Chi-Cong Vu

Jérôme Weiss

Olivier Plé

David Amitrano

The potential impact of size effects on compressive strength for the estimation of the Young's modulus of concrete

Keywords: size effect, static elastic modulus, dynamic elastic modulus, compressive strength, concrete

 for estimating the elastic modulus from the compressive strength can lead to incorrect designs.

Introduction

Compressive strength and modulus of elasticity are used as the fundamental material parameters in the design and assessment of concrete buildings [START_REF] Zhou | Effect of coarse aggregate on elastic modulus and compressive strength of high performance concrete[END_REF]. While the compressive strength is used for calculating structures following the requirements of ultimate limit states, the elastic modulus is used for estimating the deformation as well as for the design of sections of structural concrete elements according to the serviceability limit state [START_REF] En | Eurocode 2: Design of concrete structures -Part 1-1 : General rules and rules for buildings[END_REF]3]. Similarly to other mechanical properties of concrete (e.g. tensile and flexural strengths), the elastic modulus is usually supposed to be tightly related to the compressive strength [START_REF] Parra | Splitting tensile strength and modulus of elasticity of self-compacting concrete[END_REF][START_REF] Neville | Properties of concrete[END_REF]. Many empirical predictions for estimating the elastic modulus from the compressive strength have been proposed for different types of concrete by several researchers (see reviews in [START_REF] Neville | Properties of concrete[END_REF][START_REF] Noguchi | A practical equation for elastic modulus of concrete[END_REF][START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF][START_REF] Nematzadeh | Compressive strength and modulus of elasticity of freshly compressed concrete[END_REF]). In most concrete design codes, the modulus of elasticity is also predicted by using empirical expressions that assume a direct dependence of this modulus on compressive strength [START_REF] En | Eurocode 2: Design of concrete structures -Part 1-1 : General rules and rules for buildings[END_REF]3,[START_REF]Béton CE-I du[END_REF]10]. Following these expressions, an increase of the compressive strength would necessarily imply an increase of the modulus of elasticity. However, as a heterogeneous material, concrete exhibits a size-dependent behavior on the compressive strength [START_REF] Bazant | Fracture and size effect in concrete and other quasibrittle materials[END_REF][START_REF] Burtscher | Size-effect experiments on concrete in compression[END_REF][START_REF] Ozbolt | Size effect in concrete and reinforced concrete structures[END_REF][START_REF] Bazant | Size Effect on Structural Strength: A Review[END_REF][START_REF] Bazant | Size effect in tensile and compression fracture of concrete structures: Computational modeling and design[END_REF][START_REF] Muciaccia | Compressive failure and size effect in plain concrete cylindrical specimens[END_REF][START_REF] Del Viso | Shape and size effects on the compressive strength of high-strength concrete[END_REF][START_REF] Weiss | Finite) size effects on compressive strength[END_REF][START_REF] Vu | Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete[END_REF]. Therefore, the sample size dependence of compressive strength would imply, from these empirical expressions, that the elastic modulus will depend as well on the size of the concrete sample. Nevertheless, in a composite material like concrete, the elastic properties are mainly affected by the volume fraction, the density and the elastic modulus of its two components which are the hydrated cement paste and the aggregates [START_REF] Neville | Properties of concrete[END_REF][START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF].

Hence, the elastic modulus of concrete is a material-dependent parameter and should not be affected by the sample size, unlike the compressive strength, as long as the sample volume is larger than the representative volume element (RVE) of the microstructure [START_REF] Kanit | Determination of the size of the representative volume element for random composites : statistical and numerical approach[END_REF]. However, this assumption has been rarely checked, and there are inconsistencies between published datasets on this matter [START_REF] Darlington | The effect of specimen size on strength and other properties in laboratory testing of rock and rock-like cementitious brittle materials[END_REF][START_REF] Lee | Effect of Cylinder Size on the Modulus of Elasticity and Compressive Strength of Concrete from Static and Dynamic Tests[END_REF]. Furthermore, to obtain a statistically significant result, a large number of mechanical tests performed on concrete specimens with different sizes and different mixtures is required. Nevertheless, such an exhaustive analysis has still not been performed until now, due to the fact that the measure of the elastic modulus and the compressive strength are generally time-and cost-consuming. As a result, it is difficult to make a strong and clear conclusion regarding the size effects on the Young's modulus of concrete.

On the other hand, concrete, a typical quasi-brittle material, does not display a linear stress-strain relation over the entire loading until failure, as strain softening is observed as the result of progressive damaging/microcracking [START_REF] Neville | Properties of concrete[END_REF][START_REF] Vu | Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete[END_REF][START_REF] Vu | Compressive Failure as a Critical Transition : Experimental Evidence and Mapping onto the Universality Class of Depinning[END_REF]. This means that the elastic modulus, a linear property, is likely not proportional to the compressive strength (i.e. the peak stress), which is related to the non-linear behavior of concrete. Although several empirical expressions linking the compressive strength and the elastic modulus have shown a reasonable agreement with some datasets for different normal-weight concrete mixtures in different studies, these formulas failed to explain results obtained for other types of concrete and/or different strength grade [START_REF] Noguchi | A practical equation for elastic modulus of concrete[END_REF][START_REF] Jurowski | Influence of selected factors on the relationship between the dynamic elastic modulus and compressive strength of concrete[END_REF][START_REF] Zhou | A fundamental study on compressive strength, static and dynamic elastic moduli of young concrete[END_REF][START_REF] Park | Modulus of elasticity of recycled aggregate concrete[END_REF]. In particular, in all the previous studies of the relationship between the compressive strength and the modulus of elasticity, the influence of sample size on the compressive strength was not taken into account.

This might cause severe shortcomings when estimating the Young's modulus from the compressive strength determined on small concrete samples in the laboratory.

All the above observations call for a re-examination of (i) potential size effects on the elastic modulus; and (ii) the expressions of elastic modulus from the compressive strength. These two points represent the main objectives of the present work.

Following this introductory section, the rest of this paper is organized as follows. The next section presents a survey of the compressive strength-elastic modulus relationships proposed in the literature, and of possible size effects on the elastic modulus. Section 3 describes the details of experimental work performed in this study. Then, the size effects on elastic properties including static and dynamic moduli as well as the correlation between them are discussed. Section 5 deals with the relationship between compressive strength and Young's elastic modulus.

Finally, some main conclusions of this study are presented in section 6.

Background

To express the modulus of elasticity in terms of the compressive strength of concrete, numerous empirical formulas have been proposed by many researchers and/or are provided in the current concrete building design codes. Most of them focus on a possible correlation between the compressive strength with the static elastic Youngs's modulus (𝑌 𝑠 ), which are measured from uniaxial compression tests. For instance, the ACI committee recommends different empirical formulas relating the elastic modulus to the compressive strength and the density of concrete.

