Landslides - Causes, Types and Effects.pdf

278 Jiří NedomaFigure 8. Coefficient of friction as a function of tangential velocity component.Since all input data, i.e. coefficients of stress-strain laws, coefficients of thermal conductivity,body and surface forces, thermal sources, initial temperature, friction coefficients,density, etc., are in general uncertain, then method of worst scenario can be applied to findthe most “dangerous” admissible input data. By uncertain data we mean input data (physicalcoefficients, right-hand sides, boundary values, friction coefficients, etc.), which cannotbe determined uniquely but only in some intervals with errors determined by their measurementerrors. The notation reliable solution denotes the worst case among a set of possiblesolutions of investigated problems, where possibility is given by uncertain input data, andthe degree of badness is measured by a criterion-functional. For more details see [25].1.2.2. Models Based on the Multibody Contact Theory in Thermo-visco-plasticityIn the second part of this study we limit ourselves to only small part of the presentedproject, to analysis of the mathematical model of hydro-mechanical problem of non-stableslopes in connection with bigger flood.Rheology and the constitutive relations:Let Ω ⊂ R N , N = 2, 3, Ω = ∪ r ι=1 Ωι , be a union of bounded domains occupiedby visco-plastic bodies with a smooth boundary ∂Ω = Γ u ∪ Γ τ ∪ Γ 0 ∪ Γ c , where Γ urepresents one part of the boundary, where the velocity is prescribed; Γ τ is the part ofthe boundary, where the loading is prescribed; Γ 0 is the part of the boundary, where thesymmetry condition or the bilateral contact condition are given and Γ c (= ∪ k,l Γ klc , Γkl c =∂Ω k ∩∂Ω l ,k≠ l, k, l ∈{1,...,r}) represents the contact boundary. Assume the Euleriancoordinate system as a spatial variable system. Any repeated index implies summationfrom 1 to N. Let t ∈ I ≡ (0,t p ), t p > 0, where t p is the duration of the geologicalprocesses. Let u n = u.n, u t = u − u n n, τ =(τ ij n j ), τ n = τ ij n j n i , τ t = τ − τ n n be thenormal and tangential components of the velocity and stress vectors.The Bingham rheology has the following property: the rock material starts to flow ifand only if the applied forces exceed a certain limit, the so-called yield limit. We speak alsoabout Bingham solid/fluid media. We assume that the rocks are incompressible.Let u =(u i ) be the material velocity, let D =(D ij ) denote the strain rate tensor and