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On the behaviour of concrete at early-ages: A multiphase description of hygro-thermo-chemo-mechanical properties
Cement and Concrete Research 116 (2019) 202–216
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Cement and Concrete Research journal homepage: www.elsevier.com/locate/cemconres
On the behaviour of concrete at early-ages: A multiphase description of hygro-thermo-chemo-mechanical properties
T
Tobias Gasch , Daniel Eriksson, Anders Ansell ⁎
KTH Royal Institute of Technology, Department of Civil and Architectural Engineering, Stockholm SE-100 44, Sweden
ARTICLE INFO
ABSTRACT
Keywords: Hydration (A) Microstructure (B) Creep (C) Shrinkage (C) Modelling (E)
Understanding the early-age behaviour of concrete is of importance for designing durable concrete structures. To contribute to the improvement of this, a hygro-thermo-chemo-mechanical model is presented that accounts for phenomena such as hydration, external and internal drying, self-heating, creep, shrinkage and fracture. The model is based on a multiphase porous media framework, using the Thermodynamically Constrained Averaging Theory (TCAT) as starting point to derive the governing equations of the system. This allows for a systematic treatment of the multiscale properties of concrete and how these develop during hydration, e.g. chemical and physical fixation of water. The proposed mathematical model is implemented within the context of the Finite Element Method (FEM), where all physical fields are solved in a fully-coupled manner. Chosen properties of the model are demonstrated and validated using three experimental results from the literature. Generally, the simulated results are in good agreement with the measurements.
1. Introduction Durability is one of the most important issues that have to be accounted for, in both the design of new constructions and in assessments of existing concrete structures. This is in turn, to a large extent, dictated by the ability of the concrete to withstand various deterioration processes such as alkali-aggregate reactions, chloride ingression, leaching and frost damage. The resistance to most of these processes is closely related to the characteristics and evolution of the porous cement paste, from casting and throughout the service life of the structure. Thus, understanding the early-age behaviour of concrete is of major importance, since it is during this period that many properties of the hardened cement paste are determined. If not cured properly, the porosity of concrete can become larger than desired, and it may exhibit significant cracking; this increases the permeability and consequently the transport of moisture, which often has a negative effect on the durability. There is an extensive base of knowledge on the early-age behaviour of concrete available in the literature, including both experimental and theoretical studies on topics such as self-heating, self-desiccation and the chemical processes involved in the hardening [1-7]. Over the years, it has become evident that to accurately predict and understand such processes there is a need to develop advanced modelling tools, which consider not only individual properties of concrete but many of the
⁎
observed phenomena in a single framework since they all interact. To this end, the interaction between physical phenomena such as temperature and mechanical stress has been widely analysed in the literature, also including the early-age behaviour [8-12]. More recently, models that also consider the effects of moisture transport and how this affects both stresses and the chemical reactions have been suggested [13-18]. However, one of the most interesting developments is the adoption of multiphase porous media theory [19] to early-age concrete by Gawin et al. [20,21] and succeeding developments on a similar framework by for example Sciumé et al. [22] and Jefferson et al. [23]. This approach allows for a more physically accurate description of the complex processes involved in the hardening of cement, their interaction and the development of the microstructure. This study aims to develop and present a three-phase porous media model for the hygro-thermo-chemo-mechanical behaviour of concrete from early-ages and beyond. It follows in the line of the works by Gawin et al. [20,21] and Sciumé et al. [22] among others, using the Thermodynamically Constrained Averaging Theory (TCAT) developed by Gray and Miller [24,25] as a starting point. While the developed model shares many features and utilizes model equations on a similar form as previous works, it aims to improve and modify certain aspects with regard to early-age concrete. This includes the overall description of the multiphase system and its development with the cement hydration. Furthermore, new and modified equations are proposed for the
Corresponding author. E-mail addresses: [email protected] (T. Gasch), [email protected] (D. Eriksson), [email protected] (A. Ansell).
https://doi.org/10.1016/j.cemconres.2018.09.009 Received 15 November 2017; Received in revised form 14 September 2018; Accepted 21 September 2018 Available online 06 December 2018 0008-8846/ © 2018 Elsevier Ltd. All rights reserved.
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Level III > 10-3 m Concrete / mortar
Level II < 10-4 m Cement paste
Level “0” C-S-H solid 10-8 -10-9 m
Level I < 10-6 m C-S-H gel
Macro pore
Aggregate
Adsorbed water Aggregate
Intra C-S-H water Inter-layer water Chemically bound water
Fine aggregates
Unhydrated clinker
CH + other hydration products
Small gel pore 1-3 nm
Large gel pore 3-12 nm
Bulk water
Fig. 1. Schematic multiscale description of concrete based on the conceptual model by Jennings et al. [30].
definition by Powers and Brownyard [1], the gel porosity only includes pores smaller than approximately 3 nm and would thus only include the SGP in Fig. 1. Concerning the state of water in these different parts of the pore space, Ulm et al. [27] pointed out that it is only water in pores outside the C-S-H particles that should be considered to behave as a fluid using a similar categorization as Jennings [28]. Water held in smaller pores should be considered as structural water; not to be confused with the chemically bound water. With this description in mind, the solid phase s of the porous media is in this study considered to consist of aggregates A, anhydrous cement C and hydration products H as well as a gel pore space G partially filled with structural water W. For simplicity, the definition of Powers and Brownyard is used for the gel porosity, which thus includes the pore and inter-layer space intrinsic to the C-S-H solids as well as the SGP, while larger pores are considered part of the capillary pore space. For use in forthcoming Section 3, where model equations of the porous media are presented, this means that water held in the gel pores (< 3 nm) will be considered as structural water, and therefore as a part of the solid phase; although this includes water in the SGP as opposed to the definition by Ulm et al. [27]. Both fine and coarse aggregates are considered parts of species A and the hydration products H include both C-S-H as well as for example ettringite and portlandite. As will be described in the following, the volume fractions of these components are considered to change with the cement hydration. The capillary pore space makes out the remaining part of the porous media, i.e. the LGP and macro pores. It is considered to be occupied by two fluid phases, a wetting phase w of bulk water W and a non-wetting gaseous phase g, which is an ideal gas mixture of dry air D and water vapour W.
constitutive and physical behaviour of the porous media to account for the complex features of early-age concrete, including the ageing of material properties, creep, shrinkage and fracture. 2. Characterization of multiphase porous media for early-age concrete In porous media theory, concrete is normally considered a threephase material consisting of two fluid phases and a homogenous solid phase [26]. However, theoretical and experimental investigations of cement paste at lower scales point to a much more complex picture [4,6,27-29]. Considering this, there is a need to revisit and improve the simplified description of in particular the solid phase used in most models to better reflect the true material, also in models intended for analysis at larger scales. This is especially important for early-age applications, and likely also in other applications related to durability. Following the multiscale description of concrete and cement paste by Ulm et al. [27], Jennings et al. [30] and Jennings [28], a schematic description of concrete at different scales is shown in Fig. 1. Starting at the level of concrete and mortar (Level III), coarse and fine aggregates are embedded in a porous cement paste. This cement paste (Level II) is, in turn, made up of macro pores, unhydrated clinker plus portlandite (CH) and other hydration products embedded in a calcium-silicate-hydrate (C-S-H) gel. On the level of the C-S-H gel (Level I), a description following the colloidal model (CM-II) by Jennings [28] is used in Fig. 1. However, for the purpose of the current study, this choice has no major impact and other descriptions of the C-S-H gel, such as that by Feldman and Sereda [31], would be equally valid. In Jennings model, the gel is thought to be made up of C-S-H solids that are particles with an internal structure and pore space referred to as globules and shown in Level “0”. These consist of sheets of reacted clinker and water in different forms as seen in Fig. 1. The structure of the gel is formed by the packing of such globules in different arrangements, leading to two distinct inter-particle pore spaces; small gel pores (SGP) and large gel pores (LGP). Considering that the purpose of this study is to formulate a mathematical model that includes the transport of water and conversion of water between different forms, a discussion of different parts of the pore space in relation to this is warranted. Categorization of the pore space in concrete is difficult and, as pointed out by Jennings [28], several definitions of the gel porosity (and consequently also the capillary porosity) can be found in the literature. With the above description of the gel by Jennings [28], pores smaller than 12 nm are considered intrinsic to the gel. However, according to the classical
2.1. Cement hydration Hydration of Portland cement consists of a set of complex chemical and physical phenomena at the microscopic level. The chemical model employed herein follows the work of Ulm and Coussy [8,32], in which this set of reactions are lumped into a single reaction described by for example the hydrate mass mhyd, which denotes the mass of chemically bound water per unit volume. For modelling purposes it is, however, more convenient to introduce the hydration degree C = mhyd / mhyd as a state variable, which is the amount of hydrate mass normalized with the hydrate mass at complete hydration mhyd . The model furthermore includes the later developments by Cervera et al. [9] and Di Luzio and Cusatis [13]. The kinetics of hydration is governed by the diffusion of free water 203
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through the hydrates that form in layers around the cement grains on the microscopic level. Acknowledging the well-known fact that hydration is thermally activated, an Arrhenius type evolution law is formulated for ξC following Ulm and Coussy [32] such that
d dt
C
= AC ( C )
w
QC , RT
(s w ) exp
Hs
Ac2 C
+
C
(
Gs
(1)
C ) exp
c
C
,
(2)
2.2. Evolution of the solid phase In earlier adoptions of porous media theory to the early-age behaviour of concrete, the solid constituents have not been considered to be dependent on the hydration process [20-22]. Herein, this is explicitly accounted for in the model equations. The development of each constituent here follows that classical model by Powers and Brownyard [1] in that their volume fractions are linearly related to the extent of cement hydration. However, other more recent models could also be considered, for example the extension of the Powers and Brownyard model to include silica fume by Jensen and Hansen [38] or the model by Königsberger et al. [39] in which the assumption of linearity is abandoned. As will be evident later in Section 3, it will be necessary to define the volume fractions of the different constituents with respect to the solid phase to close the mathematical model. Hence, volume fractions will be referred to the total volume of the solid phase, not the total volume as is usually done [1,5,34]. Following the nomenclature used in TCAT [24,25], the sought volume fractions are referred to as is where i ∈ (A, C, H, G). Using the model of Powers and Brownyard [1], we can define these as
= A
Cs
= C
a/c A
a/c A
+
+
(
(
a/ c knw N
+
N
W
)
C
+
1
+
kgw W
)
C
+
1 C
knw N
N
a/c A
+
(
C
kgw
+
C
W
kgw
=
) )
knw
1
, (6)
C
C
+
N
+
kgw W
)
C
+
1
. (7)
C
3.1. Conservation of mass The considered porous media involves a total of seven mass balances, one for each species in each phase. However, as is usually done for concrete, these are arranged to reduce the number of mass balances in the final model formulation to three: one for the solid species (A, C, H), one for water W and one for dry air D [26]. Starting with the mass balance of all solid species, it is arranged to express the change of the total solid volume fraction
,
t
(4)
C
C
1 knw
kgw
knw
+
In this work a mechanistic approach [40] is adopted to derive the general model equations of the porous media by considering the conservation of its mass, momentum and energy. This is done with the Thermodynamically Constrained Averaging Theory (TCAT) developed by Gray and Miller [24,25] as a starting point and follows a similar procedure as in previous work on adapting multiphase porous media theory to early-age concrete, see for example the works by Gawin et al. [20,21] and Sciumé [41]. For the sake of brevity, only the final form of the macroscopic conservation equations used in the model definition are presented in the following. As is usually done in multiphase descriptions of concrete, see for example [26], it is here considered to be a three-phase material consisting of a solid phase s, a wetting phase w and a non-wetting gaseous phase g. The composition of each phase was discussed in Section 2, where especially the break down of s into species A, C and H as well as a gel pore space will have an impact on the general model equation if compared to existing models in the literature. It must also be pointed out that an assumption of small displacements of the solid is made throughout this study. Consequently, no difference is made between the spatial and material reference frames and all equations are, hence, written in their partial derivative form.
where aw is a model parameter. It was found by Norling-Mjörnell that aw = 4 gave good agreement with experimentally evaluated values of βw and is thus be used throughout this study.
As
C
3. Conservation equations for early-age concrete
(3)
(s w ) = ( s w ) a w ,
1
In Eq. (4) to Eq. (7), the term within brackets in the denominator is the total volume of the solid phase at a certain degree of hydration. The main parameters that enter the equations are the aggregate-to-cement ratio a/c, the mass of chemically bound water per mass hydrated cement knw (≈ 0.23 g/g [34]) and the mass of water held in the gel pores per mass hydrated cement kgw (≈ 0.19 g/g [34]). Furthermore, ρi is the intrinsic density of species i, but it should be noticed that ρN is the density of the chemically bound water, here set to 1350 kg/m3 [34]. From Eq. (4) to Eq. (7) it is clear that the total volume of the solid phase increases with the hydration degree ξC. However, since ρW/ρN < 1, the solid phase occupies a smaller volume than that of the reactants, i.e. the proposed microstructural model directly accounts for the chemical shrinkage associated with cement hydration. Lastly, it should be pointed out that the w/c ratio is used to define the initial volume occupied by the solid phase in the multiphase system, but does not directly affect the composition of the solid phase.
where Ac1, Ac2 and ηc are model parameters controlling the reaction rate. The asymptotic degree of hydration C describes the fraction of the cement available for hydration. Its value depends highly on the waterto-cement ratio w/c and is here estimated using the empirical formula by Pantazopoulou and Mills [33]. The thermal activation of the reaction is controlled by the Arrhenius term, where QC is the hydration activation energy that is normally close to 500 kJ/kg, T is the temperature and R is the universal gas constant. The influence of moisture saturation on the cement reaction is well-known, where the reactions effectively cease at relative humidities below approximately 0.75 [1,5,34,35]. To account for this in a simple manner, an empirical factor βw is introduced in Eq. (1). Different forms have been proposed for βw, most often depending on the pore relative humidity [13,20,35,36]. However, it is here recognized that hydrates mainly precipitate in the water-filled parts of the capillary pore space (pores > 3 nm), and βw is thus assumed to be a function of the degree of capillary saturation s w . An empirical expression proposed by Norling-Mjörnell [37] is used w
a/c
W
C
C
( +(
A
where the normalized affinity AC controls the rate of the reaction. The affinity is here defined using the analytical function proposed by Cervera et al. [9]
AC ( C ) = A c1
=
s
where
,
Ss
(5)
1
=
=
s
Ss
vs
t
Ss
+
(
s Ss v s)
+ k nw m 0C
d dt
C
,
(8)
is the velocity of the solid phase and
A As
+
C Cs
+
H Hs .
(9)
It should be mentioned that all intra phase motions of species are 204
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Table 1 Physical properties of species in the multiphase model used in the current study.
Liquid water Gas mixture Aggregates (granite) Anhydrous cement Hydration products
Species i
Phase α
Thermal conductivity λi
Thermal expansion αi
Reference density
Cpia
Specific heat
Bulk modulus Ki
W W+D A C H
w,s g s s s
4181 J/(kg⋅K) 1005 J/(kg⋅K) 790 J/(kg⋅K) 840 J/(kg⋅K) 1550 J/(kg⋅K)
0.591 W(m⋅K) 0.025 W(m⋅K) 2.5 W(m⋅K) 0.29 W(m⋅K) 1.8 W(m⋅K)
207⋅ 10−6 1/K − 25⋅ 10−6 1/K 50⋅ 10−6 1/K 35⋅ 10−6 1/K
1000 kg/m3 − 2750 kg/m3 3150 kg/m3
2.2 GPa − 45 GPa 105.2 GPa 14.9 GPa
( sw
W)
+
t
(
s Gss Ws W )
d C + dt w W s (s v)+ ( s g Wg v s) +
k nw m0C + +
( sw
+
w v w, s)
t
( sg
+
=(
t +
Dg )
( sg
+
Dg uDg )
( sg
Dg v g , s)
+
= 0,
( sg
+ (s w
W
+ s g g) .
(13)
As for the linear momentum balance, the energy conservation of the porous media is governed a by single model equation obtained through summation over all species. To arrive at this equation, it is assumed that all phases are in local thermodynamic equilibrium and recalling the assumptions introduced in Section 3.2, it follows that any contributions from viscous dissipation and mechanical work are negligible [40]. Also, any advective transfer of energy with the solid phase is considered negligible. Finally, we express the energy balance in terms of enthalpy to obtain a model equation on a similar form to the classical heat equation:
(10)
where v w, s is the relative velocity of w with respect to s. Furthermore, Wg is the vapour concentration and uWg is the diffusive velocity of W in g. The volume fraction of the capillary pore space saturated with liquid water is given by s w , while the volume fraction of the gel pore space occupied by water is described by sWs . Lastly, it is recognized that any mass exchange of water between phases cancels out following the summation over all phases to obtain Eq. (10). The influence of dry air in the gel pore space is considered negligible, hence, it is only present in the capillary pore space and its mass balance is given as
( sg
W s Ws Gs) s
+
3.3. Conservation of energy
Wg v g , s)
( s GssWs W v s) ( s g Wg uWg) = 0,
Ss
It should be mentioned that in some other adoptions of multiphase porous media theory to early-age concrete, Eq. (12) is introduced in its rate-form [20,22]. This difference will, of course, affect the final system of equations and its numerical implementation.
