An improvement in the method of correcting indirect radial strain measurements during triaxial strength tests in rocks

The present article provides an improvement in the method to correct indirect strain measurements in triaxial compressive strength tests through axial displacement and hydraulic fluid volume change measurements. The improvement focused on reducing the parameters of the formula proposed for indirect volumetric strain in the original method, thereby facilitating the development of a simpler formula in which the radial strain depends on only two parameters: the initial volume of the rock specimen and the volume changes of the hydraulic fluid for each instant. The comparison between the improvement proposed, and original method resulted in a mean absolute difference of 0.003.• This improvement does not depend on the axial strain, unlike the original method, which requires correcting the indirect axial strain measurements before correcting the indirect radial strain measurements.• This improvement can be useful for research on the stress-strain behavior of intact rock under laboratory conditions, such as in the study of the post-peak state.


Method details
The mechanical behavior of intact rock materials has been widely studied, particularly under laboratory conditions.This behavior can be studied through uniaxial and triaxial compressive strength tests to determine Young's modulus and Poisson's ratio, which are considered the main deformability properties of the intact rocks [ 1 ].
According to the methods suggested by the International Society for Rock Mechanics and Rock Engineering (ISRM) [ 2 ], there are two instruments to measure strain: strain gauges that obtain direct strain measurements, and Linear Variable Differential Transducers (LVDT) that obtain indirect strain measurements.However, there is no clarity on the appropriate use of these instruments.It is important to highlight that a significant difference is observed between the strain gauge and LVDT measurements [ 3 , 4 ].This contrast can be attributed to strain phenomena associated with the different interfaces existing on the test setup.It is noteworthy that the strain measured by the LVDTs is significantly greater than that obtained using strain gauges [ 5 , 6 ].Considering these observations, a distinct requirement emerges for a methodology that ensures the dependable application of corrections to the indirect strain measurements.
Early correction of indirect volumetric strain was documented by Farmer [ 7 ], who proposed a formula to determine the volumetric strain ( ε v ) during triaxial compressive strength tests using both the hydraulic fluid volume changes measured in Hoek's cell and the axial displacement measured by LVDT ( Eq. ( 1) ).However, this method only considers the Young's modulus of steel to consider its deformation behavior.
where ΔV and V are the volume increment and initial volume of the rock specimen, respectively, f is the compressibility factor of the hydraulic fluid, V 0 is the volume of hydraulic fluid displaced, r is the radius of Hoek's cell steel bases or rams, F is the axial force, E is the Young's modulus of steel and l is the axial displacement.Subsequently, Alejano et al. [ 3 ] proposed a method for correcting indirect strain measurements.This method is based on an energy approach to correct the axial strain measured indirectly by LVDT and volume changes of the hydraulic fluid measured during the triaxial compressive strength test ( Eq. ( 2) ).
where ΔV rad is the radial (or lateral) volume increment, r 0 steel is the initial radius of the steel bases, h 0 sp is the initial height of the rock specimen, ε i 1 is the corrected axial strain at each instant (in mstr.),V 0 sp is the initial volume of the rock specimen, and there is an infinitesimal term, that can be disregarded.
It was assumed that the axial stress magnitudes applied to the rock specimens were not sufficiently high to produce significant changes in the radius of the steel bases for all instants of the triaxial compressive strength tests, also this radius is very similar to the rock specimen radius ( Eq. ( 3) ).
where r i steel is the radius of the steel bases for each instant, r 0 sp is the initial radius of the rock specimen, υ steel is the Poisson's ratio of steel, E steel is the Young's modulus of steel, and σ i 1 is the axial stress at each instant.According to the above, the initial volume of the steel bases ( V 0 steel ), the volume at each instant ( V i steel ), and the volume change at each instant ( ΔV i steel ) of the steel bases inside the Hoek's cell can be considered as: where h 0 steel is the initial distance between lower and upper platen, h sleeve is the height of the plastic sleeve, and Δh i sp , h i sp are the height increment and height of the rock specimen at each instant, respectively.
On the other hand, the volume increment of the rock specimen at each instant ( ΔV i sp ) depends on the height and radius change of the rock specimen ( Δh i sp and Δr i sp , respectively) during the triaxial compressive strength test, such that: where ε i V is the volumetric strain, ε i 1 is the axial strain, and ε i 3 is the radial strain of the rock specimen at each instant.
Therefore, the volume increment of the hydraulic fluid at each instant ( ΔV i rad ) for a triaxial compressive strength test can be expressed as: Finally, the radial and volumetric strains of a rock specimen at each instant during a triaxial strength test can be determined indirectly ( Eq. ( 9) , ( 10) ).
Fig. 1.Volumetric stress-strain curves were obtained from strain gauges and the different approaches of a triaxial strength test, please note that the method of Alejano et al. [ 3 ] and the proposed method mostly overlap.
Fig. 2. Direct (strain gauges) and indirect (using the proposed approach) axial stress-radial strain curves of a triaxial strength test.The proposed improvement for the method proposed by Alejano et al. [ 3 ] is simpler, and the radial strain depends only on the initial volume of the rock specimen and the volume change of the hydraulic fluid at each instant, unlike the methods of Farmer [ 7 ] and Alejano et al. [ 3 ].
Corrections made using the Farmer's equation ( Eq. ( 1) ) were less precise than those of the other two approaches, whereas the current improvement did not show significant differences compared to the method proposed by Alejano et al. [ 3 ] ( Fig. 1 ).
The proposed improvement also yielded radial strain measurements comparable to those obtained using strain gauges ( Fig. 2 ).The Poisson's ratios calculated in the range of 20 % to 40 % of the maximum strength [ 12 ] of each test ( Fig. 2 ) were calculated using the proposed approach and the original method, as shown in Table 1 .The mean absolute difference (without outliers) between the proposed improvement and the original method is 0.0026 ( Fig. 3 ).
It is important to note that outliers were excluded using the interquartile range (IQR) method to calculate the mean ( Fig. 3 ).Although there is a possibility that variations in test execution or rock characteristics could have influenced the results, the overall trend of the data justifies the exclusion of these outliers.Moreover, the proposed approach is specifically applicable to triaxial compressive strength tests that use water as the hydraulic fluid, as it is an incompressible fluid [ 2 ].However, this approach has only been validated on Blanco Mera granite.Therefore, further research would be beneficial to ascertain the applicability and universalization of this approach to all rock types.

Table 1
Mean Poisson's ratios at each diameter and confinement.