Modelling of hygro-thermal behaviour of concreteat high temperature with thermo-chemical and mechanical material degradationD.Gawin a ,F.Pesavento b ,B.A.Schreflerb,*aDepartment of Building Physics and Building Materials,Technical University of L odz,Al.Politechniki 6,93-590L o dz,Poland bDepartment of Structural and Transportation Engineering,University of Padua,via Marzolo 9,35131Padua,ItalyReceived 19June 2002;received in revised form 22October 2002AbstractA mathematical model for analysis of hygro-thermal behaviour of concrete as a multi-phase porous material at high temperatures,accounting for material deterioration,is presented.Full development of the model equations,starting from the macroscopic balances of mass,energy and linear momentum of single constituents is presented.Constitutive relationships for concrete at high temperature,including those concerning material damage,are discussed.The classical isotropic non-local damage theory is modified to take into account the mechanical-and the thermo-chemical concrete damage at high temperature.The final form of the governing equations,their discretised FE form,and their numerical solution are presented.The results of two numerical examples,concerning fire performance of 1-D and 2-D HPC structures,are discussed.Ó2003Elsevier Science B.V.All rights reserved.Keywords:Concrete ;High temperature;Thermo-chemical damage;Mechanical damage;Numerical model;Hygro-thermo-mechanical behaviour1.IntroductionSince the second half of the 1990s there is a renewed interest in the modelling of concrete at high and very high temperature.Introduction of new high and ultra high performance concretes required their as-sessment as far as spalling is concerned.The BRITE EURAM Project ‘‘HITECO’’(HIgh TEmperature COncrete)[8],was aimed at this purpose and has yielded very interesting results [29].Further,increased recurrence of tunnel fires in Europe,often connected with losses of life,always with destruction of the concrete vaults and walls and causing heavy economical losses,have evidenced the dramatic need for an*Corresponding author.E-mail addresses:gawindar@ck-sg.p.lodz.pl (D.Gawin),pesa@caronte.dic.unipd.it (F.Pesavento),bas@caronte.dic.unipd.it (B.A.Schrefler).0045-7825/03/$-see front matter Ó2003Elsevier Science B.V.All rights reserved.doi:10.1016/S0045-7825(03)00200-7Comput.Methods Appl.Mech.Engrg.192(2003)1731–1771/locate/cma1732 D.Gawin et al./Comput.Methods Appl.Mech.Engrg.192(2003)1731–1771D.Gawin et al./Comput.Methods Appl.Mech.Engrg.192(2003)1731–177117331734 D.Gawin et al./Comput.Methods Appl.Mech.Engrg.192(2003)1731–1771D.Gawin et al./Comput.Methods Appl.Mech.Engrg.192(2003)1731–17711735 upgrading of existing tunnels and introduction of new standards.Again the need for a new modelling capacity for concretes under such extreme conditions(temperatures exceeding1200°C for considerable time spans)has been evidenced.The EU Project UPTUN(cost-effective sustainable and innovative upgrading methods forfire safety in existing tunnels),involving about40partners,is aimed at that purpose.Finally,the modelling of aging in concrete Nuclear Power Plant Structures requires also to take into account concrete under high(and cyclic) temperatures.This aspect is currently investigated within the EURATOM project MÆCENAS.It is commonly accepted[4–7,17–19]that all the above situations cannot satisfactorily be modelled by considering thermo-mechanical processes alone.Heat and mass transfer within the concrete body(and on its boundary)has to be taken into account,together with phase changes and dehydration.The model has hence to be a chemo-hygro-thermo-mechanical one.We have been(and are)involved in all three of the above mentioned research projects and have developed such a model.This model stems from early work of Gawin et al.[16],and Gawin and Schrefler[20]and has been further developed over the years[17].Initially its applicability was limited by critical temperature of water.This limit has now been overcome[19]. Mechanical damage alone was initially considered.A coupling of permeability with mechanical damage has been introduced in[18].When simulating a real experimentalfire situation in a motorway tunnel,it turned out that mechanical damage alone was not sufficient.Thermally induced deterioration due to strains at material meso-scale and due to concrete dehydration,called thermo-chemical damage(and a modification of stress–strain curve)is also needed.This is here introduced in a framework of the isotropic non-local damage theory.Since the theory underlying the overall model is scattered in several papers,a full devel-opment of the model equations and their FE discretization is also presented.A discussion of the choice of the state variables and ensuing constitutive relations follows.Finally,two examples showing the importance of material deterioration on hygro-thermal and mechanical performance of concrete structures at high temperature are shown and discussed.Recently several works devoted in particular to the thermal spalling phenomenon in concrete at elevated temperature has been published,e.g.