2.2 Description of experimental configurations

The list of concrete grades, maximum aggregate size, types of experimental set-up (hydraulic press or SHPB device), dimensions of samples and confining cells used in several works are gathered in Table 1. It can be remarked that the sample diameter is usually about five times larger than the maximum aggregate size. The length to diameter ratio of concrete sample is between 1 and 2. In addition, the length of the confining cell slightly exceeds the sample length in order to keep the sample inside the cell in particular when a dynamic loading is applied (SHPB device). The outer diameter is chosen as a compromise between a sufficient stiffness of the vessel (diameter large enough) and a good sensibility of strain measurements on the external surface of the vessel (small enough diameter). For this purpose, the outer diameter to inner diameter ratio varies from 1.4 [28] to 2.8 [26] with a predominance of value around 2 [6, 30–35].

Two examples of device used for quasi-static and dynamic testing are represented in Figure 4. The set-up represented in Figure 4(a) was used in [6, 31, 32] to investigate the quasi-static confined behavior of particle-reinforced cement composites. The loading capacity of the press, 1 MN, provides a maximum axial stress about 1400 MPa in the concrete sample, with a sample diameter of 30 mm. The compression plugs are fixed to the lower and upper compression plates. The extensometer that was used to measure the change of sample height is attached to two flasks directly screwed into the compression plugs. Four strain gauges are glued on the outer surface of the confining cell.

An experimental SHPB set-up, also called Koslky's apparatus, can be used to perform dynamic QOC tests. The SHPB experimental technique, widely used today, was pioneered by Kolsky [36]. It consists in a striker, an input bar and an output bar (Figure 4(b)). In [16, 30, 34], the striker, the input bar and the output bar are 80 mm in diameter, their length is, respectively, 2.2, 6, and 4 m, and the elastic limit of these elements, made of high-strength steel, is 1200 MPa. When the striker hits the free end of the input bar, a compressive incident wave is generated in the input bar. Once the incident wave (εið Þt ) reaches the specimen, a transmitted pulse (εtð Þt ) develops in the output bar whereas a reflected pulse (εrð Þt ) propagates in the opposite direction in the input bar. These three basic waves (cf. Figure 4(c)), recorded by strain gauges glued on the input and the output bars, are used to


\* MB50: microconcrete, M1: mortar without silica fume, (M1M, M1Sph): particle-reinforced mortar without silica fume, M2: mortar with silica fume, (M2S, M2M, M2Sph): particle-reinforced mortar with silica fume, R30A7: siliceous aggregate ordinary concrete, HSC: siliceous aggregate high-strength concrete, LC: limestone aggregate ordinary concrete.

calculate the input and output forces and the input and output velocities at the specimen faces according to the following equations (without taking into account the wave dispersion phenomena that cannot be neglected with large-diameter

Investigation of the Quasi-Static and Dynamic Confined Strength of Concretes by Means…

DOI: http://dx.doi.org/10.5772/intechopen.89660

Experimental set-up employed for quasi-static and dynamic QOC tests. (a) Hydraulic press used in [6, 31, 32]. (b) Sketch of SHPB system used in [16, 30, 33]. (c) Example of experimental data obtained in [30].

FinðÞ¼ t AbEbð Þ εiðÞþt εrð Þt FoutðÞ¼ t AbEbð Þ εtð Þt

(6)

Hopkinson bars) [36]:

Figure 4.

73

#### Table 1.

List of concretes, experimental set-ups, dimensions of sample and confining cell considered in several works.

Investigation of the Quasi-Static and Dynamic Confined Strength of Concretes by Means… DOI: http://dx.doi.org/10.5772/intechopen.89660

2.2 Description of experimental configurations

Compressive Strength of Concrete

to 2.8 [26] with a predominance of value around 2 [6, 30–35].

glued on the outer surface of the confining cell.

References Concretes\* (Maximum aggregates

[6, 31, 32] M1, M1M, M1Sph, M2, M2S, M2M,

\*

72

Table 1.

size in mm)

[28, 30] MB50 (5) Press,

M2Sph (6)

[16] MB50 (5) Press,

[34] R30A7 (8), HSC (8) SHPB,

[35] R30A7, LC (10) SHPB,

HSC: siliceous aggregate high-strength concrete, LC: limestone aggregate ordinary concrete.

