ResearchofTransitionalFailureModeasDamageEvolutionin …downloads.hindawi.com/journals/ace/2020/8864074.pdf · sandstone would show a deviation between diﬀerent sizes. When the

Research ArticleResearch of Transitional Failure Mode as Damage Evolution inRock Wall

Xiao-guang Li12 Changhong Li1 Yuan Li 1 and Pu-jin Zhang3

1Department of Civil Engineering University of Science and Technology Beijing Beijing 100083 China2Dongcheng District Committee of the Communist Youth League Beijing 100007 China3China Tiejian Real Estate Group Co Ltd Beijing 100039 China

Correspondence should be addressed to Yuan Li sbfqp126com

Received 13 May 2020 Revised 3 August 2020 Accepted 14 August 2020 Published 7 September 2020

Academic Editor Guoqing Cai

Copyright copy 2020 Xiao-guang Li et al+is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

+e stress condition of tunnel surrounding rock mass is complex +e stress concentration of in situ brittle rock mass caused byexcavation results in localized damage evolution parallel to the free face which is called surface instability+e rock wall shows thetransition characteristics of the failure mode with the distance from the surface to the depth Low strength surface instability andtransition failure modes of the tunnelrsquos rock wall are common in deep condition but cylindrical specimens cannot simulate stressstate of rock wall surface well in conventional rock mechanics tests +is paper conducted the indoor experimental study of thebiaxial stress state and studied the surface instability of samples An indoor test device for the simulation of transitional surfacefailure of the rock wall was developed+rough a biaxial stress loading test on the rectangular rock sample the damage process andcrack development of rock samples were analyzed and the law of stress and strain related to the failure mode transition wascharacterized as well Based on test results and strength analysis an explanation of the failure theory and its corresponding modelare proposed based on the maximum strain strength theory Furthermore this paper concludes that the failure mode of surfaceinstability presents transition feature from brittle to ductile with the increase of distance from the surface to depth

1 Introduction

With the construction of tunnel engineering petroleumindustry and disposal tunnel of nuclear wastes in the past40 years the deformation and failure characteristics of deepbrittle hard rocks have been extensively investigated andstudied Excavation in deep hard rock causes a suddenrelease of high intensity energy stored in the rock masswhich plays an important role in inducing surface spallingand rock burst [1ndash3] It has been widely accepted that thespalling and rock burst near the surface of rock wall is atypical brittle failure [4 5] W f Brace et al studied theentire process of brittle-shear fracture under compressionand proposed that the failure process of brittle rock can bedivided into the following stages [6 7] fracture closurelinear elastic deformation fracture initiation stable frac-ture propagation fracture coalescence development ofunstable fractures failure and fracture mode after peak

strength +e initial stress (σci) and fracture damagingstress (σcd) can be determined according to the stress-strain relationship of uniaxial compression test +e failureof deep-buried brittle hard rock is an unconventionalbrittle failure that has the dual mechanism of brittle failureand shear failure Recent research has mainly focused onthe brittle-shear properties of high-stress rock massHoekndashBrown proposed that the brittle rock mass willtransform into the ductility when enough confiningpressure is applied [8] HoekndashBrown failure criterionprovides an acceptable estimate of the peak strength forshear failure but a cutoff has been added for tensile con-ditions However the criterion does not adequately explainthe progressive coalition of tensile cracks and the finalshearing of the specimens at higher confining stress [9]Parterson conducted experiments on the marble at roomtemperature and proved that the rock deformation be-havior changed from brittleness to ductility with the

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8864074 12 pageshttpsdoiorg10115520208864074

increase of pressure [7] Mogi has published similar ex-perimental results and pointed out that the brittle-to-ductile transformation is usually related to rock strength[10] On the other hand some studies have shown that theoverall strength of the rock increases with the increase ofdepth and the rock becomes more brittle along with theincreasing depth which has been verified in the fieldmonitoring of the Mine by experimental tunnel and thedeep underground laboratory at Auspar [11] Rock massdamage at great depths near underground openings is oftenof shear-tensile failure character [12] To further study thefailure mode transformation of the surrounding rock alongthe vertical inward direction of the chamber wall and thetensile failure of the rock wall and to simulate the bidi-rectional stress state of the chamber a chamber wall failuresimulation instrument was developed to study the tran-sitional failure mode of the rock wall [13] Maximumprincipal strain theory and maximum shear stress theoryare two failure strength theories which are often used in thefailure analysis of rock and rock mass +e difference be-tween them is failure mode which is controlled by ex-tensional strain or shear stress Barton and Shen [14]mentioned that many scholars pointed out that the im-portance of axial splitting in response to the compressivestress fields when neighbouring surfaces were available toallow the ldquolateralrdquo extensional strain to cause extensionalfracturing Stacey [15] showed that for a material thatshowed linear deformation behaviour the onset of failureand the depth of failure could be related to a considerationof extensional strain Based on the failure features analysisthe maximum principal strain formula is proposed for theinitiation of surface instability calculation in this paper

2 Test Methods for the Transitional FailureSimulator of the Rock Wall

21 Manufacture of the Sample Chamber Referring to thedesign idea of the surface instability instrument developedby Kao [16] to consider the geometric factors (plane shape)and the biaxial stress state of the surrounding rock surface oftunnel techniques such as endoscopy laser speckle tech-nology LVDT technology digital image technology andacoustic emission technology are utilized to measure andlocate the internal and external deformation of the speci-mens Compared with the traditional triaxial test rock wallsurface damage is progressive this would be involved in theresearch on the transformation of failure mode and thegeometric factors [17] Meanwhile the axis displacementsurface displacement internal displacement cracking signalpressure data and other relevant information require to beobtained Furthermore a servo loading test should be car-ried out to simulate the stable failure process of the surfacerock of deep tunnel which is in cooperation with a lateraldisplacement-control technique +e instrument is pre-sented in Figure 1

To ensure the measurement accuracy the LVDT sensorwas calibrated to obtain a linear range of around 5mmwhich was used as the measurement range in the testFigure 2 shows the calibrating data as follows

22UCSandTriaxialCompressionTest Uniaxial and triaxialcompression tests were performed on 23 circular cylindersincluding 14 sandstone specimens and 9 granite specimens+e specimens were prepared with diameters of 30mm50mm and 100mm to evaluate the size dependence of thestrength+e loading program was in accordance with ISRMstandards and tested within a servo-hydraulic system ofTAW-2000 Failure specimens are shown in Figure 2 andthe results are listed in Table 1 +e number of images isdefined as ldquorock type-size-confining pressure-IDrdquo For ex-ample S3060-0-1 means the first sandstone specimen with30 mmtimes 60 mm size and 0 confining pressure

It can be concluded that the strength of this kind ofsandstone would show a deviation between different sizesWhen the diameters of specimen change from 30mm to100mm the uniaxial compression strength increases from40MPa to 5935MPa If the size is larger than 50mm theUCS changes a little and the trend is the same as the findingsof Chen Yu et al [18] Peng Chen and Zhiwei Zhou [19] andChuangye Wang and Xiaoya Du [20]

