TECHNICAL NOTE

On the Ratios between Elastic Modulus and Uniaxial CompressiveStrength of Heterogeneous Carbonate Rocks

V. Palchik

Received: 18 March 2010 / Accepted: 29 July 2010 / Published online: 12 August 2010

� Springer-Verlag 2010

Abstract The ratios MR = E/rc for 11 heterogeneous

carbonate (dolomites, limestones and chalks) rock forma-

tions collected from different regions of Israel were

examined. Sixty-eight uniaxial compressive tests were

conducted on weak-to-strong (5 MPa\rc\100 MPa) and

very strong (rc [ 100 MPa) rock samples exhibiting wide

ranges of elastic modulus (E = 6100–82300 MPa), uniaxial

compressive strength (rc = 14–273.9 MPa), Poisson’s ratio

(m = 0.13–0.49), and dry bulk density (q = 1.7–2.7 g/cm3).

The observed range of MR = 60.9–1011.4 and mean value

of MR = 380.5 are compared with the results obtained by

Deere (Rock mechanics in engineering practice, Wiley,

London, pp 1–20, 1968) for limestones and dolomites, and

the statistical analysis of MR distribution is performed.

Mutual relations between E, rc, q, MR for all studied rocks,

and separately for concrete rock formations are revealed.

Linear multiple correlations between E on the one hand and

rc and q on the other for Nekorot and Bina limestone and

Aminadav dolomite are obtained. It is established that the

elastic modulus and MR in very strong carbonate samples

are more correlated with q-rc combination and ea max,

respectively, than in weak to strong samples. The relation

between MR and maximum axial strain (ea max) for all

studied rock samples (weak-to-strong and very strong) is

discussed.

Keywords Uniaxial compressive test � Elastic modulus �Maximum axial strain � Rock density

1 Introduction

For a successful solution of different problems in rock

engineering, two important mechanical parameters,

namely, uniaxial compressive strength (rc) and elastic

modulus of rock (E) should be estimated correctly. It is

well known that the elastic modulus increases with

increasing uniaxial compressive strength; and there are

different empirical relations between rc and E obtained for

greywackes, limestones, agglomerates, dolomites, chalks,

sandstones, and basalts (Deere 1968; ACI 1989; Palchik

1999; Al-Shayea 2004; Gokceoglu and Zorlu 2004; Sonmez

et al. 2004; Vasarhelyi 2005; Palchik 2007; Ocak 2008). The

values of empirical coefficients involved in rc-E depen-

dencies, mathematical forms of these dependencies, and

values of squared regression coefficient (0.57 \ R2 \ 0.98)

vary with rock formation. Some of these empirical relations

are presented elsewhere (Palchik 2007).

Deere (1968) has established ranges of ratios MR = E/rc

for ten different types of intact rocks (diabase, granite,

basalts, limestones and dolomites, sandstones, shales,

quartzite, gneiss, marble and schist) and mean values of MR

in each of these ten rock types. In particular, the value of

MR for carbonate rocks (limestone and dolomites) ranges

from 250 to 700 with the mean of 420. However, the mean

value of MR cannot be used to define reliable relations

between rc and E in each of rock samples, in particular, in

heterogeneous carbonate rocks, where rock samples exhibit

different mechanical properties even within the same rock

formation. Therefore, a study of the influence of mechan-

ical properties on the value of MR is required.

The objective of this study was to examine the values of

MR = E/rc for different heterogeneous carbonate rocks

(dolomites, limestones and chalks) exhibiting a wide range

of uniaxial compressive strength (14 \ rc \ 273.9 MPa)

V. Palchik (&)

Department of Geological and Environmental Sciences,

Ben-Gurion University of the Negev,

P.O. Box 653, 84105 Beer-Sheva, Israel

e-mail: vplachek@bgu.ac.il

123

Rock Mech Rock Eng (2011) 44:121–128

DOI 10.1007/s00603-010-0112-7

and elastic modulus (E = 6,100–82,300 MPa), and to

define how uniaxial compressive strength, dry bulk density,

and maximum axial strain (at a uniaxial compressive

strength) influence these MR values.