Two examples of device used for quasi-static and dynamic testing are represented in Figure 4. The set-up represented in Figure 4(a) was used in [6, 31, 32] to investigate the quasi-static confined behavior of particle-reinforced cement composites. The loading capacity of the press, 1 MN, provides a maximum axial stress about 1400 MPa in the concrete sample, with a sample diameter of 30 mm. The compression plugs are fixed to the lower and upper compression plates. The extensometer that was used to measure the change of sample height is attached to two flasks directly screwed into the compression plugs. Four strain gauges are

An experimental SHPB set-up, also called Koslky's apparatus, can be used to perform dynamic QOC tests. The SHPB experimental technique, widely used today, was pioneered by Kolsky [36]. It consists in a striker, an input bar and an output bar (Figure 4(b)). In [16, 30, 34], the striker, the input bar and the output bar are 80 mm in diameter, their length is, respectively, 2.2, 6, and 4 m, and the elastic limit of these elements, made of high-strength steel, is 1200 MPa. When the striker hits the free end of the input bar, a compressive incident wave is generated in the input bar. Once the incident wave (εið Þt ) reaches the specimen, a transmitted pulse (εtð Þt ) develops in the output bar whereas a reflected pulse (εrð Þt ) propagates in the opposite direction in the input bar. These three basic waves (cf. Figure 4(c)), recorded by strain gauges glued on the input and the output bars, are used to

[26, 27] Mortar (2), concrete (16) Press D50 � 100 D140 � 106

SHPB

SHPB

Press

Press

MB50: microconcrete, M1: mortar without silica fume, (M1M, M1Sph): particle-reinforced mortar without silica fume, M2: mortar with silica fume, (M2S, M2M, M2Sph): particle-reinforced mortar with silica fume, R30A7: siliceous aggregate ordinary concrete,

List of concretes, experimental set-ups, dimensions of sample and confining cell considered in several works.

Set-ups Sample diameter � length

> D30 � 40 D50 � 40

Press D30 � 40 D55 � 46

D29 � 40 D60 � 45

D40 � 50 D80 � 60

D40 � 50 D80 � 60

Cell outer diameter � length

> D50 � 50 D70 � 50

The list of concrete grades, maximum aggregate size, types of experimental set-up (hydraulic press or SHPB device), dimensions of samples and confining cells used in several works are gathered in Table 1. It can be remarked that the sample diameter is usually about five times larger than the maximum aggregate size. The length to diameter ratio of concrete sample is between 1 and 2. In addition, the length of the confining cell slightly exceeds the sample length in order to keep the sample inside the cell in particular when a dynamic loading is applied (SHPB device). The outer diameter is chosen as a compromise between a sufficient stiffness of the vessel (diameter large enough) and a good sensibility of strain measurements on the external surface of the vessel (small enough diameter). For this purpose, the outer diameter to inner diameter ratio varies from 1.4 [28]

Experimental set-up employed for quasi-static and dynamic QOC tests. (a) Hydraulic press used in [6, 31, 32]. (b) Sketch of SHPB system used in [16, 30, 33]. (c) Example of experimental data obtained in [30].

calculate the input and output forces and the input and output velocities at the specimen faces according to the following equations (without taking into account the wave dispersion phenomena that cannot be neglected with large-diameter Hopkinson bars) [36]:

$$\begin{cases} F\_{in}(t) = A\_b E\_b(\varepsilon\_i(t) + \varepsilon\_r(t)) \\ F\_{out}(t) = A\_b E\_b(\varepsilon\_t(t)) \end{cases} \tag{6}$$

$$\begin{cases} V\_{in}(t) = -c\_b(\varepsilon\_i(t) - \varepsilon\_r(t)) \\ V\_{out}(t) = -c\_b(\varepsilon\_t(t)) \end{cases} \tag{7}$$

where Ab is the area of the input and output bars, Eb corresponds to their Young's modulus and cb is the speed of a 1D wave propagating in these bars (cb <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi Eb=ρ<sup>b</sup> <sup>p</sup> ), considering <sup>ρ</sup><sup>b</sup> as the density of the bars. Finally, the mean axial stress and nominal axial strain in the sample can be deduced:

$$
\overline{\sigma}\_{axial}(t) = \frac{F\_{in}(t) + F\_{out}(t)}{2A\_{\mathcal{S}}} \tag{8}
$$

$$
\overline{\varepsilon}\_{axial}(t) = \int\_0^t \frac{V\_{out}(u) - V\_{in}(u)}{h\_S} du \tag{9}
$$

where hS is the sample length. In addition, the elastic limit of the plugs must be significantly higher than the maximum axial stress reached during the dynamic tests so the elastic shortening of the plugs can easily be subtracted from the total measured shortening.

