Dynamic Problem Of Elasticity Theory Research Articles

Fluctuations of the ground surface at blasting operations on tunnel structures

The paper considers the fluctuations of the ground surface during blasting operations on tunnel structures. To estimate the vibration levels of the soil surface, under the explosive action inside a cylindrical cavity, we will consider the plane problem of elasticity theory as a first approximation. The main purpose of the work is to determine the stress-strain state of pipelines under the influence of seismic waves. As a result of this impact, longitudinal and transverse waves will propagate in the ground. To solve the problem, the reciprocity theorem is applied, for dynamic problems of elasticity theory and the results obtained are compared by wave dynamics methods. The obtained solution makes it possible to determine the vibration levels of the soil surface that occur during explosive impacts inside the cylindrical cavity of the elastic half-space. The results can be used to assess the vibration levels of the soil surface that occur in the structures of shallow tunnels laid by drilling and blasting.

Компьютерное моделирование нестационарных упругих волн напряжений в консоли и десятиэтажном здании при фундаментальном воздействии в виде функции

The aim of the work is to consider the problems of numerical simulation of seismic safety of a console and a ten-storey building with a base in the form of an elastic half-plane under non-stationary wave influences. Modeling of tasks of the transition period is an actual fundamental and applied scientific task. Methodology. To solve a two-dimensional plane dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements is used. Based on the finite element method in displacements, an algorithm and a set of programs have been developed for solving linear planar two-dimensional problems that allow solving problems with non-stationary wave effects on complex systems. The algorithmic language Fortran-90 was used in the development of the software package. Результаты. Results. The problem of the effect of a plane longitudinal wave in an elastic half-plane in the form of four trapezoids and in the form of two half-periods of a sinusoid is considered to assess the physical reliability and mathematical accuracy. A system of equations consisting of 8016008 unknowns is solved. The problem of the effect of a plane longitudinal elastic wave in the form of a Heaviside function on a console with a base (the ratio of width to height is one to ten) is considered. A system of equations consisting of 16016084 unknowns is solved. The problem of the effect of a plane longitudinal elastic wave in the form of a Heaviside function on a ten-storey building with a base in the form of a half-plane is considered. A system of equations consisting of 16202276 unknowns is solved. Contour stresses and components of the stress tensor are obtained in the characteristic areas of the problem under study. Based on the conducted research, the following conclusions can be drawn. Elastic contour stress on the sides of the console and a ten-story building are almost a mirror image of each other, that is, antisymmetric. The console and supporting structures of the building work like a beam during seismic action, that is, if there are tensile stresses on one side, then compressive stresses on the other. Bending waves mainly prevail on the contours of the console and supporting structures of the building under seismic influence.

Mathematical modeling of bending stress waves in an aboveground oil pipeline under unsteady seismic action

The problem of numerical modeling of bending waves in an aboveground oil pipeline under nonstationary seismic action is studied. To solve the unsteady dynamic problem of elasticity theory with initial and boundary conditions the finite element method was applied. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasi-regular approach to solving a system of linear ordinary differential equations of the second order in displacements with initial conditions and to approximation of the studied domain is proposed. The technique is based on the schemes: point, line and plane. The area under study is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the area under study is divided into linear finite elements with two nodal points. The algorithmic language Fortran-90 was used in the development of the software package. The problem of the effect of a plane longitudinal wave in the form of six triangles on an elastic half-plane to assess physical reliability and mathematical accuracy is considered. A system of equations consisting of 8 016 008 unknowns is solved. The calculation results are obtained at characteristic points. A quantitative comparison with the results of the analytical solution is taken. Furthermore, the problem of the impact of a plane longitudinal seismic wave at an angle of 90 degrees to the horizon on an aboveground oil pipeline is considered. The seismic impact is modeled as a Heaviside function, which is applied at a distance of three average diameters from the edge of the pipe. The calculation results were obtained at the characteristic points of the object under study. A system of equations consisting of 32 032 288 unknowns is solved. Bending waves prevail in the problem under consideration.

