Moscow University Mechanics Bulletin最新文献

Free oscillations of an orthotropic conical shell of finite length are considered. This is a problem of the 80s–90s of the last century. Most problems of solid mechanics are described by elliptic equations that have smooth solutions, and, therefore, the development of algorithms that take into account this smoothness is relevant. The paper presents a modern algorithm without saturation and considers specific calculations that show its high efficiency.

We formulate a nonlinear Maxwell-type constitutive equation for shear deformation of polymers in flow state or polymer viscoelastic melts and solutions which takes into account interaction of deformation process and structure evolution, namely, influence of the kinetics formation and breakage of chain cross-links, agglomerations of molecules and crystallites on viscosity and shear modulus and deformation influence on the kinetics. The constitutive equation is governed by an increasing material function and six positive parameters. We reduce it to the set of two nonlinear autonomous differential equations for two unknown functions (namely, stress and relative cross-links density) and prove existence and uniqueness of its equilibrium point and that its coordinates depend monotonically on every material parameter and on shear rate. We derive general equations for model flow curve and viscosity curve and prove that the first one increases and the second one decreases while the shear rate grows. Thus, the model describes basic phenomena observed for simple shear flow of shear thinning fluids.

In this paper, we study the effect of elastic constraint on the deformation of a dilatant material when an irreversible shear causes a change in volume. Two cases are considered: the constraint is caused by elastic ties external to the body and by an elastic core in the dilatant material itself. The first case is considered within the framework of a model problem, whereas the second case is considered by the problem of torsion of a round bar where the outer plastically deformable layers are compressed by the inner elastic core. The Drucker–Prager criterion is used as a yield criterion in the torsion problem.

Stability to small perturbations of two-layered parabolic flow in a plane channel is analyzed. The dispersion relation between a disturbance wavelength and its growth rate is valid in the whole range of wavenumbers and for moderately large Reynolds numbers. The results coincide with known asymptotic theory conclusions. Besides, a new effect for flows not only with viscosity stratification but also with density stratification is revealed. The agreement with experimental data is acceptable.

The approach to mathematical modeling of complex loading processes is based on the two ideas given by A.A. Il’yushin. One of them is called the Il’yushin three-term formula and sets the type of the differential dependence that connects the stress and strain deviator vectors in two- or-three-dimensional complex loading processes The second idea determines the type of the five-dimensional deformation trajectory of constant curvatures. The development of these ideas led to a new constitutive equation and to a new approach to mathematical modeling of complex loading processes. For the analysis of complex loading processes with deformation trajectories of zero curvature, Vasin’s material functions were introduced. These functions are at the center of the mathematical model. They are used for the representations of functionals and formulas for dissipative stresses and for an explicit representation of the stress vector. In this paper the features of applying the new approach to the processes with constant curvature trajectories are studied.

In this paper, a variational principle of Lagrange in the micropolar theory of elasticity for transversely isotropic and centrally symmetric material is formulated. The Ritz method and piecewise-polynomial serendipity shape functions are used to obtain the components of the tensor-block stiffness matrix and a system of linear equations.

The coupled unsteady vibrations of an underground pipeline and elastic soil caused by an inclined fall of a plane seismic wave are studied. The coupled self-similar problems are formulated. An analytical solution of the external problem for the soil is obtained. This solution leads to a theoretical expression for the force of interaction between the pipeline and the soil, for which only empirical relations were previously available. Solutions for pipeline in supersonic and subsonic cases demonstrate significantly different behavior, which should be taken into account during earthquake resistance calculations.

The problem of motion of a rigid body with a fixed point in a free molecular flow of particles is considered. It is shown that the equations of motion of this body generalize the classical Euler–Poisson equations of motion of a heavy rigid body with a fixed point, and they are represented in the form of the classical Euler–Poisson equations in the case when the surface of the body in a flow of particles is a sphere. The existence of first integrals in the considered system is discussed.

One kind of a descent of a heavy finned body in resisting medium is considered. It is shown that the gliding mode is possible for which blades are located in a horizontal plane. The stability of such modes of gliding is studied. Trajectories of gliding are constructed for various initial speeds.

From the standpoint of the linearized stability theory, two eigenvalue problems for the Orr–Sommerfeld equation with two groups of boundary conditions having a certain mechanical meaning are considered. The stability parameter, which is a real part of the spectral parameter, is estimated on the basis of the integral relations method operating with quadratic functionals. The technique of the method involves the application of the Friedrichs inequality for various classes of complex-valued functions. Using the minimizing property of the first positive eigenvalues in the corresponding problems, the values of the constants in some Friedrichs inequalities are increased, which entails the strengthening of the stability sufficient integral estimates for plane-parallel shear flows in a plane layer.