How Water-Aggregate Interactions Affect Concrete Creep: Multiscale Analysis

Concrete hydration is generally regarded as a process from which the aggregate, being chemically inert, is fully excluded, and which therefore takes place exclusively in the cement paste, where water reacts with cement grains to form hydrates. Correspondingly, concrete hydration models such as the famous Powers–Acker–Hansen model (Powers and Brownyard 1946; Acker and Ulm 2001; Jensen and Hansen 2001) are typically built on evolving volume fractions of cement clinker, water, and hydrates in the cement paste, and, considering the cement paste compartment as a thermodynamically closed system, all these volume fractions can be traced back to the hydration degree and to the (initial) water:cement mass ratio. By contrast, the volume fractions of cement paste and aggregate remain constant at the hierarchical level of concrete. In addition to other applications, such hydration models have been a particularly appropriate basis for the development of multiscale mechanics models for concrete, whether related to elasticity (Bernard et al. 2003; Hellmich and Mang 2005; Sanahuja et al. 2007), poroelasticity (Ulm et al. 2004; Ghabezloo 2010), viscoelasticity (Scheiner and Hellmich 2009), or strength (Pichler and Hellmich 2011; Pichler et al. 2013).

All these models have been experimentally validated to different levels of precision, so that on the one hand, multiscale continuum mechanics has become an accepted theoretical tool in cement and concrete research, yet on the other hand, the field is still open for improvements. The latter is true in particular for the very challenging topic of concrete creep, which spans several orders of time magnitude, from the scale of minutes (Vandamme and Ulm 2009; Delsaute et al. 2012; Boulay et al. 2012; Vandamme and Ulm 2013; Zhang et al. 2014; Irfan-ul-Hassan et al. 2016) to that of several days (Bažant et al. 1976; Tamtsia et al. 2004; Rossi et al. 2012), weeks (Tamtsia and Beaudoin 2000; Laplante 2003; Atrushi 2003; Briffaut et al. 2012), months (Rossi et al. 1994; Zhang et al. 2014), or even years (Bažant et al. 2011, 2012; Zhang et al. 2014).

This paper shows that the challenge in the multiscale modeling of concrete creep probably does not lie so much in finding the appropriate micromechanical representation of the material, but rather in the reliable estimation of the evolving volume fractions of the material constituents, used by the corresponding micromechanics models as input. In this context, the authors (1) abandoned the aforementioned assumption of the cement paste being a thermodynamically closed system, and (2) explicitly introduced water migration from the interaggregate space into the aggregate, as well as back-suction of water from the aggregate into the hydrating (and therefore water-consuming) cement paste.

The paper is organized as follows. A simple mathematical model for water migration into and from the aggregate was formulated in the section on Modeling Hydration-Dependent Water Migration to and from Aggregate. Based on the initial water:cement mass ratio, the hydration degree, and two newly introduced quantities—the water uptake capacity of the aggregate and the water-filling extent of the cement paste voids—this model provides the volume fractions of water, cement clinker, hydrates, and aggregate within concretes and mortars with water-absorbing aggregate. These volume fractions then were used in a micromechanical model for mortar and concrete creep, upscaling cement paste behavior, as quantified in ultrashort-term tests by Irfan-ul-Hassan et al. (2016), to the mortar and concrete level, as detailed in the section “Creep Homogenization of Mortars and Concretes.” Corresponding micromechanical model predictions were then compared with 32 newly performed ultrashort-term creep tests of two different mortars and two different concretes, which were made from the same cement but differ in water:cement and aggregate:cement mass ratios, in the section “Comparison of Ultrashort Creep Experiments and Corresponding Micromechanics Predictions: Identification of Water Absorption Capacities of Quartz Aggregate and of Paste Void-Filling Extent.” It is checked whether this comparison would allow for identification of one value each for (1) the water uptake capacity of aggregate; and (2) the cement-specific void-filling extent by water sucked out from the aggregate. In addition, air entrapment during mixing of concrete is considered.

