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FE Analysis of Thermo-Hydromechanical Behaviour of Concrete at High Temperature
© 2001 Elsevier Science Ltd. All rights reserved. Computational Mechanics - New Frontiers for New Millennium S. Valliappan and N. Khalili, editors.
827
FE ANALYSIS OF THERMO-HYDROMECHANICAL BEHAVIOUR OF CONCRETE AT HIGH TEMPERATURE B.A. Schrefler1, D. Gawin2, C.E. Majorana1, F. Pesavento1 1
Dipartimento di Costruzioni e Trasporti, Universitá degli Studi di Padova, Via Marzolo 9, Padova, 35131, Italy department of Building Physics and Building Materials Technical University of Lodz, 1. Politechniki 6, Lodz, 93-590, Poland
ABSTRACT In the present work an innovative model for analysing the behaviour of concrete in high temperature environment is presented. In such conditions prediction of concrete performance during and after fire exposure is of great practical importance, e.g. in tunnels, as recent fires in major European tunnels have demonstrated. This is particularly true for High Performance and Ultra High Performance concretes, which have been often used in the last decade in nuclear engineering applications, tall buildings and in tunnels, because of their higher tendency to spalling experience. It is now well established that phenomena which take place in concrete exposed to high temperatures, cannot be studied by means of purely diffusive models. In the model presented in this paper concrete is treated as partially saturated porous material where hydration-dehydration, evaporation-condensation, adsorption-desorption phenomena and non-linearities due to temperature and pressure take place. Two applications of this model to an ordinary and HP concrete structure will be shown.
KEYWORDS Concrete, High temperature, Thermo-hydro-mechanical, High performance, Damage INTRODUCTION When concrete is exposed to high temperature a rather complex analysis is required to deal with the coupled heat and mass transfer that can occur, involving both liquid mass transfer and vapour mass transfer and related mechanical effects (Bazant & Kaplan, 1996). In such severe conditions in terms of temperatures and pressures, assessment of concrete performance is of great interest in nuclear engineering applications, in safety evaluation in tall buildings and in tunnels (Bazant & Thonguthai, 1978). In particular as far as tunnels are concerned, recent major fires in key European tunnels (Channel, Mont-Blanc, Great Belt Link, Tauern) emphasised the serious hazards they present in human and economic terms. In these accidents, in the affected sections, tunnels presented extensive damage to the concrete elements. Part of the concrete lining was almost completely removed by spalling. Spalling, which may be explosive, is mainly due to different coexisting coupled processes, such as
828 thermal (heat transfer), chemical (dehydration of cement paste), hygral (transfer of water mass, in liquid and vapour form) and mechanical processes. In general, moisture transport may include airvapour mixture flow due to forced convection, free convection, and infiltration through cracks and pores, vapour transport by diffusion, flow of liquid water due to diffusion, capillary action, or gravity, and further complications associated with phase changes due to condensation/evaporation, freezing/thawing, ablimation/sublimation, and adsorption/desorption. Movement of air and water through the concrete is accompanied by significant energy transfer, associated with the latent heat of water and the heat of hydration and dehydration. At temperature largely above critical point of water important chemical transformations of components of concrete take place. These situations are much more dangerous in the case of High Performance and Ultra High Performance concrete because of their peculiar features. Hence, for concrete, particularly at high temperature, one cannot predict heat transfer only from the traditional thermal properties (i.e., thermal conductivity and volumetric specific heat) and, similarly, it is not possible to predict mass transfer considering this phenomenon as purely diffusive. In fact, in a typically phenomenological approach moisture and heat transport are described by diffusive type differential equations with temperature- and moisture content- dependent coefficients. The model coefficients are determined by inverse problem solution, i.e. using known results of experimental tests, to obtain the best agreement between theoretical prediction and experimental evidence (e.g. in the sense of least square method). Models of such a kind usually give very accurate predictions when applied to phenomena similar to those used to adjust model parameters and often very inaccurate for different situations (e.g. for similar, but not exactly the same materials, different element sizes, different moisture and temperature ranges, etc.). In other words, they are very accurate for interpolation and rather poor for extrapolation of the known experimental results. Another shortcoming of this approach is the very non-linear character of equation coefficients, which requires performing of numerous and time consuming experimental tests. Moreover various physical phenomena are lumped together and model parameters often have no clear physical interpretation. MATHEMATICAL MODEL In the model presented here, concrete is considered as a partially saturated porous material (Gawin & Schrefler, 1996) consisting of a solid phase, two gas phases and three water phases. The theoretical framework is based on averaging theories (Whitaker, 1977, Bear, 1988, Bear & Bachmat 1986, 1889, Hassanizadeh and Gray 1979, 1980 and Lewis & Schrefler 1998). In particular the approach followed consists in a modified averaging theory. The developments starts at the microscale and balance equations for phases and interfaces are introduced at this level and then averaged for obtaining macroscopic balance equations. Constitutive laws are directly introduced at macroscopic level. Refinements such as non-linearities due to temperature and pressures, hydration-dehydration, evaporation-condensation, adsorption-desortpion, phenomena are considered, (Gawin et al., 1998, Gawin et al., 1999). Different physical mechanisms governing the liquid and gas transport in the pores of partially saturated