Loading-Rate Dependency of Young’s Modulus for Class I and Class II Rocks

Understanding the time-dependent behavior of rocks is important for ensuring the long-term stability of underground structures. Aspects of such a time-dependent behavior include the loading-rate dependency of Young’s modulus, strength, creep, and relaxation. In particular, the loading-rate dependency of Young’s modulus of rocks has not been fully clariﬁed. In this study, four diﬀerent types of rocks were tested, and the results were used to analyze the loading-rate dependency of Young’s modulus and explain the underlying mechanism. For all four rocks, Young’s modulus increased linearly with a tenfold increase in the loading rate. The rocks showed the same loading-rate dependency of Young’s modulus. A variable-compliance constitutive equation was proposed for the loading-rate dependency of Young’s modulus, and the calculated results agreed well with measured values. Irrecoverable and recoverable strains were separated by loading-unloading-reloading tests at preset stress levels. The constitutive equations showed that the rate of increase in Young’s modulus increased with the irrecoverable strain and decreased with increasing stress. The increase in the irrecoverable strain was delayed at high loading rates, which was concluded to be the main reason for the increase in Young’s modulus with an increasing loading rate.

e loading-rate dependency of the strength has been comprehensively studied through uniaxial compression and indirect tensile tests [9] and shear and triaxial compression tests [10]. In particular, a 15-year creep test was performed on Tage tuff at a stress level of 30% [11]. ese studies showed that understanding the time-dependent behavior of rocks is important for ensuring the long-term stability estimating the lifetime of underground structures [12]. e basic characteristics and theory of relaxation have been used to explain strain hardening and softening [13]. Some scholars have suggested that creep and relaxation are not fully independent; Fukui et al. [14] analyzed the coupling characteristics of creep and relaxation to develop the concept of generalized relaxation.
However, the loading-rate dependency of Young's modulus of rocks has not been fully clarified. In a previous study, the data scatter for the loading-rate dependency of Young's modulus was reduced by normalizing Young's modulus at 50% of the peak strength against Young's modulus at 10% of the peak strength, which was first obtained at a predetermined loading rate. For Sanjome andesite, Young's modulus was shown to increase by 2% with a tenfold increase in the loading rate; meanwhile, Young's moduli of marble, granite, and sandstone decreased slightly with an increasing loading rate [15]. Only a few studies have explained the microcrack evolution, associated deformation, and strength properties of rocks with various strain rates [16]. Xu and Dai [17] showed that Young's and shear moduli exhibited some loading-path dependency under quasistatic loading but were insensitive to the loading path at high loading rates. Both the strength and Young's modulus of the rock-like specimens strongly depended on the strain rate. Yan et al. [18] numerically studied the rate-dependent cracking behavior of single-flawed rock specimens and found that the initiation angle and fracture process zone are significantly affected by the strain rate.
In this study, four rocks were tested under uniaxial compression, direct tension, and indirect tension to evaluate Young's moduli at different loading rates. e values of Young's moduli from loading-unloading tests in literature [15] were statistically analyzed, and the irrecoverable and recoverable strains were separated [19]. e results were used to develop constitutive equations that represent the loading-rate dependency of Young's modulus and clarify the underlying mechanism. [20] divided rocks into two classes based on the complete stressstrain curve in the postfailure region, as shown in Figure 1. Class I rocks always have a negative slope in the postfailure region, whereas Class II rocks always have a positive slope in the postfailure region.

