JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 3, pp. 505-532, Warsaw 2006 MODELLING DAMAGE PROCESSES OF CONCRETE AT HIGH TEMPERATURE WITH THERMODYNAMICS OF MULTI-PHASE POROUS MEDIA Dariusz Gawin Department of Building Physics and Building Materials, Technical University of Łódź, Poland e-mail: gawindar@p.lodz.pl Francesco Pesavento Bernhard A. Schrefler Department of Structural and Transportation Engineering, University of Padua, Italy e-mail: pesa@dic.unipd.it; bas@dic.unipd.it In this paper, the authors present a mathematical description of a multi- phasemodel of concrete based on the second law of thermodynamics. Explo- itation of this law allowed researchers to obtain definitions and constitutive relationships for several important physical quantities, like capillary pres- sure, disjoining pressure, effective stress, considering also the effect of thin films of water. A mathematical model of hygro-thermo-chemo-mechanical phenomena in heated concrete, treated as a multi-phase porous material, has been formulated. Shrinkage strains were determined using thermodyna- mic relationships via capillary pressure and area fraction coefficients, while thermo-chemical strainswere related to thermo-chemicaldamage. In themo- del, a classical thermal creep formulation has beenmodified and introduced into the model. Results of numerical simulations based on experimental te- sts carried out at NIST laboratories for two types of concrete confirmed the usefulness of themodel in the prediction of the time range, duringwhich the effect of concrete spalling may occur. Key words: concrete, multiphase, damage, thermal creep 1. Introduction During heating of concrete, several complex interacting physical and chemi- cal phenomena take place in pores which causes significant changes of its 506 D. Gawin et al. inner structure and properties (Bazant and Kaplan, 1996; Phan, 1996, Phan et al., 1997; [3]). They can lead to a decrease in the load-bearing capaci- ty or other important service features of concrete structures, especially du- ring a rapid and/or prolonged increase of ambient temperature. A mode of material damage, very specific for concrete at high temperatures, is the so called thermal spalling. This phenomenon has been studied, both experi- mentally and theoretically, for many years, e.g. Bazant and Kaplan (1996), Bazant and Thonguthai (1978), England and Khoylou (1995), Phan (1996), Phan and Carino (2002), Phan et al. (1997, 2001), Sullivan (2001), Ulm et al. (1999a,b), [3], but its physical causes are still not fully understood. The reason for such a situation are inherent technical difficulties associa- ted with testing of concrete elements at high temperature, especially when changes of physical properties and several parameters are to be measured si- multaneously, the temperature and pressure fields, moisture content, strength properties, intrinsic permeability etc., during experiments at controlled conditions. Within recent years, the authors of the paper have developed amathema- tical model of concrete at high temperatures which considers the porous and multi-phase nature of concrete (Gawin et al., 1999, 2002a,b, 2003, 2004), be- haviour of moisture above the critical point of water (Gawin et al., 2002a), load induced thermal strain (Gawin et al., 2004) as well as cracking and thermo-chemical deterioration of concrete (Gawin et al., 2003) and related changes ofmaterial properties like, for example, intrinsic permeability (Gawin et al., 2002b). The model has been successfully validated against published results of several experimental tests (Gawin et al., 2003, 2006) showing its sefulness for understanding and predicting of concrete performance at high temperatures. In this paper, we present the development of model equations taking into account both bulk phases and interfaces, and exploration of the second law of thermodynamics,whichallows us todefine several quantities used in themodel like capillary pressure, disjoining pressure or effective stress, and obtain some thermodynamic restrictions imposed on evolution equations describing mate- rial deterioration. Then, we present, as an example of the model application, numerical simulations of experimental tests performed at the NIST laborato- ries for two types of concrete (Phan, 1996; Phan and Carino, 2002; Phan et al., 1997, 2001), of which only one experienced thermal spalling at prevailing conditions. Modelling damage processes of concrete... 