An experimental study of tensile stress and deformation in an anisotropic rock

Anisotropy is a very common condition in rock mass; it can be due to different factors and directly affects the failure mechanisms affecting the rock mass. For example, metamorphic rocks that are foliated, sedimentary rocks that are stratified or volcanic formations with alternating layers. Despite the existence of several studies related to anisotropy those specifically addressing the tensile strength of anisotropic rocks are quite limited. The present study is focused on the determination of the deformability, compressive and tensile strength of anisotropic rocks. It must be highlighted that the mechanical behavior of the anisotropic rock mass is dependent on the angle between the inclination of planes of weakness (e.g. foliation) and the direction of the load. Assuming a vertical load, the two extremes of tensile strength are 0°(horizontal) and 90° (vertical). A series of laboratory tests has been done in anisotropic sandstone (lithic arkose), from Burgos (Spain), including uniaxial compressive strength tests, direct tensile strength tests, and diametric compression (Brazilian tests). The tests were carried out with strain gauges that allowed estimating the elastic modulus. To determine the anisotropic direction, ultrasonic pulse wave velocity tests were also performed. The variation of strength and deformability as a function of anisotropy is analyzed, as well as the variation of elastic behavior in tensile and compressive.


Introduction
The most common anisotropic rocks are usually transversely isotropic, due to their distinct layered structure that originates from their special deposit sediments and the sedimentary environment (Ma et al., 2018).
There are many studies about the variation of compressive strength as a function of anisotropy.However, there is a lot less about the tensile behavior in a transversely isotropic rock.In addition, tensile strength studies, due to simple sample preparation and cost, typically do not perform Direct Tensile Strength Tests (DTS), only the Brazilian test (BT) which is an indirect tensile test.2018) present a review of the works about tensile strength based on the Brazilian test.They highlight that depending on the inclination of the discontinuity different mechanic can occurs, and not necessary the failure begins in the center of the disc, condition required to considerate the Brazilian test valid.However, in both extremes (0º and 90º), the Brazilian test could provide valid results.Liao et al. (1997) performed one of the few studies about the variation of the tensile strength as a function of the foliated planes with direct tensile test.They concluded that the stress-strain behavior of argillite is anisotropic; therefore, the magnitude of the elastic constant depends on the inclination angle.Also observed was that the tensile strength varied in more than 10 times if the foliation angle was parallel or perpendicular to the load.

Ma et al. (
Using also DTS test results, Liu et al. (2022) proposed a new phenomenological anisotropic tensile strength criterion, based on Kangding slate results test.In this case, the variation of the tensile strength can double if the discontinuities are parallel or perpendicular to the load.
In the present study the tensile strength of an anisotropic sandstone is test, the variation of strength and deformability as a function of anisotropy is analyzed, as well as the variation of elastic behavior in tensile and compressive conditions.

Methodology
A series of laboratory tests has been done on anisotropic sandstone (lithic arkose), from Burgos, Spain.Being remarkable that the mechanical behavior of the anisotropic rock mass is dependent on the angle between the inclination of planes of weakness (e.g.foliation) and the direction of the load.Assuming a vertical load, the two extremes of tensile strength are 0°(perpendicular) and 90° (parallel) (Figure 1).Based on the anisotropy classification according to ultrasonic wave velocity, VA, (Tsidzi, 1997) presented in terms of Eq. ( 1), the rock is considerate highly anisotropic, showing a VA = 21%.Table 1 presents the values of P-wave obtained in the three directions.An important detail in tensile tests on anisotropic rocks is that the loading direction on the Brazilian tests is not the same as in DTS test.In the first one, the load is applied diametrically, and in the other axially.With which, the samples cut for the same cylinder are tested in the opposite directions (Figure 2).The BT is carried out in two ways, with force control, and with displacement control, the results are presented in two different graphics.In Table 2 the number of test carried out per type is shown.BT -Displacement control 90 9

Tensile test
With β = 0º, σt = 0.45 MPa, while for β = 90º, σt = 0.80 MPa, which demonstrates the clear dependence of the tensile strength as a function of the inclination of the foliation planes.(Figure 3) Figure 4 shows the results obtained with the Brazilian test.It is observed that despite the value of the tensile strength is overestimated based on the results of DTS test, they also vary depending on β.It is also observed that with load control (Figure 4a), the dispersion ratio is lower, but the average result is somewhat higher than with displacement control (Figure 4b).In both case of BST, in all tests the failure occurs in the center of the sample (parallel to the application of the load).While in the case of DTT the failure is perpendicular to load application (Figure 5).

Compressive test
As expected, the tensile strength with β = 0º is little higher than β = 90º (Figure 6).Duveau and Shao (1998) have studied how this variation changes as a function of the confinement pressure, and have observed that the greater the confinement, the smaller the difference between both values.It is highlighted that this study adopts β = 0º when the load is applied perpendicular to the foliation planes.Some authors, mainly focused on compressive strength, used β = 0º when the load is parallel to the foliation planes.

Elastic modulus
Concerning the tensile and compression elastic modulus obtained, as expected, Ec is greater than Et (Figure 7).During the tests, it was observed that the strain paths obtained by the strain gauges were much more stable in the compression test than in the tensile tests.
The Et ratio with respect to β is higher than observed in Ec, it occurs because β = 0º and β = 90º are the extreme tensile strengths, while in the case of UCS, the lowest value is found with the β ≈ 45º.

Conclusions
This study presents preliminary results of the variation of tensile strength and deformability in an transversely isotropic sandstone.Based on the results it can be concluded that:  The highest tensile strength is obtained with the discontinuities parallel to the load (β = 90º), while the maximum value of the UCS is obtained with the discontinuities perpendicular to the load (β = 0º). The value of the estimated tensile strength with the BT (by both methods) is higher than that obtained by the DTT.Using displacement control the results are lower than when applying load control. It is highlighted that the elasticity modulus varies in compression and tension, being higher in compression.Also, it is dependent on the orientation of the foliation.As future work, this study can be extended by performing laboratory tests considering different foliation angles and other rock types.It is developed in order to improve the knowledge of the mechanical behavior of anisotropic rocks.

Figure 2 .
Figure 2. (a) Scheme of load direction in DTT and BST.(b) Drilled block.

Figure 6 .
Figure 6.Uniaxial compressive strength as a function of β.

Table 1 :
P-wave velocity in relation to the direction.To estimate the tensile strength the, direct tensile strength test (DTS test) and the Brazilian test (BT) are performed.Most of the tests are performed with samples of 49 mm in diameter.The DTS tests were performed on samples with a length to diameter ratio of 2.5 based on Muñiz-Menéndez and Pérez-Rey (2023) recommendation.

Table 2 :
Summary of laboratory test carried out.