Effects of 3D deformation and nonlinear stress-strain relationship on the Brazilian test for a transversely isotropic rock

Juhyi Yim,Yoonsung Lee,Seungki Hong,Ki-Bok Min,*

a R&D Center,Hyundai Engineering and Construction,75,Yulgok-ro,Jongno-gu,Seoul,Republic of Korea

b Department of Energy Systems Engineering and Research Institute of Energy and Resources,Seoul National University,1,Gwanak-ro,Gwanak-gu,Seoul,08826,Republic of Korea

Keywords:Anisotropy Transversely isotropy Brazilian test Indirect tensile strength Nonlinearity 3D modeling

ABSTRACT To improve the accuracy of indirect tensile strength for a transversely isotropic rock in the Brazilian test,this study considered the three-dimensional (3D) deformation and the nonlinear stress-strain relationship.A parametric study of a numerical Brazilian test was performed for a general range of elastic constants,revealing that the 3D modeling evaluated the indirect tensile strength up to 40% higher than the plane stress modeling.For the actual Asan gneiss,the 3D model evaluated the indirect tensile strength up to 10%higher and slightly enhanced the accuracy of deformation estimation compared with the plane stress model.The nonlinearity in stress-strain curve of Asan gneiss under uniaxial compression was then considered,such that the evaluated indirect tensile strength was affected by up to 10% and its anisotropy agreed well with the physical intuition.The estimation of deformation was significantly enhanced.The further validation on the nonlinear model is expected as future research.

1.Introduction

The tensile strength is important to be evaluated in rock engineering application.Although assuming the tensile strength of the rock to be zero is a common practice for safe design,it is inappropriate for the application primarily related to tensile failure of rock (Nova and Zaninetti,1990).The tensile strength needs to be considered to evaluate underground structure stability,especially for the mine roof(Nova and Zaninetti,1990;Kim et al.,2019;Zhang and Chen,2019).In drilling and rock excavation,the brittleness of rock should be estimated,and some useful indices for brittleness require the tensile strength(Yarali and Kahraman,2011),which can affect the support design related to economic benefits (Diederichs and Kaiser,1999).The tensile strength is a vital factor in designing hydraulic fracturing along with fracture toughness(Haimson and Cornet,2003;Ma et al.,2018;Yang et al.,2019;Cong et al.,2022).Therefore,the Brazilian test has been commonly performed as an indirect tensile strength test which was invented to overcome the difficulties in reliable implementation of direct tensile strength test(Fairhurst,1964;Nova and Zaninetti,1990;Wei and Chau,2013).

Previous studies have focused on analyzing the elastic behavior of the Brazilian test under various physical conditions to evaluate the maximum tensile stress at the center of the Brazilian disc.The plane stress condition was generally adopted in some analytic solutions to simplify the problem to two dimensions,as listed in Table 1.With an isotropic material,Hertz derived the simple solution in 1883 by assuming the line load (Timoshenko and Goodier,1982),and it was suggested as International Society for Rock Mechanics and Rock Engineering (ISRM) suggested method for determining indirect tensile strength of the Brazilian test(Bieniawski and Hawkes,1978).Hondros (1959) derived the solution of the stress field considering the uniformly distributed surface load within contact angle,2α.The non-uniform load or friction at the loading contact between the jaw and the disc can cause the local stress concentration,but their effects on the stress at the center of the disc are limited(Lavrov and Vervoort,2002;Yu et al.,2006;Markides et al.,2012;Kourkoulis et al.,2013;Guerrero-Miguel et al.,2019).Considering the anisotropic deformability,the general solution of the two-dimensional (2D) disc under the diametral loading was derived based on functions of complex variables(Lekhnitskii,1968),and it was applied to determine elastic constants or indirect tensile strength of a transversely isotropic rock (Chen et al.,1998;Claesson and Bohloli,2002).Claesson and Bohloli (2002) proposed the approximation of this solution of normal stresses at the center of the disc under the line load to obtain the explicit expression.Recently,analytical approach,such as displacement function approach (Wei and Chau,2013),and numerical approach,such as three-dimensional(3D)finite element method(Yu et al.,2006;Liao et al.,2019),for an isotropic material revealed that the plane stress condition is not adequate to the Brazilian test because the stress is not uniform along the direction normal to the cross-section and the 3D shape and deformation influenced the stress field.In addition,as various rock properties are dependent on the stress state (Zhang et al.,2016;Gehne and Benson,2017;Martyushev et al.,2019,2023),the deformability is also stress-dependent,causing a nonlinear stress-strain relationship under uniaxial compression.Due to the nonlinear stressstrain relationship of some artificial isotropic materials,the estimated maximum tensile stress at the center of the disc,which was regarded as an indirect tensile strength of the Brazilian test,can increase compared with that by the simple solution of Hertz(1883),Kurguzov and Demeshkin (2019).However,the Brazilian test for a transversely isotropic rock has not been investigated with focus on the 3D deformation and the nonlinear stress-strain relationship.

