Dynamic Equations Of Elasticity Research Articles

The Block-Element Method in Expansion of the Solutions of Complex Boundary-Value Problems in Mechanics

An exact solution in the first quadrant of a plane boundary-value problem of the second kind for the dynamic Lame elasticity equations is constructed, probably for the first time, and is expanded in terms of solutions of boundary-value problems for the Helmholtz equation. These solutions are presented in the form of packed block elements.

Response of a plate in turbulent channel flow: Analysis of fluid–solid coupling

The paper performs simulation of a rectangular plate excited by turbulent channel flow at friction Reynolds numbers of 180 and 400. The fluid–structure interaction is assumed to be one-way coupled, i.e, the fluid affects the solid and not vice versa. We solve the incompressible Navier–Stokes equations using finite volume direct numerical simulation in the fluid domain. In the solid domain, we solve the dynamic linear elasticity equations using a time-domain finite element method. The obtained plate averaged displacement spectra collapse in the low frequency region in outer scaling. However, the high frequency spectral levels do not collapse in inner units. This spectral behavior is reasoned using theoretical arguments. The resonant vibration is stronger at the third natural frequency than at the first natural frequency. We explain this behavior by comparing the fluid and solid length scales. We further study the sources of plate excitation using a novel formulation. This formulation expresses the average displacement spectrum of the plate as an integrated contribution from the fluid sources within the channel. Analysis of the sources reveals that at the plate natural frequencies, the contribution of the fluid sources to the plate excitation peaks in the buffer layer. The corresponding wall-normal width is found to be ≈0.75δ. The integrated contribution of the overlap and outer regions together to the plate response is comparable to that from the buffer region for Reτ=180 and exceeds the buffer region contribution for Reτ=400. We analyze the decorrelated features of the sources using spectral Proper Orthogonal Decomposition (POD) of the net displacement source. We enforce the orthogonality of the modes in an inner product with a symmetric positive definite kernel. The dominant spectral POD mode contributes to the entire plate excitation. The contribution of the remaining modes from the different wall-normal regions undergo destructive interference resulting in zero net contribution. The envelope of the dominant mode further shows that the intensity of the sources peaks in the buffer region and the wall-normal width of the sources extend well into the outer region of the channel.

Cauchy Problem for Dynamic Elasticity Equations

We consider the problem on the analytic continuation of the solution of the system of vibration equations in elasticity theory in a spatial domain based on the values of the solution and the stresses on part of the boundary of this domain, i.e., a Cauchy problem. The problem is ill posed. If the part of the domain on which the Cauchy data are given is real analytic, then the problem has a local solution by the Cauchy–Kovalevskaya theorem. The special structure of the vibration equation is used to obtain explicit global solvability conditions and construct approximate solutions.

3D Vibration Analysis of Combined Shells of Revolution

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies and the mode shapes of combined hemispherical–cylindrical shells of revolution with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2D) conventional thin shell theories or higher-order thick shell theories, the present method is based upon the 3D dynamic equations of elasticity. Mathematically, minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangent to the shell mid-surface. Strain and kinetic energies of the combined shell of revolution with and without a top opening are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for the combined shells of revolution with or without a top opening, which are completely free and fixed at the bottom of the combined shells. The frequencies from the present 3D Ritz method are compared with those from 2D thin shell theories by previous researchers. The present analysis is applicable to very thick shells as well as very thin shells.

On three-dimensional elastodynamic problems of one-dimensional quasicrystals

ABSTRACTIn this paper, dynamic elasticity equations for one-dimensional (1D) quasicrystals (QCs) with arbitrary system of anisotropy are considered. Fundamental solutions (FSs) of the phonon–phason displacements, displacement speeds, and stresses arising from pulse point sources are computed. New existence, uniqueness and stability estimate theorems are obtained for dynamic elasticity equations in 1D QCs with the initial conditions (ICs). As a computational example, FS components are computed for orthorhombic and triclinic structures in 1D QCs.

Free vibration analysis of shallow and deep ellipsoidal shells having variable thickness with and without a top opening

A three-dimensional (3-D) method of analysis is presented for determining the natural frequencies and the mode shapes of hemi-ellipsoidal domes having non-uniform thickness with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2-D) conventional thin shell theories or higher-order shell theories, the present method is based upon the 3-D dynamic equations of elasticity by the Ritz method. Mathematically minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangential to the shell mid-surface. Potential (strain) and kinetic energies of the hemi-ellipsoidal dome having variable thickness with and without a top opening are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the Legendre polynomials is increased, the frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for a variety of shallow and deep hemi-ellipsoidal domes having variable thickness of five values of aspect ratios with and without a top opening, which are completely free and fixed at the bottom. The frequencies from the present 3-D analysis are compared with those from other 3-D analysis and a 2-D thin shell theory.