507 2. Mathematical model of concrete at high temperature considered as a multi-phase porous material The equations of the model are written by considering concrete as a multi- phase porous material, i.e. a medium in which pores are filled by more than one fluid. In the present case, solid skeleton voids are filled partly with liquid water and partly with a gas phase. Below the critical temperature of water, Tcr, the liquid phase consists of physically bound water and capillary water, which appears when the degree of water saturation exceeds the upper limit of the hygroscopic region, Sssp. Above the temperature Tcr, the liquid phase consists of boundwater only. In the whole temperature range, the gas phase is a mixture of dry air and water vapour, which is a condensable gas constituent for temperatures T 0.99, in the temperature range up to 800◦C. In heated concrete, above temperatures of about 105◦C, there start seve- ral complex endothermic chemical reactions, generally called concrete dehy- dration, e.g. Bazant and Kaplan (1996). They cause thermal decomposition of the cement matrix, and at higher temperatures also of the aggregate (de- pending on its type and composition), what results in a considerable chemical shrinkage of the cementmatrix andusually expansion of the aggregate. Due to these contradictory, or at least incompatible, behaviour of the concrete com- ponents, cracks of various dimensions are developing when temperature incre- ases, [3], causing an additional change of concrete strains (usually expansion). It is practically impossible to separate the effects of dehydration and cracking processes during experimental investigations of thermal dilatation of concrete at high temperatures, hence they are considered jointly as thermo-chemical strains, which in an incremental form are given by Gawin et al. (2004) dεtchem =βtchem(V )I dV (4.3) where βtchem(V ) is the material function to be determined experimentally. It is expressed in terms of the thermo-chemical damage parameter, V , given by Eq. (3.36), which was physically justified in Gawin et al. (2004), taking also into account the irreversible nature of these strains. During first heating, mechanically loaded concrete exhibits greater strains as compared to the load-free material at the same temperature. A part of them originates just from the elastic deformation due to mechanical load, εel(σ̃ s e,T), and it increases during heating because of thermo-chemical and mechanical degradationof thematerial.The timedependentpartof strainsdue to transient thermal processes, εtr(σ̃ s e,T), is generally referred to as thermal creep or Load Induced Thermal Strain (LITS) (Khoury, 1995). Its physical nature is rather complexanduptodaynot fullyunderstood, thus themodelling is usually based on results of special experimental tests, which are performed at constant heating rate for various (but constant during a particular test) levels of material stresses, see e.g. Gawin et al. (2004), Khoury (1995), [3]. The transient thermal strains, εtr, modelled for a multi-axial stress sta- te similarly as in Nechnech et al. (2001), Pearce et al., 2003, Thelandersson (1987), can be obtained from the following incremental formula (Gawin et al., 2004) dεtr = βtr(V ) fc(Ta) Q : σ̃se dV (4.4) Modelling damage processes of concrete... 