- Email: [email protected]
Comments to the paper “An application of a damage constitutive model to concrete at high temperature and prediction of spalling” by Rosen Tenchev and Phil Purnell [Int. J. Solids Struct. 42 (26) (2005) 6550–6565]
International Journal of Solids and Structures 44 (2007) 4234–4237 www.elsevier.com/locate/ijsolstr
Discussion
Comments to the paper ‘‘An application of a damage constitutive model to concrete at high temperature and prediction of spalling’’ by Rosen Tenchev and Phil Purnell [Int. J. Solids Struct. 42 (26) (2005) 6550–6565] Dariusz Gawin a
a,1
, Francesco Pesavento
b,*
, Bernhard A. Schrefler
b
Department of Building Physics and Building Materials, Technical University of Ło´dz´, Al. Politechniki 6, 93-590 Ło´dz´, Poland b Department of Structural and Transportation Engineering, University of Padua, via Marzolo 9, 35131 Padova, Italy Received 16 March 2006; accepted 13 October 2006 Available online 19 October 2006
Abstract The assumption that pressures of water and gas in concrete at high temperature are equal one to another and its theoretical consequences are discussed. The results of hygro-thermal simulations performed by Tenchev and Purnell are analysed and compared to the Authors’ results based on the assumption about local thermodynamic equilibrium. 2006 Elsevier Ltd. All rights reserved. Keywords: Concrete; High temperature; Pressure
Tenchev and Purnell have extended their finite element model of coupled thermo-hygro-mechanical behaviour of concrete at high temperature (Tenchev et al., 2001) with a new model of damage and then applied it for prediction of concrete spalling. Several experimental and theoretical studies show that such an analysis should be based on possibly reliable data on hygro-thermal state and stresses of concrete, in particular temperature distribution changes, pore pressure build-up, stored elastic energy and principal stresses, see e.g. (Phan et al., 1997; Gawin et al., 2006). Unfortunately, the hygro-thermal model used here is based on the assumption which does not take into account some important features of thermodynamics of capillary-porous media. This influences significantly the results of the presented numerical analyses, as will be explained below. During the development of a mathematical model of hygro-thermo-mechanical phenomena, the number of variables is greater than the number of available equations (Gray and Schrefler, 2001) hence some additional assumptions are necessary. Hypothesis about local thermodynamic equilibrium is commonly used for this pur-
*
Corresponding author. Tel.: +39 049 827 5588; fax: +39 049 827 5604. E-mail addresses: [email protected] (D. Gawin), [email protected] (F. Pesavento), [email protected] (B.A. Schrefler). 1 Tel.: +48 42 6313560; fax: +48 42 6313556. 0020-7683/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2006.10.013
D. Gawin et al. / International Journal of Solids and Structures 44 (2007) 4234–4237
4235
pose what implies that temperatures of all material components at the same ‘‘physical’’ point are equal, and equilibrium on the water–gas interface (curved water meniscus) is described by the Kelvin equation (Gray and Schrefler, 2001). Such an assumption is somewhat disputable for heated concrete, especially for higher heating rates, but it is commonly used in modelling (Phan et al., 1997; Gawin et al., 2003) and showed to give results in good agreement with the experiments, e.g. (Phan et al., 1997; Gawin et al., 2002, 2003, 2006; Witek et al., 2006). But Tenchev and Purnell, instead of this, assumed that gas pressure and liquid pressure are equal, PL = PG (Tenchev et al., 2001), what has several important consequences. For a capillary-porous material, such as concrete, this means that capillary pressure PC = PG PL, equals to zero, what is physically true only at full saturation of pores with liquid water, i.e. when RH = 100%. At normal and elevated temperature conditions relative humidity in concrete is usually lower, hence the corresponding capillary pressures can be considerable, e.g. for RH = 50%: PC = 93.78 MPa at temperature T = 25 C, and PC = 125.77 MPa at T = 100 C. As a result, the gradients of water pressure during drying, especially at high temperatures, are usually much higher (even 2–3 orders of magnitude) than the gradients of gas pressure at the same conditions. Hence, the resulting advection mass flux of water, with the assumption that PL = PG, cannot be properly calculated. In our opinion this effect cannot be correctly accounted for just by assuming that water relative permeability KL = 0.01, as the Authors did in their hygro-thermal model (Tenchev et al., 2001). The water