Cylindrical and Spherical Shear Waves in Nonhomogeneous Isotropic Viscoelastic Media

1973 ◽  
Vol 51 (10) ◽  
pp. 1091-1097 ◽  
Author(s):  
T. Bryant Moodie
Keyword(s):  
Equations Of Motion ◽  
Shear Waves ◽  
Three Dimensions ◽  
Radial Coordinate ◽  
Shear Stresses ◽  
Progressive Wave ◽  
Elastic Solids ◽  
Linearized Equations ◽  
Viscoelastic Media ◽  
Isotropic Media

This paper is concerned with the propagation of viscoelastic shear waves in nonhomogeneous isotropic media. Herein we develop formal methods of solving the linearized equations of viscoelastodynamics in two and three dimensions for nonhomogeneous Maxwell solids whose properties depend continuously on a single radial coordinate. These methods are developed for the linearized equations of motion formulated in terms of shear stresses, and are based on Cooper's and Reiss' extension to linear homogeneous viscoelastic media of the Karal–Keller technique. Shearing stesses are applied to the boundaries of cylindrical and spherical openings in the viscoelastic media, and formal asymptotic wave front expansions of the solutions are obtained. In both cases a modulated progressive wave that propagates with variable velocity is obtained. The modulation depends on the moduli of rigidity and viscosity, whereas the velocity depends only on the modulus of rigidity. When the viscosity parameter in our Maxwell element tends to infinity, the results reduce to the known results for nonhomogeneous elastic solids.

Oscillating Progressive Shear Waves in Nonhomogeneous Viscoelastic Solids

1973 ◽  
Vol 51 (21) ◽  
pp. 2287-2294 ◽  
Author(s):  
T. Bryant Moodie
Keyword(s):  
Asymptotic Expansion ◽  
Small Amplitude ◽  
Equations Of Motion ◽  
Shear Waves ◽  
Variable Velocity ◽  
Progressive Wave ◽  
Viscoelastic Media ◽  
Oscillatory Shearing ◽  
Leading Term ◽  
Formal Asymptotic Expansion

The equations of motion for cylindrical and spherical shear waves in nonhomogeneous, isotropic, "standard", viscoelastic media with continuous radial variations are derived. Oscillatory shearing tractions are applied to the boundaries of cylindrical and spherical openings in unlimited viscoelastic media. The propagation of small-amplitude waves is studied, and formal asymptotic expansions of the solutions are obtained. In both cases (cylindrical and spherical), the leading term of the formal asymptotic expansion represents a modulated, oscillating, progressive wave propagating with variable velocity. The modulation depends on the moduli of rigidity and viscosity, whereas the velocity depends on the moduli of rigidity only. Application of our results to the propagation of shear waves in both finite and infinite viscoelastic plates is discussed.

General linearized equations of wave propagation in strained elastic solids of cylindrical shapes using a simple perturbation method

2002 ◽  
Vol 216 (6) ◽  
pp. 683-690
Author(s):  
A Sinaie ◽  
A Ziaie
Keyword(s):  
Wave Propagation ◽  
Perturbation Method ◽  
Equations Of Motion ◽  
Particle Displacement ◽  
Elastic Solids ◽  
Cylindrical Coordinates ◽  
Linearized Equations ◽  
Practical Applications ◽  
Linear Partial Differential Equations ◽  
Static Part

The equations of particle motion in an elastic isotropic stressed medium are first derived in Cartesian coordinates and then transformed into cylindrical coordinates. The three components of the equations of motion are non-linear partial differential equations and cannot be of use in practical applications. However, noting that the particle displacement is composed of a small dynamic part superimposed on a large static part, these equations are linearized via a simple perturbation method. The linearized equations are presented in closed form. They contain variables, which may be measured and experimented upon in practice, in the field of acoustoelasticity.

scholarly journals Stress field formulation of linear electro-magneto-elastic materials

2019 ◽  
Vol 24 (12) ◽  
pp. 3806-3822
Author(s):  
A Amiri-Hezaveh ◽  
P Karimi ◽  
M Ostoja-Starzewski
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Governing Equations ◽  
Field Equations ◽  
Elastic Solids ◽  
Elastic Materials ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Sufficient Set