2 Laboratory testing and test results

Sixty-eight uniaxial compressive tests were performed

in the Rock Mechanics Laboratory at the Ben-Gurion

University. A sophisticated test equipment (load frame,

TerraTek system) operates under hydraulic closed-loop

servo-control with the maximum axial force of 1.4 MN and

load frame stiffness of 5 9 109 N/m (Hatzor and Palchik

1998; Palchik and Hatzor 2004). The axial strain cantilever

set has a 10% strain range, and the radial strain cantilevers

have a strain range limit of 7%, with linearity of 1% for the

full scale in both sets. The samples were loaded in com-

pression at a constant strain rate of 10-5/s at the ambient

temperature.

The NX (D = 54 mm) sized cylindrical rock samples

having the ratio L/D = 2 were used in the tests according

to ISRM (1981). The samples were ground to the planeness

of 0.01 mm and cylinder perpendicularity within 0.05

radians. All samples were oven-dried at the temperature of

110�C for 24 h.

The test results are summarized in Tables 1 and 3. Table 1

includes rock samples which may be described according to

the ISRM suggested methods (ISRM 1981) as very strong

(rc [ 100 MPa), whereas Table 2 presents weak-to-strong

samples (5 MPa \ rc \ 100 MPa according to ISRM

(1981)). All the samples presented in Tables 1 and 2 exhibit

maximum axial strain ea max B 1%. The rock samples hav-

ing ea max [ 1% are presented in Table 3. The elastic mod-

ulus (E) and Poisson’s ratio (m) were calculated using linear

regression along the linear portion of the stress–strain curve.

The dry bulk density (q) was calculated as M/V, where M and

V are oven-dried mass and bulk volume of a cylindrical

sample, respectively. Porosity (n, %) was defined using

values of dry bulk density (q) and specific gravity of the

solids (Gs = 2.7–2.8 g/cm3): n = [1-(q/Gs)] 9 100%.

3 Statistical analysis of MR values

The graph shown in Fig. 1 presents values of MR = E/rc in

each of all 68 studied rock samples. The value of MR is

between 60.9 and 1,011.4 with the mean of 380.5. There

Table 1 Observed mechanical properties of very strong (rc [ 100 MPa) carbonate rock samples at ea max B 1%

Rock i Sample q (g/cm3) m n (%) ea max (%) rc (MPa) E (MPa) MR

AM 1 Ad2 2.75 0.16 2 0.178 117.4 70,000 596.3

2 Ad4 – – – 0.132 102 82,300 806.9

3 Ad6 2.63 – 6.4 0.41 171 51,460 300.9

4 Ad43 2.65 0.27 5.4 0.475 273.9 64,000 233.7

5 Ad80 2.62 0.28 6.4 0.316 174.2 58,500 335.8

BIN 6 Bina1 2.63 0.23 6.1 0.326 187.2 60,450 322.9

7 Tb2-12 2.43 0.24 13.2 0.567 104.2 25,300 242.8

8 Tb3-2 2.59 0.26 7.5 0.362 139 41,830 300.9

9 Tb3-4 2.47 0.25 11.8 0.408 100.4 32,130 320

SAK 10 Sakh2 2.46 – 12.1 0.325 129.8 47,800 368.3

11 Sakh3 2.6 – 7.1 0.289 144.6 57,300 396.3

YG 12 Yn1 2.34 0.24 16.4 0.387 112 33,290 297.2

13 Ca3541 2.57 0.19 8.2 0.346 173.9 54,000 310.5

14 Ca5631 2.56 0.21 8.6 0.529 186 46,400 249.5

NK 15 Gn2-1b 2.48 0.23 11.4 0.431 177 47,000 265.5

16 Gn2-4a 2.43 0.25 13.2 0.33 141 44,600 316.3

17 Gn2-5b 2.49 0.25 11.1 0.364 162 48,600 300

18 Gn3-2a 2.45 0.24 12.5 0.361 150 44,800 298.7

19 Gn3-2c 2.42 0.25 13.6 0.395 163 44,400 272.4

20 Gn3-3a 2.47 0.28 11.8 0.394 178 49,000 275.3

KSH 21 Gir1 2.64 0.28 5.7 0.4 172.4 54,420 315.7

i number of sample, q dry bulk density, m Poisson’s ratio, n porosity, ea max maximum axial strain (at rc), rc uniaxial compressive strength, Eelastic modulus, MR ratio between E and rc, – there are no data, AM Aminadav dolomite, BIN Bina limestone, SAK Sakhnin limestone, YG Yagur

dolomite, NK Nekorot limestone, KSN Kiryat—Shmone limestone

122 V. Palchik

123

Table 2 Observed mechanical properties of weak-to-strong (5 MPa \rc \ 100 MPa) carbonate rocks samples at ea max B 1%