Table 1List of analytic or approximate solutions of stresses at the center of the Brazilian disc under different physical conditions and input parameters.P is the applied force in x′′-direction,not in radial direction.

The indirect tensile strength of the Brazilian test can be regarded as the maximum tensile stress at the center of the disc when the tensile failure splitting the disc was initiated at the center.It was shown that tensile failure occurs at the center of the disc with large contact angle 2αover 20°even for transversely isotropic rocks if the modified Hoek-Brown failure criterion was assumed (Aliabadian et al.,2019).Depending on the failure criterion or loading conditions such as friction,the stress concentration around the contact can occur and may lead to the failure (Fairhurst,1964;Kourkoulis et al.,2013;Guerrero-Miguel et al.,2019).However,failure around the contact owing to local stress concentration may not necessarily result in the crack splitting the disc (Liao et al.,2019).For a transversely isotropic rock containing a weak plane,shear failure can occur instead of tensile failure if the load of the Brazilian test is applied oblique to the strike of the layer(Dan et al.,2013;Tan et al.,2015;Ma et al.,2018).Therefore,experimental factors,such as the loading direction and the contact angle,must be taken into account while performing the Brazilian test for a transversely isotropic rock to consider the indirect tensile strength as the maximum tensile stress at the disc center.

However,compared with the indirect tensile strength of the Brazilian test,the inherent tensile strength can be lower considering the crack propagation from the center (Rocco et al.,1999) or higher considering the Griffith’s failure criterion (Fairhurst,1964).Some researchers have proposed the critical extension strain criterion as a failure criterion for the Brazilian test(Stacey,1981;Li and Wong,2013).Although these are important topics,the focus of this is to determine a more convincing indirect tensile strength for the Brazilian test in terms of the elastic behavior.

This study aims to evaluate the indirect tensile strength and the deformation of the Brazilian test for a transversely isotropic rock considering 3D deformation and nonlinear stress-strain relationship.A 3D numerical simulation is employed to account for the 3D deformation,and the nonlinear stress-strain relationship was incorporated in numerical model by utilizing the strain-dependent elastic moduli which models the initial part of the stress-strain curve under uniaxial compression.The anisotropy was only considered in the elastic model with transversely isotropic elastic constants.The laboratory tests were conducted for Asan gneiss to show how those mechanisms actually affect the indirect tensile strength and the deformation of the Brazilian test.

2.Theoretical background

Three Cartesian coordinates were defined in Fig.1 to express the physical behaviors and properties in material,cylindrical core,and Brazilian test.A local system,xyz,is defined to represent the material properties and the isotropy plane which is parallel toxzplane.The elastic constants are defined as

Fig.1.Definition of parameters,local coordinate,and two global coordinates of (a) material,(b) cylindrical core,and (c) Brazilian test for a transversely isotropic rock.

whereEiis the elastic modulus ini-direction,Gijis the shear modulus inij-plane,andvijis the Poisson’s ratio defined as the ratio ofj-direction strain toi-direction strain under uniaxial stress inidirection.

