Important aspects in the formulation of solid-fluid debris-flow models. Part I. Thermodynamic implications

This article points at some critical issues which are connected with the theoretical formulation of the thermodynamics of solid-fluid mixtures of frictional materials. It is our view that a complete thermodynamic exploitation of the second law of thermodynamics is necessary to obtain the proper parameterizations of the constitutive quantities in such theories. These issues are explained in detail in a recently published book by Schneider and Hutter (Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context, 2009), which we wish to advertize with these notes. The model is a saturated mixture of an arbitrary number of solid and fluid constituents which may be compressible or density preserving, which exhibit visco-frictional (visco-hypoplastic) behavior, but are all subject to the same temperature. Mass exchange between the constituents may account for particle size separation and phase changes due to fragmentation and abrasion. Destabilization of a saturated soil mass from the pre- and the post-critical phases of a catastrophic motion from initiation to deposition is modeled by symmetric tensorial variables which are related to the rate independent parts of the constituent stress tensor

Important aspects in the formulation of solid-fluid debris-flow models. Part II. Constitutive modelling

This article is the continuation of Part I: ‘Thermodynamic Implications' of a article with the same title. Knowledge of the content/results of Part I, Hutter and Schneider (Continuum Mech. Thermodyn., 2009) or Schneider and Hutter (Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context, 2009), is assumed. The intention is to see whether (i) well-known formulations of binary mixture models can be derived from the thermodynamic model, (ii) classical hypo-plasticity is deducible from the frictional evolution equation and (iii) the popular assumption of pressure equilibrium is justified. To this end, we ignore mass and volume fraction interaction rate densities, restrict considerations to isothermal processes, ignore higher order non-linearities in the constitutive relations and use the principle of phase separation. These assumptions transform the equilibrium stresses, heat flux and interaction forces to considerably simplified forms. Furthermore, the analysis shows that classical hypo-plasticity can be reconstructed with the introduction of a new objective time derivative for the stress-like variable. Non-equilibrium contributions to the stresses and interaction forces are also briefly discussed. It is, finally, shown that the assumption of pressure equilibrium precludes the application of frictional stresses in equilibrium. This unphysical assumption is, therefore, replaced by a thermodynamic closure condition that is more flexible and less restrictive. It allows for frictional stresses in thermodynamic equilibrium and, therefore, is sufficiently general for applications to mixture theorie

The Trouble with Trebles: What Violates G.S. 75-1.1?

At first glance the North Carolina Unfair and Deceptive Trade Practices Act appears to be a broad, almost unconstitutionally vague statute. Its federal counterpart, the Federal Trade Commission Act, evoked similar responses when it was first enforced. Like the FTC Act, North Carolina General Statute § 75-1.1 has taken shape through judicial interpretation and legislative modification. (North Carolina General Statutes hereinafter referred to as G.S.). As this process has proceeded over the last decade or so, many aspects of the scope and application of the statute have been determined. No general answer, however, has been given to the question of just what does violate the statute. The boundary between a simple breach of contract, rendering one liable for at most simple damages, and an unfair trade practice, rendering one liable for treble damages and attorney\u27s fees, remains ill-defined. The significance of the question is clear, both to the used car dealer and his customer arguing over an $800 automobile, and to the businessman whose$8,000,000 deal falls through. This problem is highlighted, but not illuminated, by the conflict of analytical processes between the Supreme Court of North Carolina and the U.S. Court of Appeals for the Fourth Circuit. This conflict is evidence of uncertainty in the objectives of the statute and uncertainty among the judiciary as to the basic desirability of the statutory remedy

On the role of grain growth, recrystallization and polygonization in a continuum theory for anisotropic ice sheets

AbstractWe outline how to incorporate microscale effects of polycrystalline ice into a continuum description. Actually, analyses of ice cores in Antarctica show that different microstructures generally produce different responses, i.e. a non-uniform distribution of c axes gives rise to anisotropic behaviour. It has been recognized that, to describe certain microstructural processes, like recrystallization or polygonization, we need a parameter able to switch them on (e.g. dislocation density or its associated lattice distortion energy). With this in mind, balance equations for a continuum theory of an anisotropic ice sheet undergoing recrystallization have been recently proposed. In this work, we examine relations for some constitutive quantities, in order to take into account the effects of grain-boundary migration, nucleation and polygonization. We check our assumptions by explicit comparison with the first 1200 m of the Byrd (Antarctica) ice core. Current literature usually gives a relation between normal grain growth and grain boundary migration rate. Here, an equation for normal grain growth which also incorporates the influence of polygonization is suggested. It is based on experimental data from the same core in Antarctica. Polygonization is a microscopic process, but here we present a continuum description of the bending stresses which promote the fragmentation of crystallites in terms of the theory of mixtures with continuous diversity

A viscoelastic damage model for polycrystalline ice, inspired by Weibull-distributed fiber bundle models. Part II: Thermodynamics of a rank-4 damage model

We consider a viscoelastic-viscoplastic continuum damage model for polycrystalline ice. The focus lies on the thermodynamics particularities of such a constitutive model and restrictions on the constitutive theory which are implied by the entropy principle. We use Müller's formulation of the entropy principle, together with Liu's method of exploiting it with the aid of Lagrange multipliers

Thermally driven interaction of the littoral and limnetic zones by autumnal cooling processes

In autumn, during the transition period, shores influence the interior dynamics of large temperate lakes by the formation of horizontal water-temperature gradients between the shallow and deep areas, whilst vertical temperature gradients are smoothed by convection due to surface cooling. A simple heat budget model, based on the heat balance of the water column without horizontal advection and turbulent mixing, allows deduction of the time-dependent difference between the mean temperature within the littoral area and the temperature in the upper mixed layer. The model corroborates that littoral areas cool faster than regions distant from shores, and provides a basis for an estimation of structure of flows from the beginning of cooling process till the formation of the thermal bar. It predicts the moment in the cooling process, when the corresponding density difference between the littoral and limnetic parts reaches a maximum. For a linear initial vertical temperature profile, the time-dependent "target depth" is explicitly calculated; this is the depth in the pelagic area with a temperature, characteristic of the littoral zone. This depth is estimated as 4/3 of the (concurrent) thickness of the upper mixed layer. It is shown that, for a linear initial vertical temperature profile, the horizontal temperature profile between the shore and the lake has a self-similar behavior, and the temperature difference between the littoral waters and the upper mixed off-shore layer, divided by the depth of the upper mixed layer, is an invariant of the studied process. The results are in conformity with field data

A viscoelastic damage model for polycrystalline ice, inspired by Weibull-distributed fiber bundle models. Part I: Constitutive models

We consider a constitutive model for polycrystalline ice, which contains delayed-elastic and viscous deformations, and a damage variable. The damage variable is coupled to the delayed-elastic deformation by a fiber bundle ansatz. We construct an isotropic theory, which can be calibrated with experimental data. Furthermore, we generalize the theory to a damage model in terms of rank-four tensors. This general model allows the evolution of anisotropic damage