Mechanics of Materials
Effect of Water Saturation on the Brazilian Tension Test of Rocks
2021 年 62 巻 1 号 p. 48-56
詳細
2021 年 62 巻 1 号 p. 48-56
Water accelerates the deformation and failure of rock and hence deteriorates the stability of rock structures on and under the ground. However, most of the previous studies examined the mechanical properties of rocks in air-dried and water-saturated conditions or the effect of water saturation on compressive strength. In this study, Brazilian tension tests were conducted with tuff, sandstone, and andesite in seven water saturation conditions between almost completely dried and water saturated. These conditions were controlled by varying the time the specimens were submerged in water and then dried. The test results showed that the Brazilian tensile strength of the tuff and the sandstone increases with a decrease of water saturation and then tends to be constant in very low water saturation conditions. In contrast, the Brazilian tensile strength of the andesite consistently increased with a decrease of water saturation. This trend for each of the rocks was also observed in the Young’s modulus estimated from the load–displacement curves in the Brazilian tension tests as well as the uniaxial compressive strength obtained in the previous studies. The applicability of the Hertz contact theory to the Brazilian tension tests with these rocks in various water saturation conditions was validated by comparing the estimated and measured Young’s modulus and by relating the Young’s modulus and the strength. The results in this study are helpful for estimating the strength and Young’s modulus of rocks in various water saturation conditions and for assessing the stability of underground structures.
It is well known that water has a significant influence on the deformation and fracture characteristics of rocks; the uniaxial compressive strength,1–7) the uniaxial tensile strength,8–11) and the Brazilian tensile strength9,12–14) are lower in wet conditions than in dry conditions. The Young’s modulus of rocks also decreases in wet conditions, and hence the stress–strain curve is smaller in wet conditions than in dry conditions under uniaxial compressive stress1,15–17) and uniaxial tensile stress.8) Time-dependent behavior such as loading rate dependence of strength and creep lifetime also changes between wet and dry conditions.18,19) The stability of rock structures such as tunnels, underground cavities, and slopes changes depending on the amount of water in the rock mass, and therefore, it is necessary to consider the rock deformation and failure in not only the above-mentioned dry and wet conditions but also various water saturation conditions.
The effect of water saturation on the deformation and fracture characteristics of rocks has been studied by many researchers. Ojo and Brook10) conducted uniaxial compression tests and uniaxial tension tests with sandstones in various water saturation conditions and reported that moisture has a greater reduction effect on the tensile strength than on the compressive strength. Hawkins and McConnell15) performed the uniaxial compression tests with sandstones and pointed out that intergranular microfracturing is more intense in the wetter samples. Lim et al.20) conducted a large number of Mode I fracture toughness tests with a synthetic rock using the semicircular specimen under three-point bending (SCB) technique and described that the fracture toughness decreases with an increase in water saturation level. Talesnick et al.13) stated that the stiffness of rocks reduces with an increase in moisture. Vásárhelyi and Ván6) estimated the sensitivity of sandstone to water content using the published data and pointed out that this sensitivity is highly dependent on the effective porosity. Yilmaz21) researched the influence of water content on the uniaxial compressive strength and elastic modulus of gypsum and described that the elastic modulus reduces about 55% from air-dried to water saturated conditions. These previous studies focused on the effect of water saturation on rock strength or some rock mechanical properties but not on various types of rock deformations. Therefore, the authors started to research the influence of water saturation on the rock deformation under compressive stress. Kataoka et al.19) and Kataoka et al.22) examined the effect of water saturation on the complete stress–strain curve and the loading rate dependence of strength, respectively, under uniaxial compressive stress. Hashiba et al.23) proposed the non-linear viscoelastic constitutive equation for representing the water- and time-dependent rock behavior under compressive stress, based on these experimental results.
Compressive stress usually acts on the surrounding rocks of tunnels and underground cavities; however, tensile stress may be applied when the tunnels and underground cavities are large in scale, complicated in shape, or internally pressurized. Tensile stress is also an important factor for the stability evaluation of rock slopes and rock foundations.24) Consequently, it is essential to understand the rock deformation and failure in various water saturation conditions under not only compressive stress but also tensile stress. In this study, the Brazilian tension tests were conducted using rock specimens in various water saturation conditions, and the effect of water saturation on the tensile strength was investigated by comparing with that on the compressive strength. In addition, the Young’s modulus was estimated from the results of the Brazilian tension tests using the Hertz contact theory, and its dependence on the water saturation was examined.
