On the load capacity and fracture mechanism of hard rocks at indentation loading

International Journal of Rock Mechanics and Mining Sciences 100 (2017) 170–176

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On the load capacity and fracture mechanism of hard rocks at indentation loading

MARK

⁎

K. Weddfelta, M. Saadatia,b, , P.-L. Larssonb a b

Atlas Copco, Örebro, Sweden Department of Solid Mechanics, KTH Royal Institute of Technology, Stockholm, Sweden

A R T I C L E I N F O

A B S T R A C T

Keywords: Stamp test Indentation Semi-brittle materials Effective volume Granite Hard rocks

The load capacity of selected hard rocks subjected to circular flat punch indentation is investigated. The compacted zone underneath the indenter is assumed to be limited and only responsible for the load transition to the rest of the material. Therefore, the theory of elasticity is used to define the stress state in a semi-infinite medium loaded by a flat punch indenter. The final load capacity is related to the formation of a sub-surface median crack that initiates due to tensile hoop (circumferential) stresses. Therefore the final failure should occur at a force level in which the hoop stress is greater than the tensile strength of the rock. Since the tensile stress is distributed over a volume of material, tensile crack failure can occur at different locations with tensile hoop stress depending on where the most critical flaw is located. Therefore, the initiation of the median crack that should be responsible for the final load capacity is treated as a probabilistic phenomenon. This process is described by Weibull theory which will be used as a failure criterion. It is assumed here that the opening of median crack triggers a final violent rupture, therefore the assumption in Weibull theory, that the final failure occurs as soon as a macroscopic fracture initiates from a microcrack is fulfilled. The effective volume is calculated for a semiinfinite medium loaded by a flat punch indenter. The material properties of Bohus granite obtained from three point bending tests are used as reference values in describing the Weibull size effect. The experimental results for the stamp load capacity of three selected hard rocks are taken from the literature. They are considered similar rocks to the reference material in this paper, which is Bohus granite. The model describes the observed load capacity with a very good accuracy for all three rocks. It is likely that the presently proposed methodology is applicable for other types of semi-brittle materials and indenter shapes.

1. Introduction The fracture system and the load capacity of brittle and semi-brittle materials loaded by an indenter, has been widely investigated in the literature.1–10 To analyze the fracture system, the focus is obviously on the formation and propagation of different types of cracks during the indentation process together with the development of a compacted zone created immediately beneath the indenter. Depending on the brittleness of the material and based on its microstructure, and the amount of defects such as pores and microcracks, the boundary between the crushed zone and the fractured zone is changed. The crushed zone is believed to develop due to high compressive stresses very early during the indentation process and then transmits the force to the rest of the material.11 The formation of different types of cracks including radial, conical and median cracks in the fractured zone, however, is connected to tensile stresses generated due to the indentation problem and is observed during the loading stage.2,3,5,7 On the other hand, the

⁎

initiation of lateral cracks, also called side cracks, is suggested to be related to the expansion of the fractured rock material under the indenter as the load is increased.1 However, the main reason for the further formation of the lateral cracks during the unloading stage is suggested to be driven by residual stresses at the boundary of the compacted zone.3,7 Different cracking patterns were observed in sharp indentation of glass both at loading and unloading stages depending on the glass composition, which reflects itself in ductility or brittleness of the material.12 The load bearing capacity of rocks during a flat punch indentation test, also called stamp test, has been historically used in the mining industry to characterize the rock properties and its response at drilling and excavation processes.1,6 The stamp strength σST , which is the mean contact pressure at failure, of rocks has been found to be affected by the indenter size and the effect of this size dependency is more pronounced in case of brittle rocks as compared to ductile rocks.1 The stamp strength together with the crater volume formed in the rock during the

Corresponding author at: Department of Solid Mechanics, KTH Royal Institute of Technology, Stockholm, Sweden. E-mail address: [email protected] (M. Saadati).

http://dx.doi.org/10.1016/j.ijrmms.2017.10.001 Received 6 December 2016; Received in revised form 22 June 2017; Accepted 13 October 2017 1365-1609/ © 2017 Elsevier Ltd. All rights reserved.

International Journal of Rock Mechanics and Mining Sciences 100 (2017) 170–176

K. Weddfelt et al.

Fig. 1. (a) Indentation with a cylindrical flat-ended punch and formation of Hertzian crack (taken from 1). (b) The contour plot shows the solution of the first principle stress in a nonfractured linear-elastic medium.