The standard ACI 318-05 [3] gives the following expression for the Young's modulus of normal-weight concrete:

𝑌 𝑠 = 0.043. 𝜌 1.5

𝑌 𝑠 = ( 𝜌 2300 ) 1.5 . (3320√𝑓 𝑐 ′ + 6900) (2)
According to EN 1992 [START_REF] En | Eurocode 2: Design of concrete structures -Part 1-1 : General rules and rules for buildings[END_REF], the relationship between the static modulus (𝑌 𝑠 ) and the compressive strength (𝑓 𝑐𝑚 ) of normal-weight concrete is described by the following equation:

𝑌 𝑠 = 22. ( 𝑓 𝑐𝑚 10 ) 0.3 (3)
where 𝑌 𝑠 is measured in GPa and 𝑓 𝑐𝑚 is in MPa. Equation (3) is suggested to be applicable up to concrete characteristic strengths of 90 MPa [START_REF] En | Eurocode 2: Design of concrete structures -Part 1-1 : General rules and rules for buildings[END_REF][START_REF] Neville | Properties of concrete[END_REF]. Both the ACI standards and EN 1992 express the static modulus of elasticity in terms of the secant modulus (𝑌 𝑠 ), but they differ in their definitions. The expression of the ACI standards (e.g. ACI 318-05 (Eq. ( 1)) and ACI 363-11 (Eq. ( 2))) are based on the specified compressive strength (𝑓 𝑐 ′ ), while EN 1992 (Eq. ( 3)) is based on the mean compressive strength (𝑓 𝑐𝑚 ) which is generally higher than the specified strength [START_REF] Neville | Properties of concrete[END_REF][START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF][START_REF] Vu | Revisiting the concept of characteristic compressive strength of concrete[END_REF]. Moreover, the secant modulus (𝑌 𝑠 ) is defined as the slope of a line drawn from a zero stress to a compressive stress equal to 0.45𝑓 𝑐 ′ in ACI 318-05 [3], and 0.4𝑓 𝑐𝑚 in EN 1992 [START_REF] En | Eurocode 2: Design of concrete structures -Part 1-1 : General rules and rules for buildings[END_REF]. As for the ACI building codes, the Architectural Institute of Japan (AIJ) propose the following equation to estimate the static elastic modulus (𝑌 𝑠 ) for normal strength concrete with the specified design strength (𝑓 𝑐 ′ ) up to 36

MPa [START_REF] Noguchi | A practical equation for elastic modulus of concrete[END_REF][START_REF]AIJ Standard for Structural Calculation of Reinforced Concrete Structures[END_REF]:

𝑌 𝑠 = 21000. ( 𝜌 2300 ) 1.5 ( 𝑓 𝑐 ′ 200 ) 0.5 (4)
In Eq. ( 4), 𝑌 𝑠 and 𝑓 𝑐 ′ are expressed in MPa and 𝜌 is in kg/m 3 . Unlike the ACI codes, Eurocode 2 and AIJ codes where the compressive strength is obtained from concrete cylinders with a height-to-diameter ratio (ℎ/𝜙 = 2), the British standard BS 8110-2 [10] uses the characteristic strength of concrete cubes at 28 days (𝑓 𝑐𝑢,28 )

to estimate the static modulus of elasticity at 28 days as follows:

𝑌 𝑠,28 = 𝐾 0 + 0.2𝑓 𝑐𝑢,28 (5)
Here 𝑌 𝑠,28 is in GPa and 𝑓 𝑐𝑢,28 is in MPa, 𝐾 0 is an empirical constant related to the elastic modulus of the aggregate and to its volume fraction [10], usually taken as 20 GPa for normal-weight concrete. In fact, the experimental value of 𝐾 0 varying from 16 to 26 GPa was observed for normal-weight concretes fabricated from different aggregate types (see reviews in [30]). This wide variation of 𝐾 0 value implies that (i) different aggregates will cause different characteristic stiffness of concrete (i.e. the elastic modulus 𝑌 𝑠,28 in Eq. ( 5)); and (ii) concretes with the same target compressive strength (i.e. strength grade) may have different elastic moduli.

Other empirical formulas relating the compressive strength to the elastic modulus can be found in other current building codes (see further details in [START_REF] Neville | Properties of concrete[END_REF][START_REF] Noguchi | A practical equation for elastic modulus of concrete[END_REF][START_REF] Nematzadeh | Compressive strength and modulus of elasticity of freshly compressed concrete[END_REF][START_REF] Park | Modulus of elasticity of recycled aggregate concrete[END_REF][START_REF] Rashid | Correlations between mechanical properties of high-strength concrete[END_REF])

and from other studies, such as those presented in Table 1. Even though these empirical formulas could provide good agreements for specific kinds of concrete mixes, they do not always give the same results and cannot cover the whole data [START_REF] Noguchi | A practical equation for elastic modulus of concrete[END_REF][START_REF] Park | Modulus of elasticity of recycled aggregate concrete[END_REF]. The major reason for this is that the mechanical properties (i.e. compressive strength and elastic modulus) of concrete are highly dependent on the qualities, specified properties and proportions of all concrete components [START_REF] Neville | Properties of concrete[END_REF][START_REF] Noguchi | A practical equation for elastic modulus of concrete[END_REF][START_REF] Nematzadeh | Compressive strength and modulus of elasticity of freshly compressed concrete[END_REF][START_REF] Park | Modulus of elasticity of recycled aggregate concrete[END_REF].

Although there is no agreement on the precise form of the relationship between the compressive strength and the static modulus of elasticity of concrete, all these previous studies agree that the static Young's modulus and the compressive strength are always positively correlated [START_REF] Neville | Properties of concrete[END_REF][START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF][START_REF] Oluokun | Elastic Modulus, Poisson's Ratio, and Compressive Strength Relationships at Early Ages[END_REF]. A similar trend is also reported in [START_REF] Lee | Effect of Cylinder Size on the Modulus of Elasticity and Compressive Strength of Concrete from Static and Dynamic Tests[END_REF][START_REF] Zhou | A fundamental study on compressive strength, static and dynamic elastic moduli of young concrete[END_REF][START_REF] Tsioulou | Combined Non-Destructive Testing (NDT) method for the evaluation of the mechanical characteristics of Ultra High Performance Fibre Reinforced Concrete (UHPFRC)[END_REF][START_REF] Bassim | Dynamic-and Static-Elastic Moduli and Strength Properties of Early-Age Portland Cement Concrete Pavement Mixtures[END_REF][START_REF] Han | Effect of temperature and age on the relationship between dynamic and static elastic modulus of concrete[END_REF][START_REF] Lee | Determination of dynamic Young's modulus of concrete at early ages by impact resonance test[END_REF] for the dynamic Young's modulus (𝑌 𝑑 ) which is determined by means of Non-Destructive Testing (NDT) methods (e.g. Ultrasonic pulse velocity (UPV) test [START_REF]Standard Test Method for Pulse Velocity Through Concrete[END_REF], Resonant frequency test [38], and other wave propagation tests [START_REF] Neville | Properties of concrete[END_REF][39][40][START_REF]Report on Nondestructive Test Methods for Evaluation of Concrete in Structures[END_REF]). A few formulas have also been proposed in the literature to describe the relationship between 𝑌 𝑑 and the compressive strength of concrete, as listed in Table 2. These empirical equations were found by converting the dependency of the static modulus on the dynamic modulus into the relationship between the static modulus and the compressive strength [START_REF] Jurowski | Influence of selected factors on the relationship between the dynamic elastic modulus and compressive strength of concrete[END_REF][START_REF] Zhou | A fundamental study on compressive strength, static and dynamic elastic moduli of young concrete[END_REF][START_REF] Lee | Evaluating the Dynamic Elastic Modulus of Concrete Using Shear-Wave Velocity Measurements[END_REF]. However, both the value of the dynamic elastic modulus and its relation with the compressive strength vary with the NDT methods [START_REF] Neville | Properties of concrete[END_REF][START_REF] Kee | Evaluation of static and dynamic residual mechanical properties of heat-damaged concrete for nuclear reactor auxiliary buildings in Korea using elasticwave velocity measurements[END_REF], the mix proportion [39], the volume content and the size of aggregates [START_REF] Jurowski | Influence of selected factors on the relationship between the dynamic elastic modulus and compressive strength of concrete[END_REF][START_REF] Zhou | A fundamental study on compressive strength, static and dynamic elastic moduli of young concrete[END_REF], the type of cement and aggregate [START_REF] Jurowski | Influence of selected factors on the relationship between the dynamic elastic modulus and compressive strength of concrete[END_REF], the water-to-cement ratio [START_REF] Zhou | A fundamental study on compressive strength, static and dynamic elastic moduli of young concrete[END_REF][START_REF] Han | Effect of temperature and age on the relationship between dynamic and static elastic modulus of concrete[END_REF][START_REF] Kumar | Pulse velocity-strength and elasticity relationship of high volume fly ash induced self-compacting concrete[END_REF], the age of concrete and curing condition [START_REF] Zhou | A fundamental study on compressive strength, static and dynamic elastic moduli of young concrete[END_REF][START_REF] Han | Effect of temperature and age on the relationship between dynamic and static elastic modulus of concrete[END_REF][START_REF] Lee | Determination of dynamic Young's modulus of concrete at early ages by impact resonance test[END_REF]39].

