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1.
2.
Shell theory equations are constructed by the method in [1] to the accuracy of quantities of the order of h*2+k, where and k = 2−4t for (h* is the relative semithickness of the shell and t is the index of the state of stress variation). Without being within the framework of the Lovetype theory, the equations obtained are compared with the Reissner-Naghdi equations. [2, 3] in which the transverse shear is taken into account, and it is shown that from the asymptotic viewpoint these latter are inconsistent. It is also shown that if the shell resists shear weakly, then from the asymptotic viewpoint the Reissner-Naghdi theory is completely well founded.The three-dimensional equations of elasticity theory are reduced to two-dimensional equations in [1] by using an asymptotic method, i.e. all members of the same order relative to the small parameter h* are taken into account at each stage of the calculations. It has been shown that without going outside the framework of the ordinary concepts of the Love-type theory of shells (in particular, without taking account of transverse shear), the shell theory equations can be constructed to the accuracy of quantities of the order of h2−2t*, but it is impossible to exceed this limit without a qualitative complication in the theory.  相似文献   

3.
We consider a method for determining the sound field in a two-dimensional layer. The method we present combines the usual method of reflected plane waves with a summation from graphs. It makes it comparatively easy to take into account the complex interference pattern due to the transformation of the various waves at the boundaries of the layer and to obtain integral relations for the sound potentials. When the layer thickness tends to infinity, the problem reduces to one concerning the reflection of sound waves at the interface of two media. We study the potentials of normal waves in the case of a harmonic source in a solid.  相似文献   

4.
We examine the case of resonance for systems close to nonlinear systems, admitting of a parametric periodic solution. Among the eigenvalues of the matrix of the system's linear part there are zero and pure imaginary ones. We have proved (under certain conditions) the absence of a periodic solution for the original system for which the generating solution is trivial.  相似文献   

5.
The plane model proposed by N. N. Verigin for a stabilized fresh water lens produced by uniform infiltration is investigated in hydrodynamic formulation in the case of equidistant horizontal slit drains. Formulas are obtained for the separation boundary, the depression curve, and characteristic dimesions of the lens.  相似文献   

6.
On the basis of results in [1], a derivation is given of the fundamental Hertz relationships for the compression of anisotropic (orthotropic) bodies which differs from [2]. It is shown that if the elastic constants satisfy some additional conditions, then the domain of contact is a circle in the compression of axisymmetric bodies along their common axes of geometric symmetry.  相似文献   

7.
In connection with the fact that failure of a structure ordinarily starts at sites of the most acute stress concentrations near cavitiies, it is of interest to determine the shape of the equally strong outlines of holes on which the technologically inevitable stress concentration would be least as compared with all other outlines.An effective exact solution of some inverse plane problems of the theory of elasticity concerning the determination of equally strong outlines of holes is proposed. A formulation of the problem is given first and the fundamental relationships are presented. Then the general problem for any number of holes in an infinite plane is reduced to a standard Dirichlet problem for the exterior of the same number of parallel slits on a parametric plane. An effective exact solution is found by this method for the case of one and two holes as well as for the case of periodic and doubly-periodic series of holes. The question of application of the solutions obtained to the theory of a minimum weight structure is considered.  相似文献   

8.
We study the dual integral equations related to the Kontorovich-Lebedev integral transforms arising in the course of solution of the problems of mathematical physics, in particular of the mixed boundary value problems for the wedge-shaped regions. We show that the solutions of these equations can be expressed in quadratures, using the auxilliary functions satisfying the integral Fredholm equation of second kind with a symmetric kernel.At present, the dual equations investigated in most detail are those connected with the Fourier and Hankel integral transforms. The results obtained and their applications are given in [1–3]. A large number of papers also deal with the theory and applications of the dual integral equations connected with the Mehler-Fock integral transform and its generalizations [4–11]., The dual integral transforms considered in the present paper belong to a more complex class than those listed above, and so far, no effective solution has been obtained for them. The only relevant results known to the authors are those in [12, 13]. In [12] a method of solving the equations (1.2) is given for a single particular value of the parameter γ = π/2, while in [13] the dual equations of the type under consideration are reduced to a solution of an infinite system of linear algebraic equations.  相似文献   