Rock i Sample q (g/cm3) m n (%) ea max (%) rc (MPa) E (MPa) MR

AD 22 Rc1 2.12 0.23 21.5 0.32 53.2 17,400 327.1

23 Rc3 2.07 0.26 23.3 0.41 51 16,000 313.7

24 Rc4 1.93 0.2 28.5 0.32 31.9 11,700 366.8

25 Rc6 2.14 0.26 20.7 0.41 63.3 19,250 304.1

26 Rc7 1.89 0.21 30 0.41 32.1 9,500 296

27 Rc8 2.11 0.2 21.9 0.37 60.3 17,300 287

28 Rc9 2.17 0.27 19.6 0.31 63.1 20,950 332

29 St1a 2.15 0.27 20.4 0.37 50.9 15,300 300.6

30 St1b 2.16 0.23 20 0.39 53.7 15,400 286.8

31 St2a 2.14 0.2 20.7 0.4 52.3 14,300 273.7

32 St2b 2.06 0.22 23.7 0.34 37.4 11,400 304.8

AM 33 Ad5 2.62 0.37 6.4 0.2 97.8 56,000 572.6

34 Ad15 2.19 0.26 21.8 0.25 67.2 29,000 431.5

35 Ad83 2.37 0.25 15.4 0.34 61.6 20,000 324.7

BM 36 Bm2 2.32 0.4 17.1 0.25 71.5 38,100 532.9

37 Bm3 1.98 0.22 29.3 0.22 45.6 21,400 469.3

BIN 38 Bina2 2.36 0.25 15.7 0.317 77 34,400 446.8

39 Bina3 2.08 – 25.7 0.362 87 28,700 329.9

40 Bina5 2.41 0.24 13.9 0.22 80 38,700 483.8

41 Bina6 2.5 0.27 10.7 0.48 89 24,800 278.7

42 Bina7 2.4 0.2 14.3 0.29 64 25,000 390.6

43 Th5-15 2.42 0.27 13.6 0.25 84 37,700 448.8

44 Th3-24 2.19 0.26 21.8 0.18 15.4 10,000 649.4

45 Th5-13 2.2 0.31 21.4 0.14 31.3 24,000 766.8

46 Tb2-5a 2.37 0.22 15.36 0.412 90.5 25,900 286.2

47 Tb3-9 2.36 0.22 15.7 0.188 44.5 25,500 574.2

48 Tb8-16 2.31 0.23 17.5 0.345 77.9 26,590 341.3

49 B1 2.59 0.4 7.5 0.19 66.5 43,100 648.1

50 B2 2.21 0.25 21.1 0.16 25 20,900 836

51 B3 2.06 0.31 26.4 0.19 35 21,000 600

52 B4 1.74 0.2 37.9 0.15 14 11,500 821

53 B5 2.33 0.25 16.8 0.29 98 35,200 359.2

54 B7 2.24 0.15 20 0.78 54.4 10,900 200.4

SOR 55 Bz5-16 2.31 0.22 17.5 0.4 78 24,300 311.5

56 Bz2-61 2.42 0.2 13.6 0.5 85.8 22,300 259.9

57 Bz2-35 2.36 0.26 15.7 0.26 32 16,200 506.3

YR 58 Yarka1 2.36 0.15 15.7 0.63 38.7 6,500 168

59 Yarka2 2.3 0.24 17.9 0.74 38.7 6,200 160.2

60 Yarka3 2.3 0.18 17.9 0.69 41 6,200 151.2

61 Yarka4 2.34 0.19 16.4 1.07 71 8,400 118.3

SAK 62 Sakh1 2.43 – 13.2 0.254 89 40,700 457.3

YG 63 Yn6 2.44 0.25 12.9 0.176 77 47,700 619.5

64 Ca5671 2.33 0.19 16.8 0.232 60 35,500 591.7

YAN 65 Yan2 2.35 0.32 16.1 0.11 35 35,400 1,011.4

i number of sample, q dry bulk density, m Poisson’s ratio, n porosity, ea max maximum axial strain (at rc), rc uniaxial compressive strength,

