thermodynamic functions for ageing viscoelasticity ...model\ note that any linear viscoelastic...

$: Thermodynamic functions for ageing viscoelasticity ...model\ note that any linear viscoelastic behavior can be described by the Maxwell chain model or ... In the case of ageing\ the$
\PERGAMON International Journal of Solids and Structures 25 "0888# 2882Ð3905

9919Ð6572:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reservedPII ] S 9 9 1 9 Ð 6 5 7 2 " 8 7 # 9 9 0 7 3 ÐX

Thermodynamic functions for ageing viscoelasticity] integralform without internal variables

Zdene³k P[ Baz³anta\�\ Christian Huetb

a Departments of Civil En`ineerin` and Materials Science\ Northwestern University\ Evanston\ IL 59190\ U[S[A[b Laboratory of Buildin` Materials\ Department of Materials\ Swiss Federal Institute of Technolo`y\ LMC:EPFL\

MX G Ecublens\ CH 09904\ Lausanne\ Switzerland

Received 5 December 0886^ in revised form 16 May 0887

Abstract

Viscoelastic materials whose creep and relaxation functions depend on the age at loading are considered[First the material is assumed to follow the solidi_cation theory\ which explains ageing of concrete by gradualdeposition of layers of newly solidi_ed non!ageing viscoelastic constituent "cement gel# on the pore walls[ Itis shown that the well!known classical Staverman and Schwarzl|s and Mandel|s formulae for the densitiesof Helmholtz free energy\ free enthalpy and dissipation in a non!ageing viscoelastic material can be gen!eralized to the case of ageing by double Stieltjes integrals over the strain or stress histories[ Their integrandscontain only the relaxation or compliance functions and their rates\ but not the internal variables nor theproperties of constituents[ The expressions obtained for the free energy and free enthalpy are subsequentlyshown to possess in general "without restriction to the solidi_cation theory# the property of a potential forthe stress or strain\ respectively[ Consequently\ if one formulates three!dimensional expressions for the freeenergy or free enthalpy on the basis of the solidi_cation theory\ one may\ conversely\ obtain constitutiveequations for ageing viscoelasticity that are consistent with continuum thermodynamics[ An expression forthe dissipated power of an ageing material is also derived[ The results should prove useful for approximatesolutions\ bounds on structural response\ and numerical solution algorithms[ Þ 0888 Elsevier Science Ltd[All rights reserved[

0[ Introduction

The thermodynamic density functions in continuum mechanics are useful for formulating restric!tions on positive energy dissipation\ revealing certain symmetries with respect to di}erent variables\establishing approximate bounds on some sti}ness and compliance characteristics in heterogeneousmaterials or structures\ determining numerical stability restrictions for computational algorithms\etc[ In viscoelasticity\ formulation of continuum thermodynamics is made di.cult by the fact that

� Corresponding author[ E!mail] z!bazantÝnwu[edu

Z[P[ Baz³ant\ C[ Huet : International Journal of Solids and Structures 25 "0888# 2882Ð39052883

one treats a material that is not in an equilibrium state[ This aspect has generated continuingcontroversy\ which can however be resolved by various ways\ e[g[\ on the basis of Coleman|stheorems^ Coleman "0853#\ Coleman and Mizel "0853#\ see also Kestin "0863\ 0868\ 0881#\ Day"0861# and Huet "0868\ 0871#[ For linear viscoelasticity without ageing\ the basic thermodynamicframework was developed mainly by Staverman and Schwarzl "0841a\ b#\ Biot "0843#\ Mandel"0853\ 0855# and Valanis "0860\ 0872#\ starting from the internal variable viewpoint\ and byColeman "0853#\ Brun "0854\ 0856\ 0858#\ Christensen "0858\ 0871# and Truesdell "0858#\ startingfrom the macroscopic variable viewpoint[

In this study\ attention is focused on viscoelasticity with ageing\ manifested by a change ofviscoelastic properties as a function of time "rather than strain\ damage\ load cycles and similarsources#[ Such ageing\ for example\ arises in Portland cement concretes due to the long!termchemical reactions of hydration of tri!calcium silicate as well as relaxation of hygro!chemicallyinduced self!equilibrated prestress of the microstructure "Baz³ant et al[\ 0886a\ b#[ It causes that thecreep or relaxation properties at the ages of one day\ one week\ one month\ one year and ten yearsare all rather di}erent[

The thermodynamic functions of an ageing viscoelastic material\ having the property that theirdi}erentiation provides the stress or strain history\ have been formulated in Baz³ant "0868\ alsopartly 0861b\ 0864#[ However\ the formulation was in terms of internal variables "partial strainsor partial stresses of the Kelvin or Maxwell chain model#\ which are not as useful as the measurable"external# state variables "strains and stresses# and cannot be determined in a unique way frompurely mechanical experiments[ The knowledge of internal variables\ by contrast\ makes it possibleto determine the overall properties\ which is the basis of the so!called integrated micromechanicsas recently developed by Huet "0882b\ 0886#[

The purpose of this work is to formulate the thermodynamic functions\ particularly the densitiesof the free energy\ free enthalpy and energy dissipation rate "or dissipated power#\ in an integralform in which no internal variables are used[ This is the property of the celebrated Staverman andSchwarzl "0841a\ b# formula for non!ageing viscoelastic materials[ We will attempt to extend thisformula to the case of ageing[ This problem was recently pointed out as an important pendingproblem by Huet "0882a#\ who discussed it on the basis of Fre�chet|s "0809# integral!type for!mulation of linear functionals and proposed two particular forms of the free energy functional\each of them being endowed with the property of a potential for materials with instantaneouselasticity[ This paper will present a new form of the free energy functional that is valid in generalwithin the framework of the solidi_cation theory " for materials exhibiting instantaneous elasticity#and exhibits the property of a potential for stress[ This form can be directly derived from micro!scopic considerations based on the solidi_cation theory\ which was developed for concrete inBaz³ant "0866# and was extended in Baz³ant and Prasannan "0878a\ b# and Carol and Baz³ant "0882#[However\ after deducing from this theory the expressions for the thermodynamic functions\ wewill show that they may have a more general validity\ not restricted to the solidi_cation theory norto concrete[

1[ StavermanÐSchwarzl and Mandel formulae for thermodynamics of non!ageing viscoelastic

materials

In non!ageing viscoelasticity\ the one!dimensional stressÐstrain relation for uniaxial stress s"t#is written as

Z[P[ Baz³ant\ C[ Huet : International Journal of Solids and Structures 25 "0888# 2882Ð3905 2884

s"t# � gt

9−

C"t−j# do"j#\ o"t# � gt

9−

F"t−j# ds"j# "0#

in which o � strain\ F"t# � compliance function\ C"t# � relaxation function\ and the integrals areStieltjes integrals ðif the derivative o¾"j# exists "no jumps#\ they can be transformed to the usual\Riemann integrals by setting do"j# � o¾"j# djŁ[ The minus label at the lower limit of the integral\9−\ indicates that the integration must begin just before t � 9\ which is necessary when the loadingstarts at time t � 9 with a jump[ The density of the isothermal Helmholtz free energy in the materialmay be expressed according to the StavermanÐSchwarzl formula as follows]

F"t# �01 g

t

9− gt

9−

C"1t−j−h# do"j# do"h# "1#

An instructive way to derive this relation is to consider the approximation of the relaxationfunction C according to the Maxwell chain model "Fig[ 0#\ for which s"t# � Sn

k�0 sk"t#\o¾ �"s¾ k:Ek#¦"sk:hk#\ k � 0\ 1\ [ [ [ \ n[ Here sk are the partial stresses of the chain "internal variables#\Ek are the elastic moduli\ and hk are the viscosities of the individual Maxwell units[ For the Maxwellchain "with relaxation times tk � hk:Ek#\ one has