Wg )
( sg
(1 + knw ) C C 1 + knw N
~ where t is the damaged total stress tensor (see the forthcoming Section 4.3.1) and g is the total body force vector. The total density ρ is calculated as
neglected in Eq. (8); this assumption should only have minor influences on the behaviour, at least after setting. The last term inside the brackets represents the chemically bound water during hydration, where m0C is the amount of cement per volume of the concrete mixture. Moreover, s / t, , it follows that s / t = since the capillary porosity = 1 which is utilized in the other model equations. In previous work on multiphase models for early-age concrete (e.g. [20,22]) water is only present in the two fluid phases. However, following the discussion in Section 2, water in the smallest pores cannot be considered to behave as a fluid and should thus not be lumped together with the more mobile water in larger pores. In our model, water held in the gel pore space is considered as part of the solid phase and is thus not contributing to the fluid flow. Hence, water is present in all three phases and its total mass balance becomes
t
i pref
( Cp )eff +
t
Hcem m 0C
T
w CWw v w, s p
q + (s w
d C + Hvap Mvap dt g g + s Cpg v g, s) T = 0,
(14)
where q is the total conductive energy flux and
( Cp )eff =
w CWw p
sw +
s AsC As p
+ s g g Cpg + +
s CsC Cs p
+
s Gss WsCWs p s HsC Hs . p
(15)
(15), Cpi
In Eq. is the isobaric specific heat capacity of species i in phase α; here considered to be constant, see Table 1. Two heat sources are present in Eq. (14) related to the heat released during hydration and the phase change of water between liquid and gas form. The heat released during hydration is controlled by the hydration enthalpy ΔHcem that depends on the composition of the cement, but is in the region of 500 kJ/kg reacted cement for normal cements [13,34]. The energy released during evaporation of water is quantified by ΔHvap, which is here set to 2257 kJ/kg evaporated water. The rate of evaporated water Mvap can be obtained from the mass balance of water in the wetting phase, see for example references [20,26,41].
Dg v s)
(11)
where s g is the volume fraction of the capillary pore space saturated with the gas mixture and v g , s is the relative velocity of g with respect to s. Furthermore, Dg is the mass concentration of dry air and uDg is the diffusive velocity of D in g. 3.2. Conservation momentum The mechanical behaviour of the porous media is governed by a single model equation obtained through the summation of the linear momentum balance of all species. As done in many applications of concrete related to durability and early-age behaviour, and so also in our model, velocities are considered small and the time scale of interest large (days), meaning that inertial forces can be neglected. Using the same argument, the forces due to mass exchanges are also small and can be omitted from the formulation [40]. This assumption yields a linear momentum balance for the entire porous media on the classical form ~ t + g = 0, (12)
3.4. Choice of state variables The above model equations include many unknowns that have to be defined, either directly as physical constants or using constitutive relationships to relate them to the state variables of the model. Hence, the choice of state variables to be used in the numerical implementation of the mathematical model warrants some attention. Starting with the obvious choices, the solid phase volume fraction s , the displacements d and the temperature T are used as state variables in Eqs. (8), (12) and (14), respectively. Also, for the total dry air content the choice is rather 205
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straightforward and the gas pressure pg is chosen as the primary variable in Eq. (11). For the mass balance of total water content in Eq. (10), however, the choice is more difficult and many alternatives can be found in the literature; for example the degree of water saturation s w , the relative humidity φ and the liquid water pressure pw. In our model we follow the work of Gawin et al. [20] and use the capillary pressure pc. Gawin et al. have for many applications shown the validity of this choice to describe the whole range of possible moisture states in concrete. They have also shown the efficiency and stability of using pc as a state variable in numerical implementations.
compressive strength. Given that the evolution of compressive strength reflects the properties of the concrete microstructure, κf will also be used to describe the development of transport properties of the early-age concrete, which are highly dependent on for example the connectivity of the pore system. This assumption in principle means that the resistance of the concrete to fluid flow depends on the curing temperature. 4.2. Hygro-thermal behavior The hygro-thermal behaviour of concrete is strongly coupled, where the transport and storage of both fluids and energy interact. The transport of both quantities follows the standard forms of Darcy's, Fick's and Fourier's laws, see Appendix A. However, as indicated, the characteristics of the mulitphase material change dramatically during hydration, which affects both the fluid storage capacity and also the transport properties. Constitutive laws that account for this are presented in the following. Given the choice of capillary pressure as a state variable, it is important to first introduce two key relationships on how pc relates to the quantities of the two fluid phases. Assuming a state of local thermodynamic equilibrium and moderate temperatures, the equilibrium state of capillary water with vapour follows the Kelvin equation
4. Constitutive and physical relationships for early-age concrete To completely describe the hygro-thermo-chemo-mechanical behaviour of early-age concrete, the governing equations presented in Section 3 must be complemented by a set of constitutive and physical relationships. Many different versions of the necessary relationships can be found in the literature and most of those used in the proposed model are taken from previously published results, either as previously applied to multiphase models or adequately modified to suit the current model formulation. Those of most importance for the early-age behaviour of concrete are presented in the following. However, for completeness, a few additional constitutive relationships that are fundamental for porous media theory are presented in Appendix A. For a more in-depth derivation and description of these, the reader is referred to previous work in the literature [20-23,26,40-42].
=
Evolution of the material properties of concrete that depend on the characteristics of the microstructure with time needs some extra attention and can be difficult to relate directly to the hydration degree. To properly describe the evolution of for example many mechanical properties, at least the phenomenon of setting needs to be accounted for, which can be accomplished using the model by De Schutter and Taerwe [43] often used in models of early-age concrete. However, an important aspect not covered in their model is how the kinetics of the reaction affects the development of the pore structure and in extension the mechanical properties [2,44,45]. To this end, a more refined model was suggested by Cervera et al. [9] to account for the effects of curing temperature. Supported by experimental evidence, they concluded that a low curing temperature leads to a more uniform pore structure than a higher temperature, which in turn leads to a higher strength at complete hydration. To describe this phenomenon, Cervera et al. [9] proposed introducing an ageing degree κf, which describes how the compressive strength evolves with time as a function of the degree of hydration and the curing temperature. This ageing model is employed here with the evolution of κf defined as f
=
TT TT
T Tref
nT
(B f
2Af
C)
d dt
C,
W
= exp
MW c p W RT
,
(17)
where the definition of relative humidity φ is introduced as the ratio between the vapour pressure and the temperature dependent vapour Wg pressure at saturation psat . Additionally, the macroscale capillary pressure is defined as the pressure difference between the non-wetting and wetting phases, which can be expressed as
4.1. Evolution of material properties
d dt
pWg psatg
pc = p g
pw ,
(18)
and derived from the second law of thermodynamics given a state of local thermodynamic equilibrium, as shown by for example Pesavento et al. [42]. The physical interpretation of capillary pressure and thus of Eqs. (17) and (18) is only valid as long as water exists in a state where it can form a capillary meniscus. However, it is shown by Lewis and Schrefler [46] and Gawin et al. [47] that the mathematical meaning of pc can be extended to include the entire moisture range. 4.2.1. Sorption isotherm Sorption isotherms are closely related to the microstructure of cement paste and in particular the pore size distribution. As discussed in Section 2, the microstructure is highly influenced by the hydration and changes significantly during the first days after casting. Thus, the definition of the sorption isotherm must be age dependent for a model that considers the early-age behaviour of concrete. In the context of multiphase porous media models, two different hydration degree dependent sorption isotherms were suggested by Sciumé et al. [22] and Chitez and Jefferson [48]. Both of these are developments of the classical expressions by van Genuchten [49] to also consider the degree of hydration. Another definition was used by Cramer et al. [50] in which the degree of water saturation was additively split into physical adsorption and capillary condensation, both parts as functions of the pore relative humidity. Here, a new definition is suggested that splits the total water saturation into water held in gel pores sWs and capillary pores s w , following the description in Section 2.2 and outlined in the governing equations presented in Section 3. The proposed saturation functions are based on the work of Norling-Mjörnell [37], who proposed and experimentally verified an analytical model for a hydration degree dependent sorption isotherm expressed in terms of moisture content and relative humidity. The model has also been successfully applied to moisture transport in early-age concrete by others using single-phase
(16)
where the first term controls the influence of the curing temperature through the model parameter nT. The temperature value TT represents the maximum temperature at which hardening of concrete occurs (≈100 °C) and Tref is a reference temperature. The second term in Eq. (16) controls the ageing with respect to the chemical reactions through model parameters Af and Bf. To reduce the number of model parameters, Di Luzio and Cusatis [14] concluded that Bf can be calculated as a function of Af , C and the hydration degree at setting 0C . Since κf is defined as a normalized strength, the evolution of the compressive strength can then be calculated as f = f f , where f is the compressive strength at C = C and T = Tref. Finally, as done in the models by De Schutter and Taerwe [43] and Cervera et al. [9], the evolution of other mechanical properties, such as tensile strength f+ and fracture energies Gf ± , is considered to be related to the development of the 206
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moisture transport models [13,15,18]. Functions of the same form as in reference [37] are used here, but reformulated in terms of degree of saturation instead of moisture content, and split into two separate variables to fit in the multiphase framework of the current study. Starting with the degree of saturation of liquid water in the gel pores, the following constitutive equation is proposed
sWs =
1
{exp [(ggs 1
10
[exp (ggs
C)
]}
10 C )]
1
network decreases rapidly, which in the mathematical model can be represented as a change in the intrinsic permeability. Using a similar formula as proposed by Gawin et al., this is here accounted for as
k = 10 Ak (1
(19)
and for the liquid water in the capillary pores, a constitutive equation on the form
sw =
10 C ) ]
exp [(gcs exp (gcs
10 C )
1 (20)
1
is proposed. Lastly, by definition it follows that = 1 The constitutive parameters ggs and gcs control the ageing described by ξC of the respective parts, where for simplicity we assume ggs = gcs = gs. While Eqs. (19) and (20) are dependent on the relative humidity φ, they can be expressed in terms of the state variables pc and T using Eq. (17). By considering the evolution of the microstructure, the degree of liquid water saturation in the entire porous media can be expressed as
sg
W stot =
s GssWs s Gs
+ sw ; +
(22)
I,
where k∞ is the intrinsic permeability at complete hydration and defines the flow characteristics in all directions. Ageing is in Eq. (22) controlled by the model parameter Ak and the ageing degree κf. As done by Sciumé [41], also the structure coefficient fs that controls the diffusivity in Fick's law is assumed to depend on the development of the internal structure of concrete during hydration on the same form as the intrinsic permeability by defining a value fs,∞. The effective permeability of the respective fluid phase is controlled by the relative permeabilities k rf that vary from 0 to 1. Commonly used definitions are those by Brooks and Corey [52] and van Genuchten [49], both of which are governed by the total degree of saturation of the pore space. To account for the changing microstructure also on this property, equations on the form proposed by Chung and Consolazio [53] for flow in concrete under elevated temperatures are adopted, where the relative permeabilities are assumed to depend on the capillary porosity ϵ and saturation s f , such that
1
,
f )k
sw.
g
(0.05 Arw )
10(0.05
Arw ) s g ,
(23)
w
(0.05 Arg )
10(0.05
Arg ) s w ,
(24)
k rw = 10 s
(21)
k rg = 10 s
although not used directly in any part of the model it is useful for evaluation purposes. The degrees of saturation described by Eqs. (19) to (21) are exemplified in Fig. 2 for different degrees of hydration and plotted in terms of capillary pressure. For comparison, the experimental results by Baroghel-Bouny et al. [51] are included in Fig. 2c. The consequence of this new definition is that during the very early-ages, the capillary system is emptied without any significant build-up of capillary forces if the sample is subjected to drying. On the contrary, water is bound much more tightly in the gel pore space, both during the early-age and beyond, which is in line with recent experimental findings [6,28,29]. Given the growth of s Gs and decrease of ϵ with hydration, the model also reflects that more water is held in the gel pore space for mature concrete.
where Arw and Arg are model parameters. Compared to Chung and Consolazio [53], it should be emphasized that it is here the capillary saturation that controls the flow of the respective phase, since no flow is assumed to occur in smaller pores. 4.3. Mechanical behaviour The mechanical behaviour of the porous media is controlled by the definition of the total stress tensor, which according to Gray et al. [54] may be expressed as
t=
s s
(25)
bP s I,
where is some measure of pressure acting in the system and b is Biot's coefficient [46,55], which relates the compressibility of the solid phase to the drained porous skeleton. The resistance of the solid matrix to external and internal loads is described by the effective stress s s . Given that it is only the solid phase that can resist deformations d, the stress-strain relationship takes the form
Ps
4.2.2. Mass fluxes The complex internal structure of concrete and its evolution highly influence the flow properties through the intrinsic permeability of the material in Darcy's law. For the early stages of hydration there is a decrease in overall porosity but, as pointed out by Gawin et al. [20] and perhaps more importantly for the flow properties, also a change in pore connectivity. The size of the largest pores of the continuous pore
s
=
were
e:
e
(e
e th
ecr),
(26)
is the fourth-order elasticity tensor defined by two material
Fig. 2. Proposed saturation functions in Eq. (19) for gel pores (a), in Eq. (20) for capillary pores (b) and in Eq. (21) for the total pore space (c) at different degrees of hydration for concrete with w/c = 0.48. Experimental results by Baroghel-Bouny et al. [51] included in (c) for comparison. 207
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constants, e.g. Young's modulus Ê and Poisson’s ratio ν. Both Ê and ν are here taken as age-independent [56]; notice that ageing of the elastic response, i.e. the measured Young's modulus, is accounted for in the definition of the creep tensor ecr. Additionally, contributions from temperature changes are given by the thermal strain tensor eth. Using the standard small strain definition, the symmetric total strain tensor e of the solid matrix relates to the state variable d as
e=
1 [ d + ( d) ]. 2
~+ = H (t ) 1
ws w s p
(28)
where sws is the fraction of the capillary pore surface in contact with W liquid water. The fraction sws is often assumed equal to stot . However, as discussed by Gawin et al. [21], this is likely not the case for a material with a large amount of fine pores as concrete, where water is also present as a thin film. Hence, for concrete such an assumption can lead to an underestimation of the shrinkage. Gawin et al. derived functions for the quantity sws from the experiments by Baroghel-Bouny et al. [51]. In our model, sws is assumed to be a function of the degree of water saturation in the gel pores, i.e. ws s
r±
±
pj ;
t =t
t+
+) t+
+ (1
)t ,
=
where A
B± =
(33)
(34)
re± r±
1 2
±
rp± re±
=
r± rp±
1 1 1
2
exp B ± (rp± 1 ldis
r±
ÊGf± (f ± ) 2
r±
if rp±
r±
rp± ,
rp±
re±
re±)/ re±
if r0±
(35)
and
+ A±
(rp±)3
3rp± + 2
6re± (rp±
1)
.
(36)
The shape of the uniaxial behaviour is thus controlled by the model parameters r0±, re± and rp± that define the initial value of r ± , the initial size of ~ ± and the value of r ± at peak strength, respectively, see [10] for more details. However, notice that due to the normalized framework, r0± = 1 is always true. In the definition of B ± , Young's modulus Ê, tensile and compressive strengths f ± and fracture energies Gf± are introduced to control the softening. Also in Eq. (36), a characteristic length ldis is introduced to ensure mesh-size objectivity of the solution using the crack-band approach [60]. Here, the value of ldis is inferred directly from the volume of the finite element in which softening occurs.