E elastic modulus, MR ratio between E and rc, – no data, AD adulam chalk, AM Aminadav dolomite, BM Beit-Meir dolomite, BIN Bina

limestone, SOR Sorek limestone, YR Yarka limestone, SAK Sakhnin limestone, YG Yagur dolomite, YAN Yanuach limestone

On the Ratios between Elastic Modulus 123

123

are twenty-three very strong samples and forty-five weak-

to-strong samples. 65 rock samples (i = 1–65) exhibit

ea max B 1.0% and only 3 samples (i = 66–68) have

ea max [ 1%. Very strong (i = 1–21, 66, 68) and weak-

to-strong (i = 22–65, 67) samples exhibit 60.9 \ MR \ 806.9

and 81.3 \ MR \ 1,011.4, respectively. Figure 1 also

shows the range of MR = 250–700 (MR mean = 420)

obtained by Deere (1968) for limestones and dolomites. As

can be seen from Fig. 1, the range of MR = 60.9–1,011.4

observed in this study is wider than the range obtained by

Deere (1968). The relative error (1) and ratio (hmin)

between minimum values of MR obtained by Deere (1968)

for limestone and dolomite and observed in this study are

the following: 1 = 121.6% and hmin = 4.11. The relative

error 1 and ratio (hmax) between the maximum values of MR

are 36.4% and 1.44, respectively.

The mean value of MR = 380.5 is similar to the

mean MR = 420 observed by Deere (1968) for limestones

and dolomites. In comparison with the range of MR =

60–1,600 obtained by Deere (1968) for ten different rock

types (including dolomite and limestone), our range of

MR = 60.9–1,011.4 is smaller.

4 Effect of mechanical properties on MR value

4.1 Relations between MR, q, rc and E

Figure 2 shows how uniaxial compressive strength influ-

ences the elastic modulus and the value of MR = E/rc for all

68 studied samples. From Fig. 2 it is clear that the elastic

modulus is partly related to uniaxial compressive strength

with small R2 = 0.5543. Figure 2 also demonstrates that

an increase in the value of rc from 14 to 274 MPa does

not influence MR value (R2 = 0.1418 is very small). The

empirical equations obtained for concrete carbonate rock

formations have different empirical coefficients and differ-

ent degrees of reliability (0.57 \ R2 \ 0.98).

Sonmez et al. (2006) and Ocak (2008) have shown that the

elastic modulus can be estimated from uniaxial compressive

strength and unit weight of the material. In this study, corre-

lation between dry bulk density q and uniaxial compressive

strength is weak: R2 = 0.5642 for all 68 studied rock samples.

The effect of dry bulk density (q, g/cm3) on E and MR

observed in this study is presented in Fig. 3. The latter shows

that the elastic modulus is partly related to dry bulk density

(R2 = 0.488): an increase in dry bulk density leads to an

increase in elastic modulus. From Fig. 3 it is clear that there

is no correlation (R2 = 0.0217 is very small) between the

value of MR and dry bulk density. Weak correlations

(R2 = 0.04–0.647) between q and MR were also obtained,

when each of 11 rock formations was considered separately.

very strong

weak-to-strong

obtained b

68 samples:

Deere (1968)

0

100

200

300

400

500

600

700

800

900

1000

1100

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67

Obs

erve

d M

R

Number of sample (i)

εa max > 1 %

n = 68min =60.9

mean =380.5max =1011.4

εa max < 1 %

y

Fig. 1 Observed values of MR

in each of 68 examined rock

samples

Table 3 Observed mechanical properties of carbonate rocks samples

exhibiting ea max [ 1%

Rock i Sample q(g/cm3)

m n(%)

ea max

(%)

rc

(MPa)

E(MPa)

MR

YG 66 Yn3 2.27 0.13 18.9 1.73 105 7,200 68.6

67 Yn4 2.1 0.13 25 1.36 75 6,100 81.3

68 Yn5 2.25 0.17 19.6 1.97 115 7,000 60.9

i number of sample, q dry bulk density, m Poisson’s ratio, n porosity,

ea max maximum axial strain (at rc), rc uniaxial compressive strength,

E elastic modulus, MR ratio between E and rc, YG Yagur dolomite

124 V. Palchik

123

Linear multiple regression analysis performed for each

of rock formations shows that the elastic modulus for some

rock formations can be presented as a composite function

of q and rc:

E ¼ aþ bqþ crc ð1Þ

where a, b, and c are regression coefficients: a = -88,440,

-147,900 and -40,640; b = 52,120, 75,180, and 24,270,

c = 41.98, 45.76, and 162.1 for Nekorot limestone, Aminadav

dolomite and Bina limestone, respectively.