Strength curves of cylindrical sandstone samples withdifferent radiuses are shown in Figure 3 +e transition offailure modes can be obtained and the samples show moreductile feature with the increasing of radial stress Similarlya set of tests for chalk were conducted by Ali Tarokh et al in2016 [21] and the results showed the transition of failuremodes (see Figure 4) Both in sandstone and granite tests theorientation of the failure planes varies from vertical tohorizontal with the increase of mean stress (see Figure 5)which is consistent with the announced results of Tarokhrsquosresearch

3 Surface Instability Test

31 Instruments and Sample Preparation +e loadingequipment used in the test was the TAW-2000 servo rocktriaxial testing machine produced by Changchun ChaoyangTest Instrument Co Ltd +e maximum axial test force is2000 kN +is test also introduced the VIC-3D system

+e selected test samples are Sichuan yellow sandstones(size 80times 90times 60mm sample numbers S8090110-1S8090110-2 S8090110-3 S8090110-4 and S8090110-5) andone granite sample (number 80times 90times 60mm H809060-1)(see Figure 6) +e sample surface was coated with a stearicacid lubricant to reduce friction [22]

32 Preparation and Operation of the Surface InstabilitySimulation Test

321 Surface Treatment of Rock Samples +e spot imagewas made by the random seal method In the test thespeckled surface of the sample needs to contact with theacrylic plate and generate intermediate principal stress andthe coating will increase the thickness of the sample [23]+erefore through the steps of surface cleaning-backgroundcoating-speckle fabrication-surface lubrication a relativelyclear and distinguishable speckle image with a uniformcoating is formed

2 Advances in Civil Engineering

322 Test Operation Procedure

Step 1 Cyclic preloading and unloading were appliedbefore the test to extrude excessive lubricant andform a uniform membrane specifically two cycleswere conducted with a preloading start from 0 kN to

100 kN at the rate of 2 kNs and then unloading to1 kNStep 2 +e experiment required to set up the imagingangle and focal length of the camera in VIC-3D systemturn on the VIC-3D system and set the frame rate of the

158

132

10

14

61

1179 3 4 5

12

Figure 1 Apparatus for simulation of surface instability 1 front wall 2 left wall 3 perforated wall 4 half windowwall 5 full-window wall6 base 7 back wall 8 wedge plate 9 FRP backing plate 10 square specimen 11 ordinary steel base plate 12 high strength bolt 13 LVDTholder 14 holder 15 top plate

10

8

6

4

2

0

Late

ral L

VD

T se

nsor

(mm

)

2 4 6 8 10 12Helical micrometer degree (mm)

Lateral LVDT sensor and calibrator reading recordLinear range

y = 1005x ndash 27118R2 = 1

(a)

10

8

6

4

2

0

Vert

ical

LV

DT

sens

or (m

m)


Vertical LVDT sensor and calibrator reading recordLinear range

y = 10008x ndash 0276R2 = 1

(b)

Figure 2 LVDTsensor calibration data (a) Horizontal LVDTsensor calibration data (red line indicates the linear range) (b) Vertical LVDTsensor calibration data (red line indicates the linear range)

Table 1 Results from compression experiments of sandstone specimens

Diameter (mm) Confining pressure (MPa) Axial stress (MPa)30 0 406830 0 391630 10 1200130 10 1269930 20 1751330 20 1708730 30 2104330 30 2325750 0 563750 10 1401350 20 1583450 30 20096100 0 5935

Advances in Civil Engineering 3

image collector to 1 framesecond until the end of theexperiment +e destruction simulation experiment isshown in Figure 7Step 3 +e test loading was divided into two steps (a)loading to 30 of the uniaxial compressive strength ofthe sample at 05 kNs load-control rate (b)+e controlmode was switched to the transverse LVDT displace-ment-control at the rate of 001mmmin +e loadingproceeded until the failure of the sample and stoppedafter the occurrence of multiple fractures

4 Results of the Surface Failure Test

41 Fracture Expansion Image of the Specimen +e80times 90times110mm cubic sandstone and 80times 90times 60mmcubic granite specimen were loaded respectively and thesample results are shown in Figure 8 In Figure 8 the rightside of the sample is the free surface It can be seen from thefigure that the failure of the sample in the bidirectional stressstate first occurs at the free surface end mainly near-verticalsplitting and then the cracks extend to the depth +e crackspacing increases as the crack angle increases

According to the definition of x y and z directions asshown in Figure 9 the imaging results (positive as tensilestrain and negative as compressive strain) of S8090110-3S8090110-4 and S8090110-5 in the critical state are shown inFigure 10 and the tensile strains at failure are 2157 με2423 με and 2401 με respectively

According to the DIC images with the increase of thepressure the deformation of the sample below side of thefree face reaches the limit value firstly and a local verticalcrack appears Afterward the cracks will not stop developingand expanding along a similar vertical direction until theentire sample is destroyed By comparing the global strainmeasured by the LVDT sensor with the local strain of theDIC image the DIC image would be more suitable for themonitor of local cracks development

42 Loading-Displacement Analysis of Rock Samples

421 Sandstone +e test results of the vertical stress-vertical strain curve and the test results of the verticalstress-transverse strain curve of cubic sandstone undertest loading are shown in Figure 11 It can be found that

D = 30mmD = 50mm

240220200180160140120100

80604020

ndash20 ndash15 ndash10 ndash5 0 5 10 15 20 25 30σ3 (MPa)

σ 1 (M

Pa)

y = 38576x + 83885R2 = 09976

y = 4735x + 82364R2 = 09969

Figure 3 Strength curves of sandstone samples

(a) (b) (c) (d) (e)

Figure 4 Photographs of failed specimens of Mons chalk A transition of failure patterns with increase in radial stress Specimen (a) showsaxial splitting observed in uniaxial compression test σ3 0 specimen (b) exhibits a shear band with angle of 67deg from σ3 15MPa specimen(c) also illustrates a shear band with angle of 55deg from σ3 25MPa specimen (d) shows a transitional failure plane with angle of 36deg fromσ3 5MPa and stage (e) clearly shows a compaction band under σ3 10MPa [21]


the failure of sandstone samples occurred when thetransverse strain reached 2000 με +e strength will con-tinue to fluctuate after the first cracking even exceedingthe first cracking strength +e first failure strength of the

sample (almost 60MPa) is consistent with the UCS(5935MPa for 100mm size specimen see Table 1) of thesame stone sample indicating that the shape of the sampledoes not influence its strength

S3060-30-2 S50100-0-1 S50100-0-2 S50100-10-1 S50100-20-1 S50100-30-1 S100200-0-1

S3060-0-1 S3060-0-2 S3060-10-1 S3060-10-2 S3060-20-1 S3060-20-2 S3060-30-1

(a)

H3060-0-1 H3060-10-1 H3060-20-1 H3060-30-1 H50100-0-1 H50100-10-1 H50100-20-1 H50100-30-1 H100200-0-1

(b)

Figure 5 Crack image of failed specimens (a) Sandstone specimens (b) Granite specimens

60m

m

90mm80mm

(a)

60m

m

90mm

80mm

(b)

