Use of stress–strain model based on Haldane's distribution function for prediction of elastic modulus

ARTICLE IN PRESS

International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524 www.elsevier.com/locate/ijrmms

Use of stress–strain model based on Haldane’s distribution function for prediction of elastic modulus V. Palchik Department of Geological and Environmental Sciences, Ben–Gurion University of the Negev, P.O. Box. 653, Beer-Sheva 84105, Israel Received 9 July 2006; received in revised form 19 September 2006; accepted 20 September 2006 Available online 22 November 2006

Abstract The stress–strain model based on Haldane’s distribution function is used to formulate a prediction model for the elastic modulus of intact carbonate rocks. The elastic modulus (E) is defined as the slope of the linear portion of stress–strain curve passing through sa and ea coordinates, where axial stress sa is related to uniaxial compressive stress (sc) and exponential function of axial strain (ea) according to the stress–strain model. The prediction model of elastic modulus obtained on this basis requires only the value of uniaxial compressive strength, and one datum point (axial stress and axial strain) within the strain range 0:05%oa pa max , where a max is the axial strain at sc. The model is well correlated with the experimental results obtained from weak to strong carbonate rocks such as dolomites, chalks and limestones having uniaxial compressive strengths less than 100 MPa, collected from different locations in Israel. r 2006 Elsevier Ltd. All rights reserved. Keywords: Carbonate rock; Stress–strain model; Elastic modulus; Uniaxial compressive strength

1. Introduction The elastic modulus is extensively used in rock engineering when deformations of different structural elements of underground storages, tunnels or mining openings must be computed. The elastic modulus can be obtained from the stress–strain response of a rock sample subjected to uniaxial compression. However, from the experimental point of view, this is not always easy. For example, as compared with compressive strength measurements carried out to determine the uniaxial compressive strength (sc), this testing procedure is much more complicated and time consuming. To overcome the difficulties encountered during determination of elastic modulus (E) by laboratory tests, some researchers [1–17] have tried to find some shortcuts to enable them to predict the elastic modulus of rocks using theoretical and empirical approaches based on the use of other mechanical properties and indices, petrographic composition, seismic data analysis, ultrasonic testing, etc. Tel.: +972 86461770; fax:+972 86472997.

E-mail address: [email protected]. 1365-1609/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2006.09.003

In particular, Deere [4], ACI [7], Palchik [9], Al-Shayea [10], Gokceoglu and Zorlu [11], Sonmez et al. [12] and Vasarhelyi [16] have shown that there is strong relation between the elastic modulus (E) and uniaxial compressive strength (sc). For example, Palchik [9], Gokceoglu and Zorlu [11] and Vasarhelyi [16] have recently established the following three equations, respectively, between the elastic modulus E (in GPa) and the uniaxial compressive strength sc (in MPa): E ¼ 0:0824sc þ 0:787 E ¼ 0:456sc þ 11:6 E ¼ 0:374sc

ðsandstone; R2 ¼ 0:756Þ, ðgreywacke; R2 ¼ 0:82Þ,

ðlimestone; R2 ¼ 0:784Þ.

(1) (2) (3)

Sonmez et al. [12] proposed the following power-law relation between E (GPa) and sc (MPa) for agglomerates: E ¼ 0:438s0:675 c

ðR2 ¼ 0:949Þ.

(4)

In this study, the relationships between E and sc are obtained from the stress–strain responses of six carbonate rocks (dolomites, chalks and limestones) collected from different locations in Israel. The relations are presented in Fig. 1. The linear correlations between E (MPa) and sc

ARTICLE IN PRESS V. Palchik / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524

515

60000 y = 993.52x - 40711 R2 = 0.9803

50000

Adulam chalk

Elastic modulus (E), MPa

Aminadav dolomite Beit-Meir dolomite y = 2672.2x0.5598 R2 = 0.5663

40000

Bina limestone Sorek dolomite

30000

Yarka limestone y = 289.78x + 827.66 R2 = 0.8959

20000

10000 y = 65.976x + 3701 R2 = 0.9685

Linear (Adulam chalk) Power (Bina limestone) Linear (Aminadav dolomite) Linear (Yarka limestone)

0 0

20

40

60

80

100

Uniaxial compressive strength (σc), MPa Fig. 1. Relationship between E and sc for six carbonate rocks collected from different parts of Israel.