Three rocks, Tage tuff, Kimachi sandstone, and Sanjome andesite, were used in this study. Tage tuff is a dacitic-rhyolitic tuff with a porosity of 30%. This tuff contains plagioclase, quartz, and a small amount of biotite and pyroxene; their particle size is 0.3–1.5 mm. Kimachi sandstone is a tuffaceous sandstone with a porosity of 24%. This sandstone contains andesite fragment, pyroxene, amphibole, plagioclase, and a small amount of quartz, potassium feldspar, and granitic fragment; their particle size is 0.5–1.5 mm. Sanjome andesite is a tuffaceous andesite with a porosity of 15%. This andesite contains hypersthene, augite, plagioclase, and magnetite phenocryst in its groundmass; their particle size is 0.3–1.5 mm. The specimens are cylindrical, 25 mm in diameter, and 13 mm in length. 50 and 43 specimens of Tage tuff and Kimachi sandstone, respectively, were made from the same rock blocks as those used in Kataoka et al.22) Due to the limited amount of rock blocks, 30 specimens of Sanjome andesite were made from a rock block different from the one used in Kataoka et al.22) The Sanjome andesite specimens used for the Brazilian tension tests in this study and for the uniaxial compression tests in Kataoka et al.22) are called A and B, respectively.
The preparation process for water saturation conditions of specimens is shown in Fig. 1. All the specimens were air-dried in a laboratory at 23°C for more than two weeks and then dried in an oven at 105°C for two days. The weight of each specimen was measured during the oven drying process, and the minimum value for each specimen was recorded as md. Next, all the specimens were removed from the oven and put into a vacuum tank. The specimens were cooled to room temperature in vacuum dry condition for one day. Then, deionized water was injected into the vacuum tank until all the specimens were submerged. The specimens were kept in submerged condition for two days, and the weight of each water-saturated specimen was recorded as mw. Some of the specimens were subjected to underwater Brazilian tension testing; the rest of the specimens were air-dried in the laboratory. After 12 hours, seven specimens of Tage tuff and Kimachi sandstone, and three specimens of Sanjome andesite were subjected to Brazilian tension testing. The weight of each of these specimens was measured as mi just before the test, and the range of water saturation w calculated via the following equation was 0.34–0.43 for Tage tuff, 0.43–0.56 for Kimachi sandstone, and 0.29–0.30 for Sanjome andesite:
| \begin{equation} w = \frac{m_{i} - m_{d}}{m_{w} - m_{d}} \end{equation} | (1) |
Preparation process for water conditions of specimens before the Brazilian tension tests.
The remaining specimens were air-dried for more than two weeks, and then some specimens were subjected to Brazilian tension testing. The water saturation w calculated via eq. (1) was 0.081–0.10 for Tage tuff, 0.24–0.29 for Kimachi sandstone, and 0.096–0.11 for Sanjome andesite. Some of the remaining specimens were immersed in deionized water for 15 seconds and then subjected to Brazilian tension testing. The water saturation w in this condition was 0.19–0.23 for Tage tuff, 0.33–0.42 for Kimachi sandstone, and 0.16–0.18 for Sanjome andesite. The remaining specimens were oven-dried for two days one more time. Six specimens for Tage tuff, three specimens for Kimachi sandstone, and four specimens for Sanjome andesite were taken out of the oven one by one and subjected to Brazilian tension testing immediately; each test was conducted in around five minutes not to change the water saturation condition of the specimen. The water saturation w was 0.00051–0.0051 for Tage tuff, 0.0040–0.0050 for Kimachi sandstone, and 0.0012–0.0091 for Sanjome andesite, which are the lowest water saturation conditions. The remaining specimens were removed from the oven and placed in a plastic container filled with desiccant until they would be cooled to room temperature. Brazilian tension tests were conducted with some of the specimens for a water saturation w of 0.012–0.017 for Tage tuff, 0.0092–0.029 for Kimachi sandstone, and 0.010–0.016 for Sanjome andesite. The last few specimens were dried in a vacuum tank for one day, covered with plastic wrap and stored in a box filled with desiccant, and then subjected to Brazilian tension testing. The water saturation w just before the tests was 0.041–0.066 for Tage tuff, 0.15–0.20 for Kimachi sandstone, and 0.050–0.059 for Sanjome andesite. The test conditions and the number of specimens are shown in Table 1.
