Extended Microprestress-Solidification Theory forLong-Term Creep with Diffusion Size Effect in

Concrete at Variable EnvironmentSaeed Rahimi-Aghdam1; Zdeněk P. Bažant, Hon.M.ASCE2; and Gianluca Cusatis, M.ASCE3

Abstract: The solidification theory has been accepted as a thermodynamically sound way to describe creep reduction due to deposition ofhydrated material in the pores of concrete. The concept of self-equilibrated nanoscale microprestress has been accepted as a viable model forthe marked multidecade decline of creep viscosity after the hydration effect becomes too feeble, and for increases of creep viscosity after anysudden change of pore humidity or temperature. Recently, however, it appeared that the original microprestress-solidification theory (MPS)predicts incorrectly the diffusion size effect on drying creep and the delay of drying creep behind drying shrinkage. Presented here is anextension named XMPS that overcomes both problems and also improves a few other features of the model response. To this end, differentnanoscale and macroscale viscosities are distinguished. The aforementioned incorrect predictions are overcome by dependence of the macro-scale viscosity on the rate of pore humidity change, which is a new feature inspired by recent molecular dynamics (MD) simulations of amolecular layer of water moving between two parallel sliding calcium-silicate-hydrate (C-S-H) sheets. The part of aging that is not caused bymicroprestress relaxation is described as a function of the growth of hydration degree, and the temperature change effect on pore relativehumidity is also taken into account. Empirical formulas for estimating the parameters of permeability dependence on pore humidity fromconcrete mix composition are also developed. Extensive validations by pertinent test data from the literature are demonstrated.DOI: 10.1061/(ASCE)EM.1943-7889.0001559. © 2018 American Society of Civil Engineers.

Author keywords: Microprestress-solidification theory (MPS); Microprestress relaxation; Drying creep; Aging viscoelasticity; Variablehumidity; Variable temperature; Scaling; Computational mechanics; Finite elements.

Introduction

Until recently, the microprestress-solidification (MPS) theory(Bažant et al. 1997a, b) appeared to give satisfactory predictionsof the creep of concrete, long-term creep included, both withoutand with simultaneous drying and temperature changes. In 2014,however, simulations by P. Havlásek at Northwestern University(in collaboration with M. Jirásek in Prague and with Z.P. Bažant)identified incorrect predictions of the effect of cross-section size onthe additional creep due to drying, and an excessive delay, behinddrying shrinkage, of the additional creep induced by drying (Bažantet al. 2014). In this paper, both deficiencies are rectified by the ex-tended microprestress-solidification theory (XMPS). Several otheramendments are also introduced to improve on previous amend-ments proposed by Jirásek and Havlásek (2014). Extensive verifi-cation by a broad range of test data is an essential objective of thispaper, due to the paramount importance of verification for a com-plex material such as concrete.

The solidification theory separates viscoelasticity of the solidconstituent, the cement gel, from the chemical aging of the hardenedcement paste caused by solidification of gel particles and character-ized by the growth of the volume fraction of hydration products. Thisseparation permits considering the viscoelastic constituent as non-aging and thus greatly simplifies mathematical formulation. Thegradual decrease of compliance with the age at loading is explainedby the growth of the volume fraction of a nonaging constituent, thecement gel or calcium silicate hydrate (C─S─H) (Bažant andPrasannan 1989a, b; Bažant and Jirásek 2017).

The solidification, however, cannot explain the marked de-crease of creep viscosities continuing even after the hydrationprogress becomes feeble. Nor can it explain the drying creep effect(e.g., Pickett effect) and transitional thermal creep. To explain thesephenomena, the concept of microprestress was conceived, resultingin the microprestress-solidification theory (Bažant et al. 1997b).

The microprestress characterizes self-equilibrated stresses at thenanoscale level. These stresses stretch and break the interatomicbonds resisting the slip of parallel C─S─H sheets and of adjacentC─S─H nanoglobules, which is believed to be the main mechanismof creep in concrete. The microprestress is considered to be theresult of disjoining pressures across the nanopores filled by ad-sorbed water layers. The micropresstress cannot be appreciably af-fected by the applied load and may be imagined as the effect of astrong but very soft prestressed spring compressed within a verystiff frame. The micropresstress is initially produced by incompat-ible volume changes in the microstructure during hydration. It thenrelaxes but later again builds up when changes of moisture contentand temperature create a thermodynamic imbalance between thechemical potentials of vapor, liquid water, and adsorbed water inthe nanopores of the cement gel.

1Graduate Research Assistant, Dept. of Civil and Environmental Engi-neering, Northwestern Univ., 2145 Sheridan Rd., CEE/A135, Evanston,IL 60208.

2McCormick Institute Professor and W.P. Murphy Professor, Civil andMechanical Engineering and Materials Science, Northwestern Univ., 2145Sheridan Rd., CEE/A135, Evanston, IL 60208 (corresponding author).Email: z-bazant@northwestern.edu

3Associate Professor, Dept. of Civil and Environmental Engineering,Northwestern Univ., 2145 Sheridan Rd., CEE/A135, Evanston, IL 60208.

Note. This manuscript was submitted on April 23, 2018; approved onJuly 24, 2018; published online on November 27, 2018. Discussion per-iod open until April 27, 2019; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Engineering Me-chanics, © ASCE, ISSN 0733-9399.

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https://doi.org/10.1061/(ASCE)EM.1943-7889.0001559https://doi.org/10.1061/(ASCE)EM.1943-7889.0001559mailto:z-bazant@northwestern.edu

The XMPS improves the modeling of drying creep, par-ticularly the dependence of macroscale viscosity on the rate ofmicroprestress. This improvement has been inspired by recentmolecular dynamics (MD) simulations of Vandamme et al. (2015)and Sinko et al. (2016, 2018). They showed that the viscosity ofcreep, associated with the rate of relative slip of parallel planarwalls or sheets of C─S─H, is greatly diminished by the presenceof a water layer between the walls, and that, furthermore, the ef-fective viscosity of slip between the solid surfaces decreases whenthe water layer moves. A useful finding is that the direction ofmovement of the water layer does not matter, which means thatdrying and wetting should have a similar effect. Although sus-pected long ago (Bažant 1970, 1972; Bažant and Chern 1985),these facts were not reflected in the original MPS model. They hap-pen to have been the cause of error in diffusion size effect on dryingcreep as well as in delay of the drying creep effect after shrinkage.

Microprestress-Solidification Theory

Within the service stress range (but with certain exceptions for un-loading and simultaneous drying), the concrete creep law can beconsidered to be linear in stress and follow the principle of super-position in time. Therefore, the creep is fully characterized by theuniaxial compliance function Jðt; t 0Þ, representing the strain intime t caused by a unit sustained uniaxial stress applied at age t 0.The triaxial generalization need not be discussed here because it iswell known how to obtain it under the assumption that the materialis isotropic (e.g., Bažant 1972, 1982; Bažant and Jirásek 2017).

