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Transport in Porous Media

, Volume 25, Issue 3, pp 351–374 | Cite as

A generalized approach to heterogeneous media

  • P. Jouanna
  • M.-A. Abellan
Article

Abstract

This paper is devoted to the description of a generalized approach developed for over-coming the fundamental difficulties that occur when the phenomenological field theory is extended from homogeneous to heterogeneous media. The origin of these difficulties lies in the kinematics of heterogeneous media where different constituents have in general different velocities at the same macroscopic point. The key idea of the generalized approach consists in choosing a reference velocity field v* independent of the movement of the matter. The consequences are examined at different levels. First of all, generalized balance relations lead to a consistent expression of the fundamental law of mechanics and of the first and second principles of thermodynamics in heterogeneous media. Unexpected consequences are linked to the objective nature of the entropy source. Moreover, this generalized approach exhibits a new set of relations between phenomenological and physical variables. Finally, it provides a common, consistent background for different existing theories derived for heterogeneous media.

Key words

thermo-hydro-mechanical physico-chemical heterogeneous phenomenological unsaturated soil heat and mass transfer soil mechanics 

Notations

aπ[resp. a*]

surface of the domain ωπ[resp. ω*] moving in the fieldvω[resp. v*]

ĉπ

mass supply to π due to other constituents per volume unit and time unit

daπ[resp. da*]

surface element of surface aπ[resp. a*]

\(\frac{{d_{\upsilon \pi \varphi } (x,t)}}{{dt}}\)

derivative of ϕ(x, t) following the movement vπ of constituent π

\(\frac{{d_{\upsilon *\varphi } (x,t)}}{{dt}}\)

derivative of ϕ(x, t) following the virtual movement v*

π, dɛπ, dɛπ

phenomenological, macroscopic and microscopic rates of strain

dω, dωπ, dω*

volume elements of the domains ω, ωπ, ω*

e

total specific internal energy of the total medium

eπ

specific internal energy of the constituent π

e

specific total energy of the constituent π

e

energy supply to π due to other constituents per volume unit and time unit

f

vectorial function such that x = f(X, t)

f*

vectorial function such that x = f*(X*, t)

F*

vectorial function such that X* = F* (x, t)

f

total external force acting on the mass unit of the total medium

fπ

external force acting on the mass unit of π

η*[s]

total influx vector of entropy through the surface element da* moving at velocity v*

ηππ]

influx vector of Qπ through the surface element daπ moving at vπ

η*ππ]

influx vector of Qπ through the surface element da* moving at v*

i*[s]

total internal volume source of entropy per unit volume of ω*

iππ]

internal source of Qπ per unit volume of the medium per time unit. This source is zero when the quantity Qπ is said to be conservative

i*ππ]

internal volume source of Qπ per unit volume of ω*

irr

irreversible part of a quantity

J*

Jacobian of the transformation f*

k

number of constituents in a set of constituents

{ie352-26}*[s]

total external volume source of entropy per unit volume of ω*

{ie352-27}ππ]

external volume source of Qπ per unit volume of the medium per time unit. If the constituent π is alone, {ie352-29}ππ] comes from the exterior of the domain. In an heterogeneous medium, {ie352-30}ππ] also includes contributions of other constituents

{ie352-31}π*π]

external volume source of Qπ perunit volume of ω*

Lππ or ππ

phenomenological coefficients in the non-equilibrium material relations.

m

external mass source of the heterogeneous medium by time unit.

m

external mass source of constituent π by time unit.

M [resp. M*]

particle in a phenomenological material [resp. virtual] domain.

nπ

external unit normal of the surface element daπ.

n*

external unit normal of the surface element da*

nPπ

power ratio of constituent π

nπ

volume ratio of constituent π

nεπ

rate of strain ratio of constituent π

nσπ

stress ratio of constituent π

nψπ

ratio relative to the macroscopic specific quantity ψ π of constituent π

N

total number of constituents in the heterogeneous medium

\(\hat P_\pi \)

momentum supply vector to π due to other constituents per volume unit and per time unit

Q

any physical quantity contained in a domain ω or ω*

Q

any physical quantity supported by constituent π in a domain ω or ω*

qπ

heat influx vector relative to constituent π through a surface element daπ moving at velocity vπ

q*π.

generalized heat influx vector relative to constituent π through a surface element da* moving at velocity v*

q*

generalized total heat influx vector relative to constituent π through a surface element da* moving at velocity v*

r

total external heat power per mass unit of the total medium and per time unit

rπ

external heat power per mass unit of constituent π and per time unit

s

specific entropy of the total medium

sπ

specific entropy of constituent π

t

time.