During the mixing of concrete or mortar, i.e., before the hydration reaction, a significant amount of water may be taken up by the aggregate. This is the case with oven-dried quartz aggregate, which is the focus of this paper. Accordingly, the total mass of water, $w$, is decomposed into that of the water in the cement paste, $w_{cp}$, and that which is absorbed in the open porosity of the quartz, $w_a$During the hydration reaction, however, part of the water which has been initially absorbed into the aggregate is sucked back into the interaggregate space, which is then occupied by the cement paste. This is because the hydration products fill less volume than their unreacted counterparts. The volume reduction during cement paste hydration [also called chemical shrinkage (Powers and Brownyard 1946; Acker and Ulm 2001; Jensen and Hansen 2001)] leads to the formation of air voids, which are partially refilled by the additional water extracted from the aggregate. This process driven by water supply from the aggregate is sometimes called internal curing (Bentz et al. 2005; Jensen and Lura 2006; Wyrzykowski et al. 2011; Zhutovsky and Kovler 2012; Justs et al. 2015). Maintaining the philosophy of the Powers model to identify linear relations between chemical reactants and products on the one hand and the degree of hydration on the other hand, the authors envision that the amount of aggregate-extracted water increases linearly with the volume of voids. The latter increases linearly with the mass of hydrates formed, which then increases, again linearly, with the degree of hydration. Hence the water content in the cement paste is linearly linked to the hydration degree as well, which is mathematically expressed as follows:In this context, $w_{cp}(\xi )$ is the mass of all the water in the cement paste in the most general understanding, i.e., both the unreacted water and that which is chemically combined to the cement clinker; in the same sense, $c$ denotes the total mass of cement, including both the unreacted cement and that which is chemically combined with water. The material constants in Eq. (2), $d$ and $k$, are the initial value of the water:cement mass ratio which is effective at the cement paste level, and the hydration-dependent (linear) increase of this effective mass ratio, respectively. The former constant can be linked to the water mass which is initially taken up by the aggregate, $w_a(0)$. This quantity is normalized by the mass of the aggregate, $a$, yielding the initial water:aggregate mass ratio in the form $w_a(0)/a$. When splitting the (nominal) water:cement mass ratio into a cement paste–specific and an aggregate-specific portion, the respective mathematical expression can be readily solved for $d$, according toTo model the suction of water from the aggregate back to the interaggregate space, which is then occupied by cement paste, cement-specific void-filling extent $\alpha$ between 0 and 1 is introduced, where 0 refers to no water filling the air voids formed during chemical shrinkage of the cement paste and 1 refers to complete filling of the air voids by water. Thereby the voids themselves evolve linearly with the hydration degree (Fig. 1). This void-filling extent $\alpha$ can be related to the back-suction–related parameter $k$ by deriving an expression for the hydration-dependent water mass which was sucked into the cement paste, through combination of Eqs. (2) and (3), yieldingand by expressing this mass as the volume of voids multiplied by the void-filling extent $\alpha$ multiplied by the mass density of water, yieldingTo relate the void volume $V_{void}$ to the initial composition of the cement paste and to the hydration degree, the void volume is considered to be equal to the volume of cement paste, $V_{cp}$, multiplied by the cement paste–related volume fraction of voids, $f_{void}^{cp}$The volume of cement paste is—in good approximation, i.e., without explicit consideration of autogeneous shrinkage—equal to the initial volumes of cement and water, i.e., $V_c$ and $V_w$, which can be expressed by the masses of cement and water as well as by the related mass densities asFinally, the volume fraction of shrinkage-induced voids in cement paste is considered to increase proportionally to the hydration degree, as quantified through the Powers–Acker–Hansen hydration model (Powers and Brownyard 1946; Acker and Ulm 2001; Jensen and Hansen 2001), evaluated for the effective initial composition of the cement paste matrix, quantified in terms of the effective initial water:cement mass fraction $w_{cp}(0)/c$ [cf. Eq. (15.4) of Pichler et al. (2009)]:where ${\rho }_{clin}=3.150\mathrm{kg}/{\mathrm{dm}}^3$ (Acker 2001), ${\rho }_{{\mathrm{H}}_2\mathrm{O}}=1.000\mathrm{kg}/{\mathrm{dm}}^3$, and ${\rho }_{hyd}=2.073\mathrm{kg}/{\mathrm{dm}}^3$ (Barthélémy and Dormieux 2003) denote the mass densities of cement clinker grains, water, and hydrates, respectively. The relation between the constant $k$ and the void-filling extent of water, $\alpha$, follows from specialization of Eq. (5) for Eqs. (6)–(8) and solving the resulting expression for $k$, yieldingEq. (9) highlights the fact that $k$, the hydration degree–related rate of the effective water:cement mass fraction of the cement paste matrix [Eq. (2)], is directly proportional to $\alpha$, the extent to which the shrinkage-induced voids in the cement paste are filled by water.