A non-local asymptotic theory for thin elastic plates

The three-dimensional dynamic non-local elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size. The integral constitutive relations, incorporating the variation of an exponential non-local kernel across the thickness, are adopted. Long-wave low-frequency approximations are derived for both bending and extensional motions. Boundary layers specific for non-local behaviour are revealed near the plate faces. It is established that the effect of the boundary layers leads to the first-order corrections to the bending and extensional stiffness in the classical two-dimensional plate equations.

3D Vibration Analysis of Hermetic Capsules by Using Ritz Method

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of a hermetic capsule comprising a cylinder closed with hemi-ellipsoidal caps at both ends. Unlike conventional shell theories, which are mathematically 2D, the present method is based upon the 3D dynamic equations of elasticity. Displacement components [Formula: see text], [Formula: see text], and [Formula: see text] in the radial, circumferential, and axial directions, respectively, are taken to be periodic in [Formula: see text] and in time, and the Legendre polynomials in the r and z directions instead of ordinary ones. Potential (strain) and kinetic energies of the hermetic capsule are formulated, and the Ritz method is used to solve the eigenvalue problem, thereby yielding upper bound values of the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Typical convergence studies are carried out for the first five frequencies. The frequencies from the present 3D method are in good agreement with those obtained from other 3D approach and 2D shell theories proposed by previous researchers.

Vibration Analysis of Shallow Spherical Domes with Non-Uniform Thickness

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies of shallow spherical domes with non-uniform thickness. Unlike conventional shell theories, which are mathematically two dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components [Formula: see text], [Formula: see text], and [Formula: see text] in the meridional, circumferential, and normal directions, respectively, are taken to be periodic in [Formula: see text] and in time, and algebraic polynomials in the [Formula: see text] and z directions. Potential (strain) and kinetic energies of the shallow spherical domes with non-uniform thickness are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3D method are compared with those from a 2D exact method, a 2D thick shell theory, and a 3D finite element method by previous researchers.

Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

We consider the dynamic elasticity equations with a locally distributed damping of Kelvin-Voigt type in a bounded domain. The damping is localized in a suitable open subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. Using multiplier techniques combined with the frequency domain method, we show that: i) the energy of this system decays polynomially when the damping coefficient is only bounded measurable, ii) the energy of this system decays exponentially when the damping coefficient as well as its gradient are bounded measurable, and the damping coefficient further satisfies a structural condition. These results generalize and improve, at the same time, on an earlier result of Liu and Rao involving the wave equation with Kelvin-Voigt damping; those authors proved the exponential decay of the energy provided that the damping region is a neighborhood of the whole boundary, and further restrictions are imposed on the damping coefficient.

Solution of two-dimensional vector wave equations via a mixed least squares method

Purpose – The purpose of this paper is to present an original mixed least squares method for the numerical solution of vector wave equations. Design/methodology/approach – The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation. Findings – To test the performance of the method, it has been implemented in original computer codes, using object-oriented logic. One model problem has been solved: propagation of Rayleigh waves. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values. Originality/value – The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications.

Frequency analysis of eccentric hemispherical shells with variable thickness

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of eccentric hemi-spherical shells of revolution with variable thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components <TEX>$u_r$</TEX>, <TEX>$u_{\Theta}$</TEX>, and <TEX>$u_z$</TEX> in the radial, circumferential, and axial directions, respectively, are taken to be periodic in <TEX>${\theta}$</TEX> and in time, and algebraic polynomials in the r and z directions. Potential and kinetic energies of eccentric hemi-spherical shells with variable thickness are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to three or four-digit exactitude is demonstrated for the first five frequencies of the shells. Numerical results are presented for a variety of eccentric hemi-spherical shells with variable thickness.

Vibrations of truncated shallow and deep conical shells with non-uniform thickness

A three-dimensional (3-D) method of analysis is presented for determining the natural frequencies of a truncated shallow and deep conical shell with linearly varying thickness along the meridional direction free at its top edge and clamped at its bottom edge. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components ur, uo and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in o and in time, and algebraic polynomials in the r and z directions. Strain and kinetic energies of the truncated conical shell with variable thickness are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated. The frequencies from the present 3-D method are compared with those from other 3-D finite element method and 2-D shell theories.

Solution of problems in dynamic elasticity via a mixed least squares method

Purpose – The purpose of this paper is to present an original mixed least squares method for solving problems in dynamic elasticity. Design/methodology/approach – The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation. The proposed theoretical formulation is given the general three-dimensional case and is tested numerically on the solution of one-dimensional wave equations. Findings – To test the performance of the method, it has been implemented in an original computer code, using object-oriented logic and written from scratch. Two one-dimensional problems from the mathematical physics, with well-known exact analytical solutions, have been solved. The numerical examples include a forced vibrating spring, fixed at its both ends and a rod, vibrating under its own weight, when one of its ends is fixed and the other is traction-free. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values. Originality/value – The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications to two and three-dimensional problems in dynamic elasticity.

