Effects of amygdale heterogeneity and sample size on the mechanical properties of basalt

Zhenjing Liu ,Chunsheng Zhng ,Chunqing Zhng ,Huin Wng ,Hui Zhou ,Bo Zhou

a School of Civil and Hydraulic Engineering,Huazhong University of Science and Technology,Wuhan,430074,China

b PowerChina Huadong Engineering Corporation Limited,Hangzhou,310014,China

c State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences,Wuhan,430071,China

d University of Chinese Academy of Sciences,Beijing,100049,China

Keywords:Amygdaloidal basalt Hard brittle rock Structural heterogeneity DFN-FDEM Mechanical properties Size-dependent effect

ABSTRACT Due to the complex diagenesis process,basalt usually contains defects in the form of amygdales formed by diagenetic bubbles,which affect its mechanical properties.In this study,a synthetic rock mass method(SRM)based on the combination of discrete fracture network(DFN)and finite-discrete element method(FDEM)is applied to characterizing the amygdaloidal basalt,and to systematically exploring the effects of the development characteristics of amygdales and sample sizes on the mechanical properties of basalt.The results show that with increasing amygdale content,the elastic modulus(E)increases linearly,while the uniaxial compressive strength(UCS)shows an exponential or logarithmic decay.When the orientation of amygdales is between 0°and 90°,basalt shows a relatively pronounced strength and stiffness anisotropy.Based on the analysis of the geometric and mechanical properties,the representative element volume(REV)size of amygdaloidal basalt blocks is determined to be 200 mm,and the mechanical properties obtained on this scale can be regarded as the properties of the equivalent continuum.The results of this research are of value to the understanding of the mechanical properties of amygdaloidal basalt,so as to guide the formulation of engineering design schemes more accurately.

1.Introduction

As a natural geological material,discontinuous structures of various scales are widely developed inside rock mass,ranging from microcracks and holes at a mesoscale to joints and fissures at a macroscale,and even faults up to several hundred metres long.At a small scale,the existence of such high heterogeneity signi ficantly affects the mechanical properties of rock,and at a large scale,it directly controls the safety and stability of the entire engineering rock mass.It has been accepted that brittle rock failure is often a dynamic evolution process from continuous to discontinuous,in which these native defects are constantly going through crack closure,initiation,propagation,and coalescence under the action of loading,eventually causing macroscopic fracture of the material(Esmaieli et al.,2010;Liu et al.,2018).

Consequently,laboratory tests and numerical simulations were used to investigate the in fluences of holes,cracks,and other defects on the fracture evolution,deformation and strength of rocks and rock-like materials.A variety of factors in fluencing rock failure mechanism,including the mechanical(un filled or filled and type of filling)and geometric(shape,length,quantity,and orientation)properties of defects,and stress conditions have been analysed(Jin et al.,2017;Aziznejad et al.,2018;Cao et al.,2018;Wang et al.,2018;Shaunik and Singh,2019;Yang et al.,2018;Zhou et al.,2018;Zhang et al.,2020).Research has suggested that the key contribution of defects is acting as stress concentration mechanism capable of producing tensile stress concentration at the tip of microcracks,resulting in local damages and failure of the samples.Hence,defects play a key role in brittle rock failure.

Considering the existence of discontinuities at various scales,the mechanical properties of a speci fic rock,such as strength and deformation,exhibit a certain scale-dependence.Hoek and Brown(1980)analysed laboratory test data and proposed an empirical size-effect formula in a normalised form,in which the uniaxial compressive strength(UCS)for rock samples with any diameter is related to that of rock samples of 50 mm in diameter.The synthetic rock mass method(SRM)is based on the numerical simulation that combines a discrete fracture network(DFN)model representing discontinuities with conventional numerical models representing the complete matrix,which can fully consider the scale and anisotropy effects of the characteristics of rock masses(Vöge et al.,2013;Zhang et al.,2015;Yang et al.,2017;Zhou et al.,2018,2020;Vazaios et al.,2018).Moosavi et al.(2018)used discrete element method(DEM)to evaluate the in fluence of microcracks with prede fined intensities and orientations on the tensile fracture behaviours of rocks,focusing on the analysis of changes in the apparent Young’s modulus,tensile strength,and fracture mode.Grif fiths et al.(2017)analysed the in fluences of the aspect ratios of pores in elliptical hole and the angle between the direction of applied stress and the major axis of pores on the mechanical behaviours of porous rock under uniaxial compression using RFPA program.Bahrani and Kaiser(2016)investigated sample size effect on the strengths of defective and intact rocks based on SRM model by integrating a particle flow code(PFC)with DFN representing defect geometries.An SRM model coupling DFN and universal distinct element code(UDEC)was employed to characterise the scaledependences of the mechanical and geometric characteristics of jointed rock masses(Farahmand et al.,2018).

As the solid part of rock mass,a rock block may contain defects with different structural characteristics,e.g.microcracks,holes and cemented joints,as shown in Fig.1,all of which may affect its mechanical properties under different loading conditions.On one hand,due to the inherent variability,it is dif ficult to use data obtained from tests on natural rock samples to reveal the in fluences of speci fic parameters.On the other hand,conventional scale-effect analysis mainly focuses on the macro-discontinuity structures at a normal engineering-scale,ignoring the mesoscale and submacroscale,while the discontinuities at different scales vary.In addition,the existing literature mainly focuses on microcracks and voids,however,until now,few studies have been conducted on basalt with amygdale defects.

