Elastic Cone Research Articles

ЦИЛИДРИЧЕСКАЯ ОБОЛОЧКА С ЗАПОЛНИТЕЛЕМ

In the article we will consider cylindrical shells, their definition and properties of cylindrical shells, as well as examples of their application. We will also consider methods for calculating cylindrical shells. A cylindrical shell is a geometric body formed by turning a rectangle around one of its sides. The result is a cylinder whose bases are two parallel and equal rectangles, and the side surface is a surface formed by rotating the rectangle around one of its sides. Cylindrical shells have two main elements – the base and the side surface. The bases of the cylindrical shell are parallel and equal rectangles, and the side surface is a surface formed by rotating the rectangle around one of its sides. A cylindrical shell with filling is considered. They are in an axisymmetric stress state. The case is considered when the filler has the shape of a hollow cylinder or an elastic cone. The results of calculations for each case are presented.

Detectability of gravity changes on the sea surface due to magma accumulation beneath submarine volcanoes

SUMMARY This study estimated the gravity change expected on the sea surface due to magma accumulation beneath submarine volcanoes. For calculation, a semi-analytical model describing the deformation of an infinite elastic cone was considered, in which a point spherical source was embedded on the axis. The expected gravity change exceeds 0.2 mGal in certain scenarios when $\delta V = 10^8\, \mathrm{m^3}$ of source volume change occurs 2 km below the summit, and the summit is shallower than 300 m below the sea surface. The steeper the slope of a submarine volcano, the greater the expected gravity change. 0.2 mGal of gravity variation can be detected with recent marine gravimeters such as cold atom interferometers. The computation method for the gravity change based on the integral transform (Mellin type) is elaborated in this paper.

Deformation of an infinite elastic cone due to a point pressure source buried on the axis: implications to volcanic deformation

SUMMARY This paper presents a semi-analytical solution describing the deformation of an infinite elastic cone due to a point pressure source buried on the axis. The solution is based on the displacement potentials developed by Love in the spherical coordinate system and is solved as a boundary value problem given on the cone surface using the techniques of Mellin transform. Numerical integration of the inverse Mellin transform yields solutions for various cone angles. The displacements calculated for the cone angle α = 90° are confirmed to agree with the well-known Mogi’s solution for the infinite half-space. The present work will be useful for interpretation of the deformation of volcanoes with steep topography and for verification of other numerical methods.

The integral equation of the impact of the elastic cone on obstacle and its approximate solution

The integral equation of the impact of the elastic cone on obstacle and its approximate solution

The dynamical stress concentration near a cone-shaped crack in a twice-truncated elastic cone

The problem on the dynamical stress concentrations near a cone-shaped crack weakening an elastic twice-truncated cone is solved in case of a steady state torsional oscillations. The problem is considered in a spherical coordinate system due to geometry of cone. The solution is found in a form of superposition of continuous solution and discontinuous one. Continuous solution is found via the use of G. Ya. Popov (2003) integral transform and reduction of the original continuous part of the problem to one-dimensional boundary value problem in vector form in the transform’s domain. The solution then was found in an explicit form. The discontinuous solution was derived with the help of Legendre and variation of Hankel integral transforms. And by solving singular integral equation with the method of orthogonal polynomials the solution of original problem is found. The wave field of an elastic twice truncated cone was investigated. The stress intensity factor near cracks edges was investigated. Also, first eigen frequencies were calculated for different crack’s geometrical characteristics.

The dynamical stress concentration near a spherical crack in a twice-truncated elastic cone

Engineering practice demands the solving of the inverse problems of fracture mechanics, with this aim, it is necessary to get an analytical solution of the corresponding direct problem. So, in this paper, research was made considering a twice-truncated elastic cone with a spherical crack inside in steady state torsional oscillation. As a result of the problem solving the eigen frequency values and dynamical stress intensity factor were investigated for different parameters of the cone and crack. The boundary resonance effect was found and analyzed; this gives the opportunity to use the derived results during nondestructive testing of the elastic body. The solution is constructed as the superposition of a continuous solution (the problem for an elastic cone without a crack under dynamical torsion), and a discontinuous one (the problem for a spherical crack). The former solution was derived with a great contribution from the G. Ya. Popov results [G. Ya. Popov, 1982]. The wave field of an elastic twice truncated cone was investigated. The eigen frequencies were calculated for different cone’s and crack’s geometrical parameters. The investigation demonstrated the fact of boundary resonance which is commonly used in the inverse problems of fracture mechanics.

