Experimental and numerical study of size effect on long-term drying behavior of concrete: influence of drying depth

Abstract

This study aims at rationalizing the analysis of drying shrinkage tests and taking better advantage of the measurements, by studying the influence of the specimen size. Three self-consolidating concrete and one vibrated concrete mixes were studied during 3 years. Drying started 24 h after casting. The tests were carried on three sizes of cylinders: Φ78, Φ113 and Φ163 mm, under the same experimental conditions (20 ± 1 °C, 50 ± 5 % relative humidity), according to RILEM recommendations. The results show that the classification of concrete depends not only on the experimental conditions, but also on the duration of the shrinkage measurement. A scientific approach based on a simple mathematical model was proposed to analyze shrinkage data. The ultimate drying shrinkage did not depend on specimen size. The existing empirical models of shrinkage and drying used in construction codes take into account the size effect only through a geometrical parameter called notional size of cross-section, assuming that concrete is an homogeneous and non-aging material. With the time variable change by the ratio of the square root of the time and the notional size of cross-section, a master curve can be found for the theoretical curves of different sizes. However, experimentally, upshifts were observed between the drying shrinkage curves for different specimens sizes, thus phenomena that occur at early age are not taken into account by current models. The drying depth notion was introduced to explain the part of drying behavior responsible for the observed difference between drying curves. The influence of this concrete layer on the long-term behavior was confirmed by a two-step modeling of the mass-loss evolution. The first one included coupled drying-hydration model. The same classification was observed numerically and experimentally, with higher drying kinetics corresponding to smaller specimens, but the experimental shift could not be reproduced. The drying depth notion was incorporated in the second model; an increase in permeability was introduced in the outer layer of the concrete specimens. The size effect on the long-term mass-loss was reproduced numerically.

Appendices

Appendix 1: Hydration part

Hydration appears in the sinks term of the mass balance equations written for the liquid water phase and can be expressed as a quantity that is dependent of hydration degree for the water component [47]:

$$q = \frac{{{\text{d}}\alpha }}{{{\text{d}}t}}\lambda C$$

(19)

where C is the initial cement content [kg/m³] and the consumed water mass per mass of hydrated cement [47]. The time derivative of the hydration degree can be written as follows:

$$\frac{d\alpha }{dt} = \frac{3D}{{R^{2} }}\frac{{\left( {1 - \alpha } \right)^{2/3} }}{{\left( {1 - \alpha^{D} } \right)^{1/3} - \left( {1 - \alpha } \right)^{1/3} }}H\left( {\phi_{l} - \phi_{l}^{cr} } \right)\frac{{\phi_{l} - \phi_{l}^{cr} }}{{\phi_{l}^{0} - \phi_{l}^{cr} }}$$

(20)

where $D,R,\alpha^{D} ,\phi_{l} ,\phi_{l}^{0} ,\phi_{l}^{cr}$ denote the diffusion coefficient of ionic species through the hydrated coating around the clinker grains, the mean radius of the initial cement grains, critical hydration degree, moisture content, initial moisture content and critical moisture content that corresponds to the lower limit under which hydration stop, respectively denotes the Heavyside function.

Porosity was actualized as follows (21):

$$\varPhi = \varPhi_{0} \left( {1 - \alpha \frac{\mu C}{E}} \right)$$

(21)

With μ, C and E the quantity of water consumed by weight of cement, the initial cement and water contents, respectively is the initial porosity (after mixing).

Finally, the actualization of the intrinsic permeability reads:

$$k_{l} = {k_{l}}_{0} \left( {\frac{\varPhi }{{\varPhi_{0} }}} \right)^{2} \left( {\frac{{1 - \varPhi_{0} }}{1 - \varPhi }} \right)^{2}$$

(22)

The relative permeabilities are assumed to be independent of the hydration degree [47].

Appendix 2: numerical procedure

In the model proposed, drying and hydration are treated in two different parts. The evolution of hydration has been determined using the programming language Python based on the work of Nguyen et al. [40, 47]. This choice has been motivated by the fact that Python ensures the link between hydration and transport programs. Drying is calculated with PyTOUGH [52] that is a Python application programming interface for TOUGH2 [53], a simulator for nonisothermal multiphase flow in fractured porous media. The numerical treatment of the simultaneous processes of phase flow and transport and hydration was handled using a sequential non-iterative split operator. Hence, for each time step defined by user the drying process was simulated. Then, the hydration program before a new drying process calculation actualized the liquid saturation and the transport properties.

For the resolution, hydration degree is calculated at time step n whereas mass and fluxes are calculated at time n + 1 (see Eq. (23)). Hence, no iteration scheme is used, leading to faster simulations.

$$\left( {\frac{d}{dt}M_{i} } \right)^{n + 1}\, = \left( {\Delta F^{i} } \right)^{n + 1}\, +\, \left( {q^{i} } \right)^{n}$$

(23)

Equation (20) is nonlinear because of the presence of the hydration degree in the right part of the equation and because the moisture content depends on the hydration degree. This ordinary differential equation was then solved by an iterative procedure until convergence.

A 1-D radially symmetric geometry is used to model the samples. A simple mesh composed of one 60-element row was used and represented the radius of the samples. The time step was 24 h because simulations were not sensible to the time step.

Appendix 3: computation of the skin’s initial porosity

Initial porosities ϕ _i were deduced from experimental determination of δ (see Table 4) by assuming the hydration does not significantly evolve in this layer where the porosity remains constant. The drying depth δ corresponds to the initial mass loss Δm ₀ for which shrinkage starts. If V _eva is the evaporated water and V _dry is the drying volume of concrete corresponding to the drying depth δ we can write (24):

$$\phi_{i} \cdot v_{dry} = v_{eva}$$

(24)

Developing Eq. (24) for a cylindrical specimen reads:

$$\left( {\pi R^{2} - \pi \left( {R - \delta } \right)} \right)^{2} h = \frac{{\Delta m_{0} }}{{\phi_{i} \rho_{e} }}$$

(25)

$\rho_{e}$: Water density (kg/m³)h The height of the specimen’s mass-loss measurement (m)

The initial porosity reads:

$$\varPhi_{i} = \frac{{\Delta m_{0} }}{{\rho_{e} \left( {\pi R^{2} - \pi \left( {R - \delta } \right)^{2} } \right)h}}$$

(26)

The gathers the values of initial porosity. The calculated porosity was based on the estimation of the drying capillary pores. Saturated micropores were not taken into account in Eq. (26). At 50 % of relative humidity, the corresponding pores radius given by Kelvin Laplace equation considering the adsorbed water is 2 nm [67]. Below this value, the pores were saturated. This fraction of concrete porosity was assumed to have no significant influence on the long-term drying behavior under our experimental conditions.

The porosity of the drying depth depends clearly on the concrete mixture. For instance the porosity of SCC-N was higher than the porosity of other concretes, and it concerned a deeper layer. The SCC-N concrete actually had higher proportion of limestone filler addition and lower cement content thus high free water content. The VC concrete showed lower initial porosity in spite of relatively high water-to-cement ratio, because of lower volume of paste.