Comparison theorems for some problems of filtration in inhomogeneous ground

The influence of variation in the inhomogeneity of the ground on the solution of filtration problems in the case of earth dams and canals is investigated. A class of problems is identified for which one can specify variations of the filtration coefficient that lead to a change in the position of the depression curve in a definite direction. Similar comparison theorems for the case of a change in the shape of particular sections of the boundary of the region were obtained in [1], in which, in particular, the change in the position of the depression curve was investigated. As examples of the application of the results, the solution for a dam with a vertical inhomogeneity of the ground is estimated in terms of the solution for a homogeneous dam with the same profile, and an axisymmetric problem is also estimated in terms of the solution of the corresponding two-dimensional problem. The obtained results can be used both in a theoretical investigation of such problems as well as with a view to obtaining estimates of a required solution in terms of known solutions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 165–169, March–April, 1981.We thank N. B. Il'inskii for constant interest in the work.     

Solution of strongly nonlinear problems in the mechanics of a solid deformable body

A new algorithm for constructing a Lagrangian formulation of the incremental theory on the basis of finite-element digitization of the equilibrium equations of strongly nonlinear mechanical systems subject to static loads is proposed. Example solutions of problems of nonlinear deformation of a solid that confirm the validity of the applied relations between increments in stresses and increments in deformations is presented. Technical University of Construction and Architecture, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 22–26, March, 1999.     

Path-independent integrals in inhomogeneous media

Summary The concept of path independent integrals introduced previously for homogeneous media is extended to the case of a crack in the interface between dissimilar half planes. Using any of the pathindependent integrals defined in this paper we obtain the stress intensity factor when the lips of the crack are loaded by identical distributions of stresses in both lips, or by concentrated forces or couples. The above integrals are extended in order to include the case of an arbitrary number of collinear cracks in the interface of two half planes. The case of periodic cracks may be derived from the above analysis. Finally, an interesting generalization may be obtained for the case where arbitrary cracks, inclusions or concentrated forces and couples, not necessarily applied at the lips of the crack, interact.

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Wegunabhängige Integrate in inhomogenen Medien

Übersicht Das Konzept wegunabhängiger Integrale, welches für homogene Medien früher eingeführt wurde, wird für den Fall eines Risses in einer Grenzfläche zwischen zwei verschiedenen Halbebenen erweitert. Durch Verwendung eines der in dieser Arbeit definierten wegunabhängigen Integrale erhalten wir den Spannungsintensitätsfaktor, wenn die Eißlippen durch an beiden Lippen identisch verteilter Spannungen oder Einzelkräfte oder Momente belastet werden. Die oben genannten Integrale sind erweitert, um den Fall einer willkürlichen Zahl von kollinearen Bissen in der Grenzfläche der Halbebenen einzuschließen. Der Fall von periodischen Rissen kann aus dieser Analysis erhalten werden. Schließlich kann eine interessante Verallgemeinerung für den Fall erhalten werden, daß willkürliche Risse, Einschlüsse oder nicht unbedingt an den Rißlippen wirkende Einzelkräfte und Momente in Wechselwirkung stehen.

     

A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media

Summary A new meshless method is developed to analyze steady-state heat conduction problems with arbitrarily spatially varying thermal conductivity in isotropic and anisotropic materials. The analog equation is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator can be employed to take the place of the fundamental solutions related to the original governing equation. Next, the particular solution is approximated by using radial basis functions, and the corresponding homogeneous solution is solved by means of the virtual boundary collocation method. As a result, a new method fully independent of mesh is developed. Finally, several numerical examples are implemented to demonstrate the efficiency and accuracy of the proposed method. The numerical results show good agreement with the actual results.This work was supported by the National Natural Science Foundation of China (No. 10472082) and Australian Research Council.     

Thermoelastic wave propagation in inhomogeneous media

Summary A novel approach to the modeling of thermoelastic wave propagation is presented, based on the thermodynamics of discrete systems. The first novelty includes the representation of integral balance laws for thermoelasticity in terms of contact quantities that describe the nonequilibrium state of elements. The next new aspect is a modification of the recently proposed wave-propagation algorithm, which is used as a tool for determining the contact quantities in a finite-volume scheme for the numerical simulation of two-dimensional thermoelastic wave propagation in inhomogeneous media. Such a modification is needed to provide the satisfaction of the thermodynamic consistency conditions between adjacent discrete elements. Results of computations for certain test problems show the efficiency and physical consistency of the algorithm. Received 11 November 1999; accepted for publication 2 May 2000     

Decaying surface waves in inhomogeneous media

Two problems on plane decaying surface waves in an inhomogeneous medium are under consideration: the problem where the waves similar to Rayleigh waves propagate in an isotropic elastic half-space that borders with a layer of an ideal incompressible fluid and the problem where the waves similar to Love waves propagate in a semi-infinite saturated porous medium that borders with a layer of an isotropic elastic medium.     

Solution of some problems of flow through fractured porous media

Problems of fluid flow through a fractured porous medium consisting of fractures and blocks with different filter characteristics are solved. The mass exchange between fractures and blocks is assumed to be proportional to the pressure difference between them. The porosity in the fractures is assumed to be negligibly small. Under these assumptions the determination of the pressure fields reduces to the integration of a system of linear differential equations. The solution is found by the operational method using the Efros theorem. The cases of oil reservoir operation by means of both galleries and wells are considered. The solutions are obtained in an analytical form convenient for calculations.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 94–102, January–February, 1995.     

On the uniqueness and stability of solutions to some continuum mechanics problems

One-dimensional shear flows of a perfect fluid, a perfect rigid-plastic body, a Newtonian viscous fluid, and a viscoplastic medium in a plane layer and the compression problem for a Euler beam are considered as examples to discuss a relation between the nonuniqueness of a solution to the basic continuum mechanics problem and the instability of the unique solution to this problem. When the basic solution is not unique, the question is raised on the mechanical interpretation of the linearized stability problem for this solution.     

Ill-posed problems in mechanics

The notion of ill-posed initial and boundary value problems for partial differential equations was introduced by Hadamard, who also presented the first example of an ill-posed problem for a specific partial differential equation. At the same time, there are numerous examples of ill-posed problems in any field of mechanics.Hadamard and some of his successors believed that any ill-posed problem has no physical meaning and such problems should not be posed.The present paper contains several examples of ill-posed problems. We show that if one deals with an applied problem, then overcoming the ill-posedness mathematically can help one to improve the structure in practice, which justifies the study of ill-posed problems.     

On wave propagation in inhomogeneous elastic media

Wave propagation in an inhomogeneous elastic rod or slab is considered. The governing equations are written in a matrix form and transformations are sought which reduce the system to a form associated with the wave equation. Integration of the system is then immediate. It is shown that such reduction may be achieved subject to a function involving the density and elastic parameters of the material adopting certain multi-parameter forms. These parameters are available for fitting to the behaviour of a variety of inhomogeneous elastic materials. A specific initial boundary value problem is solved by utilising the present method.     

Plane SH-waves in harmonically inhomogeneous elastic media

This paper considers the oblique propagation of a plane SH-wave in an inhomogeneous elastic medium whose material properties vary harmonically with a space variable. Assuming a small deviation in the harmonic variation, the method of multiple scales is applied and approximate solutions are obtained for three types of the medium.Effects of oblique propagation of SH-waves on a resonant frequency and on an unstable region of wave propagation are discussed. Effects on the reflection coefficient at a stress-free boundary when the wave propagates in a semi-infinite space are also discussed.