Analytical solution of the elastic impact problem for a cone over a half-space

Using the main provisions of the theory of H. Hertz on the mechanical impact of solids, the dynamic interaction of an elastic cone with an elastic half-space bounded by a flat surface is considered. The case is investigated when the axis of the cone of rotation is perpendicular to the boundary of the half-space, and the initial point of contact of the bodies is the vertex of the cone. To describe the local deformations of bodies in the zone of their interaction, the well-known solution of the axisymmetric static contact problem of the theory of elasticity, constructed by I. Ya. Shtaermann, is used. The problem of the collision of bodies is reduced to a second-order differential equation with quadratic nonlinearity. Two forms of analytical solution of this nonlinear Cauchy problem are obtained. The first uses the Ateb-sine, and the second uses the elliptical cosine. The equivalence of the obtained forms of solutions is established, that is, the possibility of replacing one form with another. To calculate the values of the Ateb-sine using the linear interpolation method, a special table is presented, and an analytical approximation by its elementary functions is proposed. The consistency of the results to which these two methods of approximate calculation of the Ateb-sine values are shown is shown. An approximate formula for calculating the values of the elliptic cosine is also derived and its reliability is confirmed. As a result of solving the impact problem, formulas were obtained that describe the change over time: the convergence of the centers of mass of the bodies, the forces of impact interaction, the radius of the circular contact area and the contact pressure. It is noted that the pressure is infinite in the center of the site, where the top of the cone is in contact with the half-space. The results are compared, to which two analytical forms of the solution result and numerical computer integration of the differential equation of compression of bodies subjected to impact. Established a good agreement of numerical results obtained in different ways. The effect of the taper angle on the main parameters of the dynamic interaction of bodies is investigated. It is shown that an increase in the angle of a solution of a conical body, which strikes, leads to a decrease in the maximum dynamic compression of bodies and the duration of their interaction, as well as to an increase in the maximum impact force, with a constant value of its momentum. A numerical example of calculation is given, where steel is taken as the material of a conical body, and rubber is used as the material of a fixed half-space. Problems of this type arise when calculating the parameters of impact of a piece of mineral raw materials on rubber-lined rolls of the vibration classifier.

Several Effects Unexplained by QCD

Several new experimental discoveries in high energy proton interactions, yet unexplained by QCD, are discussed in the paper. The increase of the cross sections with increasing energy from ISR to LHC, the correlation between it and the behavior of the slope of the elastic diffraction cone, the unexpected increase of the survival probability of protons in the same energy range, the new structure of the elastic differential cross section at rather large transferred momenta (small distances) and the peculiar ridge effect in high multiplicity inelastic processes are still waiting for QCD interpretation and deeper insight in vacuum.

Symmetry breaking in indented elastic cones

Motivated by simulations of carbon nanocones (see [S. P. Jordan and V. H. Crespi, Theory of carbon nanocones: Mechanical chiral inversion of a micron-scale three-dimensional object, Phys. Rev. Lett. 93 (2004) 255504]), we consider a variational plate model for an elastic cone under compression in the direction of the cone symmetry axis. Assuming radial symmetry, and modeling the compression by suitable Dirichlet boundary conditions at the center and the boundary of the sheet, we identify the energy scaling law in the von Kármán plate model. Specifically, we find that three different regimes arise with increasing indentation [Formula: see text]: initially the energetic cost of the logarithmic singularity dominates, then there is a linear response corresponding to a moderate deformation close to the boundary of the cone, and for larger [Formula: see text] a localized inversion takes place in the central region. Then, we show that for large enough indentations minimizers of the elastic energy cannot be radially symmetric. We do so by an explicit construction that achieves lower elastic energy than the minimum amount possible for radially symmetric deformations.

Fine structure of the diffraction cone: From the ISR to the LHC

Following earlier findings, we argue that the low-[Formula: see text] structure in the elastic diffractive cone, recently reported by the TOTEM Collaboration at [Formula: see text] TeV, is a consequence of the threshold singularity required by [Formula: see text]channel unitarity, such as revealed earlier at the ISR. By using simple Regge-pole models, we analyze the available data on the [Formula: see text] elastic differential cross section in a wide range of c.m. energies, namely those from ISR to LHC8, obtaining good fits of all datasets. This study hints at the fact that the non-exponential behavior observed at LHC8 is a recurrence of the low-[Formula: see text] “break” phenomenon, observed in the seventies at ISR, being induced by the presence of a two-pion loop singularity in the Pomeron trajectory.

