An Analytical Study of Failure of Transversely Isotropic Rock Discs Subjected to Various Diametrical Loading Configurations

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ScienceDirect Procedia Engineering 191 (2017) 1194 – 1202

Symposium of the International Society for Rock Mechanics

An Analytical Study of Failure of Transversely Isotropic Rock Discs Subjected to Various Diametrical Loading Configurations Z. Aliabadiana, G.F. Zhaob, A.R. Russella,* a

b

School of Civil and Environmental Engineering, The University of New South Wales, Australia State Key Laboratory of Hydraulic Engineering Simulation and Safety, School of Civil Engineering, Tianjin University, Tianjin 300072, China

Abstract The indirect (Brazilian) tensile strength test is commonly used in rock engineering and rock mechanics. Most of the research on how to interpret Brazilian test is only valid for isotropic rock. The load may be applied to disc shaped samples through flat platens or curved jaws and the different load configurations give rise to subtle but important differences to stress distributions throughout samples during testing and at the onset of failure. To reliably obtain the tensile strength accurate stress distributions, especially at the location of the initial crack, must be determined. However, most real rocks exhibit a certain type of anisotropy known as transverse isotropy due to the presence of preferred directions of grains, beddings, microcracks or pores. This study focuses on the failure modes when transverse isotropy is present. Previous studies for transverse isotropy indicate that the exact location of the initial crack is at the disc centre. These studies, however, ignore the effect of the load contact configuration on the stress distribution. This paper overcomes this limitation and applies Amadei’s analytical solution to assess the effects of the load contact area on the stress distribution. The stress distribution is coupled with a transverse isotropy failure criterion in a mechanically consistent way to find the exact location of initial crack. It is shown that the location of the initial crack depends on the transverse isotropy orientation and load configuration. © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license Published by Elsevier Ltd. This (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017. Peer-review under responsibility of the organizing committee of EUROCK 2017 Keywords: Brazilian disc; Anisotropic rocks; Analytical solution; Crack initiation

* Corresponding author. Tel.: +61-293-855-035; fax: +61-293-856-139. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017

doi:10.1016/j.proeng.2017.05.295

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Nomenclature φ Bedding orientation 2α Loading angle p Pressure applied across disc ' E, E Young’s moduli, along the transversely isotropic plane and normal to it ' -,Poisson’s ratios, lateral strain in the transversely isotropic plane under stress acting parallel and normal to it ' Shear modulus in plane normal to transversely isotropic plane G σx, σy, τxy Normal and shear stress components σ 1, σ 3 Major and minor principal stresses kφ Anisotropy parameter mi Material constant σcφ Uniaxial compression strength for a bedding orientationɔ. 1. Introduction The Brazilian test is a diametrical compression test that is used as an indirect tensile strength measurement. At first, the test was proposed by Carneiro [1] and Akazawa [2] to obtain tensile strength of concrete. In 1978, the International Society for Rock Mechanics (ISRM) officially presented Brazilian test as a standard method to indirectly determine the tensile strength of rock like materials (ISRM 2007). Nowadays the Brazilian test is widely used to determine tensile strength of rock like materials because of its simple specimen preparation, experimental performance and data reduction [3–4]. In the test, a thin disc is diametrically compressed. The corresponding stress field throughout the specimen is highly non-uniform. Many studies of Brazilian tests have been conducted on isotropic rock like material [5–8] while under different tectonic conditions, rocks naturally show anisotropy due to the presence of preferred direction of grains or beddings and oriented microcracks or pores [9]. The rock anisotropy can affect the stability of mining and civil projects such as slope and borehole stability, underground excavations, gas and oil exploration, etc. [10–13]. Among different types of rock anisotropy, transversely isotropic with a predominant orientation of planar anisotropy is very common. Generally, in nature, there are different origins of transversely isotropic rocks, which are due to layering or bedding, foliation, fissuring, jointing and fracturing, especially in sedimentary rocks, sequential layers are observed. To date, many papers have studied the transverse isotropy of rocks under Brazilian test loading conditions and evaluated the anisotropy effect on elastic properties, stress distributions, strength and fracture pattern through theoretical, experimental and numerical methods [14–19]. Despite these studies a few questions have remained unsolved. For example, how does the Brazilian load configuration affect the failure mechanism? Based on standard Brazilian test load configurations, load is applied through flat platens or curved jaws and the contact length on the sample is different in each. The contact length significantly influences stress intensity in the load contact region and also the stress distributions. It is noteworthy that the standard Brazilian tensile strength formula has been suggested for an isotropic material based on plane elastic theory in which the contact length effect on stress distribution has been ignored [20]. Even though the investigations of test samples subjected different load configurations for isotropic discs are quoted in some studies [21–24], this issue has not been addressed for transversely isotropic discs. This study uses an analytical method to bridge this gap for transversely isotropic discs. Hondros [25] solved analytically the stress distribution for isotropic discs under diametrical load. In other analytical studies a few researchers tried finding the effect of rock anisotropy on stress measurement [26–28]. Among them, Lekhnitskki [27] clearly presented the stress-strain relationship for an anisotropic disc under diametrical load through the complex function of stress. Pinto [28] evaluated Hondros’ method on schist discs and considered that the stress concentration of the disc centre is the same as that for an isotropic disc. This assumption was not completely acceptable for all bedding orientations of transversely isotropic discs. Later Pinto’s work was revised by Amadei [29]. Amadei et al. [30], Amadei [31], and Chen et al [32] showed application of Lekhnitskki’s relationship for finding indirect tensile strength of transversely isotropic disc. In this study, the analytical method of Lekhnitskki [27], Amadei [31] and Chen et al. [32] is used to figure out the stress distribution of transversely

