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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:
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
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=pS+pF,
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 so-called Gassmann-type relations. Derivation of those relations shows that the method of micro-macro 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 above-described 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]).
References:
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