concrete are clearly distinguished, i.e. capillary water and gas flows driven by their pressure gradients, adsorbed water surface diffusion caused by saturation gradients, as well as air and vapour diffusion driven by vapour density gradients, (Gawin et al., 1998, 1999). Concrete damaging effects arising from coupled hygro-thermal and mechanical interaction are considered by use of the isotropic non-local damage theory and a further coupling between intrinsic permeability and mechanical damage has been introduced to take into account the changes of material microstructure. Moreover improvements, regarding the possibility to simulate the behaviour of the material at temperatures which largely exceed the critical point of water and the real behaviour of gases present in the pores of concrete, i.e. the gases are treated as real gas, have been recently introduced in the model (Gawin et al., in prep.). The final mathematical model consists of four balance equations: mass conservation of dry air, mass conservation of the water species (both in liquid and gaseous state, taking phase changes, i.e. evaporation - condensation, adsorption - desorption and hydration - dehydration
829 process, into account), enthalpy conservation of the whole medium (latent heat of phase changes and heat effects of hydration or dehydration processes are considered) and linear momentum of the multiphase system. They are completed by an appropriate set of constitutive and state equations, as well as some thermodynamic relationships, (Lewis & Schrefler, 1998). The governing equations of the model are expressed in terms of the chosen state variables: gas pressure pg, capillary pressure pc, temperature T and displacement vector of the solid matrix u and are directly derived from Lewis & Schrefler (1998), where the constitutive law for the solid skeleton density and water density are experimentally determined. In particular the density of solid skeleton has to respect the solid mass conservation equation which is not a basic equation of the model. The governing equations of the model proposed, considering negligible both the inertial forces and the convective heat flux related to solid phase and taking into account the Bishop's stresses (Schrefler & Gawin, 1996), are the following: Dry air mass conservation equation: n ^—s - -ß - s(ß.l-n)S (1 -g n)Sg ^ — - + SgV · v s +
V a/ +
pa 1
i +
dt
p"v
J
*+ (1)
+ ±V-(nSgP^)
+
dps dThydT 9Thydr dt
q-n)Sg Ps
mdehydr p° «
Water species (liquid + vapour) mass conservation equation: w n n( ov)) \P o - P
+
/)
ftT - ßPswg — 4Λ . -I-
(pvSg+pwSw)V.vs+Sgn
?£ +
v 9S
+V ·J^ + V · (nS gP v ) + V · (nS w P w v w s ) + v
S
q-n){Sgp +p»Sw)
0P
S
P
pvSg + pwSw - ps ~
ps
^hydr m
(2)
dThydr _ dt
dehydr
Energy conservation equation (enthalpy balance):
(pC» h ΊΓ + ^c^m
° ' ) ·vT-
+ p c v9S
-V-(Ae/VT)= =
~mphase
^ ^ phase +
(3) m
dehydr^dehydr
Linear momentum equation (equilibrium equation): div{a' - l(p9 - Swpc)} + pg = 0
(4)
830 In equations (l)-(3), for the hygroscopic region of moisture content, the liquid phase is the bound water, thus forS
/=*, AHphase = AHadson>
(5)
while for the capillary region it is assumed that only capillary water can move, hence forS>Sssp
l=w, AHphase = AHvap
(6)
The time derivative of porosity has been eliminated from equations (l)-(3) by summing the mass conservation equation of the solid phase with the mass conservation equations for dry air, water vapour, capillary water and adsorbed water. Then, the mass balances of liquid water and vapour have been summed, allowing to eliminate the mass source term related to phase change. Further, it has been taken into account that hydration results in an increase of solid volume, thus also decrease of porosity according to "hydr
dt
~
d_ dt
Am
hydr
(7)
thus only two from the three material parameters describing the solid phase, i.e. φ,ρβ and ^(Am
¥ r
) are independent.
For the closure of the model a set of thermodynamic and constitutive relationships are needed. In the present work only a part of constitutive laws will be shown in detail, in particular as far as damage mechanics is concerned. For further information see Gawin et al. (1998), Gawin et al. (1999), Schrefler & Gawin (1996) and Lewis & Schrefler (1998). Thermodynamics and constitutive relationships For fluid phases the multiphase Darcy's law has been applied as constitutive equation, thus for capillary water we have: rfvm
= -=-£■ ( V p » - V P c - p«b)
(8)
while for the gas phase holds: kr9k
(Vp')
(9)
where k is intrinsic permeability tensor krg and ¡C are relative permeabilities of the gaseous and liquid phases, vws and vgs are the velocities of capillary water and gaseous phase relative to the solid phase and the vector b indicates the specific body force term (normally corresponding to the acceleration of gravity), if and rf stand for volumetric fraction of liquid and gas respectively. For the description of the diffusion process of the binary gas species mixture (dry air and water vapour) Fick's law is applied:
831 J9ga=-p9O9gagradÍEl) Jf
Κ μ ) . =-p9O9wgrad\P_\
(10)
and O9gw is the effective diffusion coefficient of vapour species in the porous medium. The layers of physically bound water (or adsorbed water) are interested by a phenomenon of mass transport which is similar to a diffusion. For sake of simplicity in the model this movement of adsorbed water has been modelled using a relationship similar to Fick's law: T¡ = -p"O»gradSt
(11)
where D¿ is the effective tensor of surface diffusion of water on the skeleton. In environmental temperature range —2-~ 0. Since the thermodynamic driven force is the capillary pressure, the above mentioned relationship has been modified in the following manner, in order to obtain a formulation containing directly a state variable (pc): 3Wd=-pWVW^3r*dpC
(12)
so the diffusion tensor multiplied by the derivative of degree of saturation of bound water on capillary pressure is a kind of equivalent diffusion tensor. Further the material is considered isotropic, hence the adsorbed water diffusion tensor can be seen as D " = Dwdl
(13)
