Moscow University Mechanics Bulletin最新文献

A plane model of a spherical robot containing a platform with a wheel is considered. Dynamic equations are derived for this robot and some assumptions concerning these equations are also discussed. A control is proposed using a multilayer neural network.

The stress-strain state arising under dynamic stretching of a homogeneous sheet of an incompressible ideally rigid-plastic material, which obeys the Mises–Hencky criterion, is studied. The lateral boundary is stressfree and the longitudinal velocities are given at the ends. The possibility of thickening or thinning of the cross section along the length of the sheet is taken into account, which simulates necking and further development of the neck. Two characteristic stretching regimes are revealed: one of them depends on the velocity at which the end sections move away from each other and the other one depends on their acceleration. For the second regime, the asymptotic integration-based analysis allows approximately determining the stress-strain state parameters.

The tensor linear anisotropic constitutive relations of incompressible viscoplastic flow connecting the stress deviator and strain rates and the following scalar relation connecting the quadratic stress invariant and the hardening function are considered. In the case of a perfect plastic material, the latter relation is an anisotropic Mises–Hencky quadratic criterion of plasticity. The mutual dependence of the fourth-rank tensors involved in tensor and scalar constitutive relations is established. As an illustration, the results are given for an orthotropic material.

The generalization of Samarskii–Sobol’ solution in the mode of heat exacerbation and localization is obtained for a quasilinear heat equation in half-space. The analogy of this solution with summer heating of moisture-saturated soil in the permafrost zone is discussed.

Unsteady oscillations of a semi-infinite underground pipeline and elastic soil caused by the propagation of a longitudinal seismic wave along the pipeline are studied. The problem is not self-similar and its solution meets significant difficulties unlike the case of an infinite pipeline. It is shown that the formulation and consideration of this problem performed earlier by Rashidov are incorrect and do not lead to its solution.

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.