The first global system,x′y′z′,in Fig.1b is defined as a system with thez′-axis equal to thez-axis and they′-axis equal to upright axis of core.Accordingly,thex′-axis is defined by the right-handed coordinate system.The coring angle,φ,represents the angle measured counterclockwise around thez-axis from they-axis to they′-axis.The cylindrical coordinate,rθy′,corresponding to the first global system,is formed by assigning thez′-axis equal to the line ofθ=0°.The second global system,x′′y′′z′′,in Fig.1c is obtained by rotating the first global system counterclockwise by the loading direction,ψ,on they′-axis,such that the load of the Brazilian test is aligned with thex′′-axis.The contact angle,2α,denotes the angle covering the contact area between the loading jaw and the Brazilian disc.In this study,it is assumed that the isotropy plane or the weak plane of a transversely isotropic rock are identical as the visual planar structure such as stratification or foliation(Min et al.,2017),and all of them are herein called as the layer.The intersection line between the layer and the cross-section is herein called as the strike of the layer.The constitutive equation of a transversely isotropic rock is defined in Eq.(2),and its axis transformation is introduced in Appendix A(Jaeger et al.,2007;Cho et al.,2012;Yim,2022;Yim et al.,2022a).

whereσand τ are the normal and shear stresses,respectively;ε is the strain;and the subscript denotes the direction and the plane where the parameter is defined.

3.Numerical and experimental tests

3.1.Numerical test

The 3D numerical modeling by the finite element code,Comsol Multiphysics,was performed to simulate the Brazilian test(COMSOL,2019).This was because the stress in the specimen of the Brazilian test should be estimated by considering various physical conditions which are difficult to be considered analytically.The test specimen was numerically reproduced in the geometric configuration identical to an actual experiment,and the load was modeled by the uniform stress boundary within the contact area as shown in Fig.1c.The strain was regarded as the value of surface average within the modeled strain gauge,and the stress was extracted from the center point on the surface of the Brazilian disc.The bottom of the specimen was modeled by a roller boundary with a small fixed area at the center to obtain the numerical solution.The indirect tensile strength was calculated using the elastic constants of Asan gneiss in Table 2,taken from Yim et al.(2022a),and the measured peak load.The simple solution by Hertz (1883) in Eq.(3) was also used to normalize parameters and illustrate the raw data of experiment (Fig.2).

Fig.2.Brazilian tests for Asan gneiss specimens with coring angle(a)over 45° and(b)below 45°.The diagonal strain gauge was ignored.

Table 2Elastic constants used for this study (Yim et al.,2022a).

wherePis the force,Dis the diameter of the Brazilian disc,andtis the thickness of the Brazilian disc.In parametric study,parametersaandbwere used instead of five elastic constants,which are defined as follows (Lekhnitskii,1968):

This was because only those two elastic parameters among five are independent for determining the stress field in 2D Brazilian model whenφ=90°.However,the layer direction (φ) can be any value,thus the axis transformation is necessary to calculate the apparentaandbon the cross-section of the Brazilian disc (Eq.(A.4)).

3.2.Laboratory test

Three blocks of Asan gneiss were cored in 0°,20°,40°,70°and 90°with respect to the normal direction of the foliation to secure NX-sized cores.Following ISRM suggested methods,22 specimens for the Brazilian tests were prepared and oven-dried for 24 h at 105°C,and the Brazilian tests were performed as shown in Fig.2(Bieniawski and Hawkes,1978).Two biaxial rosettes of strain gauge with a grid length of 10 mm and width of 3 mm were affixed on both sides of cross-section of the Brazilian disc to measure the normal strains in the directions parallel and normal to the direction of the load.To analyze the linear elastic behavior,the strains were sampled at the compressive stress range of 20-35 MPa,which contained half of the value for the maximum compressive stress at the center of the Brazilian disc generated under the peak load(Fig.3).

