Calculation of vibrations of thin plates by partitioning into elastically connected undeformable units

A mathematical model of a thin plate in the form of a system of elastically connected undeformable rectangular units is proposed and substantiated. With necessary additions, the basic statements of the method can be extended to the case of a thick plate. Novosibirsk State Engineering University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 159–168, September–October, 1998.     

Stability of annular orthotropic plates

It is assumed that the orthotropy of the plate material is rectangular or polar, and a uniformly distributed, external compressive or tensile load is applied to the internal boundary of the plate. Stability is analyzed by the Ritz method with the use of Alfutov-Balabukh and Bryan energy criteria. Diagrams of the critical external load and buckling modes as a function of the plate dimensions are given. Novosibirsk State Technical University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 166–170, March–April, 2000.     

Mathematical model of a heterogeneous medium consisting of a matrix and spherical inclusions

A mathematical model of a heterogeneous medium consisting of an elastoplastic matrix and elastic spherical inclusions is constructed. The model takes into account plastic zones, which appear in the vicinity of the inclusions. It is shown that when the effect of plastic zones is taken into account, the averaged “moduli” of volume compression, shear, and yield strength depend not only on the volume concentrations of the inclusions, but also on the mean pressure in the medium. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 170–178, July–August, 1999.     

Solution of the problem of optimal cut in an elastic beam

The Kirchhoff model of an elastic beam with a transverse cut is considered. The nonpenetration condition proposed by A. M. Khludnev is formulated at the edges of the cut. The equilibrium model of a beam with a restriction on the cut is written in the form of a variational inequality. An analytical solution is obtained with the use of the projection operator. The problem of choosing optimal cuts is formulated for the criterion of minimum opening. Conditions for determining the extremum shapes of the beam are obtained and an example of the solution of the problem is given. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 149–157, September–October, 1999.     

Oblique incidence of surface waves on an elastic plate

Oblique incidence of small-amplitude waves on an elastic plate floating on a free liquid surface is studied. The reflection and transmission factors of the waves and the vertical displacements and strains of the plate are determined. It is shown that all these characteristics depend greatly on the incidence angle and frequency of the wave and the width of the plate. Approximate solutions for the reflection and transmission factors are obtained. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 62–68, July–August, 1999.     

Eigenoscillations near a plate in a channel

Acoustic eigenoscillations of a gas near a plate in a rectangular channel, i.e., the eigenfrequency of oscillations as a function of the chord length and the position of the plate in the channel, and the form of the eigenfunctions are studied in a two-dimensional formulation. A mathematical model of eigenoscillations near a plate in a channel has been proposed and substantiated, and the dependence of the eigenfrequency of oscillations on the geometric parameters is studied numerically with the use of this model. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 78–90, March–April, 1998.     

Equilibrium problem of a plate with an oblique cut

The nonpenetration condition for a plate with an oblique cut is proposed. The variational formulation of the equilibrium problem and the equivalent formulation in the form of a boundaryvalue problem are obtained. The analytical solution is given for a one-dimensional case (a beam with a cut), and the qualitative properties of this solution are studied. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 164–174, March–April, 1998.     

Diffraction of surface waves on an inhomogeneous elastic plate

The oblique incidence of small-amplitude waves on an elastic semi-infinite composite plate floating on the free surface of finite-depth water is studied. The front part of a constant-width plate is highed to the basic part and has characteristics different from those of the basic part. The reflection and transmission coefficients of the waves and the vertical displacements of the plate are determined. It is shown that the heterogeneity of the plate material exerts a strong effect on surface-wave diffraction. Methods for decreasing the elastic deformations of the basic part of the plate are proposed. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 42–48, July–August, 2000.     

Numerical and asymptotic study of the two-dimensional problem of the hydroelastic behavior of a floating plate in waves

The problem of the behavior of a floating elastic plate in waves is solved numerically. The normal mode method is used. For a fluid of finite depth, the hydrodynamic coefficients are obtained in explicit form. Numerical results are compared with experimental data for the stress distribution in the plate and also with numerical results of other authors. The results are in good agreement for not very short waves. For incident waves whose wavelength is comparable with the length of the plate, a long-wave approximation of the solution is proposed. Within the framework of this approximation, the solution is given in analytical form. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 90–96, March–April, 2000.     

Complex-potential method in the nonlinear theory of elasticity

For materials characterized by a linear relation between Almansi strains and Cauchy stresses, relations between stresses and complex potentials are obtained and the plane static problem of the theory of elasticity is thus reduced to a boundary-value problem for the potentials. The resulting relations are nonlinear in the potentials; they generalize well-known Kolosov's formulas of linear elasticity. A condition under which the results of the linear theory of elasticity follow from the nonlinear theory considered is established. An approximate solution of the nonlinear problem for the potentials is obtained by the small-parameter method, which reduces the problem to a sequence of linear problems of the same type, in which the zeroth approximation corresponds to the problem of linear elasticity. The method is used to obtain both exact and approximate solutions for the problem of the extension of a plate with an elliptic hole. In these solutions, the behavior of stresses on the hole contour is illustrated by graphs. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 133–143, January–February, 2000.     