(MPa) (E ¼ 289.78sc+827.66, R2 ¼ 0.8959; E ¼ 993.52sc 40771, R2 ¼ 0.9803; and E ¼ 65.976sc+3701, R2 ¼ 0.9685) are obtained for Adulam chalk, Aminadav dolomite and Yarka limestone, respectively; and a powerlaw correlation (E ¼ 2672.2s0.5598 , R2 ¼ 0.566) is obtained c for Bina limestone. However, the use of these empirical equations is limited, since, empirical constants can only be used for collected carbonate rocks. In addition, although the elastic modulus (E) is increased with increasing uniaxial compressive strength (sc), these relations have different degrees of reliability, having R2 values ranging from 0.57 to 0.98. Sonmez et al. [17] indicated that it is impossible to construct a general empirical equation to predict elastic modulus of intact rock by using unique input such as particularly sc. To overcome this limitation, they developed an artificial neural network-based chart, which considers sc and unit weight (g) as input to predict elastic modulus of intact rock. In this paper, development of a new model to prediction of elastic modulus of carbonate rocks with high reliability is attempted. This is achieved using the stress–strain model proposed by Palchik [1] based on Haldane’s distribution function. In this stress–strain model, the axial stress sa is expressed in terms of uniaxial compressive strength (sc) and an exponential function of the axial strain (ea) at sa. The model proposed by Palchik [1] is given in detail in subsequent section. In this case, the slope of the linear portion of stress–strain curve (average elastic modulus) passing through sa and ea co-ordinates is also expressed in terms of sc and an exponential function of ea. This leads to the formulation of elastic modulus prediction model based on uniaxial compressive strength (sc), and one datum point

(i.e., one pair of sa and ea) within the strain range 0:05%oa pa max , where a max is the axial strain at sc. The model is only applicable to carbonate rocks having strength sco100 MPa. 2. Stress–strain model based on Haldane’s distribution function Palchik [1] suggested an analytical model based on Haldane’s distribution function for predicting the stress– strain relationship over the whole pre-failure strain range   1  e2a sa ¼ s c , (5) 1  e2a max where sa (MPa) is the axial stress, ea (%) is the axial strain at the axial stress sa, sc (MPa) is the uniaxial compressive strength, and ea max (%) is the maximum axial strain at sc. The notations are explained in Fig. 2a. This model implies that 1  e2ai 1  e2a1 1  e2a2 1  e2an ¼ ¼ ;...;¼ , sai sa1 sa2 san

(6)

where i ¼ 1,2,y, n are the number of points on stress– strain curve where the axial stress and axial strain were measured; n is the final point of pre-failure strain range, where sc and ea max at sc were measured, i.e., ean ¼ ea max and san ¼ sc; sai ¼ sa1, sa2, y, sc are axial strains measured in points 1, 2, y, n, respectively; eai ¼ ea1, ea2, y, ea max are axial strains measured at sa1, sa2, y, sc, respectively, at points 1,2,y, n, respectively. This stress–strain model allows one to predict the stress– strain relationship (sa and ea coordinates) over the whole pre-failure strain range for weak to strong carbonate rocks

ARTICLE IN PRESS V. Palchik / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524 σc = σa max

516

stress-strain curve

Axial stress σa

failure

data points

linear portion of stress-strain curve (elastic stage)

εa elast1

E =Δσa / Δεa

Δσa Δεa εa elast 2

(a)

Axial strain εa

0

εa max

1 0.9 0.8 0.7

σa / σc

0.6 0.5 0.4 0.3 0.2 0.1 0 0.0

0.1

0.2

0.3

εa max:

(b)

0.4 0.5 0.6 Axial strain (εa), %

0.7

0.50%

0.75%

0.25%

0.8

0.9

1.0

1%

100 AD5 (E = 56000 MPa)

90

Bina limestone Aminadav dolomite Adulam chalk

Axial stress (σa), MPa

80 70

B1(E = 43100 MPa) RC9 (E =20500 MPa) AD83 (E = 18000 MPa)

60

B7 (E = 10900 MPa)

50 40 data points

30

RC7 (E = 9500 MPa)

20 10 0 0

(c)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Axial strain (εa), %

Fig. 2. Stress–strain curves: (a) stress–strain relationship, where sc ¼ sa max is uniaxial compressive strength; ea max is maximum axial strain at sc; ea elast1 and ea elast2 are onset and end of elastic stage (linear portion) of stress–strain curve, respectively; E ¼ Dsa/Dea is elastic modulus; (b) relationship between normalized stress and strain according to stress–strain model (Eq. (5), [1]) based on Haldane’s distribution function; (c) observed stress–strain curves for carbonate samples rc7, rc9, ad5, ad83, b1 and b7 exhibiting E ¼ 9500, 20 500, 56 000, 18 000, 43 100 and 10 900 MPa, respectively.