Fig.3.Typical stress-strain curves of Asan gneiss specimens for(a)uniaxial compression test(redrawn from Fig.12 in Yim et al.(2022a))and(b)Brazilian test.Thick line indicates the range of compressive stress from 20 MPa to 35 MPa.(1 microstrain=10-6 strain).

Fig.4.The 3D deformation with normalized strain distribution in y′′-direction.The displacement in y′′-direction is magnified to make the maximum displacement similarly depicted (b=1,v1=v2=0.25,ψ=90°,φ=90°,2α=20°,D/t=2,εy′′,norm=εy′′πDtE1/(2P),E1=33 GPa,P=10 kN).

The load of the Brazilian test was applied in the direction normal or parallel to the strike of the foliation having coring angle over or below 45°,respectively (ψ=0°,φ＞45°orψ=90°,φ≤45°).The curved loading jaw was used to ensure the failure initiated from the center of the disc by obtaining the large contact angle,which was measured as approximately 20°on average in this study.Although the load can be applied obliquely to the strike of the layer to cause the shear failure,this study limited the loading direction(ψ=0°or 90°) to ensure the tensile failure avoiding shear failure.Therefore,in this study,the maximum tensile stress at the center of the Brazilian disc surface under the applied load and the peak load was defined as “indirect tensile stress” and “indirect tensile strength,”respectively.

4.Results and discussion

4.1.Effect of 3D deformation in Brazilian test

The stress field in the Brazilian test is influenced by the 3D deformation,herein called as“3D deforming effect.”To be specific,the 3D deforming effect indicates the stress change from the plane stress model to the 3D model.The 3D deforming effect is not negligible for the Brazilian test,and it can result in the evaluated indirect tensile strength up to 15%higher than that determined by the plane stress model for isotropic materials(Yu et al.,2006;Wei and Chau,2013).

The 3D deforming effect is attributed to the deformation normal to the cross-section (dy′′).In the plane stress model,the force equilibrium on the plane,expressed as Eq.(B.1),is solved with the displacement on the same plane without the displacement normal to the plane,as shown in Eqs.(B.2) and (B.3).Therefore,the deformation normal to the cross-section can occur under the plane stress condition,but it does not influence the in-plane stress field.However,bulging in a direction normal to the cross-section generates the additional tensile stress in the plane comparing with the plane stress model by stretching the surface and vice versa.To physically explain the 3D deforming effect for a transversely isotropic material,the following approximate relationships using the elastic constants transformed into the coordinate of the crosssection are introduced in this study:

Eq.(5) represents the relationship using the convexity as the quadratic derivative in order to express how the displacement normal to the cross-section affects to the 3D deforming effect in analogy to the plate theory (Shimpi and Patel 2006).Eq.(6) is the relationship of Poisson’s effect in order to show which factor determines the displacement normal to the cross-section.Those relationships are not theoretically complete owing to the ignorance of minor effects,such as distributed stress alongy′′-direction(Yu et al.,2006),but the results presented in Figs.5 and 7 support their validity in conditions of the Brazilian test in this study.

