The article solves a three-dimensional dynamic problem of elasticity theory for modeling the propagation of seismic waves in a heterogeneous geological environment, which is a model of a field with an area of about 200 km ^2 , using the spectral element method at different approximation orders. A detailed description of the problem and a numerical method for solving it are given. A qualitative analysis of the modeling results is given, as well as an estimate of the costs of computational time and memory at different elements orders. A justification is given for the 8th order of the spectral element as optimal under the conditions of this problem, and a technique for qualitative assessing the optimal order in similar situations is provided.
The Poincaré–Steklov operator for an elastic layer is considered, mapping normal stresses into normal displacements on a part of its boundary. It is shown that for calculating the transfer function of this operator of a three-dimensional boundary value problem for a functionally graded layer, the computational algorithm can be used that was previously developed for building the transfer function of the specified operator of a plane strain problem for a functionally graded strip.
We consider a nonlinear ordinary differential system of the third order—the Hodgkin–Huxley model with Soto–Alexandrov modifications. The system is reduced to a second-order system and an algebraic equation in the vicinity of the equilibrium point which corresponds to the Andronov–Hopf bifurcation. This results in the transition to the second-order model with the value refinement
A linear system of equations is considered, which is reduced by a point transformation to a nonlinear system of equations of shallow water over an inclined bottom. A complex-valued family of solutions to a linear system of equations that are invariant with respect to a one-parameter group of transformations is obtained. A two-parameter family of solutions to a nonlinear system of equations describing the arrival and reflection of a wave from the shore is identified. The effects of splash and wave rollback are detected. Examples of such solutions are given.
The axisymmetric flow of a thin layer of viscous liquid down the outer surface of a vertical cylinder in the field of gravity is considered. The process is modeled using the Kapitsa–Shkadov approach. Wave modes with step waves characterized by the presence of a film thickness drop are investigated. It is found that the existence regions of step wave solutions are strips.
The Il’yushin deformation plasticity theory describes a material particle behavior by means of geometrical properties of its loading path. Any path can be classified by the degree of difference between it and the proportional loading paths. Loading paths with a moderate curvature and a small torsion are the most complicated of such classes having the corresponding stress–strain relations. In the present paper, the constitutive relations for moderate curvature and torsion loading paths are proposed and examined experimentally.
In traditional calculations of phase equilibrium of mixtures, iterative methods are applied. The equation of state is used to calculate chemical potentials, and formulas (for example, Wilson’s ones) are used to calculate the initial values of equilibrium constants. However, they must be consistent. The article proposes a method of such matching. It is shown that the mismatch coefficient is greater for heavier component. Temperature ranges are found in which the matching procedure is especially necessary. The approach allows constructing a thermodynamically consistent system and eliminating some nonphysical solutions and numerical instabilities. The proposed thermodynamic matching procedure adjusts the formula to the equation and thereby improves the quality of initial approximations.
A modified Keller–Miksis equation, taking into account radial oscillations of a gas bubble covered by an anisotropic shell and located in the carrier liquid is presented. Numerical calculations are performed. The influence of the shell anisotropy on the radial oscillations of a gas bubble in an external acoustic field is analyzed. In a particular case, a comparison of the theory with available experimental data is given.
The paper develops a finite deformation extension of the standard linear solid model (the three-constant viscoelastic model) model. We demonstrate that, unlike this model describes limited creep, compared to the model based on the elementary Maxwell model.
The paper considers the effect of possible hardening of a sample with a crack with additional compression forces along the crack. This effect was first experimentally discovered by a group of scientists led by Z. Bažant in the study of a normal separation crack and has not yet received a proper explanation. In this paper, instead of the classical singular solution, we propose to consider a nonlocal regular solution for a normal separation crack, in which the stress components are regular and determine the stress concentration. In this case it becomes possible to assess the maximum loads using traditional strength criteria. It is shown that this approach allows modeling the Bažant effect and predicting its manifestation for various materials through the scale parameter in regular solutions.