(30)
4.3.2. Thermal expansion The rate of thermal expansion is assumed to be caused by a volumetric thermal expansion coefficient αs of the solid phase. To account for the continuous solidification of reaction products and the changing microstructural properties, αs is here determined at the current material state. Making use of the mass conservation of the solid phase, αs can be identified from the total density of the solid phase ρs, see first term in Eq. (13). Introducing this definition of αs, the rate of thermal strains is given as
to separate the tensile (+) and compressive (−) components of t. In Eq. (30), tj is the jth principal stress and pj its corresponding directional vector. Furthermore, ⊗ denotes the tensorial product and the Macaulay brackets ⟨−⟩ the positive parts operator. The damaged stress tensor entering in the momentum balance (12) is then defined as
~ t = (1
A ± re± 1
3
t j pj
re , f
implying that r ± is the largest value of ~ ± reached by the material. Lastly, the damage variables ω ± are directly defined as functions of the current value of r ± , also using the function proposed by Cervera et al. [10]. For both tension and compression this function is given as
4.3.1. Damage When dealing with durability issues related to cement and concrete, cracking is an important effect that needs to be accounted for. Many different techniques are available in the literature, see e.g. [57], where it in this work is chosen to adopt the theory of continuum damage mechanics. Given that we are considering concrete during early-ages, it is important to treat damage in relation to ageing in a thermodynamically consistent manner. To this end, a model based on the work by Cervera et al. [10] is used. The formulation introduces a stress split
j=1
[ 3J2 + a I1 + b t1 ]
a
(32)
i (0, t )
where aws is a model parameter. This assumption could physically be motivated by the presence of thin films also in the capillary space as long as the smaller pores in the gel are saturated in the homogenized system.
t+ =
1 1
re+ , f
r ± = max [r0±, max (~i±)]
(29)
= (sWs )aws ,
a
[ 3J2 + a I1 + b t1 ]
where I1 and J2 are the first invariant and second deviatoric invariant of the undamaged stress tensor t, respectively. Furthermore, t1 and t3 are the maximum and minimum principal stresses and H (−) denotes the Heaviside step function. Parameters aω and bω control the shape of the failure surface and are functions of the age dependent compressive and tensile strengths f ± . The same form of failure criterion was used by Saloustros et al. [59] in a similar damage model, but not using the normalized framework, where the definitions of these two model parameters can be found. Lastly, re± are model parameters that define the initial damage thresholds. Damage growth is described by the normalized stress-like internal variables r ± that define the current damage threshold. Evolution of these internal variables can be written on the explicit form [10]
(27)
ws g s )p ,
+ (1
1
~ = H( t ) 3
In the poromechanical model formulation, the second term in Eq. (25) acts as an internal load that causes volumetric changes, i.e. shrinkage or swelling, due to change in for example moisture conditions. Also in the work of Gray et al. [54], they showed that if neglecting contributions from surface tensions and assuming that surface and phase averages of pressure are equal, P s takes the form
Ps =
1
(31)
where the two damage parameters ω+ and ω− account for tensile and compressive damage, respectively. Damage is then characterized in a normalized equivalent stress space as introduced by Cervera et al. [10], which allows comparison of different stress states also for different values of the ageing degree κf. 0 so that thermodynamically Using this framework ensures that ± consistent results are obtained also during ageing [10]. To characterize the limits of the elastic domain, two non-dimensional scalar measures are introduced, one for tension ~+ and one for compression ~ . These are given on the form proposed by Lubliner et al. [58] for a plasticity based model, and written as
s d d eth = I T= dt 3 dt
s 1 d I T. s 3 T dt
(37)
4.3.3. Creep Creep is described by the Microprestress-Solidification (MPS) theory developed by Bažant and co-workers over the last decades [36,61-65]. This model has previously been incorporated into a multiphase porous media framework by Gawin et al. [21] and used by several authors throughout the years [10,14,66] to describe concrete creep in multifield 208
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models, including early-age applications. It is here adopted in a form similar to the original model, but with minor reformulations to the solidification process and the evolution of microprestress. The total creep tensor is in the MPS theory split into two parts, one viscoelastic part ecr,v that mainly describes short-term creep and one viscous part ecr,f, which describes long-term creep. In rate-form, this strain split is described as
d d d d ecr = ecr,v + ecr,f = r + dt dt dt v dt
r
:(
f
s s ),
(t )
n f
(t ) (38)
d dt
t
t
q2 ln 1
t
0.1
0
0
:
d ( dt
s s )dt
,
(40)
1 f (S )
1
= cpS p 1,
S s
f (S )
(41)
1 d Cs dt
= 0,
(43)
(44)
= kc pc ,
f
+
1 µc pref
p d c p ( µc f ) p 1 dt
s
q4
= 0,
(45)
p pref
1
T0p 1
MW pW R
p 1
,
(46)
where T0 is a reference temperature. Using Eq. (45), the model to calculate the viscosity ηf requires three model parameters (p, μc and q4) to be identified, of which p is set to 2 in this study to keep the correct dimensionality of the model. Jirásek and Havlásek [65] identified an appropriate initial condition to Eq. (45) as ηf(t0) = t0/q4, where t0 is the age of the concrete at the start of the analysis. A significant difference in the current formulation compared to the original MPS formulation and its later developments is that s enters Eqs. (38) and (40), and not the total stress t. This modification to the MPS creep model was suggested by Gawin et al. [20], who argued that in a mechanistic description of shrinkage in a porous media model, the internal load P s causing shrinkage must also contribute to the creep. Hence, what is measured as macroscopic shrinkage in an experiment is actually, in the present model, accounted for by an instantaneous and recoverable deformation due to changes in P s , and in addition a creep deformation following the effective stresses caused by P s . Thus, the creep model is active also if the concrete is externally unloaded. This coupling is discussed further by Gawin et al. [68] and has also been considered in combination with other creep models by for example Sciumé et al. [22]. Also Ulm et al. [69] discussed the creep-shrinkage coupling, especially in relation to autogenous shrinkage and basic creep.
where c and p are model parameters with p > 1. The microprestress S in Eq. (41) should be interpreted as a macroscopic average of the actual microprestress at the individual creep sites. The evolution of S with time is then given as a relaxation in the creep sites governed by ηf(S), but is also increased by changes in . This can be described by a differential equation [36,64]
1 d S+ Cs dt
1
µc = µs
where = Ê e . The creep function is taken on the same form as in the original paper by Bažant and Prasannan [61] where λ0 = 1 day is usually assumed [61] and t − t′ is the loading time duration. Lastly, q2 is a model parameter that can be identified from experiments, or from the B3 [56] or B4 [67] prediction formulas. The long-term irrecoverable part of creep is described by the Microprestress theory [36], which explains it as the shear slip in localized and overstressed atomic bonds across gel pores (creep sites) that breaks and reforms. The high tensile stress in these bonds, referred to as the microprestress S, is built up during the rapid stages of hydration and then relaxes with time. However, at later stages, it is also affected by the disjoining pressure due to hindered adsorbed water in these small pores. On a macroscopic scale this process can be modelled as a viscous flow, where the non-constant viscosity ηf governs the creep rate, see Eq. (38). As argued by Bažant et al. [36], ηf is most suitably described as a power function of S on the form 1
RT ln , MW
where also the derivative of Eq. (44) has been substituted into the new equation. The model parameter μc should be identified from experiments, and the reference pressure pref is introduced only so that μc for the standard choice p = 2 obtains the dimensionality of fluidity (Pa−1s−1) and can be set to 1 atm. Parameter q4 can be identified from experiments or from the B3 [56] or B4 [67] prediction formulas. It should be noticed that μc is not identical to the similar parameter μs suggested by Jirásek and Havlásek [65]. Comparing the two, it can be found that
where nκ is a model parameter. The age-independent behaviour of the gel is assumed to follow standard linear viscoelasticity
=
C1
where kc = C1/ρW. To calculate the evolution of the viscosity ηf thus involves four model parameters (c, p, Cs, kc) and two equations. However, Jirásek and Havlásek [65] showed that these can be reduced and also that S can be eliminated from the model. Following a similar procedure, we can derive a new expression for ηf that replaces Eqs. (41) and (42) with
(39)
,
=
with C1 being a constant. Parameter 1 is the value of at hygroscopic saturation (φ = 1). However, since only the rate of is used in Eq. (42), this value is not needed. Using Eq. (17), this can also be expressed in terms of the capillary pressure pc as
where also the explicit definition of these two parts according to the MPS theory is introduced. The variable ψr was introduced by Bažant et al. [64] to account for general temperature and moisture conditions. The viscoelastic part is described by the Solidification theory [61] in which it is hypothesized that the solid particles of the load bearing C-SH gel have non-ageing properties, and ageing of the solid matrix is instead described by the growth of the gel during hydration, referred to as solidification. This is described by the function v, here taken as a function of the ageing degree on the form
v ( f) =
1
5. Remarks on final system of equation and numerical implementation In summary, the hygro-thermo-chemo-mechanical behaviour of concrete is described by a system of eight model equations and state variables:
(42)
in which Cs is a model parameter. The variable ψs was introduced by Bažant et al. [64] to account for general temperature and moisture conditions. Furthermore, depends on capillary tension, surface tension, crystal growth pressure and disjoining pressure, and, due to the small scale of interest, water in all phases and pore spaces can be assumed to be in local thermodynamic equilibrium [36]. This means that responds instantly to changes in chemical potentials in the capillary pore system, e.g. due to temperature or humidity changes. Given this, Bažant et al. [64] proposed that, under constant S, the actual microprestress is
• Cement hydration in Eq. (1) with ξ , • The mass conservation of solid species in Eq. (8) with , • The mass conservation of water in Eq. (10) with p , • The mass conservation of dry air in Eq. (11) with p , • The three components of the linear momentum conservation in Eq. (12) with d, • The conservation of energy in Eq. (14) with T. C
s
c
g
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For a complete closure of the mathematical model, initial conditions are needed for all eight governing equations and for Eqs. (10) to (12) and (14) boundary conditions also need to be defined. The latter are defined as either Dirichlet type by constraining the respective state variable, or as Neumann type by prescribing the total mass flux of water w D Jtot and dry air Jtot as well as the surface traction ^t and heat flux qT normal to the boundary. Additionally, convective boundary conditions are frequently used for Eqs. (10) and (14), here defined using the amWg bient vapor concentration amb and temperature Tamb such that W Jtot = fW (
qT = f T (T W
term mechanical parts of the model; the second on the moisture transport and the third on creep and shrinkage as well as tensile cracking. The intentions of the examples are both to demonstrate the capabilities of the proposed model, and to identify its limitations. Although many physical mechanisms are considered, there is surely still a need to investigate and evaluate the applicability of the constitutive relationships adopted. To this end, it is also important to keep in mind that additional physical couplings could be necessary to introduce. However, for a complete analysis of the model, more examples than those allowed within the scope of this work are certainly required.
Wg amb ),
Wg
Tamb),
6.1. Hydration and ageing
(47)
T
where f and f are the mass and heat transfer coefficients, respectively. This initial-boundary value problem needs to be solved numerically, which is here done using the Finite Element Method (FEM) as implemented in the general FE code Comsol Multiphysics [70]. Setting up the discretized system of equations follows standard procedures of the FEM (see e.g. references [46,71]), starting with conversion of the pointwise model equations to their weak-form and application of the Galerkin method. All constitutive and physical equations presented in Sections 2.2 and 4 and Appendix A are then inserted into the weak-form PDEs and after introduction of the FE discretization, the matrix system to be solved takes the general form
where u contains the nodal descriptions of all state variables as well as κf and ηf. The non-linear system matrices C, K and f include contributions from all governing equations and Eqs. (16) and (45). Variables s, C, f and ηf are in this study assembled into the overall system of equations using discontinuous Lagrange shape functions, but could also be updated separately at individual Gauss point since their governing equations contain no spatial derivatives. Time integration of Eq. (48) follows the fully implicit backward differentiation formula (BDF), using an adaptive order. The non-linear equations obtained at each time step are linearized internally by the software and solved using a damped version of the Newton-Raphson method [70]. The assembled system of equations is fully-coupled, resulting in that the derived system Jacobian contains all off-diagonal terms that appear from coupling of the governing equations. This is a particularly attractive attribute of the employed solution procedure for the strongly coupled physical and chemical fields. The definition of the stress tensor includes inelastic strain contributions (ecr,v, ecr,f, eth) given on a rate-form that need to be integrated in time. This is preferably done separately from the integration of Eq. (48) in the incremental update procedure. For the creep strains, the non-ageing viscoelastic part of the creep model in Eq. (40) is first discretized in time by a Kelvin-Voigt chain with ten units. Properties of the individual units are given by the formulas derived by Jirásek and Havlásek [72] for the log-power creep function in Eq. (40). Integration of the creep strains (ecr,v and ecr,f) is then done using an algorithm derived from the exponential algorithm suggested in the original MPS papers [62,63]. In short, the integration utilizes the analytical solution to the two parts of Eq. (38), given the assumption of constant stress and strain rates in each time increment. This leads to a closed-form algorithm that gives accurate results also for very large time steps while maintaining computational efficiency. Update of the thermal strains eth is done using a mid-point integration algorithm to solve Eq. (37).
The first example concerns an experimental series performed by Khan et al. [73] to examine the early-age compressive stress-strain behaviour of different concretes and curing conditions. The specimens used consist of concrete cylinders with dimensions 100×200 mm that, after unmolding at 24 h from casting, where cured either in adiabatic conditions in a water bath or moisture sealed and cured in an ambient temperature of 21 °C. Three different concrete mixtures were tested, of which the mixture with an approximate 28-day compressive strength of 30 MPa is analysed in the following. The mixture, referred to as C-30, has a w/c = 0.5 and a/c = 5.16 and a cement content of 355 kg/m3. The numerical model is set up using axial symmetry, and since no spatial variations are expected it consists of a single finite element using linear shape functions for all state variables. Values of all model parameters used are summarized in Table 2. The same experimental series has been analysed previously in the literature by for example Cervera et al. [10] who did not consider the influence of moisture, and Di Luzio and Cusatis [14] using a single-phase moisture transport model and a microplane model to account for fracture. The chemical model is first calibrated using temperature measurements from the adiabatic curing (results not presented), before determining the parameters of the ageing model in Eq. (16) utilizing the measured compressive strength. The simulated compressive strength development is shown for both adiabatic and isothermal conditions in Fig. 3a, which is in good agreement with the experimental results for both curing conditions. In Khan et al. [73], the stress-strain response was measured for several ages, while the simulated response in Fig. 3b is only shown for an age of 1, 7 and 28 days. The response up to peak load is in excellent agreement with the measured for both curing conditions, although it should be noticed that no measurements where reported at 28 days for adiabatic curing. The simulated compressive softening response can qualitatively reproduce the measured response in that the concrete becomes more brittle with age. However, the results are only in close agreement just after the peak load. This could point to a deficiency in the ageing model and how it currently describes the evolution of the peak strength in relation to the fracture energy. The compressive fracture energy is in the presented results calculated as Gf = f3/2 Gf, , where the exponent controls the evolution of the softening response. The cause of this discrepancy should be investigated further as it could also have other causes, for example many model parameters, such as those related to the creep model, are currently assumed due to a lack of data. To exemplify the microstructural model presented in Section 2.2, the development of volume fractions of the porous media is shown in Fig. 3c, including all species of the solid phase. The influence of the curing conditions is clearly noticeable as is the small self-desiccation, where parts of the capillary pores are emptied of water.
6. Applications
6.2. Moisture transport
To show how the features of the presented mathematical model are able to describe and predict the behaviour of early-age concrete, three laboratory scale examples from the literature are solved and discussed in the following. The first focuses on the chemical, ageing and short-
The second example focuses on the ability of the model to describe the movement of moisture in concrete by studying the experiments performed by Kim and Lee [74]. The experimental program consisted of specimens with dimensions 100×100×100 mm to study self-
C (u)
d u + K (u) u = f (u), dt
(48)
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Table 2 Model and material parameters used in the numerical applications. Section 6.2 Section 6.1 Hydration model constant, Eq. (2) Hydration degree at setting, Eq. (16) Ageing model constant, Eq. (16) Ageing constant, Eq. (22) Biot's coefficient, Eq. (25) Sorption isotherm, Eqs. (20) and (19) Relative permeability, Eqs. (24) and (23) Model parameter, Eq. (29) Damage model parameter, Eq. (32) Damage model parameter, Eq. (33) Damage model parameter, Eq. (35) Damage model parameter, Eq. (35)
Solidification function, Eq. (39) Creep function, Eq. (38) Microprestress parameter, Eq. (45) Age-independent Young's modulus Age-independent Poission's ratio Asymptotic tensile strength
Asymptotic compressive strength
Asymptotic tensile fracture energy Asymptotic compressive fracture energy Asymptotic intrinsic permeability Asymptotic structure coefficient
M
H 8⋅ 10 1⋅ 10−3 5 0.1
1⋅ 107 1⋅ 10−5 8 0.075
0.1 0.38 4 0.25 10.5 22.5 1 1
0.5 0.3 3.6 0.25 10.0 23 1 −
0.4 0.3 4.75 0.25 11 30 1 −
1.1 0.3 3.25 0.35 8.5 22.5 10 1
1
7
−
−
5
f+
1 MPa
0.2 3.2
0.2 −
0.2 −
0.2 4.4
f
MPa
34.5
−
−
44
J/m2
6500
−
−
4400
1/h 1 1 1
1.25⋅ 10 1⋅ 10−4 7.5 0.15
Af nT Ak b gs Arw, Arg aws
1 1 1 1 1 1 1 1
C 0
re+ re rp+ rp
nκ q2 q4 μc
Ê ν
Gf,+
Gf,
k∞ fs,∞
1 1
4 1
1 1/MPa 1/MPa 1/MPa⋅s GPa
1 30⋅ 10−6 1.8⋅ 10−6 8.0⋅ 10−13 37
J/m2
120
m2 1
5.0⋅ 10−22 0.01
desiccation and 200×100×100 mm to study external drying of three different concretes: low strength (L), medium strength (M) and high strength (H). To limit the current numerical study, the L series is omitted since it shows much less self-dessication than the other two concretes. The concretes in the M and H series have a w/c of 0.40 and 0.28, a a/c of 4.14 and 3.15, a cement content of 423 and 541 kg/m3, and a 28-day compressive strength of 53 and 76 MPa, respectively. All specimens were unmolded after 24 h and moist cured for an additional 3 or 28 days. After curing, specimens intended to study self-desiccation were sealed on all sides, while for the drying specimens one of the 100×100 mm sides were exposed to achieve a uniaxial flow. All specimens were stored at an ambient relative humidity of 0.5 and a temperature of 20 °C. For the drying specimens, the relative humidity was
6
Section 6.3
9⋅ 10 1⋅ 10−4 6.5 0.1
Ac1 Ac2 ηc
6
− −
1 68⋅ 10−6 5.4⋅ 10−6 8.0⋅ 10−13 37
−
4.0⋅ 10−22 0.001
6
− −
1 23⋅ 10−6 2.9⋅ 10−6 8.0⋅ 10−13 59
−
1.25⋅ 10−22 0.001
2.5 1
0.3 40⋅ 10−6 1.5⋅ 10−6 1.0⋅ 10−13 40
145
1.0⋅ 10−22 0.01
measured at 3 cm, 7 cm and 12 cm from the exposed surface. The same experimental series has been analysed previously in the literature by Chitez and Jefferson [48], among others. A difficulty in setting up the numerical model, is how to describe the moist-curing period in a good manner so that satisfactory initial conditions are obtained when drying/sealing commences. In order to achieve this while maintaining a computationally efficient model, an assumption of axial symmetry is made, although prism specimens were used for the experiments. This was done in order to better describe the moisture uptake during curing compared to a plane 2D-model, since the surface area exposed to the ambient air can be defined correctly in both phases of the test by calculating an equivalent radius. During curing, the convective type boundary condition in Eq. (47) is used on all
Fig. 3. Simulations of the compressive strength development at different curing conditions (a) and the stress-strain response at different ages (b). Markers in (a–b) show the experimental data by Khan et al. [73]. Development of the microstructure with time is shown in (c) as additive volume fractions, where solid lines show adiabatic conditions and dashed lines isothermal conditions. 211
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1
Relative humidity [1]
Relative humidity [1]
1 0.9 0.8 3cm 7cm 12cm Sealed
0.7
0.9 0.8 3cm 7cm 12cm Sealed
0.7 0.6
0.6 0
50
100
150
200
0
250
50
100
150
200
250
Time since casting [days]
Time since casting [days]
(a)
(b)
Fig. 4. Simulated relative humidity at different depths from the drying surface, for medium strength (M) concrete (a) and high strengh (H) concrete (b) compared to measurements by Kim and Lee [74]. Dashed lines and open markers correspond to 3 days while solid lines and filled markers correspond to 28 days of moist-curing.