A stress-based approach to the analysis of linear electro-magneto-elastic materials is proposed. Firstly, field equations for linear electro-magneto-elastic solids are given in detail. Next, as a counterpart of coupled governing equations in terms of the displacement field, generalized stress equations of motion for the analysis of three-dimensional (3D) problems Are obtained – they supply a more convenient basis when mechanical boundary conditions are entirely tractions. Then, a sufficient set of conditions for the corresponding solution of generalized stress equations of motion to be unique are detailed in a uniqueness theorem. A numerical passage to obtain the solution of such equations is then given by generalizing a reciprocity theorem in terms of stress for such materials. Finally, as particular cases of the general 3D form, the stress equations of motion for planar problems (plane strain and Generalized plane stress) for transversely isotropic media are formulated.

Elastic Waves Originating at the Surface of a Spherical Opening in Nonhomogeneous Isotropic Media

1972 ◽  
Vol 50 (19) ◽  
pp. 2359-2367 ◽  
Author(s):  
T. Bryant Moodie
Keyword(s):  
Formal Method ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Radial Distance ◽  
Dimensional Case ◽  
Linearized Equations ◽  
General Law ◽  
Isotropic Media ◽  
Displacement Potentials ◽  
The One

This paper is concerned with the propagation of three-dimensional waves in nonhomogeneous isotropic elastic media. Herein we develop a formal method for the solution of the linearized equations of elasticity for nonhomogeneous media whose properties depend continuously on the radial distance from a point. The method is developed for the equations of motion formulated in terms of displacements, velocities, and stresses. It is based on a method developed by Karal and Keller for the asymptotic solution of the linearized equations of elasticity formulated in terms of displacements and displacement potentials, and is a generalization of results obtained for the one-dimensional case by Cooper and for the two-dimensional case by the author.The method is applied to the propagation of dilatational waves generated by a radial input velocity in the form of a unit function applied to the surface of the hole in a medium whose properties depend on the radial distance from the center of the hole according to a general law. Also, we apply the method to the propagation of purely distortional waves generated by a stress applied at the hole and corresponding to a tangential impulse symmetrical about a diameter of the hole.

scholarly journals Re-Evaluating the Classical Falling Body Problem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 553 ◽  
Author(s):  
Essam R. El-Zahar ◽  
Abdelhalim Ebaid ◽  
Abdulrahman F. Aljohani ◽  
José Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Body Problem ◽  
Inertial Frame ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Rotating Frame ◽  
Dimensional Model ◽  
Three Dimensional Model

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.

AN INVESTIGATION ON THE CHARACTERISTICS OF BENDING, BUCKLING AND VIBRATION OF NANOBEAMS VIA NONLOCAL BEAM THEORY

2014 ◽  
Vol 11 (06) ◽  
pp. 1350085 ◽  
Author(s):  
SOUMIA BENGUEDIAB ◽  
ABDELWAHED SEMMAH ◽  
FOUZIA LARBI CHAHT ◽  
SOUMIA MOUAZ ◽  
ABDELOUAHED TOUNSI
Keyword(s):  
Scale Effects ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Small Scale ◽  
Shear Stresses ◽  
Nonlocal Timoshenko Beam Theory ◽  
Walled Carbon Nanotubes

In the present study, a nonlocal hyperbolic shear deformation theory is developed for the static flexure, buckling and free vibration analysis of nanobeams using the nonlocal differential constitutive relations of Eringen. The theory, which does not require shear correction factor, accounts for both small scale effects and hyperbolic variation of shear strains and consequently shear stresses through the thickness of the nanobeam. The equations of motion are derived from Hamilton's principle. Analytical solutions for the deflection, buckling load and natural frequency are presented for a simply supported nanobeam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory and Reddy beam theories. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes.

