energy dissipation-based method for strength...

ResearchArticleEnergy Dissipation-Based Method for Strength Determination ofRock under Uniaxial Compression

M M He 12 F Pang2 H T Wang2 J W Zhu3 and Y S Chen2

1State Key Laboratory of Eco-Hydraulics in Northwest Arid Region Xirsquoan University of Technology Xirsquoan 710048 China2Shaanxi Key Laboratory of Loess Mechanics and Engineering Xirsquoan University of Technology Shaanxi 710048 China3Xirsquoan Research Institute of China Coal Research Institute Xirsquoan 710077 China

Correspondence should be addressed to M M He 807658619qqcom

Received 1 May 2020 Revised 28 July 2020 Accepted 31 July 2020 Published 13 August 2020

Academic Editor Fabio Rizzo

Copyright copy 2020 M M He 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 energy conversion in rocks has an important significance for evaluation of the stability and safety of rock engineering In thispaper some uniaxial compression tests for fifteen different rocks were performede evolution characteristics of the total energyelastic energy and dissipated energy for the fifteen rocks were studied e dissipation energy coefficient was introduced to studythe evolution characteristics of rock e evolution of the dissipation energy coefficient for different rocks was investigated elinear interrelations of the dissipation energy coefficients and the yield strength and peak strength were exploredemethod wasproposed to determine the strength of rock using the dissipation energy coefficients e results show that the evolution of thedissipation energy coefficient exhibits significant deformation properties of rock e dissipation energy coefficients linearlyincrease with the compaction strength but decrease with the yield strength and peak strength Moreover the dissipation energycoefficient can be used to determine the rock burst proneness and crack propagation in rocks

1 Introduction

Geotechnical engineering problems in civil mining andpetroleum engineering practices [1] the mechanical prop-erties of rock have become the most common measurementin most rock mass classification systems [2 3] In the field ofunderground engineering rocks undergo the complex en-ergy change which includes energy input accumulationdissipation and release [4 5] Consequently the study ofenergy conversion in rock under the loading process has asignificant influence on the stability and safety analyses inrock engineering projects

e deformation and failure process of rock is thegradual damage evolution process driven by energy It wouldbe better to describe the deformation and failure of rocksfrom the perspective of energy [6] Some researchers havecarried out experimental studies and numerical simulationson the variation of energy in the deformation and failureprocess of rocks Based on numerical simulations Ju et al [7]built a three-dimensional finite element model of the porous

rock-like medium and investigated mechanisms of defor-mation and failure and energy dissipation rule of porousrock-like media Joints play an important role in the me-chanical behavior of rockmass and the mechanical behaviorof joints changes with the change of scale Wang et al [8]studied the effect of scale on the mechanical properties ofrock by means of numerical simulation Meng et al [9]experimentally studied the characteristics of energy inuniaxial cyclic loading and unloading compression ofsandstone rock under six different loading rates and pro-posed that the energy evolution of rock is closely related tothe axial loading stress rather than the axial loading rateWang and Cui [10] revealed the relation between energychange and confining pressure during the process ofsandstone damage and characteristics of energy storage andenergy dissipation in different deformation stage Huangand Li [11] found that the magnitude of initial confiningpressure and the unloading rate significantly influence rockstrain energy conversion during unloading e effects ofwater content on the energy evolution of red sandstone have

HindawiShock and VibrationVolume 2020 Article ID 8865958 13 pageshttpsdoiorg10115520208865958

also been investigatede experimental results indicate thatthe saturation process greatly reduces the magnitude andrate of energy release [12] Ma et al [13] studied the energydissipation mechanism of the rock deformation processunder the influence of freeze-thaw cycles Li et al [14]established a damage model for the fractured rock based onenergy dissipation According to the full stress-strain curveof rock a new evaluation method of rock brittleness is putforward on the basis of energy evolution [15 16] Chen et al[17] carried out uniaxial compression tests on coal-rockcomposite specimens e strength macroscopic failureinitiation and failure characteristics of coal-rock compositespecimens were studied e abovementioned researchesmainly focus on characteristics of energy under differentconditions or concentrate on a specific energy (strain en-ergy) However a few studies have emphasized the rela-tionship between energy and strength of rock during therock deformation and failure process [18 19]

In this work the uniaxial compression tests were con-ducted for different rocks to study the energy conversioneevolution of the total input energy elastic energy and dis-sipation energy was systematically analyzed during the de-formation and failure process of rock A new energyparameter dissipation energy coefficient was introducedepurpose is to clarify the energy evolution characteristicsBesides the linear relationship between the dissipation energycoefficient and the strength of rock was also established

2 Preparation of the Sample andExperimental Methodology

21 Preparation of the Sample e rock specimens in thisexperiment were mudstone physicochemical slate schistlimestone gneiss sandstone porphyrite dolomite shalemetamorphic sandstone marble quartz schist quartzite di-orite and granite ey were collected from a quarry inShaanxi China All specimens were drilled and processed intoφ50mmtimes 100mm (diametertimes height) standard cylindersenonparallel between the two end faces of the samples waslt0005mme measurement error was within 03mm whenthe diameter was measured along the height of the samples[20] Table 1 presents the basic parameters of the test samples

22 Experimental Apparatus and Scheme e uniaxialcompression tests were conducted on theWDT-1500 testingsystem (Figure 1) which consists of four parts the loadingsystem measuring system power system and control sys-tem [21 22] e WDT test system can provide the axialloading capacity of 1500 kN confining pressure of 80MPaand frequency of 10Hz e WDT-1500 multifunctionalmaterial testing machine used in this paper has a large ri-gidity e control system adopts a DOLI full digital servocontroller imported from Germany which has the charac-teristics of high precision and stability It can control notonly the load but also the displacement and strain and canobtain a good rock full stress-strain curve At the beginningof each test the initial load is 1 kN which ensures fullcontact between the sample and the compression plate

en the loading rate was set to 05mmmin and theloading was applied continuously until the failure occurredDuring the test the data collection interval was 02 s

23 Calculation of Energy It is assumed that the deforma-tion and failure process of rock under the external load is noheat loss [23]e total energy can be calculated according tothe first law of thermodynamics as follows

U Ue

+ Ud (1)

where U is the total energy Ue is the elastic energy and Ud isthe dissipated energy e elastic energy generally stored inrock and the dissipated energy caused the plastic defor-mation and crack propagation in the specimen Figure 2illustrates the relationship between the elastic energy and thedissipated energy of rock elements under uniaxial com-pression conditions [24] As shown in Figure 2 the area ofthe shadow represents the elastic strain energy and the redarea represents the dissipated energy Each energy wascalculated as follows [25]

U 1113946ε1

0σdε1 (2)

Ue

σ2

2E (3)

Ud

U minus Ue (4)

where σ and ε1 are axial stress and strain respectivelye deformation and failure process of rock involves the

complex energy conversion To analyze the process of theenergy evolution the dissipation energy coefficient was intro-duced which is the ratio of the dissipated energy to the elasticenergy at any time during the deformation process of rock [26]

λ Ud

Ue (5)

λv Ud

v

Uev

(6)

where Udv Ue

v and λv are the dissipated energy elasticenergy and dissipated energy coefficient of the compactionpoint respectively

λu Ud

u

Ueu

(7)

where Udu Ue

u and λu are the dissipated energy elasticenergy and dissipated energy coefficient of the yield pointrespectively

λc Ud

c

Uec

(8)

where Udc Ue

c and λc are the dissipated energy elastic en-ergy and dissipated energy coefficient of the yield pointrespectively λv λu andλc are the dissipation energy coeffi-cient at three different feature points respectively ey