Mathematical modeling of stress waves under concentrated vertical action in the form of a triangular pulse: Lamb’s problem

The aim of the work. The problem of numerical simulation of longitudinal, transverse and surface waves on the free surface of an elastic half-plane is considered. Methods. To solve the non-stationary dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements was used. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasiregular approach to solving a system of second-order linear ordinary differential equations in displacements with initial conditions and to approximating the area under study is proposed. The method is based on the schemes: point, line and plane. The study area is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the study area is divided into linear end elements with two nodal points. The Fortran-90 algorithmic language was used in the development of the software package. Results. Some information is given about numerical modeling of elastic stress waves in an elastic half-plane with a concentrated wave action in the form of a Delta function. The estimated area under study has 12 008 001 nodal points. A system of equations consisting of 48 032 004 unknowns is solved. The change of elastic contour stress on the free surface of the half-plane at different points is shown. The amplitude of Rayleigh surface waves is significantly greater than the amplitudes of longitudinal, transverse, and other waves with a concentrated vertical action in the form of a triangular pulse on the surface of an elastic half-plane. After surface Rayleigh waves, a dynamic process is observed in the form of standing waves.

Modeling of the dynamic rail deflection using elastic wave propagation

There is a class of tasks that requires considering the dynamics not only for rolling stock but also for the response of the railway track. One of the directions of railway transport development, which encourages the transition to fundamentally new dynamic models of the railway track, is undoubtedly an increase in traffic speed. To solve such problems, the authors applied a model of the stressed-strained state of a railway track based on the dynamic problem of elasticity theory. The feature of this model is the calculation of dynamic stresses and deformations induced by the spread of elastic waves through the objects of the railway track. Based on the mathematical modeling of stress propagation in the under-rail basis, authors have shown the influence of various objects of a railway track on the formation of the outline of the front of the elastic wave and determined the main time intervals. Furthermore, the authors propose the following analytical method, which, in addition to the soil's physical and mechanical properties, considers the properties of the ballast as a layer that transmits pressure to the roadbed and takes an active part in the formation of the interaction space.

ЧИСЛЕННЫЙ АНАЛИЗ ДИНАМИКИ ТРЕХМЕРНЫХ АНИЗОТРОПНЫХ ТЕЛ НА ОСНОВЕ НЕКЛАССИЧЕСКИХ ГРАНИЧНЫХ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ

The application of non-classical approach of the boundary integral equation method in combination with the integral Laplace transform in time to anisotropic elastic wave modeling is considered. In contrast to the classical approach of the boundary integral equation method which is successfully implemented for solving three-dimensional isotropic problems of the dynamic theory of elasticity, viscoelasticity and poroelasticity, the alternative nonclassical formulation of the boundary integral equations method is presented that employs regular Fredholm integral equations of the first kind (integral equations on a plane wave). The construction of such boundary integral equations is based on the structure of the dynamic fundamental solution. The approach employs the explicit boundary integral equations. The inverse Laplace transform is constructed numerically by the Durbin method. A numerical solution of the dynamic problem of anisotropic elasticity theory based on the boundary integral equations method in a nonclassical formulation is presented. The boundary element scheme of the boundary integral equations method is built on the basis of a regular integral equation of the first kind. The problem is solved in anisotropic formulation for the load acting along the normal in the form of the Heaviside function on the cube face weakened by a cubic cavity. The obtained boundary element solutions are compared with finite element solutions. Numerical results prove the efficiency of using boundary integral equations on a single plane wave in solving three-dimensional anisotropic dynamic problems of elasticity theory. The convergence of boundary element solutions is studied on three schemes of surface discretization. The achieved calculation accuracy is not inferior to the accuracy of boundary element schemes for classical boundary integral equations. Boundary element analysis of solutions for a cube with and without a cavity is carried out.