Vibrations of Complete Hollow Spheres with Variable Thickness

AbstractA three-dimensional (3D) method of analysis is presented for determining the free-vibration frequencies of complete hollow spherical shells of revolution with variable thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based on the 3D dynamic equations of elasticity. Displacement components ur, uθ, and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r- and z-directions. Potential (strain) and kinetic energies of the complete hollow spheres are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper-bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the complete hollow spheres. Comparisons are also made between the f...

Vibration analysis of shallow or deep, complete parabolic shells with variable thickness

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of shallow or deep, clamped, complete parabolic shells having variable thickness along the meridional direction by the Ritz method. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies from the present analysis are compared with those from the finite element method based 2-D thin shell theory. Nondimensional frequencies are presented for a variety of shallow or deep parabolic shells having variable thickness.

ДЕКОМПОЗИЦИЯ СИСТЕМ УРАВНЕНИЙ МЕХАНИКИ СПЛОШНЫХ СРЕД 1. УПРУГОСТЬ, ТЕРМОУПРУГОСТЬ И ПОРОУПРУГОСТЬ

The work is devoted to the development of decomposition methods for systems of linear partial differential equations that arise in continuum mechanics, in particular, in the theory of elasticity and thermoelasticity and poro-elasticity. These methods are based on the decomposition (splitting) of systems of coupled equations into several independent equations. The decomposition significantly simplifies the qualitative study and interpretation of the most important physical properties related to three-dimensional equations and allows an effective study of their wave and dissipative properties. Moreover in certain cases the decomposition makes it possible to find exact analytical solutions of the corresponding boundary and initial-boundary value problems, and greatly simplifies the application of numerical methods, allowing us to use the appropriate routines for simpler equations and independent subsystems. In the first part of the work various systems of equations, including equations of elasticity theory in the form of Tedone and in the form of Beltrami–Donati–Michell are given, their dynamic generalizations are proposed, and various forms of the equations of classical and hyperbolic thermoelasticity as well as the equations poroelasticity are described. A number of historical facts, which are directly related with the considered questions and weakly reflected in Russian literature,are presented. Various types of decomposition and their generalizations are described. The representation of solutions of dynamical systems of equations resulting from the toroidal-poloidal decomposition, as well as the decompositions of Green–Lame, Cauchy–Kovalevski–Somigliana, Naghdi–Hsu–Chandrasekharaiah, and Teodorescu types are discussed in details. Special attention is given to their static analogues. A generalization of the representation of Savin for the dynamic equations of elasticity is obtained. The representations in curvilinear coordinates, in particular, the representations of Boussinesq, Timpe, Love, Michell, and Muki types are given. The bibliographical references to the original papers are listed.

Vibration of plate on foundation with four edges free by finite cosine integral transform method

The analytical solutions for the natural frequencies and mode shapes of the rectangular plate on foundation with four edges free is presented by using the finite cosine integral transform method. In the analysis procedure, the classical Kirchhoff rectangular plate is considered and the foundation is modelled as the Winkler elastic foundation. Because only are the basic dynamic elasticity equations of the thin plate on elastic foundation adopted, it is not need prior to select the deformation function arbitrarily. Therefore, the solution developed by present paper is reasonable and theoretical. In order to illuminate the correction of formulations, the numerical results are also presented to comparing with that of the other references.

Free vibration analysis of shallow spherical dome by three-dimensional Ritz method

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of a shallow spherical dome. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components , , and uz in the meridional, circumferential, and normal directions, respectively, are taken to be periodic in θ and in time, and the algebraic polynomials in the φ and z directions. Potential and kinetic energies of the shallow spherical domes are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3-D method are compared with those from a 2-D exact method, a 2-D thick shell theory, and a 3-D finite element method.

Vibrations of deep and shallow hemi-spheroidal domes with non-uniform thickness having a top cut-out

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of hemi-spheroidal domes with non-uniform thickness having a top cut-out. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The edges of the shell may be free or may be subjected to any degree of constraint. Displacement components ur, uθ, and uz in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic simple polynomials in the r and z directions. Potential (strain) and kinetic energies of the hemi-spheroidal domes with non-uniform thickness having a top cut-out are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the algebraic polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for shallow and deep hemi-spheroidal domes with non-uniform thickness having a top cut-out of five values of aspect ratios, which are free at the inner edge and free or fixed at the bottom of the domes. The frequencies from the present 3-D Ritz method are compared with those from other 3-D analysis and a 2-D thin shell theory by previous researcher.