permeability of concrete can in fact be lower than its gas permeability, even 2–3 orders of magnitude, but due to the so called Klinkenberg effect (Bamforth, 1987), which is of importance particularly for materials with a very low intrinsic permeability and exposed to low gas pressures. Another consequence of the aforementioned assumption is that mass fluxes of gas and water are always oriented in the same direction, what is not always the case, as for example in our Fig. 3 (at a distance of 0.13–0.19 m) where the gradients and hence the fluxes direction can be easily deduced. The results of numerical simulations, shown in the paper under discussion in Figures 7(a and c) and 8(a and c), indicate that at temperatures lower than the critical point of water TCR = 374.15 C, gas pressure and liquid pressure are greater than the saturation vapour pressure, due to the assumption that PL = PG. From a P–T state diagram follows that bulk water in these conditions is thermodynamically stable only in liquid phase and it can exist in gaseous phase only for a very short period of time. Hence one should expect that no vapour is present in concrete pores in such conditions and the pores are fully saturated with liquid water. But the results of the aforementioned simulations, after an appropriate elaboration, as described below, show that the relative humidity of concrete was there visibly smaller than 100%RH, what is in contradiction with thermodynamics: compare vapour pressure with vapour pressure at saturation in our Fig. 1. The consequences of the physically inadmissible assumption are also visible in the results of simulations concerning temperature, liquid water content and gas pressure, Figures 7(a–c) and 8(a–c) in the paper under discussion. With these results, it is possible to calculate the partial pressures of vapour and air, PV and PA,
GAS PRESSURE [Pa]
2.0E+06
t= 30 min.
1.5E+06
1.0E+06 Sat. Vap. pressure
5.0E+05
Vap. pressure Air pressure Gas pressure
0.0E+00 323.15 423.15 523.15 623.15 723.15 823.15 923.15 TEMPERATURE [K]
Fig. 1. Space distribution of gas pressures: vapour-, air- and total value at time instant t = 30 min obtained with the numerical model of Tenchev et al. (2001).
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D. Gawin et al. / International Journal of Solids and Structures 44 (2007) 4234–4237
contributing to the total gas pressure, PG. First, using the data about temperature and water content for the sorption isotherms, one can find the relative humidity values which together with the temperatures (and hence the saturation vapour pressures) allows for determining the vapour pressure, PV. The air pressure can be found as PA = PG PV. An example of such an analysis for time instant t = 30 min, concerning the distributions of pressures of vapour, air and gas in function of temperature is shown in our Fig. 1, together with the saturation vapour pressure. As can be seen, in this case the air pressure had a dominant contribution to the total gas pressure, especially in the surface zone with higher temperatures. To analyse better this surprising result we have performed simulations for the same example, but using the mathematical model of Gawin et al. (1999, 2002, 2003) which assumes local thermodynamic equilibrium between water and vapour. We used the few available material properties from Tenchev and Purnell and have completed the set with those of a different concrete. Scope of the exercise is to show that the thermodynamics constrains are respected. The results of our simulations are shown in our Fig. 2 in a similar form as those of Fig. 1. First of all, one can observe considerably lower gas pressures, and a dominant contribution of vapour to the total gas pressure. Then, vapour pressure is greater than zero also in the surface heated zone, what could be expected taking into account considerable vapour mass transport towards the element surface (remind that in Fig. 1 there is no gradient of the vapour pressure close to the surface). The space distributions of water and gas pressures close to the heated surface at time instant t = 10 min, obtained from our simulations are shown in Fig. 3, showing that these not only are not equal, as assumed in paper under discussion, but are of different sign as well (i.e.
t= 30 min.
GAS PRESSURE [MPa]
4.0E+05
3.0E+05 Sat. Vap. pressure Vap. pressure
2.0E+05
Air pressure Gas pressure
1.0E+05
0.0E+00 323.15 423.15 523.15 623.15 723.15 823.15 923.15 TEMPERATURE [K]
Fig. 2. Space distribution of gas pressures: vapour-, air- and total value at time instant t = 30 min obtained with the numerical model of Gawin et al. (1999, 2002, 2003).
0.0E+00
t = 10 min.