519 where fc(Ta) is the concrete compressive strength at the room temperature, and Q is a fourth order tensor defined as follows Qijkl =−γδijδkl+ 1 2 (1+γ)(δikδjl+ δilδjk) (4.5) with γ being a material parameter determined from the transient creep test. The material function βtr(V ) must be experimentally determined (Gawin et al., 2004). Due to the rreversible character of transient thermal strains, the function is expressed in terms of the thermo-chemical damage parameter, V , rather than temperature, T , which was previously used in some models, e.g. Nechnech et al. (2001), Pearce et al., 2003, Thelandersson (1987). Application of the proposed model of the thermal strains for the C-90 concrete [3] for different levels of compressive loads (including the load-free case), gave satisfactory predictions of strains, both the total ones and LITS (Gawin et al., 2004). 5. Final form of governing equations Taking into account the constitutive relationships obtained from the explo- itation of entropy inequality and neglecting the presence of interfaces, from the general set of field equations, it is possible to formulate a complete model for the analysis of the thermo-hygro-chemical and mechanical behaviour of concrete exposed to a high temperature. To describe uniquely the state of concrete at high temperatures, we need 4 primary state variables, i.e. gas pressure, pg, capillary pressure pc, tempera- ture, T , and displacement vector, u, as well as 3 internal variables describing the advancement of dehydration and deterioration processes, i.e. the degree of dehydration, Γdehydr, chemical damage parameter, V , andmechanical dama- ge parameter, d. Using the mentioned damage parameters, the total damage parameter, D, can be calculated. Themodel consists of 7 equations: 2mass balances (continuity equations), enthalpy (energy) balance, linearmomentumbalance (mechanical equilibrium equation) and3evolution equations.Thefinal formofmodel equations, expres- sed in terms of primary state variables are listed below, after introducing the constitutive relationships. The full development of the equations is presented in Gawin (2000), Gawin et al. (2004), Pesavento (2000). ➢ Mass balance equation of dry air (involving the solid skeletonmass balance) takes into account both diffusive and advective air flow aswell as variations of 520 D. Gawin et al. porosity caused by the dehydration process and deformations of the skeleton. It has the following form −n DsSw Dt −βs(1−n)Sg DsT Dt +Sgdivv s+ Sgn ρga Dsρga Dt + 1 ρga divJgag + (5.1) + 1 ρga div(nSgρ ga v gs)− (1−n)Sg ρs ∂ρs ∂Γdehydr DsΓdehydr Dt = ṁdehydr ρs Sg ➢ Mass balance equation of water species (involving the solid skeleton mass balance) considers diffusive and advective flow of water vapour, mass sour- ces related to phase changes of vapour (evaporation-condensation, physical adsorption – desorption and dehydration), and variations of porosity caused by the dehydration process and deformations of the skeleton, resulting in the following equation n(ρw−ρgw) DsSw Dt +(ρwSw+ρ gwSg)αdivv s −βswg DsT Dt +Sgn Dsρgw Dt + +divJgwg + div(nSwρ w v ws)+ div(nSgρ gw v gs)+ (5.2) −(ρwSw+ρ gwSg) (1−n) ρs ∂ρs ∂Γdehydr DsΓdehydr Dt = ṁdehydr ρs (ρwSw+ρ gwSg−ρ s) with βswg defined by βswg =βs(1−n)(Sgρ gw+Swρ w)+nβwSwρ w (5.3) ➢ Enthalpy balance equation of the multi-phase medium accounting for con- ductive and convective heat flow and heat effects of phase changes and the dehydration process, can be written as follows (ρCp)eff ∂T ∂t +(ρwC w p v w+ρgC g pv g)gradT − div(χeff gradT)= (5.4) =−ṁvap∆Hvap+ṁdehydr∆Hdehydr where χeff is the effective conductivity found from experiments (ρCp)eff = ρsC s p +ρwC w p +ρgC g p (5.5) ∆Hvap =H gw −Hw ∆Hdehydr =H w −Hws Modelling damage processes of concrete... 