Hence,aiming at rock blocks at laboratory scale,amygdaloidal basalt sampled from the Baihetan Hydropower Station was taken as the research object in this study.An SRM model based on a combination of DFN and finite-discrete element method(FDEM)was presented to characterise the rock,and the in fluences of amygdale heterogeneity and sample size on the equivalent mechanical characteristics of basalt were explored.First,the distribution characteristics of the major axis length,orientation,and minor axismajor axis ratio of amygdales were statistically analysed,which were used to establish DFNmodels.Then,they were embedded into the FDEMnumerical software to create SRMmodels.On this basis,a series of numerical tests under uniaxial compression was conducted to analyse the in fluences of amygdale heterogeneity(speci fic geometric parameters)on the mechanical properties of rocks separately,and to explore the size-dependences of geometric and mechanical properties of SRM models.

2.Structural characteristics of amygdaloidal basalt

Baihetan Hydropower Station,which is the largest hydropower plant being constructed globally,is located on Jinsha River lower reaches between Yunnan and Sichuan Provinces in southwest China(Fig.2a).Geologically,the engineering site is located in the basalt formation of Emeishan with main lithologies of cryptocrystalline and amygdaloidal basalt.Besides geological structures with large scales,including dislocation zones,faults,joints,and cracks,which are exposed in basaltic rock mass,amygdaloidal structures at a mesoscale are common in the basalt(Zhang et al.,2013;Dai et al.,2016;Xia et al.,2019).

Fig.1.Discontinuities at different scales in the rock mass and scale-dependence of rock mass properties.REV is the representative element volume.

Fig.2.Physical morphologies and microscopic structures of amygdaloidal basalt at Baihetan hydropower station:(a)Location of rock material,(b)Drilling cores from underground powerhouse caverns,(c)Basalt samples with varied geometric parameters of amygdales,and(d)A microscopic image of a thin section of amygdaloidal basalt.

As a kind of volcanic extrusive rock,basalt is a typical heterogeneous and usually contains heterogeneity in the form of holes and amygdales with varying size,shape,and abundance.The rock is cyan grey,dense and hard(Fig.2b).Vesicular textures are volcanic rock textures in which rocks are pitted with several cavities(known as vesicles)that form on the surface of the Earth in a phenomenon called extrusion.Amygdale is another related texture,where secondary minerals such as calcite,quartz,and chlorite fill vesicles.The complexity during the diagenesis process of basalt leads to great variability in the geometric parameters of amygdales,for instance,the content of amygdales in rocks may be in the range of about 0 to about 1,the size spans several orders of magnitude(generally from a few tens of microns to millimetres or even centimetres),and shape can vary from slender to spherical,as shown in Fig.2c.This huge heterogeneous structure is bound to exert an important effect on the failure mechanisms and physicomechanical characteristics of rock(Baud et al.,2014;Heap et al.,2016).

2.1.Meso-structural characteristics of amygdaloidal basalt

Fig.2d shows the orthogonal polarisation of amygdaloidal basalt.One can observe that there are many amygdales scattered thereon,accounting for about 20%-25%of the rock,and they are generally round and elliptical,with a characteristic size of 0.6-4 mm.They are mainly composed of chlorite mineral,with local quartz fillings.Matrix mineral composition mainly includes labradorite,cryptocrystalline,pyroxene,and a trace amount of chlorite.Labradorite is self-shaped lath with a major axis diameter range of 0.02-0.3 mm,and pyroxene is fine granular with a particle size range of 0.01-0.04 mm.Moreover,black vitreous material is a cryptocrystalline aggregate left uncrystallized during lava eruption.

In order to understand the physical properties of basalt,the density and longitudinal wave velocity of amygdaloidal basalt and cryptocrystalline basalt(without amygdales)for better comparison were tested,as shown in Fig.3.It can be seen that the density of amygdaloidal basalt is basically in the range of 2650-3000 kg/m3,with an average value of about 2800 kg/m3,and the longitudinal wave velocity is mainly in the range of 3000-5000 m/s,with an average value of about 3900 m/s.The density of cryptocrystalline basalt is basically at 2750-3000 kg/m3,with a mean value of about 2900 kg/m3,and the wave velocity is basically at 4000-5000 m/s,with a mean value of about 4500 m/s.Compared with cryptocrystalline basalt,amygdaloidal basalt has a larger density and wave velocity distribution range,and the density and wave velocity values are signi ficantly reduced,which re flects the defect structures in rock,showing a poor uniformity.It can be seen that amygdale structures with strong randomness and large dispersion inside basalt signi ficantly affects its physical properties.The inhomogeneity of rock itself is bound to impose an important impact on its mechanical properties,leading to the discreteness and volatility of test results.To a certain extent,this increases the difficulty of testing and analysis.Therefore,it is necessary to conduct numerical analysis.

2.2.Statistics pertaining to amygdales

Fig.3.Test results of(a)density and(b)sound wave velocity of basalt at Baihetan hydropower station.

To construct the DFN model of amygdaloidal basalt(similar to DFN),the statistical analyses of the speci fic geometric parameters,such as density,orientation,and size of amygdales,were described to obtain the corresponding probability distribution models.This information can be used as the input parameters for generating DFN models(Lei et al.,2017).Since computed tomography(CT)scanning is expensive and the geometric analysis of amygdale morphology is dif ficult,scanning of the interior of suf ficient amygdaloidal basalt samples is impossible;hence,measurement of amygdale statistics and geometric parameters was performed on only the surface of a large number of standard cylindrical samples,and the two-dimensional(2D)distribution characteristics were used to describe the three-dimensional spatial characteristics in the interior of the rock.Since most of the amygdales are approximately elliptical in two dimensions,they were simpli fied as equivalent ellipses.The geometric parameters obtained mainly include major axis orientationθ(considered as the angle between vertical axis and major axis),major axis lengthl,and the length ratio of minor to major axes of ellipseα.