ARTICLE IN PRESS V. Palchik / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524

(sco100 MPa) when the uniaxial compressive strength and maximum axial strain are known. The relationship between the normalized stress (sa/sc) and axial strain (ea) according to the stress–strain model (Eq. (5)) for four different levels of ea max ¼ 0.25%, 0.5%, 0.75%, and 1% is presented in Fig. 2b. The use of Eq. (5) for the development of elastic modulus model prediction is explained in Section 4. 3. Test procedure followed in the study and test result The uniaxial compressive tests were conducted on 34 carbonate rock samples listed in Table 1 at the Rock Mechanics Laboratory of the Ben-Gurion University. The NX (D ¼ 54 mm) -sized cylindrical rock samples having the ratio L/D ¼ 2 were used in the tests considering the

Table 1 Summary of test results Rock

Adulam chalk

Sample

ea max ea sc (MPa) (%) ea

elast1/ max

ea ea

max

Eobs (MPa)

elast2/

rc1 rc3 rc4 rc6 rc7 rc8 rc9 st1a st1b st2a st2b

53.2 51 31.9 63.3 32.1 60.3 63.1 50.9 53.7 52.3 37.4

0.32 0.41 0.32 0.41 0.41 0.37 0.31 0.37 0.39 0.4 0.34

0.055 0.149 0.119 0.244 0.176 0.17 0.158 0.251 0.175 0.228 0.206

0.07 0.507 0.513 0.498 0.505 0.522 0.594 0.505 0.562 0.66 0.609

17 400 16 000 11 700 19 250 9500 17 300 20 500 16 200 15 400 14 300 10 700

Aminadav dolomite ad5 ad15 ad83

97.8 67.2 61.6

0.2 0.25 0.34

0.21 0.071 0.093

0.57 0.408 0.253

56 000 29 000 18 000

Beit-Meir dolomite

bm2 bm3 bina5 bina6 bina7 th5-15

71.5 45.6 80 89 64 84

0.25 0.22 0.22 0.48 0.29 0.25

0.088 0.118 0.235 0.083 0.097 0.228

0.384 0.545 0.59 0.421 0.424 0.544

38 100 21 400 38 700 24 800 25 000 37 700

Bina limestone

th3-24 th5-13 b1 b2 b3 b4 b5 b7 bz5-16 bz2-61a yar1

15.4 31.3 66.5 25 35 14 98 54.4 78 85.8 38.7

0.18 0.14 0.19 0.16 0.19 0.15 0.29 0.78 0.4 0.5 0.63

0.161 0.229 0.068 0.09 0.168 0.123 0.223 0.113 0.15 0.114 0.203

0.589 0.519 0.414 0.364 0.55 0.673 0.638 0.264 0.553 0.3 0.527

10 000 24 000 43 100 20 900 21 000 11 500 35 200 10 900 24 300 22 300 6500

yar2 yar3 yar4

38.7 41 71

0.74 0.69 1.07

0.077 0.243 0.108

0.38 0.591 0.37

6200 6200 8400

Sorek dolomite

Yarka limestone

sc: observed uniaxial compressive strength; ea max: observed maximum axial strain; ea elast1/ea max and ea elast2/ea max: observed normalized axial strains at the onset and the end of the linear portion of stress–strain curve, respectively; Eobs: observed elastic modulus (slope of the linear portion of stress–strain curve).