#### 2.3 Data processing of QOC tests

A processing technique was proposed in [29] to measure the radial stress in the sample taking into account the barreling deformation of the confining ring and the shortening of the concrete sample during the test. Before this approach was proposed, the closed-form solution of an elastic ring subjected to a uniform internal pressure along its whole length was usually considered but this method can lead to a strongly erroneous estimation of the radial stress in the tested sample. Later, several improvements of the processing methodology have been proposed as listed hereafter. The main advances to process the data of QOC tests are summarized in Table 2.

First, the relation between the radial stress in the specimen and the hoop strain measured thanks to the strain gauge glued on the outer surface of the confining cell must be deduced. The methodology used in [31] is explained in Figure 5. The constitutive behavior of the confining ring was first identified by means of quasi-static tensile tests performed with small samples extracted from a sacrificed ring. Next, this constitutive behavior was introduced in a numerical simulation involving only the confining vessel. Last, two calculations were conducted with the Abaqus/Standard FE code considering an internal pressure which is continuously increased up to 400 MPa, and which is applied to the inner surface of the ring along heights of 40 mm in a first

calculation and 34 mm in a second calculation, these heights corresponding to the initial and final lengths of the concrete sample. From these calculations the relations:

Methodology to determine the function σradial hð Þ <sup>¼</sup>40mm ¼ f <sup>40</sup>ð Þ εθθ [31]. (a) Extraction of small samples from a confining cell. (b) Tensile tests and identification of the elastoplastic behavior for each sample. (c) Numerical simulation of the vessel subjected to an internal pressure of 360 MPa on 40 mm (upper part: Hoop strain contours, lower part: Radial stress contours), (d) identification of the relation between the radial stress and the

Investigation of the Quasi-Static and Dynamic Confined Strength of Concretes by Means…

DOI: http://dx.doi.org/10.5772/intechopen.89660

θθ and <sup>σ</sup>radial hð Þ <sup>S</sup>¼34mm <sup>¼</sup> <sup>f</sup> <sup>34</sup> <sup>ε</sup>

ð Þ z¼0,ext θθ <sup>þ</sup>

the data of hoop strains placed at two locations on the cell (ε

2

<sup>1</sup> � <sup>ε</sup>axial <sup>1</sup> <sup>þ</sup> <sup>ε</sup>axial

<sup>α</sup><sup>0</sup>

so, assuming a uniform radial stress in the sample, the radial stress was computed knowing the current height of the sample (hS) according to the following linear

In a similar way, the (small) radial strain in the sample was deduced based on

0ε ð Þ z¼0,ext ð Þ z¼0,ext

hSð Þ�t 40 <sup>34</sup> � <sup>40</sup> <sup>f</sup> <sup>34</sup> <sup>ε</sup>

θθ <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>ε</sup>axial

θθ were deduced,

ð Þ z¼0,ext θθ (10)

> ð Þ z¼18,ext θθ ):

ð Þ z¼18,ext θθ (11)

ð Þ z¼0,ext θθ , ε

2 <sup>3</sup> <sup>α</sup><sup>18</sup> 20ε

σradial hð Þ <sup>S</sup>¼40mm ¼ f <sup>40</sup> ε

interpolation:

outer hoop strain.

Figure 5.

σradialðÞ¼ t

εradialðÞ¼ t

75

2 3 ð Þ z¼0,ext

hSðÞ�t 34 <sup>40</sup> � <sup>34</sup> <sup>f</sup> <sup>40</sup> <sup>ε</sup>


#### Table 2.

Main advances proposed to process the data of QOC tests.

Investigation of the Quasi-Static and Dynamic Confined Strength of Concretes by Means… DOI: http://dx.doi.org/10.5772/intechopen.89660

Figure 5.

VinðÞ¼� t cbð Þ εiðÞ�t εrð Þt VoutðÞ¼� t cbð Þ εtð Þt

<sup>p</sup> ), considering <sup>ρ</sup><sup>b</sup> as the density of the bars. Finally, the mean axial

where hS is the sample length. In addition, the elastic limit of the plugs must be significantly higher than the maximum axial stress reached during the dynamic tests so the elastic shortening of the plugs can easily be subtracted from the total

