# Concrete cracking risk analysis at early age by means of a ring test

Agathe Bourchy, Boumediene Nedjar, Jean-Michel Torrenti

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Agathe Bourchy, Boumediene Nedjar, Jean-Michel Torrenti. Concrete cracking risk analysis at early age by means of a ring test. Construction and Building Materials, 2025, 482, pp.141446. 10.1016/j.conbuildmat.2025.141446. hal-05085530

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Submitted on 26 May 2025

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# Concrete cracking risk analysis at early age by means of a ring test

Agathe Bourchy<sup>a,†</sup>, Boumediene Nedjar<sup>b,†</sup>, Jean-Michel Torrenti<sup>c,d,*</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Concrete

Restrained shrinkage

Early age

Modelling

Risk of cracking

## ABSTRACT

Early-age cracking in massive concrete structures is a critical issue affecting durability and maintenance costs. This study introduces the BT-Ring test, an adaptation of the Schlitter ring test, designed to evaluate stress evolution due to restrained thermal and autogenous deformations. The test replicates in-situ conditions, enabling a comprehensive assessment of cracking risk by means of a complete analytic solution for the stresses generated during the test. Experimental results reveal an initial compressive stress phase followed by tensile stress build-up, with cracking occurring when the mean orthoradial stress exceeds the concrete's tensile strength. A delay between predicted and observed cracking highlights variability in tensile strength and possible size effects. Numerical modelling aligns well with experimental data, confirming the test's reliability in stress prediction when all the parameters of the model are correctly characterized. Future research should focus on refining numerical models by incorporating damage-based approaches to account for heterogeneities and size effects. Additionally, applying this methodology to various concrete formulations, including low-carbon mixtures, would enhance its applicability. The BT-Ring test, combined with advanced modelling, presents a promising approach for optimizing concrete formulations and mitigating early-age cracking risks in large-scale structures.

## 1. Introduction

Concrete cracking control at early age is a major issue in massive structures development because of the risk of reduced durability [1–5] and the cost of the repairs. Indeed, thermo-hydro-chemo-mechanical (T-H-C-M) behaviour of cement at early age influences the durability of concrete by potentially creating tensile stresses and inducing cracking when the deformations are restrained [6–9]. Concrete strains can have several sources: in massive structures the two main sources are: the exothermic hydration reactions (thermal deformations), and the isothermal volume variations due to hydration reactions (autogenous shrinkage). When temperature rises in massive concrete structures, the setting happens at high temperature which leads to a thermal shrinkage that is added to autogenous shrinkage. If the deformations are restrained, the increase of temperature after setting generates a compression, and when the temperature decreases tensile stresses are generated because of the evolution of the stiffness of concrete with hydration that implies a non-symmetrical evolution of the stresses with temperature. There is then a risk of cracking if these generated stresses become close to the concrete tensile strength (Fig. 1) [10].

The risk of early age cracking depends on several factors [11–15] as the heat generated by hydration [16–24], the development of the Young modulus [25–32], the compressive and tensile strength [26,33–35], the early age shrinkage, including autogenous, drying and due to thermal deformations [36–43], and the early age tensile creep [10,44–50] in restrained concretes. With the advent of low-carbon concretes and the multiplication of ingredients in concretes [51–54], it is not easy to accurately predict these factors and their evolution in advance. When mitigation techniques are used, modelling can be used [55–57]. However, estimating the risk of cracking by modelling, which depends on all these factors requires many tests -hence time- is not always applicable given construction deadlines.

Another way of estimating this risk is to carry out a test involving all these phenomena as in practice. This test must reproduce in-situ conditions, i.e. the possibility of a rise in temperature and a partial restraint of deformations. Several tests exist to measure the restrained thermal shrinkage: linear – as Temperature Stress Testing Machine (TSTM) apparatus [58–65] and its derivative like the mini TSTM [66], using a plate [67] or with a ring [68–72]. The advantage of ring testing lies in the simplicity of the test, where concrete deformation is restrained by

<footnote><sup>a</sup> Université Gustave Eiffel, Materials and Structures Department, 14-20 boulevard Newton, Champ-sur-Marne 77420, France</footnote>

<footnote><sup>b</sup> Université d'Evry Paris-Saclay, LMEE, 40 rue du Pelvoux, Evry cedex 91020, France</footnote>

<footnote><sup>c</sup> ESITC Paris, 79 Av. Aristide Briand, Arcueil 94110, France</footnote>

<footnote><sup>d</sup> Univ Gustave Eiffel, Cerema, UMR MCD, F, Marne-la-Vallée 77454, France</footnote>

<footnote>* Corresponding author at: ESITC Paris, 79 Av. Aristide Briand, Arcueil 94110, France.</footnote>

<footnote>E-mail address: jean-michel.torrenti@univ-eiffel.fr (J.-M. Torrenti).</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 1. Changes in stresses at early age in the case of restrained boundary conditions, adapted from [10].</figcaption></figure>

the ring without external intervention. It has been mostly used to investigate cracking sensitivity, restrained shrinkage, creep and stress relaxation of cement-based materials. These tests could be performed with an active or a passive ring [8,73–79]. In case of passive ring tests, rings can be traditional-circular [80,81]; elliptical [82,83]; eccentric or square-eccentric [84] or dual ring tests [85–87]. In case of active ring tests, the concrete is submitted to tensile stresses due to the rings expansion [88,89], submitted to plastic restrained shrinkage [90], or the test is performed in thermal conditions, where the temperature of the core is continuously controlled [87,91–94].

Such ring test variations is impacting the degree of restraint (DR) as calculated in Eq. (1) [95].

$$DR(\%) = 100 \cdot \frac{A_{rc}E_{rc}}{A_{rc}E_{rc} + A_{cm}E_{cm}} \quad (1)$$

$A_{rc}$, cross-sectional area of restraining core, (mm²),
$E_{rc}$, modulus of elasticity of restraining core, (GPa),
$A_{cm}$, cross-sectional area of cementitious material, (mm²),
$E_{cm}$, modulus of elasticity of cementitious material, (GPa).

Several factors have been tested with such apparatus: the cement formulation [96,97], the water to cement ratio [98,99], the effect of fibres [100–103], the effect of recycled aggregates [104,105], and validated with statistical analysis and machine learning approach [106, 107].

Schlitter et al. have proposed a cheaper and more efficient ring test for evaluating the stress development due to restrained volume change during concrete hydration Fig. 2a. This test uses an invar dual concentric ring [92] thermally regulated with copper tubes as presented on Fig. 2b. The invar rings have a lower coefficient of linear thermal expansion -CTE (1.3.10⁻⁶ °C⁻¹) than concrete (12.10⁻⁶ °C⁻¹). So, when the temperature increases the external ring restraints the deformation generating a compressive stress (because of the difference between the CTEs).

Then, when the temperature decreases, the internal ring restraints the deformation and generates a tensile stress. But Schlitter’s ring test has only a width of 38 mm for the tested sample and only mortars where tested.

In this paper, a ring test based on Schlitter’s one is adapted to test concretes. This ring is used to generate stresses that are representative of stresses occurring in massive structures when the deformations are restrained. Thanks to this ring test, the early age behaviour of a concrete in which cracking occurs is studied. After an analytical analysis of the test, a thermo-mechanical modelling of the test is done with the parameters determined in companion characterization tests of the concrete, and compared to the experimental results.

## 2. Materials and methods

### 2.1. Materials tested

The risk of cracking is increased when the heat generated by hydration is important. A concrete denoted by R1bis is tested. Its formulation is given in Table 1 [108].

Table 2 presents the concrete characteristics. The hydration heat is measured with quasi-adiabatic calorimetry test (QAC) according to the standard EN 12390-14 [109], with Q41 and Q144 corresponding respectively to the hydration heat of concrete at 41 hours and 144 hours and Tmax is the maximal temperature. Slump is measured according to standard EN 12350-2 [110] and density ρ according to standard EN 12350-6 [111]. Finally, CTE denotes the coefficient of thermal expansion.