Specimen Description. Wawersik and Fairhurst
Four types of rocks were tested: Tage tuff (Class I), Ogino tuff (Class I), Emochi andesite (Class II), and Jingkou sandstone (Class II). Tage tuff was obtained from Tochigi Prefecture in Japan and mainly comprised albite, feldspar, and small amounts of calcite. Ogino tuff was obtained from Fukushima Prefecture in Japan and mainly comprised zeolite, mullite, clay, and small amounts of plagioclase and biotite. Emochi andesite was obtained from Fukushima Prefecture in Japan and mainly comprised plagioclase, pyroxene, and a small amount of biotite. Jingkou sandstone was obtained from Chongqing in China and mainly comprised quartz, feldspar, vermiculite, and muscovite. Table 1 lists the basic physical and mechanical parameters of the rocks. Tage tuff and Ogino tuff had a higher saturated water content than Emochi andesite and Jingkou sandstone but a lower peak strength. e specimens were obtained by drilling blocks of rock that were then cut into cylinders with diameters of 25 mm and heights of 50 mm for the uniaxial compressive and uniaxial direct tensile tests. e specimens for the indirect tensile tests (i.e., Brazilian splitting tests) had a diameter of 25 mm and a height of 12.5 mm. To ensure accurate data, the specimens were polished by a 55-C0201/C polishing machine (Controls Group, Milan, Italy) according to International Society for Rock Mechanics and Rock Engineering (ISRM) standards to ensure flatness, verticality, and parallelism. e surfaces of the polished specimens had a flatness of ≤0.01 mm. More than eight specimens of each rock type were employed for each testing condition. e scatter in the test data was addressed by calculating the average value from the eight specimens.

Control Method.
e strain rate-controlled method was used to test Class I rocks (i.e., Tage tuff and Ogino tuff), as shown in Figure 1. e stress-feedback method [21] was used to test the Class II rocks (i.e., Emochi andesite and Jingkou sandstone). e control formula is given by where t is the time, f (t) is a function of time, C is the loading rate, E is Young's modulus of the rock, ε and σ are the strain and stress, respectively, and α is the ratio of the stress feedback.
e Class II rocks were tested at α � 0.3 with variable-resistance technology, in which a loading cell and linear variable differential transformer were used to collect stress and strain signals in the servo-controller. e values of the strain and stress were controlled by the variable resistances VR 1 and VR 2 , respectively. e ratio between VR 1 and VR 2 is represented by β. α is the ratio of the stress feedback and is expressed as α � k•β, where k is a constant. e stress signal value is α•β/E, and the strain signal value is ε. A constant linear combination of the stress and strain (i.e., ε − α•β/E) was used in the Add/Sub amplifier as the feedback signal in a closed loop.
is allows the stress-feedback method to obtain a precise value for Young's modulus.
All stress-strain curves were obtained at loading rates of C � 1 × 10 −6 , 1 × 10 −5 , 1 × 10 −4 , and 1 × 10 −3 s −1 for uniaxial compression; 1 × 10 −5 , 1 × 10 −6 , and 1 × 10 −7 s −1 for direct tension; and 5 × 10 −6 , 5 × 10 −5 , 5 × 10 −4 , and 5 × 10 −3 mm/s for indirect tension. For the uniaxial direct tensile tests, the specimens were prepared as follows: the top end of the specimen was liberally coated with epoxy resin, and the specimen was affixed to the surface of the upper plate and allowed to set for 24 h. e bolt portion of the plate was then screwed into the ram head, and the bottom end of the specimen was affixed to a similar lower plate. en, a slight load was placed on the specimen for 24 h prior to the tensile test.

Modulus Calculation Method.
For uniaxial compressive tests, the tangent modulus E was used to represent Young's modulus. For the uniaxial direct tensile tests, the initial modulus E DT was used to represent Young's modulus; this is the slope of the line passing through the origin point and tangent to the prefailure curve. For the uniaxial indirect tensile tests (Brazilian splitting tests), the splitting modulus E TT was obtained as follows: where E TT is the splitting modulus, F is the axial force, S ABCD is the area of the meridional plane, D v is the displacement, and D d is the specimen diameter. Figure 2 shows the meridional plane of the indirect tensile test specimens. Figure 3 shows the stress-strain curves of the Class I rocks under uniaxial compression, and Figure 4 shows the corresponding stress-strain curves of the Class II rocks. e Class I and Class II rocks showed the same loading-rate 2 Shock and Vibration dependency. Figure 5 shows the stress-strain curves and stress-irrecoverable strain curves of Tage tuff (Class I) under uniaxial direct tension. e irrecoverable strain was separated from the slope of the unloading curve. ese curves showed a clear loading-rate dependency. Figure 6 shows the force-displacement curves of Tage tuff (Class I) and Jingkou sandstone (Class II) under indirect tension. e load sharply dropped at the peak, but a loading-rate dependency can clearly be observed. e stress-strain curves of the four rocks at different loading rates were used to analyze further the loading-rate dependency of Young's modulus.