Figure 6 Rock samples (a) Sandstone sample (b) Granite sample


422 Granite +e results of the vertical stress-verticalstrain curve and the vertical stress-transverse strain curve ofcubic granite are shown in Figure 12 It can be found that thefailure of granite samples occurred when the transversestrain reached 4500+e strength will increase again after thefirst cracking even exceeding the first cracking strength

43Analysis ofTransitional FailureMode Failure of rock is acomplex process and failure mechanisms which would showa transition feature according to stress level and loadingcondition [24 25] Previous research proposed that an in-crease in effective pressure shifts the macroscopic failuremode from brittle to ductile [26] According to the difference

(a) (b)

Figure 7 Test systems (a) Rock wall chamber (b) Damage measurement processFr

ee fa

ce

Free

face

Free

face

Free

face

Free

face

S8090110-1 S8090110-2 S8090110-3

S8090110-5S8090110-4

(a)

Free

face

(b)

Figure 8 Damage images of samples (a) Sandstone damage results (b) Granite damage results


of microcracks propagation Charles Derek Martin [27]divided the stress-strain curves of brittle rock into five regionswhich represent the process of microcracks propagation andindicated that the initial step in the failure process is slidingalong grain boundaries with the wing or axial cracks de-veloping much later in the failure process +e transition offailure modes will cause the change of the orientation of thefailure plane which decreases from 90deg to 0deg with the increasein mean stress [21] From the analysis of fracture angle cracksspacing failure location and stress-strain relationship curvesshown in Figures 8ndash12 it can be seen that the first fracture ofthe samples occurs close to the free surface which could bethought as a type of tensile failure As the pressure continuallyrises shear failure appears and a sliding surface can be foundin deeper location of samples After the fractures are fullydeveloped it can be found that the transitional failure modefrom tensile failure to shear failure occurs from the free face tothe depth which simulates the stress and strain characteristicsof the rock in the biaxial stress state and reproduces the stressstate of the tunnel rock wall In the uniaxial and triaxial testsin the laboratory the failure mode of the sample is a com-bination of tensile and shear failure while the first failure ofthe rock in the surface instability chamber is in tensile failuremechanism however the failure mode has gradually trans-formed to shear mode as the increase of the failure depth

5 Research of Tunnel Wall Strength Based onthe Maximum Principal Strain Theory

Under uniaxial compression or biaxial loading the failuremode of the sample is mainly tensile failure rather than shearfailure at first so the strength is different from the valuepredicted by the MohrndashCoulomb criterion In explainingthese destruction phenomena many scholars have putforward different viewpoints Martin proposed the CWFSmodel [28] and emphasized that damage developmentcaused that as friction was mobilized in the sample and led tothe reduction of cohesion [29] Martino and Chandlerthought that the damage zone where the stress redistributionhappened was too small to permanently change rockproperties [30] Brace and Gramberg have first providedexplanations on the tensile crack under uniaxial compres-sion condition [31 32] that is because the tensile strain is

caused by tensile damage Stacey attempted to use the theoryof maximum principal strain to forecast the brittle failuremechanism of deep massive quartzite in South Africa hesuggested that for those materials with obvious linear de-formation behaviors the failure could be considered to relateto the tensile strain [15]

Since the failure mode of the surrounding rock is tensilefailure it cannot be explained by the MohrndashCoulomb cri-terion Based on the maximum principal strain theoryproposed by Stacey and Nick Barton and generalizedHookersquos law the following failure conditions can beobtained

ε1 or ε2 or ε3 εc

ε3 1E

σ3 minus μ σ1 + σ2( 11138571113858 1113859

ε2 1E

σ2 minus μ σ1 + σ3( 11138571113858 1113859

ε1 1E

σ1 minus μ σ2 + σ3( 11138571113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where E is the elastic modulus μ is Poissonrsquos ratio valueσ123 stands for principal stresses and ε123 means principalstrains If the calculated extensional strain was greater thanthe critical strain then spalling would occur So if μ (σ1 + σ2)gt σ3 then (negative) extensional strain will occur

According to the generalized Hookersquos law

σ2 μ σ1 + σ3( 1113857 (2)

By iterative substitution of equation (2) in (1) the fol-lowing can be obtained

ε1 1E

1 minus μ21113872 1113873σ3 minus μ(1 + μ)σ11113960 1113961 (3)

+erefore a simplified maximum principal strain theoryequation can be obtained

σ1 1 minus μμ

σ3 minusEεc

μ(1 + μ) when εc lt 0( 1113857 (4)

In the test the minimum principal stress is 0 for free faceand equation (4) yields

σ1 minusEεc

μ(1 + μ) (5)

Elastic modulus E and Poissonrsquos ratio μ can be calculatedaccording to the UCS test +e parameters of sandstonesamples are 707GPa and 024 respectively From equation (5)the critical principal strain εc can be given as minus2526times10minus3

(negative value means tensile strain) Initial fracture createsstrain values measured from LVDTsensor and DIC image (themeasured value is 2300 and 2400 respectively) which areconsistent with εc so it can be considered that the initial failuremode of sandstone samples is a tensile failure and the maxi-mum principal strain theory has good applicability to explainthe surface failure of the rock wall

X

Y

Z

Figure 9 VIC-3D system direction definition


exx [1]-Lagrange

exx [1]-Lagrange

00240

00210

00185

00158

00130

00103

00075

00048

00020

ndash00017

ndash00039

ndash00061

ndash00082

ndash00104

ndash00126

ndash00148

ndash00169

ndash00191

(a)

exx [1]-Lagrange

exx [1]-Lagrange

00035

00014

ndash00007

ndash00028

ndash00049

ndash00069

ndash00090

ndash00111

ndash00132

00428

00371

00313

00256

00198

00141

00083

00026

00032

(b)

exx [1]-Lagrangeexx [1]-

Lagrange00270

00236

00203

00169

00135

00101

00068

00033

00000

00039

00020

00000

ndash00020

ndash00039

ndash00059

ndash00078

ndash00100

ndash00117

(c)

Figure 10 Imaging results when DIC processes critical damage (positive as tensile strain negative as compressive strain) (a) S8090110-3critical strain DIC map (unit cm) (left) X-direction displacement field and (right) Y-direction displacement field (b) S8090110-4 criticalstrain DICmap (unit cm) (left)X-direction displacement field and (right) Y-direction displacement field (c) S8090110-5 critical strain DICmap (unit cm) (left) X-direction displacement field and (right) Y-direction displacement field


Since the failure of the sample is not only shear failure inthe uniaxial case the strength value of the rock is higher inthe uniaxial case when the MohrndashCoulomb criterion is usedto predict it (see Figure 13) However considering the planestrain problem under biaxial loading especially the strengthfeature of the rock wall because the specimen failure iscaused by tensile failure the maximum principal straintheory is more in line with the actual situation (seeFigure 13)

+e maximum principal strain theory can also be suit-able for the calculation of uniaxial compression strengthwhen the stretching lineation strain failure takes place (seeFigures 3 and 8) Li Yuan et al proposed a brittle-shear

mixing strength criterion and considered rock failure asbrittle fracture and shear failure [33] +e coefficient b wasproposed to describe the transition from brittle failure to ashear failure caused by constant confining pressure see thefollowing