517

method suggested by ISRM [18]. The samples were ground to the planeness of 0.01 mm and cylinder perpendicularity within 0.05 radians. All samples were oven dried at a temperature of 110 1C for 24 h. The load frame used in this study (TerraTek system, model FX-S-33090) operates under hydraulic closed-loop servo-control with the maximum axial force of 1.4 MN and load frame stiffness of 5  109 N/m. A short data acquisition interval (timeo3 s) was used to provide accurate determination of axial stress and axial strain coordinates over the whole pre-failure strain range for the studied rock samples. Load frame and sample with radial and axial cantilever sets, physical properties of the studied carbonate rocks and their microstructural characteristics are described in details elsewhere [1,19–23]. The stress–strain curves for each of the carbonate rock samples given in Table 1 were recorded. A typical axial stress–strain curve is presented in Fig. 2a. The linear portion of the axial stress–strain curve represents the average elastic stage of rock deformation under uniaxial compression. The average elastic modulus (E) was determined as the slope of the linear portion of stress–strain curve as shown in Fig. 2a. Here, E ¼ Dsa/Dea, where Dsa and Dea are changes in axial stress and strain, respectively; ea ¼ ea elast1 and ea ¼ ea elast2 are the onset and the end, respectively, of the elastic stage of stress–strain curve. The observed values of the elastic modulus, uniaxial compressive strength and maximum axial strain varies between 6200 and 56 000 MPa, 14 and 97.8 MPa, and 0.14% and 1.07%, respectively, for the studied rock samples are presented in Table 1. The axial strains were normalized by dividing maximum axial strains (Table 1, Fig. 3a): 0.055o(ea elast1/ea max)o0.251 and 0.253o (ea elast2/ea max)o0.7. For example, Fig. 2c presents stress–strain curves for: two samples (rc7 and rc9) exhibiting, respectively, minimum (9500 MPa) and maximum (20 500 MPa) values of E for Adulam chalk; two samples (ad83 and ad5)—minimum (18 000 MPa) and maximum (56 000 MPa) values of E for Aminadav dolomite; two samples (b7 and b1)—minimum (10 900 MPa) and maximum (43 100 MPa) values of E for Bina limestone. Linear portions defined on the observed stress–strain curves for each of the carbonate rock samples (Table 1) are marked by thick vertical lines in normalized axial strain (ea/ea max) co-ordinates in Fig. 3b. The latter also presents three points: ea(0.125) ¼ 0.125ea max, ea(0.25) ¼ 0.25ea max and ea(0.5) ¼ 0.5ea max, which are represented by horizontal dashed lines. It is noteworthy that all the observed linear portions in Fig. 3b pass through at least two of the three points mentioned above. Indeed, 23 linear portions (samples rc1, rc3, rc4, rc6-rc9, st1a, st1b, st2a, st2b, ad5, bm3, bina5, th5-15, th3-24, th5-13, b3-b5, bz5-16, yar1 and yar3) pass through the two points of ea(0.25) and ea(0.5); other 11 linear portions (samples ad15, bm2, bina6, bina7, b1, b2, b4, b7, bz2-61a, yar2 and yar4)—through two

ARTICLE IN PRESS V. Palchik / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524

518 0.8 0.7

εa elast2 /ε a max

0.6

εa /ε amax

0.5 0.4 0.3

Elastic region

0.2 0.1

εa elast1/εa max yar 4

yar 3

yar 2

yar 1

bz2-61a

b7

bz5-16

b5

b4

b3

b2

b1

th5-13

th3-24

bina7

th5-15

bina6

bm3

bina5

bm2

ad83

ad5

ad15

st2b

st2a

st1b

rc9

st1a

rc8

rc7

rc6

rc4

rc3

rc1

0

Sample Name

(a) 0.8

Adulam chalk

Aminadav dolomite

0.7

Bina limestone

Yarka limestone

Beit-Meir dolomite

Sorek dolomite

0.6 0.5

εa /εamax

0.5 0.4 0.3 0.25 0.2 0.125 0.1

(b)

yar 4

yar 3

yar 2

yar 1

bz2-61a

bz5-16

b7

b5

b4

b3

b2

b1

th5-13

th3-24

th5-15

bina7

bina6

bina5

bm3

bm2

ad83

ad15

ad5

st2b

st2a

st1b

st1a

rc9

rc8

rc7

rc6

rc4

rc3

rc1

0

Sample Name

Fig. 3. Observed locations of linear portions (elastic regions) of stress–strain curves in normalized axial strain (ea/ea ea max and ea elast2/ea max values presented; (b) with three points (ea/ea max ¼ 0.125, 0.25 and 0.5) presented.

points of ea(0.125) and ea(0.25). Four (samples rc1, rc4, bm3 and b4) of 23 linear portions mentioned above pass not only through two points of ea(0.25) and ea(0.5), but also through the third point of ea(0.125). Thus, the above-mentioned three points are evaluated as the representative coordinates of the linear portion (elastic stage) of stress–strain curve for carbonate rocks and, therefore, these points may be used to predict the slope (elastic modulus) of the linear portion in the following section.