Loading-Rate Dependency of Young's Modulus under
Uniaxial Compression. Figure 7  ese results show that Young's modulus increased linearly with each tenfold increase in the loading rate for both Class I and Class II rocks, and the same loading-rate   dependency was observed for Young's modulus of all rocks. However, Class II rocks showed greater increases in Young's modulus than Class I rocks. is may be related to rock strength, rock gaps, and weak internal surfaces.

Loading-Rate Dependency of Young's Modulus under
Uniaxial Tension. Figure 9 shows the loading-rate dependency of Young's modulus for Tage tuff (Class I) in direct tension. As noted in Section 2.3, the initial modulus E DT was taken as Young's modulus.
e relationship between Young's modulus and loading rate is nearly linear on the double-logarithmic plot with a correlation coefficient of 0.9959. Figure 9 also shows the loading-rate dependency of Young's modulus in indirect tension, which is represented by the splitting modulus E TT from equation (2). e splitting modulus and loading rate showed a linear relationship in the double-logarithmic plot with a correlation coefficient of 0.9309. us, Young's modulus of Class I rocks under uniaxial direct and indirect tension increased linearly with a tenfold increase in the loading rate. For Tage tuff, the ratio K 1 between Young's modulus and initial modulus was 1.086, and the ratio K 2 between Young's modulus and splitting modulus was 7.983. e ratio K can be used to quantitatively analyze the loading-rate dependency of Young's modulus under different loading conditions and is important for predicting the engineering properties of rock masses.

4.1.
Variable-Compliance Constitutive Equation. Figure 10 shows the nonlinear Maxwell model that was used to simulate the loading-rate dependency of Young's modulus. e following constitutive equations were proposed: where λ is the variable compliance. Its initial value is the reciprocal of the spring modulus with the parameter ranges of α 1 > 0, α 3 > 0, +∞ > m 1 > −∞, +∞ > m 3 > −∞, n 1 ≥ 1, and n 3 ≥ 1. When n 1 � n 3 � 1, the model reflects the behavior of materials with Newtonian viscosity. When n 1 ≥ 1 and n 3 ≥ 1, the model is accurate for materials with ordinary viscosity. In addition, t is the time; ε and σ are the strain and stress, respectively; and ε 1 and ε 3 are the irrecoverable and recoverable strains, respectively. Equation (7) is obtained from equations (3)-(6) to give Young's modulus at a preset stress level: A ′ � a 1/m 1 +1 1 where C is the loading rate, A′ and n′ are constants, and σ psl is the preset stress level.