C"t−j# � sn

k�0

Ek e−

t−j

tk "2#

The Helmholtz free energy per unit volume of the material may be expressed as a sum of theenergies of all the springs of the Maxwell chain at time t^

F"t# � sn

k�0

s1k "t#1Ek

"3#

Noting that sk"t# � Ðt9 Ek exp ð−"t−j#:tkŁ do"j#\ one gets

F"t# � sk

01Ek 0g

t

9−

Ek e−

t−j

tk do"j#1 0gt

9−

Ek e−

t−h

tk do"h#1 "4#

Fig[ 0[ "a# Maxwell chain model and "b# Kelvin chain model for linear viscoelasticity[


�01 g

t

9− gt

9− 0sk Ek e−

1t−j−h

tk 1 do"j# do"h# "5#

Here the last expression in parentheses is readily recognized to be equal to the relaxation functionof the argument 1t−j−h[ This proves the last expression to be equivalent to "1#[

Lest one might think that the foregoing proof might be restricted to one particular rheologicalmodel\ note that any linear viscoelastic behavior can be described by the Maxwell chain model orthe Kelvin chain model as closely as desired[ So the Maxwell chain model is equivalent to theKelvin chain model or any other model consisting of springs and dashpots\ and even to moregeneral models as explained later[ This was proven by Roscoe "0849#\ Staverman and Schwarzl"0841b# and Biot "0843#^ see also Mandel "0855#[

The density D of the dissipated power\ henceforth abbreviated as the dissipation\ is

D"t# � s"t#o¾"t#−Fþ "t# � s"t#o¾"t#−01

o¾"t# gt

9−

C"1t−j−t# do"j#

−01

o¾"t# gt

9−

C"1t−t−h# do"h#−gt

9− gt

9−

Cþ "1t−j−h# do"j# do"h# "6#

� −gt

9− gt

9−

Cþ "1t−j−h# do"j# do"h# "7#

where the superior dot denotes the derivatives with respect to the time variable t[ This expressioncan of course\ be also obtained by summing the viscous work rates in all the dashpots[

By similar calculations\ one can verify that for materials exhibiting instantaneous elasticity"which is not the case\ e[g[\ for the Kelvin unit#]

1F"t#1o"t#

� s"t# "8#

This shows that\ for such materials\ F is indeed a potential for the stress[For creep under constant stress s9\ Staverman and Schwarzl gave also the expression of the free

energy density of the Maxwell chain in terms of the compliance function F"t#[ In the presentnotation\ this expression may be written as

F"t# � ðF"t#−01F"1t#Łs9s9 "09#

Using integration by parts in "1#\ Mandel "0855# extended this expression to general histories inthe following form]

F"t# �01 g

t

9− gt

9−

ð1F"t−j#−F"1t−j−h#Ł ds"j# ds"h# "00#

� s"t#o"t#−01 g

t

9− gt

9−

F"1t−j−h# ds"j# ds"h# "01#


The complementary energy density F� "or the negative of the Gibbs free energy density# is de_nedas the Legendre transform of the free energy\ i[e[

F�"t# � s"t#o"t#−F"t# "02#

Substituting "01#\ one gets

F�"t# �01 g

t

9− gt

9−

F"1t−j−h# ds"j# ds"h# "03#

The dissipation is found to be given as

D"t# � gt

9− gt

9−

Fþ"1t−j−h# ds"j# ds"h# "04#

As shown in Mandel "0855# and\ with a new proof\ by Huet "0881#\ these forms of ther!modynamic functions are not restricted to a Maxwell chain\ and not even to any particularrheological "springÐdashpot# model[ They hold true for internal variables of any kind providedthe associated rate equations are _rst!order di}erential equations with constant coe.cients[ Huet|snew proof\ in particular\ shows that\ for such internal variables\ the kernel with two argumentsappearing in the quadratic term of the Fre�chet "0809# expression for the free energy shoulddegenerate in a function of the sum of its arguments\ a property that is veri_ed here[ Thisresult was obtained using the two!dimensional symbolic calculus already applied to nonlinearviscoelasticity by Huet "0862\ 0874#[

2[ Viscoelasticity with ageing

In the case of ageing\ the classical linear viscoelastic equations relating the histories of strain o

and uniaxial stress s read in VolterraÐStieltjes form "Volterra\ 0776\ 0848# as follows]

o"t# � gt

9

J"t\ t?# ds"t?#\ s"t# � gt

9

R"t\ t?# do"t?# "05#

Here J"t\ t?# � compliance function for uniaxial stress\ i[e[\ the strain at age t caused by a unitconstant uniaxial stress applied at age t? ¾ t\ and R"t\ t?# � relaxation function for uniaxial stress\i[e[ the uniaxial stress at age t caused by a unit constant strain applied at age t? ¾ t^ t � 9 denotesthe instant when the solid is _rst formed\ e[g[\ when the concrete mix sets "changes from ~uid tosolid#[ These functions now depend separately on t and t?\ rather than just on the time lag t−t?[The functions R"t\ t9# and J"t\ t?# are related by a Volterra integral equation of the second kindwith parameter t9\ which is known to always have\ for a _xed t9\ a unique solution[ Unlike non!ageing viscoelasticity\ the lower limit of the time integral in "05# need not be written as 9− becausea jump of stress or strain at t � 9 is physically impossible^ t � 9 is the moment at which the materialbegins to harden and thus cannot sustain a _nite shear stress as yet[ Equation "05# extendsBoltzmann|s "0763# superposition principle to linear viscoelastic materials with ageing[

The linear viscoelastic stressÐstrain relation with ageing was set into operator form by Mandel"0846\ 0863a# "whose work was extended to matrix and tensor integro!di}erential operators by


Huet\ 0863#[ These operators\ which are non!commutative\ have been applied to the drying ofmoist clays "Huet\ 0869b# and to composite structures "Huet 0869a\ 0879#[ The correspondingnumerical aspects have been handled by means of an improved version of the numerical algorithminitially developed by Baz³ant "0861a#[

As emphasized by Huet "0882#\ an important aspect that has been missing is the correspondingthermodynamic formalism\ which would generalize the classical formalism established for theelastic case and for the viscoelastic case without ageing[ The present paper aims at _lling this gap\exploiting at _rst for physical insight the solidi_cation theory of Baz³ant "0866#^ see also Baz³antand Prasannan "0878a\ b#\ Carol and Baz³ant "0882#\ and Baz³ant et al[ "0886a\ b#[ The validity ofthe solidi_cation model and its application to concrete was justi_ed in these works\ and so it neednot be discussed here[ After recalling the main features of this model\ we simply derive from acontinuum thermodynamics view the consequences of the assumptions underlying this model[Then we study the inverse problem showing that the free energy expression deduced from thesolidi_cation model has the property of a potential[

3[ Basic concepts of solidi_cation theory

In the initial development of the solidi_cation theory\ the objective was to obtain a physicallyreasonable form of functions J and R characterizing the age dependence of material properties ina manner that does not imply the properties of the constituents of the material to change withtime[ The ageing can in fact be caused by a change of concentrations of the constituents[ The bestparadigm is the growth of the mass of cement hydration products per unit volume of Portlandcement concrete\ which appears to be the main "albeit not the sole# physical mechanism for ageingof the time!hardening type exhibited by this complex material[ In this solidi_cation process\ thesolid hydration products gradually _ll the pore space originally occupied by water and by the partof the anhydrous cement that has been dissolved[ At temperatures below 099>\ the e}ect of thissolidi_cation process on the overall material sti}ness prevails over the e}ect of dissolution of theanhydrous cement grains during hydration[ The mathematical consequences of this mechanism ofage dependence of creep are captured by the solidi_cation theory "Baz³ant\ 0866^ Baz³ant andPrasannan\ 0878a\ b^ Carol and Baz³ant\ 0882#[