max)

coordinates: (a) with ea

elast1/

4. Development of model for prediction of average elastic modulus Slopes of linear portions passing through two points ea (0.125) and ea (0.25); two points ea (0.25) and ea (0.5); and two points ea (0.125) and ea (0.5) can be defined, respectively, as follows: E1 ¼

Dsa sað0:25Þ  sað0:125Þ ¼  100, Da að0:25Þ  að0:125Þ

(7)

ARTICLE IN PRESS V. Palchik / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524

E2 ¼

Dsa sað0:5Þ  sað0:25Þ ¼  100, Da að0:5Þ  að0:25Þ

(8)

að0:5Þ ¼ 0:5a max , respectively, in Eq. (5) [1] sað0:125Þ ¼

Dsa sað0:5Þ  sað0:125Þ ¼  100. E3 ¼ Da að0:5Þ  að0:125Þ

519

sc ð1  e0:25a max Þ , 1  e2a max

(10)

(9) sað0:25Þ ¼

sað0:5Þ ¼

sc , ð1 þ e0:5a max Þð1 þ ea max Þ

sc . 1 þ ea max

(11)

(12)

Then Eqs. (10)–(12) can be rewritten as pffiffiffiffi sc ð1  4 AÞ sað0:125Þ ¼ , 1  A2

(13)

σc = σamax

where E1, E2 and E3 are elastic moduli, Dsa and Dea are changes in axial stress and strain, respectively, sa(0.125), sa(0.25) and sa(0.5) are axial stresses at the axial strains ea(0.125), ea(0.25) and ea(0.5), respectively, as shown in Fig. 4. In order to determine the elastic moduli according to Eqs. (7)–(9), it is necessary to know the values of axial stresses sa(0.125), sa(0.25) and sa(0.5). The stresses sa(0.125), sa(0.25) and sa(0.5) can be obtained by inserting the values of að0:125Þ ¼ 0:125a max , að0:25Þ ¼ 0:25a max and

Axial stress (σa)

σa(0.5)

stress-strain curve

Linear portions passing through two points:

εa (0.125) and εa (0.25) (slope E1) εa(0.25) and εa(o.5) (slope E2)

σa(0.125)

σa(0.25)

εa(0.125) and εa (0.5) (slope E3)

0

0.125εa max 0.25εa max

0.5εa max

εa max

Axial strain (εa )

Fig. 4. Elastic moduli E1, E2 and E3 defined as slopes of linear portions passing through two points ea(0.125) and ea(0.25); two points ea(0.25) and ea(0.5); and two points ea(0.125) and ea(0.5), respectively.

ARTICLE IN PRESS V. Palchik / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524

520

sað0:25Þ ¼ sað0:5Þ ¼

sc pffiffiffiffi , ð1 þ AÞð1 þ AÞ

each of the studied rock samples (Table 1) as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE 1j  E j mean Þ2 þ ðE 2j  E j mean Þ2 þ ðE 3j  E j mean Þ2 , Dj ¼ m1 (20)

(14)

sc , 1þA

(15)

where A ¼ ea max .

(16)

where Dj is standard deviation of the mean elastic modulus in the jth sample; j ¼ 1,2,y,k is the number of studied samples (k ¼ 34); E1j is the elastic modulus E1 calculated by Eq. (17) for the jth sample; E2j is the elastic modulus E2 calculated by Eq. (18) for the jth sample; E3j is the elastic modulus E3 calculated by Eq. (19) for the jth sample, Ej mean is the arithmetic mean of three moduli E1j, E2j and E3j in the jth sample; m ¼ 3. Therefore, it is assumed that the average elastic modulus can be determined as the mean value of three elastic moduli E1, E2 and E3

Here, the maximum axial strain ea max is measured in percentages. Inserting Eqs. (13)–(15) into Eqs. (7)–(9), the following expressions are obtained for E1, E2 and E3 in terms of sc and ea max: pffiffiffiffi! 8sc 1 1 4 A pffiffiffiffi  E1 ¼  100, (17) 1A a max ð1 þ AÞ 1 þ A   4sc 1 pffiffiffiffi  100, 1 E2 ¼ a max ð1 þ AÞ 1þ A pffiffiffiffi! 8sc 1 4 A  100. E3 ¼ 1 1A 3a max ð1 þ AÞ