2 Shock and Vibration

reflect the changes in the primary and secondary status ofelastic energy and dissipated energy during the process ofdeformation and failure of rock respectively

3 Test Result

31 Stress-Strain Curves Characteristics e deformationand failure process of rock has four stages the compactionstage elastic deformation stage yield stage and failure stage[27]

(1) Compaction stage the stress-strain curve is non-linear which results from the closure of some pri-mary microcracks and pores under the action ofinitial pressure [28]

(2) Elastic deformation stage the stress-strain curve islinear and the slope of linear portions is the elasticmodulus

(3) Yield stage as the stress increases the stress-straincurves gradually depart from the linear It is due tothe fact that the sample gradually transformed fromelastic deformation to elastic-plastic deformation

(4) Failure stage when the peak strength is reachedthe sample shows macroscopic fracture Moreoverthe postpeak curves present two different failurecharacteristics (Figure 3) One shows an apparentbrittle failure characteristic (such as granite dio-rite quartzite limestone gneiss dolomite shalemetamorphic sandstone and quartz schist) andthe other shows an obvious ductile failure char-acteristic (such as mudstone physicochemicalslate schist porphyrite sandstone and marble)[29 30]

32 Characteristics of Total Energy e deformation andfailure process of rock also is the process of the generationexpansion connection and slip of the microcracks egeneration of new cracks needs absorption of energy and thefriction between the crack surfaces dissipate energy Actu-ally the process of rock deformation and failure is theprocess of energy accumulation and dissipation [31]

σ

Ud

Ue

E

o ε1

Figure 2 Relationship between the elastic strain energy anddissipated energy of rock under uniaxial compression [24]

Table 1 Rock types and basic parameters

Rock types Density Moisturecontent

Waterabsorption

Compressivestrength

Elasticmodulus

Tensilestrength

Poissonrsquosratio

ρ(gcm3) ω() ωa() σc(MPa) E(GPa) σt(MPa) u

Mudstone 250 236 330 3253 421 437 024Physicochemical slate 272 087 128 3248 406 428 024Schist 275 036 050 4111 453 475 023Limestone 263 029 047 3715 472 432 024Gneiss 272 022 074 3852 481 425 024Sandstone 267 189 273 5564 675 624 023Porphyrite 260 157 235 6577 746 687 022Dolomites 270 035 071 9157 1035 879 021Shale 268 052 112 9003 978 863 020Metamorphicsandstone 274 003 073 9966 1164 973 020

Marble 268 004 075 10595 1225 1054 019Quartz schist 276 014 072 12703 1495 1183 019Quartzite 281 008 070 16436 1929 1256 018Diorite 280 011 071 20744 2475 1428 017Granite 285 005 035 26158 2965 1536 016

Figure 1 e WDT-1500 multifunctional material testingmachine

Shock and Vibration 3

Figure 4 illustrates the relationships between the totalenergy (U) and axial stress (σ) for different rocks underuniaxial tests As shown in Figure 4 the total energy non-linearly increases with the applied stress in the initialcompaction stage In the elastic deformation stage the totalenergy linearly increases with the increase in stress and thegrowth rate obviously seems to remain constant Afterentering the yield stage the total energy continues to in-crease with the stress but the growth rate gradually de-creases In the failure stage the total energy shows twodifferent trends which results from 15 different rocks withtwo different types of failure characteristics [32 33]

33 Characteristics of Elastic Energy Figure 5 depicts therelationship between the elastic energy and stress for dif-ferent rocks under the uniaxial compression condition eelastic energy hardly increases in the microcracks com-paction stagee reason is that the closure of internal cracks

and pores dissipates a great number of energy and only asmall amount of energy is converted into the elastic energye elastic energy of gneiss and sandstone is around 15 kJm3 and 20 kJm3 respectively (Figures 5(a) and 5(b)) eelastic energy increases linearly with the stress and thegrowth rate reaches the maximum in the elastic stage It wasdue to a stable elastic deformation generated in the sample Itmakes a large amount of the energy transformation into theelastic energy After entering the yield stage the cracks inrock began to develop and expand e elastic energycontinues to increase with the stress but the growth rate ofthe elastic energy gradually decreases e elastic energyreaches the maximum value (the storage limit value) at thepeak point and the maximum elastic energy of shale andmarble is around 320 kJm3 and 430 kJm3 respectively(Figures 5(b) and 5(c)) In the failure stage the elastic energyaccumulated in the sample is released rapidly which makesthe internal cracks of the sample expand rapidly and thesample failed [34]

σ (M

Pa)

0

10

20

30

40

50

60

02 04 06 08 10 12 14 16 18 2000ε1 ()

MudstonePhysicochemical slateSchist

LimestoneGneiss

(a)

σ (M

Pa)

0

20

40

60

80

100

120

02 04 06 08 10 12 14 16 18 2000ε1 ()

SandstonePorphyriteDolomites

ShaleMetamorphic sandstone

(b)

MarbleQuartz schistQuartzite

DioriteGranite

σ (M

Pa)

0

50

100

150

200

250

300

02 04 06 08 10 12 14 16 18 2000ε1 ()

(c)

Figure 3 Stress-strain curves for different rocks under the uniaxial compression condition


34 Characteristics of Dissipated Energy e energy dissi-pation is the main factor leading to the internal damage ofrocks [35] Figure 6 shows variation of the dissipated energywith stress for different rocks e dissipated energy rapidlyincreases in the compaction stage e reason is that moreenergy is consumed for the closed friction of the initial cracksand pores in rocks and the creation of new cracks in thestructural redistribution In the elastic stage the growth rateof the dissipated energy is gradually reduced and the dissi-pated energy is relatively low is was due to the generatedrecoverable elastic limit by the sample and it hardly producednew microcracks and internal friction e total energy wasbasically converted into elastic energy and the dissipatedenergy was maintained at a stable level [36 37] e values oflimestone and metamorphic sandstone are stabilized at

around 10 kJm3 and 15 kJm3 respectively (Figures 6(a) and6(b)) After entering the yield stage the dissipated energystarts to steadily increase and the growth rate graduallyincreases is is because the rock deformation transformedfrom linear elasticity into nonlinear elasticity and themicrocracks inside the rock began to expand rapidly andgenerate new cracks In the failure stage the dissipated energysharply increases and exceeds the elastic energy in the end

35 Characteristics of the Dissipation Energy Coefficiente dissipation energy coefficient λ is defined as the ratio ofthe dissipated energy to the elastic energy Figure 7 shows thevariation of the dissipation energy coefficient with stress fordifferent rocks e dissipation energy coefficient has four

U (k

Jmiddotmndash3

)


LimestoneGneiss

0

50

100

150

200

250

300

350

400

10 20 30 40 50 600σ (MPa)

(a)U

(kJmiddotm

ndash3)



20 40 60 80 100 1200σ (MPa)

0

100

200

300

400

500

600

(b)

U (k

Jmiddotmndash3

)


DioriteGranite

50 100 150 200 250 3000σ (MPa)

0

200

400

600

800

1000

1200

1400

1600

1800

(c)