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isotropic disc under different load configurations. The stress distribution was combined with a transversely isotropic failure criterion in a consistent way to find the exact location of the initial point of failure, the point where a crack is most likely to occur first. Finally, the results of different load configurations for various bedding orientations are compared with each other. 2. Analytical stage 2.1. Formulation of transversely isotropic disc under diametrical load First, we consider a uniform transversely isotropic disc with diameter D = 2R and thickness t under a uniform pressure acting over a strip having a width which approaches zero for flat platens and to 2α for curved jaws. O (x, y, z) is a Cartesian coordinate system according to Fig. 1 (a). The angle φ is defined as the anticlockwise angle between the x-axis and the transversely isotropic plane. Another coordinate system O (xʹ, yʹ, zʹ) is introduced, accounting for the direction of bedding and the direction normal to bedding orientation. A boundary force per unit area (Qn) is defined force along the perimeter of the disc that is a function of θ and it is assumed to be in equilibrium. Xn and Yn are its components in x and y directions. For this problem, it is assumed that the amount of the surface force is negligible across the disc thickness, and disc deformation is small. Also, there is an elastic symmetry plane parallel to the middle plane of the disc, so the condition of plane stress can be considered for the transversely isotropic condition [27, 32].

Fig. 1.The problem geometry (a) curved jaws and (b) flat platens.

The elastic stress-strain relations in O (x, y, z) system are given in general form in Eq. (1):

ªHx º «H » « y» «¬J xy ¼»

ª a11 «a « 12 «¬ a16

a12 a22 a26

a13 º ªV x º

« » »« » a66 »¼ ¬«W xy ¼»

a26 » V y

(1)

The compliance components a11, a12, …, a66 are not only dependent on elastic properties in the O (xʹ, yʹ, zʹ) coordinate but also on the angle φ. The transversely isotropic materials are dependent on five elastic properties ' including E and E' (the Young’s moduli are along the transversely isotropic plane and normal to it (- , - ) the Poisson’s ratios are defined as lateral strain in the transversely isotropic plane under stress acting parallel and ' normal to it) and G (shear modulus in plane normal to transversely isotropic plane). The compliance components ' ' ' a11, a12, …, a66 are dependent on E , E , - and G are not related to - [32] as follows:

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sin M

cos M

4

a11

E

'



sin 2M 2

a12

sin 2M

4



4

2



E 1 E

4 1



'



E

1 G

'





§1 ¨ ©G X

'

E

'

'

E

sin 2M ¨

a16

cos M E

sin 2M

4

'

2



4

E

a26

§ cos sin 2M ¨ © E

a66

§1 ©E

2

sin 2M ¨ 2

'

4

2

sin M



M

'

sin M 2



E 

1



E

2X G

'

'

'

4

'

4

a22

· ¸, ¹

'

 cos M  sin M ,

§ sin M cos M · § 1  ¸¨ E ¹ © 2G © E 2

2X



§1 ¨ ©G

'

'





'

'

E

'

2X E

· § 1 ¸¨ ¹ © 2G

X



· ¸ cos 2M , ¹

· ¸, ¹

'

'

X

'

E

'

· ¸ cos 2M , ¹

· cos 2M . ¸ ¹ G 2

'

(2)

Eq. (1) and the components of the Cauchy stress tensor Eq. (3) (where F is Airy’s stress function) are substituted into in the strain compatibility equation Eq. (4), and the differential equation Eq. (5) is achieved.

w F 2

Vx

wy

w Hx

2

,V y



2

w F wx

4

2

w J xy

wx

wxwy

w F 2

, W xy



wxwy

(3)

2

2

4

a22

wx

w Hy 2

2

wy

2

w F

w F 4

 2a26

wx wy 3

(4)

  2a12  a66

w F

w F

4

wx wy 2

4

2

 2a16

wxwy

w F 4

3

 a11

wy

4

0 (5)

According to Lekhnitskii [27], Eq. (5) has four roots that are complex (which have non-zero imaginary parts). Two of the roots are conjugate to the others. Eq. (5) can be rewritten as: a11 P  2a16 P   2a12  a66 P  2a26 P  a22 4

3

2

0

(6)

In this study, the two roots with positive imaginary parts are defined. Lekhnitskii [27] expressed the relations between the analytical functions Ø1 (z1) and Ø2 (z2) and of the complex variables z1 = x + μ1y and z2 = x + μ2y with the first derivatives of Airy’s stress function, F :

wF wx

2 Re >‡ 1  z1  ‡ 2  z2 @ ,

wF wy

2 Re > P1‡1  z1  P2‡ 2  z2 @ .

Therefore, the general formulations of stress components are [32]:

(7)

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Vx W xy

2 Re ª¬‡ 1'  z1  ‡ 2'  z2 º¼ ,

2 Re ª¬ P12‡ 1'  z1  P22‡ 2 '  z2 º¼ , V y 2 Re > P1‡1 '  z1  P2‡ 2 '  z2 @ .

(8)

Airy’s stress function should satisfy the boundary conditions for external force components Xn, Yn in Eq. (9):

wF wx

wF

s

 ³Yn ds  c1 ,

s

³X ds  c .

wy

0

n

2

(9)

0

in which s is the length along the disc contour and Xn and Yn are considered as Fourier series, so Eq. (9) will be: s



N

s



a0  ¦ am eimD  a m e  imD ,

 ³ Yn ds  c1

³ X n ds  c2

m 1

0

0

N





b0  ¦ bmeimD  b me imD . m 1

(10)

where a0 and b0 are arbitrary constants and am, bm and their conjugates a m ,b m are related to Fourier series of Xn and Yn [27, 32]. The general form of Ø1 (z1) and Ø2 (z2) was given by Lekhnitskii [27]:

‡ 1  z1

f

A0  A1 z1  ¦Am P1m  z1 ,

‡ 2  z2

m 2

f

B0  B1 z 2  ¦Bm P2 m  z 2 . m 2

(11)

m m 2 2 ª§ z · §z · º z1 · z1 · § § 2 2 1 1 «¨  ¨ ¸  1  P1 ¸  ¨  ¨ ¸  1  P1 ¸ » , P1m  z1 ¸ ¨R ¸ » ©R¹ 1  i P1 m «¬¨© R © R ¹ ¹ © ¹ ¼

1

m m 2 2 ª§ z º · § · z z z § 2 · 1 P2  2  § 2 · 1 P2 ». 2 «  P2 m  z 2 ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ 2 2 ¸ ¨R ¸ » ©R¹ 1  i P2 m «¬¨© R © R ¹ ¹ © ¹ ¼

1

(12)

Am, Bm and their conjugates Am , B m can be expressed in am, bm and and a m ,b m . By substituting Eq. 11, 12 into Eq. 10 and based on Eq. 10, σx, σy, τxy are obtained. Also, the loading angle (2α) is very small thus the pressure (p) applied across it is identical to W / αDt, where w is the force applied. Furthermore, the stress components for every point inside the disc are given as:

Vx

W

S Dt

qxx , V y

W

S Dt

q yy ,W xy

W

S Dt

qxy .

