Elastic Body Research Articles

Propagation of skew-symmetric unsteady shear waves from thick-walled shell in elastic space

Problems of propagation and diffraction of nonstationary waves in elastic bodies are of great theoretical and practical importance in such fields of science and technology as aircraft construction, shipbuilding, seismic exploration of minerals, seismic resistance of structures, and many others. The paper considers the problem of the propagation of skew-symmetric unsteady shear waves from a thick-walled spherical shell in elastic space. To solve the problem, the integral Laplace transforms in dimensionless time, and the method of incomplete separation of variables was used. In the image space, the problem is reduced to an infinite system of linear algebraic equations, the solution of which is sought in the form of an infinite exponential series. Formulas for the displacement vector components and the stress tensor are obtained. The transition to the originals is carried out using the theory of residues. Numerical experiments have been carried out, the results of which are presented in graphs. The obtained results of the work can be used in geophysics, seismology, and design organizations in the construction of structures and in the design of underground reservoirs.

Experimental study of the effectiveness of stress reduction near V-shaped notch filled with a certain material

A variant of healing a V-shaped notch in an elastic body using an elastic material to fill the notch cavity is considered. The filling material changes the stress singularity at the apex of the V-shaped notch. The degree of changes depends on the correlation of mechanical and geometric parameters of the elastic filler with the mechanical parameters of the base material. Previously, this problem was comprehensively investigated based on the comparative analysis of the results of numerical modeling of the stress singularity exponents for a closed bimaterial wedge and a homogeneous wedge with stress-free edges at different ratios of the opening angles and elastic moduli of the homogeneous parts of the composite wedge, and at different values of Poisson's ratios. It was found that for a closed bimaterial wedge, the stress singularity in the symmetrical (with respect to the bisector of the opening angle) stress state disappears when the value of the Poisson's ratio of the material of one part (with a smaller opening angle) reaches 0.5. By means of numerical models it has been shown that the singulaity of stresses and, as a consequence, stress concentration can be eliminated even when the values of the elasticity modulus of the filling weakly compressible material is several orders of magnitude lower than the elasticity modulus of the base material. This finding made it possible to eliminate the stress singularity at the apex of the V-shaped notch when its cavity was filled with a material, whose Poisson's ratio is close to 0.5. In this work, an experimental study was carried out to substantiate and verify the effectiveness of reducing the stress concentration near V-shaped notch by filling its cavity with a material whose Poisson's ratio is close to 0.5.

On identification of prestress level in elastic bodies

On identification of prestress level in elastic bodies

LONGITUDINAL TRANSIENT VIBRATIONS OF A FINITE MOMENT ELASTIC ROD

Non-stationary longitudinal vibrations of a moment elastic rod of finite length are investigated. To describe the motion of the rod, the system of equations of the general model of moment elastic thin bodies is used without additional hypotheses. The equations of this model take into account longitudinal movements, changes in the angle of independent microrotation, as well as transverse compression of the rod. The rod material is assumed to be homogeneous and isotropic. The system of equations of motion is supplemented by physical relations that describe the relationship of displacements, changes in angles and transverse compression with forces. In contrast to classical models, in addition to normal forces, additional force factors arise in a moment rod. They are: additional moments, moment cutting forces, moments of moment stresses. Accordingly, in addition to the elastic constants of the material, additional physical parameters of the medium are taken into account, which are necessary when taking into account moment effects in the material. The conditions of generalized hinged support are used as boundary conditions at the ends of the rod. The initial conditions are assumed to be zero. To construct the solution, expansions of the desired functions and the external load into trigonometric Fourier series are used. Substituting these expansions into the original relations leads to a system of equations for the coefficients of time-dependent series. To solve it, the integral Laplace transform a in time is used. As a result, expressions for the required coefficients of expansion series in the image space are found. Each of these expressions is the sum of three products. The factors in these products are the Laplace images of the coefficients of the Fourier expansions for the load and for the influence functions. Influence functions are fundamental solutions (Green's functions) of the problem under study. The original coefficients of the series for the influence functions are found analytically using residues. The final expressions for the coefficients of the expansion series of solutions have the form of convolutions in time. The cores of these integral representations are the original coefficients of the series for the influence functions. As an example, the response of a moment elastic rod to the action of a non-stationary axial load is considered. The results obtained are illustrated graphically. The practical convergence of expansion series is estimated.