Consequently, the empirical expressions of the correlation between the compressive strength and the dynamic elastic modulus are generally specific to concrete mixes and/or distinct NDT methods used for investigating the 𝑌 𝑑 -value [START_REF] Neville | Properties of concrete[END_REF][START_REF] Lee | Effect of Cylinder Size on the Modulus of Elasticity and Compressive Strength of Concrete from Static and Dynamic Tests[END_REF]39].

As mentioned above, to evaluate the correlation between the compressive strength of concrete and the elastic properties determined from the NDT methods (i.e. dynamic modulus of elasticity), the relationship between the compressive strength and the static elastic modulus as well as the relationship between the static and dynamic elastic moduli are required [START_REF] Jurowski | Influence of selected factors on the relationship between the dynamic elastic modulus and compressive strength of concrete[END_REF]. Consequently, a considerable number of studies has been performed to analyze the correlation between 𝑌 𝑠 and 𝑌 𝑑 [START_REF] Neville | Properties of concrete[END_REF]39].

All these investigations highlighted that the static modulus of elasticity (𝑌 𝑠 ) is significantly smaller than the dynamic elastic modulus (𝑌 𝑑 ) [START_REF] Neville | Properties of concrete[END_REF][START_REF] Lee | Effect of Cylinder Size on the Modulus of Elasticity and Compressive Strength of Concrete from Static and Dynamic Tests[END_REF][START_REF] Bassim | Dynamic-and Static-Elastic Moduli and Strength Properties of Early-Age Portland Cement Concrete Pavement Mixtures[END_REF]. To account for the relationship between 𝑌 𝑠 and 𝑌 𝑑 , several empirical formulas have been proposed in building codes. A linear correlation is generally proposed, while few studies suggest a non-linear expression (e.g. Lambert function [START_REF] Han | Effect of temperature and age on the relationship between dynamic and static elastic modulus of concrete[END_REF]) (see Table 3).

Several influence factors on the 𝑌 𝑠 -𝑌 𝑑 relation have been observed such as the loading rate in the static test [START_REF] Shkolnik | Effect of nonlinear response of concrete on its elastic modulus and strength[END_REF], the NDT methodology [START_REF] Lee | Effect of Cylinder Size on the Modulus of Elasticity and Compressive Strength of Concrete from Static and Dynamic Tests[END_REF], the aggregate volume content and maximum size of coarse aggregate [START_REF] Zhou | A fundamental study on compressive strength, static and dynamic elastic moduli of young concrete[END_REF]. On the other hand, the cement type and concrete age do not seem to have a significant influence [START_REF] Han | Effect of temperature and age on the relationship between dynamic and static elastic modulus of concrete[END_REF].

Expressions (1) to (5) above represent various versions of a direct dependence of the Young's modulus on the compressive strength of concrete. Following these expressions, size effects on the compressive strength would necessarily imply a size-dependence of the elastic modulus of concrete. This is in contradiction with the general assumption that the modulus of elasticity, a linear property of a material, is actually sample size-independent. However, this assumption has not yet been clearly examined and/or confirmed, and there are contradictions between reported data in the literature. For instance, by analyzing 214 pairs of static Young's modulus values for two sizes of normal-weight concrete specimens (100x200 and 150x300 mm cylinders) collected from the existing literature, Rashid et al. [START_REF] Rashid | Correlations between mechanical properties of high-strength concrete[END_REF] observed that the 𝑌 𝑠 values obtained from the smaller-size samples are in general larger than those obtained from larger-size samples. An opposite result was however observed for high-performance concrete in [START_REF] Baalbaki | Influence of specimen size on compressive strength and elastic modulus of high-performance concrete[END_REF]. In [START_REF] Lee | Effect of Cylinder Size on the Modulus of Elasticity and Compressive Strength of Concrete from Static and Dynamic Tests[END_REF][START_REF] Lee | Evaluating the Dynamic Elastic Modulus of Concrete Using Shear-Wave Velocity Measurements[END_REF], also comparing the 𝑌 𝑠 values of these two sizes of cylinders, Lee and co-workers did not find any clear sample size effect on both the static and dynamic elastic moduli of normal weight concrete. It is worth noting, however, that a precise assessment of size effects from only two different sample sizes is difficult. Performing uniaxial compression tests on concrete cylindrical specimens of five different diameters ranging from 63.5 to 300 mm but with a fixed heigh-to-diameter ratio (ℎ/𝜙 = 2), Darlington et al. [START_REF] Darlington | The effect of specimen size on strength and other properties in laboratory testing of rock and rock-like cementitious brittle materials[END_REF] observed that there is no evidence for a strong size effect on the static Young's modulus. Despite

Experimental work

To address the main questions of the present research, a testing plan was designed to explore the size effects on the compressive strength and on the elastic modulus of concrete, as shown in the flowchart of Fig. 1. According to the recommendations in [START_REF] Bazant | Fracture and size effect in concrete and other quasibrittle materials[END_REF][START_REF] Bazant | Size Effect on Structural Strength: A Review[END_REF][START_REF] Shah | Size-effect method for determining fracture energy and process zone size of concrete[END_REF][START_REF] Carpinteri | Scale effects in uniaxially compressed concrete specimens[END_REF], concrete specimens of the same composition to determine the dynamic modulus of elasticity. For this purpose, 36 P-wave velocity measurements (section 3.3) were conducted on intact concrete specimens.

In addition, a series of 527 uniaxial compression tests (section 3.4), including the 36 samples previously used for P-wave velocity tests, were performed on all these concrete specimens. These destructive tests allowed to collect the values of both compressive failure strength and static modulus of elasticity for the different concrete samples. While the static modulus of elasticity is shown here for the first time, the compressive strength data have been used in [START_REF] Vu | Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete[END_REF] to analyze the size effects on compressive strength. In the present paper, all these data are used to examine: (i) the size effects on both static and dynamic elastic moduli of concrete;

(ii) the correlation between these two estimations of the Young's modulus; and (iii) their relationships with the compressive strength.

Materials and concrete mixtures

In Coarse (C)). The maximum aggregate size, 𝑑 𝑔 , for the F-, M-, and C-concrete mixtures were correspondingly 3.15 mm, 16 mm and 25 mm (see Fig. 2b). The water-to-cement (𝑊/𝐶) ratio was kept unchanged for all specimens in each individual concrete mixture. The details of the mix proportions for the three concrete mixtures are given in Table 4.