σ1 b C1σ3 + C2( 1113857 +(1 minus b) D1σ3 + D2( 1113857 (6)

b emσ3+ n

(7)

where (C1σ3 + C2) represents the brittle fracture strength(D1σ3 + D2) is the shear strength C1 C2 D1 and D2 are thematerial strength coefficients obtained by the experiments b

70

60

50

40

30

20

10

0

Vert

ical

stre

ss (M

Pa)

0 2 4 6 8 10 12 14 16 18 20 22Vertical strain (10ndash3)

S8090110-3S8090110-4S8090110-5

(a)

Vert

ical

stre

ss (M

Pa)

Transverse strain (10ndash3)

80

70

60

50

40

30

20

10

0 1 2 3 4 5 6

S8090110-1S8090110-2S8090110-3

S8090110-4S8090110-5

0

(b)

Figure 11 Stress versus strain curves of the cubic sandstone samples (a) Vertical stress-vertical strain curve of the sandstone sample (b)Vertical stress-transverse strain curve of the sandstone sample

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)

0 1 2 3 4 5 6 7 8Vertical strain (10ndash3)

H809060-1

(a)

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)

00 01 02 03 04 05Transverse strain (10ndash3)

H809060-1

(b)

Figure 12 Stress versus strain curves of the cubic granite samples (a) Vertical stress-vertical strain curve of the granite sample (b) Verticalstress-strain curve of the granite sample


indicates the transition coefficient from brittle failure toshear failure ranging from 0 to 1 m and n are parametersfitted out by data

If equation (6) is used to explain the transitioning offailure mode with the stress state changing the results areshown in Figure 13 MohrndashCoulomb strength line could befitted by the higher stresses data and the maximum principalstrain strength line could be calculated by E μ and UCS of30mm size sample mentioned in Table 2 But according tothe test condition of the uniaxial and triaxial compressiontest equation (2) should be changed to

σ2 σ3 (8)

+e deductive program is the same as equations (3) to(5) it can be given that

σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)

Because the average UCS value is 3992MPa the relatedcritical strain can be calculated by equation (9) asminus1355times10minus3+e strength line of εc is illustrated in Figure 13as the red line Equation (6) is fitted by the test data shown inFigure 13 C1 and C2 are confirmed by slope and intercept ofequation (9) and D1 and D2 by the parameters ofMohrndashCoulomb strength theory which is fitted in Figure 13Fitted and calculated parameters are given in Table 2 and thecorrelation coefficient arrives at 09961

+e test data and transition trend of sandstone samplescan further illustrate the reason why the in situ strength oftunnel wall is lower than the predicted strength value by

MohrndashCoulomb theory that is the tensile failure is notconsidered which leads to a higher predicted strengthvalue

6 Conclusion

Experimental study of surface instability of samples wascarried out An indoor test device for the simulation oftransitional surface failure of the rock wall was developed+rough a biaxial stress loading test on the rectangular rocksample the damage process and crack development of rocksamples were analyzed and the law of stress and strain re-lated to the failure mode transition was characterized as well+e following conclusions can be deducted

(1) +e surface fracture of the rock wall is a transitionalfailure mode It is found in the simulation test thatunder the condition of biaxial loading with thecontinuous rise of pressure the crack caused bytensile failure first appears at the end of the freesurface and followed by shear failure +e experi-ment simulates the stress-strain characteristics underthe in situ stress state of the rock wall +is phe-nomenon is also applicable to granite and it isconcluded that tensile failure is the main reason whythe wall strength of both shallow and deep tunnel islower than indoor strength

(2) +e maximum principal strain theory is applied toexplain the tensile failure of samples Under the con-dition of biaxial stress the maximum principal straintheory was deduced Additionally test data were used

250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)

Test dataTransition strength curveFitted M-C strength lineFitted max prin strain strength line

Figure 13 Comparison of three strength theories

Table 2 Parameters fitted and calculated from data of specimen with 30mm diameter

Parameter E (GPa) μ C1 C2 (MPa) D1 D2 (MPa) m (MPaminus1) n RValue 707 024 317 3992 490 7467 minus05759 0 09961


to verify the correctness of the maximum principalstrain theory to explain the tensile failure of the sample

(3) Transitional failure mode leads to the conversionof strength criterion Considering the transitionalfailure mode of the rock wall the surface fractureneeds to refer to the theory of maximum principalstrain As the failure gradually evolved to thedepth the mode changes from tensile failure toshear failure +e theory of maximum principalstrain may no longer be applicable andMohrndashCoulomb criterion or other failure criteriaare acceptable

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Dr NF Deng is acknowledged for his comprehensive treat-ment of the literature+is researchwas funded by theNationalKey RampD Program of China (no 2018YFE0101100)

References

[1] G Su T Li and J Jiang ldquoExperimental study of influence ofsupport failures on rockbursts under true-triaxial conditionrdquoAdvances in Civil Engineering vol 2018 Article ID 210545120 pages 2018

[2] J Pan S Liu and S Wang ldquoA new theoretical view ofrockburst and its engineering applicationrdquo Advances in CivilEngineering vol 2018 Article ID 4683457 12 pages 2018

[3] Q Jiang X-T Feng T-B Xiang and G-S Su ldquoRockburstcharacteristics and numerical simulation based on a newenergy index a case study of a tunnel at 2500 m depthrdquoBulletin of Engineering Geology and the Environment vol 69no 3 pp 381ndash388 2010

[4] F Pei H Ji and J Zhao ldquoEnergy evolution and AE failureprecursory characteristics of rocks with different rockburstpronenessrdquo Advances in Civil Engineering vol 2020 ArticleID 8877901 12 pages 2020

[5] X Sun Li Gan and C Zhao ldquoNumerical investigation of gob-side entry retaining through precut overhanging hard roof tocontrol rockburstrdquo Advances in Civil Engineering vol 2018Article ID 8685427 10 pages 2018

[6] W F Brace and E G Bombolakis ldquoA note on brittle crackgrowth in compressionrdquo Journal of Geophysical Researchvol 68 no 12 pp 3709ndash3713 1963

[7] M S Paterson ldquoExperimental deformation and faulting inwombeyan marblerdquo Geological Society of America Bulletinvol 69 no 4 pp 465ndash467 1958

[8] Z Lian L Tian and W Wang BrownWaits for UndergroundRock Engineering Metallurgical Industry Press BeijingChina 1986

[9] E Hoek and C D Martin ldquoFracture initiation and propa-gation in intact rock-a reviewrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 6 no 4 pp 287ndash300 2014

[10] J P Meyer and J F Labuz ldquoLinear failure criteria with threeprincipal stressesrdquo International Journal of Rock Mechanics ampMining Sciences vol 12 no 40 p 180 2012

[11] B G Tarasov and M F Randolph ldquoFrictionless shear at greatdepth and other paradoxes of hard rocksrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 6 no 1pp 316ndash328 2017

[12] V V Makarov M A Guzev V N Odintsev andL S Ksendzenko ldquoPeriodical zonal character of damage nearthe openings in highly-stressed rock mass conditionsrdquoJournal of Rock Mechanics and Geotechnical Engineeringvol 8 no 2 pp 164ndash169 2016

