
==== Front
MethodsX
MethodsX
MethodsX
2215-0161
Elsevier

S2215-0161(24)00369-8
10.1016/j.mex.2024.102918
102918
Earth and Planetary Science
An improvement in the method of correcting indirect radial strain measurements during triaxial strength tests in rocks
Martínez-Bautista E. edison.martinez@ce.ucn.cl
a⁎
Ibarra-González D. a
Arzúa J. ab
González-Fernández M.A. b
Alejano L.R. b
a Department of Metallurgical and Mining Engineering, Universidad Católica del Norte, Chile
b CINTECX. GESSMin Group. Department of Natural Resources and Environmental Engineering, Universidade de Vigo, Spain
⁎ Corresponding author. edison.martinez@ce.ucn.cl
15 8 2024
12 2024
15 8 2024
13 10291817 1 2024
14 8 2024
© 2024 The Authors. Published by Elsevier B.V.
2024

https://creativecommons.org/licenses/by/4.0/ This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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.

Graphical abstract

Image, graphical abstract

Keywords

Rock testing
Compressive strength test
Radial strain
Method name

Improved Indirect Radial Strain Correction for Triaxial Tests
==== Body
pmcSpecifications tableSubject area:	Earth and Planetary Sciences	
More specific subject area:	Rock Mechanics – Laboratory.	
Name of your method:	Improved Indirect Radial Strain Correction for Triaxial Tests.	
Name and reference of original method:	Alejano, L. R., Estévez-Ventosa, X., González-Fernández, M. A., Walton, G., West, I. G., González-Molano, N. A., & Alvarellos, J. (2021). A Method to Correct Indirect Strain Measurements in Laboratory Uniaxial and Triaxial Compressive Strength Tests. Rock Mechanics and Rock Engineering, 54(6), 2643–2670. https://doi.org/10.1007/s00603-021-02392-4
ISRM (2007). The complete ISRM suggested methods for rock characterization, testing and monitoring; 1974−2006. Suggested Methods for Determining the Uniaxial Compressive Strength and Deformability of Rock Materials Compressive Strength and Deformability of Rock Materials, 628.	
Resource availability:	There are no special resources.	

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 (εv) 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.(1) εv=▵VV(%)=100V[fV0−(πr2−FE)l]

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, V0 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)).(2) εv=▵VV=ΔVrad−π(rsteel0)2hsp0ε1i1000−infinit.Vsp0

where ΔVrad is the radial (or lateral) volume increment, rsteel0 is the initial radius of the steel bases, hsp0 is the initial height of the rock specimen, ε1i is the corrected axial strain at each instant (in mstr.), Vsp0 is the initial volume of the rock specimen, and there is an infinitesimal term, that can be disregarded.

The present article proposes an improvement in the calculation of indirect volumetric strain measurements proposed by Alejano et al. [3]. The proposed improvement was developed from the triaxial test results of 71 intact rock specimens of Blanco Mera granite, a moderately researched granite [[8], [9], [10], [11], [12], [13]], at different confinements of 0.2, 2.5, 5, 7.5, 10, 12.5, and 15 MPa and diameters of 38, 54, and 84 mm.

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)).(3) rsteeli=rsteel0(1+υsteelEsteelσ1i)≈rsteel0=rsp0

where rsteeli is the radius of the steel bases for each instant, rsp0 is the initial radius of the rock specimen, υsteel is the Poisson's ratio of steel, Esteel is the Young's modulus of steel, and σ1i is the axial stress at each instant.

According to the above, the initial volume of the steel bases (Vsteel0), the volume at each instant (Vsteeli), and the volume change at each instant (ΔVsteeli) of the steel bases inside the Hoek's cell can be considered as:(4) Vsteel0=π(rsteel0)2hsteel0

(5) Vsteeli=π(rsp0)2(hsleeve−hspi)

(6) ΔVsteeli=−π(rsp0)2Δhspi

where hsteel0 is the initial distance between lower and upper platen, hsleeve is the height of the plastic sleeve, and Δhspi, hspi 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 (ΔVspi) depends on the height and radius change of the rock specimen (Δhspi and Δrspi, respectively) during the triaxial compressive strength test, such that:(7) εVi=ΔVspiVsp0=ε1i+2ε3i⇒ΔVspi=Vsp0ε1i+2Vsp0ε3i=π(rsp0)2Δhspi+2πrsp0hsp0Δrspi

where εVi is the volumetric strain, ε1i is the axial strain, and ε3i is the radial strain of the rock specimen at each instant.

Therefore, the volume increment of the hydraulic fluid at each instant (ΔVradi) for a triaxial compressive strength test can be expressed as:(8) ΔVradi=ΔVspi+ΔVsteeli=2πrsp0hsp0Δrspi

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)).(9) ε3i=Δrspirsp0=ΔVradi2Vsp0

(10) εvi=ΔVradi+Vsp0ε1iVsp0

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).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. 1

The proposed improvement also yielded radial strain measurements comparable to those obtained using strain gauges (Fig. 2).Fig. 2 Direct (strain gauges) and indirect (using the proposed approach) axial stress-radial strain curves of a triaxial strength test.