[37,38,43,44].As pointed out in[43],there are two main reasons of the phenomenon:build-up of pore pressure and restrained thermal dilatation.The latter hypothesis was theo-retically justified in[43]to be the most important reason of thermal spalling and used for analysis of the ‘‘Chunnel’’fire[44].In the AuthorsÕopinion both of the above mentioned reasons of the phenomenon, acting together,are of importance for the thermal spalling occurrence,in particular for the explosive one,as shown by the experimental tests carried out by Phan et al.[38].At this point it is worth to discuss some limitations of the recently proposed,macroscopic phenome-nological theory by Ulm et al.[43].Effect of temperature on concrete strength is modelled there as chemo-plastic softening within the framework of chemo-plasticity,accounting explicitly for the dehydration of concrete and its cross-effects with deformation and temperature.In the model most material properties are practically unique functions of temperature only,even if formally they depend on hydrate mass and plastic softening parameter.Moisture transfer and evaporation within the heated concrete are not taken into account,hence effect of pore pressure cannot be considered.Effect of cracking on concrete behaviour is neglected.Most of these shortcomings are overcome by the theory presented in this paper.2.Macroscopic balance equationsThe balance equations are written by considering concrete as a multi-phase material.The solid skeleton voids arefilled partly by liquid water and partly by a gas phase.Below the critical temperature of water,T cr,the liquid phase consists of bound water,which is present in the whole range of moisture content,and capillary water,which appears when degree of water saturation exceeds the upper limit of the hygroscopic region,S ssp.Above the temperature T cr the liquid phase consistsof bound water only.In the whole temperature range the gas phase is a mixture of dry air and water vapour (condensable gas constituent for T <T cr ).The model equations are obtained by means of the hybrid mixture theory (HMT)originally proposed by Hassanizadeh and Gray [25–27]and then applied for geomaterials in general by Lewis and Schrefler [30]and Schrefler [42],and for building materials by Gawin [14].For the sake of brevity,only the final form of the macroscopic conservation equations is given here.The full development of the model equations,starting from the local,microscopic balance equations with successive volume averaging,are presented in [14,30,35,42].The general form of the macroscopic,volume averaged mass conservation equation of the p -phase is [30,42]:D pq pD tþq p div v p ¼q p e p ðq Þ;ð1Þwhere q p is apparent density (related to the whole volume of the medium),v p the velocity and q p e p ðq Þthe volumetric mass source,super or subscript p refers to the p -phase.This mass balance equation has the following form for the solid skeleton [14]:D sq sD tþq s div v s ¼À_mdehydr ;ð2Þwhere _mdehydr is mass source of liquid water (and corresponding skeleton mass sink)related to the cement dehydration process.After application of the relation between the phase averaged density,q p ,and the intrinsic phase averaged density,q p ,[25–27,30]:q p ¼g p q pð3Þwith g p being the volume fraction of the p -phase,and after some simple transformations,Eq.(2)can be rewritten asð1Àn Þq s D sq s D t ÀD sn D t þð1Àn Þdiv v s¼À_m dehydr q s:ð4ÞThe volume averaged mass conservation equation for liquid water (capillary and physically adsorbed)hasthe following form:D wq wD tþq w div v w ¼_mdehydr À_m vap ;ð5Þwhere _mvap is the vapour mass source caused by the liquid water evaporation or desorption (for low values of the relative humidity inside the material pores).It is worth to underline that for the liquid water we have two source terms.By introducing the water relative velocity and material derivative of water density,with respect to the skeleton,the latter equation can be written as follows:D sq wD tþv ws Ágrad q w þq w div ðv s þv ws Þ¼_mdehydr À_m vap ð6Þwhere