Therefore the final failure should occur at a force level in which the hoop stress is greater than the tensile strength of the rock. For the tensile strength, the Weibull size effect is adopted and the strength of the rock is scaled using an effective volume. The effective volume is calculated based on the part of the material that is subjected to a positive hoop stress. Therefore, the initiation of the median crack, responsible for the final load capacity, is treated as a probabilistic phenomenon. The experimental results for the stamp load capacity of selected hard rocks are taken from the literature and a fairly close agreement is obtained with the model predictions. Finally, it should be mentioned that a possible application of the present results concerns percussive drilling. As a first approximation then, dynamic effects could be addressed by changing the explicit values on the tensile strength accounting for rate effects.

stamp test is used for design of efficient drilling tools and also for the prediction of the drilling rate.6 Substantial efforts were made to predict the load capacity of the rock, during a stamp test, based on elasticity theory together with different failure criteria.5 It was observed that cracks and microcracks are formed in the rock prior to the final failure. While the well-known Hertzian crack is formed due to the first principal stress during loading, the crack propagates in a stable manner and the crack front becomes concealed in the region with intense microcracking.5 Therefore, the load capacity and final failure was connected to the initiation of a sub-surface median crack due to the tensile hoop (circumferential) stress, which is the second principal stress except centrally below the indenter. However, the suggested model is not able to predict the load capacity of the rock in the indentation test with an acceptable accuracy.5 The force-penetration response of rocks under sharp indenters has also been widely investigated and the fragmentation mechanism is explained.13,14 The sharp conical indentation test has the advantage of self-similarity in the problem compared to the flat punch indentation of different sizes but this will also lead to a situation where plastic (irreversible) deformation is introduced immediately at contact. Weibull statistics and the weakest link theory has been widely used to define the failure probability of a structure and to explain the statistical scatter in the strength of identical specimens.15,16 Furthermore, the statistical theory of size effect based on the Weibull model has also been applied to describe the size effect on the tensile strength of brittle and semi-brittle material with a random distribution of flaws and defects. One of the assumptions in this theory is that total failure occurs as soon as a macroscopic fracture initiate from a microcrack. Another assumption is the absence of any characteristic length and therefore the material should not contain sizable inhomogeneities.17 Hence, the application of the Weibull theory and Weibull size effect on brittle and semi-brittle material should be made with extra caution and the fulfillment of the assumptions should be checked carefully. In this work, the focus is towards an investigation of the load capacity of selected hard rocks subjected to circular flat punch indentation. The compacted zone underneath the indenter is assumed to be limited and only responsible for the load transition to the rest of the material. Therefore, the theory of elasticity is used to define the stress state in a semi-infinite medium loaded by a flat punch indenter. A tensile failure criterion is used and it is assumed that the final load capacity is related to the formation of a sub-surface median crack at the symmetry line. The median crack initiates due to tensile hoop stresses.

2. Elastic stress field from a rigid flat punch For most types of rocks, the indenter, which is made from hardened steel or hard metal, can be considered as relatively rigid. The elastic modulus is at least a few times higher than the rock. Therefore, in this context the stress field from the indentation is studied using the solution for a rigid punch in an elastic medium, for which the stress state is comprehensively given e.g. in 18. Early during a stamp test with a flat punch, i.e. at comparably low force compared to that of the final rupture, high tensile radial stresses form close to the surface just outside the rim of the indenter. These stresses lead to shallow ring cracks around the indenter and gradually form the well-known Hertzian crack. Below the surface, but still outside its periphery, tensile stress (first principal stress) extends downwards in a circular shape, reaching inwards below the indenter, but at a lower value. As the indenter force increases, however, subsurface tensile cracks may be expected to form also in these places (Fig. 1). Eventually the region with high enough tensile stress to cause failure encompasses also the region centrally below the punch. Here both hoop and radial stress components is first principal (they are equal) and the direction of the crack can then be expected to extend downwards and side wards. This type of crack has been named a median crack. In an elastic treatment the depth at which this occurs depends only on the Poisson's ratio and the size of the contact zone, for typical values (ν ≈ 0.2 − 0.3) at about 2–3 times the radius of the punch. It is suggested here that when the median crack opens it triggers the final violent rupture that is associated with the indentation test, 171

