pressure distribution, load capacity, coefficient of flow flux and coefficient of friction. Practical value of micropolar effects is shown, so micropolar fluid theory further closes to engineering application.     

Line integral representations for the traction problem of the elastic halfspace and their application to a study of the singular effects of load jumps

Line-integral representations for the solution of the elastostatic traction boundary-value problem of the half space are derived for the case of polyharmonic surface loading. For load regions of essentially arbitrary shape, bearing uniform tractions, these line-integral representations are applied to the analysis of the stress singularities present at the edges of the load region.

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Resume On dérive des représentations par intégrales curvilignes de la solution du problème aux conditions limites de traction élasto-statique dans le demi espace.On applique, pour des régions de charge de forme essentiellement arbitraire, ces représentations par intégrales curvilignes à l'analyse des singularités de tension qui apparaissent aux bords de la région de charge.

     

Generalized Fourier transform and its application to the volume integral equation for elastic wave propagation in a half space

In the present study, a generalized Fourier transform for time harmonic elastic wave propagation in a half space is developed. The generalized Fourier transform is obtained from the spectral representation of the operator derived from the elastic wave equation. By means of the generalized Fourier transform, a volume integral equation method for the analysis of scattered elastic waves is presented. The proposed method is based on the Krylov subspace iteration technique. During the iterative process, the discrete generalized Fourier transform is used, where the derivation of a huge and dense matrix from the volume integral equation is not necessary.     

Asymptotic analysis of integral equations for a great interval and its application to stellar radiative transferAnalyse asymptotique d'une équation intégrale sur un grand intervalle et son application au transfert radiatif stellaire

We consider the integral form of the radiative transfer equation over a large interval. This equation describes the radiative transfer of energy in a star. The asymptotic expansion of the solution is constructed and justified. The method of asymptotic partial decomposition of domain is applied. Numerical results are discussed. To cite this article: G. Panasenko et al., C. R. Mecanique 330 (2002) 735–740.