The Brazilian tension tests were conducted using a 10 kN servo-controlled testing machine. The load applied to a specimen in the diametrical direction was measured with a strain-gage-type load cell, and the displacement of the loading platen was measured with a linear variable differential transformer (LVDT). The displacement rate was controlled 0.001 mm/s during the test.
The solid lines in Figs. 2, 3, and 4 are the load–displacement curves obtained from the Brazilian tension tests with the three rocks in the seven water saturation conditions. The two load–displacement curves with the largest and the smallest slope in each of the water saturation conditions were adopted in these figures, because the scatter in each condition can be easily observed. Figs. (a) and (b) show the results in the four lower water saturation conditions and the three higher water saturation conditions, respectively. The load–displacement curves of Tage tuff in both Fig. 2(a) and Fig. 2(b) are slightly concave upward at the beginning of loading and then almost straight. Just before reaching the peak points, the curves are concave downward. The maximum load at the peak points decreases about 70% when w changes from 0.0051 to 1.0. The slope at the straight part of the curves has a reduction trend with an increase of w. The displacement at the peak points is in the range of 0.12 to 0.20 mm, and after the peak points, the load decreases sharply and then gradually in all the water saturation conditions. As shown in the photographs after the tests, each of the specimens is separated into two parts by a failure plane. Some specimens showed inclined failure planes or wedge-shaped fractures near the contact portion with the loading platens; however, differences of the failure pattern depending on the water saturation conditions were not obvious. The load–displacement curves of Kimachi sandstone shown in Fig. 3 have a concave upward part at the beginning of loading, a straight part, and a concave downward part just before the peak points, in a similar manner to those of Tage tuff. The maximum load decreases about 60% when w changes from 0.0040 to 1.0. The slope at the straight part of the curves has a reduction trend with an increase of w. The displacement at the peak points is in the range of 0.08 to 0.16 mm, and just after the peak points, the load decreases abruptly and then gradually in all the water saturation conditions. As shown in the photographs after the tests, each of the specimens is separated into two parts by a failure plane. Some specimens showed arc-shaped failure planes or wedge-shaped debris near the contact portion with the loading platens; however, differences of the failure pattern depending on the water saturation conditions were not obvious in a similar manner to Tage tuff.
Load–displacement curves obtained from the Brazilian tension tests with Tage tuff. The solid lines are the experimental results. The dashed lines were calculated by eqs. (5)–(7). The photographs show the failure pattern of the specimens after the tests. (a) Curves for the low water saturation conditions. (b) Curves for the high water saturation conditions.
Load–displacement curves obtained from the Brazilian tension tests with Kimachi sandstone. The solid lines are the experimental results. The dashed lines were calculated by eqs. (5)–(7). The photographs show the failure pattern of the specimens after the tests. (a) Curves for the low water saturation conditions. (b) Curves for the high water saturation conditions.
Load–displacement curves obtained from the Brazilian tension tests with Sanjome andesite. The solid lines are the experimental results. The dashed lines were calculated by eqs. (5)–(7). The photographs show the failure pattern of the specimens after the tests. (a) Curves for the low water saturation conditions. (b) Curves for the high water saturation conditions.
The load–displacement curves of Sanjome andesite shown in Fig. 4 are slightly concave upward at the beginning of loading and almost straight until the peak points. The maximum load decreases about 55% when w changes from 0.0012 to 1.0. The curves in the higher water saturation conditions are located inside the curves in the lower water saturation conditions, which indicates that both the slope of the curves before the peak points and the displacement at the peak points have a significant reduction with an increase of w. The load decreases sharply just after the peak points in all the water saturation conditions. The photographs after the tests show that each of the specimens from the lower oven-dried to water-saturated conditions is separated into multiple parts by a failure plane and the wedge-shaped fractures near the contact portion with the loading platens.