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

Table 1Diameter (mm)	Confinement (MPa)	Mean Poisson's Ratio (-)	
Proposed Approach	Alejano et al. (2021)	
38	0.2	0.148	0.122	
38	2.5	0.227	0.227	
38	5.0	0.240	0.241	
38	7.5	0.250	0.360	
38	10.0	0.177	0.176	
38	12.5	0.266	0.266	
38	15.0	0.126	0.128	
54	0.2	0.296	0.292	
54	2.5	0.181	0.177	
54	5.0	0.171	0.166	
54	7.5	NaN	0.217	
54	10.0	0.186	0.182	
54	12.5	0.418	0.415	
54	15.0	0.164	0.161	
84	0.2	0.261	0.275	
84	2.5	NaN	0.181	
84	5.0	0.404	0.420	
84	7.5	0.108	0.120	
84	10.0	0.229	0.260	
84	12.5	NaN	NaN	
84	15.0	NaN	0.231	

Fig. 3 Absolute differences of Poisson's ratio of tests.

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.

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the author(s) used Paperpal in order to improve writing. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.

CRediT authorship contribution statement

E. Martínez-Bautista: Conceptualization, Data curation, Writing – original draft. D. Ibarra-González: Data curation, Writing – original draft. J. Arzúa: Supervision, Funding acquisition. M.A. González-Fernández: Methodology, Supervision. L.R. Alejano: Resources, Methodology, Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The authors do not have permission to share data.

Acknowledgments

The authors acknowledge their gratitude for the partial funding of this research, provided by 10.13039/501100002850 Fondecyt Iniciación Project No. 11190065 from the National Agency for Research and Development (ANID) of Chile.

Related research article: Alejano, L.R., Estévez-Ventosa, X., González-Fernández, M.A., Walton, G., West, I.G., González-Molano, N.A., & Alvarellos, J. (2021). A Method to Correct Indirect Strain Measurements in Laboratory Uniaxial and Triaxial Compressive Strength Tests. Rock Mechanics and Rock Engineering, 54(6), 2643–2670. https://doi.org/10.1007/s00603-021-02392-4
==== Refs
References

1 Feng X.-T. Rock Mechanics and Engineering Volume 2: Laboratory and Field Testing 1st ed. 2016 CRC Press 10.1201/9781315364254
2 ISRM Suggested methods for determining the uniaxial compressive strength and deformability of rock materials Ulusay R Hudson JA The Complete ISRM Suggested Methods For Rock characterization, Testing and monitoring: 1974−2006 2007 Prepared by the Commission on Testing Methods, ISRM Ankara, Turkey 2007
3 Alejano L.R. Estévez-Ventosa X. González-Fernández M.A. Walton G. West I.G. González-Molano N.A. Alvarellos J. A method to correct indirect strain measurements in laboratory uniaxial and triaxial compressive strength tests Rock Mech. Rock Eng. 54 6 2021 2643 2670 10.1007/s00603-021-02392-4
4 Alejano L.R. Arzúa J. Castro-Filgueira U. Kiuru R. Scale Effect of Intact Olkiluoto Gneissic Rocks Through Uniaxial Compressive Testing and Geophysical Measurements 2018 Posiva Oy Report 2018–13. Finland
5 Perbawa A. Gramajo E. Finkbeiner T. Santamarina J.C. Rock triaxial tests: global deformation vs local strain measurements—implications Rock Mech. Rock Eng. 54 7 2021 3527 3540 10.1007/s00603-021-02389-z
6 Perbawa A. Gramajo E. Finkbeiner T. Santamarina J.C. Global vs Local Strain Measurements in Triaxial Tests-Implications 2019
7 Farmer I.W. Engineering Behaviour of Rocks 1983 10.1007/978-94-009-5978-1
8 González-Fernández M.A. Estévez-Ventosa X. Alejano L.R. Masoumi H. Size-dependent behaviour of hard rock under triaxial loading Rock Mech. Rock Eng. 56 8 2023 6009 6025 10.1007/s00603-023-03367-3
9 Muñoz-Ibáñez A. Herbón-Penabad M. Delgado-Martín J. Alejano-Monge L. Alvarellos-Iglesias J. Canal-Vila J Hydrostatic, strike-slip and normal stress true triaxial hydrofracturing testing of Blanco Mera Granite: breakdown pressure and tensile strength assessment Geomech. Geophys. Geo-Energy Geo-Resour. 9 1 2023 10.1007/s40948-023-00564-w
10 Muñoz-Ibáñez A. Delgado-Martín J. Juncosa-Rivera R. Size effect and other effects on mode I fracture toughness using two testing methods Int. J. Rock Mech. Min. Sci. 143 2021 10.1016/j.ijrmms.2021.104785
11 Walton G. Alejano L.R. Arzua J. Markley T. Crack damage parameters and dilatancy of artificially jointed granite samples under triaxial compression Rock Mech. Rock Eng. 51 6 2018 1637 1656 10.1007/s00603-018-1433-1
12 Quiñones J. Arzúa J. Alejano L.R. García-Bastante F. Mas Ivars D. Walton G. Analysis of size effects on the geomechanical parameters of intact granite samples under unconfined conditions Acta Geotech. 12 6 2017 1229 1242 10.1007/s11440-017-0531-7
13 Arzúa J. Alejano L.R. Dilation in granite during servo-controlled strength tests Int. J. Rock Mech. Min. Sci. 61 2013 43 56 10.1016/j.ijrmms.2013.02.007