Longitudinal wave propagation in an extended cylindrical body with external Coulomb friction

The propagation of elastic longitudinal waves in an extended cylindrical body located inside an unstrained body and interacting according to the Coulomb law is considered in the article. The problem is studied in a two-dimensional statement; therefore, the friction force (i.e., the interaction conditions) is included in the system of equations as a boundary condition. The Coulomb friction force arises due to the deformation of a cylindrical body. The reliability of numerical calculations is substantiated by solving test cases and comparing the calculations with experimental results. The numerical results obtained are presented in the form of graphs and analyzed. It is shown that the parameters (stresses and strains) of waves propagating in an elastic cylindrical body with external Coulomb dry friction decay with distance. The mechanism for reducing the stress-strain state and wave parameters is explained by the consumption of elastic energy to overcome the friction force that occurs on the contact surface. The results of the two-dimensional problem are also compared with the results of a similar problem in the one-dimensional theory, where the friction force enters directly into the equations of motion. The deviations of the results of the one-dimensional theory are up to 8-15% depending on the accepted values of the friction coefficient, i.e. the violation of the plane section hypothesis taken in one-dimensional calculations amounts to 15%. With a decrease in the radius of a cylindrical body, these deviations are reduced.

Substantiation of promising weed control methods in organic farming

Puprose. Determination of promising means of mechanization of weed control, which will allow growing tilled and vegetable crops in the organic farming system without the use of herbicides. Methods. Monographic, analytical, abstract-logical, structural modeling, design and manufacture of alternative technical means for the implementation of effective methods of weed control in the organic farming system. Results. Systematized and analyzed the main methods of weed control in the system of organic farming both in the world and in Ukraine. Alternative methods of weed control are: mechanical (mechanical removal, sprinkling), agrotechnical (mulching, crop rotation), biological, thermal (fire cultivation, hot water irrigation, steam treatment) and electrical (electric pulse, laser). In a separate group, it is necessary to single out precautionary measures that eliminate the possibility of introducing weed seeds to the fields from different sources. These include seed cleaning, proper manure storage, weed control outside the fields, compliance with quarantine measures, etc. Promising means of mechanization for weed control are cultivators with polyurethane finger-star-shaped working bodies for mechanical removal of weeds in the aisles and rows of crops and cultivators with installed equipment for fire cultivation. During fire cultivation, when the temperature of heating the soil does not exceed 70 °C, the proteins in the plant cells of weeds in the stages of “thread” and cotyledons are irreversibly damaged. In the phase of 4–6 leaves, at the same temperature, up to 90 % is destroyed, and in a later phase of development – up to 75 % of weeds. A study was made of the process of weed destruction by polyurethane finger-star-shaped working bodies, depending on the hardness and type of weeds on the experimental site (quinoa, amaranth, mouse), as a result of which from 80 % to 95 % are destroyed. Conclusions. The systematization and analysis of weed control methods in the system of organic farming was carried out, as a result of which the most promising means of mechanization and methods of weed control in vegetable and tilled crops were identified. It has been established that the integrated use of modernized elastic finger-star working bodies and fire cultivation for weed control in protective zones and rows of vegetable and tilled crops is effective. The design of a modern domestic row cultivator “FIGHTER” allows you to mount promising mechanical working bodies for weed control and equipment for fire cultivation. Keywords: weed control, organic farming, systematization of methods, mechanized vegetable cultivation, agrotechnical methods, creation of technical means, fire cultivation, rotary working bodies.

IMPACT/CONTACT OF ELASTIC BODY ON A MOVING FOUNDATION

We study numerically the dynamic impact/contact of an elastic body on a moving foundation using the mid-point algorithm. Stability results are presented when foundation is decreasing. Numerical sim­ulations on two-dimensional problems are included and we show that the energy is absorbed in the case of decreasing foundation compared to the fixed one.

Stress Distribution Near Sharp and Rounded V-Notches In Anisotropic Elastic Body Under Antiplane Strain

Stress Distribution Near Sharp and Rounded V-Notches In Anisotropic Elastic Body Under Antiplane Strain

Strain correlation functions in isotropic elastic bodies: large wavelength limit for two-dimensional systems.