where D^ is the bound water diffusive coefficient. Indeed, the moist air in the pore system is usually assumed to be a perfect mixture of two ideal gases, i.e. dry air and water vapour. Hence the ideal gas law and Dalton's law, relating the partial pressure, Pg^ of species π, the mass concentration, /#*, of species π in the gas phase and the absolute temperature, Θ, is used. Improvements, regarding the possibility to treat the components of gas phase as real gases, are recently introduced in the model. The concrete damage at high temperature (cracks development) is considered, following the model of scalar isotropic damage by Mazars (1984) and Mazars & Pijaudier-Cabot (1989). In this model, the material is supposed to behave elastically and to remain isotropic, and it is assumed that only the elastic properties of the material are affected by damage. A "modified effective stress" σ taking into account the damage D (0
(14)
832 p^=-AQ(l-D):ee:Ee(9)
(15)
where py/e is a scalar thermodynamic potential (p is the density of the material). Taking into account different performance of concrete in traction and in compression, i.e. coupling of two types of damage Dt and Dc (Mazars, 1984), the final expression for the stress has the following form: Αε ( i - 4 ) * o + βχρ[5<(ε-*0
(16)
\E
where i=t,c. The parameters At, Ac, Bt, Bc are characteristics of the material and temperature dependent. Λ^ is the initial value of the hardening/softening parameter Κ(Ό), which satisfies the principle of maximum de Saint-Venant's strain (loading function) f(e,A,KQ)
=
(17)
e-K(D)
where the equivalent strain ε is defined as follows
ε = νΣ(<εϊ>+)2
;
( < X >+=
\x\ + x\
(18)
ε{ being the principal strains. NUMERICAL MODEL The system of governing equations of the model (l)-(4), after application of the finite element method for discretization in space and backward finite difference scheme for temporal derivatives, similarly as in Gawin et al. (1999), may be written in the following concise form:
*'Κ + 1 ) = ^(*„ +1 )-
Δί
- + K ! . ( x n + 1 ) x „ + 1 - f ¿ ( x n + 1 ) = 0,
(19)
with: Kgc
K„
K cc
κ,,
0
99
Cgc
*ct
0
0
C cc
KH
0
0
ctc
0
0
K
9t
\c
\c
\c
"■im
"■ii.r
"-tit
\ζ
C
%
Cgu
cci Ccu c« ctu 0
(20)
0
"·ιν
where n is the time step number and At - the time step length. The equation set (19) is solved by means of a monolithic Newton-Raphson type iterative procedure. A two stage solution strategy has been applied at every time step to take into account damage of concrete (Gawin et al., 1999). Moreover, because of different physical meaning of the capillary pressure pc and different nature of the physical phenomena above the temperature Tcn a special 'switching' procedure has been adopted (Gawin et al., in prep.). NUMERICAL EXAMPLES Two examples, the first one concerning an experimental-numerical comparison, the second one a cylindrical specimen subjected to fire will be shown.
833 The main aim of the experimental work described in this section is to obtain a progressive measure of temperatures and pores pressure fields, in a prismatic specimen subjected to a quasi-unidirectional thermal load. The resulting distributions, in terms of measured physical parameters, have been used for a validation of constitutive laws implemented in the numerical code. This test has been carried out in the framework of BHP2000 (Kalifa at al., 1999). The thermal load is applied on one face of a prismatic concrete specimen (30x30x12 cm3), whereas the lateral faces of the specimen are heat-insulated with porous ceramic blocks. Heating is provided by means of a radiant heater (up to 5 kW and 600°C) that covers the surface of the specimen, placed 3 cm above it. The temperature of the radiant heater was 600 °C. The specimens were instrumented with 6 gauges placed at casting, for simultaneous pressure and temperature measurements. Five of them were positioned in the central square zone of 10x10 cm2, at 10, 20, 30, 40 and 50 mm from the heated surface respectively. The free volume of the gauge is around 250 mm3. This may alter slightly the measured pressure values. The experimental tests were carried out on a particular mixes corresponding to a high performance concrete in the class of 100 MPa, whose characteristic parameters at ambient temperature are shown in Table 1. Initial conditions are pg=101325 Pa, i.e. atmospheric pressure, T=298.15 K and pc=9.53e07 Pa corresponding to a relative humidity equal to 50%. As far as boundary conditions are concerned, average values for heat and mass exchange coefficients, supposed constant during simulation, have been chosen: for the heated side: 0^=18 W/m2K and ßc=0.019 m/s, not heated side: Oc=8.3 W/m2K and ßc=0.009 m/s. Hence, the cold side is not considered as adiabatic. The heat flow is unidirectional, so the discretization corresponds to a ID problem. As the Figure la-b clearly show, during the test rather high peak values of pore pressure have been reached. Others tests have been performed in more or less recent past, and almost all authors noted both the high values of pore pressure recorded during the tests, often greater than saturating water vapour pressure, and the role played by water dilatation. Numerical results present a good agreement with experimental ones, in terms of both temperature and pressure. In particular thermal field is lower in the numerical simulation. This is probably due to the choice of boundary conditions and the conductivity of the material, which may not be well set. TABLE 1 CHARACTERISTIC PROPERTIES OF CONCRETE AT 20°C Parameter Porosity Intrinsic permeability Apparent density Specific heat Thermal conductivity Young's modulus Poisson's coefficient Compression strength
Symbol
n[-J
k[m 2 ] P [kg/m3] C [J/kgK] λ [W/mK] EGPa v[-] fc [MPa]
Ordinary 0.13
io- 17
2585 940 1.70 33 0.20 30
Value
M100 0.1 IO'19 2590 948 1.67 44.2 0.18 102
The second example deals with a cylindrical specimen heated according to standard ISO-Fire curve. Mixed radiative-convective boundary conditions have been used and the exchange coefficients are Oc=18 W/m2K and ßc=0.019 m/s. Features of concrete are reported in Table 1 (ordinary concrete). Figure 2 a,b,c,d shows distribution of temperature, damage, relative humidity and vapour pressure at 10 minutes. Thermo-diffusion is clearly evident from the results (Fig. 2). It is interesting to note that the peak value of vapour pressure is lower than the corresponding value in the previous calculation, despite the faster heating (Fig. 2d). This is due to the internal material microstructure, which is less dense in the case of ordinary concrete.