v p s means the p -phase relative velocity with respect to the skeleton.1736 D.Gawin et al./Comput.Methods Appl.Mech.Engrg.192(2003)1731–1771After some algebraic transformations and application of (3)with g w ¼nS w ,we have D sn D t þn q w D sq w D t þn S w D sS w D t þ1S w q w div ðnS w q w v ws Þþn div v s¼_m dehydr À_m vap S w q w:ð7ÞIn order to eliminate the time derivative of porosity,D sn =D t ,from the latter equation,we sum it up with (4)and obtain the mass conservation equation of liquid water and solid skeleton as follows:ð1Àn Þq s D sq s D t þdiv v s þn q w D sq w D t þn S w D sS w D t þ1S w q w div ðnS wq w v wsÞ¼_m dehydr À_m vap S w q w À_m dehydr q s :ð8ÞThe macroscopic volume averaged mass conservation equation of dry air,[30,42]D gaq gaþq ga div v ga ¼0ð9Þafter changing the material time derivative D gaq ga =D t into D gq ga =D t and decomposition of the dry air velocity into the diffusional,u ga ¼v ga Àv g ,and advectional (i.e.related to the centre of gravity of the whole gas phase),v g ,components,[30,42],can be rewritten asD gq gaD t þdiv J ga d þq ga div v g¼0;ð10ÞwhereJ ga d ¼q ga u ga¼nS g q ga u ga ð11Þis the diffusive mass flux of dry air molecules in the gas phase.It is reminded that there is no source term related to the dry air.Eq.(10)is now transformed in a similar way as done for the liquid water balance,i.e.material time derivative and relative velocity with respect to the solid are introduced,relation (3)with g ga ¼n ð1ÀS w Þis applied and the resulting equation,divided by q ga ð1ÀS w Þ,is summed up with the solid skeleton mass balance (4).Finally,we obtain the following equation:n S g D sS g D t þn q ga D sq ga D t þ1S g q ga div J ga dþ1S g q ga div ðnS g q ga v gs Þþ1Àn q s D sq s D t þdiv v s¼À_m dehydr q s :ð12ÞThe macroscopic mass balance of the water vapour [14,30,42]D gwq gwD tþq gw div v gw ¼_mvap ;ð13Þcan be presented,similarly to the equation for the dry air,in the following form,D g q gwD tþdiv J gw d þq gw div v g¼_m vap ;ð14Þwhere the diffusive mass flux of vapour molecules in the gas is defined as [14,30,42]J gw d ¼q gw u gw¼nS g q gw u gw :ð15ÞOne should remember,that the gas phase is an ideal binary gas mixture of dry air and water vapour,hence [14,30,42]J gw d ¼ÀJ gad :ð16ÞD.Gawin et al./Comput.Methods Appl.Mech.Engrg.192(2003)1731–17711737By applying the same transformations as for the dry air mass balance and summing(14)with(4),we obtain the following equation:n S g DsS gD tþnq gwDsq gwD tþ1S g q gwdiv J gwdþ1S g q gwdivðnS g q gw v gsÞþ1Ànq sDsq sD tþdiv v s¼À_m dehydrq sþ_m vapS g q gw:ð17ÞWe do not have any constitutive relationship for the mass source term,_m vap,appearing in the latter equation and in(8),but we can use one of the two mass balance equations to eliminate this source term from the another one.The macroscopic,volume averaged enthalpy balance equation for the p-phase,after neglecting some terms related to viscous dissipation and mechanical work,caused by density variations due to temperature changes and caused by volume fraction changes,has the following general form[30,42]:q p C pp DpT pD t¼q p h pÀdiv~q pþq p R pHÀq p e pðqÞH p;ð18Þwhere C pp is the specific isobaric heat,~q p the heatflux,q p h p the volumetric heat sources,q p R pHthe termexpressing energy exchange with the other phases(including mechanical interaction and excluding enthalpy exchange due to mass sources),H p the specific enthalpy,of the p-phase.In concrete at high temperature all heat sources,except those related to phase changes and dehydration process,can be neglected.We assume here that all phases of the material are locally in thermodynamic equilibrium,hence their temperatures are the same,T p¼Tðp¼s;w;gÞ.These temperatures may however vary throughout the domain.Summing up the enthalpy balances for all the phases of the medium,taking into account the mass sources,specific for the particular constituents,and transforming all material time derivatives into those with respect to the solid skeleton,as well as having in mind that[30,42]:X p qpR pH¼0;ð19Þone obtains the following enthalpy balance equation for the whole medium:ðq C pÞeff DsTD tþðq w C wpv wsþq g C gpv gsÞÁgrad TÀdiv~q¼À_m vap D H vapÀ_m dehydr D H dehydr;ð20Þwhereðq C pÞeff ¼q s C spþq w C wpþq g C gp;~q¼~q sþ~q wþ~q g;D H vap¼H gwÀH w;D H dehydr¼H wÀH ws:ð21ÞAbove,H ws is the specific enthalpy of the chemically bound water,D H vap the specific enthalpy of evapo-ration and D H dehydr the specific enthalpy of dehydration.Hygro-thermal phenomena in concrete,even at high temperature,are relatively