Dynamic Theory Of Elasticity Research Articles

Vibration Analysis of a Circular Tunnel With Jointed Liners Embedded in an Elastic Medium Subjected to Seismic Waves

This paper presents a frequency domain analysis of a circular tunnel with piecewise liners subjected to seismic waves. In our model, the surrounding medium of the tunnel is considered as a linear elastic medium and described by the dynamic elasticity theory, while piecewise liners and connecting joints are treated as curved beams and described by a curved beam theory. Scattered wave field in the surrounding elastic medium are obtained by the wave function expansion approach. The governing equations for vibrations of a curved beam are discretized by the general differential quadrature method. We use domain decomposition methods to establish the global discrete dynamic equations for piecewise liners. Boundary least squares collocation methods, based on the continuity conditions of stresses and displacements between surrounding soil and the piecewise liners, are used to determine the response of the liners and the surrounding medium. Numerical results conclude that the presence of the joints significantly changes the distributions of the tunnel internal force, and dramatically increase shear forces and moment of the tunnel liners around joints.

The layered element method in the dynamic theory of elasticity

A semi-analytical approach is proposed for the numerical analysis of the dynamic behaviour of elastic layered systems and waveguides with internal and surface nonuniformities. The approach is based on representing the reflected field in the form of an expansion in fundamental solutions for the layered structure as a whole. Unlike classical boundary elements, which represent the fundamental solutions for a homogeneous elastic space, layered elements identically satisfy the boundary conditions on all outer and inner plane-parallel boundaries. Hence, to find the unknown expansion coefficients it is necessary to satisfy boundary conditions only on obstacles, that assumes the use of many fewer elements than in the boundary element method, using which it is necessary to arrange them along all the outer and inner boundaries of the region considered. Moreover, each layered element exactly describes the wave structure of the solution, which is particularly convenient when solving problems of the transmission and reflection of travelling waves in open waveguides with obstacles. A brief description of an algorithm for constructing layered elements is given and examples of the use of the layered element method for solving two-dimensional and three-dimensional problems of diffraction in an open waveguide, and also for calculating the dynamic characteristics of bounded structural elements, made of laminated materials, are presented.

Anisotropic bodies with inhomogeneities: The case of a set of cracks

We use the Betti theorem to obtain the integral equations of the dynamic theory of elasticity for a multilayer convex body with an arbitrary elastic anisotropy of layers containing plane infinitely thin cracks. The systems of integral equations relate the displacement jumps to the stresses on the crack lips and are stated numerically in terms of Fourier transforms. For the case of plane-parallel layers with a set of plane cracks on the interfaces between the layers, we propose a simple numerical-analytic method for constructing the Fourier symbol, i.e., the matrix of the kernel of the system of integral equations. The method is stable for an arbitrary combination of continuous and discontinuous conditions on the layer boundaries. Numerical examples are given for a packet of four heterogeneous anisotropic layers.

Determining linear vibration frequencies of a ferromagnetic shell

The problems of determining the roots of dispersion equations for free bending vibrations of thin magnetoelastic plates and shells are of both theoretical and practical interest, in particular, in studying vibrations of metallic structures used in controlled thermonuclear reactors. These problems were solved on the basis of the Kirchhoff hypothesis in [1–5]. In [6], an exact spatial approach to determining the vibration frequencies of thin plates was suggested, and it was shown that it completely agrees with the solution obtained according to the Kirchhoff hypothesis. In [7–9], this exact approach was used to solve the problem on vibrations of thin magnetoelastic plates, and it was shown by cumbersome calculations that the solutions obtained according to the exact theory and the Kirchhoff hypothesis differ substantially except in a single case. In [10], the equations of the dynamic theory of elasticity in the axisymmetric problem are given. In [11], the equations for the vibration frequencies of thin ferromagnetic plates with arbitrary conductivity were obtained in the exact statement. In [12], the Kirchhoff hypothesis was used to obtain dispersion relations for a magnetoelastic thin shell. In [5, 13–16], the relations for the Maxwell tensor and the ponderomotive force for magnetics were presented. In [17], the dispersion relations for thin ferromagnetic plates in the transverse field in the spatial statement were studied analytically and numerically.

On axisymmetric motion of an incompressible elastic medium under impact loading

The method of matched asymptotic expansions was employed to obtain approximate solutions to the one-dimensional boundary-value problems of nonlinear dynamic elasticity theory of impact loading on the surface of a cylindrical cavity of an incompressible medium that causes antiplane motion or torsion of the medium. The expansion of the solution in the near-front region is based on solutions of evolution equations different from the equations for quasi-simple waves.

