Modern Issues in Non-Saturated Soils
-
Upload
others
-
View
2
-
Download
0
Embed Size (px)
Citation preview
Mahir Sayir - Zurich Wilhelm Schneider - Wien
The Secretary General of CISM Giovanni Bianchi - Milan
Executive Editor Carlo Tasso - Udine
The series presents lecture notes, monographs, edited works and
proceedings in the field of Mechanics, Engineering, Computer
Science
and Applied Mathematics. Purpose of the series is to make known in
the international scientific and technical community results
obtained in some of the activities
organized by CISM, the International Centre for Mechanical
Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECTURES - No. 357
.~
~~~ ~ ~
EDlTEDBY
UNIVERSITY OF PADUA
SPRINGER-VERLAG WIEN GMBH
Le spese di stampa di questo volume sono in parte coperte da
contributi deI Consiglio Nazionale delle Ricerche.
This volume contains 118 illustrations
This work is subject to copyright.
All rights are reserved,
whether the whole or part of the material is concemed
specifically those of translation, reprinting, re-use of
illustrations,
broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
© 1995 by Springer-Verlag Wien
Original1y published by Springer-Verlag Wien New York in 1995
In order to make this volume available as economically and as
rapidly as possible the authors' typescripts have been
reproduced in their original forms. This method unfortunately
has its typographical limitations but it is hoped that they in
no
way distract the reader.
DOI 10.1007/978-3-7091-2692-9
PREFACE
This monograph is a revised version oJthe notesJor the Advanced
School lectures given at the International Centre Jor Mechanical
Sciences in Udine, September 19-23, 1994. The aim oJthe course was
to provide a review oJ some oJ the most significant achievements in
research on non-saturated soUs, a material oJ apre-eminent
importance in civU engineering, agriculture and environmental
engineering.
A phenomenological point oJ view based on Jictitious continuity oJ
the different constituents is adopted throughout this work, even if
reJerences to lower scale considerations may be made in different
parts oJ the text. The different steps oJ the modelling oJ
non-saturated soUs are reviewed in Chapters 1 to 4. Chapters 5 to
11 are devoted to the analysis oJ a number oJ case studies.
In the introductory Chapter 1, non-saturated soUs are considered in
the broader context oJ heterogeneous media. Balance, equilibrium
and non equilibrium equations are presented using a generalized
approach and in the light oJ the most recent achievements in
irreversible thermodynamics. This approach covers all possible
coupled phenomena in a single coherent model but leads to a complex
set oJ equations. In conclusion, an automatic procedure is proposed
Jor overcoming this complexity.
The behaviour laws proposed in Chapter 2 are based on the concept
oJ yield surJaces as classically used in saturated soUs. These laws
are Jirst extended to the case oJ non-saturated soUs and then to
non-isothermal conditions. The effects oJ non-saturation and
temperature are incorporated in a generalized Cam-clay model, the
simplest member oJ the Jamily oJ critical state Jormulations. Some
selected comparisons between model predictions and experimental
results are presented.
Examples oJ experimental studies in the laboratory Jor studying
coupled phenomena are presented in Chapter 3.
Numerical methods Jor solving complex thermo-hydro-mechanical
coupling equations are presented in Chapter 4. Basic concepts and
different strategies Jor discretization are examined and different
numerical techniques are described. Fundamental problems oJ
consistency, stability and convergence in linear and non-linear
situations are discussed at length using a particular set oJ
governing equations and a finite element method with estimation oJ
numerical errors.
Finally, a broad review of applications covers domains such as
water resources, road engineering, heat storage and consolidation,
embankment dams, seismic behaviour and strain localisation in
partially saturated soil, radioactive disposal and natural thermal
reservoir.
Acknowledgements
The presentation of this edited series of lectures is the fruit of
team work by the various authors, first within G.R.E. C. O.
"Geomateriaux europeen" and then in the European network
A.L.E.R.T., research institutions founded and managed at the
instigation of Professor Darve, aided efficiently by his advisers
Messrs Hicher and Reynouard and with the institutional backing from
the C.N.R.S.
It would not have been possible to complete this joint research
work without the facilities provided by the International Centre
for Mechanical Sciences in Udinefor both the organisation ofthe
lectures and the publication of the monograph. The editors warmly
thank all the organisers of the Centre, and especially the Rector,
Professor Kaliszky, for their competence and hospitality.
Finally the constant help of M-A. Abellan has been deeply
appreciated throughout the course and during preparation of the
final manuscript.
For all the authors, A. Gens
P.Jouanna B. A. Schrefler
CONTENTS
Page
Preface
Chapter 1 Generalized Approach to Heterogeneous Media by P. Jouanna
and M-A. Abellan
...................................................................
1
Chapter2 Constitutive Laws by A. Gens
..................................................................................................
129
Chapter 3 Experimental Studies by D. Bovet, P. Jouanna, E. Recordon
and C. Saix ................................... 159
Chapter4 Numerical Solutions of Thermo-Hydro-Mechanical Problems by
B.A. Schrefler and L. Simoni
.................................................................
213
Chapter 5 Forced and Natural Convection by D. Bovet and E. Recordon
....................................................................
277
Chapter 6 Large Scale Road Test by E. Recordon
..........................................................................................
283
Chapter7 Non Saturated Consolidation Under Thermo-Hydro-Mechanical
Actions. An in Situ Heat Storage Facility in a Clayey Silt by M-A.
Abellan, P. Jouanna and C. Saix
................................................. .301
Chapter 8 Construction and Impoundement of an Earthdam. Application
ofthe Coupled Flow-Deformation Analysis of Unsaturated Soils by E.
Alonso and F. Batlle
.........................................................................
357
Chapter 9 Large Strain Static and Dynamic Hydro-Mechanical Analysis
of Porous Media by E.A. Meroi and B.A. Schrefler
...............................................................
397
Chapter 10 Importance of Boundary Conditions. A Radioactive Storage
Case by L. Simoni
...............................................................................................
449
Chapter 11 Regional Problems: Vertically Averaged Modelling. Abano
Thermal Problem by L. Simoni
...............................................................................
................. 461
Annex Some Phenomenological Models of Polyphasic Soils by D. Bovet
.................................................................................................
475
Chapter 1
ABSTRACT
This first chapter is devoted to the description of a coherent and
synthetic generalized phenomenological approach to heterogeneous
media, which overcomes former difficulties and incoherences.
Generalized balance equations are written following an arbitrary
virtual movement, independent of the movement of the matter.
Classical state relations are extended to pseudo-state relations to
take hysteretic phenomena into account. Non-equilibrium relations
deduced from the analysis of the entropy source include
physico-chemical reactions and are revised according to the
generalized approach. Linking phenomenological variables and
physical variables is performed by complementary balance relations.
Finally a systematic guide is presented for handling such a complex
modelling, with a rationallisting of variables and associated
relations. It covers all couplings between complex thermo-hydroÂ
mechanical and physico-chemical phenomena as encountered in modern
engineering practice.
2 P. Jouanna and M-A. Abellan
1.0 INTRODUCTION
Situation of Chapter 1
This first chapter is a guide for modelling complex situations as
encountered in modem engineering practice.
_The physics of complex media covers different chemical species
within a given material domain submitted to coupled stresses
related to physico-chemical, mechanical and heat exchanges. For
instance, rock, water, air, and various other chemical species like
oxides, ~1ts, etc. exist in a soil. These chemical species can
occur in different phases. The most common example is water which
can exist as ice, liquid or vapour. A constituent will be defined
as a chemical species in a given phase. For example, liquid water
and-water vapour are considered as two different
constituents.
The study of ilcterogeneous media can be envisaged at different
scales from atomic to macroscopic. Distinctions can be made between
three levels of modelling:
• the first level consists of describing phenomena as they appear
under a microscope. Up to now, this description is essentially
visual and modelling at this scale is not a common practice. This
level is beyond the scope of the present study;
• a second point of view consists of awarding average physical
properties to each constituent, such as average specific mass,
partial press ure, etc. within the volume occupied by each
constituent in an elementary representative volume element. Such
average physical {>roperties will be assumed to be known through
the use of techniques such as homogenization, averaging,
ete.;
• finaIly, the phenomenological point of view assumes the matter of
each constituent to be extended to the total volume occupied by the
heterogeneous medium, leading to a fictitious continuum where
cIassic mathematical tools can be used. The phenomenological
approach, discussed in the present chapter, is used in most
engineering applications, as shown in Chapters 5 to 12.
State of art and questions
(i) The first comprehensive study of heterogeneous media was
performed by Truesdell & Toupin [1960] as an extension of the
cIassical field theory developed for homogeneous media. In this
study, the kinematics of heterogeneous media was based on the
barycentrie velocity used as a referenee and thus was essentially
adapted to mixtures where different eonstituents have velocities of
the same order of magnitude. When the baryeentrie velocity looses
its physical meaning, for example in the presence of fluids and one
porous solid, the movement of the fluids can be referred to the
solid. However, new difficulties arise when different solids are
present or when the solid chosen as a reference disappears due to
some physico-chemical reaction, etc.
(ii) Moreover, incoherences appear in the definition of total
stresses or fluxes, when constituents have different velocities.
These incoherences can be hidden by different misleading
procedures, such as working on the total medium only or defining
the so-called "inner" parts of stresses or fluxes ignoring
diffusion terms.
Generalized Approach to Heterogeneous Media 3
(iii) A complete, rationallisting of material relations and
associated variables is rarely clear in complex
thermo-hydro-mechanical and physico-chemical modelling. Moreover
the case of hysteretic phenomena remains a difficulty.
(iv) Another important problem consists of establishing a rigorous
link between the average physical variables and the
phenomenological variables.
(v) Finally the complexity of heterogeneous media induces
difficulties for managing computation with a great number of
equations and variables and for tackling the related problems of
consistency, stability and convergence.
Propositions
• Section 1.1: Kinematics of heterogeneous media
To overcome the difficulties above in (i), a totally arbitrary
velocity field v* is proposed as the reference movement for
studying the kinematics of heterogeneous media.
• Section 1.2: Generalized expression of balance equations
The choice of a virtual reference velocity field v*, apriori
independent of the movement of the matter, leads to rewriting
balance equations with a common reference for all constituents.
This generalized theory of heterogeneous media overcomes
inconsistencies the mentioned in (ii) above.
• Section 1.3: State and non-equilibrium relations
State and non-equilibrium relations are given according to a
rational procedure to overcome the difficulties mentioned in (iii).
Different possible formulations of state relations are presented
and extended to pseudo-state relations to consider hysteretic
phenomena. Non equilibrium relations are then deduced from the
entropy source and extended to physico chemical reactions using
generalized formalism. Finally the consequences of the introduction
of a virtual reference velocity field v* are investigated.
• Section ].4: The link between phenomenological and average
physical variables
A rigorous link is established between phenomenological and average
physical variables in order to overcome difficulties mentioned in
(iv). It leads to more physical variables than phenomenological
variables. An extra set of balance relations is presented to insure
the complete determination of all variables.
• Section 1.5: Modelling variables and equations
To face the complexity of modelling as emphasized in (v), a
rationallisting of variables and relations is proposed. A
systematic procedure for obtaining a complete set of elementary
equations is given in the most general situation of an
heterogeneous medium. Tbe set to be considered as governing
equations is then discussed. In conclusion, a systematic guide is
given for developing modelling in heterogeneous media.
4 P. Jouanna and M-A. Abellan
1.1 MOVEMENT
Strict1y speaking, two constituents never coexist s at the same
physical point. However in phenomenological fiction, the matter of
any constituent is assumed to be extended over the whole volume
occupied by the heterogeneous medium. Thus, the different extended
constituents are assumed to coexist at the same geQmetrical
point.
Within a phase, any constituent is considered to be physically
present at any point and the phenomenological velocity field of
each constituent coincides with the physical velocity field. When
different phases are present, the velocity field of each
constituent has to be exteöded in the region where it does not
physically exist and the phenomenological velocity field appears as
a mathematical extension of the physical velocity field to the
whole space.
As mentioned in the introduction, the only rational solution to
avoid fundamental inconsistencies consists in choosing an arbitrary
reference velocity field for studying the kinematics of an
heterogeneous medium in order to preserve symmetry between the
constituents. The generalized approach to heterogeneous media
proposed in Chapter 1 relies on a totally arbitrary velocity field
- referred to as the virtual velocity field v* - chosen as the
reference movement.