Figure 4 Variation of total energy with stress for different rocks


evolution stages and three distinct characteristic points(Figure 7(d))

e compaction stage (OA) the point A is called thecompaction point which also is the first characteristic pointof the curve e stress of point A is defined as compactionstrength e dissipation energy coefficient λ rapidly in-creases with stress It is due to the fact that the closure andfriction of the microcrack in the rock consume most of theenergy during the initial compaction stage and only anextremely small amount of energy was converted into theelastic energy erefore the dissipation energy coefficient λrapidly increases with stress

e elastic deformation stage (AB) the point B iscalled the yield point which is the second characteristicpoint of the curve e dissipation energy coefficient λ

decreases as the stress increases and reaches the minimumvalue at the yield point B e reason is the microcracks inthe rock are completely closed at this stage and the totalenergy is basically converted into the elastic energyMoreover there is almost no generation or propagation ofnew cracks at this stage and the energy dissipation isrelatively lowerefore the dissipation energy coefficientλ decreases

e yield stage (BC) the point C is called the peak pointwhich is the third characteristic point of the curve edissipation energy coefficient λ slowly increases as the stressincreases It is due to the new microcracks gradually gen-erated inside the rock Moreover the number of microcracksincreases with increase in stressis result in a rapid growthin dissipated energy and the slow growth of the elastic

Ue (

kJmiddotm

ndash3)

0

50

100

150

200

10 20 30 40 50 600σ (MPa)


LimestoneGneiss

(a)Ue (

kJmiddotm

ndash3)

0

100

200

300

400

20 40 60 80 100 1200σ (MPa)



(b)

Ue (

kJmiddotm

ndash3)

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)

Figure 5 Variation of elastic energy with stress for different rocks


energy ence the dissipation energy coefficient λ slowlyincreases as the stress increases

e failure stage (CD) the dissipation energy coeffi-cient λ dramatically increases e reason lies in thefollowing two aspects On the one hand the generation ofthe macroscopic cracks makes the elastic energy to bereleased continuously On the other hand the crackspropagation is accelerated and the relative slip betweenthe particles becomes larger which make the dissipatedenergy to increase rapidly

It can also be seen that the primary and secondary statusof elastic energy and dissipated energy vary during the fourstages In the initial stage of loading Ue and Ud increaseslowly but the increment of Ud is greater than the

increment of Ue Ud dominates at this stage In the elasticstage Ue and Ud increase and the increment in Ud issmaller than that of Ue Ue predominates in the energyconversion process In the plastic deformation stage Ud

increases rapidly while Ue decreases e dominant one isstill elastic energy In the failure stage the elastic energystored in the rock is almost converted into dissipatedenergy At this stage the dissipated energy plays an im-portant role which causes macroscopic damage andstrength loss of the sample

According to Table 2 the relationships between thedissipation energy coefficients and strength at three dif-ferent characteristic points are studied (Figure 8) eresults show that the dissipation energy coefficient linearly

Ud (

kJmiddotm

ndash3)


LimestoneGneiss

0

20

40

60

80

100

120

140

160

10 20 30 40 50 600σ (MPa)

(a)Ud (

kJmiddotm

ndash3)



0

100

200

300

400

20 40 60 80 100 1200σ (MPa)

(b)

Ud (

kJmiddotm

ndash3)


DioriteGranite

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)

(c)

Figure 6 Variation of dissipated energy with stress for different rocks


increases with an increase in compaction strength(Figure 8(a)) However the dissipation energy coefficientlinearly decreases with yield strength and peak strength(Figures 8(b) and 8(c)) e relationships can be expressedwith the linear fitting functions Figure 8 also lists the fittingfunctions and the correlation coefficient (R2) respectivelye correlation coefficient R2 is 097 094 and 094 re-spectively It shows that there is a strong linear relationshipbetween the dissipation energy coefficients and strength inthree different characteristic points

σv σu and σc are defined as compaction strength yieldstrength and peak strength respectively (σuσc) is defined asthe characteristic stress and (λuλc) are defined as the char-acteristic dissipation energy coefficients e relationship be-tween the characteristic stress and the characteristic dissipationenergy coefficient was studied (Figure 9) e characteristicdissipation energy coefficient linearly increases with thecharacteristic stresse fitting function is expressed as follows

λu

λc

143782σu

σc

minus 064269 (9)

e correlation coefficient R2 of the fitting functions is091 It shows there is a strong linear relationship betweenthe characteristic stress and the characteristic dissipationenergy coefficient

4 Discussion and Application

e stress-strain relationship can describe the defor-mation and failure processes of rocks but it also hascertain limitations in some aspects [38] For example it isdifficult to accurately divide the four stages from thestress-strain curve According to equations (4)ndash(7) thedissipation energy coefficient-stress curves of rocks areobtained and three different feature points are deter-mined as shown in (Figure 7(d)) e dissipation energy


LimestoneGneiss

λ

00

05

10

15

20

25

30

10 20 30 40 50 600σ (MPa)

(a)



λ

00

05

10

15

20

25

30

35

40

10 20 30 40 50 60 70 80 90 1000σ (MPa)

(b)

λ

0

1

2

3

4

5

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)

Peak point

Yield pointCompaction point

O A

B

C

D

λ

0

20

40

60

80σ

(MPa

)

03 06 09 12 1500ε1 ()

0

1

2

3

4

5

σλ

(d)

Figure 7 Variation of the dissipation energy coefficient with stress for different rocks


λv

Linear fittingλv

y = 05531x + 10581R2 = 097

0

1

2

3

4

5

6

2 4 6 8 100σv (MPa)

(a)

λu

Linear fittingλu

y = ndash000174x + 040491R2 = 094

00

01

02

03

04

05

06

50 100 150 200 2500σu (MPa)

(b)

Linear fittingλc

y = ndash00025x + 06091R2 = 094

λc

00

02

04

06

08

50 100 150 200 250 3000σc (MPa)

(c)

Figure 8 Relationships between stress and the dissipation energy coefficients at three different characteristic points (a) Curve of thedissipation energy coefficients for the compaction point (b) Curve of the dissipation energy coefficients for the yield point (c) Curve of thedissipation energy coefficients for the peak point

Table 2 Strength and dissipation energy coefficients at three different characteristic points

Rock typesCompaction point Yield point Peak point

σv (MPa) λv σu (MPa) λu σc (MPa) λc

Mudstone 106 140 3177 041 3253 054Physicochemical slate 103 131 3101 039 3248 055Schist 115 176 3459 031 4111 053Limestone 109 143 3274 032 3715 054Gneiss 151 193 3522 035 3852 053Sandstone 165 203 4960 032 5564 051Porphyrite 185 223 5559 029 6577 048Dolomites 289 277 8670 025 9157 035Shale 281 265 8444 028 9003 039Metamorphic sandstone 306 288 9166 022 9966 033Marble 335 314 10040 021 10595 029Quartz schist 409 358 12284 017 12703 022Quartzite 523 405 15679 012 16436 016Diorite 644 454 19318 007 20744 01Granite 766 498 22970 003 26158 005


coefficient curves of rocks under uniaxial compressionconditions are accurately divided into four stages by threedifferent feature points It approximates the N shapedcorresponding to the four stages of the stress-straincurve By comparative analysis two types of curves thefour stages of the stress-strain curve can be accuratelydivided (Figure 7(d))

It is a difficult task to accurately determine yield strengthof rocks Usually this value can only be approximatedwhich is typically 085sim09 of the peak strength [39] eintroduction of the dissipation energy coefficient provides anew idea for calculating the yield strength of the rock edissipation energy coefficients linearly increase with theyield strength (Figure 8(b)) and the fitting function can beexpressed as follows

λu minus000174σu + 040491 (10)

e yield strength of rocks can be obtained which isbased on equations (2)ndash(8) and (10)

e ratio of the total energy of the prepeak to the elasticenergy at the peak is defined as the modified brittlenessindex [40] which can be expressed as follows