A processing technique was proposed in [29] to measure the radial stress in the sample taking into account the barreling deformation of the confining ring and the shortening of the concrete sample during the test. Before this approach was proposed, the closed-form solution of an elastic ring subjected to a uniform internal pressure along its whole length was usually considered but this method can lead to a strongly erroneous estimation of the radial stress in the tested sample. Later, several improvements of the processing methodology have been proposed as listed hereafter. The main advances to process the data of QOC tests are summarized in Table 2. First, the relation between the radial stress in the specimen and the hoop strain measured thanks to the strain gauge glued on the outer surface of the confining cell must be deduced. The methodology used in [31] is explained in Figure 5. The constitutive behavior of the confining ring was first identified by means of quasi-static tensile tests performed with small samples extracted from a sacrificed ring. Next, this constitutive behavior was introduced in a numerical simulation involving only the confining vessel. Last, two calculations were conducted with the Abaqus/Standard FE code considering an internal pressure which is continuously increased up to 400 MPa, and which is applied to the inner surface of the ring along heights of 40 mm in a first

[26, 28] Radial stress estimated based on the close-form solution of an elastic ring subjected to a uniform internal

[29] Radial stress calculated considering the barreling deformation of the cell and the sample shortening [31] Radial stress calculated as in [29] and taking into the non-homogeneous and elastoplastic behavior of the

[30–32] Estimation of the error due to friction and to the interface product by applying the processing

[30, 32] Proposition of two methods to estimate the internal friction based on strain measurements on the outer

[16] Improvement of the data processing method taking into account the internal friction and the sample/cell symmetry defect based on strain measurements on the outer surface of the cell

Validation with tests applied to aluminum alloy samples

methodology to the data of numerical simulations of QOC tests

FinðÞþt Foutð Þt 2AS

Voutð Þ� u Vinð Þ u hS

where Ab is the area of the input and output bars, Eb corresponds to their Young's modulus and cb is the speed of a 1D wave propagating in these bars

(7)

(8)

du (9)

�

stress and nominal axial strain in the sample can be deduced:

εaxialðÞ¼ t

σaxialðÞ¼ t

ðt 0

(cb <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi Eb=ρ<sup>b</sup>

Compressive Strength of Concrete

measured shortening.

2.3 Data processing of QOC tests

References Advances in the processing methodology

pressure

cell

Table 2.

74

surface of the cell

Main advances proposed to process the data of QOC tests.

Methodology to determine the function σradial hð Þ <sup>¼</sup>40mm ¼ f <sup>40</sup>ð Þ εθθ [31]. (a) Extraction of small samples from a confining cell. (b) Tensile tests and identification of the elastoplastic behavior for each sample. (c) Numerical simulation of the vessel subjected to an internal pressure of 360 MPa on 40 mm (upper part: Hoop strain contours, lower part: Radial stress contours), (d) identification of the relation between the radial stress and the outer hoop strain.

calculation and 34 mm in a second calculation, these heights corresponding to the initial and final lengths of the concrete sample. From these calculations the relations: σradial hð Þ <sup>S</sup>¼40mm ¼ f <sup>40</sup> ε ð Þ z¼0,ext θθ and <sup>σ</sup>radial hð Þ <sup>S</sup>¼34mm <sup>¼</sup> <sup>f</sup> <sup>34</sup> <sup>ε</sup> ð Þ z¼0,ext θθ were deduced, so, assuming a uniform radial stress in the sample, the radial stress was computed knowing the current height of the sample (hS) according to the following linear interpolation:

$$
\overline{\sigma}\_{radial}(t) = \left(\frac{h\_{\rm S}(t) - \mathbf{34}}{4\mathbf{0} - \mathbf{34}}\right) f\_{40}\left(e^{(x=0,\text{ext})}\_{\theta\theta}\right) + \left(\frac{h\_{\rm S}(t) - 4\mathbf{0}}{\mathbf{34} - 4\mathbf{0}}\right) f\_{\rm 34}\left(e^{(x=0,\text{ext})}\_{\theta\theta}\right) \tag{10}
$$

In a similar way, the (small) radial strain in the sample was deduced based on the data of hoop strains placed at two locations on the cell (ε ð Þ z¼0,ext θθ , ε ð Þ z¼18,ext θθ ):

$$\overline{\varepsilon}\_{\text{radial}}(t) = \frac{2}{3} \left( \mathbf{1} - \overline{\varepsilon}\_{\text{axial}} \left( \mathbf{1} + \frac{\overline{\varepsilon}\_{\text{axial}}}{2} \right) \right) a\_0^0 e\_{\theta\theta}^{(x=0,\text{ext})} + \frac{\left( \mathbf{1} + \overline{\varepsilon}\_{\text{axial}} \right)^2}{3} a\_{20}^{18} e\_{\theta\theta}^{(x=18,\text{ext})} \tag{11}$$

where α<sup>0</sup> <sup>0</sup> and α<sup>18</sup> <sup>20</sup> are coefficients identified from the same calculations.