In the absence of significant plastic and viscoplastic strains thatmay arise at very high confining pressures, the normal strain ofconcrete can be decomposed as follows (Fig. 1):

ϵ ¼ ϵa þ ϵv þ ϵf þ ϵsh þ ϵT ð1Þ

where ϵa = instantaneous strain; ϵv = viscoelastic strain; ϵf =flow strain (purely viscous strain); ϵsh = shrinkage strain; and ϵT =thermal strain. The instantaneous strain, which is the strain appear-ing immediately after applying uniaxial stress σ, may be written

ϵa ¼ q1σ ð2Þ

Because the retardation spectrum of concrete creep extendssmoothly to load durations t̂ ≪ 10−4 s, it is convenient to definethe instantaneous compliance as an asymptotic extrapolation ofshort-time creep curves for near-zero load duration (i.e., for t̂ → 0).Such an extrapolation has the advantage that the age effect on thetrue mean instantaneous compliance happens to be negligible (theevidence for this fact is quite scattered but deviations are not sys-tematic). This property was demonstrated by Bažant and Osman(1976) and Bažant and Baweja (1995a, b) by considering the mea-sured compliances for load durations t ranging from 0.3 s to 1 week.They obtained optimum fits of the compliance values measured fordifferent t 0 using a smooth formula of the type Jðt; t 0Þ ¼ q1 þ ct̂n,where t̂ ¼ t − t 0 is load duration. Then, by optimizing the fit of the

data for various loading ages t 0, they obtained q1, and found thatthe q1 values for various t 0 were nearly the same [as also pointedout by Bažant and Jirásek (2017), Fig. 3.5].

Therefore, similar to B3 and B4 models (Bažant and Baweja1995a; Bažant et al. 2015), the instantaneous compliance (i.e., itsasymptotic value for t̂ → 0) is here considered as age-independent,which brings about a significant simplification. Introducing an em-pirical factor p1 depending on the cement type, the instantaneouscompliance q1 is expressed

q1 ¼p1E28

ð3Þ

where E28 = conventional elastic modulus at age 28 days (which,according toModel B4, corresponds to the loading duration of about0.001 day, or 1.44 min, whereas in previous Model B3, it was0.01 day).

The viscoelastic strain ϵv, which originates in the solid gel ofC─S─H, may be described by a relation of the same type as in theB3 and B4 models [Bažant and Baweja 1995a; Bažant et al. 2015;Bažant and Jirásek 2017, Eq. (3.11)]

ϵv ¼ σfq2Qðt; t 0Þ þ q3 ln½1þ ðt − t 0Þn�g ð4ÞFunction Qðt; t 0Þ was derived in a differential form by asymp-

totic matching by Bažant and Prasannan (1989a) and Bažant andJirásek [2017, Eq. (3.10)]. Its integration leads to a binomial inte-gral that cannot be expressed in closed form. However, in numericalstructural analysis in time steps, the integral is not needed and iseven useless if the pore humidity or temperature varies. IfQðt; t 0Þ isdesired, it can be easily evaluated numerically.

Function Qðt; t 0Þ is age-dependent and its value decreasesas concrete ages. Previous studies neglected the dependence ofQðt; t 0Þ on the growing degree of hydration, αðtÞ, and simply con-sidered it as a function of loading time t 0. This degree matters whenthe temperature or pore relative humidity, h, varies. The hydrationreaction speeds up as the temperature increases and slows down ash decreases, as does αðtÞ. In traditional normal concretes (withoutsilica fume), hydration at room temperature virtually stops whenh < 0.75. In modern concrete with silica fume and low water tocement ratio (w/c) and silica fume, the pore humidity in sealed spec-imens can drop as low as 0.65 because of self-desiccation. Besides,external drying causes nonuniform humidity profiles evolving intime.

To capture these effects, the actual time t needs to be replacedby the equivalent time θ, which is a function of hydration degree.To calculate θ, a relation of the same type as in the solidificationtheory is used and is calibrated using the experimental results on thehydration reaction reported by Bentz (2006)

θðαÞ ¼�0.28w=c

�αuα

− 1��−4=3

ð5Þ

where α ¼ αðtÞ is a function of time (Rahimi-Aghdam et al. 2017);w=c = water/cement ratio of concrete mix (by weight); αu =ultimate hydration degree in sealed condition (which depends onw=c); and

αu ¼ 0.4þ 1.45ðw=c − 0.17Þ0.8 ð6ÞUsing the equivalent time, one can modify the expression for

the evolution of Qðt; t 0Þ [Bažant and Prasannan 1989a, Eq. (17)] byreplacing the actual time t with the corresponding equivalent time θ

dQðt; t 0Þdt

¼�

λ0θ½αðtÞ�

�m nζn−1λ0ð1þ ζnÞ

ð7ÞFig. 1. Rheological model.

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where λ0 ¼ 1 day; m ¼ 0.5; n ¼ 0.1; and ζ ¼ t − t 0. This relationneeds to be integrated to find Qðt; t 0Þ

Qðt; t 0Þ ¼Z

t

t 0

�λ0

θ½αðtÞ��

m nζn−1λ0ð1þ ζnÞ

dt ð8Þ

The integral Qðt; t 0Þ in Eq. (8) cannot be expressed in a closedform, but its numerical evaluation is fast. An approximate asymp-totic matching formula for Qðt; t 0Þ was developed by asymptoticmatching by Bažant and Prasannan [1989a, Eq. (20)]. Appendix IIIgives its generalization to the equivalent time. This explicit formulais very accurate, and its use reduces the demand for computer time.

The shrinkage strain ϵsh is here understood as a pointwise ei-genstrain, whereas in the B3 and B4 models (Bažant and Baweja1995a; Bažant et al. 2015), it represents the average shrinkage ofthe whole cross section of a long beam or slab. In addition, contraryto Model B4, the autogenous shrinkage is here not separated fromthe drying shrinkage of the cross section because both are causedby a pore humidity drop (Rahimi-Aghdam et al. 2017). The shrink-age strain is approximately proportional to the relative humiditychange, whether caused by external drying or self-desiccation(Bažant and Jirásek 2017). Therefore, in step-by-step analysis, ateach integration point of each finite element, the rate of shrinkagestrain may be calculated

ϵ̇sh ¼ k0αu − α0α − α0 ḣ ð9Þ

where ḣ ¼ dh=dt is rate of humidity change; k0 = empirical con-stant; and α0 ¼ 0.9αset. Likewise, the thermal strain rate reads

ϵ̇T ¼ kTṪ ð10Þwhere Ṫ ¼ dT=dt is rate of temperature change; and kT = empiricalconstant.

To complete the creep law, one finally needs the rate of flowstrain ϵf, which is discussed next.

Evolution of Flow Strain and Microprestress

The flow strain is modeled by a viscous flow element coupled inseries to the solidifying Kelvin chain, as schematized in Fig. 2.For the flow element portrayed, it is imagined that the bonds acrossthe slip plane bridge the nanopore filled by hindered adsorbed water.They are subjected to two kinds of stress: (1) macroscopic appliedstress σ causing shear slip, which acts in the figure horizontally; and(2) tensile microprestress S, which acts transversely (vertically in thefigure). The rate of strain in the flow element is considered to be

ϵ̇f ¼σηM

ð11Þ

where ηM = macroscale viscosity. In the original MPS model, forsimplicity, the viscosity was assumed to be the same for both thenanoscale and macroscale and was in both cases assumed to bea function of microprestress, S.

This assumption was obviously a simplification because themacroscale viscosity must depend on several other phenomena aswell. In particular, it must depend on the flow of water within thepores—the mesopores or capillary pores, and (mainly) the nano-pores. The adsorbed water flow along the nanopores accelerates asthe rate of pore humidity change increases.