The present section recalls the fundamentals of kinematics in
continuum mechanics applied to heterogeneous media with emphasis
the role of the virtual velocity field v*.
1.1.1 Definitions of movement
• Movement defined in Lagrange variables
One material particle M of a material domain is identified by its
position X at time to. The Lagrange description of the movement is
known if the geometrical vector x occupied by M at time t is given
by a vectorial function f :
(1.1) x = f(X,t)
Independent variables (X,t) are referred as material or Lagrange
variables. The material displacement U(X,t) of a particle M is
defined in Lagrange variables by :
(1.2) x - X = U(X,t) = f(X,t) - X
• Movement defined in Euler variables
One material particle M can also be identified by the set of
independent variables (x,t) called Euler variables. The Euler
description of the movement gives the position X occupied by the
particle in the reference configuration in function of (x,t) by a
vectorial function F :
(1.3) X = F(x,t)
Generalized Approach to Heterogeneous Media 5
The displacement u(x,t) of a particle M is defmed in Euler
variables by
(1.4) x - X = u(x,t) = x - F(x,t)
• Relations between Lagrange and Euler descriptions
Lagrange and Euler descriptions are complementary and links between
the two points of view can be established assurning functions f and
F to be continuous and assurning the existence of inverse functions
f-1 and F-l such that :
(1.5) F= f-1 and f = F-l
• Material and virtual movements
One can define a virtual movement in Lagrange variables by :
(1.1a) x =f*(X*,t)
where the virtual particle M* is identified at time t by its
position X* in a virtual reference configuration. Conversely, in
Euler variables:
(1.3a) X* = F*(x,t)
In the following, the concept of virtual movement, which covers the
material movement as a special case, is systematically used.
1.1.2 Functions of the m6vement and their derivatives
A Functions of the movement
Quantities linked to material particles M in the physical space can
be represented by a function <p(x,t) in Euler coordinates in the
instantaneous configuration. The same quantity can be considered as
a function «II*(X*,t) in Lagrange coordinates at point X* of the
reference configuration, as obtained by the inverse function
F*(x,t) :
(1.6) <p(x,t) = <p(f*(X*,t),t) == «II*(X*,t) "i/ t
However, these equivalent functions cp and «11* do not possess the
same partial derivatives.
B Partial derivatives of a function «II*(X* ,t)
• Differential of a function «II*(X*,t)
A quantity q being representedby a function «II*(X*,t) in Lagrange
variables, its variation &! due to some arbitrary variation
(dX*, dt) around (X*, t) can be written as :
(1.7) &! == B«II*(X*, t, dX*, dt) = BclI*(X*,t) I x* dt +
B«II*(X*,t) I t dX*
6 P. Jouanna and M-A. Abellan
The variation Oq can be estimated along any space-time path, Le.
for any arbitrary set (X*, t, dX*, dt). However, special paths can
be defined at X* or t constant. In particular, Lagrange variables
make it easy to follow the variations of q along the movement of
one
-particle identified by a given position X * in the reference
configuration. The notation
o<l>*(X*,t) I X* [resp. o<l>*(X*,t) I t] refers to the
variation of <l>*(X*,t) at constant X*
[resp. at t constant].
If ö<l>*(X* ,t) I X* and ö<l>*(X* ,t) I t are functions
of (X* ,t), continuous at all necessary
orders, which do not depend on the values or signs of dX* or dt,
they are called partial derivatives and Oq is said to be a total
differential, written dq and defined by :
(1.8) dq = d<l>~~:,t) dX* + d<l>*~~*,t) dt
Total differentials are the key tool in classical thermodynamics,
which excludes hysteretic
phenomena where o<l>*(X*,t) I X* and o<l>*(X*,t) I t
are functions depending on the signs
of dX* or dt. Hereafter, assumptions required for using total
differentials are assumed to be valid, except when explicitly
mentioned.
• Gradient of a function <l>*(x* ,t)
At a given time t, between two particles a distance of dX* from X*,
öq is given by :
(1.9) öq = öq(X*, t, dX*, dt = 0) = Q<ll*(X*,t) I t dX*
Tbe partial derivative of a function <l>*(X* ,t) with respect
to X*, for a given value of t, is the gradient of q in the Lagrange
description of the movement and is noted Grad <l>*(X* ,t)
:
(1.10) :-'Xd * (<l>*(X*,t» I == Grad <l>*(X*,t) o t =
constant
(i) Une element
At a given time t, the infinitesimal vector dx in the instantaneous
configuration corresponding to the infinitesimal vector dX* in the
reference configuration around point
X* is given by application of (1.9) with the definition (1.10), for
<p(x,t) "" x and <l>*(X*,t) "" f*(X*,t) :
(1.11) dx = G.r..wI. f*(X* ,t) dX*
Generalized Approach to Heterogeneous Media 7
(ii) Volume element
The detenninant of !i.Dulf*(X*,t) in Lagrange variables gives the
ratio between a volume dro in the instantaneous configuration with
respect to its image dn* in the reference configuration around
particle X*.
(1.12) dro
detGradf*(X*,t) = dn*==J*(X*,t)
J*(X* ,t) is called the Jacobian of the function f*(X* ,t). 1t is
an invariant and does not depend on the frame of reference.
(iii) Surface element
The surface element dA * in the reference configuration built on
vectors dXa *, dXb * is
given by the vectorial product (dXa* A dXb*). The surface element
da = (dxa A dxb) built in the instantaneous configuration on dxa,
dxb corresponding to dXa*, dXb* by the function f*(X* ,t) is given
by :
(1.13) da = J*(X*,t)[!irwlTf*(X*,t)]-l dA*
with G.md.T f* : transposed gradient of G.md. f*.
(iv) Strain tensor in Lagrange variables or Green-Lagrange
tensor
The Green-Lagrange strain tensor is defined by :
(1.14) .G*(X*,t) = ~ [(.GwlTf* .Gwlf*)-1]
with I: unit tensor
(v) Green-Lagrange tensor versus the dis placement expressed in
Lagrange variables
The Green-Lagrange tensor can be expressed in function ofthe
displacement U*(X*,t) :
(1.15) 2 .G*(X*,t) == [.Gwl U*(X*,t)]T +.GouI. U*(X*,t)]
+ [Grad U*(X* ,t)]T [!i.Dul U*(X* ,t)]
-Time derivative of a function ~*(X*,t)
The variation of the quantity q supported by the particle M
identified by X*, following this particle M in its movement, is
given by :
(1.16) Bq = öq(X*, t, dX*= 0, dt) = ö~*(X*,t) I X* dt
8 P. Jouanna and M-A. Abellan
The partial derivative of cI>*(X* ,t) with respect to t, when it
exists, is noted :
(1.17) ~ (cI>*(X*,t» I X* = ddt (cI>*(X*,t» or ~*(X* ,t) U~ =
constant
When a eommon frame is used as assumed here, this derivative is
equal to the material derivative as defined in a general situation
[Truesdell & Toupin, § 72, p. 337, footnote 1]. However the
term "material" derivative is not appropriate when the movement is
virtual and for this reason the denomination derivative following
the movement will be preferred. The derivative following the
movement is extremely important. In particular if the quantity q is
assumed to be the position x of the particle X* at time t, the
first order and second order derivatives following the movement of
the function f*(X* ,t) lead to the definition of the
velocity vector V* and the aeceleration vector r* of the particle
X* : a d •
(1.18) V*(X*,t) == at (f*(X*,t» I X* == dt (f*(X*,t» ==
f*(X*,t)
a2 d2 •• (1.19) r*(x*,t) == at2 (f*(X*,t» I x* == dt2 (f*(X*,t» ==
f*(X*,t)
C Partial derivatives of a function <p(x,t)
• Differential of a funetion <p(x,t)
The variation &J. due to some arbitrary variation (dx, dt) at
point (x,t) is given by :
(1.20) Oq = Oq(x, t, dx, dt) = o<p(x,t) I x dt + o<p(x,t) I t
dx
The notation o<p(x,t) I x [resp. o<p(x,t) I t] refers to the
variation of <p(x,t) at x constant
[resp. at t constant]. If Oq is assumed to be a total differential,
o~(x,t) I xand o<p(x,t) I t are
partial derivatives: a<p(x,t) a<p(x,t)
(1.21) Bq = dq(x, t, dx, dt) = ax dx+ at dt
The variation Bq of q represented by <p(x,t) can be estimated
along any thermodynamical path for any arbitrary variations
(dx,dt). One path of special interest consists in observing
variations Bq with time at a given geometrie al point x. -
• Gradient of a function <p(x,t)
At a given time t, between two material particles at a distance of
dx from position x in the instantaneous configuration, &J. is
given by :
(1.22) Bq = öq(x, t, dx, dt = 0) = o<p(x,t) I t dx
Generalized Approach to Heterogeneous Media 9
If the partial derivative of <p(x,t) with respeet to x exists,
it is noted :
(1.23) a
(i) Line element
At a given time t, the infinitesimal vector dX* in the referenee
eonfiguration around point X* eorresponding to the infinitesimal
veetor dx in the instantaneous configuration is given
by application of (1.22) and (1.23) for q :; X* and <p(x,t) :;
F*(x,t) :
(1.24) dX* = 2nld. F*(x,t) dx
(ii) Volurne element
The determinant of &rWl F*(x,t) gives the ratio between a
volume dn* in the referenee
configuration with respect to its volume dro in the instantaneous
configuration :
(1.25) det ~ F*(x,t) = ~~* == j(x,t) = J*C~*,t)
(iii) Surface element
(1.26) dA * = det u.ruI. F*(x,t) [u.ruI.T F*(x,t)]-l da
(iv) Strain tensor in Euler variables or Euler-Almansi tensor
The Euler-Almansi strain tensor is defined by :
(1.27) 1
il(X,t) = "2 [1- u.rutTF*(x,t) 2nld. F*(x,t)]
(v) Strain tensor in function 01 the displacement expressed in
Euler variables
The Euler-Almansi strain tensor ean be expressed in funetion of the
displacement u(x,t) :
(1.28) 2 il(x,t) = [wut u(x,t)]T + wut u(x,t)] - [wuI. u(x,t)]T
[wuI. u(x,t)]
• Time derivative of a function <p(x,t)
The variation Bq of q represented by <p(x,t) at a given
geometrie al point x is given by :
(1.29) Bq = oq(x, t, dx= 0, dt) = o<p(x,t) I x dt
10 P. Jouanna and M-A. Abellan
If the partial derivative of cp(x,t) with respect to t exists, it
is noted :
(1.30) a a at cp(x,t) I x = constant == at cp(x,t)
(i) Velocity and acceleration
Partial derivative (1.30), for q = x, does not lead to the
expression of the velocity V* as
defined by (1.18). However it is possible to express the velocity
V* [resp. acceleration i*]
by a function v(x,t) [resp. y(x,t)] defined by :
(1.31) V*(X*,t) == V*(F*(x,t),t) == v(x,t)
(1.32) i*(X* ,t) == i*(F*(x,t),t) == ,,(x,t)
(ii) Rate 0/ strain
(1.33) 1
D Derivatives of cp(x,t) with respect to Lagrange variables
Using (l.1a), any function cp(x,t) can be considered as a function
cp(f*(X*,t),t).
• Gradient with respect to X* of a function cp(x,t)
The derivative of the function cp(f*(X*,t),t) at t constant, with
respect to X*, can be
defined with the help of definition (1.10) as Grad cp(x,t) :
(1.34) a
Grad cp(x,t) = Grad cp(f*(X*,t),t) == ax* cp(f*(X*,t),t) I t =
constant
• Derivative of a function cp(x,t) following the rnovernent
The differential dq of a quantity q represented by cp(x,t) is given
by :
d d ( ) acp(x,t) d acp(x,t) d q = cp x,t = ~ t + dX x
The derivative of the function cp(f*(X* ,t),t) at X* constant is
obtained following particle M in its movement. The infinitesimal
vector dx is in that case equal to v* dt and the differential
expression above becomes :
Generalized Approach to Heterogeneous Media 11
d () dq>(x,t) d dq>(x,t) * d q> x,t I X = dt t + dx v
t
Thus the derivative of q>(x,t) following the movement v* is
noted dV*dt(X,t) and defined as :
(1.35) dv*q>(x,t) dq>(x,t) [ d ( )] *( ) dt = dt + gra q>
x,t v x,t
The tenn [grad<p(x,t)] v*(x,t) is called the convection
tenn.