Strain correlation functions in two-dimensional isotropic elastic bodies are shown both theoretically (using the general structure of isotropic tensor fields) and numerically (using a glass-forming model system) to depend on the coordinates of the field variable (position vector r in real space or wavevector q in reciprocal space) and thus on the direction of the field vector and the orientation of the coordinate system. Since the fluctuations of the longitudinal and transverse components of the strain field in reciprocal space are known in the long-wavelength limit from the equipartition theorem, all components of the correlation function tensor field are imposed and no additional physical assumptions are needed. An observed dependence on the field vector direction thus cannot be used as an indication for anisotropy or for a plastic rearrangement. This dependence is different for the associated strain response field containing also information on the localized stress perturbation.

A mixed operator approach to peridynamics

<abstract><p>In the present paper we propose a model describing the nonlocal behavior of an elastic body using a peridynamical approach. Indeed, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. In particular, the regional fractional Laplacian is used as the nonlocal operator. Moreover, a combination of the fractional and classical Laplacian operators is used to obtain a better description of the phenomenological response in elasticity. We consider models with linear and nonlinear perturbations. In the linear case, we prove the existence and uniqueness of the solution, while in the nonlinear case the existence of at least two nontrivial solutions of opposite sign is proved. The linear and nonlinear problems are also solved by a numerical approach which estimates the regional fractional Laplacian by means of its singular integral representation. In both cases, a numerical estimation of the solutions is obtained, using in the nonlinear case an approach involving a random variation of an initial guess of the solution. Moreover, in the linear case a parametric analysis is made in order to study the effects of the parameters involved in the model, such as the order of the fractional Laplacian and the mixture law between local and nonlocal behavior.</p></abstract>

Mathematical modelling of elastic vibrations arising from four-body impact interaction

This paper presents a mathematical model of elastic vibrations arising from shock interaction in a four-body system. The resulting model has two applications: the study of linear elastic interaction and the study of nonlinear elastic interaction. The linear elastic body model is presented in analytical form. In the transition to nonlinear interaction, it is difficult to represent the model in analytical approximation. The application of numerical algorithms has made it possible to obtain solution of a nonlinear problem with a specified accuracy sufficient for the simulated process. The model is intended for the description of the behavior of the device for measuring the modulus of elasticity of soil by the stamp method. An algorithm is developed to obtain the displacement and acceleration of the falling load and the loading die. Under laboratory conditions, an experiment with a test rig capable of measuring the acceleration of a loading die during shock-elastic interaction with the falling load is conducted. The measuring part is a three-axis accelerometer sensor connected to a four-channel analog-digital converter, which transmits the data to a mobile device with the developed specialized software. The sampling frequency of the measuring system is 12 kHz for each of the four channels. Polyurethane was used as a nonlinear element of the system, the approximate stiffness of which was determined experimentally by measuring deformations under a given load. When comparing the experimental results with the simulation results, the numerical model parameters were optimized by an algorithm based on gradient descent, which was based on maximizing the correlation coefficient between the theoretical and experimental data.

Optimal Design of a Six‐Dimensional Force Sensor with Three‐Beam Structure

A six‐dimensional force sensor with a three‐beam structure was designed, and its strain and stress distribution under various load conditions were analyzed based on finite element simulation technology. According to the strain distribution, the Wheatstone bridge scheme of the sensor is designed, and the output voltage and sensitivity formulas in each direction are deduced. The optimal patch location was determined according to the strain distribution on the centerline of the strain beam. Finally, the size parameters of the elastic body were optimized based on the response surface method and multiobjective genetic algorithm, which greatly improved the sensitivity of the sensor in all directions. It can provide a useful guidance for the optimal design of the six‐dimensional force sensor with a three‐beam structure.