Concrete samples

In this study, four different sizes of concrete cylinders (𝜙 × ℎ = 40 × 80 mm; 70 × 140 mm; 110 × 220 mm and 160 × 320 mm) were considered.

For a given concrete mixture, about 44 samples of each sample diameter (𝜙), were produced and used for uniaxial compression tests (section 3.4). The concrete mixing, casting and curing conformed to the procedure described in the standard

NF EN 206-1 [53].
After weighting all raw composition materials, cement and aggregates are firstly blended for dry mix. After that, a small quantity of water is added and mixed for a while. Finally, remaining water is added and continuously mixed before pouring into the concrete molds. All the concrete samples were cast in cylindrical cardboard molds (see Fig. 3a), and then compacted on an external vibrating table combined with an internal vibrator for improving the consolidation of the samples. After compaction, the tops of the samples were smoothly finished by using a trowel and then covered by polyethylene sheets.

All concrete specimens were kept in their molds for 48 hours after casting in a moisture room in accordance with the recommendation of the regulation NF EN 12390-2 [54]. After this period, the concrete cylinders were taken out of their molds and continuously cured by immersing them in a water basin at a temperature of approximately 20 o C for 2 months. At the age of 56 days, all the specimens were cut by diamond grinding discs to ensure both the ends to be perpendicular to the sides of the sample and avoid flexural stresses upon loading. After sawing, the concrete specimens were kept again in the water basin for an additional month and left exposed to atmosphere in the laboratory until the testing day (see Fig. 3b). In accordance with the recommendation of the regulation NF EN 12390-3 [55],

compression tests should be carried out after a minimum age of 28 days. In this research, all the compression tests were performed five months after concrete casting.

P-wave velocity measurements

According to ASTM C597-02 [START_REF]Standard Test Method for Pulse Velocity Through Concrete[END_REF], the dynamic Young's modulus of concrete can be estimated from the P-wave velocity, 𝑉 𝑝 , of a compression wave travelling through an elastic concrete body, the Poisson's ratio, 𝜐, and the density of the concrete sample, 𝜌, as follows:

𝑌 𝑑 = 𝑉 𝑝 2 [ 𝜌(1 + 𝜐)(1 -2𝜐) 1 -𝜐 ] (6)
In this study, based on the principles of ultrasonic testing described in ASTM C597-02 [START_REF]Standard Test Method for Pulse Velocity Through Concrete[END_REF], the transit time, Δ 𝑡 , of P-waves through the concrete sample was measured by using an Acoustic Emission (AE) equipment (see Fig. 4c and Fig. 5a).

The velocity, 𝑉 𝑝 , was then calculated by dividing the wave path, ℎ, (the height of concrete sample) over the transit time, Δ𝑡, as 𝑉 𝑝 = ℎ/Δ𝑡. These P-wave velocity measurements were non-destructive tests and carried out on intact concrete specimens prior to uniaxial compression tests (section 3.4). For this work, a total of 36 specimens (three specimens for a given size and type of concrete) were tested.

In order to detect the arrival time of P-wave of concrete, a pair of piezoelectric transducers (AE sensors) with frequency bandwidth of 20-1200kHz was used.

These two transducers were fixed to the ends of concrete cylinder by applying a coupling silicon paste (see Fig. 4c). In this study, Pencil-lead break (PLB) (Hsu-Nielsen source [56]) was used as an artificial method of generating acoustic emission (AE) signals. The PLBs occurring very close to one of the two transducers, caused stress waves that propagated through the concrete sample and then recorded by the remaining transducer (see Fig. 4c and Fig. 5a). Five different PLB tests were conducted around each AE sensor, for a total of 10 values of P-wave velocity collected for each concrete sample. The transit time, Δ𝑡, in each PLB test was computed as the difference of arrival time of the P-wave between the two AE sensors. The details of these AE measurements have been presented elsewhere [START_REF] Vu | Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete[END_REF].

It is noteworthy that the S-waves were hardly distinguishable from the P-waves on the waveforms recorded by the AE system [57,[START_REF] Ohtsu | Acoustic Emission (AE) and Related Non-destructive Evaluation Techniques in the Fracture Mechanics of Concrete: Fundamentals and Applications[END_REF]. Consequently, the measurement of the shear modulus, and so of the Poisson's ratio (𝜈), was not possible from the AE measurements carried out in this study. As reported in [START_REF] Neville | Properties of concrete[END_REF]39,[START_REF] Anson | The effect of mix proportions and method of testing on Poisson's ratio for mortars and concretes[END_REF][START_REF] Sideris | Estimation of ultimate modulus of elasticity and Poisson ratio of normal concrete[END_REF][START_REF] Petro | Detection of delamination in concrete using ultrasonic pulse velocity test[END_REF], the value of Poisson's ratio for normal-weight concrete is generally in a range 0.20 to 0.25. Moreover, available data do not show any significant impact of various factors (e.g. 𝑊/𝐶 ratio, curing condition, aggregate gradation) on Poisson's ratio [START_REF] Neville | Properties of concrete[END_REF][START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF]. Similarly, published data show no evidence of shape or size effects on Poisson's ratio [START_REF] Darlington | The effect of specimen size on strength and other properties in laboratory testing of rock and rock-like cementitious brittle materials[END_REF]39,[START_REF] Anson | The effect of mix proportions and method of testing on Poisson's ratio for mortars and concretes[END_REF]. In this study, an averaged value of 𝜈 = 0.22 was assumed as the value of Poisson's ratio for all our concretes to estimate the dynamic modulus of elasticity (𝑌 𝑑 ) from Eq. ( 6).

Uniaxial compression test

After completing the nondestructive tests (section 3.3), static mechanical properties including the compressive strength and the static modulus of elasticity of concrete specimens were determined from a series of uniaxial compression tests.

These loading tests were carried out following the procedure described in the regulations NF EN 12390-3 [55]. In this work, two load-controlled compression machines (of different stiffness and loading capacity) were used because the concrete samples were of different sizes. These two machines were calibrated and certified by the French Accreditation Committee (COFRAC) [START_REF] Vu | Size effects on the mechanical behavior and the compressive failure strength of concrete: an extensive dataset[END_REF] and their specific properties conformed to the requirements in the regulation NF EN 12390-4 [START_REF]Essais pour béton durci -Partie 4 : Résistance à la compression -Caractéristiques des machines d'essai[END_REF].

For the three large sample sizes (cylinders of 70 mm, 110 mm, and 160 mm in diameter), a compression machine (Machine A, Fig. 4a) with a loading capacity of 3000 kN was used. This machine is 2.9 stiffer than the largest specimens. The other compression machine (Machine B, Fig. 4b) with a loading capacity of 300 kN was used for the smallest samples (cylinders of 40 mm in diameter). The stiffness of this machine is 3.5 times that of the 40x80-mm specimens. The loading rate applied for all compression tests was 0.5 MPa per second and kept unchanged from the beginning of loading until the sample was broken down. When the load fell below 50% of the peak load, loading and data recording were automatically stopped.

During the test, the load (𝐹), the stress (𝜎 = 4𝐹/𝜋𝜙 2 ) and the axial displacement measured directly from the test, comprises not only the true axial displacement of the specimen (Δ 𝑠𝑝 ), but also the elastic deformation of the loading frame (Δ 𝑓𝑟 ).

For this reason, to obtain the value of Δ 𝑠𝑝 , the elastic deformation (Δ 𝑓𝑟 ) must be eliminated from the measured displacement (𝛿). In the present work, we performed a calibration test on an Aluminum sample of known elastic modulus to determine the relationship between the loading deformation (Δ 𝑓𝑟 ) and the applied load (𝐹).