For the Brazilian test with an isotropic rock,as Wei and Chau(2013) reported,the 3D deforming effect increases as the ratio of thickness to diameter or the Poisson’s ratio increase,but it is independent of the elastic modulus.This is because those three factors can increase they′′-direction displacement in Eq.(6)while the elastic modulus is canceled out by combining Eqs.(5) and (6) and has no effect on the stress field of the plane stress model.However,for the Brazilian test with an anisotropic rock,the elastic modulus also affects the 3D deforming effect.Figs.4 and 6 show how changes in elastic parameters affect the strain distribution throughout the specimen,and Figs.5 and 7 demonstrate how the 3D deforming effect is related to the deformation which depends on elastic parameters.In the case ofψ=90°andφ=90°(Figs.4 and 5),for example,asa(=Ex′′/Ez′′) increases with almost consistent surface deformation,the 3D deforming effect can be reduced enough to have a negative value according to Eq.(5) (Fig.5a).Another example is shown in Fig.6,whereψ=0°andφ=45°,the certain case can even invert the sign of the displacement normal to the cross-section compared with other general cases,which in turn inverts the sign of convexity on the surface (Fig.7).This can occur because compressive and tensile stresses are applied simultaneously and the contribution of each to they′′-direction displacement depends on the elastic modulus in each direction,as shown in Eq.(6).In addition,the in-plane stresses,σx′′,2Dandσz′′,2D,are the function of anisotropic elastic constants transformed into the plane(Yim et al.,2022b).Therefore,the 3D deforming effect for an anisotropic rock is complexly related to the elastic constants transformed into the coordinate of the cross-section of the Brazilian disc,such that various combinations of elastic constants and layer direction must be considered.

Fig.5.(a)The difference of the indirect tensile stress between 2D and 3D models against Eq.(5)divided by Eq.(3);(b)The distribution of normalized displacement in y′′-direction along z′′-axis and x′′-axis (b=1,v1=v2=0.25,ψ=90°,φ=90°,2α=20°,D/t=2,dy′′,norm=dy′′πDE1/(2P),E1=33 GPa,P=10 kN).

Fig.6.The 3D deformation with normalized strain distribution in y′′-direction.The displacement in y′′-direction is magnified(a=1,v1=v2=0.1,ψ=0°,φ=45°,2α=20°,D/t=2,εy′′,norm=εy′′πDtE1/(2P),E2=33 GPa,P=10 kN).

Fig.7.(a)The difference of the indirect tensile stress between 2D and 3D models against Eq.(5)divided by Eq.(3);(b)The distribution of normalized displacement in y′′-direction along z′′-axis and x′′-axis (a=1,v1=v2=0.1,ψ=0°,φ=45°,2α=20°,D/t=2,dy′′,norm=dy′′πDE1/(2P),E2=33 GPa,P=10 kN).

The parametric study was performed on the elastic parametersa,bandv1,and two orientation parametersφandψas plotted in Fig.8.The ranges for each parameter were chosen to include cases of general transversely isotropic rocks,as shown in Fig.C.1.According to the parametric study,the 3D deforming effect for the transversely isotropic material can range from-40% to 40% under the assumption of identical Poisson’s ratios.For a horizontally layered case (φ=0°),the 3D deforming effect is independent of shear modulus represented bybwhile its magnitude increases with an increase ina.For one general example (a=2,b=1),the plane stress model can underestimate the indirect tensile strength by up to 20%because it ignores the effect of the bulging cross-section on the stress,but the relative error can be less than 10% if the load is applied parallel to the strike of the layer.Therefore,the 3D deforming effect can increase up to 40% owing to the anisotropic deformability.

Fig.8.Relative error of indirect tensile stress calculated by 2D analytic solution (Lekhnitskii,1968) in comparison with 3D modeled indirect tensile stress (2α=20°,D/t=2).

The 3D deforming effect also depends on the diameter-tothickness ratio because the deformation in they′′-direction and its quadratic derivative alongz′′-axis are proportional to the thickness.As the Brazilian disc thickens,the 3D deforming effect is deepened regardless of its sign.In the range of diameter-tothickness ratio from 1.75 to 2.5,there can be an approximate 10%change in stress when compared with the indirect tensile stress with a diameter-to-thickness ratio of 2 (Fig.9).

Fig.9.The indirect tensile stress ratio between 2D and 3D models against the diameter-to-thickness ratio(a=1,v1=v2=0.1,ψ=0°,φ=45°,2α=20°,E1=33 GPa,P=10 kN).