It may be noted that there exist other possible ageing mechanisms such as the growth ofmicrocracking and microcrack healing\ as well as bonding or debonding within the solid skeleton" for hardened cement gel suggested by Baz³ant and Prasannan\ 0878a#[ However\ they do notprevail in the hardening stage\ and would be incompatible with the linearity of the stressÐstrainrelation in "05#\ which is experimentally well established for the service stress range of concrete[Such processes may be described by the general theory of heterogeneous materials extended tonon!ageing viscoelasticity with damage by Huet "0884a\ 0886#[ There is nevertheless one otherageing mechanism that must be expected to occur in the hardening stage of concrete and iscompatible with the aforementioned linearity[ It is the relaxation of localized high stress peaks inthe microstructure of cement gel\ recently described by the microprestress!solidi_cation theory"Baz³ant et al[\ 0886a\ b#[ Consideration of this physical mechanism is beyond the scope of thispaper[

The layers of cement gel that have already solidi_ed carry a _nite stress\ s\ and thus help to resist


Fig[ 1[ Model for progressive solidi_cation of layers of non!ageing viscoelastic material from a solution in the pore[

the externally applied load[ However\ the layers of cement gel are logically considered to be stress!free "s � 9# at the moment they solidify[ This is an essential feature of the solidi_cation theoryproposed by Baz³ant "0866#[

The stress!free initial condition of the freshly solidi_ed constituent layers may be simply capturedby the model shown in Fig[ 1[ In this model "Baz³ant\ 0866#\ the subsequently deposited layers ofthe solidi_ed constituent are assumed to be coupled in parallel\ subjected to the same macroscopicstrain increments do\ and to carry normal stress parallel to the layers[ This strain uniformityassumption is advantageous for keeping the solidi_cation model simple[

4[ Mathematical formulation of solidi_cation theory

Let v"u# represent the volume of the solidi_ed matter "cement gel or hydration products\ in thecase of Portland cement concrete# per unit volume of the material at age u "obviously 9 ¾ v ³ 0#[This function\ which can be calculated from the initial concentrations of cement and wateraccording to the chemical reaction kinetics and can be determined by _tting experimental data"Baz³ant and Prasannan\ 0878a#\ is assumed to be given[ A layer of in_nitesimal volume dv"u# isthe layer deposited at age u of the material[ In this manner\ a one!to!one relationship between theage u and the layer is introduced[ Thus\ instead of spatial coordinates\ each constituent layer islabeled by the time u at which it solidi_ed[

The solidifying constituent is considered to be a non!ageing linear viscoelastic material char!acterized by the compliance function F"j# or the associated relaxation function C"j#\ wherej � t−t? � duration of unit constant stress or strain[

The stress s at age t × u in the layer dv"u# that solidi_ed at age u at which the volume of solidi_edmatter within a unit representative volume of the material was v"u# may be expressed as]


s"u\ t# � gt

u−

C"t−t?# do"t?# "06#

The integration starts just before time u because a sudden strain jump can occur at time u at whichthe non!ageing constituent already exists\ having precipitated before[ Because the layers of thesolidi_ed matter are\ in this one!dimensional model\ assumed to be mutually coupled in parallel"Fig[ 0# and thus subjected to the same strain increment do\ the stresses s in all the layers dv"u# aresuperimposed and must have the macroscopic stress s as their resultant^ i[e[

s"t# � gt

u

s"u\ t# dv"u# "07#

In the case of concrete\ all the known test results can be described adequately by assuming thatthere is only one solidi_cation process in which the cement grains are dissolved in water and thensolidify from the solution as the hydration product "Baz³ant and Prasannan\ 0878b#[ But in realitythere are several separate chemical processes involved in hydration[ So it is of interest to formulatea more general theory involving several separate solidi_cation processes characterized by volumeincrements dvm"u# "m � 0\ [ [ [ \ n# "obviously 9 ¾ Sm vm ³ 9#[ In that case\ eqns "06# and "07# may begeneralized as

sm"u\ t# � gt

t?�u

Cm"t−t?# do"t?#\ s"t# � sn

m�0 gt

u�9

sm"u\ t# dvm"u# "08#

where the subscript m refers to layers of the same constituent m solidifying at di}erent times\ andm � 0\ 1\ [ [ [ n labels di}erent constituents[ Here it is assumed that the layers dvm"u# of the solidi_edmatter for all the separate solidi_cation process "or separate constituents# are subjected to thesame macroscopic strain o[

Substitution of the _rst equation into the second furnishes]

s"t# � sn

m�0 gt

u�9 gt

t?�9

Cm"t−t?# do"t?# dvm"u# "19#

Interchanging the order of integration and taking into account that all Cm are independent of u

gives

s"t# � sn

m�0 gt

t?�9

Cm"t−t?# gt?

u�9

dvm"u# do"t?#\ or s"t# � gt

t?�9

sn

m�0

vm"t?#Cm"t−t?# do"t?# "10#

This proves that the solidi_cation theory provides a constitutive equation that indeed has the formof "05#\ with an overall relaxation function of the form

R"t\ t?# � sn

m�0

vm"t?#Cm"t−t?# "11#

Since Cm and vm cannot be negative\ we see that R"t\ t?# cannot become negative "contrary to what


happens in numerical computation of the relaxation function from some models of the creepfunction with ageing#[ Furthermore\ the last expression automatically satis_es the inequalities]

R"t\ t?# − 9\1R"t\ t?#

1t¾ 9\

1R"t\ t?#1t?

− 9\1R"t\ t?#

1t?− −

1R"t\t?#1t

\11R"t\ t?#

1t 1t?¾ 9 "12#

because

vm"t# − 9\ v¾m"t# − 9 "13#

and\ for non!ageing viscoelasticity\

Cm"t# − 9\ Cþm"t# ¾ 9\ CÝm"t# − 9 "14#

The _rst and second inequalities in "12# can be deduced from a thermodynamic formulationbased on an internal variable framework "Huet\ 0881#[ The third inequality in "12# is implied bythe second and the fourth[

Upon replacing t with new variable t?¦D and integrating over D\ one _nds the fourth inequalityin "12# to be equivalent to the inequalities

$1R"t?¦D\ t?#

1t? %D�const

− 9\ or R"t¦D\ t?¦D# − R"t\ t?# for D × 9 "15#

These inequalities mean that the material hardens "rather than softens# as it ages[The last inequality in "12# means that two relaxation curves of ageing material for two di}erent

ages t? of initial strain imposition cannot diverge from each other "which is the property ofsolidi_cation theory and the B2 model\ Baz³ant and Baweja\ 0884#[ A similar non!divergencerestriction also applies\ in the case of solidi_cation theory\ to the compliance function "Baz³antand Prasannan\ 0878a\ b#[ The non!divergence restriction in "12#\ however\ is dictated by themacroscopic thermodynamic restrictions\ that is\ the second law of thermodynamics\ only in theparticular case of the solidi_cation theory\ but not in general[ There exists an example of an ageingspringÐdashpot model "Baz³ant and Kim\ 0867# that does not conform to the solidi_cation theoryand violates "12#\ yet is thermodynamically admissible because the springs and dashpots obey therelations s¾ � E"t#o¾ and s � h"t#o¾ in which E"t# and h"t# are positive monotonically increasingfunctions[ However\ it is doubtful that such a springÐdashpot model is realistic[

Generalizing the results of Staverman and Schwarzl\ and Mandel\ we will now derive from thesolidi_cation theory the expressions of the thermodynamic functions that correspond to theseformulae[

5[ Free energy density of solidifying material under uniaxial stress

To formulate thermodynamics of a solidifying material\ we now deal with the non!ageingconstituents and take into account the variation in their concentrations[ Because the solidi_edconstituent itself is not ageing\ we may apply to it the StavermanÐSchwarzl formula "1#[ Thus thecontent of the Helmholtz free energy in layer dvm"u# is