BIM U

Ue (11)

where BIM is one of the methods to determine the pronenessof rock burste judging index is listed in Table 3 [41] Aftera simple derivation the modified brittleness index has acertain relationship with the dissipation energy coefficient

λc BIM minus 1 (12)

A new criterion for rock burst proneness with index λccan be proposed as follows

0le λc le 02 Strong rockburst

02lt λc le 05Medium rockburst

05lt λcWeak rockburst

⎫⎪⎪⎬

⎪⎪⎭ (13)

e rock burst proneness can be divided into threecategories according to a new criterion (Equation (13))Mudstone physicochemical slate schist limestone gneissand sandstone are weak rock burst Porphyrite dolomiteshale metamorphic sandstone marble and quartz schisthave medium rock burst proneness Quartzite diorite andgranite have strong rock burst proneness e results ob-tained by using the new rock burst criterion are consistentwith the actual rock burst proneness

e rate of change of the dissipation energy coefficient(K) is the derivatives of the dissipation energy coefficient tothe axial strain which can be expressed as follows

K dλdε

(14)

where K is the rate of change of the dissipation energycoefficient Figure 10 depicts the relationship between therate of change of the dissipation energy coefficient (K) andaxial strain under the uniaxial compression condition Asshown in Figure 10 at the end of the compaction phase therate of change of the dissipation energy coefficient has amutation E (the value of K presents positive and negativealternating transformation) At this time initial cracks andpores of the sample were closed Subsequently at the end ofthe elastic deformation phase there is a mutation F (thevalue of K presents negative and positive alternatingtransformation) in the rate of change of the dissipationenergy coefficient It indicates that new cracks begin to crackand expand erefore the mutation E is used as the cri-terion for the initial crack closure of the rock and themutation F is used as the criterion for the new crackpropagation of the rock

In this paper the conventional uniaxial compressiontests for 15 different rocks were conducted e evolutionlaws of the elastic properties dissipative energy and dissi-pative energy coefficient were studiede innovation of thispaper is to put forward a new calculation method of rockstrength (yield strength) and to investigate the relationshipbetween strength and the dissipation energy coefficientMoreover rock is a collection of minerals e connectionbetween mineral composition and particles is closely relatedto the mechanical properties and energy characteristics ofrock [42] erefore further work mainly focuses on theinfluence of the internal structure of rock on the energy

y = 143782x ndash 064269R2 = 091

Linear fittingλuλc

λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc

Figure 9 Relationships between the characteristic stress and thecharacteristic dissipation energy coefficient

Table 3 e judging for rock burst using the BIM value

BIM value Rock burst tendencyBIMgt 15 Weak rock burst12ltBIMle 15 Medium rock burst10leBIMle 12 Strong rock burst


evolution taking into account the influence of mineralcomposition fracture number opening fracture dip angletrace length and other factors and studies the evolution lawof the elastic energy dissipative energy and dissipativeenergy coefficient of rock

5 Conclusions

According to the energy principle the energy characteristicsand evolution of the dissipation energy coefficient for dif-ferent rocks under the uniaxial compression condition wereinvestigated e main conclusions are as follows

(1) e energy evolution characteristics of differentrocks are basically similar at the prepeak but havesignificant differences at the postpeak

(2) Energy evolution provides a good reflection of therock deformation and failure process especiallythe dissipation energy coefficient It increases at

first then decreases to a minimum value increasesslowly and finally increases rapidly Its evolutionstage corresponds closely to the deformationstages of rocks under the uniaxial compressioncondition

(3) e dissipation energy coefficients linearly increasewith the compaction strength but decrease with theyield strength and peak strength

(4) e characteristic dissipation energy coefficientslinearly increase with the characteristic stress

(5) e dissipation energy coefficient can be used notonly to accurately divide the rock deformation stageand calculate the yield strength of rocks but also as anew rock burst tendency criterion

(6) e rate of change of the dissipation energy coeffi-cient shows the mutation E and the mutation Frespectively e mutations indicate the initial poreclosure and new crack propagation of the rock

Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0

Mutation FMutation E

ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)

Figure 10 e curve of the rate of change of the dissipation energy coefficient (K) (a)eK formarble (b)eK for shale (c)eK for sandstone


sample respectively e mutation E and the mu-tation F are used as the initial pore closure and newcrack propagation criterion

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 they have no conflicts of interest

Authorsrsquo Contributions

MM carried out the lab work participated in data analysiscarried out sequence alignments participated in the designof the study and drafted the manuscript F Pang carried outthe statistical analyses and critically revised the manuscriptH Wang and Y Chen collected field data and critically re-vised the manuscript and J Zhu conceived the studydesigned the study and helped draft the manuscript Allauthors gave final approval for publication and agreed to beheld accountable for the work performed therein

Acknowledgments

e financial support provided by the National NaturalScience Foundation of China (Grants nos 11902249 and11872301) is greatly appreciated is study is sponsored bythe Natural Science Basic Research Plan in Shaanxi Provinceof China (2019JQ-395 and 17JS091)

References

[1] C H Zhu and N Li ldquoRanking of influence factors and controltechnologies for the postconstruction settlement of loess high-filling embankmentsrdquo Computers and Geotechnics vol 1182020

[2] M M He Z Q Zhang J Ren et al ldquoDeep convolutionalneural network for fast determination of the rockstrengthparameters using drilling datardquo International Journal of RockMechanics and Mining Sciences vol 123 2019

[3] M He N Li Z Zhang X Yao Y Chen and C Zhu ldquoAnempirical method for determining the mechanical propertiesof jointed rock mass using drilling energyrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 116pp 64ndash74 2019

[4] Y W Li M Long L H Zuo W Li and W C ZhaoldquoBrittleness evaluation of coal based on statistical damage andenergy evolution theoryrdquo Journal of Petroleum Science andEngineering vol 172 no 1 pp 753ndash763 2018

[5] F Gong J Yan S Luo and X B Li ldquoInvestigation on thelinear energy storage and dissipation laws of rock materialsunder uniaxial compressionrdquo Rock Mechanics and RockEngineering vol 52 no 11 pp 4237ndash4255 2019

[6] H P Xie L Y Li Y Ju R D Peng and Y M Yang ldquoEnergyanalysis for damage and catastrophic failure of rocksrdquo ScienceChina Technological Sciences vol 54 no 1 pp 192ndash209 2011

[7] Y Ju H Wang Y Yang Q Hu and R Peng ldquoNumericalsimulation of mechanisms of deformation failure and energydissipation in porous rock media subjected to wave stressesrdquo

Science China Technological Sciences vol 53 no 4pp 1098ndash1113 2010

[8] X Wang W Yuan and Y Yan ldquoScale effect of mechanicalproperties of jointed rock mass a numerical study based onparticle flow coderdquo Geomechanics and Engineering vol 21no 3 pp 259ndash268 2020

[9] Q Meng M Zhang L Han H Pu and T Nie ldquoEffects ofacoustic emission and energy evolution of rock specimensunder the uniaxial cyclic loading and unloading compres-sionrdquo Rock Mechanics and Rock Engineering vol 49 no 10pp 3873ndash3886 2016

[10] Y Wang and F Cui ldquoEnergy evolution mechanism in processof Sandstone failure and energy strength criterionrdquo Journal ofApplied Geophysics vol 154 pp 21ndash28 2018