International Journal of Rock Mechanics and Mining Sciences 100 (2017) 170–176

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the first principal in general. However, it is well recognized that in the region where the median crack is expected, the hoop stress component is very similar to the first principal stress, and at (ρ = 0) they are even identical. Since the tensile stress is distributed over a volume of material, tensile crack failure can occur at different locations with tensile hoop stress depending on where the most critical flaw is located. Therefore, the initiation of the median crack that should be responsible for the final load capacity is treated as a probabilistic phenomenon. This process is described by Weibull theory which will be used as a failure criterion. It is assumed here that the opening of median crack triggers a final violent rupture, therefore the assumption in Weibull theory, that the final failure occurs as soon as a macroscopic fracture initiates from a microcrack is fulfilled.

sometimes with large chips being released. This is taken as the definition of the so-called “load capacity” of the rock material for this particular type of test. It will be shown in this paper that the load capacity correlates well with a failure criterion based on the hoop stress field at relatively large depth below the indenter, i.e. conceptually the opening of the median crack. Assuming rotational symmetry with the non-dimensional cylindrical coordinates ρ = r / a and ζ = z / a (with the origin at the center of contact region in undeformed configuration), where a is the radius of the contact zone which in this case equals the indenter radius, the hoop stress inside the elastic medium yields 18:

ν⋅sin

σθ (ρ , ζ , ν , F , a)

=−

F πa2

( ) − 1 − 2ν ⎛1 − ϕ 2

2ρ2

R

⎜

⎝

ζ 1+ζ 2 ϕ R ⋅sin ⎛ ⎞ ⎞ + 2ρ2 R ⎝ 2 ⎠⎠ ⎟

ϕ ⋅sin ⎛θ − ⎞ where ρ , ζ ≠0 2⎠ ⎝

3. Experimental results

(1)

Experimental results from indentation tests with a flat punch are taken from three different publications. Wagner et al.1 published results for several different types of rocks using four different punch sizes. Cook et al.5 gave values for a Sierra granite using three different punch sizes. Likewise Wijk6 presented results for Bohus granite, marble and sandstone. Out of these experiments, the results (stamp strength σST , average load capacity Fexp and its standard deviation sexp ) for three hard and brittle rocks are selected (see Table 1). The similarity between the norite and Bohus granite types of rocks is manifested in the load capacities recorded at indenter sizes 2 mm (≈ 1.9 mm) and 4 mm(≈ 3.8 mm) where the results are very close, which is mainly due to the fact that they have similar Poisson's ratio. The size effect discussed by Wagner et al.1 can be clearly seen as the decrease in average contact pressure needed for chipping as the indenter size increases. Cook et al.5 used an innovative method to confine their test specimen with the aim of simulating an infinite half-space by matching the stiffness of the confinement steel ring to that of the rock. That confinement also introduced a radial pressure that was believed to be about 27.2 MPa in compression. The stress field was superimposed on the stress field produced by the indentation test. Cook et al.5 were not able to match neither a Mohr–Columb nor a tensile failure criterion to the observed data. Using the formulation of a rigid punch instead of a uniform contact pressure, the constant 64.3 (in Eq. (18) in 5) is changed to 90, which makes the model prediction of the stamp strength closer to the experimental ones at very low confining pressure and farther at higher confining pressure region (see Fig. 11 in 5). Furthermore, it is likely that the value in 5 of the tensile strength, T0 = 10.3MPa , was determined using a larger stressed volume than what is the case during the indentation test. In the framework of this paper, the tensile strength is treated as being size-dependent and typically of the order of 18–22 MPa for the effectively loaded volume produced by their punch sizes.

In Eq. (1) R, ϕ and θ are defined by:

(ρ2 + ζ 2−1)2 +4ζ 2

R=

2ζ ⎞⎟ where ϕ ∈ [0,π ] and ϕ 2 2 = π ϕ = atan ⎛⎜ 2 ρ +ζ =1 2 2 ⎝ ρ + ζ −1 ⎠

1 π θ = atan ⎛⎜ ⎞⎟ and θ ζ = 0 = 2 ⎝ζ ⎠ In such an elastic formulation the stress inside the material is linearly proportional to the applied indentation force F and the stress is made non-dimensional with respect to the average contact pressure pm :

pm =

F π⋅a2

(2)