521 and ṁvap =−ρ wSwdiv ∂u ∂t +βswρ w∂T ∂t + −ρwn ∂Sw ∂t − div [ ρw kkrw µw (−gradpg+ gradpc+ρwg) ] + (5.6) − [ ṁdehydr+(1−n) ∂ρs ∂Γdehydr ∂Γdehydr ∂t ]ρwSw ρs +ṁdehydr with βsw =Sw[(1−n)βs+nβw] and the dehydrated water source is proportional to the dehydration rate ṁdehydr = kbΓ̇dehydr (5.7) kb is amaterial parameter related to chemically boundwater anddependenton the stoichiometry of chemical reactions associated to the dehydration process. ➢ Linear momentum conservation equation of the multi-phase medium has the following form div(σse−αp s I)+ρg=0 (5.8) where the effective stress σse is given by σ s e =(1−d)(1−V )Λ0 : (εtot−εth−εtchem−εtr) (5.9) with εth being the thermal strain given by (4.2), εtchem – thermo-chemical strain (4.3), and εtr – transient thermal strain (4.4). ➢ Dehydration process evolution law, considering its irreversibility, has the form Γdehydr(t)=Γdehydr(Tmax(t)) (5.10) where Tmax(t) is the highest temperature reached by concrete up to the time instant t. The constitutive relationship Γdehydr(T) can be obtained from the results of thermo-gravimetric (TGorDTA) tests, using the definition of the dehydra- tion degree bymeans of mass changes during concrete heating Γdehydr(T)= m(T0)−m(T) m(T0)−m(T∞) (5.11) 522 D. Gawin et al. where m(T) is themass of concrete specimenmeasured at the temperature T during TG tests, T0 and T∞ are temperatures when the dehydration process starts and finishes.We assumed here T0 =105 ◦C and T ∞ =1000◦C. ➢ Thermo-chemical damage evolution equation, obtained from (3.36) on the basis of the experimental results, takes into account the irreversible character of material structural changes andmay be written as V (t)=V (Tmax(t)) (5.12) ➢ Mechanical damage evolution equation, of the following form d(t) = d(ε̃(t)) (5.13) is expressed in terms of the equivalent strain, ε̃, given by equations of the classical non-local isotropic damage theory (Chaboche, 1988; Kachanov, 1958; Mazars, 1984, 1989; Pijaudier-Cabot, 1995). The parameters of the theory can be determined from the results of strength tests (’stress-strain’ curves) at various temperatures and the application of first Eq. (3.36), as was done in Gawin et al. (2003). Model equations (5.1)-(5.13) are completed by appropriate constitutive equations and thermodynamic relations, seeGawin (2000), Gawin et al. (1999, 2002a, 2003, 2004, 2006). Anumerical solution to themodel equationswith thefinite elementmethod (Lewis and Schrefler, 1998; Zienkiewicz and Taylor, 2000) is presented and discussed in detail in Gawin (2000), Gawin et al. (2004, 2006), Pesavento (2000). 6. Numerical simulation of the NIST test at 450◦C A comparison between numerical results, obtained bymeans of themodel de- scribed in theprevious sections, and experimental results fromthe compressive tests carried out at NIST laboratories (Phan, 1996; Phan and Carino, 2002; Phan et al., 1997, 2001), will be presented here. This demonstrates the capa- bility of the numerical model to assess the risk of thermal spalling in concrete exposed to elevated temperatures. Themain aimof the tests carried out atNIST laboratories was to evaluate the thermo-mechanical behaviour of four different HPCs with different mix Modelling damage processes of concrete... 523 compositions exposed to high temperatures.Cylindrical specimens, all of them with a diameter of 100mm and height of 200mm, were tested using different test methods and target temperatures. Herein, our attention is focused on unstressed heating tests to the final target temperature of 450◦C for concretes indicated as MIX-1 (with fc = 98MPa, w/c = 0.22, k0 = 2 · 10 −19m2) and MIX-2 (fc = 82MPa, w/c = 0.33, k0 = 2 · 10 −18m2). All MIX-1 specimens and only one of four MIX-2 specimens experienced explosive spalling during the test at these conditions. During the test, the heating ratewas initially equal to 5K/minandheating was stopped when the temperature in the centre of the specimen was within 10K of the target temperature T , and the difference between the surface and centre temperatures of the concrete specimen was less than 10K. For further details concerning the mix compositions and tests procedures (set- up, temperature control, instrumentation