Fm"u\ t# dvm"u# � $01 g

t

m�u gt

j�u

Cm"1t−h−j# do"j# do"h#% dvm"u# "16#

The total Helmholtz free energy of the solidi_ed constituent per unit volume of the material is thesum of the Helmholtz free energies in all the layers\ i[e[

F"t# � sn

m�0 gt

u�9

Fm"u\ t# dvm"u# "17#

Substitution of "16# now provides]

F"t# �01 g

t

u�9 $gt

h�u gt

j�u

sn

m�0

Cm"1t−h−j# do"h# do"h#% dvm"u# "18#

Let us now try to reduce the triple integral in this equation to a double integral so as to eliminatethe variables vm[ To this end\ note _rst that\ due to the symmetry of the kernel Cm with respect tothe two variables j and h\ the non!ageing StavermanÐSchwarzl formula in "16# can be written intwo equivalent forms corresponding to the partitioning of the square integration domain into twotriangles "Fig[ 2a and b\ respectively#^

Fm"u\ t# � gt

h�u gt

j�h

Cm"1t−h−j# do"j# do"h# "29#

� gt

h�u gh

j�u

Cm"1t−h−j# do"j# do"h# "20#

Fig[ 2[ Various triangular and rectangular integration domains used in integration of the relaxation function[


These are in fact new\ nonsymmetric\ forms of the StavermanÐSchwarzl formulae[ To verify thatthese two forms are equal\ one may interchange the order of integrations in the _rst form "29# andthen switch the dummy variables j and h[ Consequently\ each of these two forms is equal to onehalf of their sum\ and thus to "16#[ In the form on the _rst line\ the second integral "over j# doesnot depend on u anymore since u is not greater than h[ So this integral is a function of t and h

only[ Therefore\ the _rst expression for Fm given by "29# may be substituted into "17#[ Thisprovides the overall free energy density in the form]

F"t# � sn

m�0 gt

u�9− gt

h�u $gt

j�h

Cm"1t−j−h# do"j#% do"h# dvm"u# "21#

Now consider in this equation the integrals over u and h\ which cover the triangular domainshaded in Fig[ 2c by means of vertical strips[ However\ the same domain can be covered by meansof horizontal strips "Fig[ 2d#\ which reveals that the triple integral in "18# is equal to the following]

F"t# � gt

h�9 $gt

j�h

sn

m�0

Cm"1t−j−h# do"j# gh

u�9−

dvm"u#% do"h# "22#

Thus the integration in u may be performed separately\ _nally furnishing the expression]

F"t# � gt

h�9 $gt

j�h

sn

m�0

vm"h#Cm"1t−j−h# do"j#% do"h# "23#

In view of "11#\ this may be recognized to have the form "Fig[ 2e#]

F"t# � gt

h�9 $gt

j�h

R"1t−j\ h# do"j#% do"h# "24#

But in "23# the order of integration may also be interchanged\ giving

F"t# � gt

j�9 $gj

h�9

sn

m�0

vm"h#Cm"1t−j−h# do"h#% do"j# "25#

which may be recognized to have the form "Fig[ 2f\ g#]

F"t# � gt

j�9 $gj

h�9

R"1t−j\ h# do"h#% do"j# � gt

h�9 $gh

j�9

R"1t−h\ j# do"j#% do"h# "26#

where in the last expression the dummy integration variables have been interchanged[Thus we have obtained two equivalent expressions for F"t#[ Interchanging again the dummy

variables and taking one half of their sum provides the following symmetric form]

F"t# �01 g

t

h�9 $gh

j�9

R"1t−h\ j# do"j#¦gt

j�h

R"1t−j\ h# do"j#% do"h# "27#

We may also write F"t# in a symmetric form involving double integration over a square domain"Fig[ 2h# with one single integrand like in the original StavermanÐSchwarzl formula]


F"t# �01 g

t

h�9 gt

j�9

r"t\ j\ h# do"j# do"h# "28#

in which

r"t\ j\ h# � sn

m�0

vm ðmin"j\ h#ŁCm"1t−j−h# "39#

This means that\ for the upper and lower triangular parts of the square integration domain in Fig[2h\ i[e[\

for j ¾ h] r"t\ j\ h# � R"1t−h\ j# "30#

for j × h] r"t\ j\ h# � R"1t−j\ h# "31#

or

r"t\ j\ h# � H"j−h#R"1t−j\ h#¦H"h−j#R"1t−h\ j# "32#

� Rð1t−max"j\ h#\ min"j\ h#Ł � Rð1t−max"j\ h#\ j¦h−max"j\ h#Ł "33#

where H denotes Heaviside step function[ Because each vm"j# is a monotonically increasing function\Sm vm"j#Cm"1t−h−j# ³ Sm vm"h#Cm"1t−h−j# if j ³ m[ Consequently\ function r may be moreconcisely de_ned as

r"t\ j\ h# � min ðR"1t−h\ j#\ R"1t−j\ h#Ł "34#

The use of {min| is a compact way to make the expressions for r symmetric with respect to j andh\ i[e[

r"t\ j\ h# � r"t\ h\ j# "35#

Function r may be called the symmetrized relaxation function[ Combining "34# and "28#\ weacquire another compact expression]

F"t# �01 g

t

h�9 gt

j�9

min ðR"1t−h\ j#\ R"1t−j\ h#Ł do"j# do"h# "36#

In contrast to the free energy expressions "24#\ "26# and "27#\ this expression as well as "28# hasalso the advantage that the integration limits do not depend on the integration variable "i[e[\ theintegration domain is a square rather than two triangles#[

Equation "36# provides the _nal result that we have been seeking*an expression for theHelmholtz free energy density in a solidifying viscoelastic material that does not involve the internalparameters vm and Cm[ This equation generalizes the StavermanÐSchwarzl formula from non!ageingto ageing linear viscoelastic materials[ Same as the classical StavermanÐSchwarzl formula\ the newformula "36# is independent of the internal variables and gives the free energy density strictly interms of the measurable "external# viscoelastic properties\ namely the relaxation function and themacroscopic strain history[


6[ Free energy as a potential for stress

Functional F is a sum of the free energies Fm dvm of non!ageing components dvm "when thesecomponents exhibit instantaneous elasticity#[ These free energies are known to be potentials forthe corresponding stresses sm"t#[ This raises the question whether\ in the case of the solidi_cationtheory\ F may represent a potential for the stress s"t#[ This means that the following property ofa potential may be expected to apply]

1F"t#1o"t#

� s"t# "37#

We will now check whether it applies for the forms of the free energy we have derived[In viscoelasticity without ageing\ this property is known to be ful_lled for materials with

instantaneous elasticity[ In that case\ the partial derivative is taken as the coe.cient of the strainderivative at time t appearing in the time derivative of the functional F\ and is sometimes calledthe {instantaneous partial derivative| of F[ Let us now consider ageing viscoelastic materials withfree energy in the symmetrized form of "36#[ We now want to check whether indeed the propertyof a potential\ expressed by "37#\ is also veri_ed if we consider a viscoelastic material for which allthe constituents exhibit instantaneous elasticity\ that is\ for which the relaxation function R"t\ t?#is bounded from above\ or the relaxation kernels Cm in "21# are bounded from above for all m[Di}erentiating "36#\ we have

1F"t#1o"t#

�01 g

t

j�9

min ðR"1t−j\ t#\ R"1t−t\ j#Ł do"j#

¦01 g

t

h�9

min ðR"1t−t\ h#\ R"1t−h\ t#Ł do"h# do"j# "38#

Now\ substituting t � j¦D and taking into account the hardening condition in "15#\ we concludethat\ in the _rst integral in "38#\ the second of the two relaxation functions is always the minimum\while in the second integral the _rst is always the minimum[ So\