The theoretical spatial distribution of the contact pressure pc is:

pc (ρ , F , a) pm

=−

σz (ρ , F , a) ζ = 0 pm

=

1 2 1−ρ2

where ρ < 1 (3)

Centrally below the indenter, at ρ = 0 , hoop and radial stress are as mentioned above equal and can be evaluated as the limit value of Eq. (1) as ρ → 0 :

σθ (ζ , ν , F , a) pm

ρ=0

=

σr (ζ , ν , F , a) pm

ρ=0

=−

ζ2 1 + 2ν + 2 4⋅(ζ +1) 2 ⋅(ζ 2+1)2

(4)

The hoop and radial stress are also the first and second principal stresses within this centerline, i.e. at ρ = 0 : σ1 = σ2 = σr = σθ . Hoop stress is compressive on the surface and compressive stress also extends down to a depth which can be seen to vary with the radial coordinate (see Fig. 2). Inside the perimeter of the punch it is roughly at ζ = 1.7 (for ν = 0.24 ) and as ρ > 1.5 the border between compressive and tensile stress goes deeper into the medium. Hoop stress has a maximum value which always occur centrally below the indenter, i.e. somewhere along the line ρ = 0 . The depth in the ζ -coordinate where this maximum occur is

ζmax = ζ

d ⎛σ (ζ , ν, F , a) ⎞ ⎜ ρ=0 ⎟=0 dζ θ ⎝ ⎠

=

4. Application of Weibull statistical approach and the effective volume of the stressed region

3 − 4ν (1 + ν ) 1 − 2ν

The probability of failure given a certain stress state is obtained by Weibull theory as

(5)

and the value of the (maximum) hoop stress at this point is

σθ (ν , F , a) ρ = 0, ζ = ζmax σθ, max (ν , F , a) (1 − 2ν )2 = = pm pm 32

⎛ − PF = 1 − e ∫V ⎝

? ? σ ? ? ⎞m ⋅ dV λ ⎠

(7)

where ° is the Macauley's bracket and the integral is taken over the tensile stress only

(6)

For ν = 0.24 , σθ, max ≈0.0085∙pm at the depth ζ ≈ 2.6. Below this point the hoop stress gradually decreases and approaches zero at greater depth. Because the load capacity is defined by the conceptual opening of a median crack at great depth centrally below the indenter, it can be expected to correlate better with the hoop stress field than with that of

0 σ =⎧ σ ⎨ ⎩

σ≤0 σ>0

(8)

The scale parameter λ and the modulus m of the Weibull distribution for granite was determined using three point bending tests on 172

International Journal of Rock Mechanics and Mining Sciences 100 (2017) 170–176

K. Weddfelt et al.

Fig. 2. (a) Hoop stress contour plot within the medium. (b) A comparison of the hoop stress at the central line below the punch for the values of Poisson's ratio being 0.20 and 0.24.

specimens of two different sizes; with an effectively loaded volume of λ= 23 mm3 and 189 mm3 respectively,20 yielding 3 9.2MPa mm and m = 21.8 Applied to the hoop stress component, the probability of failure or the probability for the initiation of the median crack is

Pθ (F , a, ν , m , λ ) = 1−e

R

(

∫∫

Pθ (F , a, ν , m , λ ) =

)

Vtot =

⎞ ⎟ ⎟ ∙ dz ∙2π ∙ r ∙ dr ⎟ ⎟ ⎠

Veff (ν , F , a, m) =

tot

∫∫ 0

(10)

(13)

0

r z ⎛ σθ a , a , ν , F , a ⎜ ⎜ σθ, max (ν , F , a) ⎜ ⎜ ⎝

(

)

⎞ ⎟ ⎟ ⋅dz⋅2π⋅r⋅dr ⎟ ⎟ ⎠

(14)

Using the non-dimensional variables ρ = r / a and ζ = z / a this becomes

Veff (ν , F , a, m)

⎛ σθ ⎟⎞ dV when σ ⎜ θ, max > 0 ⎝ σθ, max ⎠

as R, Z →∞

0

m

m

∫V

Z

∞ ∞

In Eq. (10), H is the stress heterogeneity factor that characterizes the effect of the load pattern on the cumulative failure probability and written in 22 as

1 Vtot

(12)

In the context of this study, it is assumed that the hoop stress is responsible for the load capacity and therefore the integral in Eq. (11) is taken for the positive hoop stress over the whole medium.