The Rotor Dynamic Coefficients of Eccentric Mechanical Face Seals

1996 ◽  
Vol 118 (1) ◽  
pp. 215-224 ◽  
Author(s):  
J. Wileman ◽  
I. Green
Keyword(s):  
Degrees Of Freedom ◽  
Reynolds Equation ◽  
Equations Of Motion ◽  
Shear Stresses ◽  
Dynamic Coefficients ◽  
Angular Deflection ◽  
Face Seals ◽  
Radial Forces ◽  
Rotor Dynamic ◽  
Additional Coupling

The Reynolds equation is extended to include the effects of radial deflection in a seal with two flexibly mounted rotors. The resulting pressures are used to obtain the forces and moments introduced in the axial and angular modes by the inclusion of eccentricity in the analysis. The rotor dynamic coefficients relating the forces and moments in these modes to the axial and angular deflection are shown to be the same as those presented in the literature for the concentric case. Additional coefficients are obtained to express the dependence of these forces and moments upon the radial deflections and velocities. The axial force is shown to be decoupled from both the angular and radial modes, but the angular and radial modes are coupled to one another by the dependence of the tilting moments upon the radial deflections. The shear stresses acting upon the element faces are derived and used to obtain the radial forces acting upon the rotors. These forces are used to obtain rotor dynamic coefficients for the two radial degrees of freedom of each rotor. The additional rotor dynamic coefficients can be used to obtain the additional equations of motion necessary to include the radial degrees of freedom in the dynamic analysis. These coefficients introduce additional coupling between the angular and radial degrees of freedom, but the axial degrees of freedom remain decoupled.

scholarly journals Relaxation process modeling in a turbulent boundary layer with nonzero free stream turbulence

1997 ◽  
Vol 3 (3) ◽  
pp. 255-265
Author(s):  
Eugen Dyban ◽  
Ella Fridman
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Process Modeling ◽  
Free Stream ◽  
Equations Of Motion ◽  
Turbulent Shear ◽  
Shear Stresses ◽  
Free Stream Turbulence ◽  
Relaxation Effects ◽  
Averaged Equations

In order to analyze the relaxation effects in a turbulent boundary layer with zero and nonzero free stream turbulence, the Reynolds-averaged equations of motion and energy are solved. As the closure of the Reynolds-averaged equations, the transport equation for turbulent shear stresses is used. The proposed approach leads to calculation of the relaxation scales in the turbulent boundary layer with zero and nonzero free stream turbulence. Results for friction coefficients, velocity profiles, shear stresses, thickness of the boundary layer and so called “superlayer” in a flat-plate turbulent boundary layer are presented. The results obtained are in agreement with those available from the experimental data.

scholarly journals Analytic construction of multi-brane solutions in cubic string field theory for any brane number

2019 ◽  
Vol 2019 (8) ◽  
Author(s):  
Hiroyuki Hata
Keyword(s):  
Field Theory ◽  
String Field Theory ◽  
Equations Of Motion ◽  
Open String ◽  
Three Dimensions ◽  
Simons Theory ◽  
Analytic Construction ◽  
Pure Gauge ◽  
Brane Solution ◽  
Chern Simons

Abstract We present an analytic construction of multi-brane solutions with any integer brane number in cubic open string field theory (CSFT) on the basis of the ${K\!Bc}$ algebra. Our solution is given in the pure-gauge form $\Psi=U{Q_\textrm{B}} U^{-1}$ by a unitary string field $U$, which we choose to satisfy two requirements. First, the energy density of the solution should reproduce that of the $(N+1)$-branes. Second, the equations of motion (EOM) of the solution should hold against the solution itself. In spite of the pure-gauge form of $\Psi$, these two conditions are non-trivial ones due to the singularity at $K=0$. For the $(N+1)$-brane solution, our $U$ is specified by $[N/2]$ independent real parameters $\alpha_k$. For the 2-brane ($N=1$), the solution is unique and reproduces the known one. We find that $\alpha_k$ satisfying the two conditions indeed exist as far as we have tested for various integer values of $N\ (=2, 3, 4, 5, \ldots)$. Our multi-brane solutions consisting only of the elements of the ${K\!Bc}$ algebra have the problem that the EOM is not satisfied against the Fock states and therefore are not complete ones. However, our construction should be an important step toward understanding the topological nature of CSFT, which has similarities to the Chern–Simons theory in three dimensions.

Sign in / Sign up

Export Citation Format

Share Document