Recently, Sinko et al. (2018) conducted MDsimulations of therate of slip between parallel C─S─H sheets loaded by constantshear stress and by transverse compression. An interstitial layer ofwater several molecules thick was inserted between the two sheetsand forced to move between these two sheets to simulate the flowof water into or out of the nanopore. It was found that the presenceof the interstitial water layer accelerates the relative sliding of theC─S─H sheets subject to constant sheer stress and, more impor-tantly, that the rate of relative sliding of the C─S─H sheets accel-erates if the interstitial water layer is made to move relative to themean velocity of the two parallel C─S─H sheets. The explanationis that the movement of the water layer changes the activation en-ergy landscape at the interface, causing the effective viscosity todepend strongly on the flow velocity. The direction of the flow,corresponding to drying or wetting, was found not to be important.

The conclusion is that the macroscale viscosity depends notonly on the microprestress, but also on the water flow through thepores, the velocity of which is determined mainly by the rate ofpore humidity change. It should, of course, be kept in mind that therate of humidity change is not the only phenomenon that can accel-erates the flow along the pores. Generally, the rate of any disruptionof thermodynamic equilibrium has a similar effect.

Because any phenomenon that causes thermodynamic imbal-ance increases the microprestress, one can imagine the rate of mi-croprestress to be a measure of the water flow rate in the nanoporeand the corresponding viscosity to depend only on the micropre-stress, S. In the rheological model of Fig. 2, this viscosity is cap-tured by adding an extra dashpot whose viscosity depends on theabsolute value of microprestress rate, jṠj. Introduction of the abso-lute value jṠj is justified by the MD simulations of Sinko et al.(2018), who found the directions of the flow along a simulatednanopore to be unimportant. As a result of these considerations,the macroscale viscosity may be introduced in the form

1

ηM¼ 1

ηnðSÞþ 1ηflow

¼ aSþ bjṠj ð12Þ

where ηn = nanoscale viscosity that is a function of the micropre-stress only; and ηflow = decrease in viscosity due to the waterflow or the rate of any other phenomenon causing thermodynamicimbalance in nanopores. More generally, one could consider1=ηM ¼ aSp1 þ bjṠjp2 but data fitting indicates no need for sucha complication.

The viscosity in Eq. (12) must increase with the age or, moreprecisely, with degree of hydration, α, a fact confirmed by manyexperiments (Ferraris 1999). Therefore, it is reasonable to considerboth parameters a and b in Eq. (12) to depend on α. For simplicity,a linear dependence may be assumed

a ¼ a0αuα

; b ¼ b0αuα

ð13Þ

where a0 and b0 = empirical constants. Combining Eqs. (12)and (13) yieldsFig. 2. Flow strain.

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1

ηn¼ αu

αa0S;

1

ηM¼ αu

αða0Sþ b0jṠjÞ ð14Þ

The next important issue is the evolution of microprestress.The microprestress, S, is imagined to characterize the averageof normal stresses acting across the slip planes with hindered ad-sorbed water layers between them. The disjoining pressure in theselayers, and thus also the microprestress, is considered to developfirst during the initial hardening of cement paste. During the initialrapid hydration, the microprestress builds up mainly as a result ofcrystal growth pressures and localized volume changes close tothe nanopores. Therefore, during the initial days of fast hydration,S depends mainly on the hydration degree, α, and is calculatedsimply as follows:

S ¼ S0 ¼ c0q4 for α < α0 ð15Þwhere c0 = empirical constant; q4 = creep law parameter inmodels B3 and B4; S0 = initial microprestress; and α0 = hydrationdegree prior to which the hydration reaction has the dominant con-trol of microprestress. The value α0 ¼ 0.6αu works well.

Later, after the volume changes due to hydration have almostceased, the changes of microprestress are controlled mainly by thechanges in the disjoining pressure, which responds with negligibledelay to changes in the capillary tension and surface tension at thesame location. According to Bažant et al. (1997a, b), the evolutionof microprestress can be assumed to obey a Maxwell-type rheologi-cal model with variable viscosity ηnðSÞ and stiffness CS

ṠðtÞCS

þ SðtÞηnðSÞ

¼ ṡðtÞCS

ð16Þ

where ṡðtÞ=CS = time rate of Maxwell model strain due to anyphenomena that may cause thermodynamic imbalance in the micro-structure (Bažant and Jirásek 2017). These phenomena are ana-lyzed next.

Temperature and Humidity Effects

The main phenomena affecting ṡðtÞ=CS are (1) temperaturechanges, and (2) humidity changes. First, the effect of temperaturewill be discussed. Its effect is complicated by interference of severalphysical mechanisms which can be described as follows:1. A temperature increase accelerates the bond breakages and

restorations.2. The higher the temperature, the faster the chemical process of

cement hydration, and thus the faster the aging of concrete.3. A temperature change alters the capillary tension, crystal growth

pressure, surface tension, and disjoining pressure, all of whichcan alter the microprestress and creep rate.

4. A temperature change alters the internal relative humidity,which either increases or decreases the creep rate.

5. A temperature increase alters the microstructure of C─S─H andof the weaker interfacial transition zone (ITZ), which usuallyweakens the concrete.The relative pore humidity affects the creep rate in the following

three main ways:1. As the relative humidity decreases, the viscosity increases and

bond breakage decelerates.2. The higher the humidity, the faster is the chemical process of

cement hydration and thus the aging of concrete, which reducesthe creep rate.

3. The evolution of humidity alters the capillary tension, crystalgrowth pressure, surface tension, and disjoining pressure, whichall change the microprestress.

To predict the creep rate correctly, one must consider theeffect of all the aforementioned mechanisms. All factors exceptMechanisms 4 and 5 due to temperature change have already beenconsidered in the original MPS model (Bažant et al. 1997b). Inaddition, those two mechanisms are significant for the case of tem-perature varying during the experiment. Consider the effect of tem-perature on the entire creep law. In models B3 and B4, this effectwas considered using a temperature-dependent time tT instead of theactual time t. Here, the same idea is used to calculate the viscoelasticcreep (the effect on flow term will be formulated separately)

tT ¼Z

t

0

βTðτÞdτ ð17Þ

βTðtÞ ¼ exp�QhR

�1

T0− 1TðtÞ

��ð18Þ

ϵvðTÞ ¼ σðq2QðtT ; t 0T ; t 0effÞ þ q3 ln½1þ ðtT − t 0TÞn�Þ ð19Þwhere T = absolute temperature; T0 = reference temperature, chosenas T0 ¼ 293 K; R = universal gas constant;Qh = activation energiesfor the hydration processes (whose values depend on the cementtype); and t 0T = value of tT at the time of loading.

The effect of the first temperature-related mechanism on theflow strain rate is formulated by decreasing the viscosity to reflectthe acceleration of bond breakage, and by decreasing CS to reflectthe acceleration of bond restorations. Both temperature effects canbe described by an Arrhenius-type equation. Because the first rel-ative humidity–related mechanism also changes the viscosity, theinfluences of both the temperature and relative humidity are con-sidered simultaneously, as follows:

ηðT; hÞ ¼ ηT0;Sat=βηðT; hÞ ð20Þ

βηðT; hÞ ¼ exp�QηR

�1

T0− 1TðtÞ

���p0 þ

1 − p01þ ð 1−h

1−h�Þnh�

ð21Þ

CsðTÞ ¼ CT0s =βCsðTÞ ð22Þ

βCsðTÞ ¼ exp�QCR

�1

T0− 1TðtÞ

��ð23Þ

where Qη;QC = activation energies for the viscosity change andCS; and empirical constants p0 ¼ 0.5; h� ¼ 0.75; and nh ¼ 2.For simplicity, one may assume the same activation energy for boththe macroscale and nanoscale viscosities. In addition, because gen-erally a temperature increase accelerates creep, the viscosity de-crease is assumed to be dominant, and it is found reasonable to setQC ¼ Qη=2.