1F"t#1o"t#

�01 g

t

j�9

R"t\ j# do"j#¦01 g

t

h�9

R"t\ h# do"h# "49#

� gt

t?�9

R"t\ t?# do"t?# � s"t# "40#

This proves that\ if the material is solidifying and exhibits instantaneous elasticity\ the free energyF"t# per unit volume of material is a potential for the stress[

Similarly\ it may be proven that the free energy expressions in "24#\ "26#\ and "27# or "28#\which do not depend on the hardening condition in "30#\ represent a potential[ For example\di}erentiation of "24# provides

1F"t#1o"t#

� gt

j�t

R"1t−j\ t# do"j#¦gt

h�9

R"1t−t\ h# do"h#


� gt

h�9

R"t\ h# do"h# � s"t# "41#

7[ Extension to more general ageing viscoelasticity

Equations "38#\ "40# and "41# indicate that if\ conversely\ the free energy is assumed at the outsetto have one of the equivalent forms in "36#\ "24#\ "26#\ and "27# or "28#\ independently of the waywe derived them\ then the viscoelastic constitutive equation with ageing will be obtained from "41#[As we will see later\ this still ensures the ClausiusÐDuhem inequality to be satis_ed provided anappropriate choice is also made for the expression of the dissipated power[ By postulating suchexpressions for the free energy and the dissipation we de_ne a class of linear ageing viscoelasticmaterials[ Introducing such a de_nition\ which is a converse of our initial approach\ we realizethat these expressions may be independent of the solidi_cation theory "although our derivationfrom the solidi_cation theory\ which followed a deductive line of reasoning and was based onspeci_c microscopic properties in addition the stressÐstrain relation\ was necessary to actuallyobtain these expressions^ they could have hardly been guessed intuitively#[

In particular\ the free energy density expressions in "36#\ "24#\ "26#\ and "27# or "28# do notdepend on the condition of non!divergence "12# which characterizes the solidi_cation theory\ andthey do not depend on any particular assumptions about the microscopic solidi_cation process[Furthermore\ these expressions for F except "36# do not depend on the hardening condition in"15#[ The only consequence of the hardening condition is to make possible the {min| condition thatmakes the expression in "36# compact and symmetric[

Consider now the case of materials possessing no instantaneous elasticity[ For a given constitu!ent\ this means that at least one unit in the Maxwell chain model consists only of a lonely dashpot"of viscosity h9m#\ i[e[\ a dashpot "a dashpot with no spring coupled to it in series\ or E9m : �#[ Inthat case\ due to the uniform strain assumption adopted in the solidi_cation theory\ it su.ces toadd to the foregoing expression the viscous stress\ provided that the corresponding singular Dirackernel is not already included in the kernel R[ This gives\ for this more general case\

s"t# �1F"t#1o"t#

¦ho¾ with h � sn

m�0

vm"t#h9m "42#

The theory of viscoelasticity developed by Mandel "0856a\ b\ 0863a\ b\ 0868# and Huet "0868\0871\ 0881# has thus been extended to the general and practically important class of ageingmaterials described by the solidi_cation theory[

It may be noted that\ except for the {min| condition\ the present expression for the free energybears some similarity to the general form suggested in eqn "24# of Huet "0882a#\ provided that thekernel R�"t−u\ t−v^ t# in that equation as a function of three "rather than two# variables is put inthe particular form R"t¦u\ t−v#[ The corresponding expression of the free energy has again theproperty of a potential and the ensuing constitutive equation for stress corresponds to a specialclass of linear viscoelasticity with ageing[ In this general context\ the advantage of the solidi_cation


theory is that it provides a realistic and practically useful form and a physical basis of a microscopicorigin for that vague and somewhat arbitrary choice of the relaxation kernel[

8[ Complementary "Gibbs# free energy of solidifying material

A similar procedure can be applied to the expression "03# giving the negative of the density ofthe free enthalpy "or Gibbs free energy# for a non!ageing material[ This leads to a similar resultbecause the symmetry properties of the non!ageing creep functions Fm of the constituents are thesame as those of the non!ageing relaxation functions Cm[ De_ning the overall complementaryenergy density F� of the solidifying material in the classical way as the negative of the free enthalpydensity\ i[e[

F�"t# � s"t#o"t#−F"t# "43#

we _nd the following _ve equivalent expressions]

F�"t# � gt

h�9 $gt

j�h

J"1t−j\ h# ds"j#% ds"h# "44#

� gt

h�9 $gh

j�9

J"1t−h\ j# ds"j#% ds"h# "45#

�01 g

t

h�9 $gh

j�9

J"1t−h\ j# ds"j#¦gt

j�h

J"1t−j\ h# ds"j#% ds"h# "46#

�01 g

t

j�9 gt

h�9

"max ðJ"1t−j\ h#\ J"1t−h\ j#Ł# ds"j# ds"h# "47#

The last expression with {max| follows by setting d � h−j and u � 1t−j¦D and noting thatageing "solidifying\ age!hardening# materials satisfy the following condition]

J"h¦D\ j¦D# ¾ J"u\ j# "D × 9# "48#

which is the counterpart of "15#[ However\ the expressions in "44#Ð"46# do not depend on thishardening condition\ nor on the solidi_cation theory with its non!divergence condition[

The expression in "47# represents a potential for the strain because]

1F�"t#1s"t#

� o"t# "59#

The proof is similar to that given before for the Helmholtz free energy density[In contrast to "37#\ these expressions for F� are not contingent upon the aforementioned

restriction on the existence of instantaneous elasticity[ This dichotomy arises from the non!dualityprinciple\ which was enunciated "without attention to ageing# by Mandel "0856a\ b\ 0863b\ 0868#in a much more general context[ The non!duality principle postulates that\ while the total stressmay depend upon the instantaneous strain rate "a viscous e}ect#\ the total strain cannot depend


upon the instantaneous stress rate[ Mandel based his postulate on the available experimentalresults[

Based on his postulate\ Mandel showed that the property of a potential expressed by "59# appliesto every material with memory\ whatever its behavior might be\ provided that its instantaneousdeformation either is elastic or vanishes[ A dependence of the strain on the stress rate would infact involve a possible singularity in the compliance kernel\ thus yielding an additional term in thecompliance function[ But for the Maxwell chain there is no such singular term\ even if an isolateddashpot "with no spring attached to it in series# is coupled in parallel with the other units of thechain "i[e[\ if one constituent is a viscous ~uid#[ In that case\ the instantaneous deformation under_nite stress is vanishing\ which corresponds to the second condition of ensuring that Mandel|snon!duality principle allows the complementary energy to have the property of a potential[Equation "59# shows that the class of linear viscoelastic behavior with ageing described by thesolidi_cation theory satis_es the thermodynamic consequence of this principle[

09[ Free energy in terms of compliance function

The free energy density can be obtained from the complementary energy by Legendre trans!formation inverse to "43#\

F"t# � s"t#o"t#−F�"t# "50#

Substituting our previous results for s and F�\ one may derive for the representative volumeelement with ageing several equivalent expressions for the free energy density F in terms of theoverall compliance function J"t\ t?#^

F"t# �01 g

t

h�9 6gh

j�9

ð1J"t\ j#−J"1t−h\ j#Ł ds"j#7 ds"h#

¦01 g

t

h�9 6gt

j�h

ð1J"t\ j#−J"1t−j\ h#Ł ds"j#7 ds"h# "51#

�01 g

t

j�9 gt

h�9

"J"t\ j#¦J"t\ h#−max ðJ"1t−j\ h#\ J"1t−h\ j#Ł# ds"j# ds"h# "52#

This obviously reduces to the classical formula when there is no ageing[ The symmetrizing {max|condition is again contingent on the hardening condition "48#\ but the expression in "51# isindependent of it and has more general validity than the initial solidi_cation model[