Determination of stress concentration near the holes under dynamic loadings

Purpose. To develop an approach for determining the stress state of plate structural elements with holes under dynamic loads with controlled accuracy. Methodology. The study was carried out on the basis of the Laplace transform and the method of integral equations. Findings. An approach to determining the dynamic stresses at the holes in the plates is proposed, which includes: the Laplace transform in the time coordinate; a numerical method for determining transformants of displacements and stresses based on the method of integral equations; finding originals on the basis of Prudnikovs formula adapted to dynamic problems of elasticity theory. The problem of determining the Laplace images for displacements is reduced to solving singular integral equations. Integral equations were solved numerically based on the approaches developed in the boundary element method. To find displacements and stresses, the Laplace transform inversion formulas proposed by Prudnikov are adapted to dynamic problems. The study on dynamic stresses at holes of various shapes was carried out. Originality. A new approach to the regularization of the Prudnikov formula for inverting the Laplace transform as applied to dynamic problems of the theory of elasticity has been developed. For its implementation: convergence of Fourier series based on pre-set stresses at the initial time is improved; the remainder is taken into account in the conversion formula. Practical value. A method has been developed for calculating the stress concentration at holes of arbitrary shape in lamellar structural elements under dynamic loads. The proposed approach makes it possible to determine stresses with controlled accuracy. The studies performed at circular and polygonal holes with rounded tops can be used in strength calculations for dynamically loaded plates. The influence of Poissons ratio on the concentration of dynamic stresses is analyzed.

Mathematical modeling of unsteady elastic stress waves in a console with a base (half-plane) under fundamental seismic action

The aim of the work is to consider the problems of numerical modeling of seismic safety of the console with the base in the form of an elastic half-plane under unsteady wave influences. Stress waves of different nature, propagating in the deformed body interact with each other. After three or four times the passage and reflection of stress waves in the body, the process of propagation of disturbances becomes steady, the body is in oscillatory motion. The problem of modeling problems of the transition period is an actual fundamental and applied scientific problem. Methods. The finite element method in displacements is used to solve the two-dimensional plane dynamic problem of elasticity theory with initial and boundary conditions. On the basis of the finite element method in displacements, an algorithm and a set of programs for solving linear plane two-dimensional problems have been developed, which allow solving problems with non-stationary wave effects on complex systems. The algorithmic language “Fortran-90” was used in the development of the complex of programs. The study area is divided by spatial variables into finite elements of the first order. According to the time variable, the study area is also divided into finite elements of the first order. Results. The problem of the influence of a plane longitudinal elastic wave in the form of a Heaviside function on a console with a base (the ratio of width to height is one to ten) is considered. The initial conditions are taken as zero. The system of equations from 16 016 084 unknowns is solved. Contour stresses and stress tensor components are obtained in characteristic areas of the problem. On the basis of the conducted researches it is possible to draw the following conclusions: the console (the ratio of width to height one to ten) is modeled with the elastic basis in the form of an elastic half-plane; the elastic contour stresses on the faces of the console are almost a mirror image of one another, that is, antisymmetric; the console under seismic action works as a rod of variable cross-section, that is, if there are tensile stresses on one face, then compressive stresses on the other; on the contours of the console under seismic action, bending waves mainly prevail.

Three-dimensional dynamic problems of elasticity theory about steady torsional oscillations of bimaterials “half-space – layer with a crack”

The three-dimensional dynamic problem of an elastic bimaterial "half-space − layer with a pennyshaped crack" is considered. The cracks surface are under time-stationary torsional loads. The problem is solved by boundary integral equations (BIE) method. Using solutions of Helmholtz potentials, the problem is reduced to a system of two BIE relatively unknown crack opening function. The dependences of the dynamic stress intensity factors mode III on the frequency of the applied load, the thickness of the layer, and the ratios of the elastic parameters of the materials body are analized.

Numerical solving dynamic problems of elastoplastic deformation of solids

Numerical schemes for solving two-dimensional dynamic problems of elasticity theory based upon several local approximations for each of the required functions are discussed. The schemes contain free parameters (dissipation constants). An explicit form of artificial dissipation of the solutions allows us to control its size and to effectively construct both explicit and implicit schemes. The principle of producing such schemes is applied to a plane dynamic problem of elasticity theory as an example. We describe a class of problems for which numerical algorithms using several local approximations for each of the required functions are constructed. Examples of solving practical problems are given.