external surfaces with an ambient relative humidity equal to 1 and a transfer coefficient fW = 1 m/s. Once curing is stopped, the assumption of axial symmetry does no longer matter since only one side is exposed to external drying, causing a uniaxial moisture flux. Thus, a 2D axisymmetric model is used with all state variables described by linear shape functions. All model parameters are summarized in Table 2. No calorimetry nor temperature data were available to calibrate the chemical model, hence its parameters had to be calibrated together with those of the sorption isotherm by fitting of the self-desiccation measurements. Once curing stopped, the boundary conditions on all W = 0 for these specimens. Although not an sides were changed so that Jtot ideal procedure, a good fit was achieved for both concrete types as shown by the dashed black lines (simulations) and open black squares (measurements) in Fig. 4. The solid black lines correspond to the specimens subjected to the longer period of moist-curing and also indicate a good agreement between simulations and measurements, shown by the filled black circles, for both concrete types. However, these results are more affected by how the moist-curing is modelled, e.g. the definition of the transfer coefficient fW during this period. For the specimens exposed to external drying, boundary conditions after the curing period were changed so that fW = 0.0001 m/s for the side exposed to drying. Furthermore, since all other sides were sealed, a W = 0 was defined on these. With the defined boundary moisture flux Jtot boundary conditions and the previously described procedure for moistcuring, the simulated decrease in relative humidity is overall in good agreement with the measured one for both concrete types as well as for both 3 and 28 days of moist-curing. However, the simulations slightly overestimate the drying close to the surface for the H concrete, as shown by the 3 cm measure point in Fig. 4b. This is likely related to the ageing of the permeability, which together with the mass transfer coefficient play important roles in determining the rate of moisture loss. Furthermore, it proved important that both the intrinsic permeability and the relative permeabilities are age dependent to accurately describe the observed drying process. This indicates that it is both the overall pore connectivity and size as well as the interaction between the two fluid phases in the pore space that change with age. The spatial distribution of moisture for the 28-day curing cases in Fig. 5 clearly shows how the model predicts that it is mainly the capillary pore space, i.e. s w , which is emptied, both during self-desiccation and drying. The gel pore space, i.e. sWs , is still after 200 days of drying very close to one in most parts of the specimen. This is in line with the findings from many recent experimental investigations aimed to characterize the state of water in concrete and cement pastes, see for example references [6,28,29]. It should here be recalled that the capillary pore space in this study is considered to include all pores larger than 3 nm following the definition by Powers and Brownyard [1],
although sometimes pores up to 12 nm are seen as part of the gel [28]. 6.3. Creep, shrinkage and tensile cracking The third example demonstrates the creep and shrinkage parts of the proposed model by looking at the recent experimental investigation by Theiner et al. [75]. Their study looked at the compressive creep as well as shrinkage of sealed and unsealed cylindrical specimens with dimensions 150×450 mm. The used concrete had a w/c of 0.44, a a/c of 4.84, a cement content of 375 kg/m3 and a 28-day compressive strength of 35.9 MPa. All specimens were kept at a temperature of 20 °C and were unmolded at an age of 24 h, after which all sides were sealed. For the drying specimens, the seal on the cylindrical part of the envelope surface was removed after 2, 7 or 28 days, and then exposed to an ambient relative humidity of 0.65. Creep specimens were loaded at the same time with a pressure equal to approximately 30% of the compressive strength at the age of loading; this corresponds to 5.5, 8.7 and 10 MPa, respectively. As for the previous example, the finite element model is set up using axial symmetry and linear shape functions for all state variables. All model parameters are given in Table 2. As for the second example, no calorimetry nor temperature data were available and the parameters of the chemical model had to be determined together with those of the sorption isotherm by making use of the autogenous shrinkage measurements. However, due to the creep and shrinkage coupling described in Section 4.3.3, the creep model also affects the simulated response for this case. The creep and shrinkage mechanisms can therefore not be looked at independently. Moreover, this is the case for creep measurements as well, for example, the earlyage creep of sealed specimens is affected by self-desiccation and can thus not be considered as true basic creep as the mechanism intended for drying creep is active. The model parameters used were thus obtained by a manual procedure, looking simultaneously at all measurements available. The simulated creep on sealed specimens is shown in Fig. 6a as total deformation, which thus includes the autogenous shrinkage. As done by Theiner et al. [75], the zero strain level is set to an age of 1 day. In comparison with the measurements, the autogenous shrinkage is overestimated at ages up to approximately 20 days, which consequently also affects the creep response. Nevertheless, the agreement is good and the model is able to describe the early-age creep and how it is influenced by the ongoing hydration process. Looking at the response of the unloaded specimens in Fig. 6b, the model is accurate in describing the first stage of drying shrinkage but is slightly off with respect to the final deformation; especially for the two specimens exposed at 7 and 28 days. It should here be stressed, however, that the typical trend observed in the drying shrinkage tests is recreated. Regarding the response of the 212
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1
Degree of saturation [1]
Degree of saturation [1]
1 0.8 0.6 0.4
sW tot sw s Ws
0.2 0
0.8 0.6 0.4
sW tot sw s Ws
0.2 0
0
5
10
15
20
0
5
10
15
Depth from drying surf. [cm]
Depth from drying surf. [cm]
(a)
(b)
20
Fig. 5. Spatial distribution of the moisture content for the M concrete (a) and H concrete (b) at different ages for the case of 28 days of moist curing. Different colors correspond to the three degrees of saturation from Eqs. (19) to (21) and markers to different times: ( ) 28 days, (○) 50 days, (◊) 100 days and (△) 200 days.
unsealed and loaded specimens in Fig. 6c, the measured and simulated response again shows good agreement. What can be noticed is that the drying creep effect is almost non-existent in the experimental data, which is also predicted by the simulations. This could be an indication that the mechanism of drying creep is active also for the sealed specimens at early-ages due to self-desiccation, as hypothesized in the proposed model framework. Lastly, to illustrate the capability of the model to describe crack formation in maturing concrete, the shrinkage test series by Theiner et al. [75] is modified in a numerical experiment aimed at studying the behaviour of the specimen under restrained conditions. This is achieved by restraining all vertical displacements at the top and bottom surfaces of the cylindrical specimen. To obtain a stable and well-defined crack, a notch is also added at mid-height of the specimen. The depth of the notch corresponds to a single finite element and it is located at the vertical symmetry plane of the model. All other model parameters are kept constant in comparison to the simulations presented in Fig. 6b. The simulated notch openings for the three different cases are presented in Fig. 7. There is a clear reduction of the notch displacement with the age of initial drying. This can of course in part be attributed to the lower observed shrinkage in Fig. 6b, but a significant part can also be explained by the increased maturity of the concrete. For example, it is clearly visible that it takes longer time for the crack to propagate through the thickness of the specimen for more mature concrete, which corresponds to the change in slope at approximately 85, 110 and
Fig. 7. Simulated notch openings from the numerical experiment of the notched and restrained specimen subjected to drying at different ages.
175 days for the three cases, respectively. To further illustrate this, the extent of tensile damage ω+ is shown in Fig. 8 for the specimen subjected to drying after 2 days. The extent of damage is shown before and after the crack propagates through the thickness as well as at the end of the analysis. Apart from the damage localized to the crack band that propagates from the notch, a field of diffuse damage is visible at the drying surface. However, once the full crack band from the notch is formed, this field is stable.
Fig. 6. Simulated total deformation during sealed and loaded (a), unsealed (b) and unsealed and loaded conditions (c) compared to experiments by Theiner et al. [75] shown by markers. All figures include the autogenous deformation (black line) for reference.
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load and its variation with the age of loading. However, the post-peak behaviour could only be accounted for qualitatively. The second example concerning moisture movement (internally and to the exterior) showed how the model can describe the ageing of transport properties and transfer of moisture between different forms. It also pointed to the importance of including the curing conditions in modelling the earlyage moisture movement; as this not only directly affects the amount of water held by the material, but also the ageing of transport properties, e.g. the intrinsic permeability. The third example on time-dependent deformations demonstrated the creep and shrinkage coupling used in the model. It highlighted a difficulty in independently calibrating model parameters when using a mechanistic description of shrinkage, in which the measured shrinkage to a large extent is described as a creep deformation caused by an internal load in the pore space. Hence, the model parameters of the creep model affect both the simulated response for externally loaded and unloaded specimens. Due to this interaction and the many parameters, one often has to resort to a manual procedure to find appropriate values of the model parameters; although prediction formulas, e.g. [56,67], can provide good initial guesses. It was also demonstrated how the model can describe crack propagation in maturing concrete by restraining the shrinkage specimen. An appealing novel feature of the proposed model is the inclusion of a microstructural model explicitly in the governing equations, which allows for tracking of individual constituents of the solid phase, as exemplified in Fig. 3c, and how these are affected by and affects the thermodynamic quantities of the system. In an extension, this feature can be generalized to track for example the volume fraction of individual hydration products, e.g., for use with micromechanics models or in modelling deterioration of the cement paste (e.g. leaching). It is also possible to include additional solid constituents in the model, for example silica fume when dealing with high-performance concretes or alkali-silica gel for modelling ASR damage. Furthermore, the explicit decomposition of the pore water into capillary and gel pore spaces together with the proposed split of the sorption isotherm, is theoretically in line with recent experimental findings [6,28,29]. Simulations of drying at a relative humidity of 0.5 showed that the model predicts how the larger capillary pores (> 3 nm) are emptied, while almost all water held in the gel is retained. This model property can be further explored to describe the fixation of pore water in a more physically accurate manner compared to the often phenomenological sorption isotherms used, i.e. by describing the fixation of water based on its location within the pore space rather than only how tightly it is bound.
Fig. 8. Simulated extent of tensile damage ω+ at different ages of the notched and restrained specimen subjected to drying 2 days after casting. Notice that the notch is located at a symmetry plane and that displacements are magnified by a factor of 200.
7. Conclusions A mathematical model of the hygro-thermo-chemo-mechanical behaviour of early-age concrete has been presented using a multiphase porous media framework, with concrete described as a three-phase material. Compared to previous models on a similar theoretical framework, e.g. [20-23], it suggests novel and modified features including:
• Quantification of different solid constituents of concrete explicitly in the multiphase framework, • Separation of different forms of water (chemically bound, gel and capillary) by including phase changes, • Age-dependent properties for the mass transport of fluids and the mechanical behaviour, • Tensile and compressive fracture of the solid skeleton using damage mechanics • Thermo-visco-elastic behaviour of the solid phase, • Creep and shrinkage coupling by means of the effective stress principle.
The model was implemented by means of the FEM and solved using a fully-coupled approach, where all state variables are solved for simultaneously. By applying the proposed model to three experimental studies from the literature, it was shown that it is able to describe many important aspects of the early-age behaviour of concrete. In the first example, studying the short-term mechanical behaviour, it was shown that the model accurately accounts for the stress-strain behaviour up to peak
Acknowledgements The research presented was carried out as a part of “Swedish Hydropower Centre -SVC”. SVC has been established by the Swedish Energy Agency, Energiforsk and Svenska Kraftnät together with Luleå University of Technology, KTH Royal Institute of Technology, Chalmers University of Technology and Uppsala University. www.svc.nu.
Appendix A A few additional constitutive and physical relationships are briefly presented in the following, see for example Lewis and Schrefler [46] for a more in-depth description. A.1. Equation of state The volumetric behaviour of all liquid and solid species is assumed to follow the linearized forms proposed in reference [46]. For the liquid water W
=
where
W ref W ref
1
W
(T
Tref ) + 3
1 (pw KW
w pref ) ,
= 1000 kg/m at Tref = 4 °C and
w pref
(A.1)
= 1 atm. For the solid species, additionally the deformation of the solid phase is accounted for and
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i
=
i ref
i (T
1
Tref ) +
1 s 1 p + s i I1s , Ki 3 K
(A.2) i
where is the first invariant of the effective stress tensor In Eq. (A.2), the definition of and K depends on the type of aggregate and cement used, with values used herein given in Table 1. Lastly, the volumetric behaviour of two gas species as well as their mixture is given by the ideal gas law s.
I1s
ig
=
Mi ig p , RT
i
i ref ,
i
(D , W ).
(A.3) ig
The molar mass Mi and the partial pressure p can be calculated using Dalton's law. A.2. Mass fluxes The volume-averaged advective flux of the two fluid phases w and g is described by Darcy's law, which, given appropriate assumptions [22,25,26,76], takes the form
krf k ( pf µf
s f v f ,s =
f g),
f
(w, g ),
(A.4)
f
where the dynamic viscosities μ are considered as temperature dependent physical properties. The diffusive mass flux of species in the gas phase is described by Fick's law of diffusion, which, given appropriate assumptions [26,76], takes the form
sg
Wg uWg
where
g DWg d
= JWg =
DWg d
Wg g
,
(A.5)
is the effective diffusivity tensor. Following for example Gawin et al. [77], this tensor can be defined as
g DWg d = s fs D v0
T T0
5/3
pref pg
I,
(A.6)
2
where Dv0 = 25.8 mm /s is the diffusivity of vapor in air at the reference temperature T0 = 0 °C and pressure pref = 1 atm. Lastly, it follows from the definition of the gas phase as an ideal gas mixture and Dalton's law that JWg + JDg = 0 . A.3. Conductive heat flux The conductive heat flux qi in each species i of each phase α is obtained using Fourier's law. Then the total heat flux q is obtained through summation such that
q=
eff I
(A.7)
T.
with the volume-averaged thermal conductivity of the porous medium calculated as eff =
sw W + s g g + s ( As A + Cs
C
+
Hs H
+
(A.8)
GssWs W )
i
Each λ in Eq. (A.8) is in principle dependent on the temperature, but are here considered as material constants, see Table 1.
