Variational irreversible thermodynamics of physical-chemical solids with finite deformation

IN. 1. .widJsuwum Vd. 14,Dp.11-903 0 PermmnPrm Ltd..1978. Printed inGmt hitin

VARIATIONAL IRREVERSIBLE THEIWODYNAMICS OF PHYSICAL-CHEMICAL SOLIDS WITH FINITE DEFORMATION M. A. Blur Royal Academy of B&iom, BNSS&,

Belgium

(Received 2l April 1978; received for publication 12 June 1978)

AbbrGA new thamodynamics of open thmochemicai systems and a variahai phciple of virtual diisip8tionarcappliadtotbafiaitcdcfonnationofasolidcouplcdto -ld8rdafusionsad cbemicplnrtioas.Av~wPIdnivatioPisobfPinedoftbehelddiflenarhlequrtionsnswelles ~squrtioaswithBeaetalized~s.Newformuhsforthe~podanewdcRnitionof thechemialpo(sndalpnpmented.Anou~isgivenofraua~ylargcfiddofrpplicrtions,s~as active tranaporlin biobgkal systems. finite ekmant methoda.phatic prop&as as anabgou to chemical rcacths, phase chatgcs sod rrcryrtalization,poroos solids, heredity and initiaUystressed solids. A new and unilkd insight is thus providedin highly divcrsifiadproblems. 1. INTRODUCTION

formulated initially in 1954-55[1,2]was extended more recently to nonlinear thermorheokgy[3]. At the same time a new approach to the thermochemistry of open systems was developed [4-6] which provides a new foundation of classical thermodynamics and avoids the traditional dilhculties and ambiguities of Gibbs’ classical treatment [71. This paper is an application of this new thermodynamics to the problem of finite deformation of a solid, with substances in solution, subject to chemical reactions and thennomokcular diffusion. Two special cases of this problem have been developed earlier. Gne of these excludes chemical reactions and is presented in the context of the similar problem fbr porous solids[8]. The other includes chemical reactions but assumes small perturbations of a solid in the vicinity of a state of equilibrium with initial stress [9]. The basic concepts of the new thermodynamics of open systems such as the thermobaric potential are briefly recalled in Section 2. Their application to chemical reactions and a new expression for the ffiity are developed in Section 3 along the lines developed earlierl4-61.This is applied in Section 4 to an open solid undergoing homogeneous deformations while coupled chemical reactions are occurring internally. The analogy with nonlinear thermoviscoelasticity is pointed out. The basic variational principle of virtual dissipation is formulated in Section 5 in the context of a deformable solid continuum with thermomokcular diffusion and chemical reactions. The differential field equations which govern the evolution of the continuum are detiued variation&y in Section 6. The foregoing results are based entirely on classical thermodynamics. As shown in Section 7, two additional axioms, one of which involves Nemst’s third principle, lead to a new defmition of the chemical potential which bypasses the need of introducing the principles of quantum statistics. A complementary form of the field equations are then derived in Section 8. Application of tire principle of virtual dissipation in Section 9 to a system described by generalized coordinates as unknowns, leads directly to Lagrangian equations for those unknowns without recourse to the field equations. In .Section 10 it is pointed out how the Lagm@an approach is particularly s&d to the analysis of biological systems as already illustrated for the treatment of active transport in biological membranes [6]. Section 11 recalls that Lagmngian methods provide the foundation of a large variety of 6nite element methods. The analogy between plasticity and chemical reactions from the standpoint of internal coordinates is brought to light in Section 12. How the problems of creep and recrystalization under stress coupled to phase changes may be treated by the present methods is briefly discussed in

The variational

SSVd. 14.No. I I-A

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Section 13. Earlier treatments of porous solids, the thermodynamics of heredity and the solid under initial stress are recalled in Sections M-16. The case of a solid under initial stress is not treated as a bifurcation and thereby is more general since it is not necessary to assume the existence of an unstressed state. 2.NEWT~ERMODYNA~IC~OFANOPENCELL

A new approach to the thermodynamics of open systems has been introduced and discussed extensively in some earlier publications[4-6]. The development has been accomplished entirely within the framework of classical procedures and without recourse to the principles of statistical mechanics. The results are directly applicable to a deformable solid with pure substances in solution as already described in the similar case of a porous solid saturated by viscous fluids[t?f. We shall briefly rederive here the essential concepts by using a slightly different and more direct approach. We start by considering a cell C, called a primary cell, which is first assumed rigid, contai&g a mixture of substances k at the temperature T. To this cell we adjoin large rigid reservoirs Cti called supply cells, each confining a pure substance k at the pressure and temperature p&Iii the same for all supply cells. It was shown that this co&i&n of u~orm values pe and T,,,for C**is required in order to avoid Gibbs’ paradoxM 51.We also adjoin to this system a large isothermal rigid reservoir at the temperature TOcalled a thermal well, ZW. The total system C,,+ C’ Clk+ TW is called a hypersystem while the subsystem C, +X’ Csk will be referred to as a collective system. We have considered reversible transforma&ms of this hypersysfem whereby masses and heat are transferred within the system by ~~o~~g extemaf wok OBthe system.‘I&s implies the use of reversible heat pumps. The supply cells C& and the thermal welI are assumed large enough so that POand TOremain constant in the process. The increase of internal energy $ of the hypersystem from a given initial state defines what we have called the collective potential of the primary cell C,,.The justi&ation for this de&&on is derived from the fact that $ is determined completely by the state variables of C,. This can be seen as follows. We shall first consider the case where the primary cell is rigid and contains a mixture of non reacting pure substances k at the temperature T. The cell C, may be a solid in which the various substances are in solution. The case of reacting substances isconaided inthenext section. ‘IIre mass mk of each substance in C, is written

where mokis the initial mass and M’ is the mass of each of the substances acquired by the open cell during a transformation. The state variables of C, are the masses Mk, and the temperature T. Since the masses &f”:are assumed to be provided entirely by the supply cells cSkt the same variables Af”: are atso the state variables of the supply cells. Therefore the stareof the c~fecfive system C, + 5 CSkis determined completelyby the variablesN’ and T. As a consequence the

increase of collective energy Le and of collective entropy Y of the system C, + 1: c3k may be expressed as functions of ML and T. We write Q = ~(~k,T) Y =

,49(Nk,T).