A method of solving evolutionary problems based on the Laguerre step-by-step transform

In [1], Mikhailenko proposed a method of solving dynamic problems of elasticity theory. The method is based on the Laguerre transform with respect to time. In this paper, we propose a modification of this approach, applying the Laguerre transform to a sequence of finite time intervals. The solution obtained at the end of one time interval is used as initial data for solving the problem on the next time interval. To implement the approach, four parameters are chosen: a scale factor to approximate the solution by Laguerre functions, an exponential coefficient of a weight function that is used for finding a solution on a finite time interval, the duration of this interval, and the number of projections of the Laguerre transform. A way to find parameters that provide stability of calculations is proposed. The effect of the parameters on the accuracy of calculations when using second- and fourth-order difference schemes is studied. It is shown that the approach makes it possible to obtain a high-accuracy solution on large time intervals.

Spatial stationary dynamic problem of elasticity theory for a plate with a through hole

Spatial stationary dynamic problem of elasticity theory for a plate with a through hole

Dynamic Reaction of a Composite Anisotropic Space with Tunnel Holes to the Action of Concentrated Shear Forces

An antiplane stationary dynamic problem of elasticity theory for a two-component anisotropic (orthotropic) space is considered, where one component of the pair is weakened by tunnel holes of arbitrary cross section. The space is subjected to the action of time-dependent harmonic shear forces concentrated along some line and shear stresses applied to the surfaces of the holes. To solve the problem, the fundamental solution for a composite space without holes is preliminarily constructed. Using integral representations for displacements in the composite space with holes, the boundary problem of elasticity theory is reduced to an integral equation of second kind, which is solved numerically by the method of quadratures.

Discretization of the dynamic problem of elasticity theory with a discontinuous solution

A discretization algorithm for initial boundary-value problems is developed for systems of two linear equations of hyperbolic type with discontinuous solutions. A Crank—Nicholson scheme is constructed for discretization of the Cauchy problem and error bounds are obtained for the approximate solution. A model example is solved.

Dynamic problem of elasticity theory for a space with a moving semi-infinite crack

Dynamic problem of elasticity theory for a space with a moving semi-infinite crack

Consideration of damping of oscillations in a viscoelastic medium in numerical methods of solving dynamic problems of elasticity theory

Consideration of damping of oscillations in a viscoelastic medium in numerical methods of solving dynamic problems of elasticity theory

Contact problems on the forced stationary vibrations of beams on an elastic strip, half-strip, and rectangle: PMM vol. 43, no. 5, 1979, pp. 902–910

Dynamic problems of elasticity theory are considered for a strip, a half-strip, and a rectangle partially reinforced by constant-stiffness beams. Questions of wave propagation, energy transfer, stress concentration under the beam ends are studied. The possibility of extending the solutions obtained to other classes of mixed problems is discussed. The solution of the inhomogeneous problem of stationary vibrations of a semi-infinite beam on a strip, and the system of piecewise-homogeneous solutions of this problem are constructed in quadratures. Analogous problems for a half-strip and rectangle, particularly periodic problems and the problem of the dynamics of one finite beam on a strip, are reduced to normal poincareKoch systems whose matrix elements decrease exponentially in the numbers of rows and columns, by the use of new generalized orthogonality relationships with a load. The method of piecewise-homogeneous solutions was used earlier only in elastostatic problems [1 – 9], the problems under consideration were also not solved by other methods.

Solution of the dynamic problem of elasticity theory in curvilinear coordinates

Solution of the dynamic problem of elasticity theory in curvilinear coordinates

A dynamic problem of elasticity theory for a rectangular prism

A dynamic problem of elasticity theory for a rectangular prism

Solving the second plane dynamic problem of elasticity theory for multiply connected domains

Solving the second plane dynamic problem of elasticity theory for multiply connected domains