New parabolized system of equations of stability of a compressible boundary layer

Based on estimates for the critical layer, a system of equations of stability of a compressible boundary layer is obtained. The system is parabolic and free from the known restriction on the step of the marching scheme related to ellipticity, which could not be eliminated within the framework of the previous method. A numerical scheme is described, and calculation results for the boundary layer on a heat-insulated plate are presented. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 63–69, January–February, 2000.     

Analyzing the viscoelastic state of a plate with elliptic or linear elastic inclusions

A method is proposed for studying the stress state of a viscoelastic multiply connected isotropic plate with aligned elastic inclusions. The viscoelastic state of a plate with a finite or infinite number of circular and linear inclusions is analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 2, pp. 88–98, February 2007. For the centenary of the birth of G. N. Savin.     

Eigenvibrations of a flexible platform floating on shallow water

The spectral problem for a flexible platform (airport) floating on shallow water is examined. The platform is modeded by a flexible plate of finite width and infinite length. A waveguide eigenmode is detected which propagates along the platform and attenuates exponentially away from it. The remaining eigenmodes are outgoing and growing exponentially away from the platform. All the eigenmodes can be excited only by external action on the platform. The behavior of the platform under external loading is examined. Lavrent'ev Institute of Hydrodynamics, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 173–181, January–February, 2000.     

Effect of friction in a contact problem for a plate with a pin

A solution is obtained for a contact problem concerning the tension of a rectangular elastic plate with a circular hole into which a rigid stationary pin has been inserted. There is a small gap between the hole and the pin, which is of circular cross section. Friction acts in the contact region in accordance with the Coulomb law. The finite-element method and the Boussinesq principle are used to determine the load that realizes a specified contact region. Two variants of boundary conditions on the contour of the hole are examined. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 184–192, July–August, 1998.     

Plane linear problem of the immersion of an elastic plate in an ideal incompressible fluid

A plane unsteady-state linear problem of the immersion of an elastic plate of finite length in an ideal incompressible weightless fluid is considered. The deflection of the plate and the velocity of its points are known at the initial moment of time. The fluid occupies the lower halfplane, and its boundary outside the plate is free. The plate which is the bottom of a structure immersed in the fluid with a constant velocity is modeled by an Euler beam. At the initial stage of immersion, when the displacement of the structure is much smaller than the length of the plate, the plate deflection and the distribution of bending stresses in it are determined. The model used allows one to estimate the maximum stresses occurring in the elastic plate during its impact on water and to predict the moment and site of its occurrence. Calculations are performed under the conditions of the experiment carried out in MARINTEX (Norway). Qualitative agreement between the numerical and experimental results is shown. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 150–160, May–June, 1999.     

Breaking of gravity waves in a neighborhood of their second critical propagation speed

Experimental data on gravity shallow-water waves generated by a vertical plate moving in a predetermined manner are given. The plate completely covers the cross section of the channel. It is found that with when the wave speed exceeds the first critical value known in hydraulics, the wave retains smoothness. Breaking of the waves begins at the second critical speed (which is about 1.3 times as high), whose value coincides with the limiting propagation speed of a solitary wave. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 52–58, March–April, 1998.     

Nonlinear effect of external low-frequency acoustics on eigen-oscillations in a supersonic boundary layer

A method of simulation and results of numerical calculation of the evolution of hydrodynamic disturbances in a supersonic boundary layer on a flat plate under the influence of external acoustic waves at Reynolds numbersRe=220–640 and Mach numberM=2 are described. The solution is constructed by the method of expansion with respect to the small parameter; the contribution of linear and quadratic terms to the solution is taken into account. The method developed allows one to estimate the admissible level of the acoustic field, which does not affect the development of eigen-oscillations in the boundary layer. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 99–105, September–October, 1999.     

Solution of a contact problem for a plate with a deformable insert

Contact problems with friction are solved for a rectangular plate with a circular hole into which a ring plate (insert) is placed with a small clearance. Two versions of contact boundary conditions are formulated. According to the proposed approximate formulation of the problem, the boundary conditions in both versions are satisfied not at the actual contact points but at specified pairs of points. Therefore, it is sufficient to determine attachment, slip, contact, and contact-free regions on just one of the contacting contours. The finite-element method and the Boussinesq principle are used to solve the problem. One of the versions of boundary conditions, compared to the other, gives smaller values for the strain energies of the plate and insert, the stress-concentration coefficient, and the lengths of attachment and contact regions. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 216–226, September–October, 1999.     

Inverse doubly periodic problem of the theory of bending of a plate with elastic inclusions

Based on the balanced strength principle, a problem of determining the optimal interference for fitting elastic inclusions into holes of an isotropic elastic plate weakened by a doubly periodic system of circular holes is solved. A closed system of algebraic equations is derived, which allows solving this problem. The resultant interference increases the load-carrying capacity of the composite plate being bent. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 153–161, July–August, 2006.     

An inverse elastoplastic problem for plates

We study an inverse elastoplastic problem of determining the residual stresses, the plasticity zone, and the external loads for a plate for known residual deflections which occur after removal of these loads and elastic unloading. Assuming that the deformation theory of plasticity is valid at the active stage of deformation, we prove the theorem of unique solution. An iterative method of solution is proposed and a variational formulation of the problem is given. Some simple examples are considered. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 186–194, July–August, 1999.