Constrained inflation of a stretched hyperelastic membrane inside an elastic cone

This paper studies the inflation and interaction mechanics of a flat circular membrane inside an elastic cone under the action of uniform gas pressure. The membrane is assumed to be a homogeneous and isotropic Mooney–Rivlin hyperelastic material, while the conical surface is taken to be a distributed linear stiffness in the direction normal to the undeformed surface. The set of coupled second order nonlinear ordinary differential equations that governs the constrained inflation mechanics is reduced to a set of four first order ordinary differential equations by change of variables. A two dimensional grid search technique using the bisection method is employed to determine the equilibrium configuration of the inflated membrane. The principal stretches and curvatures have been obtained which exhibit some interesting trends. It is observed that the limit point instability can completely disappear (even in the case of neo-Hookean membrane material model) when an inflating membrane interacts with a constraining surface. Most remarkably, pre-stretching the membrane can revive the occurrence of the limit point instability in certain cases leading to a softening behavior. This counterintuitive effect appears to be a shadow of the stretch induced softening behavior observed recently in literature.

Torsion of a Truncated Conically Layered Elastic Cone

We obtain the exact solution of the problem of torsion of an elastic conically layered truncated cone. The elastic constants of its layers undergo jumpwise changes on the conical surfaces. The external conical surface is loaded by tangential torsional stresses. The end face of the cone is either restrained or free of loads. The conditions of perfect contact between the layers of the cone are satisfied. The solution of the problem is constructed with the use of integral transforms applied directly to the equation of equilibrium in torsion. We construct a recursive dependence between the constants of solutions of the differential equation of our problem for the neighboring layers. This gives us a possibility to make the number of equations used to determine the constants of solutions independent of the number of layers. The method applied for the solution is presented in detail for the special cases of stratification for which the dependence of stresses on the ratio between the shear moduli of the neighboring layers is established.

The torsion of the conical layered elastic cone

An infinite conic layered cone which represents the sequence of conical adjacent funnels, inserted one into another, is considered. Each of these funnels differs by their shear modulus. Conditions of an ideal contact are fulfilled between the adjacent conical surfaces. An external conical surface is loaded by tangent torsion stress. An exact solution of the corresponding one-dimensional boundary problem is constructed in the domain of Mellin’s transformation, which is applied directly to the torsion equation. A formulation of boundary conditions in matrix form allows to obtain recurrent-type relations for unknown constants of a general solution for each layer. The proposed solving method leads to the exact solution which is independent of the layers’ number. Mellin’s transformation is inversed to finish the construction of the formulas for displacements and stress. The detailed elaboration of the proposed approach is done for two particular cases: a cone without stratification and a two-layered cone.

The axisymmetric mixed problem of elasticity theory for a cone clamped along its side surface with an attached spherical segment

The axisymmetric mixed problem of the stress state of an elastic cone, with a spherical segment attached to the base, is considered. The side surface of the cone is rigidly clamped, while the surface of the spherical segment is under a load. By using a new integral transformation over the meridial angle the problem is reduced in transformant space to a vector boundary value problem, the solution of which is constructed using the solution of a matrix boundary value problem. The unknown function (the derivative of the displacements), which occurs in the solution, is determined from the approximate solution of a singular integral equation, for which a preliminary investigation is carried out of the nature of the singularity of the function at the ends of the integration interval. Subsequent use of inverse integral transformations leads to the final solution of the initial problem. The values of the stresses obtained are compared with those that arise in the cone for a similar load, when sliding clamping conditions are specified on the side surface of the cone (for this case an exact solution of this problem is constructed, based on the known result).