00[ Energy dissipation of solidifying material in terms of relaxation function

According to "7#\ the dissipation D is expressed in forms similar to the free energy except thatthe relaxation rate appears instead of the relaxation kernel[ Therefore\ the same procedure asbefore may be followed[ In terms of the overall relaxation function\ this gives the following fourequivalent forms]


D"t# � −1 gt

h�9 $gt

j�h

Rþ"1t−j\ h# do"j#% do"h# "53#

D"t# � −1gt

j�9 $gj

h�9

Rþ"1t−j\ h# do"h#% do"j# "54#

D"t# � −gt

h�9 $gt

j�h

Rþ"1t−j\ h# do"j#¦gh

j�9

Rþ"1t−h\ j# do"j#% do"h# "55#

D"t# � −gt

h�9 gt

j�9

r¾ "t\ j\ h# do"j# do"h# "56#

Here Rþ"t\ t?# � 1R"t\ t?#:1t\ and the correct term in the {min|!conditions has been picked by heedingthe condition "15#\ which gives

for j ¾ h] r¾ "t\ j\ h# � Rþ"1t−h\ j# "57#

for j × h] r¾ "t\ j\h# � Rþ"1t−j\ h#[ "58#

Both expressions for Rþ"t\ t?# correspond to the same time lag\ 1t−j−h[ BecauseRþ"t\ t?# � Sm vm"t?#Cþm"t−t?# and Rþ"t¦D\ t?¦D# � Sm vm"t?¦D#Cþm"t−t?# ¾ Rþ"t\ t?# for D − 9\ andbecause functions vm"t# are increasing and Cþm ³ 9\ we conclude that the correct value of r¾ cor!responds to the maximum of the two Rþ"t\ t?# values[ From this\ or directly from "15#\ it followsthat D may also be written as

D � −gt

h�9 gt

j�9

max ðRþ"1t−h\ j#\ Rþ"1t−j\ h#Ł do"j# do"h# "69#

The foregoing expressions can also be derived directly from the expression "36# for the freeenergy density using the general expression for the isothermal dissipated power in terms of thetotal power and the "isothermal# rate of change of the free energy density[ This furnishes

D"t# � s"t#o¾"t#−Fþ "t# � s"t#o¾"t#−01

o¾"t# gt

9

min ðR"1t−j\ t#\ R"1t−t\ j#Ł do"j#

−01

o¾"t# gt

9

min ðR"1t−t\ h#\ R"1t−h\ t#Ł do"h#−gt

h�9 gt

j�9

r¾ "t\ j\ h# do"j# do"h# "60#

� s"t#o¾"t#−o¾"t# gt

9

R"t\ t?# do"t?#−gt

h�9 gt

j�9

r¾ "t\ j\ h# do"j# do"h# "61#

� −gt

h�9 gt

j�9

r¾ "t\ j\ h# do"j# do"h# × 9 "62#

which coincides with "56#[


In view of "11#\ the dissipation inequality is automatically satis_ed because Rþ"t\ t?# �1R"t\ t?#:1t ³ 9 if the material is solidifying\ i[e[ age!hardening "or if it is non!ageing#[ Conversely\if one departs from the condition of positiveness of dissipation\ it follows that Rþ"t\ t?# ³ 9\ andthus R is proven to be a monotonically decreasing function of t[ So\ the expressions we obtainedfor the free energy and the dissipation conform to the ClausiusÐDuhem inequality in a local form[

Conversely\ if the free energy is chosen a priori in the form of one of the equivalent expressions"36#\ "24#\ "26#\ and "27# or "28# without reference to any microstructural model\ taking thedissipation in one of the foregoing equivalent forms is a necessary and su.cient condition toobtain a thermodynamically consistent model[

01[ Energy dissipation of solidifying material in terms of compliance function

According to "04#\ the dissipation D may be expressed in forms similar to the free enthalpydensity\ except that the compliance rate appears instead of the compliance kernel[ Therefore\ thesame procedure as before may be applied[ In terms of the macroscopic compliance function J\ thisleads to the following four equivalent forms]

D"t# � 1 gt

h�9 $gt

j�h

Jþ"1t−j\ h# ds"j#% ds"h# "63#

� 1 gt

h�9 $gh

j�9

Jþ"1t−h\ j# ds"j#% ds"h# "64#

� gt

h�9 $gh

j�9

Jþ"1t−h\ j# ds"j#¦gt

j�h

Jþ"1t−j\ h# ds"j#% ds"h# "65#

� gt

j�9 gt

h�9

"max ðJþ"1t−j\ h#\ Jþ"1t−h\ j#Ł# ds"j# ds"h# "66#

where Jþ"t\ t?# � 1J"t\ t?#:1t[ The last expression with {max| follows by setting d � h−j\ t � 1t−j

and noting that ageing "solidifying\ age!hardening# materials satisfy for positive d the inequality

Jþ"t\ j¦d# ¾ Jþ"t−d\ j# "67#

With this\ all the basic thermodynamic functions of ageing viscoelasticity have been expressedwithout using internal variables[ Thus\ all of the StavermanÐSchwarzl theory extended by Mandelhas now been generalized to the case of ageing[

02[ Generalization to isotropic materials under triaxial stress

Although\ for the sake of brevity\ our formulation has been restricted to uniaxial stress\ ageneralization to triaxial stress and strain is possible with a more general version of the theory


obtained upon choosing the free energy a priori in the form of one of the equivalent expressions"36#\ "24#\ "26#\ and "27# or "28# and making use of the aforementioned properties of a potential[This is true even for anisotropic materials\ in which case the ordinary products need to be replacedby appropriately contracted tensor products[ The special case of isotropy will now be explained insome detail[

For isotropic materials\ the viscoelastic behavior may be decoupled into volumetric and devi!atoric parts\ characterized by the volumetric and deviatoric relaxation functions RV"t\ t?# andRD"t\ t?#[ The total Helmholtz free energy density may be written as a sum of volumetric anddeviatoric parts^ F � FV¦FD[ To make a symmetric expression possible\ we may _rst supposeboth RV"t\ t?# and RD"t\ t?# to satisfy the hardening conditions "15#[ Then if one assumes thevolumetric and deviatoric components of the free energy to be in the form of "36#\ which have theproperty of a potential for stress and have kernels bounded from above "i[e[\ exhibiting noinstantaneous inelastic strain#\ one obtains\ by a similar procedure as before\

FV"t# �21 g

t

h�9 gt

j�9

min ðRV"1t−h\ j#\ RV"1t−j\ h#Ł doV"j# doV"h# "68#

FD"t# �01 g

t

h�9 gt

j�9

min ðRD"1t−h\ j#\ RD"1t−j\ h#Ł doDij "j# doD

ij "h# "79#

where subscripts i\ j refer to components in Cartesian axes xi\ i � 0\ 1\ 2^ oV � okk:2 � volumetricstrain\ and oD

ij � oij−dijoV � deviatoric strain[

Generalizing any one of the equivalent expressions "24#\ "26#\ and "27# or "28#\ one obtainsalternative expressions that do not require the hardening condition to be met^

FV"t# � 2 gt

h�9 gh

j�9

RV"1t−h\ j# doV"j# doV"h# "70#

� 2 gt

h�9 gt

j�h

RV"1t−j\ h# doV"j# doV"h# "71#

and

FD"t# � gt

h�9 gh

j�9

RD"1t−h\ j# doDij "j# doD

ij "h# "72#

� gt

h�9 gt

j�h

RD"1t−j\ h# doDij "j# doD

ij "h# "73#

Then\ in view of the property of free energy as a potential\ the constitutive equation may be writtenin the classical form used for the isotropic linear viscoelastic materials with ageing]

sij"t# � 2dij gt

j�9

RV"t\ j# doV"j#¦gt

j�9

RD"t\ j# doDij "j#


Furthermore\ according to the ClausiusÐDuhem inequality\ the dissipation may be expressed asD � DV¦DD in which DV and DD are given by one of the following equivalent forms\

