### Important thermodynamic aspects in the formulation of solid-fluid debris flow models (dense and particle laden)

**Hutter, K ***(Darmstadt)*

Friday 09 January 2009, 11:45-12:10

Seminar Room 1, Newton Institute

#### Abstract

The present literature on continuous modeling of static and dynamic solid-fluid interactions is fraught with different forms of balance laws that are based on mixture and multi-phase theories (balance laws with a priori estimates for constituent pressure) respectively. They are characterised by different postulates of the partial stresses and on this basis, it is claimed eg that the multi-phase concept is superior to the mixture theory concept. It is my conjecture that such claims are premature without a proper thermodynamic analysis and a comparison of the final field equations. We perform a thermodynamic analysis for a mixture of n solid-fluid constituents with thermo-visco-elastic properties and account for rubbing frictional effects by an internal symmetric internal tensor variable, which models hypo-plastic behaviour. Volume fraction dependence is accounted for by a scalar internal variable, mixture saturation is taken into account by a constraint condition and incompressibility is interpreted as preserving the true of the respective constituents. The thermodynamic analysis is conducted with Muller's form of the entropy principle. The analysis also requires a number of ad hoc assumptions to be able to deduce definite results. These results deliver an explicit expression of the Gibbs relation and the entropy flux and yield, via the integrability conditions, explicit forms for the contituent equilibrium stresses, constituent interaction forces, entropy and equilibrium heat flux. These expressions are determined by the prescription of the Helmholtz free energy, Gibbs free energy, the saturation constraint variable and the extra entropy flux. The structure of the formulae shows how these equilibrium variables depend via partial derivatives of the free energy upon the independent constitutive variables. Three different pressure terms arise: thermodynamic pressures are the responses to constituent density variations, configuration pressures those to volume fraction dependences and saturation pressure that to the saturation constraint. It follows from these results that the hypothesis of pressure equilibriium, according to which the mixture pressure is distributed among the constituents with the weights of the volume fractions, is almost never correct except when the hypo-plastic friction and constituent volume fraction dependences are ignored as independent constitutive variables. Non-equilibrium stresses and interaction forces are dominantly governed by viscous postulates via stress parameterisations known in rheology of polymers and foods, and Darcy type relations for the interaction forces. The work suggests that debris flow modeling for solid-fluid mixtures along arbitrary terrain should be anticipated by a thermodynamic analysis to guarantee correctness of the equation.

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