E Derivatives of <I>*(X*,t) with respect to Euler
variables
Using relation (1.3a), a function <I>*(X*,t) can b:!
considered as depending on (x,t) and written
<I>*(F*(x,t),t).
• Gradient with respect to x of a function <I>*(X*,t)
The gradient with respect to x of <I>*(X*,t) can be defined
using (1.23) as follows :
(1.36) a
grad <I>*(F*(x,t),t) == Ha <I>*(F*(x,t),t» I x t =
constant
• Time derivative of a function <I>*(X*,t)
The time derivative of <1>* (X*,t) at x = constant can be
eXPressed using (1.30)
(1.37) a<l>*(x* ,t) = a (<I>*(F*(x,t),t» dt I x =
constant dt I x = constant
(i) Relation between gradient operators in Lagrange aniJ Euler
coordinates
Let q>(x,t) and <I>*(X* ,t) be the equivalent expressioM
of a given quantity in Euler and Lagrange coordinates in a
mathematical transfonnation f*(X*,t). Gradients defined by (1.10)
and (1.23) are linked by the following relation:
(1.38) Grad <I>*(X*,t) = grad q>(x,t)!irwl f*(X*,t)
(ii) Divergence operator 0/ the ve/ocity
(1.39) j*(X*,t) d· *(f*(X*» J*(X*,t) = IV v ,t ,t [Different from
Div V*(X*,t)]
(iii) Expression o/the divergence 0/ a product
(1.40) div q>v* = q> div v* + [gradq>] . v*
12 P. Jouanna and M-A. Abellan
1.1.3 Integrals
A Different possible integrations
At a given time t, the quantity q ean represent the density of some
measurable, i.e.
extensive, quantity Q(t) eontained in a given domain 00* oceupied
by the matter in the instantaneous eonfiguration. Q(t) is obtained
by integrating q represented by a funetion cp(x,t) over the domain
00*. The domain 00* is generally a volume ; when this domain is
a
surfaee or a line, the notation 00* will be replaeed by a* or
1*.
(1.41) Q(t) = J cp(x,t) doo*
00*
This integration ean be also performed in the refe;renee
eonfiguration on a domain n* which is the image of 00*. aeeording
to the inverse virtual transformation f*(X*,t)-l. The above
integral (1.41) beeomes after (1.6) and (1.12) :
(1.42) Q(t) = J !p(f*(X*,t),t) J*(X*,t) dn* = J ~*(X*,t) J*(X*,t)
dn*
The advantage of (1.41) is to operate in the instantaneous
eonfiguration, whieh has a
physieal meaning ; however the domain 00* is moving. The advantage
of (1.42) is to operate on a fixed domain n*, defined in the
referenee eonfiguration linked to v*. Both representations have
their own advantages.
B Time derivative of an integral in a fixed domain
If the integration domain 00* is a fixed domain roo , integration
of the quantity q'" cp(x,t), at time t, is given by :
(1.43) , Q(t) = J cp(x,t)dooo
000
The time derivative of Q(t) ean be readily transformed into an
integral of a time derivative, because the domain of integration is
fixed.
(1.44) dr Q(t) = ~ J cp(x,t) droo = J ~ cp(x,t) droo 000 000
The notation ddt' is used to refer to the velocity field of the
integration domain, here
v(x,t) = O.
C Time derivative of an integral in a moving domain
• Volume integrals and their time derivatives
If the integration domain ro* is moving in a velocity field v*, the
integral Q( t) is defined as previously in the instantaneous
configuration. Using (1.42), this integration can be transformed
into an integration on the fixed domain n* :
(1.45) Q(t) = f cp(x,t) doo* = f cp[f*(X* ,t),t] J*(X* ,t) dn* 00*
n*
The time derivative of Q(t) is obtained using the same role as for
expression (1.43), the fixed domain being now n* :
(1.46) dV* Q(t) = f i;[ddt cp[f*(X*,t),t] J*(X*,t)] dn* dt n
*
dv* Q(t) = f (J*(X* t) ~n[f*(X* t) t)] + J*(X* t) ocp[f*(X* ,t),t]
of*(X* ,t) dt n * '01"1'" , 'of*(X* ,t) ot
+ cp[f*(X*,t),t] ! J*(X*,t)} dn*
dv* f (1.47) dtQ(t) = (ocp~,t) + [grad cp(x,t)] v*(x,t) + cp(x,t)
div v*(x,t) }doo*
00*
(1.47a) dv* Q f (dcp(x,t) . * * dt (t) = dt + cp(x,t) dlV v (x,t)}
doo
00*
or using (1.35) and (1.40), relation (1.47a) gives the following
fundamental formula :
(1.48) dch* Q(t) = f (ocp(X,t) . at + dlV [cp(x,t) v*(x,t)]}
doo*
00*
• Surface integrals
In a virtual movement, ifthe quantity q is a surfaee density, the
integral Q(t) is defined in the instantaneous configuration on the
surfaee a* in the velocity field v*. The differential surfaee
element is the veetor da* = n* da*, n* being the unit normal and
da* the geometrie al surfaee element. The integration ean also be
performed in the referenee configuration on domain A* with dA* = N*
dA*, N* being the external unit veetor of surfaee A* at point X*
and time t. Integrant transformation uses identity (1.6) and
relation (1.13) :
(1.49) Q(t) = f cp(x,t)da* a*
= f cp[f*(X* ,t),t]J*(X* ,t)L!i.rwlT' f*(X*,t)]-1 N*dA * A*
The time derivative of Q(t), following movement v* ean be obtained
by derivation of an integral estimated on the fixed domain A *
:
. (1.50) '!h* Q(t) = ~t f cp[f*(X*,t),t] J*(X*,t)[!irwl.T !*(X*
,t)]-1 N*dA * A*
= f ~t (cp[f*(X*,t),t] J*(X*,t)L!i.rwlT f*(X*,t)]-1 }N*dA* A*
Mter eomputation, the derivative is expressed in Euler variables as
folIows:
dv* f (1.51) CitQ(t) = a*
{acp~,t) + div[ cp(x,t)v*(x,t)]-cp(x,t) &DUlT v*(x,t)}
n*(x,t)da*
• Line integrals
If the quantity q is a linear density, the integral Q(t) is defined
in the instantaneous configuration on the physicallinear domain 1*
in the velocity field v*. The differentialline element is the
veetor d1*= n* dl*, n* being the unit veetor of line 1* and dl* is
the length of the line element. The integration ean also be
performed in the referenee eonfiguration on domain L* with dL* = N*
dL*, N* being the unit veetor ofline L* at point X* and time t.
Integrant transformation takes into aecount identity (1.6) and
relation (1.11) :
(1.52) Q(t) = f <p(x,t) dl* = f 1* L *
<p[f*(X* ,t),t] !iJ:wI. f*(X* ,t) dL*
Generalized Approach to Heterogeneous Media 15
The time derivative of Q(t) following the movement v*(x,t) can be
obtained by the rule for the derivation of an integral estimated on
the fixed domain L* :
(1.53)
(1.54)
(1.55)
<!h* Q(t) = :t f <p[f*(X*,t),t] .Gradf*(X*,t) N*dL* L*
dv* Q(t) = f ddt {<p[f*(X* ,t),t] .Grad f*(X* ,t)} N*dL * dt L
*
dV*Q(t) = f {(ddt <p[f*(X*,t),t]) .GDUl f*(X* ,t) dt L*
+ <p[f*(X* ,t),t] :t.GDUl f*(X* ,t)} N*dL *
After computation, this derivative is expressed in Euler variables
by :
dv* f a (1.56) Tl Q(t)=
{dt<I>(x,t)+[grad<p(x,t)]v*(x,t)+<p(x,t) ~v*(x,t)}
n*(x,t) dl* 1*
D Relations between integrals on different varieties
The general Poincare relation states that integration of Bq in a
domain qt of a given
variety is equivalent to the integration of q on the boundary Bqt
of the domain qt.
(1.57) < Bq, '" > = < q, B", >
The explicit expression of Poincare's relation (1.57) applied at
time t for a volume gives :
(1.58) J 00*
div<p(x,t) doo* = J <p(x,t) n*(x,t) da* a*
The explicit expression of Poincare's relation applied at time t
for a surface gives :
(1.59) f rot <p(x,t) n*(x,t) da* = f <p(x,t) dl* a * 1*
16 P. Jouanna and M-A. Abellan
1.2 BALANCE RELATIONS
The balance relations express the variation of a given extensive
quantity Q stored within a domain 00 in relation with the external
contributions supplied through the surface a of the
domain 00 or supplied direct1y to points within the domain oo. In
thermo-hydro-mechanics, these balance relations are the following
fundamental principles : mass balance (mass conservation), momentum
balance (fundamental equation of mechanics), energy balance (first
principle of thermodynamics) and entropy balance (second principle
of thermodynamics).
For an homogeneous medium, the domain 00 used for writing these
balance relations is classically a domain following the medium in
its movement v. Such a point ofview presents irremediable
difficulties in the case of an heterogeneous medium where the
quantity Q can be supported by different constituents 1t, each of
them following its own movement with its own velocity field V1t
.
To overcome this basic difficulty, which leads to inconsistencies
in the classical field theory of heterogeneous media, it is
proposed here to write balance relations in a generalized
form following a non-material domain 00* moving in an arbitrary
virtual velocity field v*.
This generalized formalism make it possible to derive all possible
cases. The classical theory of homogeneous media is obtained using
v* = v. For an heterogeneous medium, any velocity field v* can be
chosen. In particular, this velocity v* can be taken equal to the
velocity V1t of one special constituent (as in the theory of porous
media where v* equals the velocity of the solid), to the
barycentric velocity (mixture theory) or any other special velocity
field.
Depending on this choice, different possibilities can be considered
for the domain 00* :
• The domain 00* may be assumed to move at velocity v.
• The domain 00* == mo may be considered as fixed in the common
frame.
• The domain 00* == COx may be assumed to move with the constituent
1t, at velocity V1t.
• The domain 00* == 00* may be assumed 10 move along any virtual
movement v* adapted to the case 10 be treated.
Generalized Approach to Heterogeneous Media
1.2.1 Synthetic balance relation for any extensive quantity Q
A Integral form of a synthetic balance relation
Al For one constituent x
(a) Euler point of view
• Integral balance relation of a quantity Q1t along the movement of
x
17
Quantity Qx relative to constituent x defined by (1.41) as the
integral of the density qx
can be expressed using the apparent mass density Px of constituent
x and the specific value
'l'x of Qx , Le. the quantity of Qx supported by the unit mass of
constituent x at point (x,t) :
(1.60)
rox rox The c1assical balance relation of a quantity Qx. following
the movement of the material
domain COn: in the velocity field Vx of the constituent x, is given
by :
(1.61)
= - f hx['I'x](x,t)nx(x,t)dax + f ~x['I'x](x,t)dCOn: + f
i-x['I'x](x,t)dCOn:
ax rox rox
hx['I'x](x,t) : influx vector of Qx through the surface element dax
movin~ at Vx
~['I'xl(x,t): extemal volume source of Qx per unit volume of the
medium per second
If the constituent x is alone, ~['I'xl comes from the exterior'of
the domain. In an
heterogeneous medium, ~['I'x] also includes contributions of()lher
constituents.
i-x['I'x](x,t): internal source of Qx per unit volume of the medium
per second. This source is equal to zero when the quantity Qx is
said 10 be conservative.
nx(x,t): external unit normal of the surface element dax at
(x,t).
Using relation (1.48), the left-hand side member of (1.61) can be
expressed as :
(1.62)
18 P. Jouanna and M-A. Abellan
• Generalized integral balance relation of Qx along a virtual
movement v*
The above expression derived following the movement of the
constituent x is not suitable for totalling the contributions of
the different constituents with different velocities vx. To
overcome this difficulty, the central idea proposed here consists
of writing the balance
relation following a virtual domain 00* moving in a virtual
velocity field v*. Such a generalized integral balance relation of
Qx is expressed by :
-z:..- ~
00*
=- f a*
h.x*['I'x](x,t)n*(x,t)da* + f ~x*['I'x](x,t)doo* + f
i.n*['I'x](x,t)doo*
00* 00*
h.x*['I'x](X,t): influx vector of Qx through the surface element
da* moving at v*.
~*['I'x](x,t) : external volume source of Qx per unit volume of
00*.
i.x*['I'x](x,t) : intern al volume source of Qx per unit volume of
00*. n*(x,t) : external unit normal of the surface element da* at
(x,t).