According to the statistics pertaining to 2500 amygdales with a major axis length≥1 mm in 18 cylindrical samples(Zhang et al.,2020),the corresponding statistical results of geometric parameters are as follows:θconforms to a uniform distribution,andlexhibits a negative exponential distribution(λ=0.25,correlation coef ficient=0.96),whereλis a parameter of the negative exponential distribution,which is often called the rate parameter.α conforms to a normal distribution(μ=0.63,σ=0.17,correlation coef ficient=0.98),whereμis the mathematical expectation of the normal distribution andσrepresents the standard deviation.Moreover,after calculation,the areal densityP20is 1.12 per cm2,and the amygdale content(areal densityP21)is 14.66%(areal densityP20is de fined as total amygdale number per sample unit area,and areal densityP21is total amygdale area per sample unit area).

3.Generation of the SRM model of amygdaloidal basalt

This section introduces the generation of the SRM model of amygdaloidal basalt combining a DFN model and an FDEM model,and the calibration process of corresponding meso-parameters.Although the actual basalt and amygdale structures are three dimensional,considering the limitations of computational capacity and time,they are treated as 2D scales.

3.1.Fundamental principles of FDEM

Combined FDEM is a numerical approach wherein a continuum mechanics principle(finite element method(FEM))is combined with a discontinuous mechanics algorithm(DEM)for the simulation of several interacting deformable solid materials(Munjiza,2004;Mahabadi et al.,2010;Lisjak and Grasselli,2014;Yan et al.,2018).The modelling domain of FDEM is classi fied using a mesh comprising 3-node triangular finite elements and 4-node cohesive crack elements embedded coincident with all adjacent triangle pair edges.Elastic deformation of intact material is modelled based on linear elasticity continuum theory by employing constant-strain triangular finite elements.The calculation of contact forces is performed among all element pairs overlapping in space.Repulsive forces along normal direction among elements in contact with each other are determined according to a distributed contact force penalty function,whereas frictional forces among contacting couples are determined by a Coulomb-type friction law.

On the basis of the displacements of relative crack wall and local stress as well as Mohr-Coulomb and maximum tensile stress failure criteria,crack elements applied for the simulation of material progressive failures could experience mode I(tensile failure),mode II(shear failure),or mixed-mode I-II(tensile-shear failure)fracturing and yielding.Because multiple mechanical models and criteria for the initiation,propagation and coalescence of cracks are taken into account in the current approach,with no prior assumptions for the modes and paths of failure,cracks can freely expand within divided grids on the basis of the state of deformation and stress,including deformation,rotation,interaction,fracture,and fragmentation.Hence,the model could take into account progressive failure processes of brittle geo-materials from continuous to intermittent and finally discontinuous.

3.2.SRM model of amygdaloidal basalt

For amygdaloidal basalt,after determining the probability distribution forms and characteristic parameters of the geometric parameters of amygdales,a DFN model of amygdales could be established,and then a hybrid DFN-FDEM model representing amygdaloidal basalt can be established,which truly and accurately re flects the meso-structural characteristics of this rock.The corresponding generation steps are described as follows:

(1)Study area dimension was first determined,and corresponding ellipse number was calculated on the basis of the amygdale density. Taking the model measuring 50 mm×100 mm(width×height)as an example,the number of ellipses is 56.

(2)According to the above statistical results of the geometric parameters of amygdales,the corresponding random variables were generated by Monte Carlo method,and the endpoint coordinates of major axis and minor axis of ellipses were calculated to generate a set of random ellipses.

(3)There was no overlapping among ellipses and ellipse-model boundaries,otherwise,new coordinates of centroid were assumed until all ellipses met the requirements.It should be noted that the random ellipse generated in this study is not a real DFN.Therefore,a similar DFN model is used,whose generation method and program are basically consistent with DFN model.

(4)The generated DFN model of amygdales was embedded into an FDEM code Irazu representing the complete matrix to construct an SRM model(DFN-FDEM)representing amygdaloidal basalt,which can consider the material heterogeneity at a mesoscale of the rock.The mechanical behaviours of the SRM model are controlled by the mechanical behaviours of rock matrix and amygdales.

The dimensions of the numerical samples of amygdaloidal basalt for calibration of meso-parameters were 50 mm×100 mm(width×height),with the side length of the triangular elements being 1 mm,and two plates of loading were set on lower and upper ends(Fig.4).A vertical displacement rate of 0.05 m/s with equal magnitudes along opposite directions at lower and upper ends was applied to achieving the axial loading(Abdelaziz et al.,2018).Various con fining stress values corresponding to laboratory experiments were employed at the right and left boundaries of model and a plane strain model with 5×10-7ms time step was adopted.Axial strain and stress were obtained through monitoring and calculating nodal displacements and forces on lower and upper loading plates,while lateral strains were determined based on monitoring and calculating node displacements at the middles of two sample sides,with approximately 10 mm width,in the same way to the measurement chain of hoop strain in the laboratory tests conducted using MTS equipment.

3.3.Calibration of meso-parameters in SRM model and numerical results

Fig.4.An SRM model of amygdaloidal basalt for calibration of meso-parameters.