[13] X Fan K Li and H Lai ldquoExperimental and numerical studyof the failure behavior of intermittent rock joints subjected todirect shear loadrdquo Advances in Civil Engineering vol 2018Article ID 4294501 19 pages 2018

[14] N Barton and B Shen ldquoRisk of shear failure and extensionalfailure around over-stressed excavations in brittle rockrdquoJournal of Rock Mechanics and Geotechnical Engineeringvol 9 no 2 pp 210ndash225 2017

[15] T R Stacey ldquoA simple extension strain criterion for fractureof brittle rockrdquo International Journal of Rock Mechanics andMining Sciences amp Geomechanics Abstracts vol 18 no 6pp 469ndash474 1981

[16] C S Kao Surface instability as damage evolution in rockPhD dissertation University of Minnesota MinneapolisMN USA 2011

[17] Z Li L Wang and Y Lu ldquoExperimental investigation on thedeformation strength and acoustic emission characteristics ofsandstone under true triaxial compressionrdquo Journal of Miningand Safety Engineering vol 36 no 1 pp 191ndash197 2019

[18] Yu Chen Y-heng Huang and P Cao ldquoSize effect experi-mental study of strength and deformation in different height-to-diameter ratio soft rocksrdquo Journal of Central South Uni-versity vol 41 no 3 pp 1073ndash1078 2010

[19] P Chen and Z Zhou ldquoSize effect experiment of rock materialwith RFRA2Drdquo Journal of Liaoning Technical University(Natural Science) vol 31 no 6 pp 842ndash845 2012

[20] C Wang and X Du ldquoComparative analysis of rock size effecttest and RFPA3D numerical simulationrdquo Industrial Mineralsamp Processing vol 47 no 2 pp 28ndash30 2018

[21] T Ali L Yuan and J F Labuz ldquoHardening in porous chalkfrom precompactionrdquo Acta Geotechnica An InternationalJournal for Geoengineering vol 12 pp 949ndash953 2017

[22] Y Peng Q Guo Z Zhang and Y Shan ldquoApplication ofbase force element method on complementary energyprinciple to rock mechanics problemsrdquo MathematicalProblems in Engineering vol 2015 Article ID 29280916 pages 2015

[23] Y Zheng and S Deng ldquoFailure probability model consideringthe effect of intermediate principal stress on rock strengthrdquoMathematical Problems in Engineering vol 2015 Article ID960973 7 pages 2015

[24] Q Jiang G Su X-T Feng G Chen M-Z Zhang and C LiuldquoExcavation optimization and stability analysis for largeunderground caverns under high geostress a case study of theChinese laxiwa projectrdquo Rock Mechanics and Rock Engi-neering vol 52 no 3 pp 895ndash915 2019

[25] Q Jiang B Yang and F Yan ldquoNewmethod for characterizingthe shear damage of natural rock joint based on 3D engraving


and 3D scanningrdquo International Journal of Geomechanicsvol 20 no 2 Article ID 6019022 2020

[26] T-F Wong H Szeto and J Zhang ldquoEffect of loading pathand porosity on the failure mode of porous rocksrdquo AppliedMechanics Reviews vol 45 no 8 pp 281ndash293 1992

[27] M Charles Derek Ae Strength of Massive Lac Du BonnetGranite Around Underground Openings Winnipeg ManitobaDepartment of Civil amp Geological Engineering University ofManitoba Winnipeg Canada 1993

[28] C D Martin ldquoSeventeenth Canadian geotechnical collo-quium the effect of cohesion loss and stress path on brittlerock strengthrdquo Canadian Geotechnical Journal vol 34no 5 pp 698ndash725 1997

[29] C D Martin and N A Chandler ldquo+e progressive fracture oflac du bonnet graniterdquo International Journal of Rock Me-chanics and Mining Sciences amp Geomechanics Abstractsvol 31 no 6 pp 643ndash659 1994

[30] J B Martino and N A Chandler ldquoExcavation-induceddamage studies at the underground research laboratoryrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 9 no 10 pp 1413ndash1426 2004

[31] W F Brace ldquoRecent experimental studies of brittle fracture ofrocksrdquo in Proceedings of the 10th US Symposium on RockMechanics American Rock Mechanics Association AustinTX USA pp 58ndash81 May 1966

[32] J Gramberg ldquo+e axial cleavage fracture 1 axial cleavagefracturing a significant process in mining and geologyrdquoEngineering Geology vol 1 no 1 pp 31ndash72 1965

[33] Li Yuan Z Yan and Q Li ldquoStrength analysis of rock materialsunder brittle shear failure moderdquo Journal of University of Scienceand Technology Beijing vol 12 no 19 pp 1364ndash1370 2012


increase of pressure [7] Mogi has published similar ex-perimental results and pointed out that the brittle-to-ductile transformation is usually related to rock strength[10] On the other hand some studies have shown that theoverall strength of the rock increases with the increase ofdepth and the rock becomes more brittle along with theincreasing depth which has been verified in the fieldmonitoring of the Mine by experimental tunnel and thedeep underground laboratory at Auspar [11] Rock massdamage at great depths near underground openings is oftenof shear-tensile failure character [12] To further study thefailure mode transformation of the surrounding rock alongthe vertical inward direction of the chamber wall and thetensile failure of the rock wall and to simulate the bidi-rectional stress state of the chamber a chamber wall failuresimulation instrument was developed to study the tran-sitional failure mode of the rock wall [13] Maximumprincipal strain theory and maximum shear stress theoryare two failure strength theories which are often used in thefailure analysis of rock and rock mass +e difference be-tween them is failure mode which is controlled by ex-tensional strain or shear stress Barton and Shen [14]mentioned that many scholars pointed out that the im-portance of axial splitting in response to the compressivestress fields when neighbouring surfaces were available toallow the ldquolateralrdquo extensional strain to cause extensionalfracturing Stacey [15] showed that for a material thatshowed linear deformation behaviour the onset of failureand the depth of failure could be related to a considerationof extensional strain Based on the failure features analysisthe maximum principal strain formula is proposed for theinitiation of surface instability calculation in this paper

2 Test Methods for the Transitional FailureSimulator of the Rock Wall

21 Manufacture of the Sample Chamber Referring to thedesign idea of the surface instability instrument developedby Kao [16] to consider the geometric factors (plane shape)and the biaxial stress state of the surrounding rock surface oftunnel techniques such as endoscopy laser speckle tech-nology LVDT technology digital image technology andacoustic emission technology are utilized to measure andlocate the internal and external deformation of the speci-mens Compared with the traditional triaxial test rock wallsurface damage is progressive this would be involved in theresearch on the transformation of failure mode and thegeometric factors [17] Meanwhile the axis displacementsurface displacement internal displacement cracking signalpressure data and other relevant information require to beobtained Furthermore a servo loading test should be car-ried out to simulate the stable failure process of the surfacerock of deep tunnel which is in cooperation with a lateraldisplacement-control technique +e instrument is pre-sented in Figure 1

To ensure the measurement accuracy the LVDT sensorwas calibrated to obtain a linear range of around 5mmwhich was used as the measurement range in the testFigure 2 shows the calibrating data as follows