U sing (l.48) this generalized balance relation can be written
under the basic expression :
(1.64) f (~[Px(X,t)'I'x(X,t)] +
div[Px(x,t)'I'x(x,t)v*(x,t)]}doo*
00*
= - f h.x*['I'x](x,t)n*(x,t)da* + f ~*['I'x](x,t)doo* + f
i.x*('I'x)(x,t)doo* a* 00* 00* .
• Comparison of the integral balance relation of a quantity Qx
along the movement of constituent x and along a virtual movement
v*
Equation (1.62) can be written :
f (~Px(X,t)'I'x(X,t)] + div[Px(x,t)'I'x(x,t)vx(x,t)] }doox
OOx
Generalized Approach to Heterogeneous Media 19
As volumes COrc and ro* are identical at time t, the preceding
volume integrals on COrc can
also be written on ro* , leading to :
f {~ [Px(x,t)'I'x(x,t)] + div[Px(x,t)'I'x(x,t)vx(x,t)]dro*
ro*
~d'l'x](x,t)dro* + f in;['I'x](x,t)dro*
ro*
f {gt [Px(x,t)'I'x(x,t)] + div [Px(x,t)'I'x(x,t)v*(x,t)]}dro*
ro*
= f div {Px(x,t)'I'x(x,t)[ v*(x,t) - vx(x,t)]} dro* - f div
hx['I'x](x,t)dro*
ro* ro*
ro* ro*
Comparison between the second members of the above expression and
(1.64) leads to :
(1.65) (a) hx*['I'x](x,t) = hx['I'x](x,t) +
Px(x,t)'I'x(x,t)(vx(x,t) - v*(x,t»
or
with the following definition of the relative velocity wx*(x,t)
:
(1.66) wx*(x,t) == vx(x,t) - v*(x,t)
20 P. Jouanna and M-A. Abellan
(b) Lagrange point of view
• Integral balance relation of a quantity Qx on the reference
domain Ox
Both members of relation (1.62) written in Euler variables along
the movement of constituent x can be transformed into Lagrange
variables. When the left-hand member is transformed using (1.47),
(1.1), (1.12) and (1.39), the ftrst member of (1.62) becomes
:
J (dehx [Px(x,t)"'x(x,t)] + Px(x,t)"'x(x,t)div vx(x,t) }drox
rox
+ P"x(f r.;(Xx,t),t)"'x(f x(Xx,t),t)dehx[Jx(Xx,t)] } dOx
= J dJtx[Px(f x(Xx,t),t)"'x(f x(Xx,t),t)Jx(Xx,t) ]dnx Ox
The speciftc value "'x of quantity Qx(t) is linked to the unit mass
of the matter and does not depend on the volume occupied by the
matter. In other words, in the reference
conftguration, a speciftc value 'P xis deftned as identical to "'x
in the instantaneous domain
With this deftnition, the remaining quantity Px(fx(Xx,t),t)Jx(Xx,t)
has the dimension of a volume mass in the reference domain. This
quantity is said to be the apparent mass density of constituent x
in Lagrange variables and is deftned by :
According to (1.67), (1.68) and (1.17) the first member of (1.62)
becomes :
= J (~[Px(Xx,t)'P x(Xx,t)] }dOx
° x •
Transforming the second member of (1.62) can be performed as
follows :
Generalized Approach to Heterogeneous Media
(i) In the first term: (1.1) and (1.13) give :
- f nx['I'x](x,t)ox(x,t)dax ax
= - f nx['I'x](fx(Xx,t),t)Jx(Xx,t)[!ioulT fx(Xx,t)]-lNx(Xx,t)dAx A
x
If the Lagrangian influx. Kx['I' x] (Xx,t) is defmed by :
this first term becomes :
21
(ii) In the second term: if the Lagrangian external volume source
"-x['I' x] (Xx,t) is defined as
(1.70) ]('x['I' x](Xx,t) == ~x['I'x](x,t)Jx(Xx,t) =
~x['I'x](fx(Xx,t),t)Jx(Xx,t)
this second term becomes with the help of (1.1) and (1.12)
f ~x['I'x](x,t)drox = f ~x['I'x](fx(Xx,t),t)Jx(Xx,t)dax O)x a
x
= f ]('x['I' x] (Xx,t)dax a x
(iü) In the third term: if the Lagrangian internal source 'Lx['I'
x](Xx,t) is defined by :
this third term becomes by the same operation
f i-x('I'x)(x,t)drox = f 1..x('I'x)(fx(Xx,t),t)Jx(Xx,t)dax O)x a
x
22 P. Jouanna and M-A. Abellan
Finally, the Lagrange integral balance relation of constituent x is
expressed by :
(1.72) f ~ [px(Xx,t)'I' x(Xx,t)]dnx n x
+ f X,x['I' x](Xx,t)dnx + f t x['I' x](Xx,t)dnx n x n x
• Integral balance relation of Qx on the virtual reference domain
Q*
The virtual velocity v* is defined using a virtual transformation
f*(X*,t) of a virtual reference domain n*. Thus relation (1.64)
written in Euler variables (x,t) along the virtual movement v* can
be written by this virtual transformation at points belonging to
the virtual reference domain n*. This fiction, where the following
notations are used, will enable to work on the same fictitious
domain for different constituents :
(1.73) 'I' x*(X* ,t) == 'l'x(x,t) = 'l'x(f*(X* ,t),t)
(1.74) px*(X*,t) == Px(x,t)J*(X*,t) = Px(f*(X*,t),t)J*(X*,t)
(1.75) H x*['I' x*](X* ,t) == fl,x*['I'x](x,t)J*(X* ,t) = fl,x*
['I'x] (f*(X* ,t),t)J*(X* ,t)
(1.76) x,x*['I' x*](X*,t) == ~x*['I'x](x,t)J*(X* ,t) =
~x*['I'x](f*(X* ,t),t)J*(X* ,t)
(1.77) t x*['I' x*](X* ,t) == i-x*['I'x](x,t)J*(X* ,t) =
i-x*['I'x](f*(X* ,t),t)J*(X* ,t)
The Lagrange integral balance relation on the virtual reference
domain n* is given by :
(1.78) f (~[Px*(X*,t)'I'x*(X*,t) ]}dn* n*
= - f Hx*['I'x*](X*,t) [!iDLd.T f*(X*,t)]-lN*(X*,t)dA* A*
+ f x,x*['I' x*](X*,t)dn* + f t x*['I' x*](X* ,t)dn* n* n*
Generalized Approach to Heterogeneous Media 23
A2 For a set of constituents
(a) Euler point of view
Writing a balance relation for a set of constituents 1t involves
fundamental difficulties due to the fact that contributions of all
constituents cannot be directly added when each constituent follows
its own movement. Mathematically speaking, different derivative
operators cannot be added when different velocity fields V1t are
followed.
However, this basic difficulty disappears if the balance relations
for the different constituents are written in function of a unique
velocity field v*. Consequently, the contributions of k
constituents given by the generalized relation (1.64) can be added
:
(1.79) ~ f a ~ {at [P1t(x,t)'I'1t(x,t)] +
div[p1t(x,t)'I'1t(x,t)v*(x,t)] }doo* n=1 00 *
n=k
*f n=k f =- L n1t*['I'1t](x,t)n*(x,t)da* + L
~*['I'1t](x,t)doo*
n=1 a n=1 00*
1t=k f +L in: * ['I'7t](x,t)doo* n=1 00*
(b) Lagrange point of view
The Lagrange balance relations for different constituents as given
by (1.72) are written on different reference domains On. Thus these
relations cannot be added. On the contrary , relations (1.78)
written on the same virtual reference domain n* can be added for a
set of k constituents :
(1.80) 'ik f (gt [P7t*(X* ,t)'P 7t*(X* ,t)] }dn* n=1 n *
n=k f = - L H 1t*['P 1t*](X* ,t)[GJ:wlT f*(X* ,t)]-l N*(X* ,t)dA *
n=l A *
n=k f + L 1t=1 n *
1t=k f 1G7t*['P 1t*](X* ,t)dn* + L n=1 n *
l 1t*['P 1t*](X* ,t)dn*
24
BI For one constituent x
(a) Euler point of view
P. Jouanna and M-A. Abellan
• Differential balance relation of a quantity Qx along the movement
of x
Integral relation (1.62) leads to the differential fonn :
(1.81) ~Px(X,t)'I'x(X,t)] + div[Px(x,t)'I'x(x,t)vx(x,t)]
(1.82) ~ [Px(x,t)'I'x(x,t)] + div[Px(x,t)'I'x(x,t)v*(x,t)]
= - dimx*['I'x](x,t) + ~x*['I'x](x,t) + i-x*['I'x](x,t)
(b) Lagrange point of view
• Differential balance relation of Qx on the reference domain
nx
Integral relation (1.72) on the reference domain On: relative to
constituent 1t leads to the Lagrange differential balance relation
of constituent x :
(1.83)
• Differential balance relation of Qx on the virtual reference
domain n *
Integral relation (1.78) on the virtual reference domain n* leads
to the following Lagrange differential balance relation of
constituent x :
(1.84) ~Px*(X* ,t)\f' x*(X* ,t)]
= - Div[Kx*[\f' x*](X* ,t)[GDulT f*(X* ,t)]-l] + x,x*(\f' x*)(X*
,t)
+ 'Lx*[\f' x*](X* ,tl
Generalized Approach to Heterogeneous Media
B2 For a set of constituents
(a) Euler point of view
The integral relation (1.79) leads to the following differential
expression:
(1.85) x=k a L {atfP1t(x,t)'I'1t(x,t)] +
div[p1t(x,t)'I'1t(x,t)v*(x,t)]}
x=1
x=k x=k x=k = - L divh1t*['I'1t](x,t) + L ~1t*['I'1t](x,t) + L
1.1t*['I'1t](x,t)
x= 1 x=1 x=1
(b) Lagrange point of view
The integral relation (1.80) leads to the following differential
expression:
(1.86)
x=k x=k = - L Div[H1t*['I' 1t*](X* ,t)\lIIwlT f*(X*,t)]-l] + L
1G1t*['I' 1t*](X* ,t)
1t=1 x=1 1t=k
+ L t 1t*['I' 1t*](X* ,t) 1t=1
C Use of the synthetic balance relations
25
It is fundamental to note that, whatever the point of view, the
specific value of any
quantity Q7t(t) is expressed by the equivalent functions 'l'7t(x,t)
== 'P 7t(X1t,t) == 'P 1t*(X* ,t) as given by (1.67) and (1.73).
Deriving the balance relations for (a) mass, (b) momentum, (c)
total energy, (d) internal energy and (e) entropy is direct1y
obtained from the above synthetic relations by setting this
specific value as equal to the following quantities :
(1.87) (a) 'l'1t(x,t) == 1
(c) 'l'1t(x,t) == et1t(x,t) == Et1t(X1t,t) == Et1t*(X* ,t)
(d) 'l'1t(x,t) == e1t(x,t) == E1t(X1t,t) == E1t*(X* ,t)
(e) 'l'1t(x,t) == Sx(x,t) == S1t(X1t,t) == S1t*(X* ,t)
Velocity of 1t
Specific entropy of 1t
The synthetic Euler balance relations (1.79) or (1.85) and the
synthetic Lagrange balance relations (1.80) or (1.86) lead to all
other balance relations, following any virtual domain, for any set
of constituents. The following developments can be considered as an
application of these synthetic relations.
26 P. Jouanna and M-A. Abellan
1.2.2 Mass balance relations
Quantity Qn(t) relative to mass is obtained by (1.60) and (1.87a)
making \j1n(x,t) = 1 :
(1.88) Qn(t) = f Pn(x,t)\j1n(x,t)d<'on = f Pn(x,t)d<'on
A Integral forms of mass balance
Al For one constituent n
(a) Euler point of view
• Integral mass balance relation along the movement of n
F1ux and source terms appearing in the Euler synthetic integral
balance relation (1.62) applied to mass along the movement of
constituent n are as follows:
(i) F1ux hn[\j1n=l] : the flux hn[1] is the mass influx of
particles of constituent n entering
the domain <On: through a surface element dan moving at velocity
Vn. As the boundary an formed by the particles of n is material,
this influx equals zero:
h n[1] = 0
(ii) Extemal volume source ~[\j1n=1]: the extemal volume source of
mass ~[\j1n=l] in the
domain <.On is due to a possible external mass source P1tm1t
plus the possible mass
contribution 'tn of other constituents by unit volume of
heterogeneous medium and time unit. This mass supply 'tn comes from
the transformation into constituent n of other constituents
existing in <On:, due to phase changes or chemical reactions.
