By Krzysztof Wilmanski (auth.), Professor Dr. Dimitrios Kolymbas (eds.)

Geomechanics is the mechanics of geomaterials, i.e. soils and rocks, and offers with interesting difficulties resembling settlements, balance of excavations, tunnels and offshore structures, landslides, earthquakes and liquefaction. This edited e-book offers fresh mathematical and computational instruments and versions to explain and simulate such difficulties in Geomechanics and Geotechnical Engineering. It incorporates a number of contributions emanating from the 3 Euroconferences GeoMath ("Mathematical equipment in Geomechanics") that have been held among 2000 and 2002 in Innsbruck/Austria and Horto/Greece.

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20 Krzysztof Wilmanski The two-point tensors PS , Pα denote the Piola-Kirchhoff partial stress tensors, bS , bα are partial body forces, εS , εα are partial densities of the internal energy, QS , Qα – partial heat fluxes, rS , rα are partial energy radiations, J is the flux of porosity, and all quantities with a hat denote productions. The balance equation of porosity (49) requires some explanation. g. [14], [15]) that the balance equation for n follows from an averaging procedure for a representative elementary volume accounting for geometrical properties of the microstructure.

If only the generalised stress appears in the yield surface then the flow is “associated” in the conventional sense. Whenever the real stress is substituted in the yield expression non-associated flow occurs. A property of the partial Legendre transformation between yield and flow is, however, that ∂d/∂σij = −λ∂y/∂σij . Thus whenever the yield surface is a function of the stress, then so is the dissipation (and vice versa). What is the meaning of the dissipation being dependent on the stress? An obvious physical interpretation is that this would serve 40 Guy T.

The fundamental assumption of a continuous modeling has the form of the following constitutive relation R = R (C) , (60) where the mapping R is assumed to be at least once continuously differentiable. The above relation is a short-cut notation for constitutive relations for each quantity of the set (56) which should be functions of variables (57). Thermodynamics in Geophysics 23 The constitutive functions (60) are said to be thermodynamically admissible if any solution of field equations satisfies identically the following entropy inequality ∂ (ρη) ˙ + H ≥ 0, + Div ρη X ∂t η = η (C) , H = H (C) .

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