The Brazilian tensile strength σt of each specimen was calculated via the following formula:
| \begin{equation} \sigma_{\text{t}} = \frac{2F_{\text{m}}}{\pi DL} \end{equation} | (2) |
| \begin{equation} \sigma_{t} = -k\log(w) + \sigma_{w}\quad (w_{\text{d}}\leq w \leq 1.0) \end{equation} | (3) |
| \begin{equation} \sigma_{t} = \sigma_{d}\quad (w<w_{\text{d}}) \end{equation} | (4) |
Figure 6 shows the relation between the normalized Brazilian tensile strength (σt/σw) and the logarithm of water saturation w for the three rocks. The results show linear relations in the range of wd ≤ w ≤ 1.0 for three rocks. The tendency for Tage tuff and Kimachi sandstone in the range of wd ≤ w ≤ 1.0 is almost similar. And the increase of tensile strength is the largest for Tage tuff. The results in Fig. 6 is consistent with the numerical relation in Table 2.
Figure 7 shows a schematic illustration of a Brazilian tension test; the disk with a diameter of D and a length of L is diametrically loaded between the two planes. Assuming that both the disk and the planes are elastic bodies, the elastic approach (displacement) Δd between the planes is calculated by the Lundberg formula in the Hertz contact theory25) as follows:
| \begin{equation} \Delta d = \frac{4F}{\pi LE'}\left\{1.8864 + \ln \left(\frac{L}{2b}\right)\right\} \end{equation} | (5) |
| \begin{equation} b = \sqrt{\frac{4FD}{\pi LE'}} \end{equation} | (6) |
| \begin{equation} 1/E' = \frac{(1 - v_{1}^{2})/E_{1} + (1 - v_{2}^{2})/E_{2}}{2} \end{equation} | (7) |
Schematic illustration of Brazilian tension test.
In this study, the disk corresponds to the rock specimen, and the planes correspond to the loading platens of the servo-controlled testing machine. To estimate the Young’s modulus E1 of each of the specimens, the Poisson’s ratio v1 was assumed to be a constant value of 0.25 because this value rarely affects the calculated result of eq. (5). The loading platens are made of steel, and hence E2 and v2 were set to be 200 GPa and 0.3, respectively. The relation between the load and displacement was calculated by eq. (5) with proper values of E1 and shown with dashed lines in Figs. 2, 3, and 4. To eliminate the effect of the initial irregular contact between the loading platens and the surface asperity of the rock specimen, the solid and dashed lines were horizontally shifted so that the dashed line passes through the origin. These figures show that the calculated dashed lines are successfully consistent with the experimental load–displacement curves before the peak points. In contrast, a little difference was observed between the dashed and solid lines at the peak points because the experimental curves are concave downward just before the peak points. These differences are the smallest for Sanjome andesite and the largest for Tage tuff. The reason is probably that the displacement at the concave downward part includes the inelastic deformation under compressive stress near the contact portion with the platens and that the compressive failure of Tage tuff is more ductile than that of Sanjome andesite. Hertz’s contact theory assumes elastic bodies. Therefore, in the Brazilian tension test, it is considered that the rock specimen is elastically deformed with small compressive failure near the contact portion in the pre-failure region and cracks rapidly grow near the peak point.
The Young’s modulus estimated from eqs. (5)–(7) is summarized in Table 1, and its relation with the water saturation w is shown with open circles in Fig. 8. The results of Tage tuff in Fig. 8(a) show large scatters but indicate that the Young’s modulus increases from 1.5 to 3.8 GPa with the decrease of w from 1.0 to 0.041 and almost constant for a water saturation less than 0.041. A similar trend is observed for Kimachi sandstone in Fig. 8(b); the Young’s modulus increases from 3.3 to 6.5 GPa with the decrease of w from 1.0 to 0.12 and is almost constant for a water saturation less than 0.12. In contrast, the results of Sanjome andesite in Fig. 8(c) show that the Young’s modulus increases from 9.7 to 19.8 GPa with the decrease of water saturation from 1.0 to 0.0012. These trends of the Young’s modulus accord with the relation between the Brazilian tensile strength and the water saturation of the three rocks shown in Fig. 5.