Thus :
. A ~n[l](x,t) = Pn(x,t)mn(x,t) + cn(x,t)
(iii) Internal source i-n['I'n=l]: the principle of mass
conservation gives i-n[l ](x,t) = O.
Thus relation (1.62) and the above flux and source terms yield the
expression of the Euler integral mass balance relation for one
constituent n following its movement Vn :
(1.89)
• Generalized integral mass balance relation along a virtual
movement v*
The flux and source tenns of the synthetic balance relation (1.64)
with 'l'x(x,t) = 1 are :
(i) Flux hx*['I'rc=1] : according to fonnula (1.65) and with hx[l]
= 0, it becomes :
hx*[1](x,t) = Px(x,t)[vx(x,t) - v*(x,t)] = Px(x,t)wx*(x,t)
27
(ii) External source ~*['I'x=l] : external volume sources Pxmx and
~ in the domain 00* are
identical to the preceding case, domains 00* and COn: being
identical at a given time t.
~x*[1](x,t) = px(x,t)mx(x,t) + ~x(x,t)
(iii) Internal source tn:*['I'x=l] : as above
tn:*[l](x,t) = 0
Hence, relation (1.64) and the above flux and source tenns lead to
the Euler generalized integral mass balance relation, for one
constituent x, following a virtual movement v* :
(1.90) f {apa~x,t) + div[Px(x,t)v*(x,t)]}dOO*
00*
= f div[Px(x,t)wx*(x,t)]doo* + f {px(x,t)mx(x,t) + ~(x,t)}doo· 00*
00*
(b) Lagrange point of view
• Integral mass balance relation on the reference domain Qx
The flux and source tenns of the synthetic relation (1.72) with 'I'
x(Xx,t) = 1 are:
(i) Flux K x['I' x= 1] : according to (1.69) and as hx[l] = 0, it
becomes
K x[1](Xx,t) = hx[1](x,t) Jx(Xx,t) = 0
(ii) External source tenn 1Grc['I' x= 1]
1Gx[1](Xx,t) = ~x[1](x,t) Jx(Xx,t) = [px(x,t)mx(x,t) +
~(x,t)]Jx(Xx,t)
Let us define :
28 P. Jouanna and M-A. Abellan
Äccording 10 (1.68) and the two definitions above, it becomes :
1\
x,7t[1](X7t,t) = P7t(X7t,t)M7t(X7t,t) + Cx(X7t,t)
(iii) Internal source term 'Lx['I'n= 1] : according to (1.71)
t 7t[l](X7t,t) = 1,7t[1](x,t)J7t(Xn,t) = 0
Thus, relation (1.72) and the above flux and source terms lead to
the Lagrange generalized integral mass balance relation, for one
constituent n, on the reference domain On
• Integral mass balance relation on the virtual reference domain n
*
The flux and source terms ofthe synthetic relation (1.78), with
'Pn*(X*,t) = I, are:
(i) Flux H,n*['I' n*= I] : according to (1.75) and nn*[1](x,t) =
Pn(x,t) wn*(x,t) it comes:
H,n*[ I](X* ,t) = h.n*[I](x,t)J*(X* ,t) = Pn(x,t)wn*(x,t)J*(X*
,t)
Defming: (1.94) W n*(X* ,t) = wn*(x,t) = wn*(f*(X* ,t),t)
and according to (1.74) it comes:
H,n*[I](X* ,t) = pn*(X* ,t)W 7t*(X* ,t)
(ii) Extemal source termx,*['Pn*= I] : according to (1.76) and the
above value of ~[I]
X,n*[I](X* ,t) = ~n[1](x,t)J*(X* ,t) = [Pn(x,t)mn(x,t) +
~(x,t)]J*(X* ,t)
Defming:
(1.95)
(1.96)
Mn*(X*,t) = mn(x,t) = mn(f*(X* ,t),t) 1\ 1\ 1\ Cx*(X*,t) =
en(x,t)J*(X*,t) = en(f*(X*,t),t)J*(X*,t)
the external source term becomes : 1\
X,n*[I](X*,t) = pn*(X* ,t)Mn*(X* ,t) + Cn*(X* ,t)
(iii) Internal source term 'Lx*['Pn*= I] : according to
(1.77)
i-x[1](x,t)J*(X*,t) = t n*[1](X*,t) = 0
Generalized Approach to Heterogeneous Media 29
Hence the Lagrange integral mass balance relation relative to
constituent n:, on the virtual reference domain n* is expressed by
:
= - f Pn:*(X* ,t)W n:*(X* ,t)LGrad.T f*(X* ,t)]-lN*(X* ,t)dA *
A*
+ f Pn:*(X*,t)Mn:*(X*,t)dn* + I Cn;*(X*,t) dn* n* n
A2 For a set of constituents
(a) Euler point of view
For a set of k constituents, it is possible to add balance
relations of mass (1.90) written on a virtual domain 00* following
a virtual movement v* chosen identical for all constituents. Thus
the Euler integral balance relation written for the set of k
constituents is :
(1.98) 'ik f (dP'iitX,t) + div[pn:(x,t)v*(x,t)]}doo* 71:=1 00
*
Special case of the total medium
For a total heterogeneous medium, with k=N constituents, the Euler
generalized differential mass balance relation can also be written
in the following equivalent form :
1 f ~(x,t)doo* = f 'ik ~(x,t)doo* = 0 71:=1 00 * 00 * 71:=1
(1.99)
(b) Lagrange point of view
Similarly, way, it is possible to add integral expressions (1.97)
written for the different constituents x on the same volume n* for
obtaining the Lagrange integral mass balance relation for a set of
k constituents :
(1.100) r,k f ~ px*(X* ,t)dn* 1t=1 n *
1t=k f = - l: px*(X*,t)Wx*(X* ,t)r.G.tild.Tf*(X*,t)]-IN*(X*,t)dA *
7t=1 A *
7t=k J 1t=k J 1\ + l: Px*(X*,t)Mx*(X*,t)dn* + l: Cx*(X*,t)dn*
1t=ln* 7t= l n*
Special case of the total medium
For a total heterogeneous medium, with k=N constituents, the
Lagrange mass balance relation can also be written in the following
equivalent fonn :
(1.101) 1t=k f 1\ f l: Cx*(X*,t)dn* = 1t=1 n * n *
B Differential forms of mass balance
BI For one constituent x
(a) Euler point of view
7t=k 1\
• Differential mass balance relation along the movement of
constituent x
The Euler differential mass balance relation following xis direcdy
obtained from (1.89) :
(1.102) ~ Px(x,t) + div[Px(x,t)vx(x,t)] = px(x,t)mx(x,t) +
~x(x,t)
• Generalized differential mass balance relation along a virtual
movement v*
The Euler generalized differential mass balance relation of
constituent x following a virtual movement v*, is obtained directly
from (1.90) :
(1.103) ~ Px(x,t) + div[Px(x,t)v*(x,t)]
= - div[Px(x,t)wx*(x,t)] + px(x,t)mx(x,t) + ~(x,t)
Generalized Approach to Heterogeneous Media 31
(b) Lagrange point of view
• Differential mass balance relation on the domain 0x
Relation (1.93) leads to the Lagrange differential mass balance
relation :
(1.104)
• Differential mass balance relation on the virtual reference
domain 0*
Relation (1.97) leads to the Lagrange generalized differential mass
balance relation:
(1.105) ~ px*(X* ,t) = - Div{ px*(X* ,t)W x*(X* ,t)[!iI:wlT f*(X*
,t)]-1 }
" + px*(X*,t)Mx*(X*,t) + Cx*(X*,t)
B2 For a set of constituents
(a) Euler point of view
The differential fonn of the Euler mass balance relation for a set
of k constituents written following the movement v* is obtained
directly from (1.98) :
(1.106) 1C=k a L (at Px(x,t) + div[Px(x,t)v*(x,t)]}
1C=1 1C=k
1C=k 1C=k " + L px(x,t)mx(x,t) + L ex(x,t)
x=l x=1
Special case of the total medium
For the total medium, with k=N constituents, relation (1.99) gives
:
(1.107) x=N L ~(x,t) = 0
1C=1
A condensed relation can be written introducing definitions of the
total apparent mass p, the total mass rate m, the barycentric
velocity VB and the barycentric relative velocity WB :
(1.108) ap~,t) + div[p(x,t)v*(x,t)] = - div[p(x,t)WB*(X,t)] +
p(x,t)m(x,t)
32
with :
(1.109)
(1.110)
(1.111)
(1.112)
Examples
m(x,t) - -- L P1t(x,t)m1t(x,t) p(x,t) 1t=1
1 1t=N VB(X,t) == -- L P1t(x,t)v1t(x,t)
p(x,t) 1t=1
P. Jouanna and M-A. Abellan
1 1t=N WB*(X,t) == VB(X,t) - v*(x,t) = -- L P1t(x,t)(V1t(x,t) -
v*(x,t»
p(x,t) 1t=1
ap~~,t) + div[p(X,t)VB(X,t)] = 0
• If the virtual velocity v*(x,t): VB(X,t) and m=ü :
ap~~,t) + div[p(x,t)VB(X,t)] = div{p(x,t)(Vß(x,t) - VB(X,t))] =
0
• If the virtual velocity v*(x,t) : va(x,t) velocity of a given
constituent "a" and m=O :
ap~~,t) + div[p(x,t)va(x,t)] = - div[p(x,t)(VB(X,t) - va(x,t))] or
~ p(x,t) + div[p(x,t)VB(X,t)] = 0
Note 0 : There is no difference in the particular case of mass
balance relations between the result obtained by the generalized or
the classical theory because the flux of mass through a material
boundary alt is equal to
zero.
(b) Lagrange point of view
The Lagrange differential mass balance relation for a set of k
constituents, is obtained from (1.100) :
(1.113) 1t=k a 1t=k L at P1t*(X*,t) = - L Div{
P1t*(X*,t)W1t*(X*,t)[.Gr.w1Tf*(X*,t)]-l}
1t=1 1t=1
Special case of the total medium
1t=k A + L (P1t*(X* ,t)M1t*(X* ,t) + C1t*(X* ,t)}
1t=1
For a total heterogeneous medium, with k=N constituents, the
Lagrange generalized differential mass balance relation can also be
written in the following equivalent form :
(1.114) 1t=N A
Generalized Approach to Heterogeneous Media 33
In this case, an overall relation can be written introducing
definitions of P *(X* ,t), M*(X*,t), VB(X*,t) and WB(X*,t) :
(1.115)
with :
(1.116)
(1.117)
(1.118)
(1.119)
~ P*(X*,t) = - Div{ P*(X*,t)WB*(X*,t)[!irwlTj*(X*,t)]-l}
+ p*(X* ,t)M*(X* ,t)
1t=N P*(X*,t) == L P1t*(X*,t)
1t=1 1t=N
M*(X* ,t) == 1 L P1t*(X* ,t)M1t*(X* ,t) P*(X*,t) 1t=1
1 1t=N VB(X*,t) == L P1t*(X*,t)V1t(X*,t)
p*(X* ,t) 1t=1
1t=N WB*(X*,t) == 1 L P1t*(X*,t)(V1t(X*,t) - V*(X*,t»
P*(X*,t) 1t=1
1.2.3 Momentum balance relations
Quantity On relative to momentum is given by (1.60) and (1.87b)
with 'l'1t == V1t :
(1.120)
Al For one constituent 1t
(a) Euler point of view
• Integral momentum balance relation along the movement of
constituent 1t
Flux and source terms appearing in the Euler synthetic integral
balance relation (1.62) applied to momentum along the movement of
constituent 1t become :
(i) Flux hx['I'1t== V1t] : the flux hx[v1t] is the momentum influx
due to the tension estimated
by the partial stress tensor .Q:1t acting on a surface element da1t
of the domain COn: moving at the velocity V7t.
n1t[ V1t](x,t) = .Q:7t(x,t)
34 P. Jouanna and M-A. Abellan
(ii) External volume source ~['I'1t == V1t] : the external volume
source ~[V1t] of momentum
in the domain COn; , per unit volume of the heterogeneous medium
and time unit, is primarily due to the external volume force field
P1tf1t. This external source also includes the
1\ momentum of the external mass source V1tP1tm1t, the momentum of
the mass supply v1tCx
and a direct momentum supply Px due to other constituents acting on
constituent 1t.