Finally, the deviatoric stress and the hydrostatic pressure were deduced based on Eqs. (2) and (3). The volumetric strain was also calculated in the following way:

$$
\overline{\varepsilon}\_{volumetric}(t) = (\mathbf{1} + \overline{\varepsilon}\_{axial})(\mathbf{1} + \overline{\varepsilon}\_{radial})^2 - \mathbf{1} \tag{12}
$$

so the hydrostatic behavior of the concrete (relation between the hydrostatic pressure and the volumetric strain) was obtained. The whole procedure that was used in [6, 30–32] is summarized on the sketch of Figure 6.

The use of a confining cell that deforms plastically during a QOC test offers the advantage of providing higher levels of measured strains on the outer surface of the ring than with a cell that remains elastic. However, it presents several main drawbacks. First, each confining cell cannot be used more often than once due to the inelastic deformation after plasticization. Second, the relation between the radial stress in the sample and the outer hoop strain does not account for any influence of the loading rate. Therefore, if the constitutive material of the cell is strain-rate sensitive (such as steel) the relation identified under static loading is no longer valid to process the experimental data of a dynamic QOC test. Therefore, a confining cell made of brass or aluminium having a much smaller strain-rate sensitivity, can be considered [37]. Third, the plasticization of confining cell limits the maximum level of internal pressure to be applied to it. Last, the effect of friction and sample/ring dissymmetry are more difficult to calibrate in the data processing. It is the reason why, in the subsequent works, confining cell made of high strength steel (σ<sup>y</sup> > 1800 MPa), which behavior remains elastic during the tests, were used [16, 34, 35]. In that case, the relation between the radial stress and the outer hoop strain (for a constant height of internal pressure) is linear. In the work developed in [16], a method was proposed to estimate the defect of symmetry (δz) represented in Figure 7(a). It is based on the difference of hoop strains εθθ (z = +3H/8) and εθθ (z = �3H/8) measured near the top and bottom surfaces of the cell:

$$\delta\_{x} = P\_{x}(h) \times \left( \frac{\varepsilon\_{\theta\theta}^{\left(x = \frac{3H}{8}\right)} - \varepsilon\_{\theta\theta}^{\left(x = -\frac{3H}{8}\right)}}{\varepsilon\_{\theta\theta}^{\left(x = \frac{3H}{8}\right)} + \varepsilon\_{\theta\theta}^{\left(x = -\frac{3H}{8}\right)}} \right),\tag{13}$$

(cf. Figure 7(b)), the plot of Figure 7(c) was obtained. Finally, the ratio of radial

(a) Definition of the defect of symmetry considered in [16]. (b) Numerical simulation of an elastic cell loaded by an internal pressure of 1GPa on 40 mm (upper part: Mises stress, lower part: Hoop strain contours). (c) Radial stress to outer hoop strain ratio as function of the height of applied pressure and the defect of symmetry.

Investigation of the Quasi-Static and Dynamic Confined Strength of Concretes by Means…

DOI: http://dx.doi.org/10.5772/intechopen.89660

<sup>¼</sup> <sup>P</sup>σð Þþ <sup>h</sup> <sup>Q</sup>σð Þ� <sup>h</sup> ð Þ <sup>δ</sup><sup>z</sup> <sup>2</sup>

where Pσ(h) and Qσ(h) are polynomial functions of degree 2 whose coefficients need to be identified for the considered cell. In addition, it was demonstrated in [30] that friction and the sample-cell interface was affecting the ratio of outer axial strain to the outer hoop strain at the cell middle point (z = 0), which provides a possible way to estimate the friction coefficient in case of a confining cell that

A validation work was developed to evaluate the accuracy and sensitivity of these processing methods to be considered for inelastic or elastic confining cells. Next, it was applied to the experimental data of QOC tests conducted with different types of concretes, mortars and high-strength concrete. The main results are

, (14)

stress to outer hoop strain can be expressed in the following way:

� <sup>σ</sup>ð Þ int radial

ε ð Þ z¼0, ext θθ

remains elastic during the QOC test.

Figure 7.

77

summarized in the next two sections.

where Pz(h) is a polynomial function of degree 1 valid for the considered cell. Based on a series of numerical simulations performed with different values of h and δ<sup>z</sup>

Figure 6. Procedure applied to process the data of the QOC test [31].