[11] D Huang and Y Li ldquoConversion of strain energy in triaxialunloading tests on marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[12] Z Zhang and F Gao ldquoExperimental investigation on theenergy evolution of dry and water-saturated red sandstonesrdquoInternational Journal of Mining Science and Technologyvol 25 no 3 pp 383ndash388 2015

[13] Q Ma D Ma and Z Yao ldquoInfluence of freeze-thaw cycles ondynamic compressive strength and energy distribution of softrock specimenrdquoCold Regions Science and Technology vol 153pp 10ndash17 2018

[14] T Li X Pei D Wang R Huang and H Tang ldquoNonlinearbehavior and damage model for fractured rock under cyclicloading based on energy dissipation principlerdquo EngineeringFracture Mechanics vol 206 pp 330ndash341 2019

[15] H Munoz A Taheri and E K Chanda ldquoFracture energy-based brittleness index development and brittleness quanti-fication by pre-peak strength parameters in rock uniaxialcompressionrdquo Rock Mechanics and Rock Engineering vol 49no 12 pp 4587ndash4606 2016

[16] I R Kivi M Ameri and H Molladavoodi ldquoShale brittlenessevaluation based on energy balance analysis of stress-straincurvesrdquo Journal of Petroleum Science and Engineeringvol 167 pp 1ndash19 2018

[17] S Chen D Yin N Jiang F Wang and Z Zhao ldquoMechanicalproperties of oil shale-coal composite samplesrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 123Article ID 104120 2019

[18] D Li Z Sun T Xie X Li and P G Ranjith ldquoEnergyevolution characteristics of hard rock during triaxial failurewith different loading and unloading pathsrdquo EngineeringGeology vol 228 pp 270ndash281 2017

[19] Y Zhang X T Feng X W Zhang et al ldquoStrain energyevolution characteristics andmechanisms of hard rocks undertrue triaxial compressionrdquo Engineering Geology vol 260pp 1ndash14 2019

[20] C E Fairhurst and J A Hudson ldquosuggested method for thecomplete stress-strain curve for intact rock in uniaxialcompressionrdquo International Journal of Mining Science andTechnology vol 36 no 3 pp 281ndash289 1999

[21] M He B Huang C Zhu Y Li and Y S Chen ldquoEnergydissipation-based method for fatigue life prediction of rocksaltrdquo Rock Mechanics and Rock Engineering vol 51 no 5pp 1447ndash1455 2018

[22] MM He Y S Chen and N Li ldquoLaboratorymeasurements ofthe mechanical damping capacities in partially saturatedsandstonerdquo Arabian Journal of Geosciences vol 12 no 14pp 1ndash7 2019


[23] H Xie L Li R Peng and Y Ju ldquoEnergy analysis and criteriafor structural failure of rocksrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 1 no 1 pp 11ndash20 2009

[24] C Yan and B H Guo ldquoCrack closure effect and energydissipation model for rocks under uniaxial compressionrdquoGeotechnical and Geological Engineering vol 38 no 4pp 621ndash629 2020

[25] M W Zhang Q B Meng and S D Liu ldquoenergy evolutioncharacteristics and distribution laws of rock materials undertriaxial cyclic loading and unloading compressionrdquo Advancesin Materials Science and Engineering vol 228 pp 270ndash2812017

[26] F Gao S P Cao K P Zhou Y Lin and L Y Zhu ldquoDamagecharacteristics and energy-dissipation mechanism of fro-zenndashthawed sandstone subjected to loadingrdquo Cold RegionsScience and Technology vol 169 pp 1ndash9 2020

[27] Z Liu C Zhang C Zhang Y Gao H Zhou and Z ChangldquoDeformation and failure characteristics and fracture evolu-tion of cryptocrystalline basaltrdquo Journal of Rock Mechanicsand Geotechnical Engineering vol 11 no 5 pp 990ndash10032019

[28] Y Zhang N Li G Lv and Z Chen ldquoQuantification of fatiguedamage at dryndashwet cycles of polymer fiber cement mortarrdquoMaterials Letters vol 257 2019

[29] Y Wang J Tang Z Dai and T Yi ldquoExperimental study onmechanical properties and failure modes of low-strength rocksamples containing different fissures under uniaxial com-pressionrdquo Engineering Fracture Mechanics vol 197 pp 1ndash202018

[30] J Zhao X-T Feng X-W Zhang Y Zhang Y-Y Zhou andC-X Yang ldquoBrittle-ductile transition and failure mechanismof Jinping marble under true triaxial compressionrdquo Engi-neering Geology vol 232 pp 160ndash170 2018

[31] Q B Zhang and J Zhao ldquoQuasi-static and dynamic fracturebehaviour of rock materials phenomena and mechanismsrdquoInternational Journal of Fracture vol 189 no 1 pp 1ndash322014

[32] D Ren D Zhou D Liu F Dong S Ma and H HuangldquoFormation mechanism of the upper triassic Yanchang for-mation tight sandstone reservoir in ordos basin-take Chang 6reservoir in Jiyuan oil field as an examplerdquo Journal of Pe-troleum Science and Engineering vol 178 pp 497ndash505 2019

[33] H Huang T Babadagli X Chen H Z Li and Y M ZhangldquoPerformance comparison of novel chemical agents formitigating water-blocking problem in tight gas sandstonesrdquoSPE Reservoir Evaluation amp Engineering vol 5 no 5 pp 1ndash92020

[34] N ZhangW Liu Y Zhang P Shan and X Shi ldquoMicroscopicpore structure of surrounding rock for underground strategicpetroleum reserve (SPR) caverns in bedded rock saltrdquo En-ergies vol 13 no 7 p 1565 2020

[35] A Taheri Y Fantidis C L Olivares B J Connelly andT J Bastian ldquoExperimental study on degradation of me-chanical properties of sandstone under different cyclicloadingsrdquo Geotechnical Testing Journal vol 39 no 4pp 673ndash687 2016

[36] Y Zhang S Cao N Zhang and C Zhao ldquoe application ofshort-wall block backfill mining to preserve surface waterresources in northwest Chinardquo Journal of Cleaner Productionvol 261 Article ID 121232 2020

[37] P Shan and X Lai ldquoAn associated evaluation methodology ofinitial stress level of coal-rock masses in steeply inclined coalseams Urumchi coal field Chinardquo Engineering Computationsvol 37 no 6 pp 2177ndash2192 2020

[38] S-Q Yang ldquoExperimental study on deformation peakstrength and crack damage behavior of hollow sandstoneunder conventional triaxial compressionrdquo Engineering Ge-ology vol 213 pp 11ndash24 2016

[39] M He Z Zhang J Zheng F Chen and N Li ldquoA newperspective on the constant mi of the Hoek-Brown failurecriterion and a new model for determining the residualstrength of rockrdquo Rock Mechanics and Rock Engineeringvol 53 no 9 pp 3953ndash3967 2020

[40] M Aubertin D E Gill and R Simon ldquoOn the use of thebrittleness index modified (BIM) to estimate the post-peakbehavior or rocksrdquo Aqua Fennica vol 232 pp 4ndash25 1994

[41] M Cai ldquoPrediction and prevention of rockburst in metalmines-a case study of Sanshandao gold minerdquo Journal of RockMechanics and Geotechnical Engineering vol 8 no 2pp 204ndash211 2016