22UCSandTriaxialCompressionTest Uniaxial and triaxialcompression tests were performed on 23 circular cylindersincluding 14 sandstone specimens and 9 granite specimens+e specimens were prepared with diameters of 30mm50mm and 100mm to evaluate the size dependence of thestrength+e loading program was in accordance with ISRMstandards and tested within a servo-hydraulic system ofTAW-2000 Failure specimens are shown in Figure 2 andthe results are listed in Table 1 +e number of images isdefined as ldquorock type-size-confining pressure-IDrdquo For ex-ample S3060-0-1 means the first sandstone specimen with30 mmtimes 60 mm size and 0 confining pressure

It can be concluded that the strength of this kind ofsandstone would show a deviation between different sizesWhen the diameters of specimen change from 30mm to100mm the uniaxial compression strength increases from40MPa to 5935MPa If the size is larger than 50mm theUCS changes a little and the trend is the same as the findingsof Chen Yu et al [18] Peng Chen and Zhiwei Zhou [19] andChuangye Wang and Xiaoya Du [20]

Strength curves of cylindrical sandstone samples withdifferent radiuses are shown in Figure 3 +e transition offailure modes can be obtained and the samples show moreductile feature with the increasing of radial stress Similarlya set of tests for chalk were conducted by Ali Tarokh et al in2016 [21] and the results showed the transition of failuremodes (see Figure 4) Both in sandstone and granite tests theorientation of the failure planes varies from vertical tohorizontal with the increase of mean stress (see Figure 5)which is consistent with the announced results of Tarokhrsquosresearch

3 Surface Instability Test

31 Instruments and Sample Preparation +e loadingequipment used in the test was the TAW-2000 servo rocktriaxial testing machine produced by Changchun ChaoyangTest Instrument Co Ltd +e maximum axial test force is2000 kN +is test also introduced the VIC-3D system

+e selected test samples are Sichuan yellow sandstones(size 80times 90times 60mm sample numbers S8090110-1S8090110-2 S8090110-3 S8090110-4 and S8090110-5) andone granite sample (number 80times 90times 60mm H809060-1)(see Figure 6) +e sample surface was coated with a stearicacid lubricant to reduce friction [22]

32 Preparation and Operation of the Surface InstabilitySimulation Test

321 Surface Treatment of Rock Samples +e spot imagewas made by the random seal method In the test thespeckled surface of the sample needs to contact with theacrylic plate and generate intermediate principal stress andthe coating will increase the thickness of the sample [23]+erefore through the steps of surface cleaning-backgroundcoating-speckle fabrication-surface lubrication a relativelyclear and distinguishable speckle image with a uniformcoating is formed





158

132

10

14

61

1179 3 4 5

12


10

8

6

4

2

0

Late

ral L

VD

T se

nsor

(mm

)



y = 1005x ndash 27118R2 = 1

(a)

10

8

6

4

2

0

Vert

ical

LV

DT

sens

or (m

m)



y = 10008x ndash 0276R2 = 1

(b)












D = 30mmD = 50mm

240220200180160140120100

80604020


σ 1 (M

Pa)

y = 38576x + 83885R2 = 09976

y = 4735x + 82364R2 = 09969


(a) (b) (c) (d) (e)





S3060-30-2 S50100-0-1 S50100-0-2 S50100-10-1 S50100-20-1 S50100-30-1 S100200-0-1

S3060-0-1 S3060-0-2 S3060-10-1 S3060-10-2 S3060-20-1 S3060-20-2 S3060-30-1

(a)

H3060-0-1 H3060-10-1 H3060-20-1 H3060-30-1 H50100-0-1 H50100-10-1 H50100-20-1 H50100-30-1 H100200-0-1

(b)


60m

m

90mm80mm

(a)

60m

m

90mm

80mm

(b)





(a) (b)


ee fa

ce

Free

face

Free

face

Free

face

Free

face

S8090110-1 S8090110-2 S8090110-3

S8090110-5S8090110-4

(a)

Free

face

(b)









ε3 1E

σ3 minus μ σ1 + σ2( 11138571113858 1113859

ε2 1E

σ2 minus μ σ1 + σ3( 11138571113858 1113859

ε1 1E

σ1 minus μ σ2 + σ3( 11138571113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



σ2 μ σ1 + σ3( 1113857 (2)


ε1 1E



σ1 1 minus μμ

σ3 minusEεc

μ(1 + μ) when εc lt 0( 1113857 (4)


σ1 minusEεc

μ(1 + μ) (5)



X

Y

Z



exx [1]-Lagrange

exx [1]-Lagrange

00240

00210

00185

00158

00130

00103

00075

00048

00020

ndash00017

ndash00039

ndash00061

ndash00082

ndash00104

ndash00126

ndash00148

ndash00169

ndash00191

(a)

exx [1]-Lagrange

exx [1]-Lagrange

00035

00014

ndash00007

ndash00028

ndash00049

ndash00069

ndash00090

ndash00111

ndash00132

00428

00371

00313

00256

00198

00141

00083

00026

00032

(b)


Lagrange00270

00236

00203

00169

00135

00101

00068

00033

00000

00039

00020

00000

ndash00020

ndash00039

ndash00059

ndash00078

ndash00100

ndash00117

(c)






σ1 b C1σ3 + C2( 1113857 +(1 minus b) D1σ3 + D2( 1113857 (6)

b emσ3+ n

(7)


70

60

50

40

30

20

10

0

Vert

ical

stre

ss (M

Pa)


S8090110-3S8090110-4S8090110-5

(a)

Vert

ical

stre

ss (M

Pa)


80

70

60

50

40

30

20

10

0 1 2 3 4 5 6

S8090110-1S8090110-2S8090110-3

S8090110-4S8090110-5

0

(b)


200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(a)

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(b)





σ2 σ3 (8)


σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)




6 Conclusion




250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)








Data Availability




Acknowledgments


References








































158

132

10

14

61

1179 3 4 5

12


10

8

6

4

2

0

Late

ral L

VD

T se

nsor

(mm

)



y = 1005x ndash 27118R2 = 1

(a)

10

8

6

4

2

0

Vert

ical

LV

DT

sens

or (m

m)



y = 10008x ndash 0276R2 = 1

(b)












D = 30mmD = 50mm

240220200180160140120100

80604020


σ 1 (M

Pa)

y = 38576x + 83885R2 = 09976

y = 4735x + 82364R2 = 09969


(a) (b) (c) (d) (e)





S3060-30-2 S50100-0-1 S50100-0-2 S50100-10-1 S50100-20-1 S50100-30-1 S100200-0-1

S3060-0-1 S3060-0-2 S3060-10-1 S3060-10-2 S3060-20-1 S3060-20-2 S3060-30-1

(a)

H3060-0-1 H3060-10-1 H3060-20-1 H3060-30-1 H50100-0-1 H50100-10-1 H50100-20-1 H50100-30-1 H100200-0-1

(b)


60m

m

90mm80mm

(a)

60m

m

90mm

80mm

(b)





(a) (b)


ee fa

ce

Free

face

Free

face

Free

face

Free

face

S8090110-1 S8090110-2 S8090110-3

S8090110-5S8090110-4

(a)