2.5E+05
-2.0E+08
2.0E+05
-4.0E+08
1.5E+05
-6.0E+08
1.0E+05
-8.0E+08 gas pressure
5.0E+04
water pressure
0.0E+00 0
WAT. PRESSURE [Pa]
GAS PRESSURE [Pa]
3.0E+05
-1.0E+09
-1.2E+09 0.005 0.01 0.015 0.02 0.025 0.03 DISTANCE [m]
Fig. 3. Space distribution of gas and capillary pressures at time instant t = 10 min obtained with the numerical model of Gawin et al. (1999, 2002, 2003).
D. Gawin et al. / International Journal of Solids and Structures 44 (2007) 4234–4237
4237
capillary water is under traction tension). We believe that all the aforementioned differences between the solutions are caused by lack in the model under discussion of thermodynamic constraints between water and vapour pressures, i.e. the Kelvin equation. Moreover, the model predicts very high gas pressures, which are much higher than those registered for similar concretes during experiments of Kalifa et al. (2000) or Consolazio et al. (1998). The pressure of 6.5 MPa at temperature of about 210 C for concrete with gas permeability of 5 · 10 17 m2, porosity of 8% and initial water content 60 kg/m3, predicted in (Tenchev et al., 2001) with the same model seems to be unrealistic. It is 2 times higher than pressure observed by Consolazio et al. (1998) for an almost fully water saturated mortar of a similar porosity, exposed to a very rapid heating. The pressures reported in the paper under discussion are also very high, taking into account its relatively high permeability and low initial moisture content assumed in the simulations. Moreover, a well-known dependence of water density on temperature is not taken into account in the discussed paper, what is of importance for temperatures exceeding 160 C (Gawin et al., 2002) and can cause a significant reduction of the pore volume occupied by gas, (Kalifa et al., 2000). This effect would cause an additional increase of air pressure, even 2–3 times as shown in (Kalifa et al., 2000). Thus one may conclude that spalling predictions based on the hygro-thermal model used in the paper, due to the controversial assumption not respecting thermodynamics of porous media, can hardly be considered as reliable. References Bamforth, P.B., 1987. The relationship between permeability coefficients for concrete obtained using liquid and gas. Mag. Concrete Res. 39 (138), 3–11. Consolazio, G.R., McVay, M.C., Rish III, J.W., 1998. Measurement and prediction of pore pressures in saturated cement mortar subjected to radiant heating. ACI Mater. J. 95 (5), 526–536. Gawin, D., Majorana, C.E., Schrefler, B.A., 1999. Numerical analysis of hygro-thermic behaviour and damage of concrete at high temperature. Mechan. Cohesive-Frictional Mater. 4, 37–74. Gawin, D., Pesavento, F., Schrefler, B.A., 2002. Modelling of hygro-thermal behaviour and damage of concrete at temperature above critical point of water. Int. J. Numer. Anal. Methods Geomechan. 26 (6), 537–562. Gawin, D., Pesavento, F., Schrefler, B.A., 2003. Modelling of thermo-chemical and mechanical damage of concrete at high temperature. Comput. Methods Appl. Mechan. Eng. 192, 1731–1771. Gawin, D., Pesavento, F., Schrefler, B.A., 2006. Towards prediction of the thermal spalling risk through a multi-phase porous media model of concrete. Comput. Methods Appl. Mechan. Eng. 195, 5707–5729. Gray, W.G., Schrefler, B.A., 2001. Thermodynamic approach to effective stress in partially saturated porous media. Eur. J. Mech. A/ Solids 20, 521–538. Kalifa, P., Menneteau, F.D., Quenard, D., 2000. Spalling and pore pressure in HPC at high temperatures. Cement Concrete Res. 30, 1915– 1927. Phan, L.T., Carino, N.J., Duthinh, D., Garboczi, E. (Eds.), 1997, Proc. Int. Workshop on Fire Performance of High-Strength Concrete, NIST Special Publication 919, Gaitherburg (MD), USA. Tenchev, R.T., Li, L.Y., Purkiss, J.A., 2001. Finite element analysis of coupled heat and moisture transfer in concrete subjected to fire. Num. Heat Transfer, Part A: Appl. 39 (7), 685–710. Witek, A., Gawin, D., Pesavento, F., Schrefler, B.A., 2006. Finite element analysis of various methods for protection of concrete structures against spalling during fire. Comput. Mech. doi:10.1007/s00466-005-0024-7.