[42] W He K Chen A Hayatdavoudi K Sawant and M LomasldquoEffects of clay content cement and mineral compositioncharacteristics on sandstone rock strength and deformabilitybehaviorsrdquo Journal of Petroleum Science and Engineeringvol 176 pp 962ndash969 2019


also been investigatede experimental results indicate thatthe saturation process greatly reduces the magnitude andrate of energy release [12] Ma et al [13] studied the energydissipation mechanism of the rock deformation processunder the influence of freeze-thaw cycles Li et al [14]established a damage model for the fractured rock based onenergy dissipation According to the full stress-strain curveof rock a new evaluation method of rock brittleness is putforward on the basis of energy evolution [15 16] Chen et al[17] carried out uniaxial compression tests on coal-rockcomposite specimens e strength macroscopic failureinitiation and failure characteristics of coal-rock compositespecimens were studied e abovementioned researchesmainly focus on characteristics of energy under differentconditions or concentrate on a specific energy (strain en-ergy) However a few studies have emphasized the rela-tionship between energy and strength of rock during therock deformation and failure process [18 19]

In this work the uniaxial compression tests were con-ducted for different rocks to study the energy conversioneevolution of the total input energy elastic energy and dis-sipation energy was systematically analyzed during the de-formation and failure process of rock A new energyparameter dissipation energy coefficient was introducedepurpose is to clarify the energy evolution characteristicsBesides the linear relationship between the dissipation energycoefficient and the strength of rock was also established

2 Preparation of the Sample andExperimental Methodology

21 Preparation of the Sample e rock specimens in thisexperiment were mudstone physicochemical slate schistlimestone gneiss sandstone porphyrite dolomite shalemetamorphic sandstone marble quartz schist quartzite di-orite and granite ey were collected from a quarry inShaanxi China All specimens were drilled and processed intoφ50mmtimes 100mm (diametertimes height) standard cylindersenonparallel between the two end faces of the samples waslt0005mme measurement error was within 03mm whenthe diameter was measured along the height of the samples[20] Table 1 presents the basic parameters of the test samples

22 Experimental Apparatus and Scheme e uniaxialcompression tests were conducted on theWDT-1500 testingsystem (Figure 1) which consists of four parts the loadingsystem measuring system power system and control sys-tem [21 22] e WDT test system can provide the axialloading capacity of 1500 kN confining pressure of 80MPaand frequency of 10Hz e WDT-1500 multifunctionalmaterial testing machine used in this paper has a large ri-gidity e control system adopts a DOLI full digital servocontroller imported from Germany which has the charac-teristics of high precision and stability It can control notonly the load but also the displacement and strain and canobtain a good rock full stress-strain curve At the beginningof each test the initial load is 1 kN which ensures fullcontact between the sample and the compression plate

en the loading rate was set to 05mmmin and theloading was applied continuously until the failure occurredDuring the test the data collection interval was 02 s

23 Calculation of Energy It is assumed that the deforma-tion and failure process of rock under the external load is noheat loss [23]e total energy can be calculated according tothe first law of thermodynamics as follows

U Ue

+ Ud (1)

where U is the total energy Ue is the elastic energy and Ud isthe dissipated energy e elastic energy generally stored inrock and the dissipated energy caused the plastic defor-mation and crack propagation in the specimen Figure 2illustrates the relationship between the elastic energy and thedissipated energy of rock elements under uniaxial com-pression conditions [24] As shown in Figure 2 the area ofthe shadow represents the elastic strain energy and the redarea represents the dissipated energy Each energy wascalculated as follows [25]

U 1113946ε1

0σdε1 (2)

Ue

σ2

2E (3)

Ud

U minus Ue (4)

where σ and ε1 are axial stress and strain respectivelye deformation and failure process of rock involves the

complex energy conversion To analyze the process of theenergy evolution the dissipation energy coefficient was intro-duced which is the ratio of the dissipated energy to the elasticenergy at any time during the deformation process of rock [26]

λ Ud

Ue (5)

λv Ud

v

Uev

(6)

where Udv Ue

v and λv are the dissipated energy elasticenergy and dissipated energy coefficient of the compactionpoint respectively

λu Ud

u

Ueu

(7)

where Udu Ue

u and λu are the dissipated energy elasticenergy and dissipated energy coefficient of the yield pointrespectively

λc Ud

c

Uec

(8)

where Udc Ue

c and λc are the dissipated energy elastic en-ergy and dissipated energy coefficient of the yield pointrespectively λv λu andλc are the dissipation energy coeffi-cient at three different feature points respectively ey



3 Test Result







σ

Ud

Ue

E

o ε1




Waterabsorption

Compressivestrength

Elasticmodulus

Tensilestrength

Poissonrsquosratio









σ (M

Pa)

0

10

20

30

40

50

60

02 04 06 08 10 12 14 16 18 2000ε1 ()


LimestoneGneiss

(a)

σ (M

Pa)

0

20

40

60

80

100

120

02 04 06 08 10 12 14 16 18 2000ε1 ()



(b)


DioriteGranite

σ (M

Pa)

0

50

100

150

200

250

300

02 04 06 08 10 12 14 16 18 2000ε1 ()

(c)






U (k

Jmiddotmndash3

)


LimestoneGneiss

0

50

100

150

200

250

300

350

400

10 20 30 40 50 600σ (MPa)

(a)U

(kJmiddotm

ndash3)



20 40 60 80 100 1200σ (MPa)

0

100

200

300

400

500

600

(b)

U (k

Jmiddotmndash3

)


DioriteGranite

50 100 150 200 250 3000σ (MPa)

0

200

400

600

800

1000

1200

1400

1600

1800

(c)








Ue (

kJmiddotm

ndash3)

0

50

100

150

200

10 20 30 40 50 600σ (MPa)


LimestoneGneiss

(a)Ue (

kJmiddotm

ndash3)

0

100

200

300

400

20 40 60 80 100 1200σ (MPa)



(b)

Ue (

kJmiddotm

ndash3)

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)









Ud (

kJmiddotm

ndash3)


LimestoneGneiss

0

20

40

60

80

100

120

140

160

10 20 30 40 50 600σ (MPa)

(a)Ud (

kJmiddotm

ndash3)



0

100

200

300

400

20 40 60 80 100 1200σ (MPa)

(b)

Ud (

kJmiddotm

ndash3)


DioriteGranite

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)

(c)





λu

λc

143782σu

σc

minus 064269 (9)





LimestoneGneiss

λ

00

05

10

15

20

25

30

10 20 30 40 50 600σ (MPa)

(a)



λ

00

05

10

15

20

25

30

35

40

10 20 30 40 50 60 70 80 90 1000σ (MPa)

(b)

λ

0

1

2

3

4

5

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)

Peak point


O A

B

C

D

λ

0

20

40

60

80σ

(MPa

)

03 06 09 12 1500ε1 ()

0

1

2

3

4

5

σλ

(d)



λv

Linear fittingλv

y = 05531x + 10581R2 = 097

0

1

2

3

4

5

6

2 4 6 8 100σv (MPa)

(a)

λu

Linear fittingλu

y = ndash000174x + 040491R2 = 094

00

01

02

03

04

05

06

50 100 150 200 2500σu (MPa)

(b)

Linear fittingλc

y = ndash00025x + 06091R2 = 094

λc

00

02

04

06

08

50 100 150 200 250 3000σc (MPa)

(c)









λu minus000174σu + 040491 (10)



BIM U

Ue (11)







⎫⎪⎪⎬

⎪⎪⎭ (13)



K dλdε

(14)



y = 143782x ndash 064269R2 = 091


λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc






5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References















































3 Test Result







σ

Ud

Ue

E

o ε1




Waterabsorption

Compressivestrength

Elasticmodulus

Tensilestrength

Poissonrsquosratio









σ (M

Pa)