The Use of Rigidity Properties in Cylindrical Shells for Noise Reduction

The present paper deals with methods aimed at developing calculation models of sound insulation of various cylindrical shells, making it possible to estimate the shell-radiated noise levels. The proposed methodology is based on the mathematical analysis of the cylindrical shell as an element that is subordinated to the equation of dynamic theory of elasticity of thin-walled constructions of variable rigidity. The work provides the corresponding controls of dynamic theory of elasticity in partial derivatives with variable coefficients. Theoretical research and calculations carried out show that sound insulation under analysis at low frequencies is quite high. A slight reduction of sound insulation at high frequencies conditioned by wave processes is also considered.

Dynamic Elasticity of Cubic Diamond

Previously, the structure of the carbon allotrope glitter has been disclosed, and a theory accompanying the structural report as to its bulk modulus at pressure predicted it would be among the hardest materials possible. The dynamic elasticity theory developed in that paper, involving the forces generated in elastic chemical bond deformations resulting from applied mechanical forces, is here applied to the cubic diamond lattice. Stresses, both lateral and axial, contribute to the bulk modulus of cubic diamond at pressure. The ultimate strength of the cubic diamond lattice, in the approximations of the dynamic elasticity theory presented in this paper, is estimated to be in excess of 1 TPa, at modest bond length deformations of about 0.1 A, and when including the zero pressure bulk modulus B 0 in the computation. In particular, the dynamic elasticity model predicts the hardest direction of cubic diamond will be for an isotropic mechanical force applied along 〈111〉 directions of the structural unit cell.

Towards a microscopic theory of the modulus of elasticity in crystalline covalent materials and a survey of potential superhard materials

The present report is an account of the generalization of the dynamic elasticity theory earlier proposed by Bucknum et al. and applied to the cubic diamond and tetragonal glitter lattices. It describes a theory of elasticity in which the elasticity moduli are based upon the microscopic constants of the various structure-types. Such microscopic constants include the force constants of the chemical bonds in the unit of pattern of the material, its associated lattice parameters, and the elastic chemical bond deformation parameters of the material. In developing the outward features of the dynamic elasticity model, it is shown that an integral over the force density in the unit cell of a given material; where the force is modeled based upon the elastic deformation forces of the chemical bonds in the unit of pattern of the material, and the volume is written as a function of the deformations taking place inside the unit cell of the material; generates the terms for calculating its modulus of elasticity at pressure, in components, that are directed along the principal axes of the unit cell. Several potential solutions to the problem of superhardness are discussed and illustrated.

Evaluation of the Dynamic Strength of Cylindrical and Conic Matrices of Finite Length for Stamping by High Explosives

The equations of two-dimensional dynamic theory of elasticity are used to study the domain of applicability of engineering one-dimensional analytic procedures developed earlier for the evaluation of the dynamic strength of cylindrical and conic matrices of finite length both with free and fastened ends subjected to stamping by high explosives. It is shown that, in view of the “buildup” effect leading, in some cases, to the violation of strength of the matrix, these procedures should be supplemented with correcting calculations performed according to the two-dimensional model for large periods of time. We propose new procedures of design-basis and verifying analyses for cylindrical and conic matrices of finite length subjected to stamping by high explosives.

Elastic constants of natural quartz.

The elastic constants of a natural-quartz sphere using resonance-ultrasound spectroscopy (RUS) are measured. The measurements of the near-traction-free vibrational frequencies of the sphere are matched with the predicted frequencies from the dynamic theory of elasticity, with optimized estimates for the elastic constants driving the differences between these sets of frequencies to a minimal value. The present computational model, although based on earlier approaches, is the first application of RUS to trigonal-symmetry spheres. Quartz shows six independent elastic constants, and our estimates of these constants are close to those computed by other means. Except for C14, after a 1% mass-density correction, natural quartz and cultured quartz show the same elastic constants. Natural quartz shows higher internal frictions.

A variational formulation and investigation of boundary-value problems of the non-linear theory of plates using an energy approach

A variational formulation of boundary-value problems of the non-linear dynamic theory of elasticity using the Hamilton functional is presented. The quasi-static boundary-value problem for thin plates is considered. The initial system of equations, in a two-dimenonal formulation, is represented in terms of generalized forces and displacements. The sufficient conditions for the existence and uniqueness of a weak solution are established.

Shift and damping of optical phonons caused by interaction with electrons

Frequency shift and damping of long-wave optical phonons caused by interaction with electrons are calculated. The equations of the dynamic theory of elasticity are considered together with the Maxwell equations and the kinetic equation for determining the deformation field, the electric field, and the distribution function of electrons. Changes in the spectrum of electrons are described using a local potential. Coulomb screening of the longitudinal electric field is taken into account.