~1t[ V1t](x,t) = P1t(x,t)f1t(x,t) + V1t(x,t)[P1t(x,t)m1t(x,t) +
~(x,t)] + P1t(x,t)
(iii) Internal source i-x['I'x= V1t] : according to the principle
of momentum conservation, this internal source is equal to
zero.
\'1t[ V1t](x,t) = 0
Thus relation (1.62) and the above flux and source terms lead to
the expression of the Euler integral momentum balance relation for
one constituent 1t following its movement V1t :
(1.121)
f Qn:(x,t)n1t(x,t)da1t a1t
Note 1: The momentum supply P1t is different from the momentum
source as defined classically [Truesdell &
Toupin, 1960, p. 567, relation (215.2)]. The expression adopted
here for this mo mentum supply P1t is linked only to the relative
movement of the constituents. The momentum due to the mass sources
is handled separately. ClassicaJ inconsistencies are thus avoided
in the writing of total energy, internal energy and entropy balance
relations.
Note 2 : The moment of momentum balance relations are not written
here because they finally lead to demonstrating the symmetry of the
stress tensor in the non-polar case, i.e. when surface and volume
couples are not present. This assurnption is used below as in
classical mechanics for the sake of simplicity. However, the
generalized theory as presented here can be readily extended to the
non-polar case.
• Generalized integral momentum balance relation along a virtual
movement
The flux and source terms, appearing in the Euler synthetic
integral balance relation (1.64) applied to momentum of constituent
1t along the virtual movement v*, are as follows :
(i) Flux hx*['I'1t== V1t] : when following the virtual domain ro*,
the classical partial stress Qx acting on the material contour of
constituent 1t moving at velocity V1t must be replaced by a
generalized partial stress Qx* acting on the surface element da*
moving at velocity v*. The synthetic formula (1.65) gives the
expression of this generalized partial stress tensor
hx*[v1t](x,t) = Qn*(x,t) where the value ofhx[v1t] is the classical
stress Q:1t:
Generalized Approach to Heterogeneous Media
(1.122) ~*(X,t) == 1I.x(x,t) + Px(x,t)vx(x,t)®(vx(x,t) -
v*(x,t»
== ~(x,t) + Px(x,t)vx(x,t)®wx*(x,t)
35
Thus the generalized partial stress ~* on a virtual surface da* is
equal to the c1assical
partial stress ~, acting on the material surface element dax, plus
the momentum influx of
particles of constituent x, moving with the velocity vx, and
entering domain 00* through its surface a* moving at the velocity
v*. It is fundamental to note [Jouanna & Abellan, 1992]
that the generalized partial stress ~* is not objective, its
defmition depending on the virtual movement v*.
(ii) External volume source ~x*['I'x== vx] : this source due to
vxPxmx, vx~x, Pxfx and ~x in the domain 00* is identical to the
source in COx because domains 00* and 00x are identical at a given
time 1.
(üi) Internal volume source i-x*['I'x= vx] : for the same reason
i-x*[vx] = i-x[vx] = 0
Hence relation (1.64) and the flux and source terms above yield the
Euler generalized integral momentum balance relation for
constituent x following a virtual movement v* :
(1.123) J (~Px(X,t)vx(X,t)] + div[Px(x,t)vx(x,t) ® v*(x,t)]
}doo*
00*
or:
00*
00*
+ div[Px(x,t)(wx*(x,t) + v*(x,t» ® v*(x,t)] }doo*
= - J {1I.x(x,t) + Px(x,t)(wx*(x,t) + v*(x,t» ® wx*(x,t)
}n*(x,t)da* a*
J A A + {Px(x,t)fx(x,t) + (wx*(x,t) + v*(x,t))[px(x,t)mx(x,t) +
c,c(x,t)] + Px(x,t) }doo*
00*
(b) Lagrange point of view
• Integral momentum balance relation on the reference domain
n1t
The flux and source tenns ofrelation (1.72) with 'l'1t(X1t,t) ==
V1t(X1t,t) become :
(i) Flux K1t['I' 1t== V 1t] : the flux K1t[V 1t](X1t,t) noted
~(X1t,t) is defined in the following as the Lagrange stress tensor.
According to (1.69), the Lagrange stress tensor, defined here, is
related to the Euler stress tensor by :
(1.125)
(ii) External source 1G1t['I' 1t== V 1t] : (1.70) and the above
expression of ~[\j11t] give :
1G1t[V 1t](X1t,t) = ~1t[ V1t](x,t)J1t(X1t,t) /\ /\
= {P1t(x,t)f1t(x,t) + V1t(x,t)[P1t(x,t)m1t(x,t) + ex(x,t)] +
P1t(x,t) }J1t(X1t,t)
=P1t(X1t,t)f' 1t(X1t,t) + V 1t(X1t,t)[P1t(X1t,t)M1t(X1t,t) +
C1t(X1t,t)] + j} 1t(X1t,t)
with the following definitions:
(1.126)
(1.127)
(iii) Internal source 'Lx['I'1t== V1t] : according to (1.71) as
i-n:[V1t](x,t) = 0 :
'L1t[V 1t](X1t,t) = 1,1t[V1t](x,t)J1t(X1t,t) = 0
Hence, according to (1.72) and the above flux and source tenns, the
Lagrange integral momentum balance relation, expressed on the
reference domain nx of the constituent 1t, is :
(1.128)
Generalized Approach to Heterogeneous Media 37
Note 3: the defInition of the Lagrange stress tensor ,l;.n(Xn,t) '"
~n(x,t)Jn(Xn,t) has been adopted to simplify the expression of the
generalized Lagrange stress tensor. Classically, the Lagrange
stress tensor is defmed by :
In this case, the fIrst tenn of the right-hand side member of
(1.l28) would be written :
f ~(Xn,t)Nn(Xn,t)dAn An
• Integral momentum balance relation on the virtual reference
domain n*
The flux and source terms of the synthetic relation (1.78), applied
with 'P x*(X* ,t) ==
Vx*(X*,t) == Wx*(X*,t) + V*(X*,t), are as follows :
(i) Flux Hx*['P x* == V x*] : according to (1.75) and the above
expression of hx*[vxJ.
Hx*[Vx*](X*,t) = hx*[v1t](x,t)J*(X*,t)
= {Q.1t(x,t) + P1t(x,t)[(W1t*(x,t) + v*(x,t» ®
w1t*(x,t)]}J*(X*,t)
Defining:
(1.129) k*(X*,t) == Q.1t(x,t)J*(X*,t) ==
Q.x(f*(X*,t),t)J*(X*,t)
it comes:
H 1t*[v 1t*](X*,t) = b:1t*(X* ,t)+P1t*(X* ,t)[(W x*(X* ,t)+ V*(X*
,t»®W x*(X* ,t)]
(ii) External source JG1t*['P x* == V 1t*] : according to (1.76)
and the above value of ~[vx],
1G1t*[V1t*](X*,t) = k.1t[V1t](x,t)J*(X*,t)
" " = {P1t(x,t)f1t(x,t)+[ W1t*(x, t)+v*(x, t)] r Px(x, t)mx(x,t)+
ex(x,t) ]+P1t(x,t) } J*(X * ,t)
= P1t*(X* ,t)1' 1t*(X* ,t)
+(W 1t*(X* ,t)+ V*(X* ,t))[px*(X* ,t)M1t*(X* ,t)+Cx*(X*
,t)]+~1t*(X* ,t)
with the definitions:
1'1t*(X*,t) == f1t(x,t) = f1t(f*(X*,t),t)
Ä " " Y1t*(X*,t) == P1t(x,t)J*(X*,t) = Px(f*(X*,t),t)J*(X*,t)
(iii) Internal source 'L1t*['P x* == V 1t*]
'Lx*[V 1t*](X* ,t) = I,x[ V1t](x,t) J*(X* ,t) = 0
38 P. Jouanna and M-A. Abellan
Hence the Lagrange integral momentum balance relation relative to
constituent 1t, on the virtual reference domain n*, is expressed by
:
(1.132) f irP1t*(X*,t)(W1t*(X*,t)+V*(X*,t»]dn* n*
-- f {[~*(X*,t)+P1t*(X*,t)(W1t*(X*,t)+V*(X*,t»®W1t*(X*,t)] -
A*
LGrwlT f*(X* ,t)]-lN*(X* ,t) }dA *
+ f {P1t*(X* ,t)f' 1t*(X* ,t)+(W 1t*(X* ,t)+ V*(X* ,t))[P1t*(X*
,t)M1t*(X* ,t) n*
A2 For a set of constituents
(a) Euler point of view
It is not possible to obtain the momentum balance for a set of k
constituents by adding up momentum balance relations (1.121)
following the movement of the different constituents. This is due
to the fact that fluxes in the second member of such relations are
evaluated on surfaces a1t following the different movements v1t. On
the contrary, adding up relations (1.123) on allconstituents 1t is
licit because fluxes relative to the different constituents are
evaluated on a single surface a*. Hence it is possible to obtain an
Euler integral momentum balance relation on a set of k constituents
as expressed by :
(1.133) ~k f a ~ (at
[P1t(x,t)V1t(x,t)]+div[p1t(x,t)v1t(x,t)®v*(x,t)] }dO)* 1t=1 0)
*
1t=k f = - I
divL~1t(x,t)+P1t(x,t)V1t(x,t)®(V1t(x,t)-v*(x,t»]dO)*
1t=1 0) *
Generalized Approach to Heterogeneous Media 39
(1.134) ~k Jf Cl ~ _ {atfP1t(x,t)(W1t*(x,t)+v*(x,t))] 1t=1 ro
*
+ div[p1t(x,t)(W1t*(x,t)+v*(x,t»®v*(x,t)] }dro*
1t=k f = - L div[~(x,t)+P1t(x,t)(W1t*(x,t)+v*(x,t»®W1t*(x,t))dro*
1t=1 ro *
1t=k f 1\ 1\ + L
{P1t(x,t)f1t(x,t)+(w1t*(x,t)+v*(x,t))[p1t(x,t)m1t(x,t)+c,r(x,t)]+p1t(x,t)
}dro* 1t=1 ro *
Special case of the total medium
With k = N constituents, an equivalent form is :
(1.135) 1t=N f 1\ f 1t=N 1\ L P1t(x,t) dro* = L P1t(x,t) dro* = 0
1t=1 ro * ro * 1t=1
(b) Lagrange point of view
Similarly, it is possible to add integral expressions (1.132)
written for the different constituents 1t on the same volume n*.