With respect to the classical Powers–Acker–Hansen model, the hydration model developed herein which also considers internal curing contains two additional quantities: (1) the water uptake capacity of the aggregate, $w_a(0)/a$; and (2) the void-filling extent of water, $\alpha$. The former quantity is involved in the expression for the initial value of the effective water:cement mass fraction [Eq. (3)], whereas $\alpha$ is involved in the mathematical expression for the evolution of the water:cement mass ratio which is effective in the cement paste, and this expression is obtained by specializing Eq. (2) for $d$ and $k$ according to Eqs. (3) and (9), respectively, and from consideration of the mass densities ${\rho }_{clin}=3.150\mathrm{kg}/{\mathrm{dm}}^3$ and ${\rho }_{{\mathrm{H}}_2\mathrm{O}}=1.000\mathrm{kg}/{\mathrm{dm}}^3$, asThis is the water:cement mass ratio which is effective at the cement paste level, and which governs the hydration reaction taking place there. Its initial value needs to be considered when quantifying the volume fractions of cement paste and aggregate in a material volume of concrete or mortar with water-absorbing aggregate; except for the use of this effective water:cement mass ratio, the latter quantification follows the standard relation given by Bernard et al. (2003) and Pichler et al. (2009), which yieldswhere ${\rho }_{agg}=2.648\mathrm{kg}/{\mathrm{dm}}^3$ is the mass density of quartz aggregate in this paper. However, it often occurs during mixing that small amounts of air are entrapped into the cement paste matrix as well. Denoting the corresponding air volume fraction by $f_{air}$, the volume fractions at the concrete or mortar level can be derived from the relationswhich imply thatThe relevance of this new water-migration model and its effect on concrete composition were tested through a creep upscaling analysis from the cement paste level to the concrete or mortar level.

The relevance of the effective water:cement mass ratio according to Eq. (10) and of the cement paste and aggregate volume fractions according to Eqs. (11) and (13), both depending on the void-filling extent $\alpha$ and the water uptake capacity of the aggregate, $w_a(0)/a$, checked by using Eqs. (10), (11), and (13) and the quantities appearing therein within a creep upscaling analysis from the cement paste to the mortar and concrete level.

The purpose of this was to determine whether the experimental results of numerous creep tests performed at the level of cement paste made from only one type of cement, and at the level of mortars and concretes made from the same type of cement but with two different types of aggregates, can be predicted by a micromechanical model which is based on only one value for the cement-specific void-filling capacity $\alpha$ and of only two aggregate-specific values for the water-absorption capacity $w_a(0)/a$.

The creep tests considered in this context all followed the protocol reported by Irfan-ul-Hassan et al. (2016). Accordingly, 3-min creep tests on the same cement paste, mortar, or concrete samples were repeated hourly. The key idea behind this protocol is that 3 min is short enough for the microstructure to remain practically the same throughout each individual test. Within 1 h, on the other hand, the hydration process in early-age cementitious systems continues in a significant manner, so that two subsequent 3-min creep tests involve remarkably different microstructures. In addition, 3 min is so short and the load levels used are so small that creep strains are—up to the very fine measurement accuracy of the displacement sensors used—fully recovered during the waiting period following each test (Irfan-ul-Hassan et al. 2016). Therefore the individual 3-min tests can be analyzed as independent experiments, i.e., there is no need to consider the entire loading history. Hence an upscaling analysis concerning cement paste, mortar, and concrete samples tested according to the aforementioned protocol can be performed in the theoretical framework of classical, nonaging microviscoelasticity (Read 1950; Sips 1951; Laws and McLaughlin 1978; Beurthey and Zaoui 2000).