The effect of the second mechanism driven by the changes oftemperature and relative humidity is already included through theacceleration of hydration reaction, and no other modification isneeded. The third and fourth mechanisms caused by temperaturechange and the third mechanism caused by relative humidity modifythe microprestress value through the changes of capillary tension,surface tension, and disjoining (or crystal growth) pressure.

All the aforementioned pressures have almost the same relationto relative humidity, and all of them are determined by changes inthe chemical potential of pore water, μ ¼ ðRT=MρlÞ ln h [ρl ¼1 g=cm3]. At each location, μ is the same in all the phases ofpore water and its change represents the ultimate driving forceof hygrothermal deformations. Therefore, and in conformity to theoriginal MPS model (Bažant et al. 1997a, b), all these pressures arecombined as one effective pressure, which can be written

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peff ¼ p0 þ CpRTM

lnðhðt;TÞÞ ð24Þ

where R = universal gas constant; M = molecular weight of water(moles); and p0 and Cp = empirical constants. To capture Mecha-nism 4 due to temperature change, the humidity is here consideredto be a function not only of time t, as in the original MPS model,but also of temperature T [which is a mechanism considered byDi Luzio and Cusatis (2013) as well]. The rate of effective pressurecan then be written

ṗeff ¼ k1�Ṫ lnðhðt;TÞÞ þ T

h∂h∂t þ

Th∂h∂T Ṫ

�ð25Þ

where the last term describes the change of relative humidity dueto the temperature change [Fig. 3(a)]. In Eq. (25), ∂h=∂T ¼ κ is thehygrothermic coefficient, introduced by Bazant (1970) and later byBažant and Najjar (1972) (Fig. 13) and known to depend stronglyon the relative humidity at which the temperature change happens[Fig. 3(b)]. Here κ is calculated based on the experimental results ofGrasely and Lange (2007).

The effect of the saturation degree on the effective pressure isnot used here directly because the major effects on the micropre-stress are those of the disjoining pressure changes on the surfacetension. They dominate in the smallest pores, which remain filledby water even at low h. This simplification, however, may causeappreciable errors at very low h. Next, these changes need to berelated to the microprestress change, which is, in turn, related tothe effective pressure change. One can use the simple relation

ṡ ∝ ṗeff ð26ÞThe last mechanism that needs to be included is the effect of

temperature change on the microstructure of C─S─H and particu-larly the interface transition zone (ITZ) (Richardson 1991; Castillo1987; Fahmi et al. 1972). Several studies showed that elevating thetemperature at early ages when the relative humidity is high altersthe microstructure of C─S─H, especially in the ITZ, such that theaverage porosity of C─S─H increases. It has been previouslyshown that the microstructure has significant impact on the macro-scopic (i.e., effective) material properties (Bostanabad et al. 2016,2018). Therefore, the microstructure change weakens the concreteand can cause sudden increase of microprestress (due to thermo-dynamic imbalance). One can formulate these effects as follows:

q1;2;3ðTÞ ¼ q1;2;3ðT0Þ½1þ cTðT − T0Þ� ð27Þ

ṡ ∝ fðhÞṪ ð28Þ

where Eq. (27) represents a weakening of microprestress andEq. (28) represents its enhancement. Function fðhÞ introduces thefact that changing the C─S─H microstructure is facilitated by ahigh relative humidity (Fahmi et al. 1972; Richardson 1991). Itssimple, empirically calibrated, form is

fðhÞ ¼ chhr ð29Þ

where r = empirical exponent set as r ¼ 3.The reality, however, is slightly more complicated. Fahmi et al.

(1972) showed that the effect of temperature on material stiffness isirreversible and is important only if the temperature rises above therange of previously experienced temperatures. Temperature fluctu-ations within that range do not have much effect.

Having formulated the mechanisms contributing to the micro-prestress, they need to be combined. All these contributions aresimply assumed to be independent and additive. Thus, one obtains

ṡ ¼ cpṗeff þ fðhÞṪ

¼ c1�Ṫ lnðhðt;TÞÞ þ T

h∂h∂t þ

Th∂h∂T

∂T∂t�þ chhvṪ ð30Þ

where cp and c1 ¼ k1cp. Consequently, the equation governing therelaxation of microprestress reads

ṠCSðTÞ

þ SηnðS;T; hÞ

¼ ṡCSðTÞ

ð31Þ

Upon inserting Eqs. (20), (22), and (30) into Eq. (31), the equa-tion for relaxation of microprestress becomes

Ṡþ CT0S

ηT0;satn ðSÞβηðT; hÞβCsðTÞ

S ¼ c1�Ṫ lnðhðt;TÞÞ þ T

h∂h∂t þ

Th∂h∂T

∂T∂t�

þ chhrṪ ð32Þ

where ηT0;satn ðSÞ ¼ 1=aSp1 ; and CT0S is a constant. As already men-tioned, it is assumed that p1 ¼ 1, and so the microprestress relax-ation equation can be simplified as follows:

Ṡþ aSβηðT; hÞβCsðTÞ

S2 ¼ c1�Ṫ lnðhðt;TÞÞ þ T

h∂h∂t þ

Th∂h∂T

∂T∂t�

þ chhvṪ ð33Þ

where aS ¼ CT0S =a. Finally, using p1 ¼ p2 ¼ 1 as before, oneobtains the rate of flow strain

(a)

(b)

Fig. 3. Change of relative humidity due to temperature change: (a) relative humidity change due to temperature change; and (b) hygrothermiccoefficient as function of initial relative humidity. (Data from Grasley and Lange 2007.)

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ϵ̇f ¼σ

ηMðS;T; hÞ¼ σðaSþ bjṠjÞβηðT; hÞ ð34Þ

Recently Refined Water Transport Model

To predict the creep rate under the influence of variable humidity, arealistic model for water transport or drying is obviously important.Over the years, many have been proposed. Some were transplantsfrom other porous material, especially soil science, but did notwork well because they ignored, or reflected poorly, two particularfeatures of concrete: (1) in concrete, there is a major distributedsink of evaporable water due to hydration (continuing for years);and (2) in normal concrete, neither the vapor phase of water nor theliquid capillary phase percolate, and a water molecule moving fromone pore to the next must pass through the hindered adsorbed phasefilling the nanopores, which caries load and behaves as part of thesolid skeleton.

Because of gradual filling of pores by deposition of hydrationproducts, the hydration sink affects the pore relative humidity muchless than it does the specific content of evaporable water. Recog-nizing this, Bažant and Najjar (1972) adopted the pore relative hu-midity as the primary variable [their model was incorporated intothe Model Code of Fédération internationale de béton (fib) (Beverly2010)]. This model was improved by Di Luzio and Cusatis (2009a, b),based on new experimental evidence. In 2018, it was improvedmore substantially by Rahimi-Aghdam et al. (2018) in three ways:(1) the humidity dependence of permeability was separated fromthe diffusivity by using a more realistic nonlinear desorption iso-therm; (2) the order-of-magnitude drop of permeability occurring atdecreasing pore humidity was made less steep than originally, andit was extended below the 50% humidity; and (3) empirical formu-las to estimate the permeability parameters from concrete strengthand composition were developed to make possible realistic esti-mates without experimental calibration of these parameters.