(18)

1 E ¼ ðE 1 þ E 2 þ E 3 Þ. 3 (19)

(21)

Taking into account Eqs. (17)–(19), (21), the average elastic modulus can be calculated as     4 8 5 E ¼ sc B C 1  D þ 100, (22) 3 3 3

Stress–strain curves obtained from weak to strong carbonate rocks having sco100 MPa are smooth and slightly concave upward, as presented in Figs. 2 and 4. The difference between the elastic moduli (slopes) E1, E2 and E3 defined on such smooth stress–strain curves is not significant (see Fig. 4) and, therefore, standard deviations (Dj) of the mean elastic modulus is not large: the minimum value of Dj ¼ 315.6 MPa at Ej mean ¼ 9180 MPa (sample th3-24) and the maximum value of Dj ¼ 2084 MPa at Ej mean ¼ 37 680 MPa (sample b5). Standard deviations (Dj) of the mean elastic modulus have been calculated for

where B, C and D are given by B¼

1 , a max ð1 þ AÞ

(23)

C¼

1 pffiffiffiffi , 1þ A

(24)

4

Parameters A, B, C and D

3.5 3 2.5 2 1.5 1 0.5 0 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Maximum axial strain (εa max), % A

B

C

D

Fig. 5. Dependences of parameters A, B, C and D on the maximum axial strain (ea

max).

1

ARTICLE IN PRESS V. Palchik / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524

D¼

1 pffiffiffiffi . 1þ 4 A

p ffiffiffiffiffiffiffiffiffiffiffi 4 ea max pffiffiffiffiffiffiffiffiffiffiffi . D¼ 1 þ 4 ea max

(25)

Inserting Eq. (16) in Eqs. (23)–(25), the following Eqs. (26)–(28) are obtained for the calculation of B, C and D in terms of the maximum axial strain: B¼

ea max , a max ð1 þ ea max Þ

(27) 900

Range of M R obtained by Deere [4] for limestones and dolomites

MR = 250

800 700

Predicted MR

(28)

Dependences of A (Eq. (16)), B (Eq. (26)), C (Eq. (27)) and D (Eq. (28)) on the maximum axial strain are presented graphically in Fig. 5. The parameters A and B decrease from 0.86 to 0.37 and from 3.6 to 0.9, respectively, with maximum axial strain (ea max) increasing from 0.15% to 1%. On the other hand, C and D are not very sensitive to the change in ea max. Standard deviations of C ¼ 0.5 and D ¼ 0.5 are 0.08 and 0.003, respectively, and are very small. Therefore, it can be assumed that CEDE0.5 when

(26)

pffiffiffiffiffiffiffiffiffiffiffi ea max pffiffiffiffiffiffiffiffiffiffiffi , C¼ 1 þ ea max

521

MR = 700

MR = 420

600 500 400

R2 = 0.9584

300 200

Range of M R obtained in this study for chalks, limestones and dolomites

100 0 0

100

200

300

400

500

600

700

800

900

Observed MR = E/σc Adulam chalk

Aminadav dolomite

Beit-Meir dolomite

Bina limestone

Sorek dolomite

Yarka limestone

Fig. 6. Comparison between the observed and predicted values of E/sc.

0.03

εa = 0.05 %

0.025

yar3 (F = 0.019, ε a max = 0.69 %)

Value of F

0.02

rc7 (F = 0.017, εa max = 0.41 %)

0.015

st2a (F = 0.011, ε a max= 0.4 % )

0.01

bm3 (F = 0.008, ε a max = 0.22 %)

0.005

b1 (F = 0.005, ε a max = 0.19 %)

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Axial strain (εa), %

Fig. 7. Values of F obtained according to Eq. (33) in all points (where sa and ea at sa were measured) of five stress–strain curves (samples yar3, rc7, st2a, bm3 and b1). F is the approximately the same in each point of stress–strain curve for concrete rock sample when 0.05%oeapea max.

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522

the maximum axial strain (ea max) ranges from 0.15% to 1%. Taking into account the fact that CEDE0.5, Eq. (22) for the calculation of elastic modulus (E) can be rewritten as E ¼ 2sc B  100.