The paper is devoted to the development of one of the methods for constructing solutions to problems of nonstationary waves in inhomogeneous viscoelastic bodies. A piecewise homogeneous structure with continuity conditions at the contact of components is accepted as the main type of inhomogeneity. The continuous heterogeneity of viscoelastic functionally graded materials is approximated by a layered homogeneous medium. The hereditary properties of the components are characterized by linear Boltzmann–Volterra relations with kernels of various types. The integral Laplace transform in time and the operation of its reversal are used. New forms of solutions for unsteady viscoelasticity problems for piecewise homogeneous bodies, convenient for numerical implementation, are obtained. The proposed approach is demonstrated on a dynamic problem for an elastic hollow sphere with a coating made of viscoelastic functionally graded material.
For the case of a simplified plane stress state and plane deformation trajectories, the constitutive relations between stresses and strains beyond the elastic limit in derivatives with respect to the parameter of tracing the loading process are obtained. The necessity of taking into account the sign of the approach angle between the tangent to the strain trajectory and the stress vector in the constitutive relations is shown.
A basic model of a cylindrical quasi-force-free solenoid for the generation of strong and superstrong magnetic fields is considered in this paper for the resulting stress-strain state under the influence of current and magnetic induction. For a single-layer model, the parameterization problem satisfying the local Maxwell equations for the magnetic quantities in poloidal and toroidal directions has been solved. Analytical minimization problem of the integral radial component of the Lorentz mechanical force through conductor thickness is performed by taking into account the introduced parameterization of the system. In the formulation of the linear elasticity theory, the quasi-static axisymmetric problem of mechanics of a deformable solid under the action of a volumetric radial force is considered analytically. Taking into account the introduced parameterization, the optimal induction ratio in the conductor from the point of view of minimizing the maximum value of the von Mises stress normalized to magnetic pressure mechanical stress is found.
To solve the problem of rescuing a multilegged robot from an emergency upside-down position, an original method is used to find optimal control of the oscillation amplitude in the vicinity of the equilibrium position for a scleronomic mechanical system with an underactuated one oscillatory degree of freedom. The control is implemented by changing the position of a specially selected group of legs relative to the robot’s body. The effectiveness of the developed control algorithm is confirmed by the results of computer simulation of the full dynamics of the robot.
We consider the inverse problem of bending an elastic circular plate when it is necessary to determine the values of the forces acting at some points and causing a given approximately values of deflection at that points.
A model of the afferent primary neuron in the form of the Hodgkin–Huxley equations modified by Aleksandrov–Soto to third-order model and two simplified second-order models is under consideration. Stationary solutions of these models, periodic solutions, stationary solutions attraction regions, and bistability intervals are compared. It is concluded that the simplified model with a functional representation of the inactivation parameter better approximates the full third-order model.
The article considers the interaction of two spherical bodies made of an anisotropic magnetizable material in an applied uniform magnetic field. The force acting on one of these bodies from the nonuniform magnetic field induced by the other body is obtained and investigated analytically. It is shown that this force can be represented as the force of interaction of two dipoles with special magnetic moments. Calculations of the magnetic force values for different directions of the body’s anisotropy vectors and the magnetic field are made. A significant effect of anisotropy on the magnitude and direction of the magnetic force between the bodies is shown.
A method of approximate solution to the characteristic equation describing the heat dissipation during oscillations of bubbles in a liquid is presented in this article. An analytical formula for the damping decrement is derived.
On the example of a mathematical model of a spring pendulum, the paper considers the algorithmic and computer implementation of the dynamic regularization method in application to the problem of restoring perturbations acting on periodic movements of the systems.
Comparison of mass distributions of unevaporated fragments of meteoroids and asteroids disrupted in the atmosphere and fragments of bodies disrupted in impact experiments modeling the destruction of asteroids in outer space is carried out. For the analytical description of the distributions, a relation is used that is derived under the assumption that the distribution density of the number of fragments over masses obeys a power law. The ranges of variation in the power index are estimated in both cases. Approximate dependences of the power index on the specific impact energy (destruction in experiments) and on the number of fragments (destruction in the atmosphere) are obtained.