Calculated Parameters.
In previous studies [10,22], a method was introduced for calculating the parameters α 1 , α 3 , m 1 , m 3 , n 1 , and n 3 , where the value of n 3 is obtained in tests at a constant or alternating loading rate. en, n 1 � n 3 was assumed for numerical simulations, and the loading-rate dependency of the strength for porous rocks and the generalized relaxation behavior of Kawazu tuff were simulated. Here, a new method is proposed for calculating the value of n 1 by using equation (7) to   Figure 11 and Tables 3-6. e calculated values agreed with the measured results. For Tage tuff (dry condition),   Table 4 indicates that the measured values of ∆E/E at stress levels of 30%, 50%, and 70% were 0.0295, 0.0371, and 0.0473, respectively, under uniaxial compression. e calculated results were 0.0375, 0.0135, and 0.0355, respectively. For Emochi andesite, Table 5 indicates that the measured ∆E/E values were 0.0335, 0.0347, and 0.0321, respectively, at the same stress levels. e calculated values were 0.0207, 0.0204, and 0.0201, respectively. In literature [15], the experimental values of ∆E/E for Sanjome andesite were 0.0190, 0.0370, and 0.0210 under dry conditions and 0.0250 under wet conditions. us, Young's modulus always increased with a tenfold increase in the loading rate regardless of the dry or wet conditions. For Jingkou sandstone, the measured ∆E/E values were 0.0192, 0.0180, and 0.0182 at stress levels of 30%, 50%, and 70%, respectively, and the calculated values were 0.0206, 0.0087, and 0.0206, respectively. e experimental value of ∆E/E for Kiwai sandstone was 0.0170 under dry conditions, which indicates that Young's modulus increased for each tenfold increase in the loading rate. However, Shirahama sandstone had a value of −0.7 under dry conditions, which indicates that Young's modulus decreased with an increasing loading rate.
Based on the results in this study and literature [15], Young's modulus generally increases with the loading rate for both Class I and Class II rocks. However, Young's modulus of Shirahama sandstone slightly decreased with an increasing loading rate, and the loading-rate dependency of Young's modulus was more obvious under wet conditions than under dry conditions. Additional test data on Young's modulus will be obtained in future research.

Discussion
In this study, loading-unloading-reloading tests were performed at preset stress levels of 30%, 50%, and 70% of the peak strength to separate the irrecoverable strain under uniaxial compression. Figure 12 shows the average values of the unloading modulus K u at stress levels of 30%, 50%, and 70%. e unloading modulus values for Tage tuff, Ogino tuff, Emochi andesite, and Jingkou sandstone were 5.55, 7.8, 12, and 11.6 GPa, respectively. For direct tension, the initial modulus was used to separate the irrecoverable strain. Tables 2-5 present the measured values of the irrecoverable strain. e correlation coefficient of Young's modulus with strength is given by (∆E/E)/(∆σ/σ). Figure 13 shows the relation between (∆E/E)/(∆σ/σ) and ε 1 /ε from Tables 3-6, which is approximated by equation (10) for rocks under uniaxial compression and tension: Equation (7) can be used to obtain e constitutive equations and test results showed that ∆E/E is related to the irrecoverable strain ε 1 and preset stress level σ psl . e rate of increase in Young's modulus increases with the irrecoverable strain, and it decreases with increasing stress.
e irrecoverable strain showed a delay at high loading rates, which may explain the increase in Young's modulus with an increased loading rate.

Conclusions
e loading-rate dependency of Young's modulus for Class I and Class II rocks was systematically studied through laboratory tests, and a variable-compliance constitutive equation was developed. e main conclusions are as follows: (1) Precise test data for Young's moduli of four rock types were obtained under uniaxial compression by improving the precision of the specimen dimensions. e initial modulus and split modulus of Tage tuff under direct and indirect tension were also obtained. Regardless of the test method, Young's modulus showed the same law of loading-rate dependency.
(2) Irrecoverable and recoverable strains were separated by loading-unloading-reloading tests at preset stress levels, where the slope of the unloading secant curve was used as the elastic constant. (3) e variable-compliance constitutive equation was used to simulate the loading-rate dependency of Young's modulus. e calculated results showed good agreement with the measured values. (4) e constitutive equations and test data showed that the rate of increase in Young's modulus increased with the irrecoverable strain and decreased with increasing stress. e irrecoverable strain showed a delayed increase at high loading rates, which may be the main reason for the increase in Young's modulus with an increasing loading rate.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.  Figure 12: Separation of irrecoverable and recoverable strains. K u is the slope of the unloading secant curves, σ psl is the preset stress level, and ε 1 and ε 3 are the irrecoverable and recoverable strains, respectively.