modern issues in non-saturated soils

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
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© 1995 by Springer-Verlag Wien
Original1y published by Springer-Verlag Wien New York in 1995
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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-hydro 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.
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)
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*
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*
(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
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
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
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 :
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 *(X*,t) with respect to Euler variables
Using relation (1.3a), a function *(X*,t) can b:! considered as depending on (x,t) and written *(F*(x,t),t).
• Gradient with respect to x of a function *(X*,t)
The gradient with respect to x of *(X*,t) can be defined using (1.23) as follows :
(1.36) a
grad *(F*(x,t),t) == Ha *(F*(x,t),t» I x t = constant
• Time derivative of a function *(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 (*(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 *(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 *(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*
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*
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(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*
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)
• 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
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)
(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 :
(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
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*
A2 For a set of constituents
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*
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
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)
• 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
B2 For a set of constituents
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
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.
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
• 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*
• 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 :
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
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
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)
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*
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
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)
• 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
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
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.
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
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
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
• 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)
(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:
(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*
• 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)
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
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*
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) *
(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 *
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
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 *
B Differential forms of momentum balance
BI For one constituent 1t
• 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)
• 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*
(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)
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
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 ) :
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*.
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
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
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
• 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
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* :
(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*
• 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)
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)
(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*
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*
With k = N constituents, an equivalent form is :
(1.168) 7t=N f I. ~(x,t)d<o* = 0 7t=1 <0 *
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*
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
• 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*
(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)
• 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*
(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)
B2 For a set of eonstituents
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

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