Here, Nekorot limestone (Fstat = 6.47), Aminadav

dolomite (Fstat = 10.19), and Bina limestone (Fstat = 13.6)

exhibit R2 = 0.812, 0.836 and 0.65, respectively.

Further, linear multiple regression analysis for E = f (rc, q)

was performed separately for weak-to-strong and very strong

samples taking into account all studied rock formations. It

is established that R2 in Eq. 1 for weak-to-strong rocks is

small (R2 = 0.338). On the other hand, very strong rocks

(when ea max B 1%) exhibit relatively large R2 = 0.793

(Fstat = 19.86, a = -171,600, b = 82,380 and c = 70.4),

R² = 0.5543

R² = 0.1418

All 68 samples

0

200

400

600

800

1000

1200

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 50 100 150 200 250 300

Ob

serv

ed M

R

Ob

serv

ed E

, MP

a

Uniaxial compressive strength (σc), MPa

E MR Poly. (E) Poly. (MR)

Fig. 2 Influence of uniaxial

compressive strength (rc) on

elastic modulus (E) and the

value of MR for all studied

samples

R2 = 0.4884800

900

1000

1100

10

20

30

40

ζ (1)

, %

χ(1) = 5670 MPa

R2 = 0.0217400

500

600

700O

bse

rved

MR

0

0

100

200

300

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.80

10000

20000

1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Ela

stic

mo

du

lus

(E),

MP

a

Dry bulk density (ρ), g/cm3

E MR Expon. (E) Log. (MR)

1 4 6 8 10 12 14 16 18 20Number of sample (i )

Fig. 3 Effect of dry bulk

density (q) on mechanical

parameters: relations between qand elastic modulus (E) and MR

for all 68 samples, and relative

(1(1), %) and root-mean-square

(v(1)) errors between calculated

(Eq.1) and observed E for very

strong rock samples

On the Ratios between Elastic Modulus 125

123

and maximum relative error 1 (1), and root-mean-square

error 1(1) (at n = 20) are 20.8–34.7% and 5,670 MPa,

respectively (see Fig. 3). These errors are less than errors

1(1) max = 37–49% and v(1) = 7,480 MPa obtained for

Aminadav dolomite samples (exhibiting the largest value of

R2 = 0.836) without taking the levels of rc in consideration.

The relative (1, %) and root-mean-square (v) errors between

the observed and calculated parameterQ

have been calcu-

lated as

1ðmÞ ¼2Q

obsðjÞ �Q

calðjÞ

���

���

QobsðjÞ þ

QcalðjÞ

� 100 ð2Þ

vðmÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

j¼1

½Q

obsðjÞ �Q

calðjÞ�2

n� 1

vuuut

ð3Þ

where (m) is the number of equation used to calculate the

parameterQ

(whereQ

is E or MR), m = 1 and 4 for

Eqs. 1 and 4, respectively; j = 1, 2, 3, … , n is the number

of tested samples;Q

obs(j) is the observed value of the

parameterQ

in jth sample;Q

cal(j) is the calculated value of

the parameterQ

in jth sample.

4.2 Relation between MR and maximum axial strain

Mathematically analyzing the observed stress–strain rela-

tions for weak-to-strong (5 MPa \rc \ 100 MPa) rock

samples, Palchik (2006) has created a stress–strain model

based on Haldane’s distribution function (Haldane 1919),

which relates the axial stress to the square of an expo-

nential function, where the exponent is axial strain (ea).

Further, the stress–strain model based on Haldane’s dis-

tribution function was used to formulate a prediction model

for the elastic modulus of intact carbonate rocks. As a

result, the analytical relation between MR and maximum

axial strain (ea max) is obtained (Palchik 2007):

MR ¼E

rc

¼ 2k

ea maxð1þ e�ea maxÞ ð4Þ

where E and rc are expressed in MPa, k = 100 is a con-

version coefficient, ea max (%) is the maximum axial strain

at rc, ra (MPa) is the axial stress, ea (%) is the axial strain

at the axial stress ra, e is Euler’s number (e = 2.718). The

notations are explained in Fig. 4. The latter shows that the

elastic modulus (E, MPa) is defined as the slope of the

linear portion of the stress–strain curve (E = k Dra/Dea

where ea is expressed in %, Dra and Dea are changes in

axial stress and strain, respectively).

