On the modeling of the dehydration induced transient creep during a heating–cooling cycle of concrete

Abstract

A thermo-hydro-mechanical coupled model is presented here in order to simulate the transient creep strain of the load induced thermal strain of concrete during a heating–cooling cycle with a concomitant applied load. In this paper, the transient creep strain is split into a drying creep component and a dehydration creep strain. Further, a dehydration variable is introduced to describe chemical transformations due to the temperature increase. It also allows to govern the occurrence of the transient creep strain. Moreover, in this approach, we consider that the dehydration creep occurs with the same kinetics as the dehydration process. This model has been implemented in the finite element code CAST3M and numerical simulations are performed to assess the capability of the model to predict transient load induced thermal behavior of concrete in the case of a heating cooling cycle.

Résumé

Un modèle thermo-hydro-mécanique couplé est présenté ici afin de simuler la composante transitoire de la déformation thermique induite sous charge au cours d’un cycle de chauffage-refroidissement. Dans cet article, le fluage thermique transitoire est décomposé en fluage de dessiccation et une composante de fluage de déshydratation. En outre, une variable de déshydratation est introduite pour décrire les transformations chimiques en raison de l’augmentation de la température. Il permet aussi de régir l’apparition du fluage thermique transitoire. En outre, dans cette approche, on considère que le fluage de déshydratation se produit avec la même cinétique que le processus de déshydratation. Ce modèle a été implémenté dans le code éléments finis et des simulations numériques sont effectuées afin d’évaluer la capacité du modèle à prédire le comportement thermique transitoire du béton dans le cas d’un cycle de chauffage refroidissement.

Appendices

Appendix

1.1 Porosity

The changes of porosity with the increase of temperature were measured for several types of concrete by [16]. Their results showed that the dependence of porosity on temperature can be approximated for concrete by a linear relationship:

$$ \phi = \phi_{0} + A_{\phi } \left( {T - T_{0} } \right) $$

where $A_{\phi }$ is a constant dependent on the type of concrete. For the experimental data for three types of concrete, the following coefficients of equation have been found by using the least-squares method [16]:

 

	Silicate concrete	Limestone concrete	Basalt concrete
$\phi_{0}$	0.06	0.087	0.0802
$A_{\phi }$ (K−1)	0.000195	0.000163	0.00017

Heat capacity

A temperature dependence of the volumetric heat capacity for the solid skeleton may be approximated by a linear relationship [31]:

$$ \left( {1 - \phi } \right)\rho^{\rm s} C_{\rm ps} = \left( {1 - \phi_{0} } \right)\rho_{0}^{\rm s} C_{\rm ps0} \left[ {1 + A_{\rm c} \left( {T - T_{0} } \right)} \right] $$

where $\left( {1 - \phi_{0} } \right)\rho_{0}^{\rm s} C_{\rm ps0}$ is the volumetric heat capacity of the solid skeleton at the reference temperature T _0, and A _c is a coefficient which takes the effect of temperature on the dry volumetric heat capacity.

The thermal capacity for the constituents gaseous in the case of perfect gas, at constant pressure, would be taken independent of temperature changes for both air and vapour heat capacity $C_{\rm pa} = 1003.5$ and $C_{\rm pv} = 1880.0$ [15]. Based on the results for the thermal capacity of the liquid water given by [32], an approximated formula has been given by [15] as follows:

$$ C_{\rm pl} = 4180 + 300 \cdot \left( {{\frac{T - 273}{T - 715}}} \right)^{2} $$

2.1 Permeability

Gawin et al. [33] proposed the following relationship to describe the compound effect of temperature, gas pressure and material damaging (crack development) on the intrinsic permeability:

$$ K\left( {T,\,\,p,\,\,\phi_{\rm M} } \right) = K_{0} \cdot 10^{{A_{\rm T} \cdot \left( {T - T_{0} } \right)}} \cdot \left( {{\frac{{p^{\rm g} }}{{p_{0}^{\rm g} }}}} \right)^{{A_{\rm p} }} \cdot 10^{{A_{\rm d} \cdot \phi_{\rm M} }} $$

where $K_{0}$ is the initial intrinsic permeability, T ₀ is the ambient temperature, $p_{0}^{\rm g}$ is the atmospheric pressure $A_{\rm T}$, $A_{\rm p}$ are material constants, $A_{{\phi_{\rm M} }}$ is a parameter associated to the damage variable $\phi_{\rm M}$.