(15)

where, qxx, qyy and qxy are the stress concentration factors at the point of interest within the disc. 2.2. Failure criterion The tensile strength of a transversely isotropic disc is dependent on stress field within the disc at the onset of failure. A range of transversely isotropic criteria [33–41] were investigated and among them the modified Hoek-Brown criterion [39] was selected for use. The characteristics of the criterion are its simple equation, only

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a few parameters are needed, and its accuracy in failure point estimation. This criterion was combined with the stress field as given in the previous section to get the location of initial crack in different bedding orientations. The aforementioned failure criterion is described in detail as follows [39]:

V1  V 3

§ · V V cM ¨ kM mi 3  1¸ V cM © ¹

o .5

(16)

where σ1, σ3 are major and minor principal stresses, φ is the bedding orientation, kφ is the anisotropy parameter, mi is the material constant, σcφ is the uniaxial compression strength for a bedding orientation φ. We introduced a new relation between σcφ and σc90 as:

V cM V c 90

GM

(17)

Bedding orientation effects can then be captured through Gφ, which is dimensionless strength factor which can be recovered from results of unconfined compression strength (UCS) tests on samples containing beddings of different orientations. The Gφ used here fits the UCS results for a Gneiss (from [39]) and is: GM

0.654M 2  0.7175M  0.5188

(18)

with φ having units of radians. The maximum amount of strength for the Gneiss is that when the bedding orientation is 90 degrees to the major principal stress (σc90) and is treated as an intrinsic constant. A measure of how much σc90 has been used up at every point in the disc can be inferred from a mobilized strength (σmφ ).

V mM



0.5 ª GM kM miV 3  (GM kM miV 3 )  4  GM V 1  V 3

«¬

2

GM

2



0.5

º »¼

2

(19)

The material has not failed where σmφ ˂ σc90Gφ. The material is at the onset of failure when σmφ = σc90Gφ. A contour plot σmφ is insightful as the maximum value of σmφ indicates where failure will initiate. 3. Results A Matlab program was developed to investigate the failure of transversely isotropic disc subjected to compressed diametrical load accounting for different load configurations. In the following analysis, to determine the effect of the load contact configurations, the strip of angle 2α is set equal to 2 degrees (to mimic flat platen loading) and 15 degrees for curved jaws as shown in Fig 1. The ratio of the diameter to thickness of a disc is 2. To assess the effect ' of transverse isotropy on stress distribution, the assumed properties are E E c 3, E G 7.5, X ' 0.25 . Contour plots showing the distributions of σmφ for flat platens and curved jaws are illustrated in Figs 2 and 3, respectively. In these figures, the results are only reliable from 0 ≤ R ≤ 0.95, because for R > 0.95 numerical errors become significant. In Table 1, the location of crack initiation for different bedding orientations are listed. For the flat platen, which has the smaller load contact area, the greatest stress concentrations are at the surfaces of the disc at the edges of the load contact areas. Cracks are likely to occur there first and then propagate to the disc centers. For orientations 15°, 30°, 45°, 60°, 75° there was also a localized maxima in the mobilised stress inside the disc (around 0.9R) near the load contacts (evident in Fig. 2b). For curved jaws, with larger load contact areas, and for bedding orientations 0°, 15°, 30°, 75°, 90°, failures initiate away from the disc centers, occurring at about 0.6R to 0.2R. For 45° and 60° failures initiate at the centers.

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Fig. 2. Distributions of σmφ for flat platens (a) 0°, (b) 45° and (c) 90° bedding orientations.

Fig. 3. Distributions of σmφ for curved jaws (a) 0°, (b) 45 and (c) 90° bedding orientations. Table 1. Location of tensile crack initiation for different load configurations. Bedding orientation (Degree)

Location of tensile crack initiation flat platens

Curved jaws

0

Load contact

0.6R

15

0.9R

0.5R

30

0.9R

0.2R

45

0.9R

Centre

60

0.9R

Centre

75

0.9R

0.4R

90

Load contact

0.5R

4. Conclusion An analytical method was used to study where failure is likely to initiate in Brazilian tests on transversely isotropic disks. The stress distributions inside the disc with two types of load configurations were studied. The following conclusions are drawn from the analyses. The stress distributions are different in discs for different

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load configurations. For very small load areas (akin to what would prevail when flat loading platens are used), failure initiates near the load contacts for all bedding orientations. For a larger load contact area (i.e. if curved jaws are used), failure may initiate near the load contacts or at the disc centers. The location depends on the bedding orientation.

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