In other words, from this calibration test, the elastic deformation of the compression machine (Δ 𝑓𝑟 ) was calculated at any applied load and then used to correct the true displacement corresponding to the deformation of the concrete sample (Δ 𝑠𝑝 ). The details of these calibration tests have been presented elsewhere [START_REF] Vu | Size effects on the mechanical behavior and the compressive failure strength of concrete: an extensive dataset[END_REF].

Determination of static elastic modulus

The static Young's modulus is generally defined as the elastic stiffness that relates to the ratio of axial stress to axial strain for a material subjected to uniaxial loading [START_REF] Neville | Properties of concrete[END_REF]. Various estimations of this static modulus from the slope of the stressstrain curve in the elastic deformation regime have been proposed, such as the tangent modulus [START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF]64], the secant modulus [START_REF] En | Eurocode 2: Design of concrete structures -Part 1-1 : General rules and rules for buildings[END_REF]3,65] or the chord modulus [66].

In this study, both the secant and tangent elastic moduli for all concrete specimens were examined. The deformation of a concrete specimen is calculated by dividing the displacement, Δ 𝑠𝑝 , (see section 3.4) by the height of specimen (𝜀 = Δ 𝑠𝑝 /ℎ).

Fig. 5b shows an example curve of the stress-strain relation for our concrete samples in a compression test.

In order to estimate the tangent elastic modulus of concrete, we defined the tangent modulus, (𝑌 𝑡 ) 𝑖 at a data point 𝑖 from a differentiation of the stress-strain curve:

(𝑌 𝑡 ) 𝑖 = 𝜎 𝑖+1 -𝜎 𝑖 𝜀 𝑖+1 -𝜀 𝑖 (7)
A 40 data points moving average of this (𝑌 𝑡 ) 𝑖 curve was then performed, and the maximum value of this moving average is taken as the tangent modulus, 𝑌 𝑡 , of the concrete sample examined (see Fig. 5b). In the present work, the secant elastic modulus, 𝑌 𝑠 was computed by the slope of a line drawn from the origin to 40% of the peak stress in accordance with the recommendation described in the standard EN 1992 [START_REF] En | Eurocode 2: Design of concrete structures -Part 1-1 : General rules and rules for buildings[END_REF].

Size effect on elastic moduli

Concrete, a typical heterogeneous material, exhibits a size-dependent behavior on the nominal compressive strength [START_REF] Bazant | Fracture and size effect in concrete and other quasibrittle materials[END_REF]. Precisely, the compressive strength decreases, in average, with increasing sample size. For the concretes used in this study, the size effect on compressive strength has also been confirmed and thoroughly analyzed elsewhere [START_REF] Vu | Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete[END_REF]. In this paper, we focus on the effect of size on the Young's modulus of concrete. This work considers both static and dynamic estimations of elastic (Young's) moduli as well as the relationship between them.

The variations of static elastic moduli 𝑌 𝑡 and 𝑌 𝑠 with the sample diameter for the three concrete groups are respectively presented in Fig. 6a and Fig. 6b. As displayed in Fig. 6c, the tangent modulus, 𝑌 𝑡 , is observed to be approximately equal to the secant modulus, 𝑌 𝑠 , regardless of sample size and concrete mixture, meaning that below 40% of the peak load, all our concretes remain essentially in the elastic domain. The mean values and the corresponding standard deviation (SD) of static elastic moduli 𝑌 𝑡 and 𝑌 𝑠 for different sample sizes of our three concrete mixtures are summarized in Table 5. As expected, the coarser materials (M-and C-concretes)

show the higher values of both the mean and the associated variability of tangent and secant moduli compared to the finer ones (F-concrete). For a given concrete mixture, the static elastic properties, including tangent and secant moduli and their standard deviations, do not exhibit any significant sample size effect.

As already presented in [START_REF] Vu | Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete[END_REF], from the non-destructive measurements (section 2.3), we observed no significant dependence of both P-wave velocities, 𝑉 𝑝 , and dynamic elastic moduli, 𝑌 𝑑 , as well as their associated variabilities, on sample size (see Fig. 11 in [START_REF] Vu | Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete[END_REF]). The absence of sample size effect on the elastic properties including the P-wave velocity, static (section 3.1), and dynamic elastic moduli means that all the concrete samples tested, including the smallest ones (40x80mmsamples), are large enough in terms of microstructural characteristics to be statistically representative of the linear (e.g. elastic) mechanical properties of the materials [START_REF] Kanit | Determination of the size of the representative volume element for random composites : statistical and numerical approach[END_REF]. The mean values of P-wave velocity (𝑉 𝑝 ) and dynamic elastic modulus (𝑌 𝑑 ) for different sample sizes of the three concrete groups are reported in Table 5.

As shown on Fig. 7, the static elastic moduli 𝑌 𝑡 and 𝑌 𝑠 are observed to be systematically smaller than the dynamic elastic modulus, 𝑌 𝑑 , regardless of concrete mixture and sample size, with a proportionality coefficient of 0.6 (𝑌 𝑡,𝑠 ~0.6𝑌 𝑑 ).

Nevertheless, it is noteworthy that the values of 𝑌 𝑡 and 𝑌 𝑠 in this study were derived from the deformation of the whole height of the concrete specimen (see section 2.4).

As recommended in [START_REF]Strain softening of concrete: Test method for measurement of the strain-softening behaviour of concrete under uniaxial compression 68[END_REF], in order to determine a proper value of the static elastic moduli 𝑌 𝑡,𝑠 , the deformation should be measured over the central third of the specimen height. According to Mansur et al. [68], the static modulus of elasticity estimated by the deformation measured from the central region of the concrete specimen is approximately 1.35 times higher than the deformation measured between the loading platens (similar to our case). Therefore, if we use the correction coefficient of 1.35 for the all values of 𝑌 𝑡,𝑠 , the linear relation in Fig. 7 becomes 𝑌 𝑡,𝑠 ~0.8𝑌 𝑑 . This is consistent with empirical relationships between static and dynamic moduli presented in [START_REF] Neville | Properties of concrete[END_REF][START_REF] Lydon | Some observations on elastic properties of plain concrete[END_REF].

The difference between static and dynamic moduli might be related to the fact that the heterogeneous characteristics of concrete influence these two moduli in different manners [START_REF] Philleo | Comparison of results of three methods for determining Young's modulus of elasticity of concrete[END_REF]. Indeed, by means of a nondestructive technique (section 2.3), the dynamic modulus of elasticity is determined prior to loading and so without microcracking induced in the concrete specimen during the test.

Consequently, the dynamic modulus refers to the genuine elastic properties of the material, while the static modulus can be affected by permanent non-linear deformations resulting from damage and microcracking [START_REF] Neville | Properties of concrete[END_REF][START_REF] Linger | Effect of stress on the dynamic modulus of concrete[END_REF]. For this reason, the dynamic modulus is considered to be approximately equal to the initial tangent modulus determined in the static test [START_REF] Neville | Properties of concrete[END_REF][START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF]. This explains why the elastic static modulus, 𝑌 𝑡 and/or 𝑌 𝑠 , is generally smaller than the elastic dynamic modulus, 𝑌 𝑑 , (or initial tangent modulus).

It should be noted that the value of Young's modulus of concrete, obtained from both static and dynamic measurements, could be affected by many other factors such as moisture content and density [START_REF] Neville | Properties of concrete[END_REF][START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF]. For our concretes, the procedure to determine the moisture content and density has been presented in details elsewhere [START_REF] Vu | Revisiting the concept of characteristic compressive strength of concrete[END_REF]. However, for a given concrete group, we did not observe any significant sample size effects on both density and moisture content (see their values in Table 5). This means that both the moisture content and density neither have a significant influence on the size-dependency of the elastic modulus of concrete, nor on the correlation between the static and dynamic elastic moduli.