of the specimens), see Phan (1996), Phan and Carino (2002), Phan et al. (1997, 2001). Initially, the specimenhad temperature T =296.15K (23◦C), gas pressure pg = 101325Pa and pore relative humidity ϕ = 50%RH. The results of our simulations are presented in Fig.1 and Fig.2. Figures 1a,b show the evolution of temperature difference, ∆T , between the surface and the centre of specimens measured during the tests and the corresponding numerical values. The accordance between the numerical and experimental results is quite good, except for the initial phase of heating of MIX-1, Fig.1a, when the experimental profile shows that the temperature dif- ference between the core and surface is practically zero for more than one hour, which was probably caused by some problemswith temperaturemeasu- rement. The samefigure shows the history ofmechanical damage parameter in themiddle of the radius obtained from the simulations. Corresponding to the maximum value of ∆T , a steep increase in the mechanical damage parame- ter d (with the maximum value dmax =∼ 75%) may be observed for MIX-1, Fig.1a, while much slower increase of d (with dmax =∼ 50%) is obtained for MIX-2, see Fig.1b. Figures 2a,b show a comparison of behaviour of tested concretes in the space domain. In particular, Fig.2a shows the comparison of gas pressure di- stributions in the radial direction at four time stations corresponding to the time range when the thermal spalling was observed during the experimental tests. The maximum values of pg are concentrated in the critical time range. Then, the pressure tends to diminish rather quickly. Gas pressures in MIX-1 are approximately two times higher than in MIX-2 which has higher perme- ability due to a higher w/c ratio. At the end of the simulation (t=300min), 524 D. Gawin et al. Fig. 1. Time evolution of the mechanical damage parameter and temperature difference in concrete specimens made of two types of the concrete: (a) MIX-1, (b) MIX-2, in a NIST experimental test it is equal to the atmospheric pressure in thewhole specimen. Figure 2b shows themechanical damage distribution along the radius.The behaviour ofMIX-1 during the ’critical stage’ is characterized by a significant increase on the da- mage parameter, which reaches the maximum value very rapidly. For MIX-2, the damage parameter does not exceed 20% which indicates much lower risk of thermal spalling than forMIX-1. The results of the presented simulations do not indicate directly the thick- ness of the spalled concrete layer which was approximately equal to 2cm for MIX-1. However, after application of the spalling indexes proposed in Gawin et al. (2006), the position of spalling occurrence can also be predicted with a sufficient accuracy. Modelling damage processes of concrete... 525 Fig. 2. Comparison of the radial distribution of gas pressure (a) andmechanical damage (b) in concrete specimensmade of two types of concrete,MIX-1 andMIX-2, during final stages of the NIST experimental test After prolonged exposure to a high temperature, the surface layer of concrete elements can be almost completely deteriorated due to stress- and temperature-induced cracking as well as thermo-chemical degradation. Such a casewill be characterised in our simulations byhighvalues of the total damage parameter, see e.g. Fig.8c in Gawin et al. (2006). If conditions favouring the previous occurrence of explosive spalling did not appear during heating, one can expect a gradual drop of the external deteriorated layer of concrete, what is sometimes called the progressive spalling. The extent of this phenomenon and its evolution in time can be estimated by means of a kind of ”critical value” of the D parameter. A reasonable value for such an analysis seems to be D∼=0.9-0.95 (Gawin et al., 2006). 