The meso-parameters used in FDEM model are classi fied as elastic parameters of triangular element(Poisson’s ratioν,elastic modulusE,and bulk densityρ),strength parameters of crack element(fracture energyGf,internal friction angleφ,cohesionc,and tensile strengthσt),and penalty values(Pn,PtandPf).Typically,fracture energyGf1of tensile cracks is determined based on fracture toughnessKIcestimated based on tensile strength derived from Brazilian disc splitting tests or obtained through three-point bending tests,and fracture energyGf2of shear cracks is approximately 10Gf1.The penalty values(Pn,PtandPf)are usually set at 10-100 times higher than elastic moduli.The model comprises two materials(amygdales and matrix),wherein the values of parameters for matrix are determined by referring to cryptocrystalline basalt(pure matrix)test results and they are reduced appropriately to give amygdale parameter values as the main component of amygdales is weaker chlorite(Liu et al.,2019).The interface between amygdale and matrix is simulated by a cohesive crack model.The calibration procedure used in the present work is an iterative process(a trial-and-error approach)which is performed through a series of numerical tests(uniaxial compression,biaxial compression and Brazilian disc splitting)to determine multiple mesoparameter values(Table 1),so that the macro-mechanical parameters of numerical simulation match the test results(Wang and Cai,2019).

Fig.5 shows the typical deviatoric stress-strain curves for amygdaloidal basalt samples exposed to various con fining stress levels from numerical simulations and laboratory tests,in which the solid line represents the numerical results,and the dotted line represents the test results.It can be seen from the dotted line that under low con fining stresses,the concave-up trend of the curve during the initial compaction stage is more obvious,which re flects the compaction response of a sample containing many initial defects(such as amygdales)in rock.Under increasing loading,the prepeak curve is linear and the post-peak stress drops rapidly,showing a signi ficant brittleness.As the con fining stress increases,the initial compaction stage of the curve almost disappears.The pre-peak curve is quasi-linear,and the ductile yield plateau and zig-zag pattern are developed close to peak stress,with the range of stress fluctuations becoming increasingly wide.In addition,the post-peak stress decreases slowly,showing a transition from brittleness to ductility,which is similar to the brittleness-ductility transformation characteristics exposed under medium and high con fining stress values for other hard brittle rock types such as marble(Liu et al.,2012;Zong et al.,2016).

It can be seen from the solid line that the initial compaction stage of curve of numerical samples is not obvious,and the mechanical behaviours of pre-peak curve show linear characteristics.The curve near the peak is zig-zag shaped,with the stress continuing to rise after a small drop,and the post-peak stress decreases.With the increase of con fining stress,curve fluctuation range around the peak is increased progressively,yield platform appears,and post-peak stress is decreased slowly,gradually transforming from brittle to ductile behaviour.In summary,the stress-strain curves drawn by laboratory tests and numerical simulations are consistent.

According to Table 2,the macro-mechanical parameters of samples derived from numerical simulations agree with laboratorytest results,indicating that numerical model and the selected meso-parameters used in the present research are reliable,based on which the numerical results can depict the main mechanical behaviours of amygdaloidal basalt.

Table 1Meso-parameters of amygdaloidal basalt in FDEM(Zhang et al.,2020).

Fig.5.Deviatoric stress-strain curves of amygdaloidal basalt samples(Zhang et al.,2020).

4.Effect of amygdale heterogeneity on mechanical characteristics of basalt

In this section,a series of numerical tests was performed based on the created SRMmodel of basalt to explore the in fluence of this structural heterogeneity on the equivalent mechanical characteristics of rock from the quantitative point of view,focusing on the analysis of the in fluences of speci fic geometric parameters,such as the numberm,sizel,orientationθ(angle between major axis direction and loading direction or vertical direction)and aspect ratio αof amygdales.

4.1.Numerical simulation scheme

Table 3 lists the numerical simulation algorithm on the basis of geometric factor changes.For the four geometric variablesm,l,α andθ,when one of them changes,other three variables remain unchanged.The typical geometric models generated by the change of each variable are shown in Fig.6,wherein the geometric dimensions are the same as those shown in Fig.4.In addition,the numerical samples were obtained on the basis of statistical findings presented in Section 2.2.Due to a high time-cost of numerical calculation in FDEM,it is impossible to conduct the calculation for many samples.In addition,to ensure the statistical reliability of the calculation results,20 realisations were generated for each numerical test scheme in the present study.

Table 2Macro-mechanical parameters of amygdaloidal basalt from laboratory test and numerical simulation(Zhang et al.,2020).

Table 3Numerical simulation scheme of in fluence of geometric parameters of amygdales on strength.“↑”indicates that a certain parameter changes,and“-”indicates that a certain parameter remains unchanged.

For each sample,the numerical simulation of uniaxial compression test was carried out,whose boundary conditions and meso-parameters used are consistent with those in the calibration model(standard sample).Based on the numerical results,the influence of the heterogeneity of amygdale structure on the equivalent mechanical properties of rock was quanti fied.

4.2.Numerical results and analysis

Peak stressσf,crack damage stressσcd,and crack initiation stress σciare characteristic stress thresholds between several important stages representing the progressive failure process of brittle rock.They are of signi ficance in the understanding of the progressivefracture process of rock and failure mechanism and predicting failure range and long-term stability of surrounding rock mass in underground engineering operations conducted under high-stress conditions.In laboratory tests on intact rock,σciis described as the stress threshold of stable crack growth,corresponding to the occurrence of micro-fractures,which is approximately 40%-60%of σf.σcdis assumed to be the stress of the appearance of unstable cracks,which corresponds to coalescence and penetration of a large number of microcracks,at around 70%-85%ofσf.σfis widely applied for establishing the envelope curves of rock strength.There are several methods used to determine these characteristic stresses.These characteristic stresses(σci,σcdandσf)of amygdaloidal basalt are estimated from the analyses of crack volume-strain curve,total volume-strain curve,and stress-strain curve of the numerical samples in this research,combined with the trend in evolution of microcrack number and lateral strain difference method(LSR),in whichσci/σfandσcd/σfare characteristic strength ratios(for details of these methods,please refer to Martin and Chandler(1994),Cai et al.(2004),Hoek and Martin(2014),and Zhao et al.(2015)).