834
0
1
2
3 Time [hours]
4
5
6
a) 4.0E+06 -,
1
1
Figure 1: Temperature distribution and gas pressure distribution in the sample during heating (numerical - bold line, experimental - thin line)
CONCLUSIONS A finite element model of concrete at high temperature based on a mechanistic approach has been presented. Concrete is considered as multiphase porous material in which adsorption-desorption, hydration-dehydration, evaporation-condensation, different fluid flows and non-linearities with temperature have been taken into account. The particular approach permits to consider the different phenomena, which take place in concrete during heating and their coupling.
835 The resulting model is appealing for prediction of thermo-hygro-mechanical behaviour of concrete structures in such severe conditions as shown with an experimental-numerical comparison and a 2D example. Phenomena concerning real behaviour of gases close to critical point of water, behaviour of concrete in a range of temperature largely above 374.15 °C, and saturation plug process, have been recently investigated and introduced in the model.
DAMAGE 0.73026
m VAP._PRESSURE •1.1171Θ+06 mi: 8.6316e+05 M 6.0921 et-05 B
3.5527e+05
■
1.0133e*05
c) d) Figure 2: temperature, damage, relative humidity and vapour pressure distribution at 10 min.
836 References Bazant, Z.P. & Thonguthai, W. (1978). Pore pressure and drying of concrete at high temperature. In J. Engng. Meek Div. ASCE 104, 1059-1079. Bazant, Z.P. & Kaplan, M.F. (1996). Concrete at High Temperatures: Material Properties and Mathematical Models. Longman, Harlow. Bear, J. (1988). Dynamics of fluids in porous media. Dover, New York, U.S.A.. Bear, J. & Bachmat, Y. (1986). Macroscopic modelling of transport phenomena in porous media, 2: applications to mass momentum and energy transfer. In Transp. in Porous Media 1, 241-269. Bear, J. & Bachmat, Y. (1990). Introduction to modelling of transport phenomena in porous media. Kluwer, Dordrecht. Gawin, D. & Schrefler, B.A. (1996). Thermo- hydro- mechanical analysis of partially saturated porous materials. In Engineering Computations 13:7, 113-143. Gawin, D., Majorana, C.E., Pesavento, F., & Schrefler, B.A (1998). A fully coupled multiphase FE model of hygro- thermo- mechanical behaviour of concrete at high temperature. In Computational Mechanics., Onate, E. & Idelsohn, S.R. (eds.), New Trends and Applications:, Proc. of the 4th World Congress on Computational Mechanics, Buenos Aires 1998, 1-19, CÍ1MNE, Barcelona. Gawin, D., Majorana, C.E., & Schrefler, B.A. (1999). Numerical analysis of hygro-thermic behaviour and damage of concrete at high temperature. In Mech. Cohes.-Frict. Mater 4, 37-/4. Gawin, D., Pesavento, F., & Schrefler, B.A.. Modelling of hygro-thermal behaviour and damage of concrete at temperature above the critical point of water, in prep.. Hassanizadeh, M. & Grav, W.G. (1979). General conservation equations for multiphase systems: 1. Averaging technique. In Adv. Water Res. 2, 131-144. Hassanizadeh, M. & Gray, W.G. (1979). General conservation equations for multiphase systems: 2 Mass, momenta, energy and entropy equations. In Adv. Water Res. 2, 191-203. Hassanizadeh, M. & Gray, W.G. Π980). General conservation equations for multiphase systems: 3 Constitutive Theory for Porous Media. In Adv. Water Res. 3, 25-40. Kalifa P., Tsimbrovska M. and Baroghel-Bouny V. (1998). High Performance Concrete at Elevated Temperatures - An Extensive Experimental Investigation on Thermal, Hygral and Microstructure Properties. Proc. of Int. Symp. on High-Performance and Reactive Powder Concretes. Aug. 16-20, 1998, Sherbrooke, Canada. Lewis, R.W., & Schrefler, B.A. (1998). The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, Wiley & Sons, Chichester. Mazars J. (1984). Application de la mecanique de V endommagement au comportament non lineaire et la rupture du beton de structure (in French), thesy de Doctorat d' Etat, L.M.T., Universite de Paris. Mazars, J. (1986). Description of the behaviour of composite concretes under complex loadings through continuum damage mechanics. In Lamb, J.P. (ed.), Proc. of the 10th U.S. National Congress of Applied Mechanics, ASME. Mazars, J. & Pijaudier-Cabot, J. (1989). Continuum damage theory - application to concrete. In J. of Eng. Mechanics ASCE 115:2, 345-365. Schrefler, B.A. &, Gawin, D. (1996). The effective stress principle: incremental or finite form?. In Int. J. for Num. and Anal. Meth. in Geomechanics 20:11, 785-815. Whitaker, S. (1977). Simultaneous heat mass and momentum transfer in porous media: a theory of drying. In Advances in heat transfer 13, Accademic Press, New York.