Like the Bažant Najjar model, the model of Rahimi-Aghdamet al. (2017) postulated that, at constant temperature, the total mois-ture flux jw is driven by the gradient of pore relative humidity h

jw ¼ −cpðhÞ∇h ð35Þwhere cp = function giving moisture permeability (kg=m · s).The condition of mass conservation of water reads

ẇt ¼ −∇ · jw þ ẇs ð36Þwhere ẇt = rate of total mass of water “per unit volume of con-crete,” ẇs = rate of water mass consumed by the chemical processof hydration, which is calculated from the rate of hydration degree,α̇, according to Rahimi-Aghdam et al.’s (2017) model. The term ẇsrepresents a distributed sink, leads to self-desiccation, and is par-ticularly important for modern concretes with low w=c or withsilica fume, or both. The self-desiccation and presence of anticlasticcapillary menisci of negative total curvature causes some concretepores to never fully fill with liquid water, even when the pore vaporpressure greatly exceeds the saturation pressure psatðTÞ (Bažantand Jirásek 2017, p. 824). The hydration degree plays a great rolein several equations (Appendix I provides more details).

For 30% ≪ h ≪ 100%, the desorption isotherm of pore watermay be realistically simplified as follows:

ḣ ¼ kðα; hÞẇt ð37Þwhere kðα; hÞ = reciprocal moisture capacity (i.e., inverse slope ofthe isotherm) (m3=kg); and α = hydration degree, growing withconcrete age. Combining Eqs. (35)–(37), one gets the governingequation of moisture diffusion in concrete:

∂h∂t ¼ kðα; hÞ∇ · ðcp∇hÞ þ

∂hs∂t ð38Þ

where the last term on the right-hand side represents the self-desiccation sink, which can be calculated using Appendix I[Eq. (64)]. The moisture permeability is calculated by an equationof the same form as in the original Bažant-Najjar model

cpðh;αÞ ¼ c1 β þ 1 − β

1þ � 1−h1−hc�r

!ð39Þ

where c1, β, hc, and r = empirical parameters. The new model byRahimi-Aghdam et al. (2018) provides equations to estimate theseparameters from the properties of concrete mix (a=c and w=c),which makes experimental calibration of these parameters for agiven concrete unnecessary; Appendix II discusses calculation ofpermeability.

Numerical Simulations and Validation by Test Data

To obtain a general model, it is important to fit the proposed modelto all the main types of experimental data that exist (which is over adozen). Many different models could fit a few types of data setsamong them, but fitting all of them, with the same parameters is theonly way to get an unambiguous result. Here, it is demonstrated forseveral important data sets from the literature dealing with variousconcretes under diverse environmental conditions.

The available data deal with the basic creep (defined as the creepat no moisture exchange), creep at different temperatures, creep andshrinkage under drying exposure, and transitional thermal creepafter a sudden change of temperature. Table 1 summarizes thecommon parameters that were used for all experiments. In addition,the new model by Rahimi-Aghdam et al. (2018) needs only thevalues of w=c and a=c, which are usually reported by the exper-imenters. The remaining calibration parameters, which are specificto each experiment, are the parameters of long-term creep model,particularly parameters q2; q3; q4, and b0 of Model B4 [adopted asstandard recommendation of Reunion Internationale des Laborato-ries et Experts des Materriaux (RILEM)]

q2 ¼ c2p2�w=c0.38

�3

ð40Þ

q3 ¼ c3p3�w=c0.38

�0.4�a=c6

�−1.1ð41Þ

q3 ¼ c4p4�w=c0.38

�2.45�a=c6

�−0.9ð42Þ

where c2, c3, and c4 = calibration parameters. Based on Model B4,p2 ¼ 0.0586, p3 ¼ 0.0393, and p4 ¼ 0.034 were set. For c2 ¼c3 ¼ c4, the equations become the same as in Model B4. The dif-ference in optimum values is not surprising because Model B4 was

Table 1. Common model parameters

Parameter Value

a0 0.005q4ch 0.035Qh=R 1,900

CT0S 1.6=q4c1 22.5q4cT 0.012

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not formulated as a pointwise constitutive law. The last parameter,b0, is simply calculated as follows:

b0 ¼ cbq4 ð43Þwhere cb = calibration parameter. Here, most parameters are de-fined as functions of concrete properties, which reduces the numberof unknown parameters to be calibrated by tests. Furthermore, thecalibration parameters used do not change over a wide range. Forinstance, c2, c3, and c4 all vary between 0.6 and 1.5 and for crudeestimates can be taken as 1. Thus, the present XMPS theory doesnot require more tedious calibration than the original MPS theory.It gives better results because the underlying phenomena are rep-resented more realistically.

Tests of Bryant and Vadhanavikkit

As mentioned in the “Introduction,” the main impetus for develop-ing the XMPS was that the original MPS theory predicted an ex-cessive delay of, and a reverse size effect on, the additional creepdue to drying, conflicting with the comprehensive tests of Bryantand Vadhanavikkit (1987). Bryant and Vadhanavikkit (1987) usedprismatic and slab specimens of different sizes, both sealed anddrying. The concrete had w=c ¼ 0.47 and a=c ¼ 5.1. During thefirst 2 days after casting, the specimens were kept in sealed molds(with no moisture exchange) at temperature T ¼ 293.15 K, andthen were exposed for 6 days to an environment of the same temper-ature and 95% relative humidity (RH). Subsequently, groups of spec-imens were exposed to drying at 60% RH or sealed, and subjectedeither to no stress (σ ¼ 0) or applied compressive stress σ ¼ 7 MPaparallel to the drying surface. The initial strain readings were takenon Day 8, before the RHwas lowered. All the sealed specimens wereprisms of size 150 × 150 × 600 mm. The drying specimens wereprisms of dimensionsD ×D × 600 mm and slabs of the same thick-nesses D, with sizes D ¼ 100, 150, 200, 300, and 400 mm.

First, the prediction of creep at sealed condition, called the basiccreep, is considered. Fig. 4(a) compares the experimental versussimulated creep values of sealed prisms for different loading times.The calibration parameters are c2 ¼ 1, c3 ¼ 1, c4 ¼ 1.1, andcb ¼ 2.5. As seen, the predictions are satisfactory.

Next, predictions of creep of exposed specimens, prisms, andslabs are considered. For each slab, among the six faces, the fourrim faces were sealed and only the two largest faces were exposed,which led to unidirectional drying across the thickness.

Fig. 4(b) compares the experimental and simulated creep valuesof a drying slab with D ¼ 150 mm, loaded at different ages. As it

can be seen, the predicted results are in good agreement. Here,no extra calibration parameter was used to model the dryingprocess, i.e., the permeability parameters were estimated fromEq. (39). It should be mentioned that the predictions usually showsome initial error at short times. This error can be due to severalphenomena. Aside from a possible measurement error, these mayinclude considering some short-time creep as instantaneous defor-mation, effect of damage, error of Model B4 in considering effect ofloading time on viscosity, and finally, error of the model itself.