Discussion
Comments to the paper ‘‘An application of a damage constitutive model to concrete at high temperature and prediction of spalling’’ by Rosen Tenchev and Phil Purnell [Int. J. Solids Struct. 42 (26) (2005) 6550–6565] Dariusz Gawin a
a,1
, Francesco Pesavento
b,*
, Bernhard A. Schrefler
b
Department of Building Physics and Building Materials, Technical University of Ło´dz´, Al. Politechniki 6, 93-590 Ło´dz´, Poland b Department of Structural and Transportation Engineering, University of Padua, via Marzolo 9, 35131 Padova, Italy Received 16 March 2006; accepted 13 October 2006 Available online 19 October 2006
Abstract The assumption that pressures of water and gas in concrete at high temperature are equal one to another and its theoretical consequences are discussed. The results of hygro-thermal simulations performed by Tenchev and Purnell are analysed and compared to the Authors’ results based on the assumption about local thermodynamic equilibrium. 2006 Elsevier Ltd. All rights reserved. Keywords: Concrete; High temperature; Pressure
Tenchev and Purnell have extended their finite element model of coupled thermo-hygro-mechanical behaviour of concrete at high temperature (Tenchev et al., 2001) with a new model of damage and then applied it for prediction of concrete spalling. Several experimental and theoretical studies show that such an analysis should be based on possibly reliable data on hygro-thermal state and stresses of concrete, in particular temperature distribution changes, pore pressure build-up, stored elastic energy and principal stresses, see e.g. (Phan et al., 1997; Gawin et al., 2006). Unfortunately, the hygro-thermal model used here is based on the assumption which does not take into account some important features of thermodynamics of capillary-porous media. This influences significantly the results of the presented numerical analyses, as will be explained below. During the development of a mathematical model of hygro-thermo-mechanical phenomena, the number of variables is greater than the number of available equations (Gray and Schrefler, 2001) hence some additional assumptions are necessary. Hypothesis about local thermodynamic equilibrium is commonly used for this pur-
*
Corresponding author. Tel.: +39 049 827 5588; fax: +39 049 827 5604. E-mail addresses: [email protected] (D. Gawin), [email protected] (F. Pesavento), [email protected] (B.A. Schrefler). 1 Tel.: +48 42 6313560; fax: +48 42 6313556. 0020-7683/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2006.10.013
D. Gawin et al. / International Journal of Solids and Structures 44 (2007) 4234–4237
4235
pose what implies that temperatures of all material components at the same ‘‘physical’’ point are equal, and equilibrium on the water–gas interface (curved water meniscus) is described by the Kelvin equation (Gray and Schrefler, 2001). Such an assumption is somewhat disputable for heated concrete, especially for higher heating rates, but it is commonly used in modelling (Phan et al., 1997; Gawin et al., 2003) and showed to give results in good agreement with the experiments, e.g. (Phan et al., 1997; Gawin et al., 2002, 2003, 2006; Witek et al., 2006). But Tenchev and Purnell, instead of this, assumed that gas pressure and liquid pressure are equal, PL = PG (Tenchev et al., 2001), what has several important consequences. For a capillary-porous material, such as concrete, this means that capillary pressure PC = PG PL, equals to zero, what is physically true only at full saturation of pores with liquid water, i.e. when RH = 100%. At normal and elevated temperature conditions relative humidity in concrete is usually lower, hence the corresponding capillary pressures can be considerable, e.g. for RH = 50%: PC = 93.78 MPa at temperature T = 25 C, and PC = 125.77 MPa at T = 100 C. As a result, the gradients of water pressure during drying, especially at high temperatures, are usually much higher (even 2–3 orders of magnitude) than the gradients of gas pressure at the same conditions. Hence, the resulting advection mass flux of water, with the assumption that PL = PG, cannot be properly calculated. In our opinion this effect cannot be correctly accounted for just by assuming that water relative permeability KL = 0.01, as the Authors did in their hygro-thermal model (Tenchev et al., 2001). The water permeability of concrete can in fact be lower than its gas permeability, even 2–3 orders of magnitude, but due to the so called Klinkenberg effect (Bamforth, 1987), which is of importance particularly for materials with a very low intrinsic permeability and exposed to low gas pressures. Another consequence of the aforementioned assumption is that mass fluxes of gas and water are always oriented in the same direction, what is not always the case, as for example in our Fig. 3 (at a distance of 0.13–0.19 m) where the gradients and hence the fluxes direction can be easily deduced. The results of numerical simulations, shown in the paper under discussion in Figures 7(a and c) and 8(a and c), indicate that at temperatures lower than the critical point of water TCR = 374.15 C, gas pressure and liquid pressure are greater than the saturation vapour pressure, due to the assumption that PL = PG. From a P–T state diagram follows that bulk water in these conditions is thermodynamically stable only in liquid phase and it can exist in gaseous phase only for a very short period of time. Hence one should expect that no vapour is present in concrete pores in such conditions and the pores are fully saturated with liquid water. But the results of the aforementioned simulations, after an appropriate elaboration, as described below, show that the relative humidity of concrete was there visibly smaller than 100%RH, what is in contradiction with thermodynamics: compare vapour pressure with vapour pressure at saturation in our Fig. 1. The consequences of the physically inadmissible assumption are also visible in the results of simulations concerning temperature, liquid water content and gas pressure, Figures 7(a–c) and 8(a–c) in the paper under discussion. With these results, it is possible to calculate the partial pressures of vapour and air, PV and PA,
GAS PRESSURE [Pa]
2.0E+06
t= 30 min.
1.5E+06
1.0E+06 Sat. Vap. pressure
5.0E+05
Vap. pressure Air pressure Gas pressure
0.0E+00 323.15 423.15 523.15 623.15 723.15 823.15 923.15 TEMPERATURE [K]
Fig. 1. Space distribution of gas pressures: vapour-, air- and total value at time instant t = 30 min obtained with the numerical model of Tenchev et al. (2001).
4236
D. Gawin et al. / International Journal of Solids and Structures 44 (2007) 4234–4237
contributing to the total gas pressure, PG. First, using the data about temperature and water content for the sorption isotherms, one can find the relative humidity values which together with the temperatures (and hence the saturation vapour pressures) allows for determining the vapour pressure, PV. The air pressure can be found as PA = PG PV. An example of such an analysis for time instant t = 30 min, concerning the distributions of pressures of vapour, air and gas in function of temperature is shown in our Fig. 1, together with the saturation vapour pressure. As can be seen, in this case the air pressure had a dominant contribution to the total gas pressure, especially in the surface zone with higher temperatures. To analyse better this surprising result we have performed simulations for the same example, but using the mathematical model of Gawin et al. (1999, 2002, 2003) which assumes local thermodynamic equilibrium between water and vapour. We used the few available material properties from Tenchev and Purnell and have completed the set with those of a different concrete. Scope of the exercise is to show that the thermodynamics constrains are respected. The results of our simulations are shown in our Fig. 2 in a similar form as those of Fig. 1. First of all, one can observe considerably lower gas pressures, and a dominant contribution of vapour to the total gas pressure. Then, vapour pressure is greater than zero also in the surface heated zone, what could be expected taking into account considerable vapour mass transport towards the element surface (remind that in Fig. 1 there is no gradient of the vapour pressure close to the surface). The space distributions of water and gas pressures close to the heated surface at time instant t = 10 min, obtained from our simulations are shown in Fig. 3, showing that these not only are not equal, as assumed in paper under discussion, but are of different sign as well (i.e.
t= 30 min.
GAS PRESSURE [MPa]
4.0E+05
3.0E+05 Sat. Vap. pressure Vap. pressure
2.0E+05
Air pressure Gas pressure
1.0E+05
0.0E+00 323.15 423.15 523.15 623.15 723.15 823.15 923.15 TEMPERATURE [K]
Fig. 2. Space distribution of gas pressures: vapour-, air- and total value at time instant t = 30 min obtained with the numerical model of Gawin et al. (1999, 2002, 2003).
0.0E+00
t = 10 min.