827
FE ANALYSIS OF THERMO-HYDROMECHANICAL BEHAVIOUR OF CONCRETE AT HIGH TEMPERATURE B.A. Schrefler1, D. Gawin2, C.E. Majorana1, F. Pesavento1 1
Dipartimento di Costruzioni e Trasporti, Universitá degli Studi di Padova, Via Marzolo 9, Padova, 35131, Italy department of Building Physics and Building Materials Technical University of Lodz, 1. Politechniki 6, Lodz, 93-590, Poland
ABSTRACT In the present work an innovative model for analysing the behaviour of concrete in high temperature environment is presented. In such conditions prediction of concrete performance during and after fire exposure is of great practical importance, e.g. in tunnels, as recent fires in major European tunnels have demonstrated. This is particularly true for High Performance and Ultra High Performance concretes, which have been often used in the last decade in nuclear engineering applications, tall buildings and in tunnels, because of their higher tendency to spalling experience. It is now well established that phenomena which take place in concrete exposed to high temperatures, cannot be studied by means of purely diffusive models. In the model presented in this paper concrete is treated as partially saturated porous material where hydration-dehydration, evaporation-condensation, adsorption-desorption phenomena and non-linearities due to temperature and pressure take place. Two applications of this model to an ordinary and HP concrete structure will be shown.
KEYWORDS Concrete, High temperature, Thermo-hydro-mechanical, High performance, Damage INTRODUCTION When concrete is exposed to high temperature a rather complex analysis is required to deal with the coupled heat and mass transfer that can occur, involving both liquid mass transfer and vapour mass transfer and related mechanical effects (Bazant & Kaplan, 1996). In such severe conditions in terms of temperatures and pressures, assessment of concrete performance is of great interest in nuclear engineering applications, in safety evaluation in tall buildings and in tunnels (Bazant & Thonguthai, 1978). In particular as far as tunnels are concerned, recent major fires in key European tunnels (Channel, Mont-Blanc, Great Belt Link, Tauern) emphasised the serious hazards they present in human and economic terms. In these accidents, in the affected sections, tunnels presented extensive damage to the concrete elements. Part of the concrete lining was almost completely removed by spalling. Spalling, which may be explosive, is mainly due to different coexisting coupled processes, such as
828 thermal (heat transfer), chemical (dehydration of cement paste), hygral (transfer of water mass, in liquid and vapour form) and mechanical processes. In general, moisture transport may include airvapour mixture flow due to forced convection, free convection, and infiltration through cracks and pores, vapour transport by diffusion, flow of liquid water due to diffusion, capillary action, or gravity, and further complications associated with phase changes due to condensation/evaporation, freezing/thawing, ablimation/sublimation, and adsorption/desorption. Movement of air and water through the concrete is accompanied by significant energy transfer, associated with the latent heat of water and the heat of hydration and dehydration. At temperature largely above critical point of water important chemical transformations of components of concrete take place. These situations are much more dangerous in the case of High Performance and Ultra High Performance concrete because of their peculiar features. Hence, for concrete, particularly at high temperature, one cannot predict heat transfer only from the traditional thermal properties (i.e., thermal conductivity and volumetric specific heat) and, similarly, it is not possible to predict mass transfer considering this phenomenon as purely diffusive. In fact, in a typically phenomenological approach moisture and heat transport are described by diffusive type differential equations with temperature- and moisture content- dependent coefficients. The model coefficients are determined by inverse problem solution, i.e. using known results of experimental tests, to obtain the best agreement between theoretical prediction and experimental evidence (e.g. in the sense of least square method). Models of such a kind usually give very accurate predictions when applied to phenomena similar to those used to adjust model parameters and often very inaccurate for different situations (e.g. for similar, but not exactly the same materials, different element sizes, different moisture and temperature ranges, etc.). In other words, they are very accurate for interpolation and rather poor for extrapolation of the known experimental results. Another shortcoming of this approach is the very non-linear character of equation coefficients, which requires performing of numerous and time consuming experimental tests. Moreover various physical phenomena are lumped together and model parameters often have no clear physical interpretation. MATHEMATICAL MODEL In the model presented here, concrete is considered as a partially saturated porous material (Gawin & Schrefler, 1996) consisting of a solid phase, two gas phases and three water phases. The theoretical framework is based on averaging theories (Whitaker, 1977, Bear, 1988, Bear & Bachmat 1986, 1889, Hassanizadeh and Gray 1979, 1980 and Lewis & Schrefler 1998). In particular the approach followed consists in a modified averaging theory. The developments starts at the microscale and balance equations for phases and interfaces are introduced at this level and then averaged for obtaining macroscopic balance equations. Constitutive laws are directly introduced at macroscopic level. Refinements such as non-linearities due to temperature and pressures, hydration-dehydration, evaporation-condensation, adsorption-desortpion, phenomena are considered, (Gawin et al., 1998, Gawin et al., 1999). Different physical mechanisms governing the liquid and gas transport in the pores of partially saturated concrete are clearly distinguished, i.e. capillary water and gas flows driven by their pressure gradients, adsorbed water surface diffusion caused by saturation gradients, as well as air