An estimation of eigenvalues of three-dimensional elastic bodies on the basis of Physics-Informed Neural Networks

An estimation of eigenvalues of three-dimensional elastic bodies on the basis of Physics-Informed Neural Networks

Periodic system of rigid inclusions in a spatial elastic wedge

A periodic contact problem (tangential contact) on a chain of thin rigid inclusions located along the edge of a three-dimensional elastic wedge of a dihedral angle in its middle half-plane (rigid sealing of the wedge faces, the axis of the chain of inclusions is parallel to the edge of the wedge) is considered. To derive the integral equation, the previously obtained fundamental solution of the problem of the action of a concentrated force inside a three-dimensional elastic wedge is used. The nucleus has latent symmetry, then it is reduced to a form with explicit symmetry. To solve the integral equation, a regular asymptotic method is used, which is effective for relatively sparse and far from the edge chains of inclusions in a three-dimensional elastic wedge. Contact characteristics are calculated for different values of dimensionless geometric parameters. Previously, similar problems were considered about one inclusion (stringer) in a strip [1] and a three-dimensional wedge [2], as well as in an unbounded elastic body [1].

Tectonic Earthquake Source Mechanism Model Based on Moment Theory

During the construction of strategic facilities, micro zoning is required to determine the peak values of terrain acceleration due to possible earthquakes. The acceleration of the soil in this case strongly depends not only on the power of the earthquake, but also on the earthquake source mechanism and the geodynamic state of the terrain. The known dislocation models of a single dipole with a moment and a pair of dipoles without a moment of the earthquake source mechanism satisfactorily describe the observed effects of the quadrant stress distribution on the Earth’s surface during earthquakes. When carrying out calculations within the framework of the theory of elasticity, the actions of the dipoles are expressed through volumetric forces. There are two known models of replacing the moments with equivalent forces: one of them is based on the equilibrium equations for an infinitely elastic space (Landau and Lifshitz,1965; Maruyama,1963), and the other is based on the representation theorem for elastic bodies, by introducing a singular internal volume, at the boundary of which there are dislocations (Vvedenskaya, 1969; Aki and Richards, 1983). Although these models involve moment effects, they themselves are derived from the momentless theory of elasticity. In our work, we propose a double dipole effect without a moment based on the moment theory of elasticity. The proposed model of the earthquake source mechanism is applied to solve the problem of stress variations in the Earth’s crust in Central Asia using the example of a particular earthquake with a simplified orientation of the rupture plane. Stress variation is understood as the difference in stresses in problems with and without an earthquake mechanism. Static stresses are obtained by solving the inverse elasticity problem with partially unknown boundary conditions. The lithosphere is a prismatic body consisting of several homogeneous blocks, the upper surfaces of which correspond to the relief of Central Asia. Verification of the results of the numerical solution is carried out by comparing the obtained stresses with previously established empirical values. As a priori stresses for solving the inverse problem, we used the solution of the elastic plane problem, the boundary conditions in which correspond to the lateral compression of the lithosphere of the region of the Indian and Arabian plates on the one hand, and the Eurasian plate on the other hand. The obtained solutions of the problem were used to analyze the geodynamic state of Central Asia. Based on the results of laboratory experiments, the unambiguity of the conclusions about the geodynamic state of the Earth’s crust (compression, extension) according to the Lode-Nadai coefficient, which are currently used by many researchers, is questioned. It is shown, contrary to earlier statements, that the values μσ = +1 and μσ = –1 can correspond simultaneously to both tension and compression cases, depending on the spatial form (ellipsoid) of the stress state. Geodynamic analysis of the Earth’s crust is carried out according to the Anderson method.

Natural vibrations of composite cylindrical shells partially filled with fluid

The article presents the results of studies of natural vibrations of circular vertical multilayer cylindrical shells completely or partially filled with a stationary compressible fluid and subjected to hydrostatic load. The behavior of the elastic structure and the fluid medium is described using the classical shell theory and the Euler equations. The effects of sloshing on the free surface of the fluid are not considered. The linearized equations of motion for shells together with the corresponding geometric and physical relations are reduced to a system of ordinary differential equations with respect to new unknowns. The acoustic wave equation is transformed to a system of differential equations using the generalized differential quadrature method. The formulated boundary value problem is solved by Godunov’s method of orthogonal sweep. The natural frequencies of vibrations are calculated based on the combination of a stepwise procedure and subsequent refinement by the method of dividing in half. The reliability of the obtained results is verified by comparison with the known numerical solutions. The dependence of low vibration frequencies on the reinforcing angle and the level of the structure filling with a fluid for simply supported rigidly clamped and cantilevered two-layer and three-layer cylindrical shells with a fluid are analyzed in detail. It is demonstrated that the possibility of changing the frequencies and vibration modes through a suitable choice of the layer arrangement scheme and the reinforcing angle of the composite material is notably determined by a prescribed combination of boundary conditions for an elastic body.

