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Cooperation with: I. Edelman (Alexander von Humboldt fellow in WIAS, Russian Academy of Sciences, Moscow), S. Foti, R. Lancellotta (Università di Torino, Italy), C. Lai (Studio Geotecnico Italiano, Milano)
Description: Aims and results of the project
The project is the continuation of the research devoted to a theoretical analysis of weak discontinuity waves on the basis of the own model (e.g., [1, 2]), as well as practical geotechnical applications particularly in a nondestructive testing of soils. Two main topics are in the process of investigation:
It has been found out that results are different in the case when the problem of monochromatic waves is formulated in the space of real wave numbers, k (initial value problems), and in the case of real frequencies, (boundary value problems). For this reason a new asymptotic approximation has been developed by I. Edelman [3, 4, 5, 6]. She has shown that the P2 wave possesses a limit of the wave length beyond which this wave cannot propagate, i.e. the frequency becomes purely imaginary. The value of this limit depends on the bulk permeability coefficient. As the existence of Stoneley waves is coupled with the existence of P2 waves these waves possess a similar property. In the work [7] this property has been illustrated by a few numerical examples and compared with the properties of solutions for real .
In the subsequent works [8, 9] the properties of bulk and surface waves defined on the space of real have been investigated for the small frequency limit and it has been found out that the singularity discovered by Edelman does not exist in this formulation. The limits for P1 and S waves are regular and they give relations for the speeds of propagation:
where are Lamé constants and the compressibility of the fluid, respectively, while are partial mass densities of the skeleton and of the fluid, respectively. These results agree with those of geophysicists. They show that the body reacts as if it was a composite of two elastic systems in a synchronized motion. Simultaneously, the limit for P2 waves is singular and it behaves as if the system were parabolic in connection with that mode which means that the speed limit of this wave goes to zero as .Similar asymptotic analysis shows that the speed of the Stoneley wave becomes zero as goes to zero. The Rayleigh wave satisfies the following relation for the speed of propagation
which is identical with the classical Rayleigh equation except that speeds of bulk waves were replaced by the lowfrequency limit counterparts.The second topic arises in connection with a transition from microscopic to macroscopic relations for granular materials. As reported in the last year a nondestructive testing of soils by means of analysis of the field data for bulk and surface waves requires the transformation of microscopic compressibility properties into macroscopic parameters. This can be achieved by analysis of three simple tests:
All these tests must satisfy the following set of equations relating a macroscopic equilibrium state to a microscopic equilibrium state.
p=p^{S}+p^{F},
where p is an external excess pressure.The definitions of three gedankenexperiments lead to a system of equations which allows to determine the dependence of macroscopic parameters and microscopic parameters on porosity. These are the socalled Gassmanntype relations. Derivation of those relations shows that the method of micromacro transition must be considerably improved in order to describe such properties of the porous material as a degree of saturation or capillarity effects. In spite of its flaws the abovedescribed method yields results which agree quite well with the field experiments. This has been demonstrated in the works of the Italian partners of the project (e.g., [10]).
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