526 D. Gawin et al. 7. Conclusions A mathematical model of hygro-thermo-chemo-mechanical phenomena in he- ated concrete, treated as a multi-phase porous material, has been formulated taking into account most of the important features of the material behaviour at these conditions. Exploitation of the second law of thermodynamics allo- wed us to obtain definitions and constitutive relationships for several impor- tant physical quantities, like capillary pressure, disjoining pressure, effective stress, considering also the effect of thin films of water. Shrinkage strains ha- ve been determined using thermodynamic relationships via capillary pressure and area fraction coefficients, while thermo-chemical strains have been related to thermo-chemical damage. A classical thermal creep formulation, based on experimental results, has been appropriately modified and introduced in the model. The results of numerical simulations, based on experimental tests for two concretes, confirmed the usefulness of themodel in the prediction of the time range when the concrete spalling may occur. Notation Aα – specific Helmholtz free energy for bulk phase α [Jkg−1] Aαβ – specific Helmholtz free energy for interface αβ [Jkg−1] aαβ – specific surface of αβ-interface [m−1] Cp – effective specific heat of porousmedium [Jkg −1K−1] Cgp – specific heat of gas mixture [Jkg −1K−1] Cwp – specific heat of liquid phase [Jkg −1K−1] D – total damage parameter [–] d – mechanical damage parameter [–] d α – strain rate tensor of bulk phase α [s−1] D gw d – effective diffusivity tensor of water vapour in dry air [m2s−1] E – Young’s modulus [Pa] êααβ – rate of mass transfer to bulk phase α from interface αβ [kgm−3s−1] ê αβ αβγ – rate of mass transfer to interface αβ from contact line αβγ [kgm−3s−1] E0 – Young’s modulus of mechanically undamagedmaterial [Pa] Eα – internal energy of bulk phase α [Jkg−1] Modelling damage processes of concrete... 527 Eαβ – internal energy of interface αβ [Jkg−1] g – gravity acceleration [ms−2] Hα – enthalpy of bulk phase α [Jkg−1] Hαβ – enthalpy of interface αβ [Jkg−1] hα – heat source in bulk phase α [Wkg−1] hαβ – heat source on interface αβ [Wkg−1] I – unit tensor [–] Jααβ – curvature of αβ interface with respect to bulk phase α [m −1] J gw d – diffusive flux of vapour [kgm−2s−1] J ga d – diffusive flux of dry air [kgm−2s−1] k – absolute permeability tensor [m2] k – absolute permeability (scalar) [m2] krπ – relative permeability of π-phase (π= g,w) [–] ṁdehydr – rate of mass due to dehydration [kgm −3s−1] ṁvap – rate of mass due to phase change [kgm −3s−1] n – total porosity (pore volume/total volume) [–] pc – capillary pressure [Pa] pg – pressure of gas phase [Pa] pw – pressure of liquid water [Pa] ps – solid skeleton pressure [Pa] pga – dry air partial pressure [Pa] pgw – water vapour partial pressure [Pa] qα – heat flux vector for bulk phase α [Wm−2] qαβ – heat flux vector for interface αβ [Wm−2] Q̂ααβ – body supply of heat to bulk phaseα from interfaceαβ [Wm −3] Q̂ αβ αβγ – body supply of heat to interface αβ from contact line αβγ [Wm−3] Q – fourth order tensor for definition of transient thermal strain [–] Sw – liquidphasevolumetric saturation (liquidvolume/porevolume) [–] Ŝ αβ αβγ – body supply of momentum to αβ-interfaces from αβγ-contact line [kgm−2s−2] sαβ – stress tensor for αβ-interface [Pam] T – absolute temperature [K] Tcr – critical temperature of water [K] Tmax – maximumtemperature attainedduringdehydrationprocess [K] t – time [s] tα – partial stress tensor of α-phase [Pa] 528 D. Gawin et al. T̂ α αβ – body supply of momentum to bulk phases from interfaces [kgm−2s−2] u – displacement vector of solid matrix [m] V – thermo-chemical damage parameter [–] vα – velocity of α-phase [ms−1] vgs – relative velocity of gaseous phase [ms−1] vws – relative velocity of liquid phase [ms−1] wαβ – velocity of interface αβ [ms−1] Greek symbols α – generic bulk phase α – Biot’s constant [–] αc – convective heat transfer coefficient [Wm −2K−1] αβ – generic interface of α- and β-phases βtchem – thermo-chemical strain coefficient [K −1] βc – convective