DV"t# � 5 gt

h�9 gh

j�9

RþV"1t−h\ j# doV"j# doV"h# "74#

� 5 gt

h�9 gt

j�h

RþV"1t−j\ h# doV"j# doV"h# "75#

and

DD"t# � 1 gt

h�9 gh

j�9

RþD"1t−h\ j# doDij "j# doD

ij "h# "76#

� 1 gt

h�9 gt

j�h

RþD"1t−j\ h# doDij "j# doD

ij "h# "77#

respectively[ From this\ the other thermodynamic functions can be obtained in similar generalizedforms[

03[ Thermodynamic functions in terms of Maxwell or Kelvin chain

Although the purpose of this paper has been to formulate the thermodynamic functions withoutusing the material properties associated with the internal variables\ expressions in terms of suchproperties are sometimes useful for computational algorithms[ In the case of Maxwell chain withage!dependent moduli Em"t# and viscosities hm"t#\ such expressions may simply be obtained bysubstituting\ into any of the foregoing expressions\

R"t\ j# � sn

m�0

vm"j#Cm"t−j#\ Cm"t−j# � E9m e

−t−j

tm "78#

where E9m are the moduli of the nonageing constituents of growing volumes vm"t#\ associated with

the individual Maxwell units of the chain[ Em"t# � E9mvm"t# and hm"t# � tmEm"t# � tmE

9mvm"t# are the

age!dependent moduli and viscosities of the ageing Maxwell chain model of the material^ seeBaz³ant "0864\ 0871#^ RILEM "0877#[ Similar expressions are available in these papers for thecompliance function J"t\ j# corresponding to an ageing Kelvin chain model[

04[ Conclusions

"0# The solidi_cation theory for ageing materials\ which represents a physically realistic model forviscoelasticity of ageing concrete caused by chemical solidi_cation on the pores wall\ allowsthe StavermanÐSchwarzl formulae for the densities of Helmholtz free energy and for thedissipated power of a linear viscoelastic material to be generalized for the case of ageing[


Mandel|s extensions of these expressions to the free enthalpy and dissipation in dual formsbased on the relaxation and compliance functions can also be generalized[

"1# The advantage of these new formulae is that they express the thermodynamic functions interms of only the measurable physical variables "the measured strain history\ and the stresshistory determined from measured load history# and measurable viscoelastic characteristics"the relaxation and compliance functions#*in other words\ that they do not involve theinternal variables such as the partial stresses of Maxwell chain model nor the characteristicsof the solidi_cation process "volume growth functions vm# and the relaxation functions of theconstituents "Cm#\ used in previous expressions for the thermodynamic functions of suchmaterials[

"2# It is found that\ aside from several other equivalent expressions\ these thermodynamic functionscan always be expressed as the double integral over strain or stress history on a square domainof a quadratic expression depending only on a symmetrized form of the relaxation function\or the compliance function\ or their rates[ The symmetrization can be reduced to taking aminimum or maximum of two relaxation or compliance functions "or their rates# with inter!changed time variables[

"3# For materials exhibiting instantaneous elasticity\ the partial derivative of the Helmholtz freeenergy density per unit volume of material with respect to the strain history yields the stresshistory[ Thus the basic property of the free energy as a potential for the stress is extended toviscoelasticity with ageing described by the solidi_cation theory[ In absence of instantaneouselasticity\ it su.ces to add the "instantaneous# purely viscous stress to the stress obtained fromthe potential[ The free enthalpy density is further proven to be a potential for the total strainhistory\ which holds true even if instantaneous elasticity is absent[ The dichotomy in theinstantaneous elasticity restrictions can be related to Mandel|s non!duality principle whosethermodynamic consequence turns out to apply also to the case of ageing[

"4# The dissipated power can be expressed in several ways as a quadratic double integral of therelaxation function rate over the strain history[ In one of these forms\ the relaxation functionin the kernel is symmetrized by the use of a minimum condition[

"5# The formulae derived are thermodynamically admissible\ satisfying the ClausiusÐDuheminequality in the local form[ This is important because the literature on concrete involvesseveral examples of ageing formulations proposed for concrete creep that have later beenshown to violate the dissipation inequality required by the second law of thermodynamics ðe[g[Baz³ant "0871#\ RILEM "0877#Ł[

"6# Although the solidi_cation model for concrete has been used as a physically instructive startingpoint\ the thermodynamic functions derived appear to have a more general validity[ Forisotropic materials characterized by the volumetric and deviatoric relaxation functions\ thesefunctions are explicitly extended to three dimensions[ The extension to the anisotropic casemay be obtained in the same straightforward fashion[

"7# The relationship between the expressions obtained here and the previous more generalexpressions of Huet "0882a# has been established[

"8# Among numerous applications that can be foreseen\ the thermodynamic functions derivedshould be useful for obtaining bounds on the response of ageing linearly viscoelastic structures"as done in Huet\ 0884b\ for the non!ageing case#[ They should further be useful for derivingapproximate solutions and for formulating numerical solution algorithms[


Acknowledgments

Funding for a visiting appointment of the _rst author at LMC\ EPFL\ Lausanne\ is gratefullyappreciated[ Further funding was provided in Switzerland by the Swiss Federal O.ce of WaterEconomy and by the Swiss National Fund for Scienti_c Research "under Contract No[ 19!34660[84#\ and in the U[S[ by the National Science Foundation "under Grant MSS!800!3365#[Thanks are due to Milan Jira�sek\ research engineer at EPFL\ for some valuable discussions[

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Baz³ant\ Z[P[\ Hauggaard\ A[B[\ Baweja\ S[\ Ulm\ F[!J[\ 0886a[ Microprestress!solidi_cation theory for concrete creep[I[ ageing and drying e}ects[ Journal of Engineering Mechanics\ ASCE 012 "00#\ 0077Ð0083[

Baz³ant\ Z[P[\ Hauggaard\ A[B[\ Baweja\ S[\ 0886b[ Microprestress!solidi_cation theory for concrete creep[ II[ Algorithmand veri_cation[ Journal of Engineering Mechanics\ ASCE 012 "00#\ 0084Ð0190[

Biot\ M[A[\ 0843[ Theory of stressÐstrain relations in anisotropic viscoelasticity and relaxation phenomena[ J[ Appl[Phys[ 14 "00#\ 0274Ð0280[

Boltzmann\ L[\ 0763[ Zur theorie der elastischen Nachwirkungen[ Sitzungsbericht\ Kaiserlich[ Akad[ Wiss[\ Wien\ Math[Naturwiss[ Klasse[ 69 "II#\ 164[

Brun\ L[\ 0854[ Sur deux expressions\ analogues a� la formule de Clapeyron\ donnant l|e�nergie libre et la puissancedissipe�e pour un corps viscoe�lastique[ Comptes Rendus Ac[ Sc[ Paris 150\ 30Ð33[

Brun\ L[\ 0856[ Thermodynamique et viscoe�lasticite�[ Cahiers du Groupe Francžais de Rhe�ologie\ SEDOCAR\ Paris\ 3\I\ 080Ð191[

Brun\ L[\ 0858[ Me�thodes e�nerge�tiques dans les syste�mes e�volutifs line�aires[ Journal de Me�canique\ Paris\ 7\ 014Ð055[Carol\ I[\ Baz³ant\ A[P[\ 0882[ Viscoelasticity with ageing caused by solidi_cation of non!ageing constituent[ Journal of

Engineering Mechanics\ ASCE\ 008 "00#\ 1141Ð1158[Christensen\ R[M[\ 0858[ Viscoelastic properties of heterogeneous media[ J[ Mech[ Phys[ Solids 06\ 12Ð30[Christensen\ R[M[\ 0871[ Theory of Viscoelasticity[ Academic Press\ New York\ pp[ 137Ð140[Coleman\ B[N[\ 0853[ Thermodynamics of materials with memory[ Arch[ Rational Mech[ Anal[ 06\ 0Ð35[