(2.2)

We have defined the collective potential ,$ as the increase of energy of the hypersystem k

C, +Z Cd f TW in the reversible ~sfo~tion. $=%+Ho

Its value is (2.3)

where HOis the heat energy acquired by the thermal well. Since the transformation is reversible

V&&ond

irwversibfe

thermodynamics

of physical-chemical

there is no entropy change of the h~~ys~rn, 9 =-

solids with finite deformation

883

C, + i Cti + ?%‘,hence HO

(2.4)

TO

and $=%-T&T

(2.5)

Since Q and 9 are functions of the state variabkzsML and T of the primary cell the collective potential 3 is also determined by the same variables. d = $CMk,T,. Because of these properties we may as the ceU enerlpy,

the ceil

entrvtpy

and

(2.6)

drop the term collective and refer to Q, 9’ and $ respectively the cefl potential, keeping in mind of course that they arc

defined here in a new way as coUcetiveconcepts. The thermodynamic function (2.5)was introduced as a fundamental potential by the author [ 1,2]in a more restricted context and applied by Mindlin to piezoelectric crystah[lOJ. According to eqn (2.1) Mk=m-mW.

(2.7)

Henec we may write 9 = ‘P1(mk,T), 9’ = y(maT),

9 =dP(m,,T)

(2.8)

as fun&m

of tbc tcmpct8turc and the total maSScSmk of the SubstanceSk in S&t&r m the cell. in order to evaluate the changes of tire thermodynamic functions B, .Y,/ a~~ociatcdwith a change of state of the open cell C, we have introduced the new key concept of t&mobaric transfer14-61dcScribcdas follows. Consider a pure substanec k in quip with the primary ceil C, through a semipermeable membrane. In this quilibrium state the substancek is at a prWurC pk and at thC SatINtemperature T as c. The preSSUrC pk Of the substance under these eon&ions is called the partial pwzwure of the substanec in the mixture. The proecss of thcrmobarie Wer of a mass d&f’ from the supply cell & to the primary cell C, is a revctsrble process by which the mass is fimt extracted from the supply celi, eompresScd and heated to the partial pressure pk and tetnperature T and then injected reversibly and a&batically through the semi-permeable membrane. The heating along this path is aceompli~hed by a reversible pump operating between ZW and dMk and injceting a differential amount of heat into dM” at each step. The process is descriid in more detail in Rcfs.[4, 51. The increase of collective energy and entropy of the system C, + i c& in this process of transfer are written

~c~ob~c

de = ;k dMk dY=&dM’

(2.9)

where gk

=

(2.10)

The diEercntial di is the increment of entropy per unit mass of substance k at each step of the thcrmobaric transfer. Similarly (2.11)

M. A. BIOT

884

is the increment of enthalpy per unit mass of each substance at each step, at the variable pressure pi variable density pi and variable temperature T’ along the path of integration. The variables gk and Sk are independent of the path of integration and have been called respectively the relative specific enthalpy and entropy of the substance in C,. In contrast with traditional ptvcedures these definitions do not involve any undetermined constants. The change in cell potential corresponding to the thermobaric transfer is according to (2.5) d.$=d%-TodY.

(2.12)

d>=&dMk

(2 13)

Substitution of the values (2.9) yields

where

has been called the themwbaric potential. From the detition of ..$ we note that & represents the external work req&ed for the thermobaric transfer of a unit mass of the particular substance. This may be verified by introducing the vahres (2.10) of 4 and & and putting 8’ = T’ - To. We obtain

(2.15) ‘The first term represents the work of the pressure on the unit mass including the negative work of extra&on from the supply ceil and the positive work of injection inte C,. The second term 8 d& is the work of the heat pump at each step along the path. Hence ch( is effectively the ext.emaI revera& work reqt&d in the them&&c transfer. If seveA masses are injected we write d$&&diUk.

(2.16)

wehaveassua#dBatnoadditionelheathrtddedtoC,draiaethereveniMein)6ctionofthe mass dML. Consider now that by using a heat pump operatitrg between C, and TW we inject reversibly into C, an amount of heat T dsr at the same time as the masses &. We obtain d%=&&U’+Tdsr

dY = 2 & dMk + dsr.

(2.17)

(2.18)

Substitution in expression (2.12) yields the increase of cell potential. (2.19) where 6 = T-T,,.

(2.200)

Again here we recognize the work 8 &T accomplished by the heat pump to inject the heat T dst into C,, operating between the temperatures To and T. The variable dsr is not a state variable. Elimination of dsT between eqn (2.19) and eqn (2.18)

Variational imvcrsible thermodynamics of pbysicalhemical

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885

yields (2.21) where

was introduced earlier as the convectivepotentinl. We may obtain the value of ,$ by integrating (2.19)along any convenient path. For example we first integrate at constant temperature T = TOthen closing the cell (dMk = 0) we raise the temperature to T. This yields 9 = B(M”,T).

(2.23)

The same path of integration may be used to integrate eqn (2.18)for dY’.We derive Y = Y(Mk,T).

(2.24)

We note that along the path of integration the partial pressures pk which appear in the difIerent.ialcoefficients are assumed to be known as functions of M” and T. By eliminating T between eqns (2.23)and (2.24)we derive B = BW’,Y). 3. NEW CHEMICAL

TIiERMODYNAtiICS

(2.25) OF AN OPEN CELL

The new concepts and results for open systems also lead to a new chemical thermodynamics[4, 51which we shall briefly outline. These fundamental results are obtained without the use of statistical mechanics. We consider again a rigid and open primary cell C, with its adjoined supply cells Cti and its thermal well TW. A chemical reaction may now take place between the various substance mixed in C,,. This chemical reaction is represented by the equation dt?lk= $r+. dt

(3.1)

where 6 is the reaction coordinate, and dmk are the masses of the various substances “produced” by the reaction. The term “produced” is understood in a generalized sense so that negative values represent substances disappearing in the reaction. Conservation of mass implies the relation $dt?lk=O

(3.2)

2

(3.3)

vk

-0.