mass transfer coefficient [ms −1] βs – cubic thermal expansion coefficient of solid [K −1] βswg – combine (solid + liquid+ gas) cubic thermal expansion coef- ficient [K−1] βsw – combine (solid + liquid) cubic thermal expansion coefficient [K−1] βtr – normalised transient thermal strain coefficient [K −1 s] βw – thermal expansion coefficient of liquid water [K −1] χwss – solid surface fraction in contact with the wetting water film [–] χeff – effective thermal conductivity [Wm −1K−1] ∆Hvap – enthalpy of vaporization per unit mass [Jkg −1] ∆Hdehydr – enthalpy of dehydration per unit mass [Jkg −1] εel – elastic strain tensor [–] εsh – shrinkage strain tensor [–] εtchem – thermo-chemical strain tensor [–] εth – thermal strain tensor [–] εtot – total strain tensor [–] εtr – transient thermal strain tensor [–] εtr – normalized transient thermal strain [–] ε̃ – equivalent strain in damage theory of Mazars [–] Γαβ – surface excess mass density of αβ–interface [kgm−2] Γdehydr – degree of dehydration [–] γαβ – macroscopic interfacial tension of αβ-interface [Jm−2] Modelling damage processes of concrete... 529 µπ – dynamic viscosity of constituent π-phase (π= g,w) [µPas] λα – specific entropy of α-phase [Jkg−1K−1] λαβ – specific entropy of αβ-interface [Jkg−1K−1] Λα – rate of net production of entropy of α-phase [Wm−3K−1] Λαβ – rate of net production of entropy ofαβ-interface [Wm−3K−1] Λ – stiffness matrix of damaged material [Pa] Λ0 – stiffness matrix of undamagedmaterial [Pa] Πw – disjoining pressure [Pa] ρ – apparent density of porousmedium [kgm−3] ρg – gas phase density [kgm−3] ρw – liquid phase density [kgm−3] ρs – solid phase density [kgm−3] ρga – mass concentration of dry air in gas phase [kgm−3] ρgw – mass concentration of water vapour in gas phase [kgm−3] Φ̂ααβ – body entropy supply to bulk phase α from interface αβ [Wm−3K−1] Φ̂ αβ αβγ – body entropy supply to interface αβ from contact line αβγ [Wm−3K−1] σ – Cauchy stress tensor [Pa] σse – Bishop effective stress tensor of skeleton [Pa] σ̃ s e – ”net” effective stress tensor of skeleton [Pa] Acknowledgments This work was carried out within the framework of the UE project ”UPTUN – Cost-Effective, Sustainable and Innovative Upgrading Methods for Fire Safety in Existing Tunnels”, No. G1RD-CT-2002 00766 and the Italian national project PRIN 2003 No. 2003084345 003 ”Damage and durability mechanics of ordinary and high performance concrete”. 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Zienkiewicz O.C., Taylor R.L., 2000,The Finite Element Method, Vol. 1: The Basis, Butterworth-Heinemann, Oxford Modelowanie procesów zniszczenia betonu w wysokich temperaturach za pomocą termodynamiki ośrodków porowatych Streszczenie Autorzy pracy prezentują równaniamodelu wielofazowegobetonuwyprowadzone na podstawie II zasady termodynamiki. Zastosowanie tej zasady pozwoliło badaczom na sformułowanie definicji i związków konstytutywnych dla kilku ważnych wielko- ści fizycznych, takich jak: ciśnienie kapilarne, ciśnienie rozklinowywujące, napręże- nie efektywne oraz na uwzględnienie efektu wywoływanego przez cienkie warstwy wody obecnej w betonie. Przedstawiono model matematyczny higro-termo-chemo- mechanicznych zjawisk zachodzących w podgrzewanym betonie traktowanym jako ośrodek porowaty. Skurcz betonu określono za pomocą równań termodynamiki opi- sujących ciśnienie kapilarne i współczynniki udziału powierzchniowego, podczas gdy odkształcenia termochemiczne skorelowano z parametrem zniszczenia chemicznego. Do rozważań wprowadzono zmodyfikowaną teorię pełzania opartą na sformułowa- niu klasycznym. Wyniki przeprowadzonych symulacji numerycznych, bazujących na badaniach doświadczalnych zrealizowanych w laboratoriach NIST na dwóch typach betonu, potwierdziły użyteczność zaprezentowanego modelu w przewidywaniu prze- działu czasu, w którymmoże dojść do termicznego odpryskiwania betonu. Manuscript received December 28, 2005; accepted for print April 4, 2006