4.2.Effect of nonlinearity in the initial curve of stress-strain relationship

The nonlinear stress-strain relationship was investigated to show its effect on the Brazilian test because the nonlinearity in Fig.3a was obvious at the stress level for the Brazilian test of Asan gneiss.Furthermore,the Brazilian test generates the various stress states in a single specimen,which corresponds to the various moduli by nonlinearity.For example,the center of the side surface is almost under uniaxial compression while both tensile and compressive stresses at the center of the cross-section act perpendicular to each other.The nonlinearity in the Brazilian test modeling was simulated by the simple nonlinear model using the directional elastic modulus linearly related with the compressive strain in corresponding direction because this model wellsimulated the measured deformation of the Brazilian test for Asan gneiss(Yim et al.,2022a).According to this model,the straindependent elastic modulus can be obtained as Eq.(7) approximating the initial part of stress-strain curve for horizontally and vertically layered cores under uniaxial compression.This model follows the linear relationship between stress and strain outside the initial part.

An inevitable error arises when nonlinearity in stress-strain relationship is approximated by linear relationship.The deformation will be underestimated if larger deformation at lower stress level is ignored by using the linear elastic modulus obtained in linear portion.Moreover,most parts of the Brazilian disc are in the low stress level,and the deformation in the direction perpendicular to the load is tensile (z′′-direction) or compressive (y′′-direction)with low magnitude,which corresponds to the lower modulus.The different deformation owing to the nonlinear modulus can incur stress change in both 2D and 3D models,but it also affects the 3D deforming effect.As Fig.10 shows,the displacement iny′′-direction and its quadratic derivative alongz′′-direction are larger in the nonlinear model than that in the linear model,so that the 3D deforming effect increases.For example,the indirect tensile stress in linear and nonlinear models in Fig.10 are 19.76 MPa and 20.46 MPa,respectively.Therefore,the 3D deforming effect and the distributed modulus by nonlinearity need to be considered to evaluate the indirect tensile strength.

Fig.10.The 3D deformation with normalized strain distribution in y′′-direction.The displacement in y′′-direction is magnified: (a) Linear model used elastic constants in Table 2,and (b) Nonlinear model adopted Eq.(7) for elastic modulus with v1,v2 and G2 in Table 2 (ψ=0°,φ=90°,2α=20°,P=40 kN,εy′′,norm=εy′′πDtE1/(2P)).

4.3.Indirect tensile strength determination for the Brazilian test

The indirect tensile strengths from Brazilian tests of Asan gneiss were calculated by four models in Table 3.The effects of anisotropy,3D deformation,and nonlinearity could be simply analyzed respectively by comparing models adjacent to each other.The average indirect tensile strength for each coring angle is plotted in Fig.11 to compare models in Table 3.The ISRM_SM(ISRM suggested method)model was a 2D isotropic linear model under line load and plane stress condition,which corresponds to Eq.(3).The 2D_Linear model used the solution by Lekhnitskii (1968) to consider the anisotropic deformability in Table 2 and surface load in addition to the ISRM_SM model.The difference of evaluated indirect tensile strength between ISRM_SM and 2D_Linear models with 0°coring angle was solely due to the loading condition difference,thus this indicates that the wide contact area reduces the evaluated indirect tensile strength under the same peak load.The additional decrease and increase of evaluated indirect tensile strength were observed forφ＜45°,ψ=90°andφ＞45°,ψ=0°,respectively.This was the result of complex combination of elastic constants and direction parameters,but its trend can be estimated using the simple solution by Claesson and Bohloli (2002) with the apparent elastic constants in Eq.(A.4) determined on the cross-section (Yim et al.,2022b).The 3D_Linear model was based on the 3D modeling of the Brazilian test to reflect the 3D deforming effect,and it increased the evaluated indirect tensile strength by up to 10%comparing with the 2D_Linear model.

Fig.11.Average indirect tensile strength determined by (a) ISRM_SM and 2D_Linear models and (b) 3D_Linear and 3D_Nonlinear models in Table 3 against the coring angle.

Table 3Information of Brazilian test models.