Propagation of normal waves through a fluid contained in a thin-walled cylinder

The properties of normal axisymmetric waves propagating through a perfect compressible fluid contained in an elastic thin-walled cylinder are investigated. The problem is solved using the complete system of equations of the dynamic theory of elasticity. The effects of interaction between elastic and fluid waves are studied within a wide frequency range. The numerical results are classified on the basis of data on the properties of partial subsystems. Partial subsystems are those for which the interaction effects are insignificant. For special cases of compound waveguides, the dispersion spectra are constructed and the kinematic and energy characteristics of normal waves are analyzed. Particular attention is given to the lowest normal wave, which has specific properties and participates in the elastic–liquid interaction over a wide frequency range.

On the Applicability of One-Dimensional Models to the Dynamic Strength Estimation for Axially Symmetric Bodies

A comparative analysis of the characteristics-difference method and the Wilkins algorithm with reference to problems of nonstationary deformation of one-dimensional cylindrical thick-walled shells has been carried out. On the basis of comparing one- and two-dimensional solutions of boundary problems of the dynamic theory of elasticity obtained by the Wilkins method, conclusions have been drawn about the validity limit of one-dimensional models for the estimation of the strength of female dies for impulse-pressure forming of closed cylindrical and conical shells.

Critical Phenomena in Cracking of the Interface Between Two Prestressed Materials. 3. Exact Solution. The Case of Equal Roots

Study is made of the critical phenomena occurring in cracking of the interface between two different materials with initial stresses. The basic relations and complex potentials for a plane problem of the three-dimensional linearized dynamic theory of elasticity are used. The exact solution is obtained for the case of equal roots (complex parameters). The basic mechanical effects are analyzed.

Critical Phenomena in Cracking of the Interface Between Two Prestressed Materials. 1. Problem Formulation and Basic Relations

Problems are formulated for critical phenomena accompanying the cracking of the interface between two different materials with initial stresses. The basic relations of the three-dimensional linearized dynamic theory of elasticity are used. Complex potentials are applied to a plane problem in the three-dimensional linearized dynamic theory of elasticity.

Critical Phenomena in Cracking of the Interface Between Two Prestressed Materials. 4. Exact Solution. The Combined Case of Unequal and Equal Roots

Study is made of the critical phenomena occurring in cracking of the interface between two different materials with initial stresses. The basic relations and complex potentials for a plane problem of the three-dimensional linearized dynamic theory of elasticity are used. The exact solution is obtained for the case where the roots are equal for the first material and unequal for the second material. The basic mechanical effects are analyzed.

CRITICAL PHENOMENA IN CRACKING OF THE INTERFACE BETWEEN TWO PRESTRESSED MATERIALS. 2. EXACT SOLUTION. THE CASE OF UNEQUAL ROOTS

Study is made of critical phenomena accompanying the cracking of the interface between two different materials with initial stresses. The basic relations and complex potentials for a plane problem of the three-dimensional linearized dynamic theory of elasticity are used. The exact solution is obtained for the case of unequal roots (complex parameters). The basic mechanical effects are analyzed.

A class of boundary-value problems in the dynamic theory of elasticity

A class of boundary-value problems of the dynamic theory of elasticity, which has not been studied to any great extent, is investigated. In these problems all the components of the displacement vector and the stress vector are specified on part of the boundary of the body (nothing is known about the components of these fields on the remaining part of the boundary). A uniqueness theorem is proved. The problem is investigated by reducing the boundary-value problem to a system of Fredholm integral equations of the first kind with smooth kernels. A scheme for the numerical determination of the unknown fields is proposed, based on a combination of the boundary-element method and the Tikhonov regularization method. A number of model examples in establishing the wave fields for an anisotropic body are considered.

Soil Spring Constants of Buried Pipelines for Seismic Design

This paper presents within the scope of the theory of static elasticity a derivation of soil spring constant in longitudinal direction of a buried pipeline in the seismic design. It is found that the spring constant depends on the shear modulus G of the soil deposits and the ratio λ of the radius of zero displacement over the radius of a buried pipeline structure. When λ increases, the soil spring constant decreases and the decreasing rate reduces remarkably when λ is greater than about 10. The ratio of soil spring constant over G is 2.7 when λ = 10. Furthermore, the dynamic effect of soil spring constants on buried pipeline structures is estimated analytically by the dynamic theory of elasticity. The soil spring constant that is derived, taking into account the boundary condition of ground surface, depends on the shear modulus G of the ground, nondimensional frequency ωb/Vs (ω = circular frequency; b = radius of buried pipeline; Vs = ground shear-wave velocity) and nondimensional depth d/b (d = depth of ...