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.

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.

On a sphere performing linear and torsional oscillations in a viscous fluid

1988 ◽  
Vol 66 (7) ◽  
pp. 576-579
Author(s):  
G. T. Karahalios ◽  
C. Sfetsos
Keyword(s):  
Boundary Layer ◽  
Viscous Fluid ◽  
Secondary Flow ◽  
Small Amplitude ◽  
Equations Of Motion ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Time Varying ◽  
Torsional Oscillations

A sphere executes small-amplitude linear and torsional oscillations in a fluid at rest. The equations of motion of the fluid are solved by the method of successive approximations. Outside the boundary layer, a steady secondary flow is induced in addition to the time-varying motion.

scholarly journals A Justification of Two-Dimensional Nonlinear Viscoelastic Shells Model

2012 ◽  
Vol 2012 ◽  
pp. 1-24
Author(s):  
Fushan Li
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Analysis ◽  
Laplace Transformation ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Nonlinear Viscoelastic ◽  
Leading Term ◽  
Viscoelastic Shells

By applying formal asymptotic analysis and Laplace transformation, we obtain two-dimensional nonlinear viscoelastic shells model satisfied by the leading term of asymptotic expansion of the solution to the three-dimensional equations.

Bifurcation Theory Applied to Oil Whirl in Plain Cylindrical Journal Bearings

1984 ◽  
Vol 51 (2) ◽  
pp. 244-250 ◽  
Author(s):  
C. J. Myers
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Equilibrium Position ◽  
Small Amplitude ◽  
Bifurcation Theory ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
The Self ◽  
Oil Whirl ◽  
Nonlinear Equations Of Motion

An analysis of the self-excited oscillations of a rotor supported in fluid film journal bearings is presented. It is shown that Hopf bifurcation theory may be used to investigate small-amplitude periodic solutions of the nonlinear equations of motion for rotor speeds close to the speed at which the steady-state equilibrium position becomes unstable. A numerical investigation supports the findings of the analytic work.

Boundary-layer development at a two-dimensional rear stagnation point

1972 ◽  
Vol 56 (1) ◽  
pp. 161-171 ◽  
Author(s):  
A. J. Robins ◽  
J. A. Howarth
Keyword(s):  
Boundary Layer ◽  
Stagnation Point ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Initial Value ◽  
Boundary Layer Development ◽  
Leading Term ◽  
Layer Development ◽  
Rear Stagnation Point ◽  
Good Agreement

This paper examines the nature of the development of two-dimensional laminar flow of an incompressible fluid at the rear stagnation point on a cylinder which is started impulsively from rest. Proudman & Johnson (1962) first examined this type of flow, andobtainedasimilarity solution of the inviscid form of the equations of motion. This solution describes the nature of the flow at large distances from the surface, for large times after the start of the motion. Here, the flow at the rear stagnation point is examined in greater detail. The solution found by Proudman & Johnson constitutes the leading term in an asymptotic expansion, valid for large times. Further terms in this expansion are now calculated, and the method of matched asymptotic expansions is used to obtain an inner solution describing the flow near the surface. A numerical integration of the full initial-value problem gives good agreement with the analytical solution.

scholarly journals On the propagation, reflection, and transmission of transient cylindrical shear waves in nonhomogeneous four-parameter viscoelastic media

1973 ◽  
Vol 8 (3) ◽  
pp. 397-411 ◽  
Author(s):  
T. Bryant Moodie
Keyword(s):  
Material Properties ◽  
Outer Boundary ◽  
Shear Waves ◽  
Infinite Plate ◽  
Radial Distance ◽  
Viscoelastic Plate ◽  
Reflection And Transmission ◽  
Viscoelastic Media ◽  
Finite Plate ◽  
Arbitrary Thickness

The purpose of this paper is to study the propagation of cylindrical shear waves in nonhomogeneous four-parameter viscoelastic plates of arbitrary thickness. The plates have a transverse cylindrical hole and their material properties are functions of the radial distance from the center of this opening. They are initially unstressed and at rest. A suddenly rising shearing traction is applied uniformly over the boundary of the opening and parallel to the faces of the plates and thereafter steadily maintained; they are otherwise free from loading. We consider both the case of a finite plate with a stress-free cylindrical outer boundary, and an infinite plate composed of two media in welded contact along a cylindrical surface symmetrical with respect to the center of the opening. We find that a reflected pulse is produced at the outer boundary of the finite plate while reflected and transmitted pulses are produced at the interface in the infinite bi-viscoelastic plate. Ray techniques are used throughout, and formal asymptotic wavefront expansions of the solution functions are obtained.

scholarly journals Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula

2008 ◽  
Vol 40 (01) ◽  
pp. 122-143 ◽  
Author(s):  
A. J. E. M. Janssen ◽  
J. S. H. van Leeuwaarden ◽  
B. Zwart
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Poisson Distribution ◽  
Integral Transformation ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Normal Distribution Function ◽  
Gaussian Form ◽  
The Central Limit Theorem ◽  
Leading Term

This paper presents new Gaussian approximations for the cumulative distribution function P(A λ ≤ s) of a Poisson random variable A λ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasi-Gaussian form, which permits evaluation in terms of the normal distribution function Φ. The quasi-Gaussian form contains an implicitly defined function y, which is closely related to the Lambert W-function. A detailed analysis of y leads to a powerful asymptotic expansion and sharp bounds on P(A λ ≤ s). The results for P(A λ ≤ s) differ from most classical results related to the central limit theorem in that the leading term Φ(β), with is replaced by Φ(α), where α is a simple function of s that converges to β as s tends to ∞. Changing β into α turns out to increase precision for small and moderately large values of s. The results for P(A λ ≤ s) lead to similar results related to the Erlang B formula. The asymptotic expansion for Erlang's B is shown to give rise to accurate approximations; the obtained bounds seem to be the sharpest in the literature thus far.

Elastic Waves

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Elastic Waves ◽  
Small Amplitude ◽  
Equations Of Motion ◽  
Plane Waves ◽  
Elastic Solid ◽  
Stress And Strain ◽  
Field Equations ◽  
Strain Stress ◽  
Thermodynamic Variables ◽  
Dynamic Variables

This chapter focuses on the mathematics of elastic waves. In the case of a continuous medium, the field equations of physics (yielding the dynamic and thermodynamic variables) arise from three conservation equations: conservation of mass, momentum and energy. For an elastic medium, these equations of motion are known as Navier equations, which give rise to a rich variety of stress waves. There are two dynamic variables in an elastic solid: stress and strain. Stress and strain are linearly related in small-amplitude deformations; this relation is expressed by Hooke's law. The chapter first introduces the basic notation for elastic waves before discussing the solutions for plane waves. It also considers surface waves and Love waves.

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