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Pesavento, B.A. Schrefler, Modelling creep and shrinkage of concrete by means of effective stresses, Mater. Struct. 40 (6) (2007) 579–591, https://doi. org/10.1617/s11527-006-9165-1. [69] F.-J. Ulm, F. Le Maou, C. Boulay, Creep and shrinkage coupling: new review of some evidence, Rev. Fr. Genie. Civ. 3 (1999) 21–37. [70] COMSOL Multiphysics ver 5.2a, COMSOL AB, Stockholm, Sweden, 2016. [71] O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, 7th ed., Butterworth-Heinemann, Oxford, UK, 2013. [72] M. Jirásek, P. Havlásek, Accurate approximations of concrete creep compliance functions based on continuous retardation spectra, Comput. Struct. 135 (2014) 155–168, https://doi.org/10.1016/j.compstruc.2014.01.024. [73] A.A. Khan, W.D. Cook, D. Mitchell, Early age compressive stress-strain properties of low-, medium, and high-strength concretes, ACI Mater. J. 92 (6) (1995), https:// doi.org/10.14359/9781. [74] J.-K. Kim, C.-S. Lee, Moisture diffusion of concrete considering self-desiccation at early ages, Cem. Concr. Res. 29 (12) (1999) 1921–1927, https://doi.org/10.1016/ S0008-8846(99)00192-1. [75] Y. Theiner, M. Drexel, M. Neuner, G. Hofstetter, Comprehensive study of concrete creep, shrinkage, and water content evolution under sealed and drying conditions, e12223–n/a, Strain 53 (2) (2017), https://doi.org/10.1111/str.12223. [76] S.M. Hassanizadeh, Derivation of basic equations of mass transport in porous media, Part 2. Generalized Darcy's and Fick's laws, Adv. Water Resour. 0309-1708, 9 (4) (1986) 207–222, https://doi.org/10.1016/0309-1708(86)90025-4. [77] D. Gawin, C.E. Majorana, B.A. Schrefler, Numerical analysis of hygro-thermal behaviour and damage of concrete at high temperature, Mech. Cohes.-Frict. Mater. 4 (1) (1999) 37–74, https://doi.org/10.1002/(SICI)1099-1484(199901) 4:1<37::AID-CFM58>3.0.CO;2-S.
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Cement and Concrete Research journal homepage: www.elsevier.com/locate/cemconres
On the behaviour of concrete at early-ages: A multiphase description of hygro-thermo-chemo-mechanical properties
T
Tobias Gasch , Daniel Eriksson, Anders Ansell ⁎
KTH Royal Institute of Technology, Department of Civil and Architectural Engineering, Stockholm SE-100 44, Sweden
ARTICLE INFO
ABSTRACT
Keywords: Hydration (A) Microstructure (B) Creep (C) Shrinkage (C) Modelling (E)
Understanding the early-age behaviour of concrete is of importance for designing durable concrete structures. To contribute to the improvement of this, a hygro-thermo-chemo-mechanical model is presented that accounts for phenomena such as hydration, external and internal drying, self-heating, creep, shrinkage and fracture. The model is based on a multiphase porous media framework, using the Thermodynamically Constrained Averaging Theory (TCAT) as starting point to derive the governing equations of the system. This allows for a systematic treatment of the multiscale properties of concrete and how these develop during hydration, e.g. chemical and physical fixation of water. The proposed mathematical model is implemented within the context of the Finite Element Method (FEM), where all physical fields are solved in a fully-coupled manner. Chosen properties of the model are demonstrated and validated using three experimental results from the literature. Generally, the simulated results are in good agreement with the measurements.
1. Introduction Durability is one of the most important issues that have to be accounted for, in both the design of new constructions and in assessments of existing concrete structures. This is in turn, to a large extent, dictated by the ability of the concrete to withstand various deterioration processes such as alkali-aggregate reactions, chloride ingression, leaching and frost damage. The resistance to most of these processes is closely related to the characteristics and evolution of the porous cement paste, from casting and throughout the service life of the structure. Thus, understanding the early-age behaviour of concrete is of major importance, since it is during this period that many properties of the hardened cement paste are determined. If not cured properly, the porosity of concrete can become larger than desired, and it may exhibit significant cracking; this increases the permeability and consequently the transport of moisture, which often has a negative effect on the durability. There is an extensive base of knowledge on the early-age behaviour of concrete available in the literature, including both experimental and theoretical studies on topics such as self-heating, self-desiccation and the chemical processes involved in the hardening [1-7]. Over the years, it has become evident that to accurately predict and understand such processes there is a need to develop advanced modelling tools, which consider not only individual properties of concrete but many of the
⁎
observed phenomena in a single framework since they all interact. To this end, the interaction between physical phenomena such as temperature and mechanical stress has been widely analysed in the literature, also including the early-age behaviour [8-12]. More recently, models that also consider the effects of moisture transport and how this affects both stresses and the chemical reactions have been suggested [13-18]. However, one of the most interesting developments is the adoption of multiphase porous media theory [19] to early-age concrete by Gawin et al. [20,21] and succeeding developments on a similar framework by for example Sciumé et al. [22] and Jefferson et al. [23]. This approach allows for a more physically accurate description of the complex processes involved in the hardening of cement, their interaction and the development of the microstructure. This study aims to develop and present a three-phase porous media model for the hygro-thermo-chemo-mechanical behaviour of concrete from early-ages and beyond. It follows in the line of the works by Gawin et al. [20,21] and Sciumé et al. [22] among others, using the Thermodynamically Constrained Averaging Theory (TCAT) developed by Gray and Miller [24,25] as a starting point. While the developed model shares many features and utilizes model equations on a similar form as previous works, it aims to improve and modify certain aspects with regard to early-age concrete. This includes the overall description of the multiphase system and its development with the cement hydration. Furthermore, new and modified equations are proposed for the
Corresponding author. E-mail addresses: [email protected] (T. Gasch), [email protected] (D. Eriksson), [email protected] (A. Ansell).
https://doi.org/10.1016/j.cemconres.2018.09.009 Received 15 November 2017; Received in revised form 14 September 2018; Accepted 21 September 2018 Available online 06 December 2018 0008-8846/ © 2018 Elsevier Ltd. All rights reserved.
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Level III > 10-3 m Concrete / mortar
Level II < 10-4 m Cement paste
Level “0” C-S-H solid 10-8 -10-9 m
Level I < 10-6 m C-S-H gel
Macro pore
Aggregate
Adsorbed water Aggregate
Intra C-S-H water Inter-layer water Chemically bound water
Fine aggregates
Unhydrated clinker
CH + other hydration products
Small gel pore 1-3 nm
Large gel pore 3-12 nm
Bulk water
Fig. 1. Schematic multiscale description of concrete based on the conceptual model by Jennings et al. [30].
definition by Powers and Brownyard [1], the gel porosity only includes pores smaller than approximately 3 nm and would thus only include the SGP in Fig. 1. Concerning the state of water in these different parts of the pore space, Ulm et al. [27] pointed out that it is only water in pores outside the C-S-H particles that should be considered to behave as a fluid using a similar categorization as Jennings [28]. Water held in smaller pores should be considered as structural water; not to be confused with the chemically bound water. With this description in mind, the solid phase s of the porous media is in this study considered to consist of aggregates A, anhydrous cement C and hydration products H as well as a gel pore space G partially filled with structural water W. For simplicity, the definition of Powers and Brownyard is used for the gel porosity, which thus includes the pore and inter-layer space intrinsic to the C-S-H solids as well as the SGP, while larger pores are considered part of the capillary pore space. For use in forthcoming Section 3, where model equations of the porous media are presented, this means that water held in the gel pores (< 3 nm) will be considered as structural water, and therefore as a part of the solid phase; although this includes water in the SGP as opposed to the definition by Ulm et al. [27]. Both fine and coarse aggregates are considered parts of species A and the hydration products H include both C-S-H as well as for example ettringite and portlandite. As will be described in the following, the volume fractions of these components are considered to change with the cement hydration. The capillary pore space makes out the remaining part of the porous media, i.e. the LGP and macro pores. It is considered to be occupied by two fluid phases, a wetting phase w of bulk water W and a non-wetting gaseous phase g, which is an ideal gas mixture of dry air D and water vapour W.
constitutive and physical behaviour of the porous media to account for the complex features of early-age concrete, including the ageing of material properties, creep, shrinkage and fracture. 2. Characterization of multiphase porous media for early-age concrete In porous media theory, concrete is normally considered a threephase material consisting of two fluid phases and a homogenous solid phase [26]. However, theoretical and experimental investigations of cement paste at lower scales point to a much more complex picture [4,6,27-29]. Considering this, there is a need to revisit and improve the simplified description of in particular the solid phase used in most models to better reflect the true material, also in models intended for analysis at larger scales. This is especially important for early-age applications, and likely also in other applications related to durability. Following the multiscale description of concrete and cement paste by Ulm et al. [27], Jennings et al. [30] and Jennings [28], a schematic description of concrete at different scales is shown in Fig. 1. Starting at the level of concrete and mortar (Level III), coarse and fine aggregates are embedded in a porous cement paste. This cement paste (Level II) is, in turn, made up of macro pores, unhydrated clinker plus portlandite (CH) and other hydration products embedded in a calcium-silicate-hydrate (C-S-H) gel. On the level of the C-S-H gel (Level I), a description following the colloidal model (CM-II) by Jennings [28] is used in Fig. 1. However, for the purpose of the current study, this choice has no major impact and other descriptions of the C-S-H gel, such as that by Feldman and Sereda [31], would be equally valid. In Jennings model, the gel is thought to be made up of C-S-H solids that are particles with an internal structure and pore space referred to as globules and shown in Level “0”. These consist of sheets of reacted clinker and water in different forms as seen in Fig. 1. The structure of the gel is formed by the packing of such globules in different arrangements, leading to two distinct inter-particle pore spaces; small gel pores (SGP) and large gel pores (LGP). Considering that the purpose of this study is to formulate a mathematical model that includes the transport of water and conversion of water between different forms, a discussion of different parts of the pore space in relation to this is warranted. Categorization of the pore space in concrete is difficult and, as pointed out by Jennings [28], several definitions of the gel porosity (and consequently also the capillary porosity) can be found in the literature. With the above description of the gel by Jennings [28], pores smaller than 12 nm are considered intrinsic to the gel. However, according to the classical
2.1. Cement hydration Hydration of Portland cement consists of a set of complex chemical and physical phenomena at the microscopic level. The chemical model employed herein follows the work of Ulm and Coussy [8,32], in which this set of reactions are lumped into a single reaction described by for example the hydrate mass mhyd, which denotes the mass of chemically bound water per unit volume. For modelling purposes it is, however, more convenient to introduce the hydration degree C = mhyd / mhyd as a state variable, which is the amount of hydrate mass normalized with the hydrate mass at complete hydration mhyd . The model furthermore includes the later developments by Cervera et al. [9] and Di Luzio and Cusatis [13]. The kinetics of hydration is governed by the diffusion of free water 203
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through the hydrates that form in layers around the cement grains on the microscopic level. Acknowledging the well-known fact that hydration is thermally activated, an Arrhenius type evolution law is formulated for ξC following Ulm and Coussy [32] such that
d dt
C
= AC ( C )
w
QC , RT
(s w ) exp
Hs
Ac2 C
+
C
(
Gs
(1)
C ) exp
c
C
,
(2)
2.2. Evolution of the solid phase In earlier adoptions of porous media theory to the early-age behaviour of concrete, the solid constituents have not been considered to be dependent on the hydration process [20-22]. Herein, this is explicitly accounted for in the model equations. The development of each constituent here follows that classical model by Powers and Brownyard [1] in that their volume fractions are linearly related to the extent of cement hydration. However, other more recent models could also be considered, for example the extension of the Powers and Brownyard model to include silica fume by Jensen and Hansen [38] or the model by Königsberger et al. [39] in which the assumption of linearity is abandoned. As will be evident later in Section 3, it will be necessary to define the volume fractions of the different constituents with respect to the solid phase to close the mathematical model. Hence, volume fractions will be referred to the total volume of the solid phase, not the total volume as is usually done [1,5,34]. Following the nomenclature used in TCAT [24,25], the sought volume fractions are referred to as is where i ∈ (A, C, H, G). Using the model of Powers and Brownyard [1], we can define these as
= A
Cs
= C
a/c A
a/c A
+
+
(
(
a/ c knw N
+
N
W
)
C
+
1
+
kgw W
)
C
+
1 C
knw N
N
a/c A
+
(
C
kgw
+
C
W
kgw
=
) )
knw
1
, (6)
C
C
+
N
+
kgw W
)
C
+
1
. (7)
C
3.1. Conservation of mass The considered porous media involves a total of seven mass balances, one for each species in each phase. However, as is usually done for concrete, these are arranged to reduce the number of mass balances in the final model formulation to three: one for the solid species (A, C, H), one for water W and one for dry air D [26]. Starting with the mass balance of all solid species, it is arranged to express the change of the total solid volume fraction
,
t
(4)
C
C
1 knw
kgw
knw
+
In this work a mechanistic approach [40] is adopted to derive the general model equations of the porous media by considering the conservation of its mass, momentum and energy. This is done with the Thermodynamically Constrained Averaging Theory (TCAT) developed by Gray and Miller [24,25] as a starting point and follows a similar procedure as in previous work on adapting multiphase porous media theory to early-age concrete, see for example the works by Gawin et al. [20,21] and Sciumé [41]. For the sake of brevity, only the final form of the macroscopic conservation equations used in the model definition are presented in the following. As is usually done in multiphase descriptions of concrete, see for example [26], it is here considered to be a three-phase material consisting of a solid phase s, a wetting phase w and a non-wetting gaseous phase g. The composition of each phase was discussed in Section 2, where especially the break down of s into species A, C and H as well as a gel pore space will have an impact on the general model equation if compared to existing models in the literature. It must also be pointed out that an assumption of small displacements of the solid is made throughout this study. Consequently, no difference is made between the spatial and material reference frames and all equations are, hence, written in their partial derivative form.
where aw is a model parameter. It was found by Norling-Mjörnell that aw = 4 gave good agreement with experimentally evaluated values of βw and is thus be used throughout this study.
As
C
3. Conservation equations for early-age concrete
(3)
(s w ) = ( s w ) a w ,
1
In Eq. (4) to Eq. (7), the term within brackets in the denominator is the total volume of the solid phase at a certain degree of hydration. The main parameters that enter the equations are the aggregate-to-cement ratio a/c, the mass of chemically bound water per mass hydrated cement knw (≈ 0.23 g/g [34]) and the mass of water held in the gel pores per mass hydrated cement kgw (≈ 0.19 g/g [34]). Furthermore, ρi is the intrinsic density of species i, but it should be noticed that ρN is the density of the chemically bound water, here set to 1350 kg/m3 [34]. From Eq. (4) to Eq. (7) it is clear that the total volume of the solid phase increases with the hydration degree ξC. However, since ρW/ρN < 1, the solid phase occupies a smaller volume than that of the reactants, i.e. the proposed microstructural model directly accounts for the chemical shrinkage associated with cement hydration. Lastly, it should be pointed out that the w/c ratio is used to define the initial volume occupied by the solid phase in the multiphase system, but does not directly affect the composition of the solid phase.
where Ac1, Ac2 and ηc are model parameters controlling the reaction rate. The asymptotic degree of hydration C describes the fraction of the cement available for hydration. Its value depends highly on the waterto-cement ratio w/c and is here estimated using the empirical formula by Pantazopoulou and Mills [33]. The thermal activation of the reaction is controlled by the Arrhenius term, where QC is the hydration activation energy that is normally close to 500 kJ/kg, T is the temperature and R is the universal gas constant. The influence of moisture saturation on the cement reaction is well-known, where the reactions effectively cease at relative humidities below approximately 0.75 [1,5,34,35]. To account for this in a simple manner, an empirical factor βw is introduced in Eq. (1). Different forms have been proposed for βw, most often depending on the pore relative humidity [13,20,35,36]. However, it is here recognized that hydrates mainly precipitate in the water-filled parts of the capillary pore space (pores > 3 nm), and βw is thus assumed to be a function of the degree of capillary saturation s w . An empirical expression proposed by Norling-Mjörnell [37] is used w
a/c
W
C
C
( +(
A
where the normalized affinity AC controls the rate of the reaction. The affinity is here defined using the analytical function proposed by Cervera et al. [9]
AC ( C ) = A c1
=
s
where
,
Ss
(5)
1
=
=
s
Ss
vs
t
Ss
+
(
s Ss v s)
+ k nw m 0C
d dt
C
,
(8)
is the velocity of the solid phase and
A As
+
C Cs
+
H Hs .
(9)
It should be mentioned that all intra phase motions of species are 204
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T. Gasch et al.
Table 1 Physical properties of species in the multiphase model used in the current study.
Liquid water Gas mixture Aggregates (granite) Anhydrous cement Hydration products
Species i
Phase α
Thermal conductivity λi
Thermal expansion αi
Reference density
Cpia
Specific heat
Bulk modulus Ki
W W+D A C H
w,s g s s s
4181 J/(kg⋅K) 1005 J/(kg⋅K) 790 J/(kg⋅K) 840 J/(kg⋅K) 1550 J/(kg⋅K)
0.591 W(m⋅K) 0.025 W(m⋅K) 2.5 W(m⋅K) 0.29 W(m⋅K) 1.8 W(m⋅K)
207⋅ 10−6 1/K − 25⋅ 10−6 1/K 50⋅ 10−6 1/K 35⋅ 10−6 1/K
1000 kg/m3 − 2750 kg/m3 3150 kg/m3
2.2 GPa − 45 GPa 105.2 GPa 14.9 GPa
( sw
W)
+
t
(
s Gss Ws W )
d C + dt w W s (s v)+ ( s g Wg v s) +
k nw m0C + +
( sw
+
w v w, s)
t
( sg
+
=(
t +
Dg )
( sg
+
Dg uDg )
( sg
Dg v g , s)
+
= 0,
( sg
+ (s w
W
+ s g g) .
(13)
As for the linear momentum balance, the energy conservation of the porous media is governed a by single model equation obtained through summation over all species. To arrive at this equation, it is assumed that all phases are in local thermodynamic equilibrium and recalling the assumptions introduced in Section 3.2, it follows that any contributions from viscous dissipation and mechanical work are negligible [40]. Also, any advective transfer of energy with the solid phase is considered negligible. Finally, we express the energy balance in terms of enthalpy to obtain a model equation on a similar form to the classical heat equation:
(10)
where v w, s is the relative velocity of w with respect to s. Furthermore, Wg is the vapour concentration and uWg is the diffusive velocity of W in g. The volume fraction of the capillary pore space saturated with liquid water is given by s w , while the volume fraction of the gel pore space occupied by water is described by sWs . Lastly, it is recognized that any mass exchange of water between phases cancels out following the summation over all phases to obtain Eq. (10). The influence of dry air in the gel pore space is considered negligible, hence, it is only present in the capillary pore space and its mass balance is given as
( sg
W s Ws Gs) s
+
3.3. Conservation of energy
Wg v g , s)
( s GssWs W v s) ( s g Wg uWg) = 0,
Ss
It should be mentioned that in some other adoptions of multiphase porous media theory to early-age concrete, Eq. (12) is introduced in its rate-form [20,22]. This difference will, of course, affect the final system of equations and its numerical implementation.
Wg )
( sg
(1 + knw ) C C 1 + knw N
~ where t is the damaged total stress tensor (see the forthcoming Section 4.3.1) and g is the total body force vector. The total density ρ is calculated as
neglected in Eq. (8); this assumption should only have minor influences on the behaviour, at least after setting. The last term inside the brackets represents the chemically bound water during hydration, where m0C is the amount of cement per volume of the concrete mixture. Moreover, s / t, , it follows that s / t = since the capillary porosity = 1 which is utilized in the other model equations. In previous work on multiphase models for early-age concrete (e.g. [20,22]) water is only present in the two fluid phases. However, following the discussion in Section 2, water in the smallest pores cannot be considered to behave as a fluid and should thus not be lumped together with the more mobile water in larger pores. In our model, water held in the gel pore space is considered as part of the solid phase and is thus not contributing to the fluid flow. Hence, water is present in all three phases and its total mass balance becomes
t
i pref
( Cp )eff +
t
Hcem m 0C
T
w CWw v w, s p
q + (s w
d C + Hvap Mvap dt g g + s Cpg v g, s) T = 0,
(14)
where q is the total conductive energy flux and
( Cp )eff =
w CWw p
sw +
s AsC As p
+ s g g Cpg + +
s CsC Cs p
+
s Gss WsCWs p s HsC Hs . p
(15)
(15), Cpi
In Eq. is the isobaric specific heat capacity of species i in phase α; here considered to be constant, see Table 1. Two heat sources are present in Eq. (14) related to the heat released during hydration and the phase change of water between liquid and gas form. The heat released during hydration is controlled by the hydration enthalpy ΔHcem that depends on the composition of the cement, but is in the region of 500 kJ/kg reacted cement for normal cements [13,34]. The energy released during evaporation of water is quantified by ΔHvap, which is here set to 2257 kJ/kg evaporated water. The rate of evaporated water Mvap can be obtained from the mass balance of water in the wetting phase, see for example references [20,26,41].
Dg v s)
(11)
where s g is the volume fraction of the capillary pore space saturated with the gas mixture and v g , s is the relative velocity of g with respect to s. Furthermore, Dg is the mass concentration of dry air and uDg is the diffusive velocity of D in g. 3.2. Conservation momentum The mechanical behaviour of the porous media is governed by a single model equation obtained through the summation of the linear momentum balance of all species. As done in many applications of concrete related to durability and early-age behaviour, and so also in our model, velocities are considered small and the time scale of interest large (days), meaning that inertial forces can be neglected. Using the same argument, the forces due to mass exchanges are also small and can be omitted from the formulation [40]. This assumption yields a linear momentum balance for the entire porous media on the classical form ~ t + g = 0, (12)
3.4. Choice of state variables The above model equations include many unknowns that have to be defined, either directly as physical constants or using constitutive relationships to relate them to the state variables of the model. Hence, the choice of state variables to be used in the numerical implementation of the mathematical model warrants some attention. Starting with the obvious choices, the solid phase volume fraction s , the displacements d and the temperature T are used as state variables in Eqs. (8), (12) and (14), respectively. Also, for the total dry air content the choice is rather 205
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straightforward and the gas pressure pg is chosen as the primary variable in Eq. (11). For the mass balance of total water content in Eq. (10), however, the choice is more difficult and many alternatives can be found in the literature; for example the degree of water saturation s w , the relative humidity φ and the liquid water pressure pw. In our model we follow the work of Gawin et al. [20] and use the capillary pressure pc. Gawin et al. have for many applications shown the validity of this choice to describe the whole range of possible moisture states in concrete. They have also shown the efficiency and stability of using pc as a state variable in numerical implementations.
compressive strength. Given that the evolution of compressive strength reflects the properties of the concrete microstructure, κf will also be used to describe the development of transport properties of the early-age concrete, which are highly dependent on for example the connectivity of the pore system. This assumption in principle means that the resistance of the concrete to fluid flow depends on the curing temperature. 4.2. Hygro-thermal behavior The hygro-thermal behaviour of concrete is strongly coupled, where the transport and storage of both fluids and energy interact. The transport of both quantities follows the standard forms of Darcy's, Fick's and Fourier's laws, see Appendix A. However, as indicated, the characteristics of the mulitphase material change dramatically during hydration, which affects both the fluid storage capacity and also the transport properties. Constitutive laws that account for this are presented in the following. Given the choice of capillary pressure as a state variable, it is important to first introduce two key relationships on how pc relates to the quantities of the two fluid phases. Assuming a state of local thermodynamic equilibrium and moderate temperatures, the equilibrium state of capillary water with vapour follows the Kelvin equation
4. Constitutive and physical relationships for early-age concrete To completely describe the hygro-thermo-chemo-mechanical behaviour of early-age concrete, the governing equations presented in Section 3 must be complemented by a set of constitutive and physical relationships. Many different versions of the necessary relationships can be found in the literature and most of those used in the proposed model are taken from previously published results, either as previously applied to multiphase models or adequately modified to suit the current model formulation. Those of most importance for the early-age behaviour of concrete are presented in the following. However, for completeness, a few additional constitutive relationships that are fundamental for porous media theory are presented in Appendix A. For a more in-depth derivation and description of these, the reader is referred to previous work in the literature [20-23,26,40-42].
=
Evolution of the material properties of concrete that depend on the characteristics of the microstructure with time needs some extra attention and can be difficult to relate directly to the hydration degree. To properly describe the evolution of for example many mechanical properties, at least the phenomenon of setting needs to be accounted for, which can be accomplished using the model by De Schutter and Taerwe [43] often used in models of early-age concrete. However, an important aspect not covered in their model is how the kinetics of the reaction affects the development of the pore structure and in extension the mechanical properties [2,44,45]. To this end, a more refined model was suggested by Cervera et al. [9] to account for the effects of curing temperature. Supported by experimental evidence, they concluded that a low curing temperature leads to a more uniform pore structure than a higher temperature, which in turn leads to a higher strength at complete hydration. To describe this phenomenon, Cervera et al. [9] proposed introducing an ageing degree κf, which describes how the compressive strength evolves with time as a function of the degree of hydration and the curing temperature. This ageing model is employed here with the evolution of κf defined as f
=
TT TT
T Tref
nT
(B f
2Af
C)
d dt
C,
W
= exp
MW c p W RT
,
(17)
where the definition of relative humidity φ is introduced as the ratio between the vapour pressure and the temperature dependent vapour Wg pressure at saturation psat . Additionally, the macroscale capillary pressure is defined as the pressure difference between the non-wetting and wetting phases, which can be expressed as
4.1. Evolution of material properties
d dt
pWg psatg
pc = p g
pw ,
(18)
and derived from the second law of thermodynamics given a state of local thermodynamic equilibrium, as shown by for example Pesavento et al. [42]. The physical interpretation of capillary pressure and thus of Eqs. (17) and (18) is only valid as long as water exists in a state where it can form a capillary meniscus. However, it is shown by Lewis and Schrefler [46] and Gawin et al. [47] that the mathematical meaning of pc can be extended to include the entire moisture range. 4.2.1. Sorption isotherm Sorption isotherms are closely related to the microstructure of cement paste and in particular the pore size distribution. As discussed in Section 2, the microstructure is highly influenced by the hydration and changes significantly during the first days after casting. Thus, the definition of the sorption isotherm must be age dependent for a model that considers the early-age behaviour of concrete. In the context of multiphase porous media models, two different hydration degree dependent sorption isotherms were suggested by Sciumé et al. [22] and Chitez and Jefferson [48]. Both of these are developments of the classical expressions by van Genuchten [49] to also consider the degree of hydration. Another definition was used by Cramer et al. [50] in which the degree of water saturation was additively split into physical adsorption and capillary condensation, both parts as functions of the pore relative humidity. Here, a new definition is suggested that splits the total water saturation into water held in gel pores sWs and capillary pores s w , following the description in Section 2.2 and outlined in the governing equations presented in Section 3. The proposed saturation functions are based on the work of Norling-Mjörnell [37], who proposed and experimentally verified an analytical model for a hydration degree dependent sorption isotherm expressed in terms of moisture content and relative humidity. The model has also been successfully applied to moisture transport in early-age concrete by others using single-phase
(16)
where the first term controls the influence of the curing temperature through the model parameter nT. The temperature value TT represents the maximum temperature at which hardening of concrete occurs (≈100 °C) and Tref is a reference temperature. The second term in Eq. (16) controls the ageing with respect to the chemical reactions through model parameters Af and Bf. To reduce the number of model parameters, Di Luzio and Cusatis [14] concluded that Bf can be calculated as a function of Af , C and the hydration degree at setting 0C . Since κf is defined as a normalized strength, the evolution of the compressive strength can then be calculated as f = f f , where f is the compressive strength at C = C and T = Tref. Finally, as done in the models by De Schutter and Taerwe [43] and Cervera et al. [9], the evolution of other mechanical properties, such as tensile strength f+ and fracture energies Gf ± , is considered to be related to the development of the 206
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moisture transport models [13,15,18]. Functions of the same form as in reference [37] are used here, but reformulated in terms of degree of saturation instead of moisture content, and split into two separate variables to fit in the multiphase framework of the current study. Starting with the degree of saturation of liquid water in the gel pores, the following constitutive equation is proposed
sWs =
1
{exp [(ggs 1
10
[exp (ggs
C)
]}
10 C )]
1
network decreases rapidly, which in the mathematical model can be represented as a change in the intrinsic permeability. Using a similar formula as proposed by Gawin et al., this is here accounted for as
k = 10 Ak (1
(19)
and for the liquid water in the capillary pores, a constitutive equation on the form
sw =
10 C ) ]
exp [(gcs exp (gcs
10 C )
1 (20)
1
is proposed. Lastly, by definition it follows that = 1 The constitutive parameters ggs and gcs control the ageing described by ξC of the respective parts, where for simplicity we assume ggs = gcs = gs. While Eqs. (19) and (20) are dependent on the relative humidity φ, they can be expressed in terms of the state variables pc and T using Eq. (17). By considering the evolution of the microstructure, the degree of liquid water saturation in the entire porous media can be expressed as
sg
W stot =
s GssWs s Gs
+ sw ; +
(22)
I,
where k∞ is the intrinsic permeability at complete hydration and defines the flow characteristics in all directions. Ageing is in Eq. (22) controlled by the model parameter Ak and the ageing degree κf. As done by Sciumé [41], also the structure coefficient fs that controls the diffusivity in Fick's law is assumed to depend on the development of the internal structure of concrete during hydration on the same form as the intrinsic permeability by defining a value fs,∞. The effective permeability of the respective fluid phase is controlled by the relative permeabilities k rf that vary from 0 to 1. Commonly used definitions are those by Brooks and Corey [52] and van Genuchten [49], both of which are governed by the total degree of saturation of the pore space. To account for the changing microstructure also on this property, equations on the form proposed by Chung and Consolazio [53] for flow in concrete under elevated temperatures are adopted, where the relative permeabilities are assumed to depend on the capillary porosity ϵ and saturation s f , such that
1
,
f )k
sw.
g
(0.05 Arw )
10(0.05
Arw ) s g ,
(23)
w
(0.05 Arg )
10(0.05
Arg ) s w ,
(24)
k rw = 10 s
(21)
k rg = 10 s
although not used directly in any part of the model it is useful for evaluation purposes. The degrees of saturation described by Eqs. (19) to (21) are exemplified in Fig. 2 for different degrees of hydration and plotted in terms of capillary pressure. For comparison, the experimental results by Baroghel-Bouny et al. [51] are included in Fig. 2c. The consequence of this new definition is that during the very early-ages, the capillary system is emptied without any significant build-up of capillary forces if the sample is subjected to drying. On the contrary, water is bound much more tightly in the gel pore space, both during the early-age and beyond, which is in line with recent experimental findings [6,28,29]. Given the growth of s Gs and decrease of ϵ with hydration, the model also reflects that more water is held in the gel pore space for mature concrete.
where Arw and Arg are model parameters. Compared to Chung and Consolazio [53], it should be emphasized that it is here the capillary saturation that controls the flow of the respective phase, since no flow is assumed to occur in smaller pores. 4.3. Mechanical behaviour The mechanical behaviour of the porous media is controlled by the definition of the total stress tensor, which according to Gray et al. [54] may be expressed as
t=
s s
(25)
bP s I,
where is some measure of pressure acting in the system and b is Biot's coefficient [46,55], which relates the compressibility of the solid phase to the drained porous skeleton. The resistance of the solid matrix to external and internal loads is described by the effective stress s s . Given that it is only the solid phase that can resist deformations d, the stress-strain relationship takes the form
Ps
4.2.2. Mass fluxes The complex internal structure of concrete and its evolution highly influence the flow properties through the intrinsic permeability of the material in Darcy's law. For the early stages of hydration there is a decrease in overall porosity but, as pointed out by Gawin et al. [20] and perhaps more importantly for the flow properties, also a change in pore connectivity. The size of the largest pores of the continuous pore
s
=
were
e:
e
(e
e th
ecr),
(26)
is the fourth-order elasticity tensor defined by two material
Fig. 2. Proposed saturation functions in Eq. (19) for gel pores (a), in Eq. (20) for capillary pores (b) and in Eq. (21) for the total pore space (c) at different degrees of hydration for concrete with w/c = 0.48. Experimental results by Baroghel-Bouny et al. [51] included in (c) for comparison. 207
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constants, e.g. Young's modulus Ê and Poisson’s ratio ν. Both Ê and ν are here taken as age-independent [56]; notice that ageing of the elastic response, i.e. the measured Young's modulus, is accounted for in the definition of the creep tensor ecr. Additionally, contributions from temperature changes are given by the thermal strain tensor eth. Using the standard small strain definition, the symmetric total strain tensor e of the solid matrix relates to the state variable d as
e=
1 [ d + ( d) ]. 2
~+ = H (t ) 1
ws w s p
(28)
where sws is the fraction of the capillary pore surface in contact with W liquid water. The fraction sws is often assumed equal to stot . However, as discussed by Gawin et al. [21], this is likely not the case for a material with a large amount of fine pores as concrete, where water is also present as a thin film. Hence, for concrete such an assumption can lead to an underestimation of the shrinkage. Gawin et al. derived functions for the quantity sws from the experiments by Baroghel-Bouny et al. [51]. In our model, sws is assumed to be a function of the degree of water saturation in the gel pores, i.e. ws s
r±
±
pj ;
t =t
t+
+) t+
+ (1
)t ,
=
where A
B± =
(33)
(34)
re± r±
1 2
±
rp± re±
=
r± rp±
1 1 1
2
exp B ± (rp± 1 ldis
r±
ÊGf± (f ± ) 2
r±
if rp±
r±
rp± ,
rp±
re±
re±)/ re±
if r0±
(35)
and
+ A±
(rp±)3
3rp± + 2
6re± (rp±
1)
.
(36)
The shape of the uniaxial behaviour is thus controlled by the model parameters r0±, re± and rp± that define the initial value of r ± , the initial size of ~ ± and the value of r ± at peak strength, respectively, see [10] for more details. However, notice that due to the normalized framework, r0± = 1 is always true. In the definition of B ± , Young's modulus Ê, tensile and compressive strengths f ± and fracture energies Gf± are introduced to control the softening. Also in Eq. (36), a characteristic length ldis is introduced to ensure mesh-size objectivity of the solution using the crack-band approach [60]. Here, the value of ldis is inferred directly from the volume of the finite element in which softening occurs.
(30)
4.3.2. Thermal expansion The rate of thermal expansion is assumed to be caused by a volumetric thermal expansion coefficient αs of the solid phase. To account for the continuous solidification of reaction products and the changing microstructural properties, αs is here determined at the current material state. Making use of the mass conservation of the solid phase, αs can be identified from the total density of the solid phase ρs, see first term in Eq. (13). Introducing this definition of αs, the rate of thermal strains is given as
to separate the tensile (+) and compressive (−) components of t. In Eq. (30), tj is the jth principal stress and pj its corresponding directional vector. Furthermore, ⊗ denotes the tensorial product and the Macaulay brackets ⟨−⟩ the positive parts operator. The damaged stress tensor entering in the momentum balance (12) is then defined as
~ t = (1
A ± re± 1
3
t j pj
re , f
implying that r ± is the largest value of ~ ± reached by the material. Lastly, the damage variables ω ± are directly defined as functions of the current value of r ± , also using the function proposed by Cervera et al. [10]. For both tension and compression this function is given as
4.3.1. Damage When dealing with durability issues related to cement and concrete, cracking is an important effect that needs to be accounted for. Many different techniques are available in the literature, see e.g. [57], where it in this work is chosen to adopt the theory of continuum damage mechanics. Given that we are considering concrete during early-ages, it is important to treat damage in relation to ageing in a thermodynamically consistent manner. To this end, a model based on the work by Cervera et al. [10] is used. The formulation introduces a stress split
j=1
[ 3J2 + a I1 + b t1 ]
a
(32)
i (0, t )
where aws is a model parameter. This assumption could physically be motivated by the presence of thin films also in the capillary space as long as the smaller pores in the gel are saturated in the homogenized system.
t+ =
1 1
re+ , f
r ± = max [r0±, max (~i±)]
(29)
= (sWs )aws ,
a
[ 3J2 + a I1 + b t1 ]
where I1 and J2 are the first invariant and second deviatoric invariant of the undamaged stress tensor t, respectively. Furthermore, t1 and t3 are the maximum and minimum principal stresses and H (−) denotes the Heaviside step function. Parameters aω and bω control the shape of the failure surface and are functions of the age dependent compressive and tensile strengths f ± . The same form of failure criterion was used by Saloustros et al. [59] in a similar damage model, but not using the normalized framework, where the definitions of these two model parameters can be found. Lastly, re± are model parameters that define the initial damage thresholds. Damage growth is described by the normalized stress-like internal variables r ± that define the current damage threshold. Evolution of these internal variables can be written on the explicit form [10]
(27)
ws g s )p ,
+ (1
1
~ = H( t ) 3
In the poromechanical model formulation, the second term in Eq. (25) acts as an internal load that causes volumetric changes, i.e. shrinkage or swelling, due to change in for example moisture conditions. Also in the work of Gray et al. [54], they showed that if neglecting contributions from surface tensions and assuming that surface and phase averages of pressure are equal, P s takes the form
Ps =
1
(31)
where the two damage parameters ω+ and ω− account for tensile and compressive damage, respectively. Damage is then characterized in a normalized equivalent stress space as introduced by Cervera et al. [10], which allows comparison of different stress states also for different values of the ageing degree κf. 0 so that thermodynamically Using this framework ensures that ± consistent results are obtained also during ageing [10]. To characterize the limits of the elastic domain, two non-dimensional scalar measures are introduced, one for tension ~+ and one for compression ~ . These are given on the form proposed by Lubliner et al. [58] for a plasticity based model, and written as
s d d eth = I T= dt 3 dt
s 1 d I T. s 3 T dt
(37)
4.3.3. Creep Creep is described by the Microprestress-Solidification (MPS) theory developed by Bažant and co-workers over the last decades [36,61-65]. This model has previously been incorporated into a multiphase porous media framework by Gawin et al. [21] and used by several authors throughout the years [10,14,66] to describe concrete creep in multifield 208
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models, including early-age applications. It is here adopted in a form similar to the original model, but with minor reformulations to the solidification process and the evolution of microprestress. The total creep tensor is in the MPS theory split into two parts, one viscoelastic part ecr,v that mainly describes short-term creep and one viscous part ecr,f, which describes long-term creep. In rate-form, this strain split is described as
d d d d ecr = ecr,v + ecr,f = r + dt dt dt v dt
r
:(
f
s s ),
(t )
n f
(t ) (38)
d dt
t
t
q2 ln 1
t
0.1
0
0
:
d ( dt
s s )dt
,
(40)
1 f (S )
1
= cpS p 1,
S s
f (S )
(41)
1 d Cs dt
= 0,
(43)
(44)
= kc pc ,
f
+
1 µc pref
p d c p ( µc f ) p 1 dt
s
q4
= 0,
(45)
p pref
1
T0p 1
MW pW R
p 1
,
(46)
where T0 is a reference temperature. Using Eq. (45), the model to calculate the viscosity ηf requires three model parameters (p, μc and q4) to be identified, of which p is set to 2 in this study to keep the correct dimensionality of the model. Jirásek and Havlásek [65] identified an appropriate initial condition to Eq. (45) as ηf(t0) = t0/q4, where t0 is the age of the concrete at the start of the analysis. A significant difference in the current formulation compared to the original MPS formulation and its later developments is that s enters Eqs. (38) and (40), and not the total stress t. This modification to the MPS creep model was suggested by Gawin et al. [20], who argued that in a mechanistic description of shrinkage in a porous media model, the internal load P s causing shrinkage must also contribute to the creep. Hence, what is measured as macroscopic shrinkage in an experiment is actually, in the present model, accounted for by an instantaneous and recoverable deformation due to changes in P s , and in addition a creep deformation following the effective stresses caused by P s . Thus, the creep model is active also if the concrete is externally unloaded. This coupling is discussed further by Gawin et al. [68] and has also been considered in combination with other creep models by for example Sciumé et al. [22]. Also Ulm et al. [69] discussed the creep-shrinkage coupling, especially in relation to autogenous shrinkage and basic creep.
where c and p are model parameters with p > 1. The microprestress S in Eq. (41) should be interpreted as a macroscopic average of the actual microprestress at the individual creep sites. The evolution of S with time is then given as a relaxation in the creep sites governed by ηf(S), but is also increased by changes in . This can be described by a differential equation [36,64]
1 d S+ Cs dt
1
µc = µs
where = Ê e . The creep function is taken on the same form as in the original paper by Bažant and Prasannan [61] where λ0 = 1 day is usually assumed [61] and t − t′ is the loading time duration. Lastly, q2 is a model parameter that can be identified from experiments, or from the B3 [56] or B4 [67] prediction formulas. The long-term irrecoverable part of creep is described by the Microprestress theory [36], which explains it as the shear slip in localized and overstressed atomic bonds across gel pores (creep sites) that breaks and reforms. The high tensile stress in these bonds, referred to as the microprestress S, is built up during the rapid stages of hydration and then relaxes with time. However, at later stages, it is also affected by the disjoining pressure due to hindered adsorbed water in these small pores. On a macroscopic scale this process can be modelled as a viscous flow, where the non-constant viscosity ηf governs the creep rate, see Eq. (38). As argued by Bažant et al. [36], ηf is most suitably described as a power function of S on the form 1
RT ln , MW
where also the derivative of Eq. (44) has been substituted into the new equation. The model parameter μc should be identified from experiments, and the reference pressure pref is introduced only so that μc for the standard choice p = 2 obtains the dimensionality of fluidity (Pa−1s−1) and can be set to 1 atm. Parameter q4 can be identified from experiments or from the B3 [56] or B4 [67] prediction formulas. It should be noticed that μc is not identical to the similar parameter μs suggested by Jirásek and Havlásek [65]. Comparing the two, it can be found that
where nκ is a model parameter. The age-independent behaviour of the gel is assumed to follow standard linear viscoelasticity
=
C1
where kc = C1/ρW. To calculate the evolution of the viscosity ηf thus involves four model parameters (c, p, Cs, kc) and two equations. However, Jirásek and Havlásek [65] showed that these can be reduced and also that S can be eliminated from the model. Following a similar procedure, we can derive a new expression for ηf that replaces Eqs. (41) and (42) with
(39)
,
=
with C1 being a constant. Parameter 1 is the value of at hygroscopic saturation (φ = 1). However, since only the rate of is used in Eq. (42), this value is not needed. Using Eq. (17), this can also be expressed in terms of the capillary pressure pc as
where also the explicit definition of these two parts according to the MPS theory is introduced. The variable ψr was introduced by Bažant et al. [64] to account for general temperature and moisture conditions. The viscoelastic part is described by the Solidification theory [61] in which it is hypothesized that the solid particles of the load bearing C-SH gel have non-ageing properties, and ageing of the solid matrix is instead described by the growth of the gel during hydration, referred to as solidification. This is described by the function v, here taken as a function of the ageing degree on the form
v ( f) =
1
5. Remarks on final system of equation and numerical implementation In summary, the hygro-thermo-chemo-mechanical behaviour of concrete is described by a system of eight model equations and state variables:
(42)
in which Cs is a model parameter. The variable ψs was introduced by Bažant et al. [64] to account for general temperature and moisture conditions. Furthermore, depends on capillary tension, surface tension, crystal growth pressure and disjoining pressure, and, due to the small scale of interest, water in all phases and pore spaces can be assumed to be in local thermodynamic equilibrium [36]. This means that responds instantly to changes in chemical potentials in the capillary pore system, e.g. due to temperature or humidity changes. Given this, Bažant et al. [64] proposed that, under constant S, the actual microprestress is
• Cement hydration in Eq. (1) with ξ , • The mass conservation of solid species in Eq. (8) with , • The mass conservation of water in Eq. (10) with p , • The mass conservation of dry air in Eq. (11) with p , • The three components of the linear momentum conservation in Eq. (12) with d, • The conservation of energy in Eq. (14) with T. C
s
c
g
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For a complete closure of the mathematical model, initial conditions are needed for all eight governing equations and for Eqs. (10) to (12) and (14) boundary conditions also need to be defined. The latter are defined as either Dirichlet type by constraining the respective state variable, or as Neumann type by prescribing the total mass flux of water w D Jtot and dry air Jtot as well as the surface traction ^t and heat flux qT normal to the boundary. Additionally, convective boundary conditions are frequently used for Eqs. (10) and (14), here defined using the amWg bient vapor concentration amb and temperature Tamb such that W Jtot = fW (
qT = f T (T W
term mechanical parts of the model; the second on the moisture transport and the third on creep and shrinkage as well as tensile cracking. The intentions of the examples are both to demonstrate the capabilities of the proposed model, and to identify its limitations. Although many physical mechanisms are considered, there is surely still a need to investigate and evaluate the applicability of the constitutive relationships adopted. To this end, it is also important to keep in mind that additional physical couplings could be necessary to introduce. However, for a complete analysis of the model, more examples than those allowed within the scope of this work are certainly required.
Wg amb ),
Wg
Tamb),
6.1. Hydration and ageing
(47)
T
where f and f are the mass and heat transfer coefficients, respectively. This initial-boundary value problem needs to be solved numerically, which is here done using the Finite Element Method (FEM) as implemented in the general FE code Comsol Multiphysics [70]. Setting up the discretized system of equations follows standard procedures of the FEM (see e.g. references [46,71]), starting with conversion of the pointwise model equations to their weak-form and application of the Galerkin method. All constitutive and physical equations presented in Sections 2.2 and 4 and Appendix A are then inserted into the weak-form PDEs and after introduction of the FE discretization, the matrix system to be solved takes the general form
where u contains the nodal descriptions of all state variables as well as κf and ηf. The non-linear system matrices C, K and f include contributions from all governing equations and Eqs. (16) and (45). Variables s, C, f and ηf are in this study assembled into the overall system of equations using discontinuous Lagrange shape functions, but could also be updated separately at individual Gauss point since their governing equations contain no spatial derivatives. Time integration of Eq. (48) follows the fully implicit backward differentiation formula (BDF), using an adaptive order. The non-linear equations obtained at each time step are linearized internally by the software and solved using a damped version of the Newton-Raphson method [70]. The assembled system of equations is fully-coupled, resulting in that the derived system Jacobian contains all off-diagonal terms that appear from coupling of the governing equations. This is a particularly attractive attribute of the employed solution procedure for the strongly coupled physical and chemical fields. The definition of the stress tensor includes inelastic strain contributions (ecr,v, ecr,f, eth) given on a rate-form that need to be integrated in time. This is preferably done separately from the integration of Eq. (48) in the incremental update procedure. For the creep strains, the non-ageing viscoelastic part of the creep model in Eq. (40) is first discretized in time by a Kelvin-Voigt chain with ten units. Properties of the individual units are given by the formulas derived by Jirásek and Havlásek [72] for the log-power creep function in Eq. (40). Integration of the creep strains (ecr,v and ecr,f) is then done using an algorithm derived from the exponential algorithm suggested in the original MPS papers [62,63]. In short, the integration utilizes the analytical solution to the two parts of Eq. (38), given the assumption of constant stress and strain rates in each time increment. This leads to a closed-form algorithm that gives accurate results also for very large time steps while maintaining computational efficiency. Update of the thermal strains eth is done using a mid-point integration algorithm to solve Eq. (37).
The first example concerns an experimental series performed by Khan et al. [73] to examine the early-age compressive stress-strain behaviour of different concretes and curing conditions. The specimens used consist of concrete cylinders with dimensions 100×200 mm that, after unmolding at 24 h from casting, where cured either in adiabatic conditions in a water bath or moisture sealed and cured in an ambient temperature of 21 °C. Three different concrete mixtures were tested, of which the mixture with an approximate 28-day compressive strength of 30 MPa is analysed in the following. The mixture, referred to as C-30, has a w/c = 0.5 and a/c = 5.16 and a cement content of 355 kg/m3. The numerical model is set up using axial symmetry, and since no spatial variations are expected it consists of a single finite element using linear shape functions for all state variables. Values of all model parameters used are summarized in Table 2. The same experimental series has been analysed previously in the literature by for example Cervera et al. [10] who did not consider the influence of moisture, and Di Luzio and Cusatis [14] using a single-phase moisture transport model and a microplane model to account for fracture. The chemical model is first calibrated using temperature measurements from the adiabatic curing (results not presented), before determining the parameters of the ageing model in Eq. (16) utilizing the measured compressive strength. The simulated compressive strength development is shown for both adiabatic and isothermal conditions in Fig. 3a, which is in good agreement with the experimental results for both curing conditions. In Khan et al. [73], the stress-strain response was measured for several ages, while the simulated response in Fig. 3b is only shown for an age of 1, 7 and 28 days. The response up to peak load is in excellent agreement with the measured for both curing conditions, although it should be noticed that no measurements where reported at 28 days for adiabatic curing. The simulated compressive softening response can qualitatively reproduce the measured response in that the concrete becomes more brittle with age. However, the results are only in close agreement just after the peak load. This could point to a deficiency in the ageing model and how it currently describes the evolution of the peak strength in relation to the fracture energy. The compressive fracture energy is in the presented results calculated as Gf = f3/2 Gf, , where the exponent controls the evolution of the softening response. The cause of this discrepancy should be investigated further as it could also have other causes, for example many model parameters, such as those related to the creep model, are currently assumed due to a lack of data. To exemplify the microstructural model presented in Section 2.2, the development of volume fractions of the porous media is shown in Fig. 3c, including all species of the solid phase. The influence of the curing conditions is clearly noticeable as is the small self-desiccation, where parts of the capillary pores are emptied of water.
6. Applications
6.2. Moisture transport
To show how the features of the presented mathematical model are able to describe and predict the behaviour of early-age concrete, three laboratory scale examples from the literature are solved and discussed in the following. The first focuses on the chemical, ageing and short-
The second example focuses on the ability of the model to describe the movement of moisture in concrete by studying the experiments performed by Kim and Lee [74]. The experimental program consisted of specimens with dimensions 100×100×100 mm to study self-
C (u)
d u + K (u) u = f (u), dt
(48)
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Table 2 Model and material parameters used in the numerical applications. Section 6.2 Section 6.1 Hydration model constant, Eq. (2) Hydration degree at setting, Eq. (16) Ageing model constant, Eq. (16) Ageing constant, Eq. (22) Biot's coefficient, Eq. (25) Sorption isotherm, Eqs. (20) and (19) Relative permeability, Eqs. (24) and (23) Model parameter, Eq. (29) Damage model parameter, Eq. (32) Damage model parameter, Eq. (33) Damage model parameter, Eq. (35) Damage model parameter, Eq. (35)
Solidification function, Eq. (39) Creep function, Eq. (38) Microprestress parameter, Eq. (45) Age-independent Young's modulus Age-independent Poission's ratio Asymptotic tensile strength
Asymptotic compressive strength
Asymptotic tensile fracture energy Asymptotic compressive fracture energy Asymptotic intrinsic permeability Asymptotic structure coefficient
M
H 8⋅ 10 1⋅ 10−3 5 0.1
1⋅ 107 1⋅ 10−5 8 0.075
0.1 0.38 4 0.25 10.5 22.5 1 1
0.5 0.3 3.6 0.25 10.0 23 1 −
0.4 0.3 4.75 0.25 11 30 1 −
1.1 0.3 3.25 0.35 8.5 22.5 10 1
1
7
−
−
5
f+
1 MPa
0.2 3.2
0.2 −
0.2 −
0.2 4.4
f
MPa
34.5
−
−
44
J/m2
6500
−
−
4400
1/h 1 1 1
1.25⋅ 10 1⋅ 10−4 7.5 0.15
Af nT Ak b gs Arw, Arg aws
1 1 1 1 1 1 1 1
C 0
re+ re rp+ rp
nκ q2 q4 μc
Ê ν
Gf,+
Gf,
k∞ fs,∞
1 1
4 1
1 1/MPa 1/MPa 1/MPa⋅s GPa
1 30⋅ 10−6 1.8⋅ 10−6 8.0⋅ 10−13 37
J/m2
120
m2 1
5.0⋅ 10−22 0.01
desiccation and 200×100×100 mm to study external drying of three different concretes: low strength (L), medium strength (M) and high strength (H). To limit the current numerical study, the L series is omitted since it shows much less self-dessication than the other two concretes. The concretes in the M and H series have a w/c of 0.40 and 0.28, a a/c of 4.14 and 3.15, a cement content of 423 and 541 kg/m3, and a 28-day compressive strength of 53 and 76 MPa, respectively. All specimens were unmolded after 24 h and moist cured for an additional 3 or 28 days. After curing, specimens intended to study self-desiccation were sealed on all sides, while for the drying specimens one of the 100×100 mm sides were exposed to achieve a uniaxial flow. All specimens were stored at an ambient relative humidity of 0.5 and a temperature of 20 °C. For the drying specimens, the relative humidity was
6
Section 6.3
9⋅ 10 1⋅ 10−4 6.5 0.1
Ac1 Ac2 ηc
6
− −
1 68⋅ 10−6 5.4⋅ 10−6 8.0⋅ 10−13 37
−
4.0⋅ 10−22 0.001
6
− −
1 23⋅ 10−6 2.9⋅ 10−6 8.0⋅ 10−13 59
−
1.25⋅ 10−22 0.001
2.5 1
0.3 40⋅ 10−6 1.5⋅ 10−6 1.0⋅ 10−13 40
145
1.0⋅ 10−22 0.01
measured at 3 cm, 7 cm and 12 cm from the exposed surface. The same experimental series has been analysed previously in the literature by Chitez and Jefferson [48], among others. A difficulty in setting up the numerical model, is how to describe the moist-curing period in a good manner so that satisfactory initial conditions are obtained when drying/sealing commences. In order to achieve this while maintaining a computationally efficient model, an assumption of axial symmetry is made, although prism specimens were used for the experiments. This was done in order to better describe the moisture uptake during curing compared to a plane 2D-model, since the surface area exposed to the ambient air can be defined correctly in both phases of the test by calculating an equivalent radius. During curing, the convective type boundary condition in Eq. (47) is used on all
Fig. 3. Simulations of the compressive strength development at different curing conditions (a) and the stress-strain response at different ages (b). Markers in (a–b) show the experimental data by Khan et al. [73]. Development of the microstructure with time is shown in (c) as additive volume fractions, where solid lines show adiabatic conditions and dashed lines isothermal conditions. 211
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1
Relative humidity [1]
Relative humidity [1]
1 0.9 0.8 3cm 7cm 12cm Sealed
0.7
0.9 0.8 3cm 7cm 12cm Sealed
0.7 0.6
0.6 0
50
100
150
200
0
250
50
100
150
200
250
Time since casting [days]
Time since casting [days]
(a)
(b)
Fig. 4. Simulated relative humidity at different depths from the drying surface, for medium strength (M) concrete (a) and high strengh (H) concrete (b) compared to measurements by Kim and Lee [74]. Dashed lines and open markers correspond to 3 days while solid lines and filled markers correspond to 28 days of moist-curing.
external surfaces with an ambient relative humidity equal to 1 and a transfer coefficient fW = 1 m/s. Once curing is stopped, the assumption of axial symmetry does no longer matter since only one side is exposed to external drying, causing a uniaxial moisture flux. Thus, a 2D axisymmetric model is used with all state variables described by linear shape functions. All model parameters are summarized in Table 2. No calorimetry nor temperature data were available to calibrate the chemical model, hence its parameters had to be calibrated together with those of the sorption isotherm by fitting of the self-desiccation measurements. Once curing stopped, the boundary conditions on all W = 0 for these specimens. Although not an sides were changed so that Jtot ideal procedure, a good fit was achieved for both concrete types as shown by the dashed black lines (simulations) and open black squares (measurements) in Fig. 4. The solid black lines correspond to the specimens subjected to the longer period of moist-curing and also indicate a good agreement between simulations and measurements, shown by the filled black circles, for both concrete types. However, these results are more affected by how the moist-curing is modelled, e.g. the definition of the transfer coefficient fW during this period. For the specimens exposed to external drying, boundary conditions after the curing period were changed so that fW = 0.0001 m/s for the side exposed to drying. Furthermore, since all other sides were sealed, a W = 0 was defined on these. With the defined boundary moisture flux Jtot boundary conditions and the previously described procedure for moistcuring, the simulated decrease in relative humidity is overall in good agreement with the measured one for both concrete types as well as for both 3 and 28 days of moist-curing. However, the simulations slightly overestimate the drying close to the surface for the H concrete, as shown by the 3 cm measure point in Fig. 4b. This is likely related to the ageing of the permeability, which together with the mass transfer coefficient play important roles in determining the rate of moisture loss. Furthermore, it proved important that both the intrinsic permeability and the relative permeabilities are age dependent to accurately describe the observed drying process. This indicates that it is both the overall pore connectivity and size as well as the interaction between the two fluid phases in the pore space that change with age. The spatial distribution of moisture for the 28-day curing cases in Fig. 5 clearly shows how the model predicts that it is mainly the capillary pore space, i.e. s w , which is emptied, both during self-desiccation and drying. The gel pore space, i.e. sWs , is still after 200 days of drying very close to one in most parts of the specimen. This is in line with the findings from many recent experimental investigations aimed to characterize the state of water in concrete and cement pastes, see for example references [6,28,29]. It should here be recalled that the capillary pore space in this study is considered to include all pores larger than 3 nm following the definition by Powers and Brownyard [1],
although sometimes pores up to 12 nm are seen as part of the gel [28]. 6.3. Creep, shrinkage and tensile cracking The third example demonstrates the creep and shrinkage parts of the proposed model by looking at the recent experimental investigation by Theiner et al. [75]. Their study looked at the compressive creep as well as shrinkage of sealed and unsealed cylindrical specimens with dimensions 150×450 mm. The used concrete had a w/c of 0.44, a a/c of 4.84, a cement content of 375 kg/m3 and a 28-day compressive strength of 35.9 MPa. All specimens were kept at a temperature of 20 °C and were unmolded at an age of 24 h, after which all sides were sealed. For the drying specimens, the seal on the cylindrical part of the envelope surface was removed after 2, 7 or 28 days, and then exposed to an ambient relative humidity of 0.65. Creep specimens were loaded at the same time with a pressure equal to approximately 30% of the compressive strength at the age of loading; this corresponds to 5.5, 8.7 and 10 MPa, respectively. As for the previous example, the finite element model is set up using axial symmetry and linear shape functions for all state variables. All model parameters are given in Table 2. As for the second example, no calorimetry nor temperature data were available and the parameters of the chemical model had to be determined together with those of the sorption isotherm by making use of the autogenous shrinkage measurements. However, due to the creep and shrinkage coupling described in Section 4.3.3, the creep model also affects the simulated response for this case. The creep and shrinkage mechanisms can therefore not be looked at independently. Moreover, this is the case for creep measurements as well, for example, the earlyage creep of sealed specimens is affected by self-desiccation and can thus not be considered as true basic creep as the mechanism intended for drying creep is active. The model parameters used were thus obtained by a manual procedure, looking simultaneously at all measurements available. The simulated creep on sealed specimens is shown in Fig. 6a as total deformation, which thus includes the autogenous shrinkage. As done by Theiner et al. [75], the zero strain level is set to an age of 1 day. In comparison with the measurements, the autogenous shrinkage is overestimated at ages up to approximately 20 days, which consequently also affects the creep response. Nevertheless, the agreement is good and the model is able to describe the early-age creep and how it is influenced by the ongoing hydration process. Looking at the response of the unloaded specimens in Fig. 6b, the model is accurate in describing the first stage of drying shrinkage but is slightly off with respect to the final deformation; especially for the two specimens exposed at 7 and 28 days. It should here be stressed, however, that the typical trend observed in the drying shrinkage tests is recreated. Regarding the response of the 212
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1
Degree of saturation [1]
Degree of saturation [1]
1 0.8 0.6 0.4
sW tot sw s Ws
0.2 0
0.8 0.6 0.4
sW tot sw s Ws
0.2 0
0
5
10
15
20
0
5
10
15
Depth from drying surf. [cm]
Depth from drying surf. [cm]
(a)
(b)
20
Fig. 5. Spatial distribution of the moisture content for the M concrete (a) and H concrete (b) at different ages for the case of 28 days of moist curing. Different colors correspond to the three degrees of saturation from Eqs. (19) to (21) and markers to different times: ( ) 28 days, (○) 50 days, (◊) 100 days and (△) 200 days.
unsealed and loaded specimens in Fig. 6c, the measured and simulated response again shows good agreement. What can be noticed is that the drying creep effect is almost non-existent in the experimental data, which is also predicted by the simulations. This could be an indication that the mechanism of drying creep is active also for the sealed specimens at early-ages due to self-desiccation, as hypothesized in the proposed model framework. Lastly, to illustrate the capability of the model to describe crack formation in maturing concrete, the shrinkage test series by Theiner et al. [75] is modified in a numerical experiment aimed at studying the behaviour of the specimen under restrained conditions. This is achieved by restraining all vertical displacements at the top and bottom surfaces of the cylindrical specimen. To obtain a stable and well-defined crack, a notch is also added at mid-height of the specimen. The depth of the notch corresponds to a single finite element and it is located at the vertical symmetry plane of the model. All other model parameters are kept constant in comparison to the simulations presented in Fig. 6b. The simulated notch openings for the three different cases are presented in Fig. 7. There is a clear reduction of the notch displacement with the age of initial drying. This can of course in part be attributed to the lower observed shrinkage in Fig. 6b, but a significant part can also be explained by the increased maturity of the concrete. For example, it is clearly visible that it takes longer time for the crack to propagate through the thickness of the specimen for more mature concrete, which corresponds to the change in slope at approximately 85, 110 and
Fig. 7. Simulated notch openings from the numerical experiment of the notched and restrained specimen subjected to drying at different ages.
175 days for the three cases, respectively. To further illustrate this, the extent of tensile damage ω+ is shown in Fig. 8 for the specimen subjected to drying after 2 days. The extent of damage is shown before and after the crack propagates through the thickness as well as at the end of the analysis. Apart from the damage localized to the crack band that propagates from the notch, a field of diffuse damage is visible at the drying surface. However, once the full crack band from the notch is formed, this field is stable.
Fig. 6. Simulated total deformation during sealed and loaded (a), unsealed (b) and unsealed and loaded conditions (c) compared to experiments by Theiner et al. [75] shown by markers. All figures include the autogenous deformation (black line) for reference.
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load and its variation with the age of loading. However, the post-peak behaviour could only be accounted for qualitatively. The second example concerning moisture movement (internally and to the exterior) showed how the model can describe the ageing of transport properties and transfer of moisture between different forms. It also pointed to the importance of including the curing conditions in modelling the earlyage moisture movement; as this not only directly affects the amount of water held by the material, but also the ageing of transport properties, e.g. the intrinsic permeability. The third example on time-dependent deformations demonstrated the creep and shrinkage coupling used in the model. It highlighted a difficulty in independently calibrating model parameters when using a mechanistic description of shrinkage, in which the measured shrinkage to a large extent is described as a creep deformation caused by an internal load in the pore space. Hence, the model parameters of the creep model affect both the simulated response for externally loaded and unloaded specimens. Due to this interaction and the many parameters, one often has to resort to a manual procedure to find appropriate values of the model parameters; although prediction formulas, e.g. [56,67], can provide good initial guesses. It was also demonstrated how the model can describe crack propagation in maturing concrete by restraining the shrinkage specimen. An appealing novel feature of the proposed model is the inclusion of a microstructural model explicitly in the governing equations, which allows for tracking of individual constituents of the solid phase, as exemplified in Fig. 3c, and how these are affected by and affects the thermodynamic quantities of the system. In an extension, this feature can be generalized to track for example the volume fraction of individual hydration products, e.g., for use with micromechanics models or in modelling deterioration of the cement paste (e.g. leaching). It is also possible to include additional solid constituents in the model, for example silica fume when dealing with high-performance concretes or alkali-silica gel for modelling ASR damage. Furthermore, the explicit decomposition of the pore water into capillary and gel pore spaces together with the proposed split of the sorption isotherm, is theoretically in line with recent experimental findings [6,28,29]. Simulations of drying at a relative humidity of 0.5 showed that the model predicts how the larger capillary pores (> 3 nm) are emptied, while almost all water held in the gel is retained. This model property can be further explored to describe the fixation of pore water in a more physically accurate manner compared to the often phenomenological sorption isotherms used, i.e. by describing the fixation of water based on its location within the pore space rather than only how tightly it is bound.
Fig. 8. Simulated extent of tensile damage ω+ at different ages of the notched and restrained specimen subjected to drying 2 days after casting. Notice that the notch is located at a symmetry plane and that displacements are magnified by a factor of 200.
7. Conclusions A mathematical model of the hygro-thermo-chemo-mechanical behaviour of early-age concrete has been presented using a multiphase porous media framework, with concrete described as a three-phase material. Compared to previous models on a similar theoretical framework, e.g. [20-23], it suggests novel and modified features including:
• Quantification of different solid constituents of concrete explicitly in the multiphase framework, • Separation of different forms of water (chemically bound, gel and capillary) by including phase changes, • Age-dependent properties for the mass transport of fluids and the mechanical behaviour, • Tensile and compressive fracture of the solid skeleton using damage mechanics • Thermo-visco-elastic behaviour of the solid phase, • Creep and shrinkage coupling by means of the effective stress principle.
The model was implemented by means of the FEM and solved using a fully-coupled approach, where all state variables are solved for simultaneously. By applying the proposed model to three experimental studies from the literature, it was shown that it is able to describe many important aspects of the early-age behaviour of concrete. In the first example, studying the short-term mechanical behaviour, it was shown that the model accurately accounts for the stress-strain behaviour up to peak
Acknowledgements The research presented was carried out as a part of “Swedish Hydropower Centre -SVC”. SVC has been established by the Swedish Energy Agency, Energiforsk and Svenska Kraftnät together with Luleå University of Technology, KTH Royal Institute of Technology, Chalmers University of Technology and Uppsala University. www.svc.nu.
Appendix A A few additional constitutive and physical relationships are briefly presented in the following, see for example Lewis and Schrefler [46] for a more in-depth description. A.1. Equation of state The volumetric behaviour of all liquid and solid species is assumed to follow the linearized forms proposed in reference [46]. For the liquid water W
=
where
W ref W ref
1
W
(T
Tref ) + 3
1 (pw KW
w pref ) ,
= 1000 kg/m at Tref = 4 °C and
w pref
(A.1)
= 1 atm. For the solid species, additionally the deformation of the solid phase is accounted for and
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i
=
i ref
i (T
1
Tref ) +
1 s 1 p + s i I1s , Ki 3 K
(A.2) i
where is the first invariant of the effective stress tensor In Eq. (A.2), the definition of and K depends on the type of aggregate and cement used, with values used herein given in Table 1. Lastly, the volumetric behaviour of two gas species as well as their mixture is given by the ideal gas law s.
I1s
ig
=
Mi ig p , RT
i
i ref ,
i
(D , W ).
(A.3) ig
The molar mass Mi and the partial pressure p can be calculated using Dalton's law. A.2. Mass fluxes The volume-averaged advective flux of the two fluid phases w and g is described by Darcy's law, which, given appropriate assumptions [22,25,26,76], takes the form
krf k ( pf µf
s f v f ,s =
f g),
f
(w, g ),
(A.4)
f
where the dynamic viscosities μ are considered as temperature dependent physical properties. The diffusive mass flux of species in the gas phase is described by Fick's law of diffusion, which, given appropriate assumptions [26,76], takes the form
sg
Wg uWg
where
g DWg d
= JWg =
DWg d
Wg g
,
(A.5)
is the effective diffusivity tensor. Following for example Gawin et al. [77], this tensor can be defined as
g DWg d = s fs D v0
T T0
5/3
pref pg
I,
(A.6)
2
where Dv0 = 25.8 mm /s is the diffusivity of vapor in air at the reference temperature T0 = 0 °C and pressure pref = 1 atm. Lastly, it follows from the definition of the gas phase as an ideal gas mixture and Dalton's law that JWg + JDg = 0 . A.3. Conductive heat flux The conductive heat flux qi in each species i of each phase α is obtained using Fourier's law. Then the total heat flux q is obtained through summation such that
q=
eff I
(A.7)
T.
with the volume-averaged thermal conductivity of the porous medium calculated as eff =
sw W + s g g + s ( As A + Cs
C
+
Hs H
+
(A.8)
GssWs W )
i
Each λ in Eq. (A.8) is in principle dependent on the temperature, but are here considered as material constants, see Table 1.
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