Wave propagation in viscoelastic

Although part of Chapter 16 is devoted to wave propagation in viscoelastic materials and some specific simple cases are studied in detail as part of the engineering applications of viscoelasticity, it is useful to mention here that there are several experimental methods to determine the dynamic response... 

Nunziato, J.W., Walsh, E.K., Schuler, K.W., and Barker, L.M., Wave Propagation in Nonlinear Viscoelastic Solids, in Handbuch der Physik, Band VIa/4 (edited by Flugge, S.), Springer-Verlag, Berlin, 1974, pp. 1-108. 

KC Valanis, S Chang. Stress wave propagation in a finite viscoelastic thin rod with a constitutive law of the hereditary type. In TC Huang, MW Johnson, eds. Developments in Theoretical and Applied Mechanics. New York Wiley, 1965. 

In this section first Biot s viscoelastic model is summarized which simulates high- and low-frequency wave propagation in water-saturated sediments by computing phase velocity and attenuation curves. Subsequently, analysis teehniques are introduced whieh derive P-wave veloeities and attenuation eoefficients from ultrasonie signals transmitted radially across sediment eores. Additional physieal properties like... 

F. Simonetti, "Lamb Wave Propagation in Elastic Plates Coated with Viscoelastic... 

It was these studies of the complex dielectric constant as a function of frequency which led to the search for euialogous methods of studying viscoelasticity hy measuring a complex viscosity or elastic modulus. The first success was observation of shear wave propagation in polymer solutions . The only theoretical treatment of wave propagation which could he found as a clue to analysis of the measurements was in a geophysical journal. From the wave propagation the complex shear modulus and its frequency dependence could be derived. 

Nonlinear soil behavior can be approximated by an equivalent linear characterization of soil dynamic properties. The method makes use of the exact continuum solution of wave propagation in horizontally layered viscoelastic materials subjected to vertically propagating transient motions (e.g., Roesset 1977). It models the nonlinear variation of soil shear modulus and... 

Some basic lamina and laminate behavioral characteristics were deliberately overlooked in the preceding discussion. Among them are plastic or nonlinear deformations, viscoelastic behavior, and wave propagation. 

This is indeed a system of three second-order differential equations. The tensor elements Cyki may be complex-valued in case of viscoelasticity. Analysis shows that the propagation can be split into three orthogonally polarized planar waves propagating along a wave vector k. Those three waves may have different propagating celerities. Phase celerity and polarization ilj are connected through Christoffel equation ... 

Analysis of these effects is difficult and time consuming. Much recent work has utilized two-dimensional, finite-difference computer codes which require as input extensive material properties, e.g., yield and failure criteria, and constitutive laws. These codes solve the equations of motion for boundary conditions corresponding to given impact geometry and velocities. They have been widely and successfully used to predict the response of metals to high rate impact (2), but extension of this technique to polymeric materials has not been totally successful, partly because of the necessity to incorporate rate effects into the material properties. In this work we examined the strain rate and temperature sensitivity of the yield and fracture behavior of a series of rubber-modified acrylic materials. These materials have commercial and military importance for impact protection since as much as a twofold improvement in high rate impact resistance can be achieved with the proper rubber content. The objective of the study was to develop rate-sensitive yield and failure criteria in a form which could be incorporated into the computer codes. Other material properties (such as the influence of a hydrostatic pressure component on yield and failure and the relaxation spectra necessary to define viscoelastic wave propagation) are necssary before the material description is complete, but these areas will be left for later papers. 

For a viscoelastic material both K and G are complex quantities. When the material sample has finite dimensions other modes of wave propagation may occur as a result of multiple scattering from the material boundaries. Mode conversion from longitudinal wave to shear wave, and vice versa, occurs on reflection at a solid boundary. For material samples in the form of thin rods or plates the modes of wave propagation are extenional waves (with speed determined by the Young s modulus, or the plate modulus), and flexural (bending) waves [8]. 

When a sinusoidal (harmonic) sound wave propagates through a viscoelastic material, the stresses and strains in the material vay sinusoidally. Eg.22 predicts, in this case, a phase lage between the stress and the strain, which leads to conversion of acoustic energy to heat. From the Fourier transform of Eg.22 it follows that the sinusoidal stress and strain are related by complex, freguency-dependent elastic moduli as follows. 

Measurements of physical properties usually encompass the whole, undisturbed sediment. Two types of parameters can be distinguished (1) bulk parameters and (2) acoustic and elastic parameters. Bulk parameters only depend on the relative amount of solid and fluid components within a defined sample volume. They can be approximated by a simple volume-oriented model (Fig. 2.2a). Examples are the wet bulk density and porosity. In contrast, acoustic and elastic parameters depend on the relative amount of solid and fluid components and on the sediment frame including arrangement, shape and grain size distribution of the solid particles. Viscoelastic wave propagation models simulate these complicated structures, take the elasticity of the frame into account and consider interactions between solid and fluid constituents. (Fig. 2.2b). Examples are the velocity and attenuation of P-and S-waves. Closely related parameters which mainly depend on the distribution and capillarity of the pore space are the permeability and electrical resistivity. 

Shear Wave Propagation. A pulse shearometer (Rank Bros.) was used to measure the propagation velocity of a shear wave through the weak gels formed by the solutions of HMHEC in dilute NaCl. The polymer concentration range studied was 0.5-2.0%. With this apparatus, the frequency of the shear wave is approximately 1200 rad s" and the strain is <10 . At this strain, n pst systems behave in a linear viscoelastic fashion, and the wave-rigidity modulus, G is... 

In the methods of the preceding section, the propagation of a shear wave into the viscoelastic liquid is not observed directly only its effect on the propagating surface is detected and measured. On the other hand, if the damping is not too severe, so that a wave train of several maxima and minima is perceptible, the wavelength and attenuation can be measured directly, as described in this section. The sample must be sufficiently large so that the waves are attenuated before they can experience reflection. 

At much higher frequencies, bulk longitudinal waves can be propagated in polymeric liquids, and these may yield indirectly some information about shear properties. Since their behavior is dominated by the bulk compressional- viscoelasticity, however, discussion of them will be postponed to Chapter 8. 

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