Note that Eq. 4 was developed by Palchik (2007) only

for weak-to-strong carbonate rock samples and, hence, very

strong rock samples were not taken in consideration. In this

section, the author wishes to examine the validity of the use

of Eq. 4 for very strong rock samples and to compare the

relations between ea max and MR for weak-to-strong and

very strong rock samples.

The observed and analytical (Eq. 4) relationships

between MR and ea max for all rock samples exhibiting

ea max B 1% are plotted in Fig. 5a. The observed values of

MR are represented by open and closed circles for weak-to-

strong and very strong samples, respectively. From Fig. 5a

it is clear that the calculated (Eq. 4) and observed values of

MR for very strong rock sample are similar.

Figure 5b presents relative (1, %) and root-mean-

square (v) errors between the calculated (Eq. 4) and

observed MR at ea max B 1%. Compare values of 1(4) and

v(4) for very strong and weak-to-strong rock samples. All

very strong samples exhibit small 1(4) = 0.08–10.8%,

Failurestress-strain relation

linear portion of axial

0

Axi

al s

tres

s σ a

, MP

a

Axial strain εa, %0 εa max

σc

Δεa

Δσa

E = k(Δσa/Δεa)

k =100

stress-strain curve

Fig. 4 Definition of elastic

modulus (E) from stress–strain

response

126 V. Palchik

123

whereas for weak-to-strong rock samples this value

reaches 20.1–24.3%. The value of v(4) = 23 for very

strong samples is less than v(4) = 43.6 for weak-to-strong

samples. Hence, parameter MR in very strong rock sam-

ples is more correlated with ea max than in weak-to-strong

samples.

5 Conclusions

The laboratory compressive tests, statistical analysis, and

empirical and analytical relations have been used to esti-

mate the values of MR = E/rc in heterogeneous carbonate

rocks. The following main conclusions are made:

• Although the mean value of MR observed in this study

(380.5 for all samples) and MR mean = 420 obtained by

Deere (1968) for limestones and dolomites are similar,

the range of MR = 60.9–1,011.4 for carbonate rocks is

wider than the range (250 \ MR \ 700) obtained by

Deere (1968). The range of MR observed in this study

does not overtop the range of 60 \ MR \ 1,600

obtained by Deere (1968) for 10 rock types (including

dolomite and limestone).

• The observations confirm that there are no general

empirical correlations (with reliable R2) between elastic

modulus (E) and uniaxial compressive strength (rc),

MR = E/rc and uniaxial compressive strength (rc),

elastic modulus (E) and dry bulk density (q), MR and

dry bulk density (q), uniaxial compressive strength (rc),

and dry bulk density (q) which could be suitable for all

studied heterogeneous carbonate rock formations.

• The elastic modulus (E) of some studied rock forma-

tions can be presented as a combined function of rc and

700

800

900

1000

1100

R

400

500

600

700

Val

ue

of

MR

100

200

300

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Maximum axial strain (εεa max),%

weak -to- strong very strong Eq.(4)

10

15

20

25

Rel

ativ

e er

ror

(ζ(4

)), %

very strong samples weak- to -strong samples

χ(4) = 23

ζ(4) = 10 %

χ(4) = 43.6

0

5

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64

Number of sample (i)

ζ(4) = 10 %

a

b

Fig. 5 Relations between

ea max B 1% and MR.

a Calculated (Eq. 4) and

observed MR; b relative (1(4), %)

and root-mean-square (v(4))

errors between calculated

(Eq. 4) and observed MR

On the Ratios between Elastic Modulus 127

123

q. The linear multiple correlations (Eq. 1) between

E and rc, q for Nekorot and Bina limestones, and

Aminadav dolomite were obtained. It is observed that

the elastic modulus in very strong carbonate samples is

more dependent on combined effect of rc and q (Eq. 1)

than in weak-to-strong rocks.

• It is observed that the parameter MR is inversely related

to the maximum axial strain (ea max). The ratio between

E and rc is exponential function of the maximum axial

strain (Eq. 4) not only in weak-to-strong, but also in

very strong carbonate rock samples. The parameter MR

in very strong rock samples is more correlated with

ea max than in weak-to-strong samples.

References

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