Compressive and tensile strengths (EN 12390-3 and 12390-6 [112, 113]) and Young modulus (EN 12390-13 [114]) are measured on 16*32 cm samples from 1 to 7 days. Values are presented in Table 3. The tensile strength value $R_t(t)$ at 28 days, equals to 3.8 MPa, is extrapolated from Eq. (2) [115], $R_i(t)$ the extrapolated tensile strength value (MPa), $R_{28}$ the 28 days tensile strength of the concrete (MPa), s a constant, i the time and α a coefficient equal to 1 for i < 28 days or to 2/3 for i ≥ 28 days. The equivalent time is calculated $t_{eq}(t)$ from Eq. (3) with $E_a$ the activation energy (J.mol⁻¹), R the perfect gas constant (8314 J.mol⁻¹. K⁻¹) and T the temperature (K).

$$R_t(t) = R_{28} \cdot \exp \left[ s \left( 1 - \sqrt{\frac{28}{i}} \right) \right]^\alpha \quad (2)$$

$$t_{eq}(t) = \int_{S=0}^{S=t} \exp \left[ \frac{E_a}{R} \left( \frac{1}{T(20)} - \frac{1}{T(s)} \right) \right] ds \quad (3)$$

<figure><img src="image_5.png" /><figcaption>Fig. 2. Schema (a) and picture (b) of the invar dual concentric ring and the gauges positions from Schlitter et al. [92].</figcaption></figure>
<table><caption>**Table 1**
R1bis concrete formulation [108].</caption><thead><tr><th>Cement_R_2012 (kg.m<sup>-3</sup>)</th><th>K<sub>2</sub>O (kg.m<sup>-3</sup>)</th><th>Water (kg.m<sup>-3</sup>)</th><th>Sand_NR_2016 (kg.m<sup>-3</sup>)</th><th>Gravel_NR 4/8 (kg.m<sup>-3</sup>)</th><th>Gravel_NR 8/12 (kg.m<sup>-3</sup>)</th><th>W/B</th></tr></thead><tbody><tr><td>410</td><td>1.75</td><td>203</td><td>864</td><td>100</td><td>829</td><td>0.46</td></tr></tbody></table>

<table><caption>**Table 2**
Fresh and thermal characteristics of R1bis.</caption><thead><tr><th>Slump (mm)</th><th>Density (g. cm<sup>-3</sup>)</th><th>Q41 (kJ. kg<sup>-1</sup>)</th><th>Q144 (kJ. kg<sup>-1</sup>)</th><th>T<sub>max</sub> (°C)</th><th>CTE (10<sup>-6</sup> °C<sup>-1</sup>)</th></tr></thead><tbody><tr><td>110</td><td>2.33</td><td>63.4</td><td>64.3</td><td>55.0</td><td>14</td></tr></tbody></table>

<table><caption>**Table 3**
Compressive and tensile strengths and Young modulus measured on 16*32 cm samples of R1bis concrete.</caption><thead><tr><th></th><th colspan="3">Compressive strengths R<sub>c</sub></th><th colspan="3">Tensile strengths R<sub>t</sub></th><th colspan="2">Young modulus E</th></tr></thead><tbody><tr><th>Age (day)</th><td>1</td><td>2</td><td>7</td><td>1</td><td>2</td><td>7</td><td>1</td><td>7</td></tr><tr><th>MPa</th><td>28.2</td><td>33.0</td><td>40.2</td><td>3.3</td><td>3.4</td><td>3.6</td><td>31,600</td><td>39,300</td></tr></tbody></table>

## 2.2. Methods

Measurement of residual stress development due to restrained volume change is done with a dual invar concentric ring - BT-Ring [116]. The invar rings have a CTE equal to $1.21 \cdot 10^{-6} \cdot {}^{\circ}\text{C}^{-1}$.

Considering the aggregates size, the radius of the outer ring is increased compared to the original ring developed by Schlitter et al.: the distance between the inner and outer ring is here equal to 75 mm in order to have three times the largest aggregates ($R_{II} = 146.0$ mm; $R_{IC} = 165.0$ mm; $R_{OO} = 259.0$ mm; $R_{OC} = 240.0$ mm; $t = 19.0$ mm and $h = 72.6$ mm). HBM quartz deformation gauges ($CTE = 0.5 \cdot 10^{-6} \cdot {}^{\circ}\text{C}^{-1}$) and thermocouples are disposed along the rings: four on the inner ring (G1 to G4) and four on the outer ring (G5 to G8) as presented on Fig. 3a. Concrete temperature is followed thank to a thermocouple dipped into fresh concrete, in the middle of the ring, between gauges G1 and G2. The deformations gauges are fixed thank to HBM X60 glue and protected

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** BT-Ring test (a) schema with the radius and pressions names used to get the analytical pression and stress expressions, pictures (b) from the bottom side once ran and (c) ready for a test with the thermal protection [116].</figcaption></figure>
with polish, rubber and aluminium paper. Deformations are measured every two minutes. The regulation of concrete temperature is ensured by water circulating through the invar rings (24 holes in the outer ring and 15 holes in the inner ring) (Fig. 3b) with a temperature profile according to the one measured in the quasi-adiabatic calorimetric (QAC) test (Figure S1 and Figure S2). Rock wool is used as a thermal insulation (Fig. 3c).

Before the test, the two invar rings, the wooden base and the plastic tubs to assure the water circulation are put together and slightly greased so there is no friction between the concrete and the rings. The water bath evolution is programmed according to the one measured in the QAC test. Following this, concrete is batched in order to be able to fill up and start the BT-Ring test within 15 minutes after the cement-water contact. It is only after the first recording of deformation for the 8 gauges that the thermal insulation with rock wool and insulatation cover is placed around the test to limit any thermal losses.

### 2.2.1. Thermal correction of the gauges

The strains measured by each gauge $ε_i$ are corrected thank to the Eq. (4) given by HBM to take into account the quartz gauge thermal deformation, where $ε_{corr}$, the corrected deformation of each gauge and $T_i$ the temperature at each gauge.

$$ε_{corr_i} = ε_i - (-16.56 + 2.4T_i - 0.0859T_i^2 + 0.000349T_i^3) + 0.639(T_i - 20) \quad (4)$$

Because desiccation shrinkage is not considered, in order to avoid concrete desiccation on the top of the sample, a curing compound is applied and then the sample is recovered with an aluminium sheet.

### 2.2.2. Temperature evolution

Temperature evolution measured during QAC test is reproduced in BT-Ring test thank to thermostatic bath. Fig. 4 shows the evolution of temperature (thermocouple measurements) in concrete, close to the deformation gauges and in the thermostatic bath. The temperature in the concrete and the invar rings gets within 5 minutes in equilibrium with the water bath. Along the invar ring (thermocouples fixed next to the gauges as specified in Section 2.2 and Fig. 3) and the concrete ring (thermocouple dipped into fresh concrete between gauges G1 and G2), less than one Celsius degree difference can be observed during the test. The temperature gradients observed on Fig. 4, even if minimal are not imposed. Of course, another temperature history could be imposed in the test. For instance, using a temperature evolution in a real structure (measured or obtained by modelling of the thermal problem) could be

<figure><img src="image_7.png" /><figcaption>Fig. 4. Temperature evolution during R1bis concrete BT-Ring test, close to the deformation gauges on the invar rings (G1-G4 in black and G5-G8 in green), ambient environment (orange), in concrete (grey) and in the thermostatic bath (blue).</figcaption></figure>

used.

Up to the time of 12 h, the temperature increases by more than 2.8 °C.h⁻¹ showing the high reactivity and fast hydration of the R1bis concrete, while between times 12 h to 17 h, the temperature still increases but with a slower rate (around 0.4 °C.h⁻¹). After 17 h, the temperature begins to decrease corresponding to a slowdown of the concrete reactivity.

### 2.2.3. Mechanical correction

Considering that the gauges are placed on the invar ring and not directly in contact with the concrete, a correction is applied to get the applied pressure on the concrete and so the stress developed into the concrete.

Considering the equilibrium in cylindrical coordinates, the independence of orthoradial stress $σ_{θθ}$ with respect to $θ$, we get Eqs. (5) and (6). First, we consider the outside ring to not apply pressure during shrinkage i.e. $P_{oo} = 0$. There is also $P_{II} = 0$.

$$\frac{\partial \sigma_{rr}}{\partial r} + \frac{1}{r} \frac{\partial \sigma_{r\theta}}{\partial \theta} + \frac{\partial \sigma_{rz}}{\partial z} + \frac{\sigma_{rr} - \sigma_{\theta\theta}}{r} = 0 \quad \text{and} \quad (5)$$

$$\frac{\partial \sigma_{r\theta}}{\partial r} + \frac{1}{r} \frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \frac{\partial \sigma_{\theta z}}{\partial z} + \frac{2\sigma_{r\theta}}{r} = 0 \quad (6)$$

$$\frac{\partial \sigma_{rr}}{\partial r} + \frac{\sigma_{rr} - \sigma_{\theta\theta}}{r} = 0 \quad (6)$$

Eq. (7) is from Saint Venant principle with regard to the elastic displacement field,

$$(\lambda + 2\mu) \cdot \overrightarrow{\text{grad}}(\text{div}(\vec{u})) - \mu \cdot \overrightarrow{\text{rot}}(\overrightarrow{\text{rot}}(\vec{u})) = 0 \quad (7)$$

where $u_θ = 0$, $u_z = 0$ and $u_r = u_r(r)$ leading to $(\overrightarrow{\text{rot}}(\vec{u})) = 0$. The integration of Eq. (7) leads to Eqs. (8) and (9), where $C_1$ and $C_2$ are constant.

$$u_r = C_1 r + \frac{C_2}{r} \quad (8)$$

$$\begin{align}
\epsilon_{rr} &= \frac{\partial u_r}{\partial r} = C_1 - \frac{C_2}{r^2} \nonumber \\
\epsilon_{\theta\theta} &= \frac{u_r}{r} = C_1 + \frac{C_2}{r^2} \quad \text{because } u_\theta = 0 \tag{9} \\
\epsilon_{zz} &= 0 \nonumber
\end{align}$$

The general equation of the stress in function of the Young modulus E, the Poisson coefficient $ν$ and the strain $ε$ can be calculated with the displacements obtained in Eq. (10) and leading to Eq. (11) and Eq. (12).

$$\sigma_{ij} = \lambda \text{tr} \epsilon S_{ij} + 2 \mu \epsilon_{ij} \text{ with } \text{tr} \epsilon = 2C_1; \quad \lambda = \frac{\nu E}{(1 + \nu)(1 - 2\nu)} \text{ and } \mu = \frac{E}{2(1 + \nu)} \quad (10)$$

$$\sigma_{rr} = 2(\lambda + \mu)C_1 - 2\mu \frac{C_2}{r^2} \quad (11)$$

$$\sigma_{\theta\theta} = 2(\lambda + \mu)C_1 + 2\mu \frac{C_2}{r^2} \quad (12)$$

Solving Eqs. (11) and (12) considering $r = R_{II}$ gives Eqs. (13) and (14).

$$\sigma_{rr} = 2(\lambda + \mu)C_1 - 2\mu \frac{C_2}{R_{II}^2} = 0 \quad (13)$$

$$\epsilon_{\theta\theta}(r = R_{II}) = C_1 + \frac{C_2}{R_{II}^2} \quad (14)$$

Combining Eqs. (12) and (13) gives the constants $C_1$ and $C_2$ in Eqs. (15) and (16).
$$C_2 = R_{II}^2 \cdot \frac{\lambda + \mu}{\lambda + 2\mu} \cdot \varepsilon_{\theta\theta}(r = R_{II}) \qquad (15)$$

$$C_1 = \frac{\mu}{\lambda + 2\mu} \cdot \varepsilon_{\theta\theta}(r = R_{II}) \qquad (16)$$

Combining Eq. (11) with Eq. (14) and Eq. (15), this leads to the pression P<sub>IC</sub> in Eq. (15).

$$P_{IC} = \sigma_{rr}(r = R_{IC}) = 2(\lambda + \mu) \cdot \frac{\mu}{\lambda + 2\mu} \cdot \varepsilon_{\theta\theta}(r = R_{II}) - 2\mu \frac{R_{II}^2}{R_{IC}^2} \cdot \frac{\lambda + \mu}{\lambda + 2\mu} \cdot \varepsilon_{\theta\theta}(r = R_{II}) \qquad (17)$$

Eq. (8) is updated to Eq. (18).

$$u_r = \frac{C_2}{r} = \frac{1}{r} \cdot R_{II}^2 \cdot \frac{\lambda + \mu}{\lambda + 2\mu} \cdot \varepsilon_{\theta\theta}(r = R_{II}) \qquad (18)$$

To get P<sub>OC</sub>, the pressure applied by concrete on the outside invar ring, Eqs. (11) and (12) are reused, leading to Eqs. (19) and (20).

$$\sigma_{rr}(r = R_{OO}) = 2(\lambda + \mu)C_1 - 2\mu \frac{C_2}{R_{OO}^2} = 0 \qquad (19)$$

$$\varepsilon_{\theta\theta}(r = R_{OO}) = C_1 + \frac{C_2}{R_{OO}^2} \qquad (20)$$

Adding (18) - 2(λ + μ)(19) allows to find the constants C<sub>1</sub> and C<sub>2</sub> shown in Eqs. (21) and (22).

$$C_2 = R_{II}^2 \cdot \frac{\lambda + \mu}{\lambda + 2\mu} \cdot \varepsilon_{\theta\theta}(r = R_{II}) \text{ with } \lambda = \frac{\nu E}{(1+\nu)(1-2\nu)} \text{ and } \mu = \frac{E}{2(1+\nu)} \qquad (21)$$

$$C_1 = \frac{\mu}{\lambda + 2\mu} \cdot \varepsilon_{\theta\theta}(r = R_{II}) \qquad (22)$$

Combining Eq. (11) with Eq. (21) and Eq. (22), this leads to the pression P<sub>OC</sub> in Eq. (23).

$$P_{OC} = \sigma_{rr}(r = R_{OC}) = 2(\lambda + \mu) \cdot \frac{\mu}{\lambda + 2\mu} \cdot \varepsilon_{\theta\theta}(r = R_{OO}) - 2\mu \frac{R_{OO}^2}{R_{OC}^2} \cdot \frac{\lambda + \mu}{\lambda + 2\mu} \cdot \varepsilon_{\theta\theta}(r = R_{OO}) \qquad (23)$$

Using Eq. (5), but applied in the concrete ring leads to Eqs. (24) and (25).

$$\sigma_{rr} = A - \frac{B}{r^2} \qquad (24)$$

$$\sigma_{\theta\theta} = A + \frac{B}{r^2} \qquad (25)$$

In the concrete ring, we get Eqs. (26) and (27), and subtract them to get the constants A and B values in Eqs. (28) and (29).

$$\sigma_{rr}(r = R_{IC}) = -P_{IC} \qquad (26)$$

$$\sigma_{rr}(r = R_{OC}) = -P_{OC} \qquad (27)$$

$$A = \frac{P_{IC}R_{IC}^2 - P_{OC}R_{OC}^2}{R_{OC}^2 - R_{IC}^2} \qquad (28)$$

$$B = (P_{IC} - P_{OC}) \cdot \frac{R_{IC}^2 R_{OC}^2}{R_{OC}^2 - R_{IC}^2} \qquad (29)$$

This gives Eqs. (30) and (31) respectively corresponding to the stress developed close to the inside and outside Invar rings.

$$\sigma_{\theta\theta}(r = R_{IC}) = \frac{P_{IC}(R_{IC}^2 + R_{OC}^2)}{R_{OC}^2 - R_{IC}^2} \qquad (30)$$

$$\sigma_{\theta\theta}(r = R_{OC}) = \frac{2P_{IC}R_{IC}^2 - P_{OC}(R_{IC}^2 + R_{OC}^2)}{R_{OC}^2 - R_{IC}^2} \qquad (31)$$

From this and reusing Eq. (27), it is possible to calculate the average stress (Eqs. (32) to (34)) in the whole concrete ring and not just close the invar rings, both inside and outside one, with A and B the coefficients in Eqs. (28) and (29).

$$\langle \sigma_{\theta\theta} \rangle = \frac{1}{V} \iiint_V \sigma_{\theta\theta} dV = \frac{1}{R_{OC}^2 - R_{IC}^2} \int_r \sigma_{\theta\theta} r dr \qquad (32)$$

$$\langle \sigma_{\theta\theta} \rangle = \frac{1}{R_{OC}^2 - R_{IC}^2} \int_{R_{IC}}^{R_{OC}} \left( Ar + \frac{B}{r} \right) dr \qquad (33)$$

$$\langle \sigma_{\theta\theta} \rangle = A + \frac{2B}{R_{OC}^2 - R_{IC}^2} \cdot \ln \frac{R_{OC}}{R_{IC}} \qquad (34)$$

### 3. Experimental results

The raw evolution of the strains is presented on Figure S3, with the strains measured on the inside Invar ring (gauges 1–4) plotted in red, and the strains measured on the outside invar ring (gauges 5–8) plotted in blue. With time, the strains evolve between 0 to −180µm.m<sup>−1</sup>, going more into negative values. Around t = 62 h, a jump on the strain variations appears on the inside ring gauges – bigger for gauges 1 and 2, meaning a crack is located between these gauges. The noisy strain evolution of gauge 5 might be due to a defect while positioning the gauge 5 on the outside invar ring. Even if presented on the thermally corrected strain evolution (Figure S4) with Eq. (2), gauge 5 is excluded from the average strain of the outside invar ring (Figure S5). For the inside invar ring, gauges 1–4 are considered for the average value. After correction, the main changes on the strain measurements are observed between 0 and 30 h, where the temperature variation is the more important.

At each time step, from the average measured strains, the stress developed in the concrete close to the inside (blue), the outside (red) invar rings and in the whole concrete ring (orange) are calculated respectively from Eqs. (30), (31) and (32) (Fig. 5a). The evolution of R1bis concrete tensile strength (green) is plotted as a comparison on Fig. 5a. The temperature history has been considered in the tensile strength evolution by means of the calculation of an equivalent time. The assumption there is that the temperature history affects only the kinetics. Due to the so-called cross-over effect, this is wrong, but at a very early age, the effect is limited (see for instance Maruyama and Lura [11]).