The generalized Lagrange integral momentum balance relation written
for a set of k constituents is :
(1.136) 1t=k f ~1 n *
~ [pn*(X* ,t)(W n*(X* ,t)+V*(X* ,t))]dn*
1t=k J =- L 1t=1 A *
(~*(X* ,t)+pn*(X* ,t)(W 1t*(X* ,t)+V*(X* ,t»®W 1t*(X* ,t)]
LGrwlTf*(X*,t)]-l }N*(X*,t)dA*
1t=k f + L 1t=1 n *
{pn*(X* ,t)f' n*(X* ,t)+(W n*(X* ,t)+ V*(X* ,t))[P1t*(X* ,t)Mn*(X*
,t)
1\ 1\ +Cn:*(X* ,t)]+P1t*(X* ,t) }dn*
Special case of the total medium: with k = N constituents, an
equivalent form is
(1.137) 1t=N f 1\ L Pn*(X* ,t)dn* = 0 n=1 n *
40 P. Jouanna and M-A. Abellan
B Differential forms of momentum balance
BI For one constituent 1t
(a) Euler point of view
• Differential momentum balance relation along the movement of
1t
Integral relation (1.121) leads to the differential form :
(1.138) ~ [P1t(x,t)V1t(x,t)] + div[p1t(x,t)V1t(x,t) ®
V1t(x,t)]
= - div !I1t(x,t) + P1t(x,t)f1t(x,t) + V1t(x,t)[P1t(x,t)m1t(x,t) +
~(x,t)] + P1t(x,t)
• Generalized differential momentum balance relation along v*
Integral relations (1.123) amI (1.124) lead to the differential
forms :
(1.139) ~P1t(X,t)V1t(X,t)] + div[p1t(x,t)V1t(x,t) ® v*(x,t)]
= - div {!I1t(x,t) + P1t(x,t)V1t(x,t) ® (V1t(x,t) - v*(x,t»}
" " + P1t(x,t)f1t(x,t) + V1t(x,t)[P1t(x,t)m1t(x,t) + en:(x,t)] +
P1t(x,t)
or:
(1.140) ~P1t(X,t)(W1t*(X,t) + v*(x,t))) + div[p1t(x,t)(w1t*(x,t) +
v*(x,t» ® v*(x,t)]
= - div{!I1t(x,t) + P1t(x,t)(W1t*(x,t) + v*(x,t» ® W1t*(x,t)}
+ P1t(x,t)f1t(x,t) + (W1t*(x,t) + v*(x,t))[P1t(x,t)m1t(x,t) +
~(x,t)] + P1t(x,t)
(b) Lagrange point of view
• Differential momentum balance relation on the reference domain
01t
Integral relation (1.128) leads to :
(1.141) ~ [P1t(X1t,t)V1t(X1t,t)] = -
Div{~1t(X1t,t)[!inl.d.Tf1t(X1t,t)]-l}
• Differential momentum balance relation on the virtual reference
domain n*
Integral relation (1.132) leads to :
(1.142) ~ [P1t*(X*,t)(W1t*(X*,t) + V*(X*,t))]
=-Div{ [~1t*(X* ,t)+P1t*(X* ,t)(W 1t*(X* ,t)+ V*(X*,t))®W 1t*(X*
,t)][!i..Dul.T f*(X* ,t)]-1 } 1\
+ P1t*(X* ,t) l' 1t*(X* ,t)+(W 1t*(X* ,t)+V*(X* ,t))[P1t*(X*
,t)M1t*(X* ,t)+ C1t*(X* ,t)] 1\
+ P1t*(X*,t)
(a) Euler point of view
The integral fonn of the balance relation (1.134) leads to the
differential fonn :
(1.143) 1t=k a L (at[P1t(x,t)(w1t*(x,t)+v*(x,t))]
1t=1
+div[p1t(x,t)(w 1t*(x,t)+v*(x,t) )®v* (x,t)] } 1t=k
= - L div{Qn(x,t)+P1t(x,t)(W1t*(x,t)+v*(x,t))®W1t*(x,t)} 1t=1
1t=k 1\ 1\ + L {P1t(x, t)f1t(x,t)+(w 1t* (x, t)+v*
(x,t))[P1t(x,t)m1t(x,t)+en(x,t)]+P1t(x,t))
1t=1
Special case of the total medium
For a total heterogeneous medium, with k = N constituents, relation
(1.135) gives :
(1.144) 1t=N 1\
L P1t(x,t) = 0 1t=1
In this case, a condensed differential relation can be written for
the total medium:
(1.145) ~ [p(X,t)VB(X,t)] + div[p(x,t)VB(X,t) ® v*(x,t)]
1t=N = - div Q*(x,t) + p(x,t)f(x,t) + L
{(W1t*(x,t)+v*(x,t))[P1t(x,t)m1t(x,t)+~(x,t)]}
1t=1
using the expression (1.111) of the barycentric velocity VB and the
following definitions of
the generalized total stress Q*(x,t ) and the total volume force
f(x,t ) :
42 P. Jouanna and M-A. Abellan
7t=N (1.146) .a*(x,t) == L (.a1t(x,t) + P1t(x,t)V1t(X,t) ®
(V1t(x,t) - V*(x,t»}
7t=1
f(x,t) == -- L P1t(x,t)f1t(x,t) p(x,t) 1t=1
The generalized total stress tensor .a*(x,t) is thus equal to the
surn of generalized partial
stresses .an* as defined by (1.122). 7t=N
(1.148) .a*(x,t) == L .an*(x,t) 7t=1
It is fundamental to note that the generalized total stress .a* is
not objective, its definition depending on the virtual rnovernent
v*. '
Note 4 : Relation (1.148) leads to a consistent definition of the
generalized total stress as the sum of the generalized partial
stresses. This solves classical difficulties, where a distinction
was made between the total stress and its internal part, this
distinction having no physical meaning [Truesdell & Toupin,
1960, p. 568, formula (215.6)].
Examples :
• If v*(x,t) '" 0 : 1t=N ft [p(X,t)VB(X,t)] = - div sro(x,t) +
p(x,t)f(x,t) + L (V7t(x,t)[P7t(x,t)m7t(x,t) + 6'n(x,t)]} 7t=1
1t=N with SI*(x,t) '" SIQ(x,t) = L ([P7t(x,t)V7t(x,t) ® V7t(x,t)] +
Qx(x,t)}
7t=1 • If the virtual velocity v*(x,t) '" VB(X,t) :
ft [P(X,t)VB(X,t)] + div[p(X,t)VB(X,t) ® vB(x,t)]
1t=N = - div 'SIvB(x,t) + p(x,t)f(x,t) + L {(W7t*(x,t) +
VB(X,t»[P7t(x,t)m7t(x,t) + t'n<x,t)]}
7t=1
1t=N with SI*(x,t) '" SIVB(x,t) = L ([P7t(x,t)V7t(x,t) ® (V7t(x,t)
- VB(X,t»] + SI7t(x,t»)
7t=1
• If the virtual velocity v*(x,t) '" va(x,t) of one given
constituent "a" :
ft [p(X,t)VB(X,t)] + div[p(x,t)VB(X,t) ® va(x,t)]
1t=N = - div SIva(x,t) + p(x,t)f(x,t) + L ((W7t*(x,t) +
va(x,t»[P7t(x,t)m7t(x,t) + t'x<x,t)]}
1t=1 1t=N
with SI*(x,t) '" SIva(x,t) = L {[P7t(x,t)V7t(x,t) ® (V7t(x,t) -
va(x,t)) + crn(x,t)]) 1t=1
These examples show that the generalized total stress tensor SI*
depends on the virtual velocity field v* of the virtuai domain
00*.
Generalized Approach to Heterogeneous Media
(b) Lagrange point of view
The Lagrange differential momentum balance relation is deduced from
(1.136) :
(1.149)
1t=k
x=k a L at [P1t*(X* ,t) (W 1t*(X* ,t) + V*(X* ,t»]
x=1
= - L Div{ [bx*(X*,t) 1t=1
+ P1t*(X*,t)(W 1t*(X* ,t)+ V*(X* ,t»®W 1t*(X* ,t)][!i.twlT f*(X*
,t)]-1 }
43
1t=k A
+ L {P1t*(X* ,t)1' 1t*(X* ,t)+(W 1t*(X* ,t)+ V*(X* ,t))[P1t*(X*
,t)M1t*(X* ,t)+Cx*(X* ,t)]} 1t=1 x=k A
+ L P1t*(X*,t) x=1
Special case of the total medium
For a total heterogeneous medium, with k = N constituents, relation
(1.137) gives:
(1.150) 1t=N 1\
L P 1t*(X,t) = 0 1t=1
In this case, a condensed differential relation can be written for
the total medium:
ap*(X*,t)VB*(X*,t)
at (1.151)
= - Div{ ~**(X*,t)[!iJ:wlTf*(X*,t)]-l }+P*(X*,t) F*(X*,t) 1t=N
1\
+ L [(W 1t*(X* ,t)+ V*(X* ,t))[P1t*(X* ,t)M1t*(X* ,t)+C1t*(X* ,t)]]
x=1
using definitions (1.116) and (1.118) and defining ~**(X*,t) and
1'*(X*,t) as :
(1.152) 1t=N
(a) ~**(X* ,t) == L {bx*(X* ,t)+P1t*(X* ,t)(W 1t*(X* ,t)+ V*(X*
,t»®W 1t*(X* ,t)} 1t=1 1t=N
== L ~1t**(X*,t) x=1
1t=N (b) f'*(X*,t) == 1 L P1t*(X*,t)f'1t*(X*,t)
P*(X*,t) 1t=1
44 P. Jouanna and M-A. Abellan
1.2.4 Total energy balance relations
Quantity Qx(t) relative to total energy is obtained by (1.60) and
(1.87c) with 'l'1t == eUt :
(1.153) Qx(t) == f P1t(x,t)'I'1t(x,t)dco1t = f
P1t(x,t)eUt(x,t)dco1t
A Integral forms of total energy balance
Al For one constituent 1t
(a) Euler point of view
• Integral total energy balance relation along the movement of
constituent 1t
F1ux and source terms appearing in the Euler synthetic integral
balance relation (1.62) applied to total energy along the movement
of constituent 1t are as follows :
(i) F1ux nx['I'X= eUt] : the flux of total energy nx[eUt]
represents the flux !I1tV1t and the heat flux vector q1t through a
surface element da1t which is moving at velocity V1t. Hence :
n1t[eUt](x,t) = .o:n;(x,t)V1t(x,t) + q1t(x,t)
(ii) External volume source ~['I'1t== eUt] : the external volume
source ~[V1t] of total energy for constituent 1t in the domain
0Jn:, per unit volume of the heterogeneous medium and by time unit,
includes :
A A - the power supply Il1t(P1tmx+Cx) due to the rate of external
mass supply P1tmx+Cx
from the outside of 0Jn:, ll1t being the specific chemical
potential of 1t. By definition, the specific chemical potentialll1t
is the chemical energy related to a unit mass.
- the kinetic power V1t[~ v1t(p1tm1t+~)] due to the momentum supply
rate v1t(p1tm1t+~) provided to constituent 1t,
- the power P1tf1tv1t of external volumeforces,
- the extemal heat power P1tr1t - and apower supply ~ from the
other constituents.
A ~1t[et1t](x,t) = ~1t(x,t)[P1t(x,t)m1t(x,t)+Cx(x,t)]
1 A + V1t(x,t) 2" V1t(x,t)[P1t(x,t)m1t(x,t)+Cx(x,t)]
A + P1t(x,t)f1t(x,t)v1t(x,t) + P1t(x,t)r1t(x,t) + ex(x,t)
(Hi) Internal volume source 1..1t['I'1t== et1t] : according to the
principle of total energy conservation, the inner source of total
energy within the constituent 1t is equal to zero.
i.x[ eUt] = 0
Generalized Approach to Heterogeneous Media 45
Hence relation (1.62) and above flux and source terms yield the
explicit expression of the Euler integral total energy balance
relation for one constituent x following Vx :
(1.154)
C1)x
J 1\ 1 1\ + (Ilx(x,t)[px(x,t)mx(x,t)+ex(x,t)] +
vx(x,t)2"vx(x,t)[px(x,t)mx(x,t)+ex(x,t)]
1\ + Px(x,t)fx(x,t)vx(x,t) + px(x,t)rx(x,t) + ex(x,t) }drox
• Generalized integral total energy balance relation along v*
The flux and source terms of the synthetic balance relation (1.64)
with 'l'x(x,t) == et7t are :
(i) Flux fa,x*['I'x== etx] : the expression of fa,x*[etx] is
obtained by a direct application of formula (1.65a) :
(1.155) fa,x*[etx](x,t) = fa,x[etx](x,t) +
px(x,t)et7t(x,t)(vx(x,t)-v*(x,t»
= .Q:x(x,t)vx(x,t) + qx(x,t) +
px(x,t)etx(x,t)(vx(x,t)-v*(x,t»
(ii) The extern al volume source is given as above by ~['I'x=
etxl.
(iii) The internal source i.n:['I'x== etx] is given as above by
i.n:['I'x](x,t) = O.