Since a chemical reaction is generally irreversible and associated with an entropy production, in order to evaluate the collective potential by the procedures outlined above, we must construct a system such that the change of state resulting from the chemical reaction may be obtained by an quivalent reversible process. Such a process may be described as follows. Consider the reaction to occur in a rigid closed adiabatic cell. ‘Ibe reaction dt produces a change of composition and temperature of the cell from state (1) to state (2). In order to accomplish the same change of state reversibly we tirst

M.A. BIOT

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bring the cell to an intermediate state (I) where the reaction is in eq~ib~um. This is obtained by using thermobaric transfers and heat pumps, while freezing the reaction. At equilibrium we allow the reaction dt to occur thus reaching the state (2’).We then bring the cell to the final state (2) again using thermobaric transfers and heat pumps. It is important to note that during this reversible process the state of the supply cells is the same in the initial and Gnal state since the masses extracted during one part of the process are restituted during the other. Hence the change in the collective system is the same as due to a reaction in the primary cell alone. The work accomplished on the hypersystem during this process is the increase djch of cell potential. We may write

d.A~= d%, - TodYlch

(3.4)

where da,,, is the increase of internal energy and dYCp,h the increase of entropy as defined above in terms of collective concepts. However when the change occurs in the adiabatic closed cell as a chemical reaction, d& = 0. Hence

where dYCbmay be interpreted as the entropy “produced” by the reaction. Following De Dander [ I 11the &nity A is deGnedby the relation

Hence

In the case of chemical eq&ihrium d.% = A = 0.

Ontheothefhaadforan~cdl~~chemicalrcactionwedenotebyddli’aad~the values (2.18)and (2.19)found previously. We write

dY = 2 A diU’ + dsr.

(3.8)

When adding a chemical reaction we find

(3.9) EIimination of dsr between these two equations yields d#= -Adb+&AdM’+6d$‘.

(3.10)

Note that the internal energy does not depend on df since for a ckaed adkhatic cell uo energy is provided to the ceil (d.@ = dsr = 0). Hence eqn (2.17)remains valid, i.e. d%-&dM”+TdsT.

(3.11)

Variationalineversibk thermodynamics of phys~~~~rnk~ solis with ilnitedeformation

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Based on these collective concepts and eqns (3.9) we have derived new expression for the heat of reaction and the afRnity[4-6].We shall briefly outline this derivation. We consider a hypersystem constituted by two rigid cells C,,C, and a thermal well ZW.The cell C,, is the primary cell and C, is a cell of composition and temperature such that the chemical reaction considered is in equilibrium, We assume a reaction d( to occur in C, while the reverse reaction -df occurs in C, As the reaction proceeds we. remove the products dmk = vkdt from C, and inject them by thermobaric transfer into C,..Using a heat pump we inject into C,, the amount of heat &r d@required to maintain its temperature constant, The temperature of C, is also maintained constant by injecting the amount of heat - gPTd& The composition and temperature hence also the pressure of the cells C, and C, do not vary. The supply cells remain unchanged since they are not involved in the process just described. If we denote by dlol and dV the change of energy of C,, and C, respectively, we may write d%+dlplq=O

(3.12)

since this is a consequence of the fact that no change occurs in the collective system C, + c- + $ Cd. Appl~ng eqn (3.11)with dM’ = - q dt we obtain

(3.13) where Z;9 is the reiative specific enthalpy of each substance in C,. We substitute these values in eqn (3.12)taking into account the relation (3.14) where ph

and Tq are the partial pressures and temperature in C,. We derive (3.15)

In this expression, obtained earIier[l, 51,&,r is a new concept caIkd intrinsic keut of reucrion.It is obtained by removing the products as the reaction proceeds at constant temperature. It is more representative of the chemical energy than the traditional concept which includes the heat of mixing as defined earlier [4,5J. We may write (3.15)in differential form as d&r = 2 vkd&

(3.16)

which generalizes completely and rigorously Kirchhof’s classical result for the heat of reaction[l2]. Consider now the entropy change of the COtieCthe system C, + & + t c&. The changes due to C, and C, are respectively d9’ and dsP,. Since the collective system does not change we write dY+dYq =O.

(3.17)

Apply the second of eqns (3.9). we write

(3.18)

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In the second equation, A = 0, since the reaction is in equilibrium. The relative specific entropy in C, is denoted by Siq. We may write the relation jk - jzq =

I

PIT

d&.

(3.19)

~&a,

Taking into account this relation after substituting the values (3.18) into eqn (3.17) we derive djk

+!k!b

Teq T ’

(3.20)

This is the new expression for the afIinity already obtained in earlier workI4, 51 Ebmination of ST between eqns (3.15) and (3.20) yields (3.21) where (3.22) The value (3.21) of the aBuity is a rigorous consequence of classical thermodynamics. We shah see below (Section 7) by the use of additional axioms how another expression may be derived in more familiar form. When several reactions occur in the primary cell C, the mass dnr of a particular substance produced by the reactions is

dm=$epdf,

(3.23)

where & are the coordinates of the various reactions. Adding the effects of each reaction in eqns (3.9) we obtain

d%

P--

dY=T

2%A,df,+&&fk+~dsT 1

e A,,d&,+e&dMk+dst

(3.24)

where Ap is the affinity for each reaction. Elimination of dsT between these two equations yields (3.25) In these expressions the state variabks are f, Mk and 9. The masses M’ added by convection are considered as distinct and independent from those drnk = 5 vt, d& produced by the reactions and which depend only on the chemical coordinates Q THERMOMECHANICSANDCHEMiCALKINETICSOFANOPENDEFORWABLECELL In the foregoing analysis we have assumed the primary ceil to be rigid. We shah now consider the cell to be deformable. In the initiaI state let it be a cube of unit size oriented along the coordinate axes xi. It may be an open cell containing masses mu of pure substances in solution at uniform temperature T. The state variables of this cell may be chosen to be the reaction coordinates I, the masaea Mk added by convection, the entropy Y and six strain components + 10 this section we consider the strain to be homogeneous. 4.

