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Phenomenological models of viscoplastic, thixotropic, and granular materials

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Abstract

A simple constitutive framework is used to review some phenomenological models for solid-fluid mixtures. Several successful descriptions of viscoplasticity assume that stress is proportional to a normalized deformation rate. The modulus in this constitutive equation evolves with time for thixotropic materials. This approach is used in most models based on scalar measures of structure. Such measures are determined by an evolution equation that is insensitive to rate reversals during shear flow. This behavior appears to be characteristic of inelastic materials such as an aqueous solution of bentonite. However, the same solution with guargum does not exhibit this response due to elasticity induced by the polymer. Models for granular media extend this constitutive framework by including compressibility effects and a scalar measure of particle interactions. The effect of fluid viscosity on particle interactions is incorporated using a mixture theory approach. Pipe flow of viscoplastic materials and shear flow of granular media are analyzed using boundary conditions that allow slip at solid surfaces.

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Abbreviations

T :

stress tensor

T g :

grain stress tensor

T f :

fluid stress tensor

P :

pressure tensor

I :

identity tensor

D :

rate of deformation

D g :

grain deformation rate

D f :

fluid deformation rate

E :

normalized rate of deformation

E g :

normalized grain deformation rate

ζ:

principal values of E

ϑ:

angle given as a function of det (E)

p :

pressure

ℳ:

modulus

ℳ0 :

equilibrium modulus

Λ:

gel strength

Λ0 :

constant gel strength

λ:

characteristic time

λ0 :

constant characteristic time

λ1 :

constant characteristic time

λ∞ :

constant characteristic time

m :

material exponent

n :

material exponent (power-law index)

p 0 :

pressure for granular materials

ℳ1 :

modulus for granular materials

µ0 :

bulk viscosity for granular materials

µ1 :

viscosity for granular materials

\(\overrightarrow \upsilon\) f :

fluid velocity

ϱ f :

fluid density

µf :

fluid viscosity

p f :

fluid pressure

\(\overrightarrow f\) :

interaction body force

\(\overrightarrow \upsilon\) g :

grain velocity

ϱ g :

grain density

d g :

characteristic grain dimension

µ g :

grain viscosity

p g :

grain pressure

e :

coefficient of restitution

v :

solid fraction

T :

granular temperature

ψ:

Helmholtz free energy function

s :

specific entropy

\(\overrightarrow Q\) :

translational energy flux

k :

conductivity

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Authors and Affiliations

  1. Shell Research Limited, Thornton Research Centre, Chester, England, UK

    A. Berker

  2. Department of Mechanical Engineering, University of Houston, 4800 Calhoun Road, Houston, Texas, USA

    W. E. Van Arsdale

  3. 3M Company, SA & CD, 3M Center, Building 236-2B-11, 55144-1000, St. Paul, MN, USA

    A. Berker

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Berker, A., Van Arsdale, W.E. Phenomenological models of viscoplastic, thixotropic, and granular materials. Rheola Acta 31, 119–138 (1992). https://doi.org/10.1007/BF00373235

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  • DOI: https://doi.org/10.1007/BF00373235

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