After a period with no stress measured (latent period of cement hydration), around t = 6 h, R1bis concrete starts developing compressive stress due to the temperature increase inducing restrained swelling by the outside invar ring, reaching a maximum of compressive stress developed around t = 11 h. From this point, R1bis concrete starts to cool down, with an evolution of autogenous and thermal shrinkage and Young modulus, inducing a decrease of compressive stress and leading to tensile stress measurement (restrained shrinkage by the inside invar ring).

At times t = 40 h and t = 50 h, the stress respectively close to the inside invar ring and in the concrete ring get higher than the tensile strength of R1bis concrete. However, the release of the stress corresponding to the crack of the concrete (jump on the stress evolution) (Fig. 5b) is only observed at t = 62 h. At this time, tensile stress respectively close to the outside and inside invar ring and in the concrete ring reaches a maximum value of 3.8 MPa, 6.0 MPa and 4.6 MPa. Even if the stress reached out close to the inside invar ring is 1.5 times higher than R1bis concrete tensile strength at 28 days (3.8 ± 0.5 MPa), it is only when the stress close to the outside invar ring reaches the same value that the concrete ring cracks. This probably means that the
<figure><img src="image_8.png" /><figcaption>Fig. 5. (a) Evolution of stress measured in R1bis concrete during BT-Ring test close to the inside ring (orange) and outside ring (yellow), and (b) picture of the crack appearing around time t = 62 h.</figcaption></figure>

concrete close to the inside invar ring is already cracked at t = 40 h, but not crossing the whole section of the concrete ring. This happens only later in time, when in average, the developed stress in the concrete gets higher than the tensile strength. The experimental delay observed when the cracking occurs might be due to the standard deviation of the tensile strength.

## 4. Thermo-mechanical modelling

In this section, the model developed in this study is presented. Several studies had been done using finite element, with 3D geometry [94,117–119] or 2D-axisymetry [117,120,121]. As working with a dual invar ring test, it has been decided to work with 2D-axisymetry. However, contrary to some authors, the influence of mechanic on the thermal behaviour, even though is it possible to neglect it.

### 4.1. Modelling conditions

On Fig. 6 is presented the BT-Ring finite element mesh used in the numerical model, with 1140 Quad8 elements with quadratic interpolation and corresponding to 3679 nodes. The thermal boundaries are taken equal to T<sub>ext</sub> = 20°C, T<sub>init_concrete</sub> = 20°C. The thermal exchange coefficient between inside/outside is equal to 3.6.10<sup>-3</sup>J.h<sup>-1</sup>.mm<sup>-2</sup>.°C<sup>-1</sup> and the QAC temperature is imposed at the middle of the invar rings.

<figure><img src="image_9.png" /><figcaption>Fig. 6. BT-Ring finite element mesh.</figcaption></figure>

For the mechanical boundary conditions, there are axisymmetric conditions, continuous displacements between concrete and invar rings and E<sub>invar</sub> = 145 GPa and v<sub>invar</sub> = 0.23.

### 4.2. Thermal modelling

The concrete thermal behaviour evolution is modelled with the heat Eq. (35), where the concrete calorific capacity C<sub>th</sub> = 2.4 J.cm<sup>-3</sup>.°C<sup>-1</sup>, the thermal conductivity k = 2 W.m<sup>-1</sup>.K<sup>-1</sup> and the hydration heat and kinetic are based on QAC experiment cf. Eqs. (36) to (40).

τ is the characteristic time (h), β is an exponent, Ã(ξ) is the hydration reaction chemical affinity (h<sup>-1</sup>), E<sub>a</sub> is the energy of activation (J.mol<sup>-1</sup>), R is the Perfect gas constant 8.314 J.mol<sup>-1</sup>.K<sup>-1</sup> and T is the concrete temperature (K). Infinite hydration degree ξ<sub>∞</sub> is predicted with Eq. (38) considering the water on binder ratio w/b [122].

$$C_{th} \frac{dT}{dt} + k \nabla^2 T = \dot{Q} \qquad (35)$$

$$\xi(t) = \xi_{\infty} \cdot \frac{Q(t)}{Q_{\infty}} \qquad (36)$$

$$Q = Q_{\infty} \cdot \exp\left(-\left(\frac{\tau}{t}\right)^{\beta}\right) \qquad (37)$$

$$\xi_{\infty} = 1 - e^{-3.3w/b} \qquad (38)$$

$$\dot{\xi} = \tilde{A}(\xi) \exp\left(-\frac{Ea}{RT(t)}\right) \qquad (39)$$

$$\tilde{A}(\xi) = \dot{Q} \frac{\xi_{\infty}}{Q_{\infty}} \exp\left(\frac{Ea}{RT(t)}\right) \qquad (40)$$

### 4.3. Mechanical modelling

The concrete mechanical behaviour evolution is modelled with an incremental constitutive relation as presented in Eq. (41), σ the strains tensor, ε the deformation tensor, C the rigidity tensor depending on E(ξ) – elastic modulus – and Poisson coefficient considered as constant, ε<sub>au</sub>, the autogenous shrinkage, ε<sub>th</sub> the thermal deformation and ε<sub>bc</sub> the basic creep deformation. As ε<sub>th</sub> is proportional to the derivate of thermal coefficient of expansion, which is considered constant, ε̇<sub>th</sub> is equal to zero in this equation.
$$ \dot{\sigma} = C(\xi)\dot{\epsilon}_{el} = C(\xi)(\dot{\epsilon} - \dot{\epsilon}_{au} - \dot{\epsilon}_{th} - \dot{\epsilon}_{bc}) \quad (41) $$

To model the behaviour of the R1bis at early age, several tests have been performed such as following the evolution of process of setting, autogenous shrinkage, dynamic and static Young modulus, basic creep and mechanical strengths.

Autogenous shrinkage is experimentally measured from fresh state thank to BTJADE test [123]. Figure S6 shows the autogenous shrinkage evolution in function of the hydration degree, experimental and modelled with Eq. (42) [124].

$$ \epsilon_{au} = 35 - 900 (\xi - 0.32)^2 \quad (42) $$

Figure S7 shows the Young modulus evolution measured with BT JASPE [125] and modelled according to Eq. (43); with $E(\xi_{\infty})$, the final Young modulus equal to 41.5 GPa, $\xi_0$, the mechanical percolation threshold equal to 0.14 [126,127] and a, a coefficient equal to 0.6.

$$ E(\xi) = E_{\infty} \cdot \left( \frac{\xi(t) - \xi_0}{\xi_{\infty} - \xi_0} \right)^a \quad (43) $$

Table 4 presents the characteristics used to connect Young modulus to hydration degree using Eqs. (38) to (40) and (43).

At the end of Young modulus continuous testing with BT JASPE, a basic creep test is done under 20 % of compressive strength at 7 days and modelled thank to Kelvin-Voigt rheological model [128–130]. It can be modelled Eq. (44) – where $\sigma$ is the applied stress at time $t_0$, C a $t_0$ independent constant and $\tau$ the $t_0$ dependent characteristic time – measured between 13 and 20 days, with 20 % of the compressive strength at 7 days applied. In this case, the basic creep is closed to 180 µm.m$^{-1}$ after 7 days of measurement.

$$ \epsilon_{bc} = \frac{\sigma}{C} \ln \left( 1 + \frac{t - t_0}{\tau(t_0)} \right) \quad (44) $$

However, because of the finite elemental code with the incremental formula, it is difficult to use it directly. Instead of it, a rheological model with 3 Kelvin-Voigt (KV) elements is used as presented in Eqs. (45) and (46) and Figure S8.

$$ \tau_{bc}^i \epsilon_{bc}^i + \left( \tau_{bc}^i \frac{k_{bc}^i(\xi)}{k_{bc}^i(\xi)} + 1 \right) \epsilon_{bc}^i = \frac{\dot{\sigma}}{k_{bc}^i} \quad \text{with } i = 1, 2 \text{ and } 3 \quad (45) $$

$$ k_{bc}^i(\xi) = k_{bc}^i_{\infty} \left( \frac{1-b}{1-b^{\frac{\xi}{\xi_{\infty}}}} \right) \quad \text{with } i = 1, 2 \text{ and } 3 \text{ and } b = 0.6 \quad (46) $$

The characteristic times $\tau_{bc}^i$ are fixed at 5, 50 and 500 h and the stiffness of each element $k_{bc}^i$ are fitted on experimental results (Table 5).

## 5. Experimental and model comparison

The thermal calculus is done according to Eqs. (36) to (40). The temperature map at the maximum peak (Fig. 7) presents a homogenous temperature within the rings and the concrete with less than 0.5°C variation. This modelled temperature evolution is in agreement with the experimental evolution, with less than 2°C difference.

Eq. (34) allows to calculate the mean orthoradial stress in a concrete section, considering the pressure applied on the concrete ring by the invar ring – both inside and outside one. Experimental results – mean

<table><caption>**Table 4**

Characteristics used to connect Young modulus to hydration degree using Eqs. (40) to (42) and (43).</caption><thead><tr><th>Hydration heat</th><th>ξ<sub>∞</sub></th><th>ξ<sub>0</sub></th><th colspan="2">Young modulus</th></tr><tr><th>Q<sub>∞</sub> (kJ.kg<sup>-1</sup>)</th><th>β</th><th>τ (h)</th><th>a</th><th>E<sub>∞</sub> (GPa)</th></tr></thead><tbody><tr><td>67.3</td><td>0.85</td><td>11.1</td><td>0.80</td><td>3.84</td><td>35.74</td></tr></tbody></table>

<table><caption>**Table 5**

Kelvin-Voigt rheological elements characteristics used for R1bis modelling as presented on Figure S9.</caption><tr><td>τ<sub>bc</sub><sup>i</sup> (h)</td><td>5</td><td>50</td><td>500</td></tr><tr><td>k<sub>bc</sub><sup>i</sup> (MPa)</td><td>443100</td><td>63830</td><td>73020</td></tr></table>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Temperature map in BT-Ring model at the thermal peak (t = 18 h).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Evolution of average experimental (orange) and modelized (black) orthoradial stress and experimental tensile strength (red) in concrete in function of time.</figcaption></figure>

orthoradial stress and tensile strength (same as in Fig. 5a) – are compared to the model mean orthoradial stress, cf. Fig. 8.

Good agreement between the experiment and the model mean orthoradial stress is observed. The mean experimental stress is close to R1bis tensile strength when cracking occurs, supporting the scenario of an opening crack at the interface concrete-inside ring and its propagation over the concrete ring section when tensile strength is exceeded. Experimentally, there is a 10 h delay between the experimental stress reaching the mean concrete tensile strength (around 50 h) and the observed cracking of the concrete around 60 h. This delay could be explained by the natural variability of the tensile strength (standard deviation equal to 0.5 MPa). This differs from findings on real structures: for instance, in the CEOS project, in a test of a restrained shrinkage massive concrete beam, cracking occurs for a stress below the mean tensile strength measured on samples [131]. And, in the next generation Eurocode 2, for the assessment of the risk of cracking a value corresponding to 80 % of the concrete tensile strength is considered [132]. The difference with our experiment could be explained by a size effect: the larger the sample size, the bigger the probability of having a defect and the lower the tensile strength [133] or by an effect of sustained load,
the so-called Rüsch effect [134]. Regarding the experimental set-up, with height and gap between the invar rings (75 mm), aggregates up to 21 mm i.e. three times smaller than the gap can be used which corresponds to usual aggregate size in concrete formulation. Between zero to 60°C, invar Young modulus is considered as constant [135], restraining properly the sample and so inducing residual stress required for such testing with a DR superior or egal at 72 % as in ASTM C 1581–09 [136]. Even though the gauges used for strain measurement have a CTE around three times lower than the invar ring (1.21.10⁻⁶ °C⁻¹ vs 0.5.10⁻⁶ °C⁻¹), the impact of the glue seems to be a more probable option.

Numerically, a possible way to improve the thermo-mechanical model and discuss the possible size effect would be to use a damage model applied to a 3-dimensional structure with distributed properties (or at least a localized defect). This type of model can describe the size effect [137] and the mechanical behaviour before the brutal opening of the crack, hence the stress development with time.

## 6. Conclusions

This study presents an adapted version of the Schlitter ring test to evaluate the early-age cracking risk in concrete under restrained thermal and autogenous deformations. The developed BT-Ring test successfully replicates in-situ conditions, providing a comprehensive assessment of stress evolution in a controlled environment. The key findings of this research can be summarized as follows:

* The BT-Ring test allows precise monitoring of stress development in concrete (and not only in mortar) subjected to thermal and autogenous shrinkage under partial restraint, with the same behaviour induced than in a massive structure and the difference skin/heart of the structure.

* BT-ring test has the advantages of needing only a small volume of concrete (less than 20 L for both the QAC test and BT-Ring test), with results expected within a week, compared to the usual methods to evaluate the risk of cracking in concrete (mechanical strengths over a month).

* Experimental results show that tensile stress progressively builds up after an initial compressive phase, with cracking occurring once the mean orthoradial stress exceeds the concrete tensile strength.

* The comparison between experimental and numerical results confirms the model's ability to predict stress evolution with good accuracy, reinforcing its relevance for assessing cracking risk.

* Unlike findings in large-scale structures, where cracking occurs at stress levels below the mean tensile strength, this study suggests that size effects could play a significant role in cracking prediction.

Future research should focus on refining the numerical model by incorporating a damage-based approach to better capture size effects and distributed material heterogeneities. Additionally, extending this methodology to different concrete mix designs, including low-carbon concretes, would enhance its applicability to modern construction materials. The BT-Ring test, combined with advanced modelling, represents a promising tool for optimizing concrete formulations and mitigating early-age cracking risks in large-scale structures.

## CRediT authorship contribution statement

**Bourchy Agathe:** Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Data curation.

**Torrenti Jean-Michel:** Writing – review & editing, Project administration, Methodology, Investigation, Formal analysis, Data curation.

**Nedjar Boumediene:** Writing – review & editing, Software, Methodology, Investigation.

## Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests.

Torrenti reports financial support was provided by Vicat SA. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This work was supported by the French Department of Ecological Transition and Territory Cohesion and Vicat company. The authors would like to thank Dr. Barnes-Davin and Laetitia Bessette from Vicat company, and the staff at Université Gustave Eiffel for their support during experimental development and testing, especially Jean-Claude Renaud, Joël Billo, Damien Suray-Faure, Jean-Daniel Simitambe and Dr. Martin.

## Appendix A. Supporting information

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.conbuildmat.2025.141446.

## Data availability

Data will be made available on request.

## References

[1] G. De Schutter, Fundamental study of early age concrete behaviour as a basis for durable concrete structures, Mater. Struct. 35 (1) (2002) 15-21.

[2] A. Bentur, K. Kovler, Evaluation of early age cracking characteristics in cementitious systems, Mater. Struct. /Mater. Et. Constr. 36 (257) (2003) 183-190.

[3] S.P. Shah, W.J. Weiss, W. Yang, Shrinkage cracking - can it be prevented? Concr. Int. 20 (4) (1998) 51-55.

[4] F. Barre, P. Bisch, D. Chauvel, J. Cortade, J.-F. Coste, J.-P. Dubois, S. Erlicher, E. Gallitre, J. Mazars, C. Rospar, A. Sellier, J.-M. Torrenti, F. Toutlemonde, Control of Cracking in Reinforced Concrete Structures: Research Project, Wiley, 2016.

[5] B. Klemczak, A. Žmij, Insight into thermal stress distribution and required reinforcement reducing early-age cracking in mass foundation slabs, Materials 14 (3) (2021) 477.

[6] E.M. Fairbairn, M. Azenha, TC CMS: Thermal Cracking of Massive Concrete Structures: State of the Art. Report of the RILEM Technical Committee 254-CMS, RILEM, 2018.

[7] K.L. Scrivener, D. Damidot, C. Famy, Possible mechanisms of expansion of concrete exposed to elevated temperatures during curing (also known as DEF) and implications for avoidance of field problems, 1999, pp. 93-101.

[8] I. Khan, A. Castel, R.I. Gilbert, Tensile creep and early-age concrete cracking due to restrained shrinkage, Constr. Build. Mater. 149 (2017) 705-715.

[9] R. Cortas, E. Rozière, S. Staquet, A. Hamami, A. Loukili, M.P. Delplancke-Ogletree, Effect of the water saturation of aggregates on the shrinkage induced cracking risk of concrete at early age, Cem. Concr. Compos. 50 (2014) 1-9.

[10] B. Delsaute, J.M. Torrenti, B. Nedjar, S. Staquet, A. Bourchy, M. Briffaut, Modeling compressive basic creep of concrete at early age, Mech. Time-Depend. Mater. 28 (1) (2024) 143-162.

[11] I. Maruyama, P. Lura, Properties of early-age concrete relevant to cracking in massive concrete, Cem. Concr. Res. 123 (2019) 105770.

[12] F.J. Ulm, O. Coussy, What is a "massive" concrete structure at early ages? Some dimensional arguments, J. Eng. Mech. 127 (5) (2001) 512-522.

[13] JCI, Guidel. Control Crack. Mass Concr. (2008).

[14] T.-H. Wen, T.Y.P. Yuen, V.K.S. Li, A.T. Yeung, A case study on early-age cracking of high-strength concrete construction by coupled thermal-mechanical analysis and field monitoring, Case Stud. Constr. Mater. 21 (2024) e03436.

[15] Z. Qu, Y. Zhang, Z. Liu, R. Si, J. Wu, A review on early-age cracking of concrete: Causes and control, Case Stud. Constr. Mater. 21 (2024) e03848.

[16] Q.D. Nguyen, M.S.H. Khan, T. Xu, A. Castel, Mitigating the Risk of Early Age Cracking in Fly Ash Blended Cement-Based Concrete Using Ferronickel Slag Sand, J. Adv. Concr. Technol. 17 (2019) 295-308.

[17] S. Uthaman, V. Vishwakarma, R.P. George, D. Ramachandran, K. Kumari, R. Preetha, M. Premila, R. Rajaraman, U.K. Mudali, Enhancement of strength and durability of fly ash concrete in seawater environments: Synergistic effect of nanoparticles, Constr. Build. Mater. 187 (2018) 448-459.
[18] K. Scrivener, B. Lothenbach, N. De Belie, E. Gruyaert, J. Skibsted, R. Snellings, A. Vollpracht, TC 238-SCM: hydration and microstructure of concrete with SCMs, 2015.

[19] K. Scrivener, R. Snellings, B. Lothenbach, A practical guide to microstructural analysis of cementitious materials, CRC Press, Taylor and Francis Group, 2016.

[20] K. Scrivener, W. Wieker, Adv. hydration Low., Ambient Elev. Temp. (1992) 1449-1482.

[21] K.L. Scrivener, P. Juilland, P.J.M. Monteiro, Advances in understanding hydration of Portland cement, Cem. Concr. Res. 78 (2015) 38-56.

[22] K.L. Scrivener, A. Nonat, Hydration of cementitious materials, present and future, Cem. Concr. Res. 41 (7) (2011) 651-665.

[23] J. Cheung, A. Jeknavorian, L. Roberts, D. Silva, Impact of admixtures on the hydration kinetics of Portland cement, Cem. Concr. Res. 41 (12) (2011) 1289-1309.

[24] K.A. Riding, J.L. Poole, K.J. Folliard, M.C.G. Juenger, A.K. Schindler, New model for estimating apparent activation energy of cementitious systems, Acids Mater. J. 108 (5) (2011) 550-557.

[25] I. Maruyama, G. Igarashi, Cement reaction and resultant physical properties of cement paste, J. Adv. Concr. Technol. 12 (6) (2014) 200-213.

[26] D.H. Nguyen, V.T.N. Dao, A novel approach to estimate the evolution of fracture energy and tensile softening curve of concrete from very early age, Eng. Fract. Mech. 196 (2018) 43-55.

[27] C. Liu, D. Xie, W. She, Z. Liu, G. Liu, L. Yang, Y. Zhang, Numerical modelling of elastic modulus and diffusion coefficient of concrete as a three-phase composite material, Constr. Build. Mater. 198 (2019) 1251-1263.

[28] J. Huang, K. Krabbenhoft, A.V. Lyamin, Statistical homogenization of elastic properties of cement paste based on X-ray micromotography images, Int. J. Solids Struct. 50 (5) (2013) 699-709.

[29] L. Hong, X. Gu, F. Lin, Influence of aggregate surface roughness on mechanical properties of interface and concrete, Constr. Build. Mater. 65 (2014) 338-349.

[30] J. Zhao, J.-J. Zheng, G.-F. Peng, A numerical method for predicting Young's modulus of heated cement paste, Constr. Build. Mater. 54 (2014) 197-201.

[31] Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials, J. Mech. Phys. Solids 11 (2) (1963) 127-140.

[32] G. Ramesh, E.D. Sotelino, W.F. Chen, Effect of transition zone on elastic moduli of concrete materials, Cem. Concr. Res. 26 (4) (1996) 611-622.

[33] A. Vollpracht, M. Soutsos, F. Kanavaris, Strength development of GGBS and fly ash concretes and applicability of fib model code's maturity function - a critical review, Constr. Build. Mater. 162 (2018) 830-846.

[34] S. Kucharczyk, M. Zajac, J. Deja, The influence of limestone and Al2O3 content in the slag on the performance of the composite cements, Procedia Eng. 108 (2015) 402-409.

[35] F.A. Oluokun, E.K. Burdette, J.H. Deatherage, Splitting tensile strength and compressive strength relationships at early ages, Acids Mater. J. 88 (2) (1991).

[36] P. Turcry, A. Loukili, L. Barcelo, J.M. Casabonne, Can the maturity concept be used to separate the autogenous shrinkage and thermal deformation of a cement paste at early age? Cem. Concr. Res. 32 (9) (2002) 1443-1450.

[37] O.M. Jensen, P.F. Hansen, Influence of temperature on autogenous deformation and relative humidity change in hardening cement paste, Cem. Concr. Res. 29 (4) (1999) 567-575.

[38] I. Chu, S.H. Kwon, M.N. Amin, J.-K. Kim, Estimation of temperature effects on autogenous shrinkage of concrete by a new prediction model, Constr. Build. Mater. 35 (2012) 171-182.

[39] D. Shen, J. Jiang, J. Shen, P. Yao, G. Jiang, Influence of curing temperature on autogenous shrinkage and cracking resistance of high-performance concrete at an early age, Constr. Build. Mater. 103 (2016) 67-76.

[40] P. Lura, O.M. Jensen, K. van Breugel, Autogenous shrinkage in high-performance cement paste: an evaluation of basic mechanisms, Cem. Concr. Res. 33 (2) (2003) 223-232.

[41] T. Kanda, H. Momose, K. Imamoto, Shrinkage Cracking Resistance of Blast Furnace Slag Blended Cement Concrete - Influencing Factors and Enhancing Measures, J. Adv. Concr. Technol. 13 (1) (2015) 1-14.

[42] G. Sant, The influence of temperature on autogenous volume changes in cementitious materials containing shrinkage reducing admixtures, Cem. Concr. Compos. 34 (7) (2012) 855-865.

[43] I. Maruyama, A. Teramoto, Impact of time-dependent thermal expansion coefficient on the early-age volume changes in cement pastes, Cem. Concr. Res. 41 (4) (2011) 380-391.

[44] I. Khan, T. Xu, A. Castel, R.I. Gilbert, Early-age tensile creep and shrinkage-induced cracking in internally restrained concrete members, Mag. Concr. Res. 71 (22) (2019) 1167-1179.

[45] R.D. Browne, R. Blundell, The influence of loading age and temperature on the long term creep behaviour of concrete in a sealed, moisture stable, state, Mat. ériaux Et. Constr. 2 (2) (1969) 133-143.

[46] Z.P. Bazant, Prediction of concrete creep and shrinkage: past, present and future, Nucl. Eng. Des. 203 (1) (2001) 27-38.

[47] Q. Zhang, R.Le Roy, M. Vandamme, B. Zuber, Long-term creep properties of cementitious ma-terials: Comparing microindentation testing with macroscopic uniaxial compressive testing, Cem. Concr. Res. 58 (2014) 89-98.

[48] X. Li, Z.C. Grasley, E.J. Garboци, J.W. Bullard, Modeling the apparent and intrinsic viscoelastic relaxation of hydrating cement paste, Cem. Concr. Compos. 55 (2015) 322-330.

[49] G.M. Ji, T. Kanstad, Ø. Bjontegaard, E.J. Sellevold, Tensile and compressive creep deformations of hardening concrete containing mineral additives, Mater. Struct. 46 (7) (2013) 1167-1182.

[50] B. Delsaute, New approach for monitoring and modelling of the creep and shrinkage behaviour of cement pastes, mortars and concretes since setting time, Thèse de l'Université Libre de Bruxelles & Université Paris-Est, 2017.

[51] Y. Zhang, T. Kim, A. Castel, T. Xu, Thermal cracking in high volume of fly ash and GGBFS concrete, Int. J. Concr. Struct. Mater. 17 (1) (2023) 65.

[52] L. Wang, H.Q. Yang, S.H. Zhou, E. Chen, S.W. Tang, Mechanical properties, long-term hydration heat, shrinkage behavior and crack resistance of dam concrete designed with low heat Portland (LHP) cement and fly ash, Constr. Build. Mater. 187 (2018) 1073-1091.

[53] I. Khan, T. Xu, M.S.H. Khan, A. Castel, R.I. Gilbert, Effect of various supplementary cementitious materials on early-age concrete cracking, J. Mater. Civ. Eng. 32 (4) (2020).

[54] S. Altoubat, T.M. Junaid, M. Leblouba, D. Badran, Effectiveness of fly ash on the restrained shrinkage cracking resistance of self-compacting concrete, Cem. Concr. Compos. 79 (2017) 9-20.

[55] J. Liu, Q. Tian, Y. Wang, H. Li, W. Xu, Evaluation method and mitigation strategies for shrinkage cracking of modern concrete, Engineering 7 (3) (2021) 348-357.

[56] A. Jędrzjejewska, F. Benboudjema, L. Lacarrière, M. Azenha, D. Schlicke, S. Dal Pont, A. Delaplace, J. Granja, K. Hájková, P. Joachim Heinrich, G. Sciумè, E. Strieder, E. Stierschneider, V. Smilauer, V. Troyan, COST TU1404 benchmark on macroscopic modelling of concrete and concrete structures at early age: Proof-of-concept stage, Constr. Build. Mater. 174 (2018) 173-189.

[57] J. Walraven, A. Bigaj-van Vliet, fib, Model Code for Concrete Structures 2010, Ernst de Son, 2013.

[58] J.P. Charron, Contribution à l'étude du comportement au jeune âge des matériaux cimentaires en conditions de déformations libres et genées, Thèse De. l'Univ. é De. Laval (2003).

[59] G. Toma, M. Pigeon, J. Marchan, B. Bissonnette, A. Delgrave, Early age agenous restrained shrinkage: stress build up and relaxation, Proc. Second Int. Res. Semin. self-dessiccation it'S. Import. Concr. Technol. (1999).

[60] R. Springenschmidt, R. Breitenbücher, M. Mangold, Development of the cracking frame and the temperature-stress testing machine, in: Springerschmidt (Ed.) E&FN spon, 1994.

[61] D.H. Nguyen, V.T. Nguyen, P. Lura, V.T.N. Dao, Temperature-stress testing machine - a state-of-the-art design and its unique applications in concrete research, Cem. Concr. Compos. 102 (2019) 28-38.

[62] H. Zhu, Y. Hu, Q. Li, R. Ma, Restrained cracking failure behavior of concrete due to temperature and shrinkage, Constr. Build. Mater. 244 (2020).

[63] L. Li, V. Dao, P. Lura, Autogenous deformation and coefficient of thermal expansion of early-age concrete: Initial outcomes of a study using a newly-developed temperature stress testing machine, Cem. Concr. Compos. 119 (2021) 103997.

[64] J. Xin, G. Zhang, Y. Liu, Z. Wang, N. Yang, Y. Wang, R. Mou, Y. Qiao, J. Wang, Z. Wu, Environmental impact and thermal cracking resistance of low heat cement (LHC) and moderate heat cement (MHC) concrete at early ages, J. Build. Eng. 32 (2020).

[65] D. Shen, Q. Huang, Y. Huang, C. Liu, J. Kang, H. Shao, Early-age cracking failure behavior of high strength concrete with temperature rising inhibitor under different temperature histories, Constr. Build. Mater. 450 (2024) 138581.

[66] M. Liang, Z. Chang, P. Holthuizen, Y. Chen, S. He, E. Schlangen, B. Savija, Efficiently assessing the early-age cracking risk of cementitious materials with a mini temperature stress testing machine, Cem. Concr. Compos. 153 (2024) 105710.

[67] N. Banthia, C. Yan, S. Mindess, Restrained shrinkage cracking in fiber reinforced concrete - a novel test technique, Cem. Concr. Res. 26 (1996).

[68] R.W. Carlson, T.J. Reading, Model of studying shrinkage cracking in concrete building walls, Acids Struct. J. 85 (1988) 395-404.

[69] S.P. Shah, M. Grzybowski, Model to predict cracking in fiber reinforced concrete due to restrained shrinkage, Mag. Concr. Res. 41 (1989) 125-135.

[70] J. Weiss, S.P. Shah, Restricted shrinkage cracking: the role of shrinkage reducing admixtures and specimen geometry, Mater. Struct. 85 (2002) 85-91.

[71] A.-A.Ao, S.Ha.T. Officials, AASHTO PP 334-08 - Standard Method of Test for Estimating the Cracking Tendency of Concrete Washington DC, USA, 2008.

[72] ASTM, ASTM C1581/C1581M-16 - Standard Test Method for Determining Age at Cracking and Induced Tensile Stress Characteristics of Mortar and Concrete under Restrained Shrinkage West Conshohocken, PA, USA, 2016.

[73] M. Briffaut, Etude de la fissuration au jeune âge des structures massives en béton: influence de la vitesse de refroidissement, des reprises de bétonnage et des armatures, Thèse De. l'Éc. Norm. Supérieure De. Cachan (2010).

[74] A. Haouas, Comportement au jeune âge des matériaux cimentaires - Caractérisation et modélisation chimio-hydro-mécanique du retrait, Thèse De. l'ENS Cachan (2007).

[75] A. Messan, Contribution à l'étude du comportement au très jeune âge des structures mince en mortier, Thèse De. l'Univ. é De. Montp. II (2006).

[76] Y. Zhang, S. Afroz, Q.D. Nguyen, T. Kim, J. Eisenträger, A. Castel, T. Xu, Analytical model predicting the concrete tensile stress development in the restrained shrinkage ring test, Constr. Build. Mater. 307 (2021) 124930.

[77] D.-Y. Yoo, J.-J. Park, S.-W. Kim, Y.-S. Yoon, Influence of ring size on the restrained shrinkage behavior of ultra high performance fiber reinforced concrete, Mater. Struct. 47 (7) (2014) 1161-1174.

[78] Y. Liu, Y. Wei, L. Ma, L. Wang, Restrained shrinkage behavior of internally-cured UHPC using calcined bauxite aggregate in the ring test and UHPC-concrete composite slab, Cem. Concr. Compos. 134 (2022) 104805.
[79] F. Kanavaris, C. Ferreira, C. Sousa, M. Azenha, Thermo-chemo-hygro-mechanical simulation of the restrained shrinkage ring test for cement-based materials under distinct drying conditions, Constr. Build. Mater. 294 (2021) 123600.

[80] R. Swamy, A. Bandyopadhyay, H. Stavrides, The Ring Method of Measuring Restricted Shrinkage in Mortar and Concrete, Cem., Concr. Aggreg. 1 (1) (1979) 13-20.

[81] J. Zhang, Y. Gao, Y. Han, J. Wang, Evaluation of shrinkage induced cracking in early age concrete: From ring test to circular column, Int. J. Damage Mech. 26 (5) (2017) 771-797.

[82] O. Oladiran, Brunel University London, Assess. Restrain Shrinkage Crack. Concr. Elliptical Rings (2014).

[83] W. Dong, X. Zhou, Z. Wu, G. Kastiukas, Effects of specimen size on assessment of shrinkage cracking of concrete via elliptical rings: Thin vs. thick, Comput. Struct. 174 (2016) 66-78.

[84] H. Zhu, H.R. Li, X.C. Zhu, On concrete restrained eccentric ring and squared eccentric ring shrinkage test methods, Constr. Build. Mater. 84 (2015) 239-244.

[85] A.B. Hossain, J. Weiss, Assessing residual stress development and stress relaxation in restrained concrete ring specimens, Cem. Concr. Compos. 26 (5) (2004) 531-540.

[86] H. Choi, M. Lim, R. Kitagaki, T. Noguchi, G. Kim, Restrained shrinkage behavior of expansive mortar at early ages, Constr. Build. Mater. 84 (2015) 468-476.

[87] K. Raoufi, J.L. Schlitter, D.P. Bentz, W.J. Weiss, Parametric assessment of stress development and cracking in internally cured restrained mortars experiencing autogenous deformations and thermal loading, Adv. Civ. Eng. (2011) 16.

[88] K. Kovler, J. Sikuler, A. Bentur, Restrained shrinkage tests of fibre-reinforced concrete ring specimens: effect of core thermal expansion, Mater. Struct. 26 (4) (1993) 231-237.

[89] V. Lamour, A. Haouas, M. Moranville, R. Schell, A new technique for characterisation of early-age cracking of mortars, in: K. Kovler, J. Marchand, S. Mindness, J. Weiss (Eds.), Proceedings of the International Symposium on Concrete Science and Engineering: a Tribute to Arnon Bentur, RILEM Publications SARL, Evanston, IL, USA, 2004.

[90] A. Messan, P. Ienny, D. Nectoux, Free and restrained early-age shrinkage of mortar: Influence of glass fiber, cellulose ether and EVA (ethylene-vinyl acetate), Cem. Concr. Compos. 33 (3) (2011) 402-410.

[91] A. Pertué, Etude expérimentale du retrait endogène, du fluauge propre et de la fissuration des matières cimentaires aux jeune et très jeune âges, Thèse De. l'Univ. é De. Nantes (2008).

[92] J.L. Schlitter, A.H. Senter, D.P. Bentz, T. Nantung, W.J. Weiss, A dual concentric ring test for evaluation residual stress development due to restrained volume change, J. ASTM Int. 7 (9) (2010).

[93] M. Briffaut, F. Benboudjema, J.M. Torrenti, G. Nahas, A thermal active restrained shrinkage ring test to study the early age concrete behaviour of massive structures, Cem. Concr. Res. 41 (1) (2011) 56-63.

[94] M. Briffaut, F. Benboudjema, J.M. Torrenti, G. Nahas, Numerical analysis of the thermal active restrained shrinkage ring test to study the early age behavior of massive concrete structures, Eng. Struct. 33 (4) (2011) 1390-1401.

[95] H.T. See, E.K. Attiogbe, M.A. Miltenberger, Shrinkage cracking characteristics of concrete using ring specimens, Acids Mater. J. 100 (3) (2003) 239-245.

[96] Z. Cheng, R. Zhao, Y. Yuan, F. Li, A. Castel, T. Xu, Ageing coefficient for early age tensile creep of blended slag and low calcium fly ash geopolymer concrete, Constr. Build. Mater. 262 (2020).

[97] D.W. Mokarem, R.E. Weyers, D.S. Lane, Development of a shrinkage performance specifications and prediction model analysis for supplemental cementitious material concrete mixtures, Cem. Concr. Res. 35 (5) (2005) 918-925.

[98] A.B. Hossain, B. Pease, J. Weiss, Quantifying early-age stress development and cracking in low water-to-cement concrete: restrained-ring test with acoustic emission, Transp. Res. Rec. 1834 (1) (2003) 24-32.

[99] R. Bloom, A. Bentur, Free and restrained shrinkage of normal and high-strength concretes, Acids Mater. J. 92 (2) (1995) 211-217.

[100] H.R. Shah, J. Weiss, Quantifying shrinkage cracking in fiber reinforced concrete using the ring test, Mater. Struct. 39 (9) (2006) 887-899.

[101] M. Briffaut, F. Benboudjema, L. D'Aloia, Effect of fibres on early age cracking of concrete tunnel lining. Part I: laboratory ring test, Tunn. Undergr. Space Technol. 59 (2016) 215-220.

[102] N. Banthia, C. Yan, S. Mindess, Restrained shrinkage cracking in fiber reinforced concrete: A novel test technique, Cem. Concr. Res. 26 (1) (1996) 9-14.

[103] S. Altoubat, D. Lange, A new look at tensile creep of fiber reinforced concrete, Acids Spec. Publ. Fiber Reinf. Concr. (2003).

[104] V. Corinaldesi, G. Moriconi, Evaluation of recycled aggregate concrete cracking through ring test, Appl. Mech. Mater. 174-177 (2012) 1475-1480.

[105] S. Sadati, K.H. Khayat, Restrained shrinkage cracking of recycled aggregate concrete, Mater. Struct. /Mater. Et. Constr. 50 (4) (2017).

[106] K. Aghae, A. Roshan, Predicting time to cracking of concrete composites under restrained shrinkage: A review with insights from statistical analysis and ensemble machine learning approaches, J. Build. Eng. 97 (2024) 110856.

[107] T.A. Do, B.-A. Le, Machine learning approach for predicting early-age thermal cracking potential in concrete bridge piers, Forces Mech. 17 (2024) 100297.

[108] R.P. Martin, F. Toutlemonde, Theoretical and experimental validation of a simple method to reproduce representative DEF-prone conditions in laboratory, Mater. Struct. 46 (8) (2013) 1245-1255.

[109] AFNOR, NF EN 12390-14 Essai pour béton durci - Partie 14: M. éthode semi-adiabatique De. D. éTermin. De. la Chal. D. égagée Par. Le. béton au cours De. Son. Process. De. durcissement (2018).

[110] AFNOR, NF EN 12350-2 Essais pour béton frais Partie 2: Essai D. 'affaissement' (2019).

[111] AFNOR, NF EN 12350-6 Essais pour béton frais Partie 6, Masse volumique, 2019.

[112] AFNOR, NF EN 12390-3 Essais pour béton durci Partie 3: R. éistance à la Compress. Des. éprouvettes (2019).

[113] AFNOR, NF EN 12390-6 Essais pour béton durci Partie 6: D. éTermin. De. la R. éistance En. Tract. Par. fendage D. 'éprouvettes' (2024).

[114] AFNOR, En. 12390-13. Essai pour béton durci - Partie 13: D. éTermin. du Modul. S. éCant. D. 'élasticité En. Compress. (2021).

[115] D.S. Atrushi, Tensile and compressive creep of early age concrete: Testing and modelling (PhD Thesis), Norwegian University of Sciences and Technology, 2003.

[116] A. Bourchy, Relation chaleur d'hydratation du ciment: montée en température et contraintes générées au jeune age du béton, These de l'Université Paris-Est, 2018.

[117] M. Azenha, Numerical Simulation of the Behaviour of Concrete at since its Early Ages, Universidade to Porto, Portugal, 2009.

[118] W. Dong, W. Yuan, X. Zhou, F. Wang, The fracture mechanism of circular/ elliptical concrete rings under restrained shrinkage and drying from top and bottom surfaces, Eng. Fract. Mech. 189 (2018) 148-163.

[119] M. Briffaut, F. Benboudjema, L. D'Aloia, Effect of fibres on early age cracking of concrete tunnel lining. Part II: numerical simulations, Tunn. Undergr. Space Technol. 59 (2016) 221-229.

[120] J.H. Moon, Shrinkage, Residual Stress and Cracking in Heterogeneous Materials, Purdue University, West Lafayette, USA, 2006.

[121] J. Zreiki, F. Bouchelaghem, M. Chaouche, Early-age behaviour of concrete in massive structures, experimentation and modelling, Nucl. Eng. Des. 240 (10) (2010) 2643-2654.

[122] V. Waller, Relation entre composition des bétons, exothermie en cours de prise et résistance en compression, Thèse De. l'Éc. Natl. Des. Ponts Et. Chauss.-. ées (1999).

[123] C. Boulay, Test ring for early age measurements of the autogenous shrinkage of a concrete, RILEM-JCI, Int. Workshop ConCrack 3 (2012) 111-122.

[124] B. Delsaute, S. Staquet, Monitoring the Viscoelastic Behaviour of Cement Based Materials by Means of Repeated Minute-Scale-Duration Loadings, in: M. Serdar, I. Gabrijel, D. Schlicke, S. Staquet, M. Azenha (Eds.), Advanced Techniques for Testing of Cement-Based Materials, Springer International Publishing, Cham, 2020, pp. 99-134.

[125] C. Boulay, S. Staquet, B. Delsaute, J. Carette, M. Crespini, O. Yazoghi-Marzouk, E. Merliot, S. Ramanich, How to monitor the modulus of elasticity of concrete, automatically since the early age? Mater. Struct. (2013).

[126] L. Stefan, F. Benboudjema, J.M. Torrenti, B. Bissonnette, Prediction of elastic properties of cement pastes at early ages, Comput. Mater. Sci. 47 (3) (2010) 775-784.

[127] J.M. Torrenti, F. Benboudjema, Mechanical threshold of cimentitious materials at early age, Mater. Struct. 38 (277) (2005) 299-304.

[128] B. Delsaute, C. Boulay, S. Staquet, Creep testing of concrete since setting time by means of permanent and repeated minute-long loadings, Cem. Concr. Compos. 73 (2016) 75-88.

[129] G.De Schutter, L. Taerwe, Degree of hydration-based description of mechanical properties of early age concrete, Mater. Struct. 29 (190) (1996) 335-344.

[130] F. Benboudjema, J.M. Torrenti, Early age behaviour of concrete nuclear containments, Nucl. Eng. Des. 238 (2008) 2495-2506.

[131] L. Buffo-Lacrière, S. Baron, F. Barré, D. Chauvel, A. Darquennes, J.P. Dubois, J. Gayete, F. Grondin, B. Kolani, H. Lançon, A. Loukili, G. Moreau, C. Rospars, A. Sellier, J.M. Torrenti, Restrained shrinkage of massive reinforced concrete structures: results of the project CEOS.fr, Eur. J. Environ. Civ. Eng. 20 (7) (2016) 785-808.

[132] A. Menga, J.-M. Torrenti, T. Kanstad, A.B.E. Klausen, Early-age cracking due to restraint: Laboratory and field investigations on the predictive capacity of the simplified method in Annex D of the future EC2, Struct. Concr. 25 (6) (2024) 4300-4323.

[133] M.R. Van Vliet, J.G. Van Mier, Experimental investigation of size effect in concrete and sandstone under uniaxial tension, Eng. Fract. Mech. 65 (2-3) (2000) 165-188.

[134] H. Rüsch, Researches toward a general flexural theory for structural concrete, Acids J. 32 (1960) 1-28.

[135] C.E. Guillaume, Invar and its applications, Nature (1904).

[136] A. International, ASTM, Standard C 1581-09. Standard Test Method for Determining Age at Cracking and Induced Tensile Stress Characteristics of Mortar and Concrete under Restrained Shrinkage, West Conshohocken, PA, USA, 2009.

[137] M. Vorechovský, V. Sadílek, Computational modeling of size effects in concrete specimens under uniaxial tension, Int. J. Fract. 154 (2008) 27-49.