Free

face

(b)









ε3 1E

σ3 minus μ σ1 + σ2( 11138571113858 1113859

ε2 1E

σ2 minus μ σ1 + σ3( 11138571113858 1113859

ε1 1E

σ1 minus μ σ2 + σ3( 11138571113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



σ2 μ σ1 + σ3( 1113857 (2)


ε1 1E



σ1 1 minus μμ

σ3 minusEεc

μ(1 + μ) when εc lt 0( 1113857 (4)


σ1 minusEεc

μ(1 + μ) (5)



X

Y

Z



exx [1]-Lagrange

exx [1]-Lagrange

00240

00210

00185

00158

00130

00103

00075

00048

00020

ndash00017

ndash00039

ndash00061

ndash00082

ndash00104

ndash00126

ndash00148

ndash00169

ndash00191

(a)

exx [1]-Lagrange

exx [1]-Lagrange

00035

00014

ndash00007

ndash00028

ndash00049

ndash00069

ndash00090

ndash00111

ndash00132

00428

00371

00313

00256

00198

00141

00083

00026

00032

(b)


Lagrange00270

00236

00203

00169

00135

00101

00068

00033

00000

00039

00020

00000

ndash00020

ndash00039

ndash00059

ndash00078

ndash00100

ndash00117

(c)






σ1 b C1σ3 + C2( 1113857 +(1 minus b) D1σ3 + D2( 1113857 (6)

b emσ3+ n

(7)


70

60

50

40

30

20

10

0

Vert

ical

stre

ss (M

Pa)


S8090110-3S8090110-4S8090110-5

(a)

Vert

ical

stre

ss (M

Pa)


80

70

60

50

40

30

20

10

0 1 2 3 4 5 6

S8090110-1S8090110-2S8090110-3

S8090110-4S8090110-5

0

(b)


200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(a)

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(b)





σ2 σ3 (8)


σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)




6 Conclusion




250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)








Data Availability




Acknowledgments


References












































D = 30mmD = 50mm

240220200180160140120100

80604020


σ 1 (M

Pa)

y = 38576x + 83885R2 = 09976

y = 4735x + 82364R2 = 09969


(a) (b) (c) (d) (e)





S3060-30-2 S50100-0-1 S50100-0-2 S50100-10-1 S50100-20-1 S50100-30-1 S100200-0-1

S3060-0-1 S3060-0-2 S3060-10-1 S3060-10-2 S3060-20-1 S3060-20-2 S3060-30-1

(a)

H3060-0-1 H3060-10-1 H3060-20-1 H3060-30-1 H50100-0-1 H50100-10-1 H50100-20-1 H50100-30-1 H100200-0-1

(b)


60m

m

90mm80mm

(a)

60m

m

90mm

80mm

(b)





(a) (b)


ee fa

ce

Free

face

Free

face

Free

face

Free

face

S8090110-1 S8090110-2 S8090110-3

S8090110-5S8090110-4

(a)

Free

face

(b)









ε3 1E

σ3 minus μ σ1 + σ2( 11138571113858 1113859

ε2 1E

σ2 minus μ σ1 + σ3( 11138571113858 1113859

ε1 1E

σ1 minus μ σ2 + σ3( 11138571113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



σ2 μ σ1 + σ3( 1113857 (2)


ε1 1E



σ1 1 minus μμ

σ3 minusEεc

μ(1 + μ) when εc lt 0( 1113857 (4)


σ1 minusEεc

μ(1 + μ) (5)



X

Y

Z



exx [1]-Lagrange

exx [1]-Lagrange

00240

00210

00185

00158

00130

00103

00075

00048

00020

ndash00017

ndash00039

ndash00061

ndash00082

ndash00104

ndash00126

ndash00148

ndash00169

ndash00191

(a)

exx [1]-Lagrange

exx [1]-Lagrange

00035

00014

ndash00007

ndash00028

ndash00049

ndash00069

ndash00090

ndash00111

ndash00132

00428

00371

00313

00256

00198

00141

00083

00026

00032

(b)


Lagrange00270

00236

00203

00169

00135

00101

00068

00033

00000

00039

00020

00000

ndash00020

ndash00039

ndash00059

ndash00078

ndash00100

ndash00117

(c)






σ1 b C1σ3 + C2( 1113857 +(1 minus b) D1σ3 + D2( 1113857 (6)

b emσ3+ n

(7)


70

60

50

40

30

20

10

0

Vert

ical

stre

ss (M

Pa)


S8090110-3S8090110-4S8090110-5

(a)

Vert

ical

stre

ss (M

Pa)


80

70

60

50

40

30

20

10

0 1 2 3 4 5 6

S8090110-1S8090110-2S8090110-3

S8090110-4S8090110-5

0

(b)


200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(a)

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(b)





σ2 σ3 (8)


σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)




6 Conclusion




250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)








Data Availability




Acknowledgments


References







































S3060-30-2 S50100-0-1 S50100-0-2 S50100-10-1 S50100-20-1 S50100-30-1 S100200-0-1

S3060-0-1 S3060-0-2 S3060-10-1 S3060-10-2 S3060-20-1 S3060-20-2 S3060-30-1

(a)

H3060-0-1 H3060-10-1 H3060-20-1 H3060-30-1 H50100-0-1 H50100-10-1 H50100-20-1 H50100-30-1 H100200-0-1

(b)


60m

m

90mm80mm

(a)

60m

m

90mm

80mm

(b)





(a) (b)


ee fa

ce

Free

face

Free

face

Free

face

Free

face

S8090110-1 S8090110-2 S8090110-3

S8090110-5S8090110-4

(a)

Free

face

(b)









ε3 1E

σ3 minus μ σ1 + σ2( 11138571113858 1113859

ε2 1E

σ2 minus μ σ1 + σ3( 11138571113858 1113859

ε1 1E

σ1 minus μ σ2 + σ3( 11138571113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



σ2 μ σ1 + σ3( 1113857 (2)


ε1 1E



σ1 1 minus μμ

σ3 minusEεc

μ(1 + μ) when εc lt 0( 1113857 (4)


σ1 minusEεc

μ(1 + μ) (5)



X

Y

Z



exx [1]-Lagrange

exx [1]-Lagrange

00240

00210

00185

00158

00130

00103

00075

00048

00020

ndash00017

ndash00039

ndash00061

ndash00082

ndash00104

ndash00126

ndash00148

ndash00169

ndash00191

(a)

exx [1]-Lagrange

exx [1]-Lagrange

00035

00014

ndash00007

ndash00028

ndash00049

ndash00069

ndash00090

ndash00111

ndash00132

00428

00371

00313

00256

00198

00141

00083

00026

00032

(b)


Lagrange00270

00236

00203

00169

00135

00101

00068

00033

00000

00039

00020

00000

ndash00020

ndash00039

ndash00059

ndash00078

ndash00100

ndash00117

(c)






σ1 b C1σ3 + C2( 1113857 +(1 minus b) D1σ3 + D2( 1113857 (6)

b emσ3+ n

(7)


70

60

50

40

30

20

10

0

Vert

ical

stre

ss (M

Pa)


S8090110-3S8090110-4S8090110-5

(a)

