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.

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

In a four-element model, which consists of two Voigt elements in series, the impulse responses of spherical waves for particle displacement, velocity, acceleration and strain are computed numerically. These impulse responses are shown to be useful in synthesis for the desired wave response through convolution, between the input of the source function and the impulse responses. The rate of decay of peak amplitude versus distance, as waves radiate out, is examined in the region close to the source which may be nuclear or dynamite explosives and found to be linear in log-log scale. The amplitude spectra of the unit impulse response are obtained through Fourier analysis, and the model is found to behave as a low pass filter. With the properly chosen elastic constants for springs in the model and damping coefficients for dashpot, it is seen that, like the earth materials, the attenuation exponent (in neper/1000 ft) versus frequency in the model is approximately linear. One of the results observed byCollins andLee [1] is fitted by the model. The model thus simulates more nearly the wave in earth materials close to the source than the Voigt solid.

On the attenuation of transient spherical waves in a four-element viscoelastic model and its implications

The inhomogeneous wave equation for a plane compressional wave propagating in a Voigt medium is solved to yield exact mathematical expressions for the medium's response to a unit impulse excitation pressure. The solutions obtained, which are Green's function of the medium, show the pulse-lengthening, attenuation, and oscillation properties of Voigt waves. It is shown that, unlike previously published solutions, particle velocities oscillate because of elastic recovery of the stressed medium.

Transient waves in a Voigt medium

Closed‐form solutions to the propagation of spherical waves from an internal steady oscillating pressure source as well as to the transient response of a viscoelastic medium to a pressure impulse are presented. The assumption that the loss factors, quantities related to the dissipative property of a viscoelastic medium, are independent of frequency is used here for investigating the wave‐propagation phenomenon due to the action of an impulse. Numerical examples for the displacement and stress functions have been used to illustrate the damping effect of a viscoelastic medium by comparing with its special case—the elastic medium. It is found that this effect shows its significance with increased traveling distance. Also, in the steady‐oscillation problems it is of importance when the vibration frequency is high, while in the transient case it causes the decrease in the peak amplitude and also a time lag in the arrival of the wavefront.

Spherical waves in viscoelastic media

The purpose of this paper is to study the propagation of cylindrical shear waves in nonhomogeneous five-parameter viscoelastic media of arbitrary thickness. The media 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 media with a stress-free cylindrical outer boundary and an infinite media 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 media while reflected and transmitted pulses are produced at the interface in the infinite bi-viscoelastic media. Ray techniques are used throughout, and formal asymptotic wave-front expansions of the solution functions are obtained.

Transient cylinderical shear waves propagation in non-homogeneous five-parameter viscoelastic media

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James E. Jamison

Papers

Series transform solutions for Voigt transients