Relation between Young’s modulus and water saturation. The open circles indicate the Young’s modulus estimated from the Brazilian tension tests in this study. The filled circles indicate the Young’s modulus obtained from the uniaxial compression tests in Kataoka et al.22) (a) Tage tuff. (b) Kimachi sandstone. (c) Sanjome andesite.
Kataoka et al.22) performed the uniaxial compression tests with Tage tuff, Kimachi sandstone, and Sanjome andesite. The water saturation conditions for the cylindrical specimens 25 mm in diameter and 50 mm in height were controlled by varying the time they were submerged in water and then dried, in a similar way to that explained in Fig. 1. However, the specimens for the uniaxial compression tests are larger than those for the Brazilian tension tests, and hence the air-drying time and the water-immersing time were set to 24 hours and 30 seconds, respectively, to realize the desired water saturation. Because of the limitation of the number of specimens, the uniaxial compression tests were not conducted in higher oven-dried conditions. The uniaxial compression tests were conducted using a 500 kN servo-controlled testing machine. The load was measured with a strain-gauge-type load cell, and the displacement was measured with an LVDT. The strain rate was alternated between 10−5/s and 10−4/s during the test, and the uniaxial compressive strengths corresponding to the two strain rates were determined from the inner and outer envelope curves of the stress–strain curve. Kataoka et al.22) discussed the relation between the water saturation and the average strength corresponding to the strain rate of 10−5/s in each of the water saturation conditions, whereas all of the strengths corresponding to the strain rate of 10−5/s were used for the comparison with the Brazilian tensile strength obtained in this study, as shown in Fig. 9.
Although the data of uniaxial compression test is not as much as that of Brazilian tension test, the similar trend can be investigated. The uniaxial compressive strength of Tage tuff increases linearly from 7.7 to 35.2 MPa when the water saturation w changes from 1.0 to 0.02. Then the strength tends to be constant as w continues to decrease. A similar result is observed for Kimachi sandstone; the strength increases from 18.7 to 56.6 MPa with a linear relation to the water saturation, when w decreases from 1.0 to 0.06, and then is constant. The result for Sanjome andesite shows that the strength increases from 68.8 to 99.9 MPa with the increase of w from 1.0 to 0.0003. The relation between the water saturation and the uniaxial compressive strength of the three rocks is almost consistent with that between the water saturation and the Brazilian tensile strength shown in Fig. 5 and approximated by eqs. (3) and (4). σt in these formulae was replaced by the uniaxial compressive strength σc, and the values of the constants in these equations were determined as shown in Table 2. Since the rock blocks of Tage tuff and Kimachi sandstone in this study are the same as those used in Kataoka et al.22) the ratios of the values obtained from σc and σt are shown in this table. These ratios for k, σw, and σd are almost the same for Tage tuff and Kimachi sandstone, which indicates that the brittleness index represented by σc/σt is around 10 and independent of w. In situ rock masses are actually in between water-saturated and air-dried conditions. In these conditions, the three rocks showed that the strength decreases with an increase of water saturation, probably due to the stress corrosion and the strength degradation of clay minerals.22) In vacuum-dried and oven-dried conditions, Tage tuff and Kimachi sandstone showed constant strength in not only Brazilian tension tests but also uniaxial compression tests, which indicates the inherent strength of these rocks. Sanjome andesite showed no constant strength in these conditions in a similar manner to Inada granite in the previous study.22) This is probably because the porosity of Sanjome andesite and Inada granite are lower than Tage tuff and Kimachi sandstone and hence the strength of the andesite and the granite was affected by small amount of water remaining in such very low water saturation conditions.