V8riaM

imvmibk

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solids with finite deformation

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There arc various ways of measuring the Rnite strain which have been the object of numerous discussions and applications by the author. We shall briefly recall the essential concepts. The homogeneous deformation and solid rotation of the unit cube are represented by the afiine transformation

where xi and 4 are the coordinates of material points before and after deformation. Struin components may be defined in various ways. Green’s tensor is

We may also use a definition introduced in 1939 by the author which avoids many of the dithcultks attached to Green’s tensor in applications and was developed extensively in a rnon~hrl31, fn this definition we first perform an aitinhc ~~~~n 4 - C&f+ ry)x/

(4.3)

putting Ed= efi followed by a solid rotation such that the total transformation is equivaknt to (4.1).The six independent values of lifdefine the strain. They are functions of 01~To the second order we derive (4.4) with

Oii .=

3

-

Of/ -

Uii).

(4.5)

Gther similar &~~ of Q may be used which are non tensorial. For cxampk in two dimensions we may write

and use q t en t12 as measure of the finite strain. To the second’order their vahres were shown to ber141 12

61 = fzll +-&I21

2

l==0g

l

-$22,@*2+

I2=~(~*2+Ll2*)+fa2,(ctn-n,,).

ad

(4.7)

When using the general notation * for this case, we put c21= 0. A large number of variationsof this

of definition are possii in two and three dimensions as indicated fl4J. The ~~~n~ thus ffied are nontensorial but in many probkms this is an advantage as &&rated by the derived nontensorial concept of slide mnduhrsfl3,14]. The deeper mn for this usefuhress is due to the fact that the local representation of stress and strain may be tailored to the physical anisotropy whether intrinsic or induced by the presence of initial stresses. type

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M. A.BIOT

The notation edifor the strain used hereafter includes any of the various definitions of strain described above. The corresponding stress components is defined by virtual work, so that

F/&j

(4.8)

represents the virtual work associated with the virtual deformation See We omit the summation sign although eij is not necessarily a tensor, with the understanding that the summation is extended to all six independent variables eb According to the de@tion of ,j its differential in the case of a deformable cell is obtained by simply adding the external work TQdeu performed by the stresses Tii under the conditions dt = dM’ = dsr = 0. By adding this term to the values (3.25) of d$ we obtain d$ = i”iide,, - $ e A,d&+$ 0

&dMk+8dSr.

(4.9)

The value of dSPremains the same as (3.24) (4.10) . .

Ehmmuion of dsr between (4.19) and (4.10) yields (4.11) We note that according to Cqns (2.10). (2.15), (2.22) and (3.21) the quantities &, g&,&, & and Ap are functions of the partial pressures pk and the temperature T. in turn p&is a function of T, q and the masses mk of each Substancein solution. This mass is given by

mk

(4.12)

where me is the initial value of mb Hence &, gh, &, & and A, may be expressed as functions of e,,, T, ML and &, The values of 9 and Y may be conveniently obtained by integrating (4.9) and (4.10) tirst at constant temperature To then heating the cell to the temperature T maintaining constant the values eri, & and ML.We obtain

(4.13) Elimination of T between these two relations yields (4.14) where 9 is now a known function of et, f, ML and 9. From the difIerential(4.11) we derive (4.15) and a fourth one which plays a special role (4.16)

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irrcversibk tImuod_ks

of physicacehemicnl solids with hite defamation

891

We denote by Q the quantities Gb ##kand 8 and by qi the corresponding variables Q, Mk and X Equations (4.15)are then written (4.17)

The quantities Qi may be considered as mixed mechanical thermodynamic driving forces which are known functions of time imposed by the environment. Equations (4.16)and (4.17)are not suthcient to determine the unknown variables qi and & because Ap is not known. The additional relations are provided by chemical kinetics by which reaction rates are given as (the dot denotes a time derivative),

& =fp(Qj,

(4.18)

mkt T).

This may be expressed in terms of eib f, M’ and S using the T s T(ejj, fp

vahc

(4.12)Of mk

Mk, 9)

aid

Writing

(4.19)

obtained by soiving for T the vahte (4.13)of Y. The rates of reaction become

Equations (4.17)and (4.20)now oonstitute a complete system for the chemical kinetics of the deformable open cell. We may write these equations in a form which corresponds to a general Lagmt@n formulation of irreversii thermodynamic systems. As already pointed out, the afhnity is a function of the partial pressures and the temperature (4.21) Since pk is a function of ey, mk, T using eqns (4.12)and (4.19)we write

E~~in~on of Spbetween eqns (4.20)and (4.22)yields Ap = spCCij9

8,

ML*9)

(4.23)

where the affinity is now expressed in terms of reaction rates 4 by what we have called a rate

facet WI, s%. Intiuciag into (4.16)the vahre (4.23)of 4 we obtain the system of differential equations

which govern the time evolution of the open deformable cell. These equations are now in the Lagra@r form introduced by the author. A particuhu case of interest is obtained by assuming that the system, while nonlinear and irreversible, is never very fai from equilibrium For such a q~~~ve~bk system Onsag& principie[l5,16f apphes to the chemioal reactions. This is expressed by writing the rate functions in the form (4.25)

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where D = ; 2

b&j,

(4.26)

Mk, Y)&

is a positive quadratic form in &, with coefficients dependent on the state of the system.

