Evaluation of zero-stress temperature and cracking temperature of high performance concrete at early ages

Assessing the risk of cracking of high performance concrete induced by restrained volume changes from early ages is of considerable significance. To estimate and control such cracking risk of high performance concrete, two characteristic temperatures, namely zero-stress temperature (Tz) and cracking temperature (Tx) are crucial. In this study, the two temperatures are investigated in-depth by both theoretical analysis and experimental studies. For predicting the evolutions of Tz and Tx from early ages, rigorous yet practical models are proposed, which crucially take the visco-elastic behaviour of concrete into account. The reliability and predictive capability of the proposed models are demonstrated through a series of comparisons between the predicted and the measured results. Based on the predicted Tz and Tx profiles, practical thermal control criteria for preventing concrete from cracking caused by restrained strain are put forward. In principle, the actual temperature (T) of concrete should be kept higher than both Tz and Tx to properly maintain the stress induced by restrained strain in compression at early ages. If T becomes lower than Tz and reduces continuously, the lower the value of T, the higher the risk of cracking of concrete induced by restrained strain. As a consequence, once the value of T reaches or becomes lower than Tx, cracking is highly likely to occur. For a given actual temperature condition, lowering Tz and Tx can mitigate the risk of the cracking of concrete. Finally, effective measures for such lowering of Tz and Tx are also proposed.


Introduction
Due to the release of hydration heat and the thermal interaction with the ambient environment after mixing, high performance concrete tends to experience considerable initial temperature rise and associated thermal expansion (see Fig. 1a and b). Then, as the hydration reaction slows down and the loss of hydration heat plays the dominant role, a cooling process commences and a corresponding thermal contraction happens [1]. In fact, besides thermal strain, considerable autogenous and drying shrinkage often take place simultaneously at early ages [2][3][4][5][6]. Since concrete structures are always subjected to a certain level of external restraint in practice, stress due to restrained strain tends to occur. For an example of a thin concrete slab with external restraint (see Fig. 1d): as the total strain including thermal strain, autogenous shrinkage, and drying shrinkage expands at an initial stage (Fig. 1b), residual compressive stress evolves after creep relaxation, as shown in Fig. 1c. Subsequently, as the total strain shrinks, the residual compressive stress reduces and transfers into tensile stress progressively. As long as the tensile stress reaches the tensile strength of concrete, early-age cracking is the result (see Fig. 1d). According to [7][8][9][10], the concrete temperatures corresponding to the zero stress and the cracking stress are the zero-stress temperature (T z ) and the cracking temperature (T x ), as marked in Fig. 1a. The two feature temperatures are of significant importance to carry out a holistic study on the cracking risk of early-age concrete induced by restrained strain.
While measurements of T z and T x of early-age concrete had been reported in the past studies [7][8][9][10][11][12], there remain no appropriate numerical models for the two temperatures. In this paper, an attempt has been made to model the evolutions of the two temperatures at early ages based on substantial previous studies performed by the authors [5,10,13]. Both the theoretical basis and practical significance of predicting T z and T x were presented and justified. Moreover, effective and practical measures for minimizing the risk of cracking of concrete caused by restrained strain are also discussed.

Theoretical basis
To determine the stress induced by externally restrained strain, a phenomenological model based on the elastic stress-strain response of concrete has been proposed in [14], as given below: where t is the age of concrete, r tot is the total stress induced by the restrained total strain (R res e tot ), and R res is the degree of restraint. e th , e as , and e d are free thermal strain, autogenous and drying shrinkage of concrete, respectively. For the strains in Eq. (1), contraction is defined as positive, and expansion as negative. E e is the effective elastic modulus that takes aging and stress relaxation (or creep) of early-age concrete into account. According to [15], the effective elastic modulus can then be expressed as: where E(t) is the elastic modulus, b(t) is an aging coefficient (of between 0.6 and 1.0) [16], and u(t) is the creep coefficient. According to the authors' research in [17,18], u(t) of early-age concrete can be approximated using a revised MC2010 creep model. Although Eq. (1) is proposed primarily for a 1D configuration, the equation is valuable for assessing the stress of concrete induced by restrained strain in reality. Generally, combined with the finite element  1) can be applied to model the stresses in 3D concrete members with complex geometries [19]. For those concrete slabs, beams, and columns which can be simplified as a 1D configuration, the equation can be used directly. In terms of the potential defect of Eq. (1), further discussion can be found in Sects. 4 and 5.