slow,hence inertial forces can be neglected.For such a case the macroscopic,volume averaged linear momentum balance equation for the p-phase has the following,general form[30,42]:div t pþq p gþq p½e pðq_rÞþ^t p ¼0;ð22Þ1738 D.Gawin et al./Comput.Methods Appl.Mech.Engrg.192(2003)1731–1771D.Gawin et al./Comput.Methods Appl.Mech.Engrg.192(2003)1731–17711739 where t p is the macroscopic stress tensor in the p-phase,g the acceleration of gravity,q p^t p the volumetric exchange term of linear momentum with other phases due to mechanical interaction,and q p e pðq_rÞthat due to phase changes or chemical reactions.These exchange terms are subject to the constraint Xq p½e pðq_rÞþ^t p ¼0:ð23ÞpAfter summing up the macroscopic linear momentum balances for all the phases and introducing the total stress tensorr¼t sþt wþt g;ð24Þtaking into account condition(23)and assuming continuity of stresses at thefluid–solid interfaces,one obtainsdiv rþq g¼0;ð25Þwhereq¼ð1ÀnÞq sþnS w q wþnð1ÀS wÞq gð26Þis the averaged apparent density of the medium.The volume averaged angular momentum balance equation shows[30,42],that for non-polar media,as moist concrete is assumed in this work,all macroscopic partial stress tensors are symmetric, t p¼ðt pÞT;ð27ÞFrom the macroscopic entropy inequalities for the medium constituents,limitations for the form of con-stitutive relationships can be deduced,ing the Colemann–Noll procedure,as done in[23].This as-sures that the constitutive relations do not violate the second law of thermodynamics.3.State variablesA proper choice of state variables for description of concrete at high temperature is of particular im-portance.From a practical point of view,the physical quantities used,should be possibly easy to measure during experiments,and from a theoretical point of view,they should uniquely describe the thermodynamic state of the medium[22].They should also assure a good numerical performance of the computer code based on the resulting mathematical model.The necessary number of the state variables may be signifi-cantly reduced if existence of local thermodynamic equilibrium at each point of the medium is assumed.In such a case physical state of different phases of water can be described by use of the same variable.When fast hygro-thermal phenomena in concrete at high temperature are analysed,the assumption is debatable, but it is almost always used in modelling.We also apply it in development of the present model.Having in mind all the aforementioned remarks,we will briefly discuss now the state variables chosen for the present e of temperature(the same for all constituents of the medium because of the as-sumption about the local thermodynamic equilibrium state)and solid skeleton displacement vector is rather obvious,thus it needs no further explanation.As a hygrometric state variable various physical quantities, which are thermodynamically equivalent,may be used,e.g.volumetric––or mass moisture content,vapour pressure,relative humidity,or capillary pressure.Analysing concrete at high temperature,one must re-member that at temperatures higher than the critical point of water(i.e.647.3K)there is no capillary(or free)water present in the pores of concrete,and there exists only the gas phase of water,i.e.vapour.Then, very different moisture contents may be encountered at the same moment in a heated concrete,ranging from full saturation with liquid water(e.g.in some nuclear vessels or in so called‘‘moisture clog’’zone in aheated concrete[12])up to almost completely dry material.For these reasons it is not possible to use,in a direct way,one single variable for the whole range of moisture contents.Instead,an appropriate StefanÕs problem could be formulated,with different state variables in zones separated by moving interfaces. However,such an approach is numerically very costly,e.g.[1,34,39],and usually avoided in practical applications,as already mentioned in[19].For description of concrete moisture state,Bazant et ed in their model[5,6]the relative humidity, but in zones fully saturated with liquid water,where pressures higher than the atmospheric one can occur,a different meaning must be given to this variable,permitting its value to be higher than one,what is physically inadmissible.Then,application of a shrinkage coefficient,relating strain changes with changes of the relative humidity,is consistent with the phenomenological approach,used in[5,6],but not with the mechanistic one,which is used in this paper.Apparently,the most natural choice for the state variable seems to be mass or volumetric moisture