Volume-weighted mixture theory for granular materials

Chen, Kuo Ching; Tai, Yih-Chin

doi:10.1007/s00161-007-0064-7

Cited by 11 publications

(11 citation statements)

References 25 publications

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“…Its simplicity also yields modelling and computational advantages. For a formal comparison of volume and mass weighted mixture concepts, see Chen and Tai [8].…”

Section: Physical Assumptionsmentioning

confidence: 99%

“…In the present model we are left with the equivalent quantities B s ,Z s and D s , but owing to the mixture character of the model the quantities ν s and ∇ν s should also arise. To disregard the latter contributions is hardly feasible and, therefore, if we want to use the representations for L and N proposed in the literature for single-body hypo-plasticity, 8 their adaption is necessary. The existing recent literature on hypo-plastic constitutive modelling and parameter identification for special choices of the operators L and N clearly point at a dominant role played by the void ratio e = (1 − ν s )/ν s .…”

Section: Frictional Part Of the Solid Stressmentioning

confidence: 99%

“…We recall the dependencies of the constitutive quantities C = {T s ,T f ,m i s } as listed in (8). Applying, now, the 'principle of phase separation', toT s | N andT f | N , these dependencies reduce tō…”

Section: Non-linear Viscous Solid and Fluid Stressesmentioning

confidence: 99%

See 2 more Smart Citations

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

Hutter¹,

Schneider²

2010

Continuum Mech. Thermodyn.

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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 theories.

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“…Its simplicity also yields modelling and computational advantages. For a formal comparison of volume and mass weighted mixture concepts, see Chen and Tai [8].…”

Section: Physical Assumptionsmentioning

confidence: 99%

Section: Frictional Part Of the Solid Stressmentioning

confidence: 99%

See 1 more Smart Citation

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

Hutter¹,

Schneider²

2010

Continuum Mech. Thermodyn.

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“…So, let λ 1 , λ 2 be the eigenvalues of the mean surface stretching, and let f 1 , f 2 be corresponding (for the moment arbitrary) orthonormal eigenvectors 3 . We shall propose three models for σ ⋆ s .…”

Section: This Yieldsmentioning

confidence: 99%

“…Finally, note that (8.10) holds, in fact, for any orthonormal vectors f 1 , f 2 in the tangent space to S, and not necessarily for eigenvectors of D S v τ . 3 The matrix (D α β ) in the representation D S vτ = D α β τα ⊗ τ β is needed to obtain the eigenvalues and eigenvectors of D S vτ, and it has been deduced in Luca et al [11].…”

Section: This Yieldsmentioning

confidence: 99%

Modeling shallow gravity-driven solid-fluid mixtures over arbitrary topography

Luca¹,

Tai²,

Kuo³

2009

Communications in Mathematical Sciences

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Abstract. The purpose of this paper is to derive modeling equations for debris flows on real terrain. Thus, we use curvilinear coordinates adapted to the topography as introduced, e.g., by Bouchut and Westdickenberg [F. Bouchut and M. Westdickenberg, Commun. Math. Sci., 2(3), 2004], and develop depth-averaged models of gravity-driven saturated mixtures of solid grains and pore fluid on an arbitrary rigid basal surface. First, by only specifying the interaction force and ordering approximations in terms of an aspect ratio between a typical length perpendicular to the topography, and a typical length parallel to the topography, we derive the governing equations for the shallow flow of a binary mixture, driven by gravitational force. In doing so, the non-uniformity through the avalanche depth of the constituent velocities and of the solid volume fraction is accounted for by coefficients of Boussinesq type. Then, the material behaviour peculiarities of both constituents properly enter the theory. One constituent is a granular solid. For its stresses we propose three models, one of them of Mohr-Coulomb type. The other constituent is a Newtonian/non-Newtonian fluid with small viscosity, obeying a viscous bottom friction condition. The final governing equations for the shallow flow of the mixture, incorporating the constitutive assumptions, are deduced, and the limiting equilibrium is then investigated.

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Nonhydrostatic granular flow over 3-D terrain: New Boussinesq-type gravity waves?

Castro-Orgaz

Hutter

Giráldez

et al. 2015

J. Geophys. Res. Earth Surf.

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Understanding granular mass flow is a basic step in the prediction and control of natural or man-made disasters related to avalanches on the Earth. Savage and Hutter (1989) pioneered the mathematical modeling of these geophysical flows introducing Saint-Venant-type mass and momentum depth-averaged hydrostatic equations using the continuum mechanics approach. However, Denlinger and Iverson (2004) found that vertical accelerations in granular mass flows are of the same order as the gravity acceleration, requiring the consideration of nonhydrostatic modeling of granular mass flows. Although free surface water flow simulations based on nonhydrostatic depth-averaged models are commonly used since the works of Boussinesq (1872, 1877), they have not yet been applied to the modeling of debris flow. Can granular mass flow be described by Boussinesq-type gravity waves? This is a fundamental question to which an answer is required, given the potential to expand the successful Boussinesq-type water theory to granular flow over 3-D terrain. This issue is explored in this work by generalizing the basic Boussinesq-type theory used in civil and coastal engineering for more than a century to an arbitrary granular mass flow using the continuum mechanics approach. Using simple test cases, it is demonstrated that the above question can be answered in the affirmative way, thereby opening a new framework for the physical and mathematical modeling of granular mass flow in geophysics, whereby the effect of vertical motion is mathematically included without the need of ad hoc assumptions.

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Volume-weighted mixture theory for granular materials

Cited by 11 publications

References 25 publications

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

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

Modeling shallow gravity-driven solid-fluid mixtures over arbitrary topography

Nonhydrostatic granular flow over 3-D terrain: New Boussinesq-type gravity waves?

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