4.2.1.Effectofthequantityofamygdales

To investigate the in fluence of amygdale number on the equivalent mechanical characteristics of basalt,the number of amygdalesmin the numerical samples is set to 14,28,56,84,112 and 168,a total of six working conditions,wherem=56 is the average value of the number of amygdales.The relationship between the mechanical properties of basalt by numerical simulation and the number of amygdales obtained is shown in Fig.7,where the scattered points represent the values of mechanical properties in different samples calculated under the same working condition,and the broken line points denote the average values of the corresponding properties of all samples calculated under the same working condition(as below).

The Young’s modulusEdecreases linearly from 43.59 GPa to 26.5 GPa with the increase in number of amygdalesm,a reduction of 40%.The UCS(peak strengthσc)shows logarithmic decay,decreasing from 350 MPa to 84.04 MPa,a reduction of 76%.Meanwhile,the decreasing trend is most obvious whenmis small and tends to be stable asmincreases.The trends inσci,σcdandσcd/σcare similar to that ofσc,whileσci/σctends to decrease signi ficantly at first and then increases slightly.In addition,with the increase inm,the overall discreteness of the calculated results decreases.In other words,as the number of amygdales increases,the overall area occupied by them also increases,and the macromechanical properties of samples tend to be uniform.

Fig.6.DFN models of amygdaloidal basalt with varied geometric parameters for uniaxial compression tests:(a)Number of amygdales(m),(b)Size of amygdales(l),(c)Aspect ratio of amygdales(α),and(d)Orientation of amygdales(θ).Twenty realisations were generated for each set of geometric parameter conditions.

Fig.7.In fluence of number of amygdales on the mechanical properties of basalt:(a)E,(b)σc,(c)σci,(d)σcd,(e)σci/σc,and(f)σcd/σc.

The relationships ofEandσcwithmobtained through leastsquares fitting are as follows:

4.2.2.Effectofthesizeofamygdales

To investigate the in fluence of amygdale size on the equivalent mechanical properties of basalt,the major axis length of amygdaleslis taken as 2 mm,3 mm,3.5 mm,4 mm,5 mm and 6 mm.This gave a total of six working conditions,among whichl=3.5 mm is the mean value of the major axis length of amygdales.The relationship between the mechanical properties of basalt as derived from numerical simulations andlis presented in Fig.8,in which the changes inEandσcis close to those shown in Fig.7.

Aslincreased,the change inEis similar to that when the parametermis changed,and the change is linear.The UCS(peak strengthσc)decreases from 350 MPa to 75.81 MPa with the increase inl,which decreases by 78%in an exponential manner.Meanwhile,the reduction rate ofσcis larger whenlis smaller.The σci,σcdandσcd/σcare consistent with the trend inσc,while theσci/σcfirst decreases and then increases.In addition,aslincreases,the global discreteness of the calculated results also decreases signi ficantly.

Through least-squares fitting,the relationships ofEandσcwithlare as follows:

4.2.3.Effectoftheaspectratioofamygdales

To better comprehend the effect of amygdale aspect ratio on the mechanical characteristics of basalt,minor-major axes ratioαis set to 0.3,0.5,0.65,0.8 and 1 giving five operating conditions,where α=0.65 is the mean value of the aspect ratio of amygdales.The relationship between the macro-mechanical properties of basalt derived from numerical simulations andαis presented in Fig.9,where the changes inEandσcis close to those shown in Figs.7 and 8.

With the increase inα,Edecreases linearly from 43.59 GPa to 32.97 GPa,with a reduction of 25%,and the UCS(peak strengthσc)undergoes logarithmic decay,decreasing from 350 MPa to 110.1 MPa,with a reduction of 70%.The trends inσci,σcdandσcd/σcare similar to that ofσc,whileσci/σcdecreases signi ficantly at first and then increases slowly.

The relationships ofEandσcwithαobtained through leastsquares fitting are as follows:Therefore,amygdaloidal basalt exhibits relatively signi ficant strength and stiffness anisotropy.

Fig.8.In fluence of amygdale size on the mechanical properties of basalt:(a)E,(b)σc,(c)σci,(d)σcd,(e)σci/σc,and(f)σcd/σc.

4.2.4.Effectoftheorientationofamygdales

The mechanical behaviour of rock with defects is largely controlled by the spatial distribution characteristics of the defects,especially when these defects are oriented in a certain direction.The macro-mechanical response of rock often shows anisotropic characteristics.To grasp the in fluence of orientation arrangement of amygdales on the macro-mechanical properties,the orientation θis taken as 0°(parallel to the loading direction),15°,30°,45°,60°,75°and 90°(perpendicular to the applied loading),giving a total of seven working conditions.The relationship between the macromechanical properties of samples obtained by numerical simulation andθis presented in Fig.10.