Coleman\ B[N[\ Mizel\ V[J[\ 0853[ Existence of caloric equations of state in thermodynamics[ J[ Chemical Physics 39\0005Ð0014[

Day\ W[A[\ 0861[ The Thermodynamics of Simple Materials with Fading Memory[ Springer!Verlag\ Berlin[Fre�chet\ M[\ 0809[ Sur les fonctionnelles continues[ Ann[ Sc[\ Paris 16\ 2\ 082Ð105[Huet\ C[\ 0869a[ Sur l|e�volution des contraintes et de�formations dans les syste�mes multicouches constitue�s de mate�riaux

viscoe�lastiques pre�sentant du vieillissement[ Comptes Rendus Ac[ Sc[ Paris 169\ 102Ð105[Huet\ C[\ 0869b[ Le se�chage des pa¼tes argileuses et la me�canique des milieux continus e�volutifs[ Bulletin du Groupe

Francžais des Argiles 12\ 28Ð46[Huet\ C[\ 0862[ Application a� la viscoe�lasticite� non line�aire du calcul symbolique a� plusieurs variables[ Rheologica Acta

01\ 168Ð177[Huet\ C[\ 0863[ Ope�rateurs matriciels en viscoe�lasticite� line�aire avec vieillissement et application aux structures vis!

coe�lastiques he�te�roge�nes[ In] Hult\ J[ "Ed[#\ Mechanics of Visco!Elastic Media and Bodies[ Springer\ Berlin\ pp[ 020Ð035[

Huet\ C[\ 0868[ Sur la notion d|e�tat local en rhe�ologie[ Science et Techniques de l|Armement\ 3\ 500Ð541[Huet\ C[\ 0879\ Adaptation d|un algorithme de Baz³ant au calcul des multilames viscoe�lastiques vieillissants[ Materials

and Structures\ Paris 02\ 63\ 80Ð87[Huet\ C[\ 0871[ Topics in thermodynamics of rheological behavior[ Rheologica Acta 10\ 259Ð254[Huet\ C[\ 0874[ Relations between creep and relaxation functions in nonlinear viscoelasticity with or without ageing[

Journal of Rheology 18 "2#\ 134Ð146[Huet\ C[\ 0881[ Minimum theorems for viscoelasticity\ Eur[ J[ Mech[ A:Solids\ 00 "4# 542Ð573[Huet\ C[\ 0882a[ Some basic tools and pending problems in the development of constitutive equations for the delayed

behaviour of concrete[ In] Baz³ant\ Z[P[\ Carol\ I[ "Eds[#\ Creep and Shrinkage of Concrete[ Spon\ London\ pp[ 078Ð199[

Huet\ C[\ 0882b[ An integrated approach of concrete micromechanics[ In] Huet\ C[ "Ed[#\ Micromechanics of Concreteand Cementitious Composites[ Presses Polytechniques et Universitaires Romandes\ Lausanne\ pp[ 006Ð035[

Huet\ C[\ 0884a[ A continuum thermodynamics approach for studying microstructural e}ects on the non!linear fracturebehaviour of concrete seen as a multicracked granular composite material[ In] Wittmann\ F[H[ "Ed[#\ FractureMechanics of Concrete and Concrete Structures\ vol[ 1[ Aedi_catio\ Freiburg\ pp[ 0978Ð0097[

Huet\ C[\ 0884b[ Bounds for the overall properties of viscoelastic heterogeneous and composite materials[ Archives ofMechanics\ Warsaw 36 "5# 0014Ð0044[

Huet\ C[\ 0886[ An integrated micromechanics and statistical continuum mechanics approach for studying the fracturebehaviour of microcracked heterogeneous materials with delayed response[ Engineering Fracture Mechanics\ specialissue on Statistical Fracture Mechanics\ in press[

Kestin\ J[\ 0863[ Entropy and entropy production^ discussion paper[ Domingos\ I[J[J[D[ et al[ "Eds[#[ Foundations ofContinuum Thermodynamics[ Macmillan\ London\ pp[ 032Ð047[

Kestin\ J[\ 0868[ A Course in Thermodynamics[ Hemisphere\ New York^ revised printing\ 1 volumes[Kestin\ J[\ 0881[ Local!equilibrium formalism applied to mechanics of solids[ Int[ J[ Solids Structures "George Herrmann

Anniversary Issue# 18 "03:04#\ 0716Ð0725[Manel\ J[\ 0846[ Sur les corps viscoe�lastiques dont les proprie�te�s de�pendent de l|a¼ge "On the viscoelastic bodies with age!

dependent properties\ in French#[ Comptes Rendus Ac[ Sc[ Paris 136\ 064Ð067[Mandel\ J[\ 0853[ Cours de Me�canique[ Ecole Polytechnique\ Paris[Mandel\ J[\ 0855[ Cours de Me�canique des Milieux Continus[ Gauthier!Villars[ Paris\ 1 volumes[Mandel\ J[\ 0856a[ Application de la thermodynamique aux milieux viscoe�lastiques a� e�lasticite� nulle ou restreinte[

Comptes Rendus Ac[ Sc[\ Paris 153\ 022Ð023[Mandel\ J[\ 0856b[ Application de la thermodynamique aux milieux viscoe�lastiques[ Archives of Mechanics\ Warsaw

1[19\ 070Ð077[Mandel\ J[\ 0863a[ Un principe de correspondance pour les corps viscoe�lastiques line�aires vieillissants[ In] Hult\ J[ "Ed[#\

Mechanics of Visco!Elastic Media and Bodies[ Springer\ Berlin\ pp[ 33Ð44[Mandel\ J[\ 0863b[ Introduction a� la Me�canique des Milieux Continus De�formables[ Scienti_c Editions of Poland\

Warsaw[Mandel\ J[\ 0868[ Variables cache�es\ puissance dissipe�e\ dissipativite� normale[ Sciences et Techniques de l|Armement\

Paris\ 42\ 3\ 414Ð427[


RILEM Committee TC 58 "0877#[ "Z[ Baz³ant\ Chairman#\ State of the art in mathematical modeling of creep andshrinkage of concrete[ In Baz³ant\ Z[P[\ Wiley\ J[ "Eds[#\ Mathematical Modeling of Creep and Shrinkage of Concrete[Chichester and New York\ pp[ 46Ð104[

Roscoe\ R[\ 0849[ Mechanical models for the representation of viscoelastic properties[ Brit[ J[ Appl[ Phys[ 0\ 060Ð062[Staverman\ A[J[\ Schwarzl\ P[\ 0841a[ Thermodynamics of viscoelastic behaviour "model theory#[ Proc[ Acad[ Sc[\ The

Netherlands 44\ 363Ð374[Staverman\ A[J[\ Schwarzl\ P[\ 0841b[ Non!equilibrium thermodynamics of visco!elastic behaviour[ Proc[ Acad[ Sc[\

The Netherlands 44\ 375Ð381[Truesdell\ C[\ 0858[ Rational Thermodynamics[ McGraw!Hill\ New York[Valanis\ K[C[\ 0860[ Irreversibility and existence of entropy[ Int[ J[ Non!Linear Mechanics 5\ 226Ð259[Valanis\ K[C[\ 0872[ Partial integrability as a basis for the existence of entropy in irreversible systems[ ZAMM 52\ 62Ð

79[Volterra\ V[\ 0776[ Sopra le funzioni che dipendono da altre funzioni[ R[ C[ Acad[ Lincei 3 "2#\ 86Ð094[Volterra\ V[\ 0848[ Theory of Functionals and of Integral and Integro!di}erential Equations "0814 Madrid Lectures#[

Dover\ New York[

thermodynamic functions for ageing viscoelasticity ...model\ note that any linear viscoelastic...

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