content,which are well defined for the whole range of temper-atures and pressures in concrete.However,this quantities are not continuous at interfaces between different materials,and are not well adapted for numerical simulations,both in fully saturated conditions and in a range of very low moisture contents.Moreover,there is not any direct,physically sound(from the mechanistic point of view)relation between moisture content and stresses.Another possible choice for the moisture state variable is vapour pressure,which,however,has no physical meaning in a medium fully saturated with water and then,it creates serious numerical problems for moisture contents close to these conditions,as shown by our extensive tests.The moisture state variable proposed by the authors in[19]is capillary pressure,which was shown to be a thermodynamic potential of the physically adsorbed water and,with an appropriate interpretation,can be also used for description of water at pressures higher than the atmospheric one[20].The capillary pressure has been shown to assure good numerical performance of the computer code[16–19],and is very convenient for analysis of stress state in concrete,because there is a clear relation between pressures and stresses [23,41].Application of capillary pressure as a state variable was avoided in some previous models,e.g.[4–7], because of theoretical problems related to its definition at the macro-scale.However,some recent works in Thermodynamics[22,23],resolved these theoretical problems.Hence,the chosen primary variables of the present model are the volume averaged values of:gas pressure,p g,capillary pressure,p c,temperature,T,and displacement vector of the solid matrix,u.For temperatures lower than the critical point of water,T<T cr,and for capillary saturation range, S>S sspðTÞ(S ssp means the upper limit of the hygroscopic moisture range,being at the same time the lower limit of the capillary one),the capillary pressure is defined asp c¼p gÀp w;ð28Þwhere p w denotes water pressure.This equation is,in reality,a constitutive relationship at thermodynamic equilibrium which can be obtained from an exploitation of the entropy inequality by means of the Cole-man–Noll method,see e.g.[22,23].For all other situations,and in particular T P T cr,when condition S<S ssp is always fulfilled(there is no capillary water in the pores),the capillary pressure only substitutes formally the water potential W c defined as:W c¼RTM wlnp gwf gws;ð29Þwhere M w is the molar mass of water,R the universal gas constant and f gws the fugacity of water vapour in thermodynamic equilibrium with saturatedfilm of physically adsorbed water[19].For physically adsorbed water at lower temperatures(S<S ssp and T<T cr)the fugacity f gws should be substituted in the definition of the potential W c(29),by the saturated vapour pressure p gws.Having in mind the Kelvin equation[24], valid for the equilibrium state of capillary water with water vapour above the curved interface(meniscus) 1740 D.Gawin et al./Comput.Methods Appl.Mech.Engrg.192(2003)1731–1771ln p gw p gws¼Àp cq wM wRT;ð30Þwe can note,that in the situations,where(29)is valid,the capillary pressure may be treated formally as the water potential multiplied by the density of the liquid water,q w,according to the relation[19] p c¼ÀW c q w;ð31ÞThanks to this similarity,it is possible to use during simulations‘‘formally’’the capillary pressure even in the low moisture content range,when the capillary water is not present in the pores.However,one should remember,that in such situations capillary pressure cannot be identified to a pressure in its normal physical meaning[19],see for example(49).4.Constitutive relationshipsAs constitutive relationships may be used:equations of state for constituents of the medium,material functions describing certain physical properties of the analysed material,as well as some physical relations betweenfluxes of extensive thermodynamic quantities and intensive thermodynamic quantities(called sometimes thermodynamic forces)which cause them.These physical relations can be obtained directly from the entropy inequality,e.g.