Relationship between compressive strength and elastic (Young's) modulus

As mentioned in section 1, there are numerous empirical expressions given in the current building codes and/or proposed in the literature to estimate the elastic modulus based on the compressive strength of concrete. In this study, only the expressions proposed in the most popular building codes (i.e. the standards EN 1992 [START_REF] En | Eurocode 2: Design of concrete structures -Part 1-1 : General rules and rules for buildings[END_REF] and ACI 318-05 [3]) are used in order to compare with our results.

As displayed in Eqs. ( 1) and (3) as well as for other equations summarized in Table 1, it is clear that these empirical formulas are dimensionally conflicting (e.g.

Pa vs. Pa 0.5 in Eq. ( 1) and Pa vs. Pa 0.3 in Eq. ( 3)). Besides this, Fig. 8, which shows the correlations between the compressive strength, 𝜎 𝑓 , and the static (𝑌 𝑡 and 𝑌 𝑠 ) and dynamic (𝑌 𝑑 ) elastic moduli for all of our concrete samples, epitomizes several problems while using such empirical relations. In this figure, we show the modulus of elasticity (static and dynamic)-compressive strength relationships for all our concrete samples. The relationships obtained using the expressions (Eqs. (

(3)) given in the codes mentioned above are also plotted in Fig. 8 by taking (𝑓 𝑐 ′ = 𝜎 𝑓 ) and (𝑓 𝑐𝑚 = 𝜎 𝑓 ). From this, we observed that:

(i) for the static modulus of elasticity, the empirical expressions are systematically much larger than our values of both 𝑌 𝑡 and 𝑌 𝑠 , regardless of concrete mixture and sample size.

(ii) for the dynamic modulus of elasticity, 𝑌 𝑑 , experimental data are below the empirical predictions for F-concrete, but above the predictions for M-and Cconcretes. In other words, for M-and C-concretes, both EN 1992 and ACI 318-05 underestimate the value of 𝑌 𝑑 while these empirical predictions overestimate it for F-concrete.

(iii) the mean value of the dynamic modulus of elasticity, 𝑌 𝑑 , seems to slightly increase with increasing compressive strength, 𝜎 𝑓 , for F-and M-concretes (see Fig. 8a andb). This is not the case for C-concrete (Fig. 8c) and, gathering the data obtained for all sample sizes for a specified concrete mixture in Fig. 8d, an opposite trend is actually observed. This means that, due to size effects on compressive strength, there is no correlation between the dynamic modulus of elasticity and the compressive strength, in contrast with the empirical expressions summarized in Table 2.

(iv) for an individual dataset (fixed concrete mixture and sample size), the correlations of compressive strength 𝜎 𝑓 with both static moduli 𝑌 𝑡 and 𝑌 𝑠 are either very weak or unsignificant (see Table 6). Merging strength data for all sample sizes of a given concrete group, the absence of correlation is clear as well (see Fig. 8a, b andc). This is expected, as we observed a size effect on strength (see in more details in [START_REF] Vu | Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete[END_REF]) but not on the elastic modulus (Fig. 8).

(v) considering now all concrete samples (all sizes, all materials), the disagreement with the empirical formulas ( 1) and ( 3) is even more striking (Fig. 8d). Indeed, increasing the aggregate size implies an increase of the elastic modulus, as expected, while F-concrete showed, in average, larger strengths than M-and Cconcrete. As explained in more details in [START_REF] Vu | Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete[END_REF], this last point is a consequence of the size effects on compressive strength, in relation with the pore structure of our hardened concrete samples.

This demonstrates the absence of significant and meaningful link between elastic modulus (either static or dynamic) and compressive strength. This, we argue, is not surprising, given that the modulus of elasticity is directly related to elastic (linear) deformation, while the compressive strength is affected by the non-linear behavior due to the progressive development of damage and microcracking [START_REF] Neville | Properties of concrete[END_REF][START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF].

For a perfect crystalline material without defects, the ultimate (tensile) strength, resulting from pure brittle failure, would be proportional to the elastic modulus [START_REF] Miannay | Fracture Mechanics[END_REF].

For a disordered material like concrete under compression, such scenario is unrealistic.

Nevertheless, it is noteworthy that the empirical expressions (e.g. Equations

(1) and (3) or those summarized in Table 1) have been proposed from experimental datasets of only one standardized size of concrete sample (e.g. cylinder of 150x300 mm in US [73]; cylinder of 160x320 mm or cube of 150x150x150 mm in many

European nations [53,[START_REF]Essais pour béton durci -Partie 1 : Forme , dimensions et autres exigences relatives[END_REF]; either cubes of 100x100x100 mm or 150x150x150 mm in other countries) with different concrete mixtures (i.e. different strength grades) [START_REF] Neville | Properties of concrete[END_REF][START_REF] Noguchi | A practical equation for elastic modulus of concrete[END_REF][START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF][START_REF] Park | Modulus of elasticity of recycled aggregate concrete[END_REF]. These empirical expressions may thus actually express the influence of some factors such as aggregate type, density, or mix proportion on both the elastic properties and the compressive strength. They might therefore provide a relatively good estimation of the Young's modulus from the compressive strength in case of normal-weight concretes, as long as the concrete sample size used to measure the strength is in conformity with these standardized sizes [START_REF] Noguchi | A practical equation for elastic modulus of concrete[END_REF]. In the present study, we examined the influence of sample size on the relationship between the modulus of elasticity and the compressive strength, while, by definition, the strength grade (i.e.

the characteristic compressive strength) of a given concrete mixture is independent of sample size. From these observations, it can be concluded that, in order to estimate the Young's modulus from the compressive strength of concrete, in many cases (e.g. F-concrete samples in this study), due to the sample size effect on the compressive strength, the use of empirical expressions such as Eq. ( 1) or Eq. ( 3) may lead to an overestimation (an unsafe result). Owing to the limited number of different concrete mixtures considered in the present work and the similarity of their characteristic compressive strength (see [START_REF] Vu | Revisiting the concept of characteristic compressive strength of concrete[END_REF] for more details), we cannot conclude that these empirical formulas could not be acceptable for other concrete mixtures (e.g. other type of concrete, other strength grade mixture). However, more generally, to measure the elastic modulus from a static load test, rather than from such empirical relations, is highly recommended and considered as being much safer for concrete structural design.

Conclusion

In this paper, from an extensive experimental study of the compressive behavior of different concrete mixtures and different sample sizes: we (i) explored a potential effect of sample size on the elastic properties (i.e. Young's modulus and P-wave velocity); and (ii) discussed the relationship between compressive strength and elastic modulus. The following main conclusions can be drawn:

(i) The coarser concrete shows a larger elastic modulus than the finer ones. This is consistent with the fact that the elastic modulus of concrete is affected by the elastic modulus of the aggregate and by the volumetric proportion of aggregate in the concrete.

(ii) The ratio of the static modulus of elasticity to the dynamic modulus for all specimens, regardless of sample size and concrete mixture, is always smaller than unity. Nevertheless, this ratio directly depends on the methodologies applied for measuring the deformation of specimens during the loading and for estimating the static elastic modulus. This means that there is no simple conversion between the dynamic modulus, which is known as the genuine elastic modulus of the material, and an estimate of the static modulus which is used in structural design.

