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Wave propagation in porous and granular materials    

Collaborator: K. Wilmanski  

Cooperation with: I. Edelman (Alexander von Humboldt fellow in WIAS, Russian Academy of Sciences, Moscow), C. Lai (Studio Geotecnico Italiano, Milano), S. Foti, R. Lancellotta (Politecnico di Torino, Italy)

Description:

Aims and results of the project

The project is devoted to a theoretical analysis of weak discontinuity waves on the basis of the own model [1], [2], as well as practical geotechnical applications particularly in a nondestructive testing of soils. Three main topics are in the process of investigation:

1.

Long wave approximations of surface waves on the vacuum/porous body and fluid/porous body interfaces,

2.

Approximate solutions by the method of propagators of the bulk and surface waves in vertically heterogeneous semispace,

3.

Estimations of porosity by means of measurements of speeds of bulk waves.

The first problem is the continuation of the research carried out in the group during the last two years. The analysis of surface waves has been performed primarily in the limit of high frequency which yields the speeds of propagation of impulses ([3]). Such an approximation has the disadvantage of not corresponding to the frequency ranges used in geotechnical applications (approx. 1 to 10 Hz in contrast to some kHZ used in our work). For this reason a new asymptotic approximation has been designed. It seems to give correct results (a preliminary report is due to appear ([4])) but some singularities for different modes of propagation still require an explanation. Let us mention that such singularities (phase speeds going to infinity for low critical frequencies of monochromatic surface waves) appear also in the analysis of classical Rayleigh waves in heterogeneous materials.

The second problem is related to the first one because it concerns surface wave solutions for heterogeneous materials. Under the assumption that material properties depend on the variable z measuring the distance from the surface of the semispace, the wave analysis leads to a differential eigenvalue problem of the following form  

$$\frac{d{\bf f}}{dz}=A\left( z\right) {\bf f},\quad {\bf f\in }\Re ^{4},\quad {\bf A\in }\Re ^{4}\times \Re ^{4},$$ (1)

where in the single component solid the vector ${\bf f}$ and the matrix ${\bf A}$ are defined as follows

$${\bf f:=}\left( U,W,f_{3},f_{4}\right) ^{T},\quad f_{3}:=\mu... ...d f_{4}:=\left( \lambda +2\mu \right) \frac{dW}{dz}+k\lambda U,$$

 

$${\bf A:}=\left( \begin{array} {cccc} 0 & k & \mu ^{-1} & 0 ... ...ight) ,\quad \zeta :=4\mu \frac{\lambda +\mu }{\lambda +2\mu }.$$ (2)

In these relations $\lambda ,\mu ,\rho$ are z-dependent material parameters, $k,\omega$ are the wave number and frequency, respectively. U,W denote the amplitudes of disturbances.

In order to solve this problem, i.e. in order to find a relation between k and $\omega$, a method of successive approximations has been developed. The method is based on the assumption that the deviation of the matrix ${\bf A}$from its average ${\bf A}_{0}$ with respect to z is a small quantity  

$$\frac{\left\Vert {\bf A-A}_{0}\right\Vert }{\left\Vert {\bf A}_{0}\right\Vert }{\bf l}1.$$ (3)

The method has been tested on the classical problem of Rayleigh waves, and it shall be applied in the future to surface waves in two-component heterogeneous systems as well as in some flow stability problems in such systems.

The third problem is related to a nondestructive testing of soils. It is attempted to develop a systematic acoustic method of in situ measurements of such quantities as the porosity of soils. The first result of this form has been obtained in [5], [6]. It is based on very simple relations between macroscopic material parameters and the porosity. One obtains the following relation between porosity n and the speeds of propagation of the longitudinal P1-wave c_P1, the longitudinal P2-wave c_P2, and the shear S-wave c_S 

$$n=\frac{\rho ^{SR}-\sqrt{\rho ^{SR2}-4\rho ^{FR}\left( \rho ... ...ight) },\quad \varphi :=\frac{2\left( 1-\nu \right) }{1-2\nu },$$ (4)

where $\rho ^{SR},\rho ^{FR}$ are real mass densities of components and $\nu$ denotes Poisson's ratio (approx. 0.155-0.20). Comparison of these results with measurements on the site of Pisa tower ([5]) shows a very good agreement.

References:

1.   K. WILMANSKI, Mass exchange, diffusion and large deformations of poroelastic materials, in: Modeling and Mechanics of Granular and Porous Materials, G. Capriz, V.N. Ghionna, P. Giovine, eds., Birkhäuser, Basel, 2002, pp. 213-244.
2.   , Thermodynamics of multicomponent continua, in: Earthquake Thermodynamics and Phase Transformations in the Earth's Interior, R. Teisseyre, E. Majewski, eds., International Geophysics Series 76, Academic Press, San Diego, 2001, chapter 25, pp. 567-656.
3.   I. EDELMAN, K. WILMANSKI, Asymptotic analysis of surface waves at vacuum/porous medium and liquid/porous medium interfaces, WIAS Preprint no. 695, 2001, in print in: Contin. Mech. Thermodyn., 14 (2002).
4.   K. WILMANSKI, Propagation of sound and surface waves in porous materials, WIAS Preprint no. 684, 2001, in print in: Arch. Mech., 54 (2002).
5.   S. FOTI, C. LAI, R. LANCELLOTTA, Porosity of fluid-saturated porous media from measured seismic wave velocity, submitted.
6.   R. LANCELLOTTA, K. WILMANSKI, On estimation of porosity in granular materials by means of acoustic measurements, in preparation.

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