Equations (4.24)are now iit aqi

=c?i

g+$=o. P

(4.27)

P

For a closed cell (Mk = 0) these equations are the same as obtained in the analysis of where the role of internal cwrdinatea is played by tlte chemical variables tp [31.

nonlineu thermoviscoebicity

5. PRINCIPLE

OF VIRTUAL

DISSIPATION

FOR CONTINUOUS

SYSTEMS

Wemayconsidtra~~acdlcctionof~primary~~s.ARimportant property of the collective potential is its additivity. Hence the collective potential of a continuum may be written

I

v= nBda

(5.1)

where Q is the domaiu b&xe deformation aud # is the cell potential per unit iuitial volume. The elementary initial volume is dfi = dx, d.xsdxs with i&al coordiaatei xi. Similarly the collective energy and entropy are

S

(5.2)

V=U-T&‘.

(5.3)

Because of eqn (2.5)we may write

We also assume that the continuum is subject to a potential force &Id such as gravity. The potential field per unit mass is a function ‘S(a) of the coordinates. If we call p the mass per unit initial volume at a displaced point % the mechanical potential energy of the continuum is G=

I P%Wda n

(5.4)

We define 9=V+G

(5.5)

as a rttixcdcdlectfue potrrrtial which embodies mixed me&&al and thermodynamic properties. We now consider the continuum to u&ergo a completely getteral transform&ion which may be irreversible. With a virtual intStesimal transformation we may write d’Alemberts principle

as &I&,+SU+SGW

(5.6)

Variational irreversible thermodynamicsof physical-cbcmical solids with finite deformation

893

where 6W is the virtual work of external forces in addition to the potential forces and &q, the virtual work of the inertia forces with generalized inertia forces 4 corresponding to generalized coordinates 4i. The variables must of course be varied subject to certain constraints which we shall specify. Elimination of U between eqns (5.3)and (5.6)yields

2 I,&j~+6P+ToSS=&W.

(5.7)

Consider now that there is no variationalflow of matteror heat across the bounduryof R. In this case 6s represents the entropy produced in R. To indicate this we write S* instead of S and eqn (5.7)becomes

h I&

+ 69 + T&S* = 6W.

(5.8)

The term TOGS*represents a virtual dissipation and qn (5.8)is the general form of the principle of oiqual dissipation[3] generalizing d’Alembert’s principle to irreversible thermodynamic systems. The principle may be written in an alternate form particularly useful for continuous systems as follows. It was shown earlier[3] that we may write 69 + T,,8S* = &RB+ n T&s*dfl

I

(5.9)

where Ss* is the entropy produced and Z&* is the virtual dissipation, both per unit initial volume. The term &@ is (5.10)

where & denotes a variation obtained by excluding the variation &* of entropy produced. The principle of virtual dissipation (5.8)thus becomes[3]. (5.11)

Notethat T is the local temperature. We have called T&* the intrinsic dissipation. We must now define the variables to be varied and the constraints which they must obey. One of the variables is the field of displacements U{of the solid. The new coordinates become 4 = xi + ut. Another field is the mass displacement vector M: of each substaace relative to the solid. It is de&d as the total mass which has flowed across a material area initially perpendicular to the xi axis and initially equal to unity. It obviously satisfies the mass conservation constraint (5.12) Summation signs are omitted for tensor-mlquantities. An equation of entropy balance is also obtained by considering the rate of increase of entropy in an arbitrary domain a. It may be written (5.13)

In this equation, h is the rate of the heat acquired per unit initial volume of Q’, S*mis the rate of entropy produced per unit initial volume which is not due to pure thermal conduction and

894

M. A. BIOT

&A@is the rate of convected entropy per unit area of the initial boundary A’. We write

6=%

(5.14) I

where I& is the rate of heat flow per unit initial area across a face initially perpendicular to Xi. By integration by parts eqn (5.13)may be transformed to (5.15) where

s’

A.

(5.17)

The domain AI’being arbitrary the integral (5.15)implies (5.18) This generalizes Meixner’sresuh[l7] which is restricted to thermal flow. Tiie integration with zero initial values yields Y=s’+s

(5.19)

where (5.20) is the entropy soyrplisd.Tbt vector SI is the total entropy diapkentent due to convection and combction. Rel#on (5.19) expresses the basic entropy baktce, while relation (5.28) is a holonomic constraint an&gnus to (5.12)for the masses. I&e rate of entropy production j* per unit initial vohne (5.16) may ah be written

l+

S* = 9 &

AfjS*'Sj*

(5.21)

where AI~is the thermal resistivity tensor of a deformed element of solid, relating fii and aT/axi. The V&bkS u,, #, &, S* and f, CO&Ctdy dstine the StotC Of the deformable SOli with thermomolecular diffusion and chemical reactions. The strain components q may be chosen according to any of the particular definitions described in !%ction4. They may be tensorial or non tensorial functions of 41. we write eij *

lij(&w)

(5.22)

with the property of invariance kder a @d rotation. For a non homogeneous deformation ail are the gradients.

(5.23)

Variational irreversible thermodynamics of physical-chemical solids with finite deformation

895

The total rate of intrinsic dissipation per unit initial volume is the positive definite expression

(5.24) where 9, is the rate function defined by (4.23)and 9 is a dissipation function

which is a quadratic function of A@ and Si with coefficients dependent on the local state. This quadratic form represents thermomolecular diffusion with the local validity of Onsager’s principle[lS, 161.The coefficients C!j represent the coupling between mass and entropy flow including entropy convection. The virtual dissipation is immediately derived from these results. It is written (5.26) 6. VARIATIONAL DIFFUSION