Zero-stress temperature
For fresh concrete, despite considerable volume change before setting (or the time-zero) [5,20,21], no stress due to restrained strain appears. After timezero, concrete starts to behave like a solid (with E e [ 0) and enables stress transfer. According to Eq. (1), to maintain r tot at negligible levels, the restraint-free e tot should be kept as close to zero as practicable since R res is generally non-zero. On that basis, the fictitious temperature profile (T z ) that keeps e tot equal to zero (and thereby always zeros r tot ) can be expressed as where T z (t) is the zero-stress temperature, a CTE (t) is the coefficient of thermal expansion (CTE). T 0 (t 0 ) is the initial reference temperature (or namely, time-zero temperature) for thermal stress and strain analysis of high performance concrete, which refers to the temperature at time-zero (t 0 ) [22]. In this study, the time-zero (t 0 ) of fresh concrete is defined as the time point when concrete materials form a stable solid skeleton to develop tensile and compressive stresses [5]. Rearranging Eq. (3), T z (t) can be determined by

Cracking temperature
Through the analysis above, the improved model to determine the total stress (Eq. (1)) induced by restrained total strain under an actual temperature profile (T) can be presented as [10]: It is clear that as long as the actual temperature (T) and T z are the same, r tot is zero. When T is higher than T z , e tot is expansive, causing compressive stress. On the contrary, if T is lower than T z , then e tot is contractive, resulting in tensile stress. Once the tensile stress reaches the actual tensile carrying capacity (k Á f t ) of early-age concrete, cracking induced by restrained strain happens. Given the cracking of concrete initiates when T reaches the cracking temperature T x , then where, f t is the tensile strength of concrete samples measured in accordance with relevant standards, and k is a coefficient for approximating the actual in-situ tensile carrying capacity of concrete. Typically, k is between 0.6 and 0.8 [23][24][25]. From Eq. (6), the cracking temperature (T x ) can be expressed as: Theoretically, T x (t) represents a critical temperature profile of concrete that designates conditions at which the initiation of cracking takes place. Once the actual concrete temperature becomes lower than T x (t), cracks propagate. Besides, it should be noted that the obtained T z and T x profiles of concrete (Eqs. (4) and (7)) are both essentially temperature-dependent, due mainly to the effects of actual temperature on the evolutions of autogenous shrinkage, effective elastic modulus, CTE, and tensile strength at early ages [26][27][28]. This indicates that the evolutions of T z (t) and T x (t) of young concrete can be a function of equivalent age.
3 Evolutions of T z and T x profiles of concrete at early ages

Concrete mix
Following the authors' previous studies [5,10,17,29], a concrete mix with water-to-binder ratio (w/b) of 0.25 (see Table 1) was studied to demonstrate the determination and evolution of T z and T x profiles in this paper.
The 28-day compressive strength of the concrete is approximately 85.0 MPa. The oxide compositions of cementitious materials and particle sizes of aggregates used in the concrete mix can be found in Tables 2 and  3.

Key properties of concrete
All key parameters of the concrete mix used in the calculations of T z and T x can be found in [5], in which the autogenous shrinkage, CTE, elastic modulus, splitting tensile strength, and time-zero are reported.
In this study, these physical properties of young concrete are modelled using equivalent age (or maturity). According to [30][31][32], the equivalent age is calculated by Eq. (8).
where E a is the activation energy (J/mol), T(t) is the temperature history of concrete, R is the ideal gas constant (8.3 J/(molÁK)), and T ref is the reference temperature (293.0 K). In this study, E a is taken as 30.0 kJ/mol, as recommended for cementitious pastes containing fly ash and silica fume in [33]. Improved empirical models for simulating each key property of young concrete as a function of equivalent age are given below. Figure 2 shows the development of autogenous shrinkage of the concrete at early ages. To match the measured autogenous shrinkage, a modified empirical model (Eq. (9)) recommended in the CEB-FIP standard [34] and Eurocode 2 [35] is used, as follows:

Autogenous shrinkage
where e Ã as is the estimated ultimate autogenous shrinkage, and t e is the equivalent age in days. By fitting the test data using a nonlinear regression method, e Ã as and t 0 can be determined and equal -390 9 10 -6 m/m and 8.5 h (* 0.35 d), respectively.

Coefficient of thermal expansion
The evolution of the CTE of early-age concrete (see Fig. 3) is modelled by the following equation: where, a Ã CTE is the estimated ultimate CTE of concrete (taken as 12.0 9 10 -6 /°C). a and b are fitting parameters (0.20 and 0.41, respectively).

Elastic modulus
A revised model (Eq. (11)) drawn from the Eurocode 2 [35] is applied to simulate the development of the elastic modulus of early-age concrete measured in [5], as shown in Fig. 4. Combing Eqs. (11) and (2), the effective elastic modlus can be obtained.
where, E Ã is the ultimate elastic modulus of concrete (* 48.0 GPa), and s and n are fitting coefficients (0.018 and 0.32, respectively). Figure 5 presents the splitting tensile strength of the concrete measured in [5] and modelled by Eq. (12). Based on the splitting tensile strength, the direct tensile strength of concrete can be estimated by multiplying by a conversion factor of 0.9 [35,36].

Tensile strength
where, f Ã t is the ultimate splitting tensile strength of concrete. p and q are fitting coefficients. In this study, f Ã t , p, and q are 7.1 MPa, 0.015, and 0.5, respectively.

Results and discussion
Inputting those key properties presented above into Eqs. (4) and (7), the T z and T x profiles of the concrete with w/b of 0.25 were obtained (see Fig. 6). For simplicity, drying shrinkage was neglected in the calculation, due to the negligible shrinkage of the concrete under sealed conditions in [5]. The temperature of concrete at the time-zero was assumed at * 20°C, and the degree of restraint was deemed as 1.0. In Fig. 6, it can be observed that the obtained T z profile increases over time, resulting mainly from the development of autogenous shrinkage of concrete at early ages. Different from the evolution of T z , the determined T x profile firstly decreases dramatically and then increases gradually. This may be due to the combined effect of the fast evolution of tensile strength and the low effective elastic modulus and CTE at early ages (as presented by Eq. (7)).
To form an in-depth understanding of the practical significance of the obtained T z and T x profiles, a random actual temperature profile (T) is assumed in Fig. 6. It can be observed that T is split into three areas by T z and T x profiles: Area A: T is larger than T z , indicating the total stress of concrete induced by restrained strain is compressive (see Eq. (5)). The higher the value of T, the larger the compressive stress; Area B: T is lower than T z but remains higher than T x , meaning that the compressive stress induced by restrained strain has transferred into tensile stress. The lower the value of T, the larger the tensile stress, and the higher the cracking risk of concrete; Area C: T is lower than T x , implying that the tensile stress induced by restrained strain exceeds the tensile carrying capacity of concrete. Cracks appear and propagate.
From the analysis above, it is clear that for a given actual temperature profile, lowering T z and T x is of considerable significance for mitigating the risk of early-age cracking [9]. According to Eqs. (4) and (7), under a specified degree of restraint, minimizing the magnitude of autogenous and drying shrinkage or reducing the time-zero temperature (T 0 ) can lower both T z and T x . Besides, increasing tensile strength or decreasing effective elastic modulus & CTE can reduce T x exclusively. In practice, modifying the deformational and mechanical properties of the concrete with a given mix design is challenging. Therefore, the most convenient measure for lowering T z and T x is to decrease T 0 , which can be practically achieved by a wide range of temperature control measures [37]. For example, concrete mixing and casting are scheduled at night or in cool weather. Or, aggregates and water used for producing concrete are cooled before mixing. In previous studies, those temperature control measures are proposed mainly for limiting the peaking temperature of concrete or mitigating the temperature rise induced by hydration heat at early ages. In this study, the potential significance of reducing T 0 for lowering T z , T x , and thereby associated cracking risk of concrete is highlighted.

Assessment of the proposed T z model
In [4,38,39], the stress of early-age concrete caused by restrained thermal strain and autogenous shrinkage (excluding drying shrinkage) was studied, as shown in Figs. 7, 8, 9. Combining the measured temperature and shrinkage data in [4,38,39] and Eq. (4), the T z profile of each concrete mix was determined, see Figs. 7,8,9. Because of the lack of data for maturity transformation, the obtained T z profiles were described as a Fig. 6 Evolutions of T z and T x of early-age concrete as a function of equivalent age function of age. Besides, Eq. (10) was used to approximate the evolution of the CTE at early ages.
In Figs. 7, 8, and 9, the characteristic T z points determined by the cross-over point of the actual temperature profile and the calculated T z profile are * 35.0, * 43.0, and * 29.0°C, respectively. But, the realistically measured T z points, corresponding to the measured zero stress in Fig. 7 -9 are * 40.5, * 50.0, and * 34.0°C, respectively. The determined T z results seem to be always lower than those correspondingly measured values, which may be due mainly to neglecting the visco-elastic response of concrete in Eq. (1). As stressed in Sect. 2, Eq. (1) is essentially built on the assumption of elastic stress-strain behaviour of concrete. However, under sustained loads, the actual stress-strain response of concrete is non-elastic [40][41][42][43], due to the existence of non-recoverable creep component (or plastic strain).

Finalized T z and T x models
To take plastic strains (e p ) of early-age concrete under sustained loads into account, the original T z model (Eq. (4)) can be revised as follows However, e p as an input parameter in Eq. (13) is hard to be determined or approximated. For convenience, Eq. (13) can be further improved by introducing a coefficient c instead of e p . The modified T z model is expressed as where, c t ð Þ is a conversion coefficient drawn from the B3 model in which it represents the visco-elastic compliance of concrete [44][45][46], as given below where, c 0 , g, and h are fitting parameters. Qualitatively, with the strength and elastic modulus developing from very early ages, the non-elastic response of concrete tends to diminish [40][41][42][43]. Accordingly, the conversion coefficient (c) reduces over time. Using nonlinear regression analysis to achieve a good fitting between the measured and predicted T z in [4,38,39], c 0 , g, and h are set to 1.25, 0.30, and 0.41, respectively (as shown in Figs. 7,8,9).
To further examine the applicability of the improved T z model with the determined c, a comparison between the measured and predicted T z results of the concrete mix with w/b ratio of 0.25 (see Table 1) is   Measured and calculated T z of early-age concrete (w/b of 0.35), adapted from [39] performed. In [10], the authors measured the T z of the concrete in accordance with modified uniaxial restrained shrinkage tests, using a Temperature Stress Testing Machine [27,47]. For obtaining multiple zerostress temperatures, a fluctuating temperature profile was imposed on a concrete specimen under fully restrained conditions from an age of 3 days onwards, as plotted in Fig. 10. In parallel, the total strain of another specimen under the same temperature history but restraint-free conditions was also measured. The two test specimens were both fully sealed by adhesive foil tape to achieve negligible moisture interaction between concrete and ambient environment. Autogenous shrinkage (AS, see Fig. 10) was obtained by subtracting thermal strain from the free total strain, as reported in [27]. Based on the determined autogenous shrinkage and CTE in [10], the T z profile of the concrete was calculated by Eqs. (4) and (14), respectively. In Fig. 10, it can be observed that the T z profile determined by Eq. (14) matches those measured T z points better than the other profile, which demonstrates the suitability of the proposed conversion coefficient c (Eq. (15)). To sum up, Eq. (15) appears to be proper for approximating the visco-elastic behaviour of early-age concrete with w/b between 0.25 and 0.40 (as shown in Figs. 7,8,9,10). However, it should be noted that those fitting coefficients (c 0 , g, and h) may vary for other concrete mixes with significantly low or high w/b.
In addition, combining Eq. (14) with Eq. (7), the revised T x model with visco-elastic behaviour taken into account can be obtained. To examine the predictive capability of the T x model, further data analysis is carried out. In [9], the stress of concrete mix with w/b of 0.385 induced by restrained strain is experimentally measured, using a rigid cracking frame. As shown in Fig. 11, the measured stress reaches zero at * 30 h, and the cracking of concrete takes place at * 105 h. Correspondingly, the T z and T x at * 30 h and * 105 h are * 53.1°C and * 23.0°C, respectively. Due to the absence of the measured autogenous shrinkage in [9], it is challenging to obtain a precise T z profile of the concrete. The T z profile given in Fig. 11 was approximated according to Eq. (14) and the measured T z at * 30 h. On that basis, the T x profile of concrete from 24 h onwards was estimated by Eq. (7). In the determination of T x , a constant basic creep coefficient of 2.0 was used to transfer the measured elastic modulus in [9] into effective elastic modulus (via Eq. (2)). Such value of basic creep coefficient is recommended by the Australian Standard (AS3600-2018) [48] for concrete with 28-day compressive strength of over 60 MPa.
In Fig. 11, the calculated T x profile shows an overall rising trend at early ages, which is consistent with the estimated T x profile in Fig. 6. From 24 h to * 103 h, the actual concrete temperature (T) is always higher than the value of the estimated T x , indicating no cracking occurs. As both T and T x profiles evolve over time, a cross-over between T and T x occurs at * 103 h, indicating that cracking takes place. The temperature at 103 h can therefore be Fig. 10 Measured and calculated T z of early-age concrete (w/b of 0.25), adapted from [10] regarded as the estimated T x value (* 25.3°C) under the given temperature condition in [9]. Compared to the realistically measured T x (* 23.0°C) in [9], the predicted outcome (* 25.3°C) appears to be quite acceptable. The prediction error is approximately 10% ( 25:3À23 23 Â 100%). Such good agreement between the predicted and measured T x points again demonstrates the validity of the proposed T z and T x models in this study.

Conclusions
In this study, the zero-stress temperature (T z ) and the cracking temperature (T x ) associated with the volume change of high performance concrete are investigated in-depth. The key contributions and findings of the research can be summarized as follows: • Both the practical significance and the theoretical basis for obtaining continuous T z and T x profiles of high performance concrete from very early ages are clarified. Unique models with the visco-elastic behaviour of concrete taken into account are proposed for predicting the evolutions of T z and T x profiles at early ages. The predictive capability of the T z and T x models is demonstrated through a series of comparisons between the predicted and the measured results; • Based on the determined T z and T x profiles, practical thermal control criteria for assessing and mitigating the risk of cracking of concrete induced by restrained volume change are proposed and justified. Ideally, the actual temperature of concrete (T) should be kept higher than both T z and T x in order to maintain the stress induced by restrained strain in compression. Once the value of T becomes lower than T z , tensile stress caused by restrained strain develops. As the value of T continues to decrease, the tensile stress rapidly approaches the tensile carrying capacity of concrete. When T is equal to or lower than T x , the risk of cracking becomes markedly high. To minimize such cracking risk, thermal control measures must be taken to (i) lower T x and (ii) ensure T being kept always higher than T x . For lowering T z and T x , in addition to mitigating the autogenous and drying shrinkage as well as increasing the tensile strength of early-age concrete, reducing the temperature of concrete at the time-zero is also beneficial. Funding Open Access funding enabled and organized by CAUL and its Member Institutions. This study was funded by the Australia Research Council project (DP 180103160). The first author also acknowledges the financial support from the Danish Hydrocarbon Research and Technology Centre, Denmark.
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Declarations
Competing interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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