(iii) For a given concrete mixture, there is no sample size effect on the elastic properties of concrete, including the static and dynamic elastic moduli. This is in full contrast with the compressive strength, which exhibits significant size effects for all the concrete mixtures.

(iv) A consequence of point (iii) above is that there is no direct proportionality between elastic modulus (a size-independent property) and compressive strength (a size-dependent property) of concrete. This raises severe concerns about the use of empirical formulas proposed in building codes to estimate the Young's modulus from the compressive strength, and consequently on the design of structural concrete elements. Consequently, the authors recommend using non-destructive methods such P-wave velocity measurements or performing uniaxial compression tests on concrete specimens to directly determine the elastic modulus of concrete.

a

  larger range of system sizes, the restricted number of concrete samples (< 10 samples for each size) tested in this study, combined with a large variability of 𝑌 𝑠 values for a given size, limits the statistical significance of the conclusion. This calls for more extensive data and further research on these possible size effects on the elastic modulus of concrete. The limited number of tests generally performed comes from the fact that this kind of experimental program is time-(e.g. due to the installation of the equipment to measure the sample deformation) and costconsuming. From this survey of the literature several important questions arise: (a) Does the elastic modulus of concrete depend on sample size? (b) Is there any influence of sample size on the relationship between the static and dynamic elastic moduli of concrete? (c) Considering well-established size effects on compressive strength [14, 15, 19] in one hand, and the proposed empirical relationships between compressive strength and elastic modulus (equations (1) to (5)), what could be the implications in terms of the estimation of elastic properties of concrete from strength data? And consequently in terms of design of concrete structural elements? Motivated by those questions, we carried out an extensive experimental program on 527 cylindrical concrete specimens of three different concrete mixes and four different sample sizes. The details of the experimental research are described in the next section.

  and of at least three different sizes with a minimum size range of 1:4 should be used to investigate the size effects on mechanical strength. In the present work, concrete cylinders with a constant height-to-diameter ratio (ℎ/𝜙 = 2) and four different sizes (𝜙 = 40, 70, 110, 160 mm) were fabricated from three different concrete mixtures based on three different aggregate sizes. The experimental study included both non-destructive and destructive tests. The aim of the non-destructive tests was

  this work, a CEM I 52.5N type Portland cement satisfying the standard NF EN 197-1 [49] was used for preparing all concrete mixtures. Natural sand and gravel were used as the fine and coarse aggregates, respectively. All the aggregates used for concrete casting were clean and dry, and their specific properties complied with the requirements in NF EN 12620 [50] and NF EN 1097-5 [51] standards. The size distribution of aggregates (shown in Fig. 2a) was investigated from a sieving analysis following NF EN 933-1 [52]. Normal tap water was used for mixing and curing of concrete samples in this study. Applying the weight method described in the regulation NF-EN 206-1 [53], we prepared three different concrete mixtures based on three different aggregate sizes (Fine (F; i.e. only sand), Medium (M),

(

  𝛿) were continuously recorded every 0.2 seconds. The displacement (𝛿) was the axial displacement of the bottom steel platen of the compression machine and measured by one Linear Variable Differential Transducer (LVDT) attached on the frame of the machine. It is noteworthy that both the sample and the compression machine are deforming upon loading. Consequently, the axial displacement (𝛿)

FiguresFig. 1

 1 Figures

Fig. 2 .

 2 Fig. 1. Experimental program

Fig. 3 .

 3 Fig. 3. Different steps of fabrication of the concrete specimens. (a) Raw constituent materials and cylindrical cardboard molds with different sizes used for casting concrete samples; (b) Concrete samples left exposed to atmosphere in the laboratory after the curing period in a water basin.

Fig. 4 .

 4 Fig. 4. Testing setup for: uniaxial compression tests ((a)-Compression machine A and (b)-Compression machine B) and (c) P-wave velocity measurements.

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .

 5678 Fig. 5. Determination methods for the elastic moduli of concrete: (a) Testing setup of the acoustic pulse velocity measurement for evaluating the dynamic elastic modulus of a concrete sample; (b)Illustration of the methods of estimation of the tangent elastic modulus (𝑌 𝑡 ), where the red curve is the stress-strain curve, the blue curve is the 5 Hz differentiated stress-strain curve (𝑑𝜎/𝑑𝜀) and the solid black curve is the 40-points moving average of the blue curve; and of the secant elastic modulus (𝑌 𝑠 ) which is calculated by the slope of a line (dashed green line) drawn from the origin to 40% of the peak stress 𝜎 𝑓 .

Table 1 .

 1 Empirical relations of static Young's modulus with compressive strength 𝑓 𝑐 ′ is the specified compressive strength in MPa, 𝜌 is the density in kg/m 3 , 𝑓 𝑐𝑘 is the characteristic compressive strength in MPa. 𝑘 1 , 𝑘 2 are the correction factor corresponding to coarse aggregates and mineral admixtures, respectively.

	Tables
	References	Expression for static Young's	Range of concrete
		modulus, 𝒀 𝒔 (MPa)	strength
	Ahmad and Shah, 1985 [75]	𝑌 𝑠 = 𝜌 2.5 (𝑓 𝑐 ′ ) 0.325	𝑓 𝑐 ′ < 84 MPa
	Kakizaki et al., 1992 [5, 76]	𝑌 𝑠 = 3.65(𝑓 𝑐 ′ ) 0.5	80 MPa < 𝑓 𝑐 ′ < 140 MPa
	CEB-FIB Mode code, 1993 [9]	* 𝑌 𝑠 = 21500𝛼 𝐸 (	𝑓 𝑐𝑘 10 ) 1/3	𝑓 𝑐𝑘 ≤ 80 MPa
	Iravani, 1996 [77]	** 𝑌 𝑠 = 4700𝐶 𝑐𝑎 (𝑓 𝑐 ′ ) 0.5	55 MPa < 𝑓 𝑐 ′ < 125 MPa
	Malaikah, 2005 [78]	𝑌 𝑠 = 2090(𝑓 𝑐 ′ ) 0.5 + 22680	45 MPa < 𝑓 𝑐 ′ < 90 MPa
	Noguchi et al., 2009 [6]	*** 𝑌 𝑠 = 𝑘 1 𝑘 2 × 1.486 × 10 -3 (𝑓 𝑐 ′ ) 1/3 𝜌 2	40 MPa < 𝑓 𝑐 ′ < 160 MPa

Notes:

* 𝛼 𝐸 is a constant that depends on the type of aggregates.

** 𝐶 𝑐𝑎 is a coefficient related to the type of aggregates. ***

Table 2 .

 2 Empirical relations of dynamic Young's modulus with compressive strength 𝑓 𝑐 ′ is the specified compressive strength in MPa, 𝜌 is the density in kg/m3, 𝑓 𝑐𝑘 is the characteristic compressive strength in MPa.* 𝑎, 𝑏 are the experimental coefficients which are found by the linear regression of ln(𝑓 𝑐 ′ ) versus ln(𝑌 𝑑 ) 𝑌 𝑑 is expressed in psi.10 6 and 𝑓 𝑐 ′ is measured by psi. § 𝑎 is the coefficient related to the volume content of aggregate and maximum coarse aggregate size; and 𝑏 is coefficient influenced by the volume content of aggregate, maximum coarse aggregate size and water-to-cement ratio. § § 𝛾 is the volume content of aggregate in the concrete mix.