DERIVATION OF FIELD EQUATIONS FOR COUPLED THERMOMOLECULAR AND CHEMICAL REACTIONS IN A DEFORMABLE SOLID CONTINUUM

We consider a deformable solid which undergoes a deformation described by the material displacement field rci.The coupled thermomolecuhu diffusion datiue to the solid is deacrii by the vectors Si and M:. ‘Ihe vector Si is the entropy displacement due to conduction and convection, while It@ is the mass displacement relative to the solid of the various substances in solution. The scalar field fp represents the distribution of chemical coordinates, and s* is the entropy produced per unit initial volume. These unknown fields are to be determined as functions of the initial coordinates xi and the time t. Equations governing these fields are readily obtained by applying the principle of virtual dissipation assuming arbitrary variations which vanish at the boundary. In this case the virtual work of external boundary forces vanishes (SW = 0) and the variational principle (5.11) is written (6.1) The variation S& is obtained by varying only cii,M’, Q and s (excluding s*). We find

According to (4.15)and (4.16)this may be written

Since p is independent of the chemical reaction we write p=&,+M’)

where po is the initial mass per unit volume. We derive

(6.4)

Id. A. 8107

8%

hence

2

hd + SW) = Tw260,~ + (pdMk v

+p

6.6)

where (6.7)

Qk”$k+g

de&m a mixed convectiveptentiai which takes into account the body force poteatipl Z

F~ywecons&theinertiafotccs.Inordertoavoidunducc~wb&doaotadd sigRi&a&ytotbcp accmcy~eistroducesonws~~~.Al~oniiccurate evahtion of the inertia forces will be found in an earlier paper dealing with @uidraaacrtad.~ soli&[8].Wes~Pruumctbot~~f~~pnduePloialytothenrrabmrinaIlillgdtfiCtime derivative of the momentum pci,of tbe solid. Per unit initial volume the virtual work of the inertia forces is

in this exprcssio~~A is the domain occupkd after deformation by an element of the solid initl&yofuaitvdume,p=mdAisthsportialQBlityofsubrtsocekafter~and Su:isthevirtualdisphcen#atdt&:substPnceincarterisncoo~.ItwasshownthW[81 (6.9) We derive for ttte virtual wprk of the inertia forces (6.10) We now sub$itute the values (5.26),(6.6)and ($10) in the principle of virtual dissipation (6.1). From relations (5.12),(5.20)and (5.23)we derive the variations (6.11) and we integrate by parts the terms in (6.1)which contains tbcse variations. In tbe final result we cancel the factors multiplying the arbitrary variations. This yields

a8

aa

z+zyo9

-&+92e,=o.

These equations along with (5.24)for S* namely (6.13) constitute a complete set of ditIerential equations for the time evolution of the variabks ui, Mt. SI, &, and s*. The value of Y is determined from equation (5.19) while M’ is derived from (5.12).

Variational 7. ADDITIONAL

irreversible

tbermodynmics

AXIOMS

of physicakkmicai

AND NEW DEFINITION

solids with finite deformation OF THE CHEMICAL

897

POTENTIAL

thermodynamic functions derived in the foregoing analysis are based exclusively on the axioms of classical thermodynamics. It is possible to proceed further and derive thermodynamic functions analogous to those associated with the axioms of quantum statistics. However in deriving these results we may bypass completely the statistical treatment by introducing two very simple axioms and by this process obtain a new definition of the chemical potential. Let us go back to expressions (3.15)and (3.20)for the affinity and the heat of reaction. We assume that we may write The

(7.1) Axiom (b)

(7.2)

Note that Axiom (b) implies Nemis’ third principlr. The lower limit of the integrals in (7.1)and (7.2)is the state of absolute zero and it is assumed that the integration may be performed as a limiting process by extrapolation. We further assume that the u+ants of integration b(0) are characteristic of the pure substances and independent of the chemical reactions. These constants are considered to be derived by measuring heats of reaction for a suflicient number of cases. In principle they may also be obtained from quantum statistics but in fact this is seldom practical. Substitution of the values (7.1)and (7.2)into expression (3.20)for the aBinity yields

A=-&k

(7.3)

where /‘k

_=

zp - nf**

(7.4)

with ik*r

s

I

PkT

0

dek+&(O)

sf’=

I

PkT dsk.

0

(7.5)

Equation (7.3)expresses the aI%@ in the traditional form with a new dejinition (7.4) of the chemical potential.

In the previous sections we have defined the entropy 9 as a co&&e concept which depends on the state of the supply cells. We may consider the particular case where the supply cells are all in the state of absolute zero temperature and extrapolate to this case the results obtained in Section 3 from classical thermobynamics. To simplify the writing consider the case of a single reaction 6. The entropy diRerential(3.9) becomes dY=?;*df+~@Wkfk+dsr A

(7.6)

where Skhas been replaced by Sl’ as expressed by (7.5).We may also write relation (3.20)in the form (7.7)

Substituting this value into (7.6).taking into account the relation d?&=dMk+v& SSVol. II. No. I I-B

(7.8

M. A. BIOT

898 we

obtain k

6

(7.9)

d~=~s’?ddmk-~d~+ds~. T On the other hand we may write

(7.10)

T dST= hp~dt + 9 hk” dM’ + hijde++ C dT. Also by definition

Using relations (7.8),(7.10)and (7.11).the value (7.9)of the entropy di&rentiai becomes (7.12) The coefl&nts 3?, kt’“,b and C are functions only of tnh q and T. Hence in@ration of (7.12)yields 9 = Y(mk, Q T)

(7.13)

as a function of the same variables. This expression is valid whether mk results from chemical reactions or convection. We may also derive the cell potential in terms of the chemical potential pk by writing the

convective potential as ‘#‘k =pk

(7.14)

-/hk

where PU is the chemical po&ntial of the substance in the supply cell. With the value (7.3)of A and taking into account relation (7.8)we write (3.10)in the form

(7.15) lf we assumethe supply ceils to be at absolute zero, this becomes

dj=$jkdmtfBdL?-

e a(0) dM’.

(7.16)

Hence the value of 3 still depends on ML unless we neglect ~(0). On the other hand if we consider a continuum with zero variation of the total mass of each substance in the domain 0,

In &MkdQ=O we

(7.17)

obtain for the variation of the total collective potential (7.18)

This value depends only on i$mkand SY’.