and vapour diffusion driven by vapour density gradients, (Gawin et al., 1998, 1999). Concrete damaging effects arising from coupled hygro-thermal and mechanical interaction are considered by use of the isotropic non-local damage theory and a further coupling between intrinsic permeability and mechanical damage has been introduced to take into account the changes of material microstructure. Moreover improvements, regarding the possibility to simulate the behaviour of the material at temperatures which largely exceed the critical point of water and the real behaviour of gases present in the pores of concrete, i.e. the gases are treated as real gas, have been recently introduced in the model (Gawin et al., in prep.). The final mathematical model consists of four balance equations: mass conservation of dry air, mass conservation of the water species (both in liquid and gaseous state, taking phase changes, i.e. evaporation - condensation, adsorption - desorption and hydration - dehydration
829 process, into account), enthalpy conservation of the whole medium (latent heat of phase changes and heat effects of hydration or dehydration processes are considered) and linear momentum of the multiphase system. They are completed by an appropriate set of constitutive and state equations, as well as some thermodynamic relationships, (Lewis & Schrefler, 1998). The governing equations of the model are expressed in terms of the chosen state variables: gas pressure pg, capillary pressure pc, temperature T and displacement vector of the solid matrix u and are directly derived from Lewis & Schrefler (1998), where the constitutive law for the solid skeleton density and water density are experimentally determined. In particular the density of solid skeleton has to respect the solid mass conservation equation which is not a basic equation of the model. The governing equations of the model proposed, considering negligible both the inertial forces and the convective heat flux related to solid phase and taking into account the Bishop's stresses (Schrefler & Gawin, 1996), are the following: Dry air mass conservation equation: n ^—s - -ß - s(ß.l-n)S (1 -g n)Sg ^ — - + SgV · v s +
V a/ +
pa 1
i +
dt
p"v
J
*+ (1)
+ ±V-(nSgP^)
+
dps dThydT 9Thydr dt
q-n)Sg Ps
mdehydr p° «
Water species (liquid + vapour) mass conservation equation: w n n( ov)) \P o - P
+
/)
ftT - ßPswg — 4Λ . -I-
(pvSg+pwSw)V.vs+Sgn
?£ +
v 9S
+V ·J^ + V · (nS gP v ) + V · (nS w P w v w s ) + v
S
q-n){Sgp +p»Sw)
0P
S
P
pvSg + pwSw - ps ~
ps
^hydr m
(2)
dThydr _ dt
dehydr
Energy conservation equation (enthalpy balance):
(pC» h ΊΓ + ^c^m
° ' ) ·vT-
+ p c v9S
-V-(Ae/VT)= =
~mphase
^ ^ phase +
(3) m
dehydr^dehydr
Linear momentum equation (equilibrium equation): div{a' - l(p9 - Swpc)} + pg = 0
(4)
830 In equations (l)-(3), for the hygroscopic region of moisture content, the liquid phase is the bound water, thus forS
/=*, AHphase = AHadson>
(5)
while for the capillary region it is assumed that only capillary water can move, hence forS>Sssp
l=w, AHphase = AHvap
(6)
The time derivative of porosity has been eliminated from equations (l)-(3) by summing the mass conservation equation of the solid phase with the mass conservation equations for dry air, water vapour, capillary water and adsorbed water. Then, the mass balances of liquid water and vapour have been summed, allowing to eliminate the mass source term related to phase change. Further, it has been taken into account that hydration results in an increase of solid volume, thus also decrease of porosity according to "hydr
dt
~
d_ dt
Am
hydr
(7)
thus only two from the three material parameters describing the solid phase, i.e. φ,ρβ and ^(Am
¥ r
) are independent.
For the closure of the model a set of thermodynamic and constitutive relationships are needed. In the present work only a part of constitutive laws will be shown in detail, in particular as far as damage mechanics is concerned. For further information see Gawin et al. (1998), Gawin et al. (1999), Schrefler & Gawin (1996) and Lewis & Schrefler (1998). Thermodynamics and constitutive relationships For fluid phases the multiphase Darcy's law has been applied as constitutive equation, thus for capillary water we have: rfvm
= -=-£■ ( V p » - V P c - p«b)
(8)
while for the gas phase holds: kr9k
(Vp')
(9)
where k is intrinsic permeability tensor krg and ¡C are relative permeabilities of the gaseous and liquid phases, vws and vgs are the velocities of capillary water and gaseous phase relative to the solid phase and the vector b indicates the specific body force term (normally corresponding to the acceleration of gravity), if and rf stand for volumetric fraction of liquid and gas respectively. For the description of the diffusion process of the binary gas species mixture (dry air and water vapour) Fick's law is applied:
831 J9ga=-p9O9gagradÍEl) Jf
Κ μ ) . =-p9O9wgrad\P_\
(10)
and O9gw is the effective diffusion coefficient of vapour species in the porous medium. The layers of physically bound water (or adsorbed water) are interested by a phenomenon of mass transport which is similar to a diffusion. For sake of simplicity in the model this movement of adsorbed water has been modelled using a relationship similar to Fick's law: T¡ = -p"O»gradSt
(11)
where D¿ is the effective tensor of surface diffusion of water on the skeleton. In environmental temperature range —2-~ 0. Since the thermodynamic driven force is the capillary pressure, the above mentioned relationship has been modified in the following manner, in order to obtain a formulation containing directly a state variable (pc): 3Wd=-pWVW^3r*dpC
(12)
so the diffusion tensor multiplied by the derivative of degree of saturation of bound water on capillary pressure is a kind of equivalent diffusion tensor. Further the material is considered isotropic, hence the adsorbed water diffusion tensor can be seen as D " = Dwdl
(13)