9.
Two-dimensional dynamic equations of thin plate vibrations are obtained from the three-dimensional dynamic equations of elasticity theory on the basis of an asymptotic method [1 – 3], Such an approach permits establishing the limits of applicability of the two-dimensional dynamic equations and the corresponding boundary and initial conditions, and indicating the means of obtaining refined results.The question of the construction of an inner state of stress of a thin plate under dynamic conditions is examined herein. The possibility of considering states of stress with distinct variability in time and in the coordinates and with a distinct relationship between the displacement intensities, is taken into account.  相似文献   

10.
The problem of compression of an elastic plane with a slit of variable width commensurate to the elastic strains is considered. The case of the origination of several contact sections of the slit edges is investigated. Adhesion of the edges hence occurs at some part of the contact area, while slip is possible at the rest of this area. A solution of the problem is obtained in quadratures by the Muskhelishvili method using the apparatus of linear conjugates of analytic functions. The stress and displacement potentials are found, the magnitudes of the contact sections and the adhesion zones are determined. A specific example is analyzed and numerical computations are carried out.The contact problem for a plane weakened by a constant-width rectilinear slit has been considered in [1 – 3].  相似文献   

11.
The separation of a system of three elasticity theory equations in the static case to a system of two equations and one independent equation for a space with a radial inhomogeneity is presented in a spherical coordinate system. These equations are solved by separation of variables for specific kinds of radial inhomogeneity. In particular, solutions are found for the Lamé coefficients μ = const, λ (ifr) is an arbitrary function, μ = μorβ, λ = λorβ.While methods of solving problems associated with the equilibrium of an elastic homogeneous sphere have been studied sufficiently [1], problems with spherical symmetry of the boundary conditions have mainly been solved for an inhomogeneous sphere [2, 3],For a particular kind of inhomogeneity dependent on one Cartesian coordinate, the equations have been separated completely in [4], A system of three equations with a radial inhomogeneity in a spherical coordinate system is separated below by a method analogous to [4].  相似文献   

12.
We determine a family of self-similar solutions of a two-dimensional problem involving the filtration of an incompressible liquid in regions with moving boundaries. Our work is based on a method developed by Galin for solving the problem of settling of water cones in a gravitational field [1 – 3]. Following this method, we reduce the problem to one of finding an analytic function of a complex variable and the time, which effects a conformal mapping of the filtration region onto a strip and satisfies a special nonlinear condition on the boundary. For the solution of a problem of this kind Galin proposed the method of successive approximations.  相似文献   

13.
Using the concepts developed in [1] we investigate, in the presence of certain restrictions, the stability of a weakly inhomogeneous state parametrically perturbed by a small random addition of white noise. We show that when the characteristic wavelength is arbitrarily small as compared with the distance over which it varies substantially, then the mechanism of formation of the eigenfunctions responsible for the stability of the state is analogous to the mechanism given in [1]. In the present case it is not the boundaries that act as reflectors, as in [1], but the points at which the condition of existence of the global eigenfunction for the homogeneous problem holds. We obtain the criterion of stability of the state in question and discuss the problem of application of the results obtained to the case in which the ratio of the characteristic wavelength to the distance over which it varies substantially, cannot be taken as arbitrarily small.  相似文献   

14.
An extension of the equivalence of “area” rule [1, 2] is presented. The rule was initially derived for stationary flows of perfect (inviscid and non-heat-conducting) gas past slender fine pointed bodies (or blunted bodies in the hypersonic flow case) whose transverse dimensions are small in comparison with their length. According to that rule the wave drag of a three-dimensional body is equal to the wave drag of an axisymmetric body with the same distribution of cross-sectional areas along the axis. The rule is extended here to stationary and nonstationary flows past nonslender bodies and to internal flows, using the procedure of averaging with respect to the angular variable of a cylindrical system of coordinates. That procedure is, strictly speaking, valid for nearly axisymmetric bodies. However the numerical solutions obtained by the authors for a fairly wide range of external and internal problems show that the generalized equivalence rule is applicable to substantially nonaxisymmetric configurations (*) (see next page).  相似文献   