After successful predictions of drying creep for the referencesize, slabs of different sizes (thicknesses) were considered. Fig. 5(a)demonstrates correct predictions of the diffusion size effect in dry-ing creep, which means that the drying creep in smaller specimens isfaster and that the final value of drying creep is bigger. In this regard,Hávlśek at Northwestern (Bažant et al. 2014) found the originalMPS model to predict, incorrectly, the opposite—a lower final dry-ing creep for smaller specimens, which contradicted the test resultsand was what motivated the development of XMPS.

Finally, consider prisms in which only the bottom and top aresealed and the four long faces are exposed to h ¼ 0.6. According tothe simplification suggested in Model B3, the prism can be ap-proximated by an effective cylinder of the same volume/surfaceratio, for which the diameter equals 1.09× the prism side. Thisapproximation, however, did not yield very good results. Therefore,the specimen was simulated in two dimensions as a prism.Exploiting symmetries, only 1=4 of the prism sufficed for analysis.The simulation results using the real prism achieved closer fits[Fig. 5(b)]. Then, different diameters of effective cylinders weretried and, interestingly, the best approximation occurred when thediameter was almost equal to the prism side.

To illustrate the diffusion size effect on the drying part of creepmore clearly, the basic creep part was subtracted. Fig. 6 shows thedrying part of creep for specimens with different sizes. As can beseen, the diffusion size effect is significant, and the XMPS theory isable to predict it.

Until now the analyes have considered the diffusion sizeeffect in drying creep but, of course, the drying shrinkage is size-dependent as well. The cause is the differences in the rate of dryingof samples with different size. Fig. 7(a) illustrates the predictedversus simulated shrinkage of specimens with different diameters.As seen, for smaller specimens the shrinkage rate is higher and theshrinkage value is larger.

Finally, the effect of considering the effective time in Eq. (5)instead of the real time is considered. Fig. 7(b) shows the resultsobtained on the basis of both the real loading time and effective

(a) (b)

Fig. 4. Experimental versus predicted results for the experiments by Bryant and Vadhanavikkit (1987): (a) sealed samples loaded at different ages;and (b) drying slabs (unidirectional drying) with D ¼ 150 mm, loaded at different ages.

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loading time for an experiment in which the loading and dryingtimes are different. It is seen that better predictions are obtainedwith the effective loading time.

All the material parameters of the present predictions are thesame for all the simulations. This cannot be achieved with otherexisting models. The same is also true for the fitting of the datafor other concretes, which follows.

Often (especially in bending), the cracking damage can make asignificant contribution to the Picket effect (Bažant and Yunping1994), defined as a difference in total creep of a drying specimenfrom the sum of the creep of an identical sealed specimen and theshrinkage of a load-free identical companion specimen. The reasonis that shrinkage of a load-free specimen is diminished by cracking,whereas in compressed creep specimens, the effect of cracking is

minimal or zero. This phenomenon is more important for thin spec-imens, for early ages, and for flexure [it was considered in the testsof Bažant and Yunping (1994), in which cracking explained almosthalf of the observed Pickett effect]. However, in the tests of Bryantand Vadhanavikkit (1987), the contribution of cracking to thePicket effect was only about 2% and thus was neglected in simu-lations. Nevertheless, the understanding of cracking contribution tothe Pickett effect calls for deeper examination in future research.

Tests of Kommendant et al.

Kommendant et al. (1976) measured creep for different ages t 0 atloading and at different temperatures. Two almost identical con-crete mixes were used. One mix (Berks) was characterized by

(a) (b)

Fig. 7. (a) Experimental versus simulated shrinkage; and (b) experimental versus predicted results using real loading time and effective loading time.

(a) (b)

Fig. 5. Experimental versus predicted results for slab and prisms with different sizes [experiments by Bryant and Vadhanavikkit (1987)]: (a) dryingslabs (unidirectional drying) with different sizes; and (b) drying prisms (four faces exposed and two faces sealed) with different sizes.

(b)(a)

Fig. 6. Experimental versus predicted results for the size effect in drying part of creep [experiments by Bryant and Vadhanavikkit (1987)]: (a) dryingslabs (unidirectional drying) with different sizes; and (b) drying prisms (four faces exposed and two faces sealed) with different sizes.

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w=c ¼ 0.381 and a=c ¼ 4.34. The second mix (York) was almostthe same, with w=c ¼ 0.384 and a=c ¼ 4.03. The test specimenswere cylinders 15.24 × 40.46 cm (6 × 16 in:) sealed against mois-ture loss. All the specimens were cured at 23°C, and 5 days prior toloading, the temperature started increasing at a constant rate of13.33°C=day until the target value of 43°C or 71°C was reached.Furthermore, for each temperature, several tests at different ages ofloading were conducted. The calibration parameters were c2 ¼ 0.93,c3 ¼ 1.85, c4 ¼ 1.18, and cb ¼ 1.25.

Fig. 8 shows the experimental versus predicted creep values forthe York mix at T ¼ 20°C. The results are in good agreement withthe experimental ones. Although the specimens were sealed, self-desiccation caused the relative humidity to decrease, which mayhave affected the microprestress value.

Next, consider the experiments at T ¼ 43°C. Fig. 9(a) presentsthe predicted creep values for different load durations and for

T ¼ 43°C. As can be seen, the predictions agree well with the testdata. In the modeling of the experiments at elevated temperatures,increased values of creep parameters q1, q2, and q3 were considereddue to the temperature-induced compliance increase [Eq. (27)].Finally, the model’s ability to predict the creep values for testswith different temperatures but the same loading time is assessed.Fig. 9(b) documents good prediction accuracy.

Tests of York et al.

York et al. (1970) conducted several basic creep and drying tests.The creep experiments were carried out on cylindrical specimens of152 × 406 cm. The concrete properties were w=c ¼ 0.43 anda=c ¼ 4.62. Unfortunately, after about 300 days, the specimens’sealing failed. Therefore, only data up to 300 days, shown by solidcircles, are fitted, and the subsequent data, shown as empty circles,are ignored. Fig. 10(a) shows the experimental versus predicted re-sults. As can be seen, until about t ¼ 300 days, the predictions areclose enough. The calibration parameters are c2 ¼ 0.8, c3 ¼ 1.3,c4 ¼ 1.18, and cb ¼ 1.25. The same activation energies are hereused for all concretes.

Tests of Russell and Corley

These drying creep tests included three ages of concrete (t ¼ 28,180, and 360 days). The specimens were cylindrical, with the diam-eter of 15.2 cm and height of 30.5 cm; w=c ¼ 0.45 and aggregatecement ratio a=c ¼ 3.95 by weight, exposed to an environmentwith henv ¼ 0.5 and temperature T ¼ 23. Drying began after 7 daysof curing in a humidity chamber with henv ¼ 1. Fig. 10(b) com-pares the experimental versus simulated results for different loadingtimes. The predictions give a close enough agreement. The calibra-tion parameters were c2 ¼ 1.1, c3 ¼ 1.1, c4 ¼ 1.1, and cb ¼ 1.25.

(a) (b)

Fig. 9. Experimental versus predicted results of Kommendant et al.’s (1976) experiment: (a) different loading times at T ¼ 43°C; and (b) differenttemperatures loaded at t 0 ¼ 28 days.

(b)(a)

Fig. 10. Experimental versus predicted results: (a) experiment by York et al. (1970) for sealed samples with different temperatures; and (b) experimentby Russell and Corley (1978) for drying samples loaded at different ages.

Fig. 8. Experimental versus predicted results of Kommendant et al.’s(1976) experiment for different loading times at T ¼ 20°C.