Vert

ical

stre

ss (M

Pa)


80

70

60

50

40

30

20

10

0 1 2 3 4 5 6

S8090110-1S8090110-2S8090110-3

S8090110-4S8090110-5

0

(b)


200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(a)

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(b)





σ2 σ3 (8)


σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)




6 Conclusion




250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)








Data Availability




Acknowledgments


References







































(a) (b)


ee fa

ce

Free

face

Free

face

Free

face

Free

face

S8090110-1 S8090110-2 S8090110-3

S8090110-5S8090110-4

(a)

Free

face

(b)









ε3 1E

σ3 minus μ σ1 + σ2( 11138571113858 1113859

ε2 1E

σ2 minus μ σ1 + σ3( 11138571113858 1113859

ε1 1E

σ1 minus μ σ2 + σ3( 11138571113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



σ2 μ σ1 + σ3( 1113857 (2)


ε1 1E



σ1 1 minus μμ

σ3 minusEεc

μ(1 + μ) when εc lt 0( 1113857 (4)


σ1 minusEεc

μ(1 + μ) (5)



X

Y

Z



exx [1]-Lagrange

exx [1]-Lagrange

00240

00210

00185

00158

00130

00103

00075

00048

00020

ndash00017

ndash00039

ndash00061

ndash00082

ndash00104

ndash00126

ndash00148

ndash00169

ndash00191

(a)

exx [1]-Lagrange

exx [1]-Lagrange

00035

00014

ndash00007

ndash00028

ndash00049

ndash00069

ndash00090

ndash00111

ndash00132

00428

00371

00313

00256

00198

00141

00083

00026

00032

(b)


Lagrange00270

00236

00203

00169

00135

00101

00068

00033

00000

00039

00020

00000

ndash00020

ndash00039

ndash00059

ndash00078

ndash00100

ndash00117

(c)






σ1 b C1σ3 + C2( 1113857 +(1 minus b) D1σ3 + D2( 1113857 (6)

b emσ3+ n

(7)


70

60

50

40

30

20

10

0

Vert

ical

stre

ss (M

Pa)


S8090110-3S8090110-4S8090110-5

(a)

Vert

ical

stre

ss (M

Pa)


80

70

60

50

40

30

20

10

0 1 2 3 4 5 6

S8090110-1S8090110-2S8090110-3

S8090110-4S8090110-5

0

(b)


200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(a)

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(b)





σ2 σ3 (8)


σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)




6 Conclusion




250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)








Data Availability




Acknowledgments


References











































ε3 1E

σ3 minus μ σ1 + σ2( 11138571113858 1113859

ε2 1E

σ2 minus μ σ1 + σ3( 11138571113858 1113859

ε1 1E

σ1 minus μ σ2 + σ3( 11138571113858 1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



σ2 μ σ1 + σ3( 1113857 (2)


ε1 1E



σ1 1 minus μμ

σ3 minusEεc

μ(1 + μ) when εc lt 0( 1113857 (4)


σ1 minusEεc

μ(1 + μ) (5)



X

Y

Z



exx [1]-Lagrange

exx [1]-Lagrange

00240

00210

00185

00158

00130

00103

00075

00048

00020

ndash00017

ndash00039

ndash00061

ndash00082

ndash00104

ndash00126

ndash00148

ndash00169

ndash00191

(a)

exx [1]-Lagrange

exx [1]-Lagrange

00035

00014

ndash00007

ndash00028

ndash00049

ndash00069

ndash00090

ndash00111

ndash00132

00428

00371

00313

00256

00198

00141

00083

00026

00032

(b)


Lagrange00270

00236

00203

00169

00135

00101

00068

00033

00000

00039

00020

00000

ndash00020

ndash00039

ndash00059

ndash00078

ndash00100

ndash00117

(c)






σ1 b C1σ3 + C2( 1113857 +(1 minus b) D1σ3 + D2( 1113857 (6)

b emσ3+ n

(7)


70

60

50

40

30

20

10

0

Vert

ical

stre

ss (M

Pa)


S8090110-3S8090110-4S8090110-5

(a)

Vert

ical

stre

ss (M

Pa)


80

70

60

50

40

30

20

10

0 1 2 3 4 5 6

S8090110-1S8090110-2S8090110-3

S8090110-4S8090110-5

0

(b)


200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(a)

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(b)





σ2 σ3 (8)


σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)




6 Conclusion




250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)








Data Availability




Acknowledgments


References





































exx [1]-Lagrange

exx [1]-Lagrange

00240

00210

00185

00158

00130

00103

00075

00048

00020

ndash00017

ndash00039

ndash00061

ndash00082

ndash00104

ndash00126

ndash00148

ndash00169

ndash00191

(a)

exx [1]-Lagrange

exx [1]-Lagrange

00035

00014

ndash00007

ndash00028

ndash00049

ndash00069

ndash00090

ndash00111

ndash00132

00428

00371

00313

00256

00198

00141

00083

00026

00032

(b)


Lagrange00270

00236

00203

00169

00135

00101

00068

00033

00000

00039

00020

00000

ndash00020

ndash00039

ndash00059

ndash00078

ndash00100

ndash00117

(c)






σ1 b C1σ3 + C2( 1113857 +(1 minus b) D1σ3 + D2( 1113857 (6)

b emσ3+ n

(7)


70

60

50

40

30

20

10

0

Vert

ical

stre

ss (M

Pa)


S8090110-3S8090110-4S8090110-5

(a)

Vert

ical

stre

ss (M

Pa)


80

70

60

50

40

30

20

10

0 1 2 3 4 5 6

S8090110-1S8090110-2S8090110-3

S8090110-4S8090110-5

0

(b)


200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(a)

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(b)





σ2 σ3 (8)


σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)




6 Conclusion




250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)








Data Availability




Acknowledgments


References








































σ1 b C1σ3 + C2( 1113857 +(1 minus b) D1σ3 + D2( 1113857 (6)

b emσ3+ n

(7)


70

60

50

40

30

20

10

0

Vert

ical

stre

ss (M

Pa)


S8090110-3S8090110-4S8090110-5

(a)

Vert

ical

stre

ss (M

Pa)


80

70

60

50

40

30

20

10

0 1 2 3 4 5 6

S8090110-1S8090110-2S8090110-3

S8090110-4S8090110-5

0

(b)


200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(a)

200

180

160

140

120

100

80

60

40

20

0

Vert

ical

stre

ss (M

Pa)


H809060-1

(b)





σ2 σ3 (8)


σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)




6 Conclusion




250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)








Data Availability




Acknowledgments


References







































σ2 σ3 (8)


σ1 1 minus μμ

σ3 minusEεc

μ εc lt 0( 1113857 (9)




6 Conclusion




250

200

150

100

50

00 5 10 15 20 25 30 35

σ1 = 490σ3 + 7467

σ3 (MPa)

σ 1 (M

Pa)

σ1 = 317σ3 + 3992σ 1 =

endash05759σ 3 (3

17σ 3 + 3993) + (1 ndash endash

05759σ 3) + (490σ 3 +

7467)








Data Availability




Acknowledgments


References







































Data Availability




Acknowledgments


References















































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