0

10

20

30

40

50

60

02 04 06 08 10 12 14 16 18 2000ε1 ()


LimestoneGneiss

(a)

σ (M

Pa)

0

20

40

60

80

100

120

02 04 06 08 10 12 14 16 18 2000ε1 ()



(b)


DioriteGranite

σ (M

Pa)

0

50

100

150

200

250

300

02 04 06 08 10 12 14 16 18 2000ε1 ()

(c)






U (k

Jmiddotmndash3

)


LimestoneGneiss

0

50

100

150

200

250

300

350

400

10 20 30 40 50 600σ (MPa)

(a)U

(kJmiddotm

ndash3)



20 40 60 80 100 1200σ (MPa)

0

100

200

300

400

500

600

(b)

U (k

Jmiddotmndash3

)


DioriteGranite

50 100 150 200 250 3000σ (MPa)

0

200

400

600

800

1000

1200

1400

1600

1800

(c)








Ue (

kJmiddotm

ndash3)

0

50

100

150

200

10 20 30 40 50 600σ (MPa)


LimestoneGneiss

(a)Ue (

kJmiddotm

ndash3)

0

100

200

300

400

20 40 60 80 100 1200σ (MPa)



(b)

Ue (

kJmiddotm

ndash3)

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)









Ud (

kJmiddotm

ndash3)


LimestoneGneiss

0

20

40

60

80

100

120

140

160

10 20 30 40 50 600σ (MPa)

(a)Ud (

kJmiddotm

ndash3)



0

100

200

300

400

20 40 60 80 100 1200σ (MPa)

(b)

Ud (

kJmiddotm

ndash3)


DioriteGranite

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)

(c)





λu

λc

143782σu

σc

minus 064269 (9)





LimestoneGneiss

λ

00

05

10

15

20

25

30

10 20 30 40 50 600σ (MPa)

(a)



λ

00

05

10

15

20

25

30

35

40

10 20 30 40 50 60 70 80 90 1000σ (MPa)

(b)

λ

0

1

2

3

4

5

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)

Peak point


O A

B

C

D

λ

0

20

40

60

80σ

(MPa

)

03 06 09 12 1500ε1 ()

0

1

2

3

4

5

σλ

(d)



λv

Linear fittingλv

y = 05531x + 10581R2 = 097

0

1

2

3

4

5

6

2 4 6 8 100σv (MPa)

(a)

λu

Linear fittingλu

y = ndash000174x + 040491R2 = 094

00

01

02

03

04

05

06

50 100 150 200 2500σu (MPa)

(b)

Linear fittingλc

y = ndash00025x + 06091R2 = 094

λc

00

02

04

06

08

50 100 150 200 250 3000σc (MPa)

(c)









λu minus000174σu + 040491 (10)



BIM U

Ue (11)







⎫⎪⎪⎬

⎪⎪⎭ (13)



K dλdε

(14)



y = 143782x ndash 064269R2 = 091


λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc






5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References

















































σ (M

Pa)

0

10

20

30

40

50

60

02 04 06 08 10 12 14 16 18 2000ε1 ()


LimestoneGneiss

(a)

σ (M

Pa)

0

20

40

60

80

100

120

02 04 06 08 10 12 14 16 18 2000ε1 ()



(b)


DioriteGranite

σ (M

Pa)

0

50

100

150

200

250

300

02 04 06 08 10 12 14 16 18 2000ε1 ()

(c)






U (k

Jmiddotmndash3

)


LimestoneGneiss

0

50

100

150

200

250

300

350

400

10 20 30 40 50 600σ (MPa)

(a)U

(kJmiddotm

ndash3)



20 40 60 80 100 1200σ (MPa)

0

100

200

300

400

500

600

(b)

U (k

Jmiddotmndash3

)


DioriteGranite

50 100 150 200 250 3000σ (MPa)

0

200

400

600

800

1000

1200

1400

1600

1800

(c)








Ue (

kJmiddotm

ndash3)

0

50

100

150

200

10 20 30 40 50 600σ (MPa)


LimestoneGneiss

(a)Ue (

kJmiddotm

ndash3)

0

100

200

300

400

20 40 60 80 100 1200σ (MPa)



(b)

Ue (

kJmiddotm

ndash3)

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)









Ud (

kJmiddotm

ndash3)


LimestoneGneiss

0

20

40

60

80

100

120

140

160

10 20 30 40 50 600σ (MPa)

(a)Ud (

kJmiddotm

ndash3)



0

100

200

300

400

20 40 60 80 100 1200σ (MPa)

(b)

Ud (

kJmiddotm

ndash3)


DioriteGranite

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)

(c)





λu

λc

143782σu

σc

minus 064269 (9)





LimestoneGneiss

λ

00

05

10

15

20

25

30

10 20 30 40 50 600σ (MPa)

(a)



λ

00

05

10

15

20

25

30

35

40

10 20 30 40 50 60 70 80 90 1000σ (MPa)

(b)

λ

0

1

2

3

4

5

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)

Peak point


O A

B

C

D

λ

0

20

40

60

80σ

(MPa

)

03 06 09 12 1500ε1 ()

0

1

2

3

4

5

σλ

(d)



λv

Linear fittingλv

y = 05531x + 10581R2 = 097

0

1

2

3

4

5

6

2 4 6 8 100σv (MPa)

(a)

λu

Linear fittingλu

y = ndash000174x + 040491R2 = 094

00

01

02

03

04

05

06

50 100 150 200 2500σu (MPa)

(b)

Linear fittingλc

y = ndash00025x + 06091R2 = 094

λc

00

02

04

06

08

50 100 150 200 250 3000σc (MPa)

(c)









λu minus000174σu + 040491 (10)



BIM U

Ue (11)







⎫⎪⎪⎬

⎪⎪⎭ (13)



K dλdε

(14)



y = 143782x ndash 064269R2 = 091


λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc






5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References

















































U (k

Jmiddotmndash3

)


LimestoneGneiss

0

50

100

150

200

250

300

350

400

10 20 30 40 50 600σ (MPa)

(a)U

(kJmiddotm

ndash3)



20 40 60 80 100 1200σ (MPa)

0

100

200

300

400

500

600

(b)

U (k

Jmiddotmndash3

)


DioriteGranite

50 100 150 200 250 3000σ (MPa)

0

200

400

600

800

1000

1200

1400

1600

1800

(c)








Ue (

kJmiddotm

ndash3)

0

50

100

150

200

10 20 30 40 50 600σ (MPa)


LimestoneGneiss

(a)Ue (

kJmiddotm

ndash3)

0

100

200

300

400

20 40 60 80 100 1200σ (MPa)



(b)

Ue (

kJmiddotm

ndash3)

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)









Ud (

kJmiddotm

ndash3)


LimestoneGneiss

0

20

40

60

80

100

120

140

160

10 20 30 40 50 600σ (MPa)

(a)Ud (

kJmiddotm

ndash3)



0

100

200

300

400

20 40 60 80 100 1200σ (MPa)

(b)

Ud (

kJmiddotm

ndash3)


DioriteGranite

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)

(c)





λu

λc

143782σu

σc

minus 064269 (9)





LimestoneGneiss

λ

00

05

10

15

20

25

30

10 20 30 40 50 600σ (MPa)

(a)



λ

00

05

10

15

20

25

30

35

40

10 20 30 40 50 60 70 80 90 1000σ (MPa)

(b)

λ

0

1

2

3

4

5

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)