Tπ,Tπ, Tπ"

temperature of constituent π

v

velocity vector of an homogeneous medium

Vb

barycentric velocity vector of the total medium

Vπ, Vπ, Vπ"

phenomenological, macroscopic and microscopic velocity vectors of π

v*

reference velocity vector

wb*}

relative barycentric velocity vector in the movement v*

wπ*

relative velocity vector (w π * = vπ − v*)

x

position of a particle in the instantaneous configuration at time t

X*

position in the reference configuration Ω* relative to the movement v*

Greek Symbols

Λππ or Λππ

phenomenological coefficients in the non-equilibrium material relations

μπ, μπ, μ"π

specific chemical potential of constituent π

π or π′

indexes of constituents

ρ

mass density of the heterogeneous medium

ρπ, ρ"π

macroscopic and microscopic mass density of π, with ρπ = 〈ρ"π〉.

<π

phenomenological or apparent mass density of constituent π

σπ

phenomenological stress tensor of constituent π

σπ

macroscopic stress tensor of constituent π(σπ = 〈σ"π〉)

σ"π

microscopic stress tensor of constituent π

σ*π

generalized partial stress tensor of constituent π

σ*

generalized total stress

ϕ

any function of the Euler coordinates (x, t)

ψπ, ψπ, ψ"π

specific value of Qπ

ω

volume domain

ωπ

volume domain following constituent π in its movement vπ

ω*

volume domain following the virtual movement v*

ω*

reference domain, image of ω* in the transformation F* (x, t)

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References

  1. 1.
    Truesdell, C. and Toupin, R.: The classical field theories, in: Flügge (ed.), Encyclopedia of Physics, Volume III/I, Principles of Classical Mechanics and Field Theory, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1960.Google Scholar
  2. 2.
    Biot, M. A.: General theory of three-dimensional consolidation, J. Appl. Phys. 12 (1941), 155–164.Google Scholar
  3. 3.
    Coussy, O., Lassabatère, T.: Comportement mécanique des milieux poreux partiellement saturés. Modélisation du retrait de dessiccation, Ecole de Mécanique des Milieux Poreux, Aussois, 30 mai–3 juin 1994, organisée par le Comité Français de Mécanique des Roches, 1994.Google Scholar
  4. 4.
    Jouanna, P. and Abellan, M.-A.: Lois de conservation généralisées en milieux hétérogènes et relativité du tenseur total des contraintes, flux total de chaleur et flux total d'entropie, C. R. Acad. Sci. Paris, t. 317, Série 11, p. 721–726, 1993.Google Scholar
  5. 5.
    Jouanna, P. and Abellan, M.-A.: Theoretical approach, in: A. Gens, P. Jouanna and B. Schrefler (eds), Modem Issues in Non-saturated Soils, Advanced School of the C.I.S.M., Udine, Italy, September 19–23 1994, Springer-Verlag, Wien, New York, 1995.Google Scholar
  6. 6.
    Abellan, M.-A.: Approche phénoménologique généralisée et modélisation systématique de milieux hétérogènes. Illustration sur un sol non-saturé en évolution thermo-hydro-mécanique et physicochimique. Tome 1 (Concepts) et Tome 2 (Illustration), Thèse de Doctorat, Université Montpellier II, France, 1994.Google Scholar
  7. 7.
    Prigogine, I.: Introduction à la thermodynamique des processus irréversibles, Dunod, Paris, 1968.Google Scholar
  8. 8.
    De Groot, S. R. and Mazur, P.: Non-equilibrium Thermodynamics, North-Holland, Amsterdam, 1969.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • P. Jouanna
    • 1
  • M.-A. Abellan
    • 1
  1. 1.DTMC, ISTEEM, URA CNRS 1767, Université Montpellier IIFrance

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