Application of the p-version of FEM to hierarchic rod models with reference to mechanical contact problems

The formulation of a system of hierarchic models for the simulation of the mechanical response of slender elastic bodies, such as elastic rods, is considered. The present work is concerned with aspects of implementation and numerical examples. We use a finite element formulation based on the principle of minimum potential energy. The displacement fields are represented by the product of one-dimensional field functions and two-dimensional director functions. The field functions are approximated by the p-version of the finite element method. Our objective is to control both the model form errors and the errors of discretization with a view toward the development of advanced engineering applications equipped with autonomous error control procedures. We present numerical examples that illustrate the performance characteristics of the algorithm.

Design of falling rock preventive countermeasure based on three-dimensional simulation

To rationally design rockfall countermeasures, detailed rockfall behavior must be predicted using numerical simulations based on the conditions of slopes and rockfalls. In this study, we perform a rockfall simulation using discontinuous deformation analysis. This method is used to analyze the motion between blocks composed of polygons of arbitrary elastic bodies; furthermore, the energy conservation law and the uniqueness/convergence of the solution are guaranteed for the entire analysis target. Additionally, the rockfall shape is reproduced in detail. Both the rockfall shape and a parameter known as the velocity energy ratio, which indicates the decay of energy at the collision time of the rockfall, significantly affect the rockfall behavior. In this study, we model the actual slope shape from data measured via airborne laser and quantitatively evaluate the effect via simulations by changing the rockfall shape and velocity energy ratio. In this paper, we consider how to use the simulation for the design of rockfall countermeasures based on the results such as the change in the amount of jumping due to the difference in slope shape, geology, and rockfall shape.

Application of neural networks to inverse elastic scattering problems with near-field measurements

<abstract><p>This paper is concerned with the application of a machine learning approach to inverse elastic scattering problems via neural networks. In the forward problem, the displacements are approximated by linear combinations of the fundamental tensors of the Cauchy-Navier equations of elasticity, which are expressed in terms of sources placed inside the elastic solid. From the near-field measurement data, a two-layer neural network method consisting of a gated recurrent unit to gate recurrent unit has been used to reconstruct the shape of an unknown elastic body. Moreover, the convergence of the method is proved. Finally, the feasibility and effectiveness of the presented method are examined through numerical examples.</p></abstract>

MODELLING AND ANALYSIS OF INITIAL STRESS STATE IN A THIN-WALL CYLINDER WITH VARIABLE CHARACTERISTICS

A linearized model of prestressed elastic body vibrations is described in terms of the nonsymmetric Piola stress tensor, and a weak statement of the original problem is given. With the help of the Lagrange variational principle, we construct a statement of the problem of steady axisymmetric oscillations of a thin-walled (shell-like) cylinder with variable material properties in the presence of an inhomogeneous prestress field. The motion equations, essential and natural boundary conditions are formulated. Integral characteristics are introduced that average the change in prestresses and material properties over the shell thickness. On the basis of the model presented, the problem on steady-state vibrations of a functionally graded prestressed cylindrical shell is numerically investigated via the shooting method; the influence of the laws of change in material modules and prestress components on dynamic characteristics (e.g., amplitude-frequency characteristics, natural vibration frequencies) is analyzed, a comparative analysis of the solutions is carried out depending on the thin-walled parameter. For the considered cylindrical shell with axial inhomogeneity, based on the acoustic sounding data, we study the inverse problem of restoring two laws of prestress inhomogeneity characterizing the nonuniform initial stress-strain state of tension or inflation. As additional information, we consider the measurement data of the displacement components on the cylinder's surface, given in a set of points at a certain frequency of steady-state oscillations; in such a case, the material characteristics are considered to be known and may depend on the axial coordinate, which makes it possible to set an arbitrary functional gradient along the axis of the cylinder. The mode of probing loading is determined by specifying the traction type and oscillation frequency. Explicit formulas for the desired laws of prestress inhomogeneity are obtained, and computational experiments are carried out to restore them. A high accuracy of reconstruction is revealed far from the ends of the cylinder when probing at a frequency in the vicinity of the first resonance.