The 3D_Nonlinear model determined the indirect tensile strength by 3D modeling with the elastic constants in Table 2 and Eq.(7).In comparison with the 3D_Linear model,the indirect tensile strength of the 3D_Nonlinear model was overestimated by up to about 10% for cases ofφ＜45°,ψ=90°,but there was no significant difference for cases ofφ＞45°,ψ=0°.Interestingly,the 3D_Nonlinear model followed the physical intuition,which have the similar indirect tensile strengths for the foliation perpendicular to the failure plane (φ=0°,70°,90°) and the decreasing indirect tensile strength with the decreasing angle between the foliation and the failure plane (φ=0°-40°) (Nova and Zaninetti,1990;Li et al.,2020).Therefore,for Asan gneiss,the nonlinearity can increase the evaluated indirect tensile strength by up to about 10%more than determined by the linear elastic model,and it can contribute to more physically convincing indirect tensile strength and its anisotropy.

4.4.Deformation of the Brazilian test

The lateral and axial strains at the center of the Brazilian disc were computed by 2D_Linear,3D_Linear,and 3D_Nonlinear models,and compared with the measured strains for Asan gneiss(Fig.12).For this analysis in Fig.12c,the strains of the Brazilian test were additionally sampled when the applied load reached the certain value causing the maximum compressive stress at the center of the Brazilian test model to reach 30 MPa.The relative difference of strains between 2D_Linear and 3D_Linear models was about 2% on average.After consideration of the nonlinear stressstrain relationship in addition to the 3D deforming effect,the accuracy of the estimated strains were greatly enhanced,achieving an average magnitude of 91%of the measured strains.Thus,effects of 3D deformation and nonlinear stress-strain relationship are important for modeling the deformation of the Brazilian test.

Fig.12.Comparison between the measured and modeled strains in axial and lateral directions at the center of the Brazilian disc using (a)2D_Linear model,(b) 3D_LInear model,and(c)3D_Nonlinear model(redrawn from Fig.17 in Yim et al.(2022a)).Measured strains were obtained as the mean of two symmetrically placed strains,and the trend line was drawn without outliers indicated by the mark with dotted line.The line y=x is drawn as a dashed line.

5.Conclusions

This study investigated the effects of 3D deformation and the nonlinear stress-strain relationship on the elastic behavior of the Brazilian test.The laboratory test for Asan gneiss was conducted to see those effects in an actual rock.While the inherent uncertainty of the Brazilian test,such as rock heterogeneity and measurement uncertainty,limits the precision of the indirect tensile strength,this study can contribute to enhance the accuracy of indirect tensile strength by eliminating the systematic error owing to the ignorance of actual physical mechanisms.This will allow more delicate hydraulic fracturing or excavation design for transversely isotropic rocks even though the relationship between the indirect and actual tensile strength requires further study.The main conclusions are drawn as follows:

(1) A number of 3D modeling of Brazilian test were performed as parametric study,and it was revealed that the plane stress condition can cause up to 40% error of the indirect tensile strength.By considering the 3D deforming effect,the indirect tensile strength of Asan gneiss increased by up to 10%and the accuracy of deformation estimation was enhanced slightly.

(2) A significant nonlinearity in stress-strain curve for Asan gneiss under uniaxial compression was observed at the low stress level where the Brazilian test was mainly performed.The nonlinear stress-strain curve was simply modeled by the elastic modulus linearly related to the compressive strain.By applying the nonlinear model,the indirect tensile strength was overestimated by up to about 10%,and its anisotropy was well agreed with the physical intuition.The accuracy of deformation estimation was greatly improved.If the nonlinearity is pronounced for a certain rock,this effect should be considered.Further validation or the study based on more rigorous nonlinear model are expected as future research.

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.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education(Grant No.2023R1A2C1004298),and a grant from the Human Resources Development program(Grant No.20204010600250) of the Korea Institute of Energy Technology Evaluation and Planning(KETEP),funded by the Ministry of Trade,Industry,and Energy of the Korean Government.

Appendix A.Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2023.06.013.