The Young’s modulus obtained from the 50% tangent of stress–strain curves in uniaxial compression tests of Kataoka et al.22) are shown with filled circles in Fig. 8. For Kimachi sandstone, the Young’s modulus estimated from the Brazilian tension tests shows a large scatter, but the values are consistent with those obtained from the uniaxial compression tests. This result indicates that the Hertz contact theory is suitable for explaining the elastic deformation before the peak point for Kimachi sandstone. For Tage tuff, although the Young’s modulus estimated from the Brazilian tension tests and that obtained from the uniaxial compression tests are slightly different, the trends are similar to each other. It is considered that the compressive failure occurred at the contact portion between the specimen and the loading platens in the Brazilian tension test. Therefore, the estimated Young’s modulus has a slight decrease due to this deformation. The rock blocks of Sanjome andesite used in this study (A) and in the previous study (B) are different; the uniaxial compressive strength of Sanjome andesite A and B is 94.6 (Ogawa26)) and 79.3 MPa (Kataoka et al.22)), respectively, in air-dried conditions. Hence, the Young’s modulus estimated from the Brazilian tension tests is compared with the data reported in other previous papers.27,28) Okubo and Nishimatsu27) conducted uniaxial compression tests for the condition of air-dried for more than two weeks and reported that the uniaxial compressive strength and Young’s modulus of their Sanjome andesite blocks are 95.2 MPa and 10.8 GPa, respectively, which are consistent with the uniaxial compressive strength (94.6 MPa) and Young’s modulus (11.2 GPa) of the Sanjome andesite A in air-dried conditions reported in Ogawa.26) Yamaguchi et al.28) conducted uniaxial compression tests for the condition of oven-dried for two days and vacuum-dried for over 30 days and reported that the uniaxial compressive strength and Young’s modulus of their Sanjome andesite blocks are 104 MPa and 16.3 GPa, respectively. It was found that the Young’s modulus values reported in these previous studies fall within the range of the results estimated from the Brazilian tension tests in Fig. 8(c), which indicates that the Hertz contact theory is applicable to the elastic deformation before the peak point in the Brazilian tension tests with Sanjome andesite.
The relation between the Brazilian tensile strength and estimated Young’s modulus is shown in Fig. 10. The results of the three rocks in various water saturation conditions are located near the straight line. Chang et al.29) reported a correlation between uniaxial compressive strength and Young’s modulus of many rocks. Rybacki et al.30) also reported that the compressive strength of shales correlates almost linearly with Young’s modulus. However, these previous researchers did not consider the effect of as much water saturation as in this study on the relation of rock strength and Young’s modulus. In previous studies, estimation methods of elastic properties, such as Young’s modulus and Poisson’s ratio, were proposed using the Brazilian tension test.31–33) The estimation method of Young’s modulus in various water saturation conditions in this study requires small amount of rock samples and no additional devices such as strain gauges and hence is easier than the previous methods. However, further discussion on the Poisson’s ratio is needed in a future study.
Three rocks, Tage tuff, Kimachi sandstone, and Sanjome andesite, were subjected to Brazilian tension testing in various water saturation conditions. The seven types of water saturation conditions were controlled by varying the time the specimens were submerged in water and then dried. The load–displacement curves obtained from the tests with the three rocks showed that both the maximum load and the slope of the curves before the peak points have a reduction trend with an increase of water saturation. The displacement at the peak points fell within similar ranges for all the water saturation conditions for Tage tuff and Kimachi sandstone, whereas the curves for the higher water saturation conditions were located inside the curves for the lower water saturation conditions for Sanjome andesite because the displacement decreased with an increase of water saturation. The relation between the Brazilian tensile strength and the logarithm of water saturation was represented by a single line for Sanjome andesite and by a bilinear line for Tage tuff and Kimachi sandstone, which indicates that the strength of the tuff and the sandstone tends to be constant for very low water saturation conditions. This relation between the strength and water saturation was also observed in uniaxial compression tests conducted in the previous study.
The Young’s modulus of each specimen was estimated from the load–displacement curve using the Hertz contact theory and found to coincide with the Young’s modulus obtained from uniaxial compression tests in the previous studies. The relation between the Young’s modulus and the water saturation for each of the three rocks was similar to the relation between the strength and the water saturation. In addition, the Brazilian tensile strength and the estimated Young’s modulus showed a linear relation, and the relation was independent of the rock types in this study. These results indicate that the Hertz contact theory is applicable to the Brazilian tension tests with these rocks for various water saturation conditions.
The test method used in this study is useful for investigating the strength and Young’s modulus of rocks for various water saturation conditions, and the results in this study are helpful for assessing the stability of underground structures. The research issue for the future is to obtain more data on the effect of water saturation on the rock deformation and failure under various stress states, such as uniaxial tension, shear, and triaxial compression.
The authors are grateful to Mr. Masato Ogawa of the University of Tokyo for his assistance in conducting the tests.