Concerning the relative permeability of fluids the following relationship [16] is proposed.

$$ k_{\rm rl} = \left[ {1 + \left( {{\frac{1 - RH}{0.25}}} \right)^{{B_{\rm l} }} } \right]^{ - 1} \,\,\cdot \,\,S^{{A_{\rm l} }} $$

where $A_{\rm l}$, $B_{\rm l}$ are constant with value from the range <1, 3>.

The gas relative permeability of concrete it can be described by the formula given by [34, 35] $k_{\rm rg} = 1 - \left( {{\frac{S}{{S_{\rm cr} }}}} \right)^{{A_{\rm g} }} \quad {\rm for} \quad S < S_{\rm cr}$where S _cr is the critical saturation value, above which there is no gas flow in the medium, A _g is a constant, which usually has value from the range <1, 3>.

Diffusivity

The effective diffusivity is given by the following expression [18]:

$$ D_{\rm eff} \left( {S^{\rm l} } \right) = f_{\rm s} \left( {\phi ,\,\,S^{\rm l} } \right)D_{\rm va} \left( {T,\,\,p^{\rm g} } \right) $$

where the structure factor f _s is given [36, 37]

$$ f_{\rm s} \left( {\phi ,\,\,S^{\rm l} } \right) = \phi_{\rm g}^{4/3} \left( {1 - S^{\rm l} } \right)^{2} = \phi^{4/3} \left( {1 - S^{\rm l} } \right)^{10/3} $$

The diffusivity of vapour in the air at temperature T and pressure p ^g is given in [38]

$$ D_{\rm va} \left( {T,\,p^{\rm g} } \right) = D_{\rm v0} \left( {{\frac{T}{{T_{0} }}}} \right)^{{A_{\rm v} }} {\frac{{p_{0}^{\rm g} }}{{p^{\rm g} }}} $$

where D _v0 = 2.58 × 10⁻⁵ [m² s⁻¹] is the diffusion coefficient of vapour species in the air at the reference temperature T ₀ = 273,15 [K] and pressure $p_{0}^{\rm g}$ = 101325 [Pa] [39]. A _v is a constant, which for the value A _v  = 1.667 gives good correlation with the experimental data concerning vapour diffusion at different temperature [40].

The saturation vapour pressure

The saturation vapour pressure $p^{\rm {vs}} \left(T \right)$, which depends only upon temperature, could be obtained from the empirical correlation. Tabulated results [32] which link the water vapour saturation pressure $p^{\rm vs}$ with temperature T could be approximated by the following formula [41]:

$$ p^{\rm vs} \left( T \right) = \exp \left( {6.4075 + {\frac{16.82669\,T}{228.73733 + T}}} \right) $$

It should be noted that this equation is taken until the critical point of water 647.15 [K] after which it is not possible to distinguish between the liquid and vapour.

Another equation can be taken for $p^{\rm vs}$ after the critical temperatures following the L-function given in [42]:

$$ p^{\rm vs} \left( T \right) = p^{\rm vs} (647.15)\left[ {L_{0} + L_{1} {\frac{T}{647.15}} + L_{2} \left( {{\frac{T}{647.15}}} \right)^{2} } \right] $$

where L ₀ = 15.8568, L ₁  = −34.1706 and L ₂  = 15.7437.

• values of different parameters in Eqs. 12–14

$\lambda_{\text{d0}}$	1.39 [W/(m K)]
$A_{\lambda }$	0.0006 [K−1]
$\phi$	0.1 [−]
$\rho_{{}}^{\rm s}$	2590 [kg/m3]
$\rho^{l}$	999.84 [kg/m3]
T 0	293.15 [K]

$$ A = 0.44\quad {\text{B = 18.6}} \cdot 1 0^{ 6} \quad (18) $$

$$ \beta_{\rm c} = 0.018\left[ {m/s} \right]\quad (49) $$