Variationalirreversiblethermodynamicsof physicakhemicai solids with finite deformation

899

8. COMPLEMENTARY FORM OF THE FIELD EQUATIONS L.et us neglect the inertia term-G,H~axi in the field equations (6.12).Physically this means that we neglect the effect of inertia forces on the diffusion process. The second and third equations (6.12)become

-$$=Xit

if

= Xi

(8.1)

where (8.2) Equations (8.1)are linear in &fFand Si. They may be solved for # may be written

me

Izr: = q

SiZGE aXi

and Si and the solution

(8.3

where 8’ is the dissipation function B expressed as a quadratic form in X: and Xi instead of # and $i. By substituting the values (8.3)into (5.12)and (5.20)we obtain

. jQk-$g$), _ - aXia (-aW ax, >

(8.4

1

On the other hand the time derivation of (4.12)yields (8.5) where f,(~, mb T) is the rate of reaction (4.18).Also from (5.19)and (5.24)substituting A,,(%, mk,T) insteadof SBep and f, for & we write (8.6) Combining equations (8.4H8.6) and adding the first group of the field equations (6.12)we obtain

(8.7) The entropy Y as well as other variabks in these equations are expressed in terms of the unknowns Us,mk and T. The time evolution of these variables is governed by the complementary form (8.7)of the field equations[6]. 9. LAGRANGIAN

EQUATIONS

The principle of virtual dissipation (5.11) may be applied to derive directly Lagran& equations which govern the evolution of complex systems described by generalixed coordinates. The fields are approximated by the expressions

M. A. BIOT

M: = AR;“tqi, xl, t)

s,=S&II,Xl, r) 5, = uqb

Xl,

a

(9.1)

wbcre qr are generalized coordina&s to be determined as functions of time. In a large number of problems the entropy produced 9* does not contriite signiflcantIyto the values of the state variables. Hence in &is case suitably chosen generalized coordinates qr are su&ient to describe the state of the system. The mixed collective potential is evaluated as 9 = e?l, f)

(9.2)

and the rate of dissipation expressed in generalized coo&nates is

(9.4) The virtual dissipation is then (9.5) The virtual work of the inertia forces is approximated as (9.6)

4=

Iog&)+.

(9.7)

special care must be excrciscd in evaluaGng the variation 69 = S# since the vtitional principle assumes that the normal cof8M:aad8SIanzeK,atthcboua&uyoffZa condition which is not obeyed by using expressions (9.1) in evaluating the variations. Hence SM! and A!$,are now discontinuous at the boundary and the terms containing SMLand SS are in&&e and yield a &ite contribu&~ at this bow&y. This contribution is easily evaluated by integra@j by parts. We derive

tm where A is the boundary of the initial domain and q its unit outward normal.

Finauy the virtual work of the surface tra&ms ft per uuit initial area is (9.9) Substitution of expressions (9.3, (9.6), (9.8) and (9.9) into the variational principle (5.11)with

Variationalirreversibletkmmdynamics of pbysicakhcmical solids with finite deformation

901

arbitrary variations 8qi yields the Lagrangian equations &+R

i

+g+i?i!!!- Qi ddi 8qi

where (9.11) is a mixed mechanical and thermodynamic driving force which represents the effect of the environment. Equations (9.10) are in the same form as derived from the general Lagrangian thermodynamics[3,6,8,9]. When the effect of entropy produced, on the state variable, is not negligible we may introduce additional generalized coordinates q: and write

s*= s*(q:,

(9.12)

q, t).

The unknowns q; are now included in the Lagrangian equations (9.10). Additional equations for q: are then obtained by writing qn (6.13) Ts* = f:

seg, + 20

(9.13)

at suitably chosen points equal in number to the number of additional coordinates q:. We may also express the generalized inertia force 4 approximately by means of the kinetic energy as follows. We integrate qn (9.6)with respect to time assuming the variations to vanish at the limits of integration. We derive (9.14) We denote by (9.15) the approximate kinetic energy and by 6F the variation of F due only to the variation of rib J5quation(9.14)may then be written (9.16) This relation beii valid for arbitrary variation &i implies (9.17) The derivative aflagi is evaluated by assuming p independent of qi. With this vahre of Zithe m quations (9.10)are written d --

89 _c,~+~D dt ( adi> 3%

I

z+z=

a@ Qi*

(9.18)

10. APPLICATION TO BIOLOGICAL SYSTEMS

Biological systems are open deformable systems exchanging matter work and energy with the environment, while chemical reactions coupled to thermomokcular difMon occur internally. Such systems of considerable complexity are eminently suited to a simplieed description

M. A.BIOT

902

by generalized coordinates whose evolution obeys the Lagrangian equations (9.18). In particular this Lagrangian formulation has been applied by the author to biological membranes with active transport[6]. Theories developed by Katchalsky[l8] and others are shown to be considerably simplified by the L,agrangian formulation which in addition attains a high degree of generality. The example treated provides an excellent illustration of the power of the method by providing an easy evaluation of the coupling coefficients between external flows through the membrane as influenced by coupled internal chemical reactions. The phenomenon is called active transport because some of the flows occur against the concentration gradient. Il.FINITEELEMENTMETHODS

The Lalyaneian formulation provides the foundation of a large variety of finite element methods, choositig as generalized coordinates, values of the field variables at the vertices of a lattice dividing the continuum into finite elements. Equations (9.1) and (9.12) may then be considered as interpolation formulas giving the values of the geld in the finite elements or in small groups of such elements. All kinds of interpolation formulas may be used, such as linear quadratic or others, leading to a large variety of techniques. 12.APPLICATIONTOPLASTICITYANDANALOGYWITHCHEMICALREACTIONS A

natural extension of the technique of internal coordinates as introduced by the author [ I31 was applied to describe plastic properties[31. It is of interest to point out that this may be achieved by including in the virtual dissipation terms of the type T&* = 9&,&q,,