(29) a

The value of A ¼ e max (Eq. (16)) can be obtained from the basic stress–strain model, Eq. (5), when one measurement in any point is available, i.e., rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sc A ¼ ea max ¼ 1  ð1  e2a i Þ. (30) sa i Inserting Eq. (30) into Eq. (26), gives the following mathematical expression for the calculation of the parameter B: B¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i . 2 ln 1  ðsc =sa i Þð1  e a i Þ 1 þ 1  ðsc =sa i Þð1  e2a i Þ

(31) Inserting Eq. (31) into Eq. (29), gives the final model for the prediction of elastic modulus (E) in terms of uniaxial compressive strength (sc), and sa i and ea i measured at one point E¼

2sc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  100,  lnð 1  sc F Þð1 þ 1  sc F Þ

bm3 and b1, respectively) only at 0.05%oeapea max, whereas at eao0.05% the difference in values of F is significant (i.e., Eq. (6) is not valid at very small strains). The same results were obtained for others studied rock samples. Thus, to predict the parameter F using Eq. (33), any datum point should be selected over the strain range 0.05%oea p ea max. One datum point (chosen arbitrarily over the strain range 0.05%oea p ea max), the value of F at this point, and the predicted value of E for the studied rock samples are presented in Table 2. The latter demonstrates that the values of ea i, sa i and F range from 0.052% to 0.731%, from 9.08 to 87.4 MPa, and from 0.0032 to 0.02, respectively. The predicted elastic modulus (Table 2) is compared with the observed elastic modulus (Table 1) of

Table 2 Values of eai and sai measured at the same point, F value and predicted elastic modulus Rock

Sample

ea i (%)

sa i (MPa)

F

Epred (MPa)

Adulam chalk

rc1 rc3 rc4 rc6

0.184 0.066 0.188 0.376

34.49 11.73 22.31 60.78

0.0089 0.011 0.014 0.0087

19 155 15 775 12 335 18 961

rc7 rc8 rc9 st1a st1b st2a st2b

0.083 0.222 0.064 0.217 0.121 0.12 0.126

9.08 40.1 15.65 34.83 21.8 19.74 16.9

0.0168 0.0089 0.0077 0.01 0.0099 0.011 0.013

9878 18 580 22 176 16 627 16 900 15 161 12 888

Aminadav dolomite

ad5 ad15 ad83

0.162 0.058 0.169

87.43 16.58 31.57

0.0032 0.0066 0.0091

57 677 26 239 18 025

Beit-Meir dolomite

bm2 bm3 bina5 bina6 bina7 th5-15

0.083 0.129 0.052 0.292 0.109 0.083

28.16 27.07 20.85 69.04 27.91 30.65

0.0054 0.0084 0.0047 0.0064 0.007 0.005

32 829 21 173 37 754 25 471 24 642 35 043

Bina limestone

th3-24 th5-13 b1 b2 b3 b4 b5 b7

0.117 0.059 0.567 0.083 0.105 0.083 0.057 0.731

10.94 13.51 21.77 16.94 22.25 9.54 20.35 53.85

0.0191 0.0083 0.0049 0.009 0.0085 0.016 0.0053 0.0143

9626 22 475 36 837 20 912 21 533 11 770 31 376 9876

Sorek dolomite

bz5-16 bz2-61a yar1

0.149 0.235 0.351

34.36 48.97 26.21

0.0075 0.0077 0.019

21 511 20 226 7524

Yarka limestone

yar2 yar3 yar4

0.526 0.136 0.663

32.2 12.83 57.28

0.02 0.0185 0.0128

6931 7729 9062

(32)

where F¼

1  e2a i . sa i

(33)

5. Discussion 5.1. Examination of the proposed prediction model Combining Eqs. (32) and (33) (at sa i ¼ sc and ea i ¼ ea max) results in an expression for predicting the ratio between E and sc MR ¼

1  100. 0:5a max ð1 þ ea max Þ

(34)

The observed value of MR ¼ E/sc obtained in this study for dolomites, limestones and chalks varies between 118 and 836. Deere [4] has established that the value of MR for limestones and dolomites is between 250 and 700, with the average value of MR ¼ 420 (Fig. 6). The comparison between the observed values of MR and predicted values of MR from Eq. (34) for all rock samples tested in this study is presented in Fig. 6. It can be shown that the observed value of MR is well correlated (R2 ¼ 0.9584) with the predicted MR obtained from the prediction model, Eq. (34). Eq. (6) suggests that the value of F, Eq. (33), is the same at any point of the axial stress–strain curve of a rock. Fig. 7 demonstrates that the F value in each of the points on the stress–strain curve is approximately the same (F ¼ 0.019, 0.017, 0.011, 0.008 and 0.005 for samples yar3, rc7, st2a,

ea i and sa i: axial strain and axial stress measured in point that was chosen arbitrarily over the strain range 0.05%oeapea max; F: parameter obtained from Eq. (33); Epred: elastic modulus predicted according to Eq. (32).