Choosing, in this context, a standard micromechanical representation for mortar and concrete (Scheiner and Hellmich 2009; Baweja et al. 1998; Bernard et al. 2003; Hellmich and Mang 2005), namely that of a composite material consisting of a (viscoelastic) cement paste matrix with (elastic) aggregate inclusions and (potentially occurring) air inclusions (Fig. 2) the (homogenized) relaxation tensor at the concrete/mortar scale, ${\mathbb{R}}_{hom}$, follows from those at the cement paste scale, ${\mathbb{R}}_{cp}$, as well as from the volume fractions of cement paste, aggregate, and (potentially occurring) air, as (Scheiner and Hellmich 2009)where * indicates Laplace–Carson (LC) transforms of the originally time-dependent quantities occurring in the standard convolution integrals of linear viscoelasticity [correspondence principle; (Read 1950; Sips 1951; Laws and McLaughlin 1978; Beurthey and Zaoui 2000)]and back-transformation of Eq. (14) from the Laplace–Carson domain to the time domain may be performed by the Gaver–Wynn–Rho algorithm (Scheiner and Hellmich 2009; Gaver 1966). The Appendix provides mathematical details of the LC-transformed homogenized bulk and shear moduli, $k_{hom}^{*}$ and ${\mu }_{hom}^{*}$; the fourth-order unity tensor $\mathbb{I}$ with its volumetric and deviatoric parts, ${\mathbb{I}}_{vol}$ and ${\mathbb{I}}_{dev}$; and the morphology tensor ${\mathbb{P}}_{sph}^{*}$. The relaxation tensors ${\mathbb{R}}_{cp}^{*}$ correspond to a power law–type creep behavior characterized by an elastic modulus $E_{cp}$, a Poisson’s ratio ${\nu }_{cp}$, a creep modulus $E_{c,cp}$, and a creep exponent ${\beta }_{cp}$

The aforementioned material characteristics at the cement-paste level all depend on the (here, effective) water:cement mass ratio and the hydration degree, as identified in the more than 500 creep tests on cement paste reported by Irfan-ul-Hassan et al. (2016) (Fig. 3). Quadratic interpolation between experimental data was used to consider the (effective) water:cement mass ratios between those which were explicitly tested (Fig. 4). This allow for remaining as close as possible to the measured properties, thereby avoiding the (small) uncertainties resulting from the application of available multiscale models. However, this was not possible for the quantification of Poisson’s ratio, which was not measured experimentally. Therefore a validated multiscale model (Pichler et al. 2008; Pichler and Hellmich 2011) was used to establish a relationship between the Poisson’s ratio and the elastic modulus (Fig. 5).

This study is devoted to aggregate consisting of quartz, with elastic bulk and shear moduli of Bass (2013)Air inclusions, if they exist, do not exhibit a solid elastic stiffness

This paper extended the Powers–Acker–Hansen hydration model to consider water migration including (1) the initial water uptake by the aggregate and (2) back-suction of this water to the cement paste matrix during hydration. The following discussion refers to phase volume evolutions predicted by the extended hydration model, to the self dessication–driven desaturation of cement pastes with and without water migration, and to the driving force for water migration from the aggregate to the cement paste matrix. The section closes with a future outlook.

This paper presented an extension of the classical Powers hydration model with respect to internal curing, and checked the relevance of the latter through micromechanical upscaling of effective water:cement mass ratio–dependent cement paste creep functions, up to the levels of mortar/concrete. Remarkably, internal curing can be considered in terms of only two additional quantities: an aggregate-specific uptake capacity, and a cement paste–specific void-filling extent. Identifying these quantities for two types of oven-dried quartz aggregate and for one type of cement allowed for satisfactory prediction of numerous ultrashort-term creep tests on two mortars and two concretes (Figs. 8–11). Such creep tests directly deliver the hydration-dependent (nonaging) creep properties, and are also valid for medium-term creep tests on very old pastes (Irfan-ul-Hassan et al. 2016). Neglecting internal curing effects, i.e., initial water uptake through the aggregate followed by back-suction of this water from the aggregate domain to that of the maturing cement paste, clearly does not allow for satisfactory micromechanical prediction of the creep properties at the mortar and concrete level, as is quantified in Table 2 and illustrated in Fig. 15, which shows predictions based on $w_a(0)/a=\alpha =f_{air}=0$ while keeping all other input variables as defined in this paper. Obviously, the creep response predicted by the micromechanical model (Fig. 2) would be too soft under these conditions. It is also illustrative to quantify the degree of hydration when the back-suction of water from aggregate to the cement paste is complete, simply because no water is left in the aggregate. To this end, $w_{cp}(\xi )/c$ in Eq. (10) is set equal to $w/c$ and the resulting expression is solved for hydration degree $\xi$Notably, the creep tests analyzed herein refer to hydration degrees smaller than ${\xi }^{*}$ [Figs. 8(b), 9(b), 10(b), and 11(b)].