It is observed that with the increase in orientationθof amygdales in the range of 0°-90°,the change trend in the macromechanical properties is not obvious.The Young’s modulusEdecreases slowly from 36.81 GPa to 35.4 GPa,i.e.Eis the largest when the amygdales lay parallel to the loading direction.The UCS(peak strengthσc),σciandσcddecrease slightly at first and then increase,all of which are the lowest whenθ=60°.The trends inσci/σcand σcd/σcrun contrary to those ofσciandσcd,which first increase and then decrease,i.e.the characteristic strength ratios are minimised when the dip angle is 90°(parallel to the loading direction).

5.Scale-dependences of geometrical and mechanical properties of amygdaloidal basalt

5.1.Numerical simulation scheme

Firstly, a 2D DFN model with suf ficiently large size(2000 mm×2000 mm)was generated based on the statistical data of distribution characteristics for the amygdales.Then,random sampling was performed within its range to generate multiple DFN models of smaller sizes,whose widthdis 25 mm,50 mm,75 mm,100 mm,150 mm,200 mm,300 mm,400 mm and 500 mm,respectively,with a constant aspect ratio of 2,as shown in Fig.11.Based on the measured amygdale content(areal densityP21),the size-dependence of the geometric parameters of amygdaloidal basalt blocks was analysed to obtain the corresponding REV.On this basis,some of the DFN models above were embedded into the FDEM to create corresponding SRMmodels of different sizes.According to the aforementioned boundary conditions and meso-parameters,a series of numerical tests under uniaxial compression was conducted on these SRM models to analyse the in fluence of sample size on the strength of amygdaloidal basalt blocks.Similarly,to eliminate this potential randomness and take into account the constraints of computing time,the SRM models were created 3-20 times at each scale;the larger the size,the smaller the number of modes.

Fig.9.In fluence of aspect ratio of amygdales on the mechanical properties of basalt:(a)E,(b)σc,(c)σci,(d)σcd,(e)σci/σc,and(f)σcd/σc.

5.2.Numerical results and analysis

5.2.1.Size-dependenceofgeometricalparameters

To assess the in fluence of sample size on geometric properties of amygdaloidal basalt blocks,the amygdale content(areal densityP21)and its coef ficient of variation(CoV)of multiple DFN models at different sizes were measured.The REV of geometric properties was determined based on the variation degree ofP21.As shown in Fig.12,the scattered points represent discrete data ofP21of different samples(20 samples)at the same size,the blue line denotes the mean of these discrete data,and the orange line denotes its CoV which is described as standard deviation-average value ratio.Also,in this study,an acceptable CoV is less than 10%(Farahmand et al.,2018).

The average value ofP21of different samples at each size is unchanged(fluctuating slightly only at small sizes),at about 14.5%,which is consistent with the data obtained from the statistical characteristics of amygdales.This indicates that no matter how large the sample size,the defect content in the sample is similar,and it does not increase gradually with increasing size.With the increase in sample size,the CoV decreases nonlinearly,and the decrease trend is more pronounced at small sizes.For the samples with small sizes(smaller than 100 mm),the CoV is more than 20%,andP21dispersion across different samples of the same size is obvious,indicating that the number of defects within the samples at these sizes cannot represent the structural characteristics of the overall rock block.When the sample sizes are 100-150 mm,the CoV decreases to 10%-15%,at which the fluctuation range ofP21becomes signi ficantly narrower.However,for samples larger than 200 mm,the CoV is below 10%,andP21tends to be stable,indicating that defect population in samples is high enough to represent the overall conditions of rock block.Hence,the geometric REV size of 200 mm was characterised on the basis of aforementioned parameter variabilities.

5.2.2.Size-dependencyofthestrength

In this section,the strength and CoV of multiple SRM models at different sizes were calculated,and the mechanical REV was obtained according to variability degree change of UCS(i.e.σc).As shown in Fig.13a,as the sample size increases,the average value and CoV ofσcof SRM models decrease,especially for smaller samples,and remain almost constant beyond a certain size.In addition,dispersion degree and fluctuation range of different samples of the same size decrease signi ficantly.For sample smaller than 100 mm,the CoV is greater than 15%,and the fluctuation range ofσcis notably wider,which indicates that a model with this dimension cannot be considered as statistically homogeneous due to the lack of the possibility of obtaining stable ranges of properties.When the sample sizes are 100-150 mm,the CoV decreases to about 10%,and the fluctuation range ofσcat this size is narrower.For samples larger than 200 mm,the CoV is always below 10%,which is below acceptable value of CoV to establish REV size,and overallσcat this size tends to be asymptotic.Therefore,the mechanical REV size of 200 mm was characterised based on the dispersion degree and fluctuation range of theσcof samples with different sizes.Theσcat the REV size is approximately 96.87 MPa,about 71%of that of the standard sample.

Fig.10.In fluence of orientation of amygdales on the mechanical properties of basalt:(a)E,(b)σc,(c)σci,(d)σcd,(e)σci/σc,and(f)σcd/σc.

Similarly,with the increase insample size,the dispersionand mean value ofσciandσcdfor different samples with the same size exhibit a similar trend to that inσc.On the contrary,the average value ofσci/σcandσcd/σcfor different samples with the same size keeps increasing withthe sizeand tends to bestable.This shows that withtheincrease in sample size,the established SRMmodels are more uniform,and the characteristic strength ratios are continuously increasing,which can represent the entire rock mass and its condition.

In summary,an ultimate REV of 200 mm for amygdaloidal basalt blocks was determined according to the size-dependence of mechanical and geometric features.The mechanical features achieved at this scale are regarded as equivalent continuum properties and may be applied as the input parameters for larger-scale rock mass simulation based on a continuum approach.