ARTICLE IN PRESS V. Palchik / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524

523

70000

60000

Predicted E, MPa

50000

40000

R 2 = 0.9757

30000

20000

10000

0 0

10000

20000

30000

40000

50000

60000

Observed E, MPa Fig. 8. Comparison between the observed and predicted (Eq. (32)) elastic modulus.

failure

Axial stress σa

0.05 % < εai < = εa max

εa = 0.05 % F (Eq. 33) εai

σai and εai

E (Eq. 32)

σc

σai

0

σc

Axial strain εa

Fig. 9. Parameters (sc, sa i and ea i at sa i) used for the prediction of elastic modulus according to the proposed model. Axial stress sa i and strain ea i are measured in any point over the strain range 0.05%oeapea max.

the studied rock samples is Fig. 8, where a good R2 ¼ 0:9757 cross-correlation is obtained.

the use of elastic modulus prediction model given by Eq. (32) in engineering practice:

5.2. Practical use of the proposed prediction model

1. Both the uniaxial compressive strength (sc) and one datum point (sa i and ea i at sa i) over the strain range 0.05%oeapea max are measured using simple load frames used in industry without recording the whole stress–strain curve. To perform one point measurement on the stress–strain curve, sophistical test equipment (where strain output is recorded rapidly with a short acquisition time interval) is not required.

As it readily follows from a simple algorithm presented in Fig. 9, the elastic modulus can be predicted from Eq. (32) when the uniaxial compressive strength (sc) and parameter F are known. To calculate F using Eq. (33), only one point measurement (sa i and ea i) over the strain range 0.05%oeapea max is needed. There are two possibilities for

ARTICLE IN PRESS 524

V. Palchik / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 514–524

2. The sc value is determined by a standard Schmidt hammer (or point load tests) performed on rock blocks. Schmidt hammer test is based on the rebound of a steel hammer from the rock surface, which is proportional to the compressive strength of the rock surface [24–26]. The values of sa i and ea i at sa i are measured using an indentation test [27,28] by forcing an indenter into the tested rock under controllable indenter displacement and applied force. However, some additional studies should be performed on the correlation between force– displacement curve obtained from indentation test and stress–strain curve obtained by test procedure suggested by ISRM [18]. Thus, the proposed model allows us to avoid the definition of the linear portion (see Fig. 2a) on the stress– strain curve, which is needed to determine the elastic modulus. The definition of the linear portion on a smooth curve for carbonate rocks having sco100 MPa is difficult, since it requires a large number of data points on the stress–strain curve (for example, see Figs. 2a and c). 6. Conclusions Elastic modulus prediction model was formulated using axial stress coordinates expressed in terms of uniaxial compressive strength (sc) and exponential function of axial strain according to the stress–strain model [1] based on Haldane’s distribution function. As a result, it is shown that the average elastic modulus (E) is partly related to the uniaxial compressive strength (sc) and to the ratio between the axial stress and exponential function of the axial strain measured at this axial stress. The prediction model proposed in this study does not involve any empirical coefficients, and in order to obtain accurate estimates for the elastic modulus of different carbonate rocks from weak to strong (sco100 MPa) (dolomites, chalks, limestones), it is necessary to know only the uniaxial compressive strength (sc), and any one stress–strain datum point within the strain range 0.05oeapea max. Acknowledgments I thank H. Sonmez for his helpful comments and remarks. References [1] Palchik V. Stress–strain model for carbonate rocks based on Haldane’s distribution function. Rock Mech Rock Eng 2006;39(3): 215–32. [2] US Bureau of reclamation. Physical properties of some typical foundation rocks. Concrete laboratory report, SP-39, 1953. [3] D’Andrea DV, Fischer RL, Fogelson DE. Prediction of compressive strength from other rock properties. Rep Invest US Bur Mines 6702, 1965.

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