2.5E+05
-2.0E+08
2.0E+05
-4.0E+08
1.5E+05
-6.0E+08
1.0E+05
-8.0E+08 gas pressure
5.0E+04
water pressure
0.0E+00 0
WAT. PRESSURE [Pa]
GAS PRESSURE [Pa]
3.0E+05
-1.0E+09
-1.2E+09 0.005 0.01 0.015 0.02 0.025 0.03 DISTANCE [m]
Fig. 3. Space distribution of gas and capillary pressures at time instant t = 10 min obtained with the numerical model of Gawin et al. (1999, 2002, 2003).
D. Gawin et al. / International Journal of Solids and Structures 44 (2007) 4234–4237
4237
capillary water is under traction tension). We believe that all the aforementioned differences between the solutions are caused by lack in the model under discussion of thermodynamic constraints between water and vapour pressures, i.e. the Kelvin equation. Moreover, the model predicts very high gas pressures, which are much higher than those registered for similar concretes during experiments of Kalifa et al. (2000) or Consolazio et al. (1998). The pressure of 6.5 MPa at temperature of about 210 C for concrete with gas permeability of 5 · 10 17 m2, porosity of 8% and initial water content 60 kg/m3, predicted in (Tenchev et al., 2001) with the same model seems to be unrealistic. It is 2 times higher than pressure observed by Consolazio et al. (1998) for an almost fully water saturated mortar of a similar porosity, exposed to a very rapid heating. The pressures reported in the paper under discussion are also very high, taking into account its relatively high permeability and low initial moisture content assumed in the simulations. Moreover, a well-known dependence of water density on temperature is not taken into account in the discussed paper, what is of importance for temperatures exceeding 160 C (Gawin et al., 2002) and can cause a significant reduction of the pore volume occupied by gas, (Kalifa et al., 2000). This effect would cause an additional increase of air pressure, even 2–3 times as shown in (Kalifa et al., 2000). Thus one may conclude that spalling predictions based on the hygro-thermal model used in the paper, due to the controversial assumption not respecting thermodynamics of porous media, can hardly be considered as reliable. References Bamforth, P.B., 1987. The relationship between permeability coefficients for concrete obtained using liquid and gas. Mag. Concrete Res. 39 (138), 3–11. Consolazio, G.R., McVay, M.C., Rish III, J.W., 1998. Measurement and prediction of pore pressures in saturated cement mortar subjected to radiant heating. ACI Mater. J. 95 (5), 526–536. Gawin, D., Majorana, C.E., Schrefler, B.A., 1999. Numerical analysis of hygro-thermic behaviour and damage of concrete at high temperature. Mechan. Cohesive-Frictional Mater. 4, 37–74. Gawin, D., Pesavento, F., Schrefler, B.A., 2002. Modelling of hygro-thermal behaviour and damage of concrete at temperature above critical point of water. Int. J. Numer. Anal. Methods Geomechan. 26 (6), 537–562. Gawin, D., Pesavento, F., Schrefler, B.A., 2003. Modelling of thermo-chemical and mechanical damage of concrete at high temperature. Comput. Methods Appl. Mechan. Eng. 192, 1731–1771. Gawin, D., Pesavento, F., Schrefler, B.A., 2006. Towards prediction of the thermal spalling risk through a multi-phase porous media model of concrete. Comput. Methods Appl. Mechan. Eng. 195, 5707–5729. Gray, W.G., Schrefler, B.A., 2001. Thermodynamic approach to effective stress in partially saturated porous media. Eur. J. Mech. A/ Solids 20, 521–538. Kalifa, P., Menneteau, F.D., Quenard, D., 2000. Spalling and pore pressure in HPC at high temperatures. Cement Concrete Res. 30, 1915– 1927. Phan, L.T., Carino, N.J., Duthinh, D., Garboczi, E. (Eds.), 1997, Proc. Int. Workshop on Fire Performance of High-Strength Concrete, NIST Special Publication 919, Gaitherburg (MD), USA. Tenchev, R.T., Li, L.Y., Purkiss, J.A., 2001. Finite element analysis of coupled heat and moisture transfer in concrete subjected to fire. Num. Heat Transfer, Part A: Appl. 39 (7), 685–710. Witek, A., Gawin, D., Pesavento, F., Schrefler, B.A., 2006. Finite element analysis of various methods for protection of concrete structures against spalling during fire. Comput. Mech. doi:10.1007/s00466-005-0024-7.