15.
The framework of the linear mechanics of liquid crystal media [1] is used to study propagation of waves in a layer of a nematic liquid crystal (NLC) on an inclined plane, in a magnetic field, for three different cases of orientation of the anisotropy axis, namely orthogonal to the inclined plane, parallel to the inclined plane and orthogonal to the plane of flow. Such orientations of the anisotropy axis are realized in practice in the course of special machining of solid surfaces [2]. Exact solutions of the equations of motion are obtained describing the steady flow of the layer, and the behavior of small plane perturbations is studied. It is shown that two types of plane waves can propagate in a layer of the nematic mesophase, namely, the surface and the orientational waves. In the case of long surface waves the formulas for the critical Reynolds number are obtained. For the orientational waves a sufficient criterion of stability of the flow in the layer is obtained for two cases. The influence of the magnetic field and of the rheological parameters of NLC on the character of propagation of the first and second type waves is investigated.From amongst the papers dealing with wave propagation in NLC, we draw the readers' attention to [3] which deals with the longitudinal, shear and torsional waves in a liquid crystal domain and obtains the corresponding dispersion relationships.  相似文献   

16.
The propagation of a wave of a finite amplitude in a medium with a nonlinearity of the second degree and negative viscosity, is examined. It is shown that in a finite time singularities appear in the solution. The exact solution of the Cauchy problem is given for a specific case. Recently the effects of negative viscosity which cause an increase in the energy of the wave motion have been studied intensively in electrodynamics, plasma physics, the Earth's atmosphere, in the theory of the circulation of the oceans and of flow in open channels [1–4], Wave amplification caused by an energy transfer from turbulent to regular motions, is possible in any medium having space-time fluctuations, provided the correlation time is sufficiently small [5, 6]. As the wave amplitude increases, nonlinear effects become important; they have been taken into account in cases where the interaction of a finite number of harmonics [2, 4] and the structure of steady motions have been examined [3].It is shown in this paper that in a medium with negative viscosity and a second degree dynamic nonlinearity, a solution of the Cauchy problem for an arbitrary “good” form of the initial perturbation, exists over a finite time interval. An example of such a solution is given.  相似文献   

17.
The general uncoupled dynamical problem of thermoelasticity for a half-space under the condition of a thermal impact with a finite rate of change in temperature on its boundary is solved by the method of principal (fundamental) functions within the framework of a generalized theory of heat conduction.An elastic steel half-space is analyzed as an illustration. The problem on thermal stresses originating in an elastic half-space due to thermal impact produced by a jump change in temperature on the boundary was first analyzed in [1]. Since the temperature change on the boundary occurs at a finite rate, it is generally impossible to realize the thermal impact considered in [1] physically. The dynamic effects in an elastic half-space under a thermal impact with finite rate of change in the temperature on the boundary have been studied in [2]. For high rates of change of the heat flux we obtain a generalized wave equation of heat conduction [3] taking into account the finite velocity of heat propagation. Hence, the solution of the ordinary parabolic heat conduction equation used in [1, 2] does not correspond to the true temperature field. The problems of [1, 2] have been examined in [4, 5], respectively, within the framework of a generalized theory of heat conduction.  相似文献   

18.
We propose one of the possible versions of the optimum control of the forced motions of elastic systems of the type of rods, plates, and shells. We apply the procedure developed to elementary problems on the transition of a freely-supported rod or plate from an initial state φ, ψ to the rest state in the least possible time T in the presence of a constraint on the forcing load. We use the elementary results of theory of the l-problem of moments of Krein [1–3].  相似文献   

19.
A nonlinear escape problem for conflict-controlled systems described by differential equations with a lagging argument is considered. The sufficient escape conditions which are realized in the class of piece wise-constant functions are obtained. The paper relates to the researches in [1 – 8] and is a continuation of [9, 10].  相似文献   

20.
The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved.The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained.Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.  相似文献   

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