where D^ is the bound water diffusive coefficient. Indeed, the moist air in the pore system is usually assumed to be a perfect mixture of two ideal gases, i.e. dry air and water vapour. Hence the ideal gas law and Dalton's law, relating the partial pressure, Pg^ of species π, the mass concentration, /#*, of species π in the gas phase and the absolute temperature, Θ, is used. Improvements, regarding the possibility to treat the components of gas phase as real gases, are recently introduced in the model. The concrete damage at high temperature (cracks development) is considered, following the model of scalar isotropic damage by Mazars (1984) and Mazars & Pijaudier-Cabot (1989). In this model, the material is supposed to behave elastically and to remain isotropic, and it is assumed that only the elastic properties of the material are affected by damage. A "modified effective stress" σ taking into account the damage D (0
(14)
832 p^=-AQ(l-D):ee:Ee(9)
(15)
where py/e is a scalar thermodynamic potential (p is the density of the material). Taking into account different performance of concrete in traction and in compression, i.e. coupling of two types of damage Dt and Dc (Mazars, 1984), the final expression for the stress has the following form: Αε ( i - 4 ) * o + βχρ[5<(ε-*0
(16)
\E
where i=t,c. The parameters At, Ac, Bt, Bc are characteristics of the material and temperature dependent. Λ^ is the initial value of the hardening/softening parameter Κ(Ό), which satisfies the principle of maximum de Saint-Venant's strain (loading function) f(e,A,KQ)
=
(17)
e-K(D)
where the equivalent strain ε is defined as follows
ε = νΣ(<εϊ>+)2
;
( < X >+=
\x\ + x\
(18)
ε{ being the principal strains. NUMERICAL MODEL The system of governing equations of the model (l)-(4), after application of the finite element method for discretization in space and backward finite difference scheme for temporal derivatives, similarly as in Gawin et al. (1999), may be written in the following concise form:
*'Κ + 1 ) = ^(*„ +1 )-
Δί
- + K ! . ( x n + 1 ) x „ + 1 - f ¿ ( x n + 1 ) = 0,
(19)
with: Kgc
K„
K cc
κ,,
0
99
Cgc
*ct
0
0
C cc
KH
0
0
ctc
0
0
K
9t
\c
\c
\c
"■im
"■ii.r
"-tit
\ζ
C
%
Cgu
cci Ccu c« ctu 0
(20)
0
"·ιν
where n is the time step number and At - the time step length. The equation set (19) is solved by means of a monolithic Newton-Raphson type iterative procedure. A two stage solution strategy has been applied at every time step to take into account damage of concrete (Gawin et al., 1999). Moreover, because of different physical meaning of the capillary pressure pc and different nature of the physical phenomena above the temperature Tcn a special 'switching' procedure has been adopted (Gawin et al., in prep.). NUMERICAL EXAMPLES Two examples, the first one concerning an experimental-numerical comparison, the second one a cylindrical specimen subjected to fire will be shown.
833 The main aim of the experimental work described in this section is to obtain a progressive measure of temperatures and pores pressure fields, in a prismatic specimen subjected to a quasi-unidirectional thermal load. The resulting distributions, in terms of measured physical parameters, have been used for a validation of constitutive laws implemented in the numerical code. This test has been carried out in the framework of BHP2000 (Kalifa at al., 1999). The thermal load is applied on one face of a prismatic concrete specimen (30x30x12 cm3), whereas the lateral faces of the specimen are heat-insulated with porous ceramic blocks. Heating is provided by means of a radiant heater (up to 5 kW and 600°C) that covers the surface of the specimen, placed 3 cm above it. The temperature of the radiant heater was 600 °C. The specimens were instrumented with 6 gauges placed at casting, for simultaneous pressure and temperature measurements. Five of them were positioned in the central square zone of 10x10 cm2, at 10, 20, 30, 40 and 50 mm from the heated surface respectively. The free volume of the gauge is around 250 mm3. This may alter slightly the measured pressure values. The experimental tests were carried out on a particular mixes corresponding to a high performance concrete in the class of 100 MPa, whose characteristic parameters at ambient temperature are shown in Table 1. Initial conditions are pg=101325 Pa, i.e. atmospheric pressure, T=298.15 K and pc=9.53e07 Pa corresponding to a relative humidity equal to 50%. As far as boundary conditions are concerned, average values for heat and mass exchange coefficients, supposed constant during simulation, have been chosen: for the heated side: 0^=18 W/m2K and ßc=0.019 m/s, not heated side: Oc=8.3 W/m2K and ßc=0.009 m/s. Hence, the cold side is not considered as adiabatic. The heat flow is unidirectional, so the discretization corresponds to a ID problem. As the Figure la-b clearly show, during the test rather high peak values of pore pressure have been reached. Others tests have been performed in more or less recent past, and almost all authors noted both the high values of pore pressure recorded during the tests, often greater than saturating water vapour pressure, and the role played by water dilatation. Numerical results present a good agreement with experimental ones, in terms of both temperature and pressure. In particular thermal field is lower in the numerical simulation. This is probably due to the choice of boundary conditions and the conductivity of the material, which may not be well set. TABLE 1 CHARACTERISTIC PROPERTIES OF CONCRETE AT 20°C Parameter Porosity Intrinsic permeability Apparent density Specific heat Thermal conductivity Young's modulus Poisson's coefficient Compression strength
Symbol
n[-J
k[m 2 ] P [kg/m3] C [J/kgK] λ [W/mK] EGPa v[-] fc [MPa]
Ordinary 0.13
io- 17
2585 940 1.70 33 0.20 30
Value
M100 0.1 IO'19 2590 948 1.67 44.2 0.18 102
The second example deals with a cylindrical specimen heated according to standard ISO-Fire curve. Mixed radiative-convective boundary conditions have been used and the exchange coefficients are Oc=18 W/m2K and ßc=0.019 m/s. Features of concrete are reported in Table 1 (ordinary concrete). Figure 2 a,b,c,d shows distribution of temperature, damage, relative humidity and vapour pressure at 10 minutes. Thermo-diffusion is clearly evident from the results (Fig. 2). It is interesting to note that the peak value of vapour pressure is lower than the corresponding value in the previous calculation, despite the faster heating (Fig. 2d). This is due to the internal material microstructure, which is less dense in the case of ordinary concrete.
