Advanced Mathematical and Computational Geomechanics by Krzysztof Wilmanski (auth.), Professor Dr. Dimitrios

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

Geomechanics is the mechanics of geomaterials, i.e. soils and rocks, and bargains with interesting difficulties similar to settlements, balance of excavations, tunnels and offshore structures, landslides, earthquakes and liquefaction. This edited booklet offers fresh mathematical and computational instruments and types to explain and simulate such difficulties in Geomechanics and Geotechnical Engineering. It features a selection 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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Thermodynamic considerations indicate that the flux J results from the diffusion (relative motion of fluid components with respect to the skeleton), and the source n ˆ describes relaxation to the thermodynamic equilibrium. Certainly we do not need all partial energy balance equations if we have to determine only one temperature field T . In such a case one relies on the bulk balance which follows from (48) by addition of equations for all components. Due to its relative complexity we will not present this equation in this work and refer the reader to the paper [18].

The transform has the property xi = ∂Y /∂yi . There is an obvious symmetry that X is also the Legendre transform of Y . A marginally more complicated case occurs when X = X (xi , ai ) and the partial Legendre transform (again defined by X + Y = xi yi ) is made to Y = Y (yi , ai ), with the ai regarded as passive variables. In this case it D. ), Advanced Mathematical and Computational Geomechanics © Springer-Verlag Berlin Heidelberg 2003 36 Guy T. Houlsby is straightforward to show that the transform has the additional property ∂Y /∂ai = −∂X/∂ai .

Although it is currently used by only a minority of those studying plasticity, it seems likely that in time convex analysis will be come the standard paradigm for plasticity theory. In the following C is a subset in a normed vector space V , usually with the dimension of Rn , but possibly infinite dimensional. The notation , is used for an inner product, or more generally the action of a linear operator on a function. The topological dual space of V (the space of linear functionals on V ) is V .

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