Investigation of the Quasi-Static and Dynamic Confined Strength of Concretes by Means… DOI: http://dx.doi.org/10.5772/intechopen.89660

Figure 7.

where α<sup>0</sup>

Figure 6.

76

<sup>0</sup> and α<sup>18</sup>

Compressive Strength of Concrete

<sup>20</sup> are coefficients identified from the same calculations. Finally, the deviatoric stress and the hydrostatic pressure were deduced based on Eqs. (2) and (3). The volumetric strain was also calculated in the following way:

<sup>2</sup> � <sup>1</sup> (12)

(z = +3H/8) and εθθ

A, (13)

(z = �3H/8)

εvolumetricðÞ¼ t ð Þ 1 þ εaxial ð Þ 1 þ εradial

why, in the subsequent works, confining cell made of high strength steel (σ<sup>y</sup> > 1800 MPa), which behavior remains elastic during the tests, were used [16, 34, 35]. In that case, the relation between the radial stress and the outer hoop strain (for a constant height of internal pressure) is linear. In the work developed in [16], a method was proposed to estimate the defect of symmetry (δz) represented in

<sup>δ</sup><sup>z</sup> <sup>¼</sup> Pzð Þ� <sup>h</sup> <sup>ε</sup> <sup>z</sup>¼3<sup>H</sup> ð Þ<sup>8</sup>

0 @

θθ � <sup>ε</sup> <sup>z</sup>¼�3<sup>H</sup> ð Þ<sup>8</sup> θθ

1

θθ <sup>þ</sup> <sup>ε</sup> <sup>z</sup>¼�3<sup>H</sup> ð Þ<sup>8</sup> θθ

<sup>ε</sup> <sup>z</sup>¼3<sup>H</sup> ð Þ<sup>8</sup>

where Pz(h) is a polynomial function of degree 1 valid for the considered cell. Based on a series of numerical simulations performed with different values of h and δ<sup>z</sup>

Figure 7(a). It is based on the difference of hoop strains εθθ

measured near the top and bottom surfaces of the cell:

Procedure applied to process the data of the QOC test [31].

used in [6, 30–32] is summarized on the sketch of Figure 6.

so the hydrostatic behavior of the concrete (relation between the hydrostatic pressure and the volumetric strain) was obtained. The whole procedure that was

The use of a confining cell that deforms plastically during a QOC test offers the advantage of providing higher levels of measured strains on the outer surface of the ring than with a cell that remains elastic. However, it presents several main drawbacks. First, each confining cell cannot be used more often than once due to the inelastic deformation after plasticization. Second, the relation between the radial stress in the sample and the outer hoop strain does not account for any influence of the loading rate. Therefore, if the constitutive material of the cell is strain-rate sensitive (such as steel) the relation identified under static loading is no longer valid to process the experimental data of a dynamic QOC test. Therefore, a confining cell made of brass or aluminium having a much smaller strain-rate sensitivity, can be considered [37]. Third, the plasticization of confining cell limits the maximum level of internal pressure to be applied to it. Last, the effect of friction and sample/ring dissymmetry are more difficult to calibrate in the data processing. It is the reason

(a) Definition of the defect of symmetry considered in [16]. (b) Numerical simulation of an elastic cell loaded by an internal pressure of 1GPa on 40 mm (upper part: Mises stress, lower part: Hoop strain contours). (c) Radial stress to outer hoop strain ratio as function of the height of applied pressure and the defect of symmetry.

(cf. Figure 7(b)), the plot of Figure 7(c) was obtained. Finally, the ratio of radial stress to outer hoop strain can be expressed in the following way:

$$-\frac{\sigma\_{radial}^{(int)}}{\varepsilon\_{\theta\theta}^{(x=0,\,\text{ext})}} = P\_{\sigma}(h) + Q\_{\sigma}(h) \times \left(\delta\_{\mathbf{z}}\right)^{2},\tag{14}$$

where Pσ(h) and Qσ(h) are polynomial functions of degree 2 whose coefficients need to be identified for the considered cell. In addition, it was demonstrated in [30] that friction and the sample-cell interface was affecting the ratio of outer axial strain to the outer hoop strain at the cell middle point (z = 0), which provides a possible way to estimate the friction coefficient in case of a confining cell that remains elastic during the QOC test.

A validation work was developed to evaluate the accuracy and sensitivity of these processing methods to be considered for inelastic or elastic confining cells. Next, it was applied to the experimental data of QOC tests conducted with different types of concretes, mortars and high-strength concrete. The main results are summarized in the next two sections.
