Том 40, № 3 (2019)

Two UNO modifications of the Godunov method for calculating waves in an elastic-plastic body are presented, both being of the second order of accuracy in space and time. The waves are governed by the differential equations in the hydrostatic pressure and the components of the velocity vector and the deviator of the stress tensor. The solid plasticity is taken into account by the von Mises condition. In the first UNO modification, the unknown functions of the governing equations are determined using the invariants reconstructed by their grid cell values. In the second modification, the invariants are not applied. Instead, the unknown functions themselves are reconstructed by their cell values. The first modification better resolves the interaction of discontinuities, whereas the second one is algorithmically simpler. The latter can be useful in simulating multidimensional problems. These features of the presented modifications are illustrated by one- and two-dimensional problems.

An axially movement of the elastic web with small transverse vibrations is considered. It is supposed that the web movement described by the model of continuous elastic membrane panel is with a constant acceleration and is under action of centripetal, Coriolis, transverse inertial and in-plane tension forces. There are presented the transformations reduced the considered defining differential equation to Gauss hypergeometric equation. The total solution of this equation is presented by the help of hypergeometric series, the analysis of panel movement with a constant acceleration and arising vibrations is performed.

Experimental results on assessing the effects of strain and stress rates on the strength of brittle building materials such as fine-grain concretes and ceramic brick are presented. Specimens of these materials were dynamically tested using the Kolsky method and its modification, the Brazilian test (or splitting test). Were obtained strain rates up to 2.5 × 10³ s⁻¹ at compression and the stress rates up to 3 × 10³ GPa/s at tension. As a result of the experiments, values of the Dynamic Increase Factor (DIF) were determined for these the materials studied. Their curves as a function of strain and stress rates were constructed. The experimental data is compared with the theoretically obtained values of DIF as a function of strain rate available in the literature for fine-grain and fiber-reinforced concretes.

The present paper is dedicated to numerical solving of three dimensional boundary-value problems in poroviscoelastic formulation. The poroviscoelastic formulation is treated as a combination of Biot’s theory of poroelasticity and elastic-viscoelastic correspondence principle. Kelvin-Voigt model and Standard linear solid model are employed in order to describe viscoelastic media properties. Boundary element method and boundary integral equation method are applied to obtain Laplace domain solution of boundary-value problem. Modified Durbin’s algorithm of numerical inversion of Laplace transform is used to perform solutions in time domain. Research is also dedicated to development of numerical modelling technique based on Boundary Element Method (BEM) in Laplace domain for solution of three dimensional transient poroviscoelastic problems. A problem of the three-dimensional poroviscoelastic prismatic solid clamped at one end, and subjected to uniaxial and uniform impact loading at another is considered. Viscosity parameter influence on dynamic responses of displacements and tractions is studied.

The supercritical bending of arches in an elastic medium loaded with a lateral distributed load or concentrated force, including arches obtained by pre-bending by means of compressing a beam, is investigated. A resolving nonlinear equation is obtained, its analytical solution and numerical solutions are given.

Within the framework of linear model with coinciding volume and shear relaxation kernels non-stationary problem for viscoelastic half space with normal displacements given on its boundary is considered. Solution representation in the form of generalized convolution of corresponding plane elasticity theory problem solution and one-dimensional viscoelasticity theory problem solution is used. These solutions are written down as convolution of boundary conditions with the kernels - surface Green functions. Time Laplace transform is used for kernels constructing. Its inversion is carried out exactly using either analytical representations of the transforms (for plane problem of elasticity theory) or asymptotic method (for one-dimensional problem of viscoelasticity). Explicit formulas for normal and shear stresses on the boundary of viscoelastic half space are obtained using the structure of Green functions. Examples of calculation show that contrary to elastic medium discontinuities of the first kind don’t exist on the wave front.

For the laminated shells on the basis of the Timoshenko model, taking into account the transversal compression used for each layer, two versions of two-dimensional equations describing geometrically nonlinear deformation for arbitrary displacements and small strains are constructed. They are based on previously proposed consistent relationships of the non-linear elasticity theory, usage of which does not lead to the appearance of “false” bifurcation solutions. The first version corresponds to the contact problem statement, in accordance with which the contact stresses into the contact points of the layers as unknowns are introduced, and the second version corresponds to the preliminary satisfaction of the kinematic coupling relations for the layers along the displacements. An example is given of the application of the derived equations for solving the linear problem of a plane stress-strain state of a straight beam under the action of normal surface forces applied to the front boundary surfaces is given.

We consider a two-dimensional unsteady coupled problem of elasticity with diffusion and preset unsteady volumetric disturbances. The mathematical model is based on a local equilibrium model of elastic diffusion. The solution is sought in integral form. Bulk Green’s functions are found via Laplace transform and Fourier transform for unbounded medium, sine and cosine transform for semi-bounded continuum, Fourier’s series for bounded continuum.