Torsion of an infinite truncated hollow elastic cone

Torsion of an infinite truncated hollow elastic cone

Dynamics of a truncated elastic cone

The problem of determining the nonstationary wave field of an elastic truncated cone with nonzero dead weight is formulated in terms of wave functions. The Laplace transform with respect to time and an integral transform with respect to time polar angle are used to reduce the problem to a one-dimensional vector problem in the transform domain. The transforms of the wave functions are expanded into series in inverse powers of the Laplace transform parameter, which makes it possible to study the wave process at the initial instants of interaction. A method is proposed to solve the problem for an elastic cone doubly truncated by spherical surfaces

The Solution of Mitchell's Problem for the Elastic Infinite Cone with a Spherical Crack

The new problem about the stress concentration around a spherical crack inside of an elastic cone is solved for the point tensile force enclosed to a cone's edge. The constructed discontinuous solutions of the equilibrium equations have allowed to express the displacements and stress in a cone through their jumps and the jumps of their normal derivatives across the crack's surface. The application of the integral transformation method under the generalized scheme has reduced the problem solving to the solving of the integrodifferential equation system with regard to the displacements' jumps. This system was solved approximately by the orthogonal polynomial method. The use of this method has allowed to take into consideration the order of the solution's singularities at the ends of an integral interval. The correlation between the crack's geometrical parameters, its distance from an edge, and the SIF values is established after the numerical analysis. The limit of the proposed method applicability is specified.

Anatomical Definition of the Subglottic Region

Background: Due to laryngeal neoplasia, as well as infectious and autoimmune diseases, the subglottic region is of great clinical relevance. However, descriptions of the subglottic structures are inconsistent. The aim of our study was to present a precise analysis of the subglottic region and derive functional and clinical conclusions. Methods: Histological, histochemical and immunohistochemical investigations as well as scanning electron microscopy were performed and combined with injection techniques applied to the subglottic region of the larynges of 33 body donors. Results: The three-dimensional extensions of the subglottic region were newly defined: the inferior arcuate line of the vocal cord was defined as the cranial border. The lower margin of the cricoid is the caudal border. Craniolaterally, the elastic cone and, further caudally, the cricoid form the border. Therefore, the definition presented comprises heretofore unnamed ventral and dorsal parts of the lower larynx. The subglottic region can be described as cylindrical, becoming smaller in the cranial direction, following the elastic cone. The ventral boundary is formed by the median part of the cricothyroid ligament (ligamentum conicum), the dorsal part by the cricoid cartilage. The walls of the subglottis are divided into three or four layers composed of collagenous and elastic fibres in which seromucous glands are embedded. Subglottic blood vessels including a tight subepithelial capillary plexus were delineated. Conclusion: The new definition of the subglottic extensions presented is helpful and essential for precise laryngeal tumour classification. The results indicate that the boundaries are unlikely to counteract tumour progression in the subglottis. Furthermore, the findings suggest that these structures contribute to temperature regulation of breath, protection against inflammation as well as breath frequency and depth-dependent mucus secretion.

The stress state of an elastic composite cone with centre of rotation at the apex

The torsion of a composite cone that has a centre of rotation at its apex is investigated in a spherical system of coordinates. A composite cone is a cone with one shear modulus, inserted into a conical funnel having another shear modulus and with ideal mechanical contact between its surface and the inner surface of the conical funnel. The auxiliary problem of a composite cone with its apex truncated by a spherical surface is considered first. The outer surface of such a conical body is not loaded, but a load that reduces to a torque is applied to its spherical surface. The auxiliary problem is reduced to a one-dimensional discontinuous boundary-value problem using a specially constructed integral transformation. The exact solution of this boundary-value problem is constructed. The limit is then taken in the solution obtained as the radius of the spherical surface tends to zero for the purpose of obtaining an exact solution of the problem of the torsion of a composite cone that has a centre of rotation at the apex.

Investigation of Foundation Vibrations Resting on a Layered Soil System

This paper presents an investigation of the dynamic response of foundations resting on a layered soil underlain by a rigid layer. Model block vibration test results are used for the investigation. For the analysis, two different methods, namely, the equivalent spring-mass-dashpot model and the cone model, are used. A simple method to estimate the equivalent stiffness of the foundations resting on any multilayered soil system is presented. Obtaining stiffness from the proposed method and using different values of the damping factor ranging between 1.5 and 10.0%, the dynamic response of a foundation resting on a layered soil system is computed. One-dimensional wave propagation in an elastic cone for the analysis of foundations resting on the elastic homogeneous half-space or layered soil is also used to compute dynamic responses of the foundations resting on different layered soil. Finally, results obtained from two analytical methods are compared with the test results. It has been observed from the comparison that the results obtained by the equivalent spring-mass-dashpot model with a damping factor of 1.5% matched well with the experimental results for all cases. Results obtained by the cone model match well with experimental results for the cases where the top layer is softer than the bottom layer.