834
0
1
2
3 Time [hours]
4
5
6
a) 4.0E+06 -,
1
1
Figure 1: Temperature distribution and gas pressure distribution in the sample during heating (numerical - bold line, experimental - thin line)
CONCLUSIONS A finite element model of concrete at high temperature based on a mechanistic approach has been presented. Concrete is considered as multiphase porous material in which adsorption-desorption, hydration-dehydration, evaporation-condensation, different fluid flows and non-linearities with temperature have been taken into account. The particular approach permits to consider the different phenomena, which take place in concrete during heating and their coupling.
835 The resulting model is appealing for prediction of thermo-hygro-mechanical behaviour of concrete structures in such severe conditions as shown with an experimental-numerical comparison and a 2D example. Phenomena concerning real behaviour of gases close to critical point of water, behaviour of concrete in a range of temperature largely above 374.15 °C, and saturation plug process, have been recently investigated and introduced in the model.
DAMAGE 0.73026
m VAP._PRESSURE •1.1171Θ+06 mi: 8.6316e+05 M 6.0921 et-05 B
3.5527e+05
■
1.0133e*05
c) d) Figure 2: temperature, damage, relative humidity and vapour pressure distribution at 10 min.
836 References Bazant, Z.P. & Thonguthai, W. (1978). Pore pressure and drying of concrete at high temperature. In J. Engng. Meek Div. ASCE 104, 1059-1079. Bazant, Z.P. & Kaplan, M.F. (1996). Concrete at High Temperatures: Material Properties and Mathematical Models. Longman, Harlow. Bear, J. (1988). Dynamics of fluids in porous media. Dover, New York, U.S.A.. Bear, J. & Bachmat, Y. (1986). Macroscopic modelling of transport phenomena in porous media, 2: applications to mass momentum and energy transfer. In Transp. in Porous Media 1, 241-269. Bear, J. & Bachmat, Y. (1990). Introduction to modelling of transport phenomena in porous media. Kluwer, Dordrecht. Gawin, D. & Schrefler, B.A. (1996). Thermo- hydro- mechanical analysis of partially saturated porous materials. In Engineering Computations 13:7, 113-143. Gawin, D., Majorana, C.E., Pesavento, F., & Schrefler, B.A (1998). A fully coupled multiphase FE model of hygro- thermo- mechanical behaviour of concrete at high temperature. In Computational Mechanics., Onate, E. & Idelsohn, S.R. (eds.), New Trends and Applications:, Proc. of the 4th World Congress on Computational Mechanics, Buenos Aires 1998, 1-19, CÍ1MNE, Barcelona. Gawin, D., Majorana, C.E., & Schrefler, B.A. (1999). Numerical analysis of hygro-thermic behaviour and damage of concrete at high temperature. In Mech. Cohes.-Frict. Mater 4, 37-/4. Gawin, D., Pesavento, F., & Schrefler, B.A.. Modelling of hygro-thermal behaviour and damage of concrete at temperature above the critical point of water, in prep.. Hassanizadeh, M. & Grav, W.G. (1979). General conservation equations for multiphase systems: 1. Averaging technique. In Adv. Water Res. 2, 131-144. Hassanizadeh, M. & Gray, W.G. (1979). General conservation equations for multiphase systems: 2 Mass, momenta, energy and entropy equations. In Adv. Water Res. 2, 191-203. Hassanizadeh, M. & Gray, W.G. Π980). General conservation equations for multiphase systems: 3 Constitutive Theory for Porous Media. In Adv. Water Res. 3, 25-40. Kalifa P., Tsimbrovska M. and Baroghel-Bouny V. (1998). High Performance Concrete at Elevated Temperatures - An Extensive Experimental Investigation on Thermal, Hygral and Microstructure Properties. Proc. of Int. Symp. on High-Performance and Reactive Powder Concretes. Aug. 16-20, 1998, Sherbrooke, Canada. Lewis, R.W., & Schrefler, B.A. (1998). The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, Wiley & Sons, Chichester. Mazars J. (1984). Application de la mecanique de V endommagement au comportament non lineaire et la rupture du beton de structure (in French), thesy de Doctorat d' Etat, L.M.T., Universite de Paris. Mazars, J. (1986). Description of the behaviour of composite concretes under complex loadings through continuum damage mechanics. In Lamb, J.P. (ed.), Proc. of the 10th U.S. National Congress of Applied Mechanics, ASME. Mazars, J. & Pijaudier-Cabot, J. (1989). Continuum damage theory - application to concrete. In J. of Eng. Mechanics ASCE 115:2, 345-365. Schrefler, B.A. &, Gawin, D. (1996). The effective stress principle: incremental or finite form?. In Int. J. for Num. and Anal. Meth. in Geomechanics 20:11, 785-815. Whitaker, S. (1977). Simultaneous heat mass and momentum transfer in porous media: a theory of drying. In Advances in heat transfer 13, Accademic Press, New York.