Peak point


O A

B

C

D

λ

0

20

40

60

80σ

(MPa

)

03 06 09 12 1500ε1 ()

0

1

2

3

4

5

σλ

(d)



λv

Linear fittingλv

y = 05531x + 10581R2 = 097

0

1

2

3

4

5

6

2 4 6 8 100σv (MPa)

(a)

λu

Linear fittingλu

y = ndash000174x + 040491R2 = 094

00

01

02

03

04

05

06

50 100 150 200 2500σu (MPa)

(b)

Linear fittingλc

y = ndash00025x + 06091R2 = 094

λc

00

02

04

06

08

50 100 150 200 250 3000σc (MPa)

(c)









λu minus000174σu + 040491 (10)



BIM U

Ue (11)







⎫⎪⎪⎬

⎪⎪⎭ (13)



K dλdε

(14)



y = 143782x ndash 064269R2 = 091


λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc






5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References



















































Ue (

kJmiddotm

ndash3)

0

50

100

150

200

10 20 30 40 50 600σ (MPa)


LimestoneGneiss

(a)Ue (

kJmiddotm

ndash3)

0

100

200

300

400

20 40 60 80 100 1200σ (MPa)



(b)

Ue (

kJmiddotm

ndash3)

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)









Ud (

kJmiddotm

ndash3)


LimestoneGneiss

0

20

40

60

80

100

120

140

160

10 20 30 40 50 600σ (MPa)

(a)Ud (

kJmiddotm

ndash3)



0

100

200

300

400

20 40 60 80 100 1200σ (MPa)

(b)

Ud (

kJmiddotm

ndash3)


DioriteGranite

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)

(c)





λu

λc

143782σu

σc

minus 064269 (9)





LimestoneGneiss

λ

00

05

10

15

20

25

30

10 20 30 40 50 600σ (MPa)

(a)



λ

00

05

10

15

20

25

30

35

40

10 20 30 40 50 60 70 80 90 1000σ (MPa)

(b)

λ

0

1

2

3

4

5

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)

Peak point


O A

B

C

D

λ

0

20

40

60

80σ

(MPa

)

03 06 09 12 1500ε1 ()

0

1

2

3

4

5

σλ

(d)



λv

Linear fittingλv

y = 05531x + 10581R2 = 097

0

1

2

3

4

5

6

2 4 6 8 100σv (MPa)

(a)

λu

Linear fittingλu

y = ndash000174x + 040491R2 = 094

00

01

02

03

04

05

06

50 100 150 200 2500σu (MPa)

(b)

Linear fittingλc

y = ndash00025x + 06091R2 = 094

λc

00

02

04

06

08

50 100 150 200 250 3000σc (MPa)

(c)









λu minus000174σu + 040491 (10)



BIM U

Ue (11)







⎫⎪⎪⎬

⎪⎪⎭ (13)



K dλdε

(14)



y = 143782x ndash 064269R2 = 091


λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc






5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References




















































Ud (

kJmiddotm

ndash3)


LimestoneGneiss

0

20

40

60

80

100

120

140

160

10 20 30 40 50 600σ (MPa)

(a)Ud (

kJmiddotm

ndash3)



0

100

200

300

400

20 40 60 80 100 1200σ (MPa)

(b)

Ud (

kJmiddotm

ndash3)


DioriteGranite

0

200

400

600

800

1000

1200

50 100 150 200 250 3000σ (MPa)

(c)





λu

λc

143782σu

σc

minus 064269 (9)





LimestoneGneiss

λ

00

05

10

15

20

25

30

10 20 30 40 50 600σ (MPa)

(a)



λ

00

05

10

15

20

25

30

35

40

10 20 30 40 50 60 70 80 90 1000σ (MPa)

(b)

λ

0

1

2

3

4

5

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)

Peak point


O A

B

C

D

λ

0

20

40

60

80σ

(MPa

)

03 06 09 12 1500ε1 ()

0

1

2

3

4

5

σλ

(d)



λv

Linear fittingλv

y = 05531x + 10581R2 = 097

0

1

2

3

4

5

6

2 4 6 8 100σv (MPa)

(a)

λu

Linear fittingλu

y = ndash000174x + 040491R2 = 094

00

01

02

03

04

05

06

50 100 150 200 2500σu (MPa)

(b)

Linear fittingλc

y = ndash00025x + 06091R2 = 094

λc

00

02

04

06

08

50 100 150 200 250 3000σc (MPa)

(c)









λu minus000174σu + 040491 (10)



BIM U

Ue (11)







⎫⎪⎪⎬

⎪⎪⎭ (13)



K dλdε

(14)



y = 143782x ndash 064269R2 = 091


λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc






5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References
















































λu

λc

143782σu

σc

minus 064269 (9)





LimestoneGneiss

λ

00

05

10

15

20

25

30

10 20 30 40 50 600σ (MPa)

(a)



λ

00

05

10

15

20

25

30

35

40

10 20 30 40 50 60 70 80 90 1000σ (MPa)

(b)

λ

0

1

2

3

4

5

50 100 150 200 250 3000σ (MPa)


DioriteGranite

(c)

Peak point


O A

B

C

D

λ

0

20

40

60

80σ

(MPa

)

03 06 09 12 1500ε1 ()

0

1

2

3

4

5

σλ

(d)



λv

Linear fittingλv

y = 05531x + 10581R2 = 097

0

1

2

3

4

5

6

2 4 6 8 100σv (MPa)

(a)

λu

Linear fittingλu

y = ndash000174x + 040491R2 = 094

00

01

02

03

04

05

06

50 100 150 200 2500σu (MPa)

(b)

Linear fittingλc

y = ndash00025x + 06091R2 = 094

λc

00

02

04

06

08

50 100 150 200 250 3000σc (MPa)

(c)









λu minus000174σu + 040491 (10)



BIM U

Ue (11)







⎫⎪⎪⎬

⎪⎪⎭ (13)



K dλdε

(14)



y = 143782x ndash 064269R2 = 091


λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc






5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References














































λv

Linear fittingλv

y = 05531x + 10581R2 = 097

0

1

2

3

4

5

6

2 4 6 8 100σv (MPa)

(a)

λu

Linear fittingλu

y = ndash000174x + 040491R2 = 094

00

01

02

03

04

05

06

50 100 150 200 2500σu (MPa)

(b)

Linear fittingλc

y = ndash00025x + 06091R2 = 094

λc

00

02

04

06

08

50 100 150 200 250 3000σc (MPa)

(c)









λu minus000174σu + 040491 (10)



BIM U

Ue (11)







⎫⎪⎪⎬

⎪⎪⎭ (13)



K dλdε

(14)



y = 143782x ndash 064269R2 = 091


λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc






5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References
















































λu minus000174σu + 040491 (10)



BIM U

Ue (11)







⎫⎪⎪⎬

⎪⎪⎭ (13)



K dλdε

(14)



y = 143782x ndash 064269R2 = 091


λ uλ

c

050

055

060

065

070

075

080

086 088 090 092 094 096 098084σuσc






5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References















































5 Conclusions









Mutation F

K

Mutation E

ndash20

ndash10

0

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(a)

K

0


ndash20

ndash10

10

20

30

40

50

60

K

02 04 06 08 10 1200ε1 ()

(b)

K


ndash20

ndash10

0

10

20

30

40

50

60

70

K

02 04 06 08 10 12 14 16 1800ε1 ()

(c)




Data Availability






Acknowledgments


References















































Data Availability






Acknowledgments


References














































energy dissipation-based method for strength...

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