Hence relation (1.64) and above flux and source terms lead to the
Euler generalized integral total energy balance relation for one
constituent x following the movement v* :
46 P. Jouanna and M-A. Abellan
(1.156) f (~ [P1t(x,t)eUt(x,t)] + div[p1t(x,t)eUt(x,t)v*(x,t)]
}doo*
00*
= - J (P1t(x,t)eUt(x,t)W1t*(x,t) + 2n;(x,t)(W1t*(x,t)+v*(x,t» +
q1t(x,t) }n*(x,t)da* a*
+ J (J.11t(x,t)[P1t(x,t)m1t(x,t)+~(x,t)] 00*
1 1\ +
(W1t*(x,t)+v*(x,t»2<W1t*(x,t)+v*(x,t»[P1t(x,t)m1t(x,t)+Cx(x,t)]
1\ + P1t(x,t)f1t(x,t)(W1t*(x,t)+v*(x,t» + P1t(x,t)r1t(x,t) +
ex(x,t) }doo*
(b) Lagrange point of view
• Integral total energy balance relation on the reference domain
Q1t
The flux and source terms of the synthetic relation (1.72), applied
with 'P 1t(X1t,t) ==
EUt(X1t,t) , are as follows :
(i) Flux K 1t['P 1t== EUt] : the flux K 1t[EUt] is derived from the
flux nx[eUt] by (1.69)
K1t[Et1t] = n1t[eUt] J1t(X1t,t) = !:.ax(x,t)V1t(x,t) + q1t(x,t)]
J1t(X1t,t)
K 1t[EUt] = b;(X1t,t)V 1t(X1t,t) + Q1t(X1t,t)
according to the definition (1.125) of ~(X1t,t) and the following
definition ofQ1t(X1t,t)
(1.157)
(ii) External volume source x,1t['Px= EUt] : according to (1.70)
and the above expression
of ~[eUt]:
X,1t[EUt](X1t,t) = ~1t[et1t](x,t)J1t(X1t,t) 1\ 1 1\ =
(J.11t(x,t)[P1t(x,t)m1t(x,t)+Cx(x,t)] + V1t(x,t) 2
V1t(x,t)[P1t(x,t)m1t(x,t)+Cx(x,t)]
1\ + P1t(x,t)f1t(x,t)V1t(x,t) + P1t(x,t)r1t(x,t) + ex(x,t)
}J1t(X1t,t)
Generalized Approach to Heterogeneous Media
with the following definitions :
Rx(Xx,t) == rx(x,t) = rx(f x(Xx,t),t) 1\ 1\ 1\ Ex(Xx,t) ==
ex(x,t)Jx(Xx,t) = ex(fx(Xx,t),t) Jx(Xx,t)
(iii) Internal volume source 'Lx['I'x= Eud : according 10
(1.71)
'Lx[Eut1(Xx,t) = 1.x[em;](x,t) Jx(Xx,t) = 0
47
Hence, according to (1.72) and the above flux and source tenns, the
Lagrange integral total energy balance relation, expressed on the
reference domain On: of the constituent x, is :
+ f (Jlx(Xx,t)[Px(Xx,t)Mx(Xx,t)+Cx(Xx,t)] Ox
1 1\ + V x(Xx,t)2V x(Xx,t)[px(Xx,t)Mx(Xx,t)+Cx(Xx,t)]
• Integral total energy balance relation on the virtual reference
domain 0*
The flux and source tenns of the relation (1.78) with 'I' x*(X*,t)
== Em*(X*,t) are :
(i) Flux H x*['I' x*== Em*] : according to (1.75) and the above
expression of h.x*[em*]
Hx*[Em*] = nx*[ etx] (x,t)J* (X* ,t)
= (!ht(x,t)(wx*(x,t)+v*(x,t» + qx(x,t) +
px(x,t)etx(x,t)wx*(x,t)}J*(X* ,t)
= ~*(X*,t)(Wx*(X*,t)+V*(X*,t» + Qx*(X*,t)
+ px*(X* ,t)Etx*(X* ,t)W x*(X* ,t)
with the following definition:
(1.162) Qx*(X* ,t) == qx(x,t)J*(X* ,t) = qx(f*(X* ,t),t)J*(X*
,t)
48 P. Jouanna and M-A. Abellan
(ii) External source term 1Gx*[\f' x*== Etx*] : according to (1.76)
and the above value of
~[et7t1 :
1Gx*[Etx*](X*,t) = k.x[etx](x,t) J*(X*,t)
" = {Ilx(x,t)[px(x,t)mx(x,t)+ex(x,t)]
1 " + (w x * (x,t)+v* (x,t) )2<w x* (x, t)+v* (x, t» [Px(x,
t)mx(x, t)+ex(x, t)]
" + Px(x,t)fx(x,t)(wx*(x,t)+v*(x,t» + px(x,t)rx(x,t) + ex(x,t)
}J*(X* ,t)
" = J.!x*(X* ,t)[px*(X* ,t)Mx*(X* ,t)+Cx*(X* ,t)]
+ (W x*(X* ,t)+ V*(X* ,t»~W x*(X* ,t)+ V*(X* ,t))[px*(X* ,t)Mx*(X*
,t)+C'\*(X* ,t)]
" + px*(X*,t)f'x*(X*,t)(Wx*(X*,t) + V*(X*,t» + px*(X*,t)Rx*(X*,t) +
Ex*(X*,t)
with the definitions:
(1.163)
(1.164)
(1.165) " " " Ex*(X*,t) == ex(x,t)J*(X*,t) =
ex(f*(X*,t),t)J*(X*,t)
(iii) Internal source term 'L*[\f' x*== Etx*] : according to
relation (1.77)
'Lx*[Etx*](X*,t) = ix[etx](x,t)J*(X*,t) = 0
Hence, the Lagrange integral total energy balance relation, on the
virtual reference domain n*, is expressed by :
(1.166)
= I * {[px*(X*,t)Etx*(X*,t)Wx*(X*,t) +
bx*(X*,t)(Wx*(X*,t)+V*(X*,t»
+ Qx*(X*,t)][G.r.wI.Tf*(X*,t)]-1 }N*(X*,t)dA*
+ ! * {J.!x*(X*,t)[px*(X*,t)Mx*(X*,t)+C\*(X*,t)]
+ (W x*(X* ,t)+ V*(X* ,t»~W x*(X* ,t)+ V*(X* ,t) )[px*(X* ,t)Mx*(X*
,t)+Cx*(X * ,t)]
1\
+ px*(X* ,t)f' x*(X* ,t)(W x*(X* ,t)+ V*(X* ,t» + px*(X* ,t)Rx*(X*
,t)+ Ex*(X* ,t) }dn*
Generalized Approach to Heterogeneous Media 49
A2 For a set of constituents
(a) Euler point of view
The Euler total energy balance relation for a set of k constituents
is directly obtained by summing up relations (1.156) along the
virtual movement v*.
(1.167)
7t=k f = - I.
{P7t(x,t)et7t(x,t)w7t*(x,t)+Q7t(x,t)(W7t*(x,t)+v*(x,t))+Q7t(x,t)}n
*(x,t)da * 7t=1 a *
+ (w 7t * (x, t)+v* (x, t) )i(W 7t *(x, t)+v* (x, t) )[P7t(x, t
)m7t(x, t)+~( x, t)] 1\
+ P7t(x,t)f7t (x,t)(W7t*(x,t)+v*(x,t)) + P7t(x,t)r7t(x,t) + en(x,t)
}d<O*
Special case of the total medium
With k = N constituents, an equivalent form is :
(1.168) 7t=N f I. ~(x,t)d<o* = 0 7t=1 <0 *
50 P. Jouanna and M-A. Abellan
(b) Lagrange point of view
For a set of k constituents, the Lagrange total energy balance
relation is obtained by summing relations (1.166) on the reference
domain 0*.
'ik f ~ [P1t*(X*,t)Et1t*(X*,t)]dO* ~1 0*
(1.169)
~k f = - l: ([P1t*(X* ,t)Et1t*(X* ,t)W 1t*(X* ,t) ~1 A*
+ Ln:*(X* ,t)(W 1t*(X* ,t)+ V*(X* ,t» + Q1t*(X* ,t)]lGr.wlT f*(X*
,t)]-l } N*(X* ,t)dA *
~k f A + l: (J..L1t*(X* ,t)[P1t*(X* ,t)M1t*(X* ,t)+C1t*(X*
,t)]
1t=1 0*
+ (W 1t*(X* ,t)+V*(X* ,t»~W 1t*(X* ,t)+V*(X*,t))[P1t*(X* ,t)M1t*(X*
,t)+C1t*(X* ,t)]
A +P1t*(X* ,t)1' 1t*(X* ,t)(W 1t*(X* ,t)+ V*(X* ,t» + P1t*(X*
,t)R1t*(X* ,t) + E1t*(X* ,t) }dO*
Special case of the total medium
For a total heterogeneous medium with k = N constituents the
Lagrange generalized differential total energy balance relation can
be written in an equivalent form :
(1.170) 1t=N f l: ~*(X*,t)dO* = 0 1t=1 o*
B Differential forms of total energy balance
BI For one constituent 1t
(a) Euler point of view
• Differential total energy balance relation along the movement of
1t
Integral relation (1.154) leads to the differential form :
(1.171) ~ [P1t(x,t)et1t(x,t)] +
div[p1t(x,t)et7t(x,t)V1t(x,t)]
+ J.l.1t(x,t)[P1t(x,t)m1t(x,t)+Cx(x,t)] +
V1t(x,t)2"V1t(x,t)[P1t(x,t)m1t(x,t)+Cx(x,t») A
+ P1t(x,t)f1t(x,t)V1t(x,t) + P1t(x,t)r1t(x,t) + ex(x,t)
• Generalized differential total energy balance relation along
v*
Integral relation (1.156) leads to :
(1.172) ~ [P1t(x,t)etn;(x,t)] + div[p1t(x,t)et1t(x,t)v*(x,t)]
= - div[p1t(x,t)et1t(x,t)W1t*(x,t) + Q:1t(x,t)(W1t*(x,t)+v*(x,t» +
q1t(x,t)]
" + 1l1t(x,t)[P1t(x,t)m1t(x,t)+en;(x,t)]
1 " + (w 1t* (x, t)+v* (x,t»2( w 1t *(x, t)+v* (x,t» [P1t(x,
t)m1t(x, t)+ en;(x, t)]
" + P1t(x,t)f1t(x,t)(W1t*(x,t)+v*(x,t»+ P1t(x,t)r1t(x,t) +
en(x,t)
(b) Lagrange point of view
• Differential total energy balance relation on the reference
domain 01t
Integral relation (1.161) leads to : a
(1.173) dt [P1t(X1t,t)Et1t(X1t,t)]
" + J..l1t(X1t,t)[P1t(X1t,t)M1t(X1t,t)+C1t(X1t,t)]
51
• Differential total energy balance relation on the virtual
reference domain n*
Integral relation (1.166) leads to :
(1.174) ~ (P1t*(X* ,t)Et1t*(X* ,t)
= - Div{[P1t*(X*,t)Et1t*(X*,t)W1t*(X*,t)
+ ~1t*(X*,t)(W1t*(X*,t)+V*(X*,t» + Q1t*(X*,t)]
LG,ra,d.Tf*(X*,t)]-l}
" + J..l1t*(X* ,t)[P1t*(X* ,t)M1t*(X* ,t)+C1t*(X* ,t)]
+ (W 1t*(X* ,t)+V*(X* ,t»~W 1t*(X* ,t)+ V*(X* ,t))[P1t*(X*
,t)M1t*(X*,t)+C1t*(X* ,t)]
" + P1t*(X* ,t)f' 1t*(X* ,t)(W 1t*(X* ,t)+V*(X* ,t» + P1t*(X*
,t)R1t*(X* ,t) + En*(X* ,t)
52 P. Jouanna and M-A. Abellan
B2 For a set of eonstituents
(a) Euler point of view
The integral fonn of the balance relation (1.167) leads to the
generalized differential fonn :
(1. 175a) 1C=k l: {i [Pn;(x,t)et1r;(x,t)] +
div[Pn;(x,t)etn;(x,t)v*(x,t)]}
1C=1 1C=k
1C=k A + l: {Jln;(x,t)[Pn;(x,t)mn;(x,t)+en;(x,t)]
1C=1 1 A
+ (w n;*(x,t)+v* (x,t) )2<wn;*(x, t)+v* (x,t)
)[Pn;(x,t)mn;(x,t)+ en;(x,t)]
A + Pn;(x,t)fn;(x,t)(wn;*(x,t)+v*(x,t» + Pn;(x,t)rn;(x,t) +
en;(x,t)}
Special ease of the total medium
For a total heterogeneous medium, with k=N constituents, relation
(1.168) gives :
(1. 175b)
1C=1
n=N iJ n=N l: { P~;t1t} = - l: (div[pnemv7t] + div(a:I1:V7t) +
divq7t}
n=1 n=1 n=N
A 1 A + l: {~7t[p7tmn;+c 7t] + V7t2"7t[P7tmn;+c ru + P7tf7tv7t +
P7tr7t}
7t=1 • If the virtual velocity v*(x,t) 55 VB(X,t) :
n=N iJ(p e ) n=N l: { ~t t1t + div[pnemvB]} = - l:
(div[pnet7t(V7rVB)] + div(a7t(w7t*+vB» + divq7t}
n=l n=1
n=N A 1 A
+ l: {~7t[p7