The method described herein, showing how to integrate internal curing events into micromechanical modeling of concrete, in particular concerning creep, can be straightforwardly extended to aging creep behavior based on earlier contributions such as those by Scheiner and Hellmich (2009) or Sanahuja (2013).

Another obvious extension concerns the deeper reasons for the dependencies of the creep properties of cement paste on the water:cement mass ratio (Fig. 4), which may be deciphered through micromechanical resolution to the level of the hydrates (Königsberger et al. 2016), or even to the level of lubricating water layers between calcium silicate sheets (Pellenq et al. 2009; Sanahuja and Dormieux 2010; Shahidi et al. 2014, 2016a, b; Vandamme et al. 2015).

Upscaling of the creep behavior to the larger scales of mortar or concrete was performed in the LC space, according to the analytical formulae described in this appendix. An isotropic fourth-order tensor, $\mathbb{G}$, can be decomposed into a volumetric part and a deviatoric part as $\mathbb{G}=G^{vol}{\mathbb{I}}^{vol}+G^{dev}{\mathbb{I}}^{dev}$, where $G^{vol}$ and $G^{dev}$, respectively, are the (scalar) volumetric and deviatoric components of the tensor, and ${\mathbb{I}}_{vol}$ and ${\mathbb{I}}_{dev}$ are the volumetric and deviatoric parts of the fourth-order identity tensor $\mathbb{I}$, defined as $I_{ijkl}=1/2({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk})$, ${\mathbb{I}}_{vol}=1/3(\mathbf{1}\otimes \mathbf{1})$, and ${\mathbb{I}}_{dev}=\mathbb{I}-{\mathbb{I}}_{vol}$, respectively, where 1 denotes the second-order identity tensor with components equal to the Kronecker delta ${\delta }_{ij}$, namely ${\delta }_{ij}=1$ for $i=j$, and 0 otherwise. They satisfy ${\mathbb{I}}^{vol}:{\mathbb{I}}^{vol}={\mathbb{I}}^{vol}$, ${\mathbb{I}}^{dev}:{\mathbb{I}}^{dev}={\mathbb{I}}^{dev}$, and ${\mathbb{I}}^{vol}:{\mathbb{I}}^{dev}={\mathbb{I}}^{dev}:{\mathbb{I}}^{vol}=0$.

The collection of analytical formulas is started with the LC-transformed Hill tensors for spherical inclusions embedded in an infinite cement matrix with quasi-elastic stiffness ${\mathbb{R}}_{cp}^{*}$, occurring in concentration and stiffness expressions of Eq. (14). The Hill tensor iswhere ${\mathbb{S}}_{sph}^{*}$ = LC-transformed Eshelby tensor of a spherical inclusion embedded in an infinite cement paste matrix. The LC-transformed Eshelby tensor ${\mathbb{S}}_{sph}^{*}$ is isotropic, and its volumetric and deviatoric components are (Zaoui 2002; Hellmich et al. 2004)

Next, the expressions for the homogenized quasi-elastic stiffness tensor ${\mathbb{R}}_{hom}^{*}$ are discussed. For mortar or concrete, insertion of the LC-transformed Eshelby tensor expressions [Eq. (52)] into Eq. (51), and further insertion of the obtained Hill tensor, together with the vanishing quasi-elastic stiffnesses of air and the available quasi-elastic stiffness of quartz [Eq. (16)], into the expression for the quasi-elastic stiffness of the homogenized mortar or concrete [Eq. (14)], yields scalar expressions for the LC-transformed bulk and shear moduliwhere $A_{\infty ,agg}^{*,vol}$, $A_{\infty ,agg}^{*,dev}$, $A_{\infty ,air}^{*,vol}$, and $A_{\infty ,air}^{*,dev}$ are denoting LC-transformed volumetric and deviatoric components of the Eshelby problem–related strain concentration tensors for quartz aggregate and air. For air, the volumetric and deviatoric components of the strain concentration tensor can be written as

For quartz, these components are

The authors cordially thank for valuable help the laboratory staff of the Institute of Mechanics of Materials and Structures, TU Wien—Vienna University of Technology. The first author also thanks the Higher Education Commission (HEC) Pakistan and the University of Engineering and Technology, Lahore, Pakistan, for their support.