	References	Expression for dynamic	Type of concrete	NDT method
		Young's modulus, 𝒀 𝒅 (GPa)
	Lee et al., 1997 [36]		* 𝑌 𝑑 = 𝑎. (𝑓 𝑐 ′ ) 𝑏	Normal weight concrete	Resonant frequency
	Malhotra et al., 2004 [39]	** 𝑌 𝑑 = 8.67 × 10 6 𝑓 𝑐 ′ 𝑓 𝑐 ′ +1550	Normal weight concrete	Resonant frequency
	Zhou et al., 2015 [25]	§ 𝑌 𝑑 = 𝑎𝑓 𝑐 0.295 + 𝑏	Normal weight concrete	Resonant frequency
	Jurowski et al., 2018 [24]	§ § 𝑌 𝑑 =	43.𝜌 1.5 .10 -6 1.5𝛾	(𝑓 𝑐 ′ ) 0.5	Normal and High strength concrete	Resonant frequency
	Kumar et al., 2019 [9]	𝑌 𝑑 = 0.4079(𝑓 𝑐𝑘 ) 1.2557	Self-Compacting Concrete (SCC)	UPV
	Notes: 1 psi = 0.00689476 MPa,

** NDT: Non-Destructive Testing UPV: Ultra-sonic Pulse Velocity

Table 3 .

 3 Empirical relationships between the static (𝑌 𝑠 ) and dynamic (𝑌 𝑑 ) elastic moduli of concrete * 𝑌 𝑠 = 𝑌 𝑑 . (1 -𝑎𝑒 -𝑏𝑌 𝑑 ) 𝑏 Both units of 𝑌 𝑠 and 𝑌 𝑑 are in GPa. 𝜌 𝑐 is the density of the hardened concrete in a unit of kg/m 3 . 𝑎 and 𝑏 are the regression coefficients. § 𝑐 and 𝑑 are the regression coefficients. 𝑐 is in a range of 0.88 to 0.97.

	References	Equation	Type of	Type of	NDT
				concrete	static elastic	method
					modulus
				Light-and
	Popovics, 1975 [79]	* 𝑌 𝑠 =	1.4 446.09𝑌 𝑑 𝜌 𝑐	Normal-weight	Secant	UPV
				concrete
	British standard BS8110-2, 1985 [10]	𝑌 𝑠 = 1.25𝑌 𝑑 -19	Normal weight concrete	Secant	Resonant frequency
	Lydon et al., 1986 [69]	𝑌 𝑠 = 0.83𝑌 𝑑	Light-and Normal-weight concrete	Secant	Resonant frequency
	Han et al., 2004 [35]			Normal weight concrete	Secant	Resonant frequency
	Zhou et al., 2015 [25]	§ 𝑌 𝑠 = 𝑐𝑌 𝑑 + 𝑑	Normal weight concrete	Secant	Resonant frequency
	Pešic et al., 2015 [80]	𝑌 𝑑 = 1.32𝑌 𝑠	Normal weight concrete	Secant	UPV
	Lee et al., 2017 [42]	𝑌 𝑠 = 0.16𝑌 𝑑 1.36 1.24 𝑌 𝑠 = 0.33𝑌 𝑑	Normal weight concrete	Secant	UPV Resonant frequency
	Bassim et al., 2020 [34]	𝑌 𝑠 = 0.84𝑌 𝑑 -0.74	Normal weight concrete	Secant	Resonant frequency
	Notes: NDT: Non-Destructive Testing
	UPV: Ultra-sonic Pulse Velocity

** **

Table 4 .

 4 Mix proportions for the three concrete groups.

	Concrete group	Water (kg/m 3 )	Cement (kg/m 3 )	W/C	Sand (kg/m 3 )	Medium (kg/m 3 ) aggregate	Coarse (kg/m 3 ) aggregate	Maximum size (mm) aggregate	Finesse modulus
	F	225	450	0.50	1350	0	0	3.15	3.24
	M	195	335	0.58	800	1065	0	16	6.95
	C	195	335	0.58	800	0	1065	25	9.21

Table 5 .

 5 Density, moisture content, P-wave velocity, static and dynamic elastic moduli of different sample sizes and different concrete groups.

	Concrete	Sample size,	Density, 𝝆	Moisture		Static elastic modulus	P-wave	Dynamic
	group	𝝓 × 𝒉	(kg/m 3 )	content, 𝒘 𝒄				velocity, 𝑽 𝒑	elastic
		(mm x mm)				(%)	Tangent modulus, 𝒀 𝒕	Secant modulus, 𝒀 𝒔	(m/s)	modulus, 𝒀 𝒅 (GPa)
								(GPa)		(GPa)
			Mean	SD Mean	SD Mean	SD Mean	SD Mean	SD Mean SD
		40x80	2204.1 20.7	5.4	0.5	17.2	3.8	16.9	3.7 3798.0 164.5	28.0 2.4
		70x140	2201.1 11.0	5.4	0.4	18.5	2.2	18.1	2.2 3808.1 142.6	27.4 2.0
	F
		110x220	2177.7 25.0	5.2	0.5	17.2	1.4	16.9	1.4 3823.7 121.7	27.2 1.8
		160x320	2153.1 25.3	5.0	0.2	17.8	1.9	18.0	2.0 3782.4 135.5	26.5 1.8
		40x80	2415.3 16.5	3.4	0.3	21.7	4.1	21.6	4.1 4343.9 211.2	39.8 3.9
		70x140	2397.4 11.4	3.4	0.3	23.6	2.4	23.4	2.4 4337.0 134.4	38.7 2.3
	M
		110x220	2386.3 18.6	3.0	0.1	21.7	1.7	21.6	1.7 4268.6 225.7	37.4 4.0
		160x320	2366.4 12.4	2.9	0.3	22.5	1.9	23.6	2.7 4283.0 187.9	37.0 3.2
		40x80	2415.1 25.2	3.4	0.4	20.0	4.9	19.5	4.8 4168.7 293.9	36.0 4.7
		70x140	2421.6 11.7	3.4	0.4	24.8	2.8	24.2	2.7 4476.2 166.0	41.5 2.7
	C
		110x220	2393.9 17.6	3.0	0.2	21.3	2.1	20.0	1.8 4359.4 262.4	38.8 4.4
		160x320	2384.6 18.0	3.1	0.2	22.2	2.8	21.7	3.1 4365.5 252.3	38.3 4.3

Table 6 .

 6 Compressive strength (𝜎 𝑓 ) and its correlation coefficient with the static moduli (𝑌 𝑡 and 𝑌 𝑠 ) for different concrete samples.

	Concrete group	Sample sizes, 𝝓 × 𝒉 (mm x mm)	Compressive strength, 𝝈 𝒇 (MPa) Mean SD	𝝈 𝒇 vs. 𝒀 𝒕	𝝈 𝒇 vs. 𝒀 𝒔
		40x80	56.4	7.9	0.42	0.42
		70x140	49.7	5.3	0.43	0.43
	F
		110x220	44.3	2.4	0.19	0.09
		160x320	41.9	2.4	0.27	0.21
		40x80	49.6	6.2	0.21	0.21
		70x140	45.2	4.2	0.38	0.38
	M	110x220	41.9	1.9	0.41	0.20
		160x320	39.2	1.8	-0.06	-0.36
		40x80	39.5	4.6	0.40	0.40
		70x140	40.0	6.3	0.48	0.48
	C
		110x220	37.1	2.7	0.45	0.15
		160x320	36.7	1.4	0.12	0.03

Acknowledgements

The authors gratefully acknowledge the support of the AGIR program from Grenoble Alpes University (UGA), France, for this research. We would also like to thank the technicians of UGA and USMB for their technical assistances.