6.Discussion

6.1.Effect of heterogeneity content on the strength

Based on the analysis in Section 4.2,the effects of the number,size and aspect ratio of amygdales on the mechanical properties of basalt are similar because they all affect the amygdale content(areal densityP21).Therefore,a relationship between the content of amygdales and rock strength was revealed and this result was compared with those from the literature.The interior of a rock mass contains defects with different structural features,such as microcracks,holes(filled holes)and amygdales,all of which may affect their mechanical behaviours.Fig.14 shows the relationship between the defect content of different rock types and the UCS.

It can be seen from Fig.14 that the best fitting curve of the relationship between simulated UCS(i.e.σc)and microcrack density in intact Lac du Bonnet(LdB)granite is an exponential decay function,in which with the increase in microcrack density from0 to 0.2 mm/mm2,σcdecreases from about 190 MPa to 100 MPa(Hamdi et al.,2015).The relationship betweenσcof the volcanic rock containing circular holes with a diameter of 0.5 mm and porosity is quasi-exponential,in whichσcdecreases from 548 MPa at porosity of 0-47 MPa at porosity of 0.4,with a strength reduction factor of 0.91.For the volcanic rock containing circular crystals(filled holes)with a diameter of 1 mm,theσcand crystal fraction show a quasilogarithmic relationship,in which theσcdecreases from 458 MPa to 337 MPa by enhancing crystal fraction from 0.02 to 0.4,with a strength reduction factor of 0.38.Therefore,increasing porosity or crystal fraction decreases volcanic rock strength on which the effect of pore fraction is greater than that of crystal fraction(Heap et al.,2016).Similar to the volcanic rock with circular crystals,the best fitting curve between amygdale content andσcof basalt in this study shows a logarithmic relationship,in which the strength reduction of basalt is more signi ficant at lower amygdale contents,and the strength tends to be stable at larger amygdale contents.With the increase in amygdale content from 0 to 0.44,theσcdecreases from about 350 MPa to as low as 75.81 MPa,with a strength reduction factor of 0.78.

Fig.11.SRMmodels of amygdaloidal basalt blocks with different sizes:(a)DFN models with different sizes,and(b)SRM model with size of 100 mm×200 mm.

Fig.12.In fluence of sample size on P21.

Therefore,defects play key roles in controlling rock strength,which decreases with increasing defect content;however,for different types of defects,such as microcracks,holes,crystals(filled holes),or amygdales,the change in strength caused by defect content differs.For defects,such as microcracks and holes,the strength decreases exponentially with increasing defects;however,when the defects are filled holes,such as crystals or amygdales,the strength decreases logarithmically with increasing defect content.In addition,the reduction inσcof rock with filled holes(including volcanic rock with crystals and amygdaloidal basalt)is smaller compared with that in porous volcanic rock.

6.2.Effect of heterogeneity angle on the strength

Various kinds of defect fabrics are found,such as bedding in sedimentary rocks,schistosity and gneissism in metamorphic rocks,and holes in volcanic rocks.These signi ficantly affect the mechanical properties of the rocks.The results in this study were compared with the experimental or numerical data of strength anisotropy in the recent literature(Fig.15).For sedimentary rocks and metamorphic rocks,the preferential orientation of defects on the horizontal axis is in accordance with the angle between schistosity or bedding and applied stress.For volcanic rocks with holes or amygdales,orientation is in accordance with the angle between major axis of amygdales or holes and applied stress.The normalised UCS(i.e.σc)on the longitudinal axis is expressed asσcθ/σc0,a ratio representing the strength anisotropy,whereσc0is de fined as theσcat a defect orientation of 0°,andσcθrepresents the σcor peak stress at other defect orientations.

For shale or gneiss,as this orientation of bedding or schistosity increases from 0°to 90°,theσcfirst decreases and then increases(with a maximum at 0°or 90°and minimum at about 45°)(Cho et al.,2012).For elliptical voids in in finite media,volcanic rocks or basalts with holes,theσccontinues to decrease with the increase in the angle between applied stress and major axis of holes from 0°to 90,i.e.the sample strength is the highest when the major axis direction of holes is parallel to the loading direction,and it is the lowest when the major axis direction of holes is perpendicular to the loading direction.This is consistent with the analysis of the concentration of stress around one elliptical hole(Bubeck et al.,2017;Grif fiths et al.,2017).Therefore,defective rocks containing bedding or elliptical pores with preferential orientation can exhibit considerable strength anisotropy.For amygdaloidal basalt,as the angle between loading direction and major axis of amygdales increases,theσcfirst decreases and then slightly increases(with a maximum at 0°and minimum at 60°).Its overall trend is slightly different from that of other rocks,and the decrease in strength is small,leading to lower strength anisotropy compared to other geologic media.This is because an amygdale is a result of pores being filled by other minerals under later geological action,which is not in complete consistency with the effects of simple bedding and pores.

Crustal rock strength anisotropy is usually assigned to the preferential alignment of the existing defect structures such as layer planes,holes or fillings inside.However,strength evolution for the angle between each of these fabric directions and applied stress is markedly different.Several mechanical anisotropy sources in crust along with their different contributions thereto indicate the signi ficance of orienting rocks either catalogued or collected in field and provide a comprehensive description for their textural heterogeneity.