(12.1)

where s4, is the variation of an internal plastic strain due to dislocation motion and 9& is a function of the local state and & The variables 411are treated as intemaI gene&i& plastic coordinates. Comparing with expression (5.26) the analogy with chemical reactions is obvious and 9&,is the tensor equivalent of the a#hity Sp, 13.PHASECHANGESANDCRYSTALIZATIONUNDERSTRESS

When the solid contains small crystal grams it may be approximated as a continuum. The state of an element of this continuum may then be defined by the external variables and by a large number of generalized internal variables which describe the microthermodynamic state of the elasnent. These variables may correspond to crystal geometry, local strains and temperatures. There may also be present a number of d&rent phases coutaining each a certain number of pure substances. The cell potential may then be expressed in terms of these external and iutemal variables, and the rate of dissipation in terms of these variables and their time derivatives. Application of the principle of virtual dissipation to this case yields field equations of the same type as (6.12) in&ding the internal coordinates of the microthermodyn_amics. Creep may result due to crystals dissolving at some points and recrystahziag at others, because of disequilibrium in the microthermodynamics. 14.APPLICATIONTOPOROUS

SOLIDS

The foregoing results are applicable to a large category of porous solids when the motion of the pore fIrrid relative to the soiid may be treated thermodynamieaily as a di&sion. This problem has been discussed in more detail earlier[8). 15.HEREDITY AS A RESULT OF THE PRESENCE OF INTERNAL COORDINATES The variational principle applied response of external csordinotis a

to a system with internal coordinates yields for the response which e&bits heredity. This approach was Mated for Ibar t48ah&i@ in M4[11 and exteudad to ttoa linear viaeoel&ity[3, 191. Esptrtions (4.24) for the mponae of a deformable cell govam a syatesn whore the Wy is the result of intomal chemical reactions. With quasi reversible reactions (4.27) it behaves as a nonlinear viscoelastic solid [3].

Variationalirreversiblethermodynamicsof physical-chemicalsolids with finite deformation

903

16. SOLID UNDER INITIAL STRESS

A particular case of the present theory is that of a solid in thermodynamic and mechanical equilibrium under initial stress. Small departures from equilibrium are then considered, with small displacements and small perturbations of the thermodynamic variables, including thermomolecular diffusion and chemical reactions. The problem has been analyzed in detall[9] and constitutes a direct application of the linear thermodynamics developed already in 1954-55[ 1, 21. A considerable simpliication results in this case due to the fact that sr, the entropy due to thermal convection, is a state variable replacing the cell entropy 9. It is also important in this case to use the definition (4.1) of the strain or definitions of the type (4.7). Green’s tensor (4.2) is not suitable because it leads to spurious complications. The theory of initially stressed solids, for isothermal or adiabatic deformations without chemical reactions or molecular diffusion, was treated extensively in a monograph [ 13). Note that the problem is not treated as a bifurcation but as (I small deviationfrom an equilibriumstate. The theory is therefore more general since no reference is required to an originally unstressed state which may not exist physically. REFERENCES 1. M. A. Biot. Theory of stress-strainrelationsin a&tropic viscoelasticityand relaxationphenomena.Z. Appl. Phys. 25, 1385-M (1954). 2. M. A. Biot, Variationalprincipks in irreversibk thermodynamicswith applicationto viscoelasticity. Phys. Rcu. 97, 1463-1469(1955). 3. M. A. Biot, Va m irreversiblethermodynamicsof nonlinear tbermorbeology.Q. AppL Math. 34, 213248 (1976). 4. hf. A. Biot. New chemical tltermodynamksof open systems, tbermobaricpotentiala new concept. Bull. Aced. R de Z?&iqr~e (Ckssu des Sciatces) 62,239-257 (1976). 5. hi. A. Blot, New fundamentalconcepts and results in thermodynamicswitb cltemical applications. Cha. Phys. 22, 183198 (1977). 6. hi. A. Biot, Variational-Lqpangkn thermodynamicsof evolution of collective chemical systems. Chem. Phys. 2% 97-115 (1978). 7. J. W. Gibbs, Thermodynamics,Vol. 1. Longmans.London (1996l. 8. M. A. Biot, Vuiatkn&La9ran& thermodynamicsof nonisotbermalfinite strain me&anics of porous solids and tbermotnokcukrdiiion. Znt.1. SoZZds Stntctunrs13.579-597 (1977). 9. M. A. Bit, Vuktknal-m irreversiblethermodynamicsof iniiy stressed soliis witb tbermomokcular diffusionand cbemkal reactions.J. Me& Phys. Sofids 25,~397 (1977,;and Errata26.29 (1978). IO. R. D. Mindlln,Equationsof bigb frequency vibrationsof tbermopkzo-ekctric crystal plates. Znt.Z. solids. Structrcrrs 10.625-637 (1974). 1I. Tb. De Dander. L’A#nZfLGautbkr-Villars.Paris (1934). 12. I. prioaline and R. Defay, Chemical Fhemtodyuamics.Longmans,London (1965). 13. M. A. Blot. Mechuuicsof ZncremmtolZkformations.Wiley, New .York (1965). 14. M. A. Biot. Buckling and dynamics of multIlayeredand laminatedplates under initial stress. Int. Z. So/ids Structuns. 10.419-451 (1974). IS. L. Onryer, R#iproecrlrelations in irreversibleprocesses-I. Phys. Rm. 37.405426 (1931). 16. L. Onsager,Reciprocalrelations in irreversibkprocesses-II. Phys. Reu. I, 2265-2279(1931). 17. J. Meiier. Zur tbermodynamikder tbermodiiusion..Ann. der Physik 39.333356 (1941). 18. A. Katciulsky a&R. Span&r. Dynamics of membraneprocesses, Q. Reu. Biophys., 127-175(1978). 19. R. A. Scbapery. A Theory of NonlinearTbermoviscoelasticltyBased on Irreverst%rk Thermodynamics.In Prtx. 5th U.S. c0ngrr.w Appl. M&I. 511-539(1966).