[22,23],and they describe some well-known laws of physics,like FourierÕs law, DarcyÕs law or FickÕs law.In our model,dry air,water vapour and their mixture are assumed to behave as perfect gases,following DaltonÕs law and the Clapeyron equation of state:p g¼p gaþp gw;ð32Þq p¼p p M p=TR;ðp¼ga;gw;gÞ;ð33Þwhere1 M g ¼q gwq g1M wþq gaq g1M a:ð34ÞThe density of water vapour calculated by means of(33)differs significantly from the results of the lab-oratory tests for temperatures higher than approximately160°C,but as shown in[19],this difference has a small effect on the results of simulations concerning high temperature performance of concrete and(33) may be used for practical problems with a sufficient accuracy.The state equation of water should take into account the considerable,non-linear decrease of water density in the temperature range close to the critical point of water,what has an important influence on hygro-thermal phenomena in concrete at these temperatures[19].The following formula[13],gives a reasonable accordance with experimental results and assures a good numerical performance of the code [19]:q w¼ðb0þb1Tþb2T2þb3T3þb4T4þb5T5Þþðp w1Àp wrÞða0þa1Tþa2T2þa3T3þa4T4þa5T5Þ;ð35Þwhere p w1¼10MPa,p wr¼20MPa,a0¼4:89Â10À7,a1¼À1:65Â10À9,a2¼1:86Â10À12,a3¼2:43Â10À13,a4¼À1:60Â10À15,a5¼3:37Â10À18,b0¼1:02Â10À3,b1¼À7:74Â10À1,b2¼8:77Â10À3,b3¼À9:21Â10À5,b4¼3:35Â10À7and b5¼À4:40Â10À10.In the above formula water is assumed to be incompressible.At lower relative humidities,usually en-countered at higher temperatures,most of the liquid water consists of the physically adsorbed water,whichis exposed to strong interaction with the solid skeleton,resulting in an increase of the water density [11].At the same time one can expect a decrease of the density with an increase of the capillary pressure (decrease of the water pressure),what is the case in heated concrete.Having in mind these two opposite trends,as well as lack of sufficient experimental data,we have assumed here incompressibility of the liquid water inside the pores of concrete.For the solid skeleton,the following form of the state equation has been assumed:q s ¼q s ðT ;p s ;tr r 0;C dehydr Þ;ð36Þwhere p s means the solid skeleton pressure [30,42],given by (47),tr r 0the first invariant of effective stress tensor,defined by (48),and C dehydr is degree of cement dehydration at high temperature.The latter process starts at a temperature about 100°C and continues with variable intensity during increase of concrete temperature.The following relation for the rate of the first invariant of the effective stress tensor has been assumed [30,42]:D s ðtr r 0ÞD t ¼3e K T div v s 0@þ1e K s D s p sD t Àb s D sT D t 1A ;ð37Þwhere b s is thermal expansion coefficent of the solid,e KT and e K s the actual values of bulk moduli for the whole medium and the solid skeleton (grains),respectively,taking into account possible influence of both the dehydration and cracking process.The inner structure of concrete pores is very complex and it contains both very narrow gel pores and much greater macro-pores and cracks (especially at higher temperatures),resulting sometimes in not continuous capillary pores.Nevertheless,at the macro-scale level,the volume averaged advective flux (i.e.caused by pressure gradients)of liquid water may be still described by Darcy Õs law [11,40],in the following form [19,30,42]:nS p v p s ¼Àk r p klp ½grad p p Àq p g ð38Þwhere k is intrinsic permeability tensor,k r p and l p ðp ¼g ;w Þdenote relative permeability and dynamicviscosity of the gaseous phase and liquid water.The fluids Õviscosities change significantly with temperature increase,what should be taken into account during analysis of concrete at high temperature.The formulae describing these changes are given in [17].During heating of concrete,its intrinsic permeability k may increase by up to 4orders of magnitude when compared to its initial value at ambient temperature [7,12].The dehydration and cracking progress with increasing temperature (cracking may be caused also by the mechanical stress at macro-scale)and result in gradual increase of the permeability value.For description of these intrinsic permeability changes,different approaches may be followed [18].In this paper a mechanistic one has been adopted,assuming that intrinsic permeability does depend not only upon temperature and moisture content,as assumed in classical phenomenological approach,e.g.[5,6],but also upon gas pressure and mechanical damage parameter,d [17,19]:k ¼k 0Á10A T ðT ÀT 0ÞÁpg p g0 A pÁ10A d d ;ð39a Þwhere d is the mechanical damage parameter,A T ,A p and A d are material constants.The term related to mechanical damage describes the effect of concrete cracking [3],and the gas pressure term the effect of cracks Õopening,on increase of the permeability.The influence of particular terms in (39a)on performance of concrete at high temperature is discussed in [16,18].The latter relation can be alternatively expressed in terms of total damage parameter,D ,defined by Eq.(54),。