6.3.In fluence of sample size on the strength

Generally,the effect of sample size refers to the effect of rock sample size with a constant ratio of slenderness,including the scale effect of different properties,such as geometry,mechanics,hydraulics and temperature.The most commonly applied size effect relation was introduced by Hoek and Brown(1980),who suggested a power function in normalised form to correlate theσcof intact rock samples with a certain diameterdto that of a sample with 50 mm in diameter,as presented in Fig.16.The equation is expressed asσcd/σcd50=(50/d)0.18,in whichσcdandσcd50are theσcvalues of the samples with arbitrary and 50 mm diameter,respectively;anddis the sample diameter(mm).The normalisedσcdecreases with the increase in sample size,which is attributed to the enhanced heterogeneity of rock by volume and greater probability of micro-defects to allow unstable propagation of cracks.When the sample contain enough defects,a constant value of strength may appear corresponding to the REV,which is the smallest volume where the test results are independent of sample size and can represent the whole rock.

However,Hoek and Brown(1980)’s formula only represents rocks with good integrity and homogeneity and may overestimate the strength of samples with micro-defects or those affected by weathering or temperature(Stavroua and Murph,2018).For the marble samples heated at the temperature of 200°C,the variation trend of normalisedσcwith sample size agrees well with Hoek and Brown(1980)’s model.On the other hand,strength decrease after thermal damage with increasing sample size is intensi fied,thus increasing model’s exponent(Guan et al.,2018).For amygdaloidal basalt,due to the presence of more defects on a mesoscale,the size effect of strength is more severe.Therefore,there is no general law applicable to the size effect of rock strength,which has a close relation with lithology,heterogeneity type,and the structure and size of fractures.

Fig.13.In fluence of SRM sample size on the strength:(a)σc,(b)σci,(c)σcd,(d)σci/σc,and(e)σcd/σc.

The size effect of the strength of rock mass is a key bridge from which to establish the relationship between the mechanical properties of rock sample and in situ rock mass,which can provide an important basis for the estimation of mechanical features of rock masses based on the test results of mechanical properties of standard laboratory samples.When the sample size increases gradually from the standard sample size in the laboratory,the size and structure of the defects in the samples constantly change,and the mechanical properties of the samples change accordingly.For the same sample size,the distribution of defects in the samples is obtained by random sampling,thus the mechanical properties of samples exhibit natural variability.Therefore,the mechanical properties mentioned here refer to the average value of a certain quantity of samples.Of course,with the increase in sample size,the discreteness in the sampling process becomes smaller,and the defect density in the samples is unchanged regardless of sampling method.Therefore,the discreteness of their mechanical properties also decreases accordingly.

7.Conclusions

In this study,aiming at the rock blocks of amygdaloidal basalt at the laboratory scale,an SRM model based on the combination of DFN and FDEM was presented to explore the in fluences of the heterogeneity of amygdale structure and sample size on the equivalent mechanical characteristics of basalt.The following conclusions are drawn:

(1)The SRM model,which is established by combining the DFN model generated from the statistical characteristics of the geometric parameters of amygdales and an FDEM model representing the intact matrix,can represent the mesostructural heterogeneity of the rock,and accurately reproduce the mechanical behaviours of amygdaloidal basalt observed in laboratory tests,such as failure modes and characteristics of stress-strain curve.

(2)The presence of amygdales plays a key role in controlling the mechanical characteristics of basalt.With the increase in amygdale content(quantity,size and aspect ratio),Eincreases linearly,while the UCS(peak strengthσc)shows a exponential or logarithmic decay.The trends inσci,σcdand σcd/σcare similar to that ofσc,whileσci/σctended to decrease signi ficantly at first and then increased slightly.

Fig.14.In fluence of defect content onσc.

(3)With the increase in amygdale orientationθin the range of 0°-90°,Edecreases gradually,with a maximum found when the amygdales lie parallel to the direction of loading.The UCS(peak strengthσc),σciandσcdfirst decrease and then increase,with high value at both ends and a low value in the middle;on the contrary,σci/σcandσcd/σcfirst increase and then decrease,being lowat both ends and high in the middle.Therefore,amygdaloidal basalt exhibits relatively signi ficant strength and stiffness anisotropy.

(4)An REV size of 200 mm for amygdaloidal basalt blocks was determined according to the scale-dependence of mechanical and geometric characteristics.The mechanical features obtained at this scale are regarded as equivalent continuum properties and may be applied as the input parameters for the simulation of large-scale rock masses based on a continuum approach.Hence,the strength of rock obtained through laboratory tests on samples with small scales cannot be applied for the estimation of the strength of rock blocks.The REV size and strength of rock blocks could be determined through scale effect analysis.

Fig.15.In fluence of defect orientation onσc.

Fig.16.In fluence of sample size onσc.

Declarationofcompetinginterest

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

Acknowledgments

This work is supported by the Key Projects of the Yalong River Joint Fund of the National Natural Science Foundation of China(Grant No.U1865203),the Key Program of National Natural Science Foundation of China(Grant No.41931286),and the China Postdoctoral Science Foundation(Grant No.2021M691147).The authors also thank Geomechanica Inc.for the use of Irazu simulation software and Dr.Omid Mahabadi for his patient guidance in using the software and valuable suggestions regarding this manuscript.

Listofsymbols

P21Areal fracture intensity

σ3Con fining stress

σ1-σ3Deviatoric stress

σcUCS

σfPeak strength

σciCrack initiation strength

σcdCrack damage strength

σci/σfInitiation strength ratio

σcd/σfDamage strength ratio

ρ Bulk density

EYoung’s modulus

ν Poisson’s ratio

φ Internal friction angle

cCohesion

σtTensile strength

KIcMode I fracture toughness

Gf1Mode I fracture energy

Gf2Mode II fracture energy

PnNormal contact penalty

PtTangential contact penalty

PfFracture penalty

ε1Axial strain

ε3Lateral strain