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Tests of Di Luzio et al.

Di Luzio et al. (2015) focused on basic and drying creep of themodern high-performance concrete used in large-span prestressedbridges. In this experimental investigation, cylindrical specimens ofdiameter 150 mm and height 360 mm were used with environmen-tal relative humidity of 50% and a temperature of 20°C. Basic anddrying creep tests were conducted starting at the ages of 2 and28 days. The concrete properties were w=c ¼ 0.37 and a=c ¼ 4.The interesting aspect about these tests is that the loading startedat the early age of only 2 days. Fig. 11(a) shows the experimentalversus predicted results for basic creep. Fig. 11(b) illustrates thesame for drying creep. As can be seen, the results are in good agree-ment with the test data and indicate that the XMPS theory is able topredict creep correctly even at early ages. The calibration param-eters were c2 ¼ 1.35, c3 ¼ 1.1, c4 ¼ 1.33, and cb ¼ 1.25.

Tests of L’Hermite et al.

The comprehensive laboratory study of L’ Hermite et al. (1965)included several different types of creep tests. Only the fits of dry-ing creep tests at different humidities are shown here because goodfits of other types of creep tests have already been demonstratedfor other data sets. The environmental relative humidities werehenv ¼ 1, 0.75, and = 0.5. The specimens were prisms 28 cm long,with a cross section side of 7 × 7 cm. The concrete mix had w=c ¼0.45 and a=c ¼ 3.95. Fig. 12 shows the experimental versus simu-lated results for different environmental relative humidity values.Again, the predictions agree well with the experiments. The cali-bration parameters were c2¼0.9, c3¼0.8, c4¼1.0, and cb¼1.25.

Tests of Fahmi et al.

In this test series (Fahmi et al. 1972), the temperature was increasedduring the test. This produced additional creep, called the transi-tional thermal creep, for both the sealed and drying specimens. Thespecimens were hollow cylinders with inner diameter 12.7 cm, outerdiameter of 15.24 cm, and length of 101.6 cm. The concrete mix hadw=c ¼ 0.45 and a=c ¼ 3.95. The observed creep curves show up-ward jumps when the temperature is raised. Fig. 13(a) compares thesimulations with the data for basic creep, and Fig. 13(b) does thesame for drying creep.

The predictions are satisfactory. To achieve them, it was im-portant to consider all the mechanisms by which the temperaturecan change the creep rate. Especially, it was important to considerthe change of relative humidity in the pores caused by the temper-ature rise. Otherwise, the predicted jump for the drying conditionwould have been much bigger than observed. In addition, it wasimportant to consider the compliance increase due temperature in-crease (30% compliance increase of sealed condition was reported).

Fig. 12. Experimental versus predicted results for the experiments byL’ Hermite et al. (1965).

(b)(a)

Fig. 11. Experimental versus predicted results for the experiments by Di Luzio et al. (2015): (a) sealed specimen loaded at different ages; and(b) exposed specimen loaded at different ages.

(a) (b)

Fig. 13. Experimental versus predicted results for the experiments by Fahmi et al. (1972): (a) specimen in fog room; and (b) drying specimen.

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The calibration parameters were c2 ¼ 0.73, c3 ¼ 1.2, c4 ¼ 0.8,and cb ¼ 1.25. It should be mentioned that these experimentswere previously simulated by Bažant et al. (2004) using a differentmethod.

Conclusions

Based on the analyses carried out, the following conclusions canbe drawn:• The XMPS eliminates two main drawbacks of the original ver-

sion: (1) the reverse diffusion size effect on the part of creep dueto drying, and (2) the excessive delay of the drying part of creepafter the drying shrinkage. Both phenomena are now predictedcorrectly. Predictions of the diffusion size effect on shrinkageare also improved.

• Different nanoscale and macroscale viscosities can now be dis-tinguished. In XMPS, the flow term is considered to be a func-tion of the macroscale viscosity, which depends on the porehumidity rate, a feature that transpired from earlier MD simula-tions showing that the apparent viscosity in sliding of two par-allel C─S─H sheets should depend on the velocity of molecularlayer of water moving between these two sheets.

• In the XMPS, the age effect on creep at variable humidity andtemperature histories is based not on an empirical effective age(or maturity), but on the effective hydration time calculated fromthe growth of hydration degree at each point of the structure(or each integration point of a finite element program).

• The temperature change effect on pore relative humidity can berealistically described by a humidity-dependent hygrothermiccoefficient [introduced by Bažant and Najjar (1971)].

• Empirical formulas for estimating the parameters of the hu-midity dependence of permeability have been developed. Theymostly obviate the need for calibrating the permeability lawby tests.

• The XMPS is fully compatible with the Model B4. Unlikethe previous version, all the material parameters can be esti-mated from Model B4, making calibration unnecessary in mostapplications.

Appendix I. Algorithm for Evolution of Hydration

1. For the cement paste or concrete with known water/cementand aggregate/cement ratios, calculate the initial volume frac-tion of cement Vc0 and water V

w0

Vc0 ¼ρaρw

ρaρw þ ρcρwa=cþ ρcρaw=cð44Þ

Vw0 ¼ρaρcw=c

ρaρw þ ρcρwa=cþ ρcρaw=cð45Þ

where ρw, ρc, and ρa = specific mass of water (in1,000 kg=m3), cement (here considered as 3,150 kg=m3),and aggregates (here 1,600 kg=m3) for gravel and sand com-bined, respectively.

2. Calculate the average cement particle size (i.e., particle radius)a0, based on the cement type. In this study, the Blaine finenessof cement, fbl, equal to 350 m2=kg was considered to corre-spond to particle radius of 6.5 μm. Also, calculate the numberof cement particles, ng, per unit volume of cement

a0 ¼ 6.5ðμmÞ350

fblð46Þ

n ¼ Vc0

43πa30

ð47Þ

3. Choose a reasonable hydration degree for setting the time αset,and the time αc at which the C─S─H barrier will be com-pleted, i.e., critical hydration degree at which a completeC─S─H barrier will form around the anhydrous cement grains(about 1 day). For a normal cement with an0 ¼ 6.5 μm andw=c ¼ 0.3 at T ¼ 20°C, the values α0set ¼ 0.05 and αc ¼ 0.36are good approximations. For specimens with different T,w=c, and cement type, reasonable values may be calculated asfollows:

αc ¼ α0cf1ðaÞf2ðw=cÞf3ðTÞ < 0.65 ð48Þ

f1ðaÞ ¼an0a

ð49Þ

f2ðw=cÞ ¼ 1þ 2.5ðw=c − 0.3Þ ð50Þ

f3ðTÞ ¼ exp�EαR

�1

273þ T0− 1273þ T

��ð51Þ

αsetα0set

¼ αcα0c

ð52Þ

4. Calculate the volume fraction of cement. Vcset, portlandite,VCHset , and gel (C─S─H plus ettringite), V

gset. Using these frac-

tions, calculate the radius of the anhydrous cement remnants,aset, and the outer radius of C─S─H barrier, zset. To describethe chemical reaction of hydration, use the volume ratiosζgc ¼ 1.52 and ζCHc ¼ 0.59 in the following equations:

Vcset ¼ ð1−αsetÞVc0 VCHset ¼ ζCHcαsetVc0Vgset ¼ ζgcαsetVc0 ð53Þ

aset ¼�Vcset43πng

�13

; zset ¼�Vcset þ Vgset

43πng

�13 ð54Þ

5. In each time step, use the hydration degree α and humidity hfrom previous step to calculate the water diffusivity Beff

Beff ¼ B0f0ðhpÞf4ðαÞ ð55Þ

f0 ¼ cf þ1 − cf

1þ�1−hpt1−h�

nh

ð56Þ

f4ðαÞ ¼ γe−γ for α ≤ α� ð57aÞ

f4ðαÞ ¼ ðβ=αsÞmeðβ=αsÞm for α > α� ð57bÞwhere

γ ¼�

ααmax

�m; β ¼ α − α� þ α�αs=αmax ð58Þ

Use αs ¼ 0.3, αmax ¼ αc=2, α� ¼ 0.75αc, and m ¼ 1.8.Furthermore, c, h�, nh, and cf are empirical parameters. In thisstudy, nh ¼ 8, h� ¼ 0.88, and cf ¼ 0.

6. In each step, calculate the radius of the equivalent contact-freeC─S─H shells, ẑ, that gives the same free surface area as theactual shell radius z would if the shell surfaces were free withno contact

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ẑt ¼zt

1þ ðzt−a0u Þ5ð59Þ

where u ¼ a0=6.4. This relation has been modified comparedwith that of Rahimi-Aghdam et al. (2017) in order to representthe multidecade continuation of hydration better (this changehas no effect on model performance up to 10 years).

7. In each time step, using pore humidity h, anhydrous cementremnant size a, and C─S─H barrier outer radius z from theprevious time step, as well as the latest diffusivity value, cal-culate the water discharge Q1t

Q1t ¼ 4πatztBeffðα; hpÞhp − hczt − at

�ẑ2

z2t

�ð60Þ

The last factor on the right-hand side, ẑ2=z2t , serves to considerthe reduction of C─S─H shell surfaces due to mutual contacts.

8. In each step, using the calculated water dischargeQ1t , calculatethe increment of cement volume, dVct , of portlandite, dVCHt ,and cement gel dVgt

Vctþdt ¼ Vct þ dVct ¼ Vct − ngQ1t ζcwdt ð61aÞ

Vgtþdt ¼ Vgt þ dVgt ¼ Vgt þ ngQ1t ζgwdt ð61bÞ

VCHtþdt ¼ VCHt þ dVCHt ¼ VCHt þ ngQ1t ζCHwdt ð61cÞ

where Vt ¼ VðtÞ, and so on; and ζcw, ζgw, and ζCHw = volumesof the cement consumed, C─S─H gel produced, and por-tlandite produced per unit volume of discharged water,respectively. These volume fractions are calculated usingζcw ¼ 1=ζwc, ζgw ¼ ζgcζcw, and ζCHw ¼ ζCHcζcw.

9. In each time step, use the calculated dVct to calculate theincrement of hydration degree dαt and cement particleradius dat

atþdt ¼ at þ dat ¼ at þ1

4πa20ngdVct ð62aÞ

αtþdt ¼ αt þ dαt ¼ αt − 34πa30ng

dVct ð62bÞ

10. At each time step, calculate the increment of gel barrier dzt

dzt ¼dVgt þ dVct

4πẑ2for αt > αc ð63Þ

11. Finally, calculate the self-desiccation increment of relativehumidity, dhst , saturation degree, dS

capt , and interparticle

porosity, dϕcapt

dhst ¼ Kh�dVct ðζbw þ ϕnpζgcÞ − dϕcapt Scapt

ϕcapt

�ð64Þ

where

dϕcapt ¼ −ðdVgt þ dVCHt þ dVct Þ þ ðϕgp − ϕnpÞζgc ð65Þwhere ϕnp = nanopore part of gel porosity in which the poresare too small to obey the Kelvin relation. These pores are as-sumed to be always saturated. In this study, 2=3 of gel poreswere assumed to be nanopores.

Appendix II. Algorithm for Determining Permeabilityand Diffusivity

Based on Bažant and Najjar’s (1972) model, which has beenembodied in the fib Model Code 2010 [Fération internationalede béton (Beverly 2010)], the equation for moisture diffusion inconcrete reads

∂h∂t ¼ kðα; hÞ∇ · ðcp∇hÞ þ

∂hs∂t ð66Þ

where kðα; hÞ = reciprocal moisture capacity (i.e., inverse slope ofthe isotherm) (m3=kg), where α is the hydration degree; cp = per-meability; and the last term on the right-hand side is a distributedsink representing the self-desiccation. Same as in the Bažant-Najjar(1972) model, the dependence of moisture permeability cp on hmay again be expressed as follows:

cpðh;αÞ ¼ c1 β þ 1 − β

1þ � 1−h1−hc�r

!ð67Þ

where c1, β, hc, and r = unknown parameters that should bedetermined based on experiments of relative humidity evolution.Usually, however, the creep and shrinkage testers do not report therelative humidity values, and so these values and the parameters areguessed, which can cause significant error. To solve this issue,Rahimi-Aghdam et al. (2018) proposed simple empirical relationsto determine these parameters based on concrete admixtures. Theyset r ¼ 2 and proposed calculating the other three parameters asfollows:

c1 ¼ 60½1þ 12ðw=c − 0.17Þ2�α=αu ð68Þ

hc ¼ 0.77þ 0.22ðw=c− 0.17Þ1=2 þ 0.15�αuα

− 1�

but hc < 0.99

ð69Þ

β ¼ cf=c1 ð70Þ

c0f ¼ 60½1þ 12ðw=c − 0.17Þ2�α=αu ð71Þ

cf ¼(c0f ðh > hsÞ0.1c0f þ 0.9c0fðh=hsÞ4 ðh < hsÞ

ð72Þ

where αu = ultimate hydration degree in sealed concrete, which is afunction of w=c; it is estimated

αu ¼ 0.46þ .95ðw=c − 0.17Þ0.6 but αu < 1 ð73Þ

Appendix III. Functions for ComplianceApproximation

Based on the definition of effective hydration time, the asymp-totic matching approximation from Bažant et al. (1997b) can begeneralized

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Qðt; t 0; t 0effÞ ¼ Qfðt 0effÞ�1þ

�Qfðt 0effÞ

zðt; t 0; t 0effÞ�

rðt 0eff Þ�− 1

rðt 0effÞ

Qfðt 0effÞ ¼�0.086

�t 0eff

1 day

�29 þ 1.21

�t 0eff

1 day

�49

�−1

zðt; t 0; t 0effÞ ¼�

t 0eff1 day

�−0.5ln

�1þ

�t − t 01 day

�0.1�

rðt 0effÞ ¼ 1.7�

t 0eff1day

�0.12

þ 8 ð74Þ

where t 0eff = effective hydration time at the time of loading, whichis calculated from the hydration degree at the time of loadingusing Eq. (5).

Acknowledgments

Partial financial support from the U.S. Department of Transporta-tion, provided through Grant No. 20778 from the InfrastructureTechnology Institute of Northwestern University, from the NSFunder Grant No. CMMI-1129449, and from Nuclear RegulatoryCommission (NRC) under Award No. NRC-HQ-60-14-FOA-0001,are gratefully appreciated. Thanks are due to Petr Havlásek, visitingresearcher at Northwestern coadvised by Milan Jirásek from CTUPrague, for discovering the discrepancy of the initial MPS theoryvis-à-vis Bryant et al.’s size effect data.

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