(9)

where the integral is taken for the positive hoop stress over the whole medium. When there is heterogeneity in the stress distribution within a body, the effective volume Veff can be written as a portion of the total volume Vtot as, see 21,

H=

⎟

∫ ∫ dz⋅2π⋅r⋅dr = π⋅R2⋅Z→∞ 0

Veff = H ⋅Vtot

⎜

Since the geometry in the indentation problem is not bounded, the total volume tends to infinity as

m

⎛ r z ∞ ∞ ⎜ σθ a , a , ν, F , a − ⎜ λ 0 0 ⎜ ⎜ ⎝ 1−e

σθ, max (ν, F , a) ⎞m −⎛ ∙ Veff (ν, F , a, m) λ ⎝ ⎠

a3

(11)

where σθ is the local hoop stress, σθ, max is the maximum hoop stress within the whole volume. Using Eq. (10) and Eq. (11), the probability of failure in Eq. (9) can be written as

∞ ∞

=

m

σθ (ρ , ζ , ν , F , a) ⎞ ⋅dζ ⋅2π⋅ρ⋅dρ ⎟ ⎝ σθ, max (ν , F , a) ⎠

∫∫ ⎛

⎜

0

0

(15)

Since both σθ (ρ , ζ , ν , F , a) and σθ, max (ν , F , a) has the same dependency (being linearly proportional) on the contact pressure pm , that variable cancels out and the effective volume becomes independent to the applied force, i.e.

Table 1 Summary of previous data from indentation tests. Norite, from1 UCS = 274 MPa ν = 0.24

Bohus granite, from6 UCS = 239 MPa ν = 0.24 (from19)

Sierra granite, from5 UCS = 170 MPa ν = 0.20

a [mm]

σST [MPa]

Fexp [kN]

a [mm]

σST [MPa]

Fexp [kN]

sexp [kN]

1.9

3200

36.3

2

2770

34.9

1.6

3 4 5

2550 2430 2130

72.2 122 167

5.2 10 16

3.8

2480

112

7.6

2100

381

15.25

1680

1227

173

a [mm]

σST [MPa]

Fexp [kN]

sexp [kN]

2.5

2000

39.3

2.1

5

1980

156

8.1

10

1650

517

27

International Journal of Rock Mechanics and Mining Sciences 100 (2017) 170–176

K. Weddfelt et al.

Veff (ν , F , a, m)

Veff

=

a3

a3

Table 2 Comparison of previous experimental results and those of the model presented.

(ν , m) = Ha (ν , m) m

∞ ∞

∫∫

=

0

0

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

σθ (ρ , pm

σθ, max pm

⎞ ⎟ ⎟ ⋅dζ ⋅2π⋅ρ⋅dρ ⎟ ⎟ ⎠

ζ , ν) (ν )

Rock

Veff 3PB ⎞ σθ, max = ⎜⎛ ⎟ σ3PB ⎝ Veff ⎠

0.407

22.7

30.5

0.407

20.6

110.7

0.407

18.7

402

0.407

17.0

1473

0.407

22.5

33.5

0.407

21.3

71.3

0.407

20.5

121.8

0.407

19.9

184.6

0.343

22.0

38.4

0.343

20.0

139.7

0.343

18.2

508

7.6 15.25 Bohus granite ν = 0.24

2 3 4 5

Sierra granite ν = 0.20

2.5 5 10

Fexp ± sexp Fmod ± smod

[kN]

36.3 ± 29.8 ± 1.7 112.5 ± 108.3 ± 6.2 381 ± 394 ± 23 1227 ± 1440 ± 82 34.9 ± 1.6 32.8 ± 1.9 72.2 ± 5.2 69.7 ± 4.0 122 ± 10 119 ± 6.8 167 ± 16 180 ± 10 39.3 ± 2.1 37.6 ± 2.1 156 ± 8.1 137 ± 7.8 517 ± 27 497 ± 28

and standard deviation of the load capacity with the model predictions, the probability of failure should be directly written as function of load capacity according to

(17)

m

Pθ (F , m , λF ) = 1−e

−⎛ F ⎞ ⎝ λF ⎠

(18)

where the new Weibull scale parameter λF is the load capacity at failure, FLoad Capacity , identified from Eq. (2), Eq. (6) and Eq. (12)

λF =

32⋅πa2⋅λ 1

(1 − 2ν )2⋅(Ha (ν , m)⋅a3) m

(19)

The expected average value and standard deviation of the model prediction of load capacity, Fmod and smod respectively is obtained from

1 2 1 2 Fmod = λF ⋅Γ ⎛1 + ⎞ and smod = λF ⋅ Γ ⎛1 + ⎞ − Γ ⎛1 + ⎞ m m m ⎠ ⎝ ⎠ ⎝ ⎠ ⎝

Ha = V eff / a3 [-]

(20)

3

5

1.5

7

5

0.

0.5

1

3

1.5

10

2

1 1.5 2 0.05

0.15

10

5

3 0.1

0.2

∞

exp (− u) u x du

(21)

The results from the prediction of the model together with the experimental ones are presented in Table 2. The model describes the observed mean value of load capacity with a very good accuracy (see Table 2). This is also true for the standard deviation of the experimental values. The major deviation for the norite and the Bohus granite cases concerns the fact that the model underestimates the load capacity of the smaller punches while it overestimates the force for the larger ones. However, in the case of Sierra granite, the load capacity behavior appears to be underestimated by the model, see Fig. 4. One possible explanation for this could be that the radial confining pressure in this test, believed to be initially about 27.2 MPa changes the load capacity behavior of Sierra granite. It should however be mentioned that the range of the tests on which these observations are made is small. This discrepancy is more visible if the stamp strength is plotted against punch radius, see Fig. 5. The best straight line fit to the experimental data (dashed lines in Fig. 5) has a power law dependency of:

2

0. 5 0.4

∫0

1

0.2

20

Γ(1+x ) =

2

0. 4

0.2

0.7

0. 3

where Γ is the Euler function of the second kind 0.2

Weibull modulus [-]

λF = FLoad Capacity

3.8

where m is Weibull modulus and σ3PB and Veff 3PB are the tensile strength and the effective volume of the material from three point bending experiment, the values for Bohus granite that are used in this paper being σ3PB = 18.7MPa , Veff 3PB = 189mm3 and m = 21.8 20. When σθ, max is known, the force level at failure, FLoad Capacity can easily be obtained using Eq. (2) and Eq. (6). Experimental values in Table 1 are given as average values of an unspecified number of indentation tests. In addition, Wijk6 reported the standard deviation values derived from his experiments. The standard deviation could also be derived from the test data given by Cook et al.5 To enable direct comparison of the experimental results for the average

5

σθmax [MPa]

1.9

Norite ν = 0.24

(16)

1/ m

25

Ha (ν, m) [-]

[MPa]

where Ha is a non-dimensional volume. The integral in Eq. (16) should be taken over an infinite medium. An arbitrary outer boundary is imposed on the integrals in this work and they are determined numerically. It can be shown that if this boundary is larger than a certain level (for the case of flat punch and material properties of ν = 0.24 and m = 21.8 is about ρ , ζ > 4a ), the result for the non-dimensional volume and accordingly the effective volume converges to a constant value. This non-dimensional volume, Ha , takes on a particular value for every combination of Poisson's ratio and Weibull modulus, see Fig. 3. It should also be mentioned that the singularity of the hoop stress at the contact perimeter in Eq. (16) has a negative sign and therefore does not make a convergence problem for the integral due to the Macauley's bracket indicating that the integration function is zero at the singularity. When the effective volume is known for the indentation problem (using the punch size together with the non-dimensional volume obtained from Fig. 3), one can obtain the corresponding hoop stress at failure during the indentation, σθ, max , using the Weibull size effect and known material properties from experiment

15

a [mm]

0.25

0.3

0.35

0.4

0.45

Poisson ratio [-] Fig. 3. Non-dimensional effective volume as function of Poisson's ratio and Weibull modulus.

174

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K. Weddfelt et al.

Fig. 4. Indentation force shown as function of punch radius. Circles represents experimental results and lines shows the range of model prediction of load capacity ± two standard deviations.

σST =

norite : a−0.301 ⎧ F Bohus granite : a−0.271 ∝ ⎨ πa2 −0.141 ⎩ Sierra granite : a

lower end this is comparable to the grain size of the rock. So it may be, e.g. that one single constant value of the Weibull module is insufficient to cover the complete range. In another word, the lack of self-similarity in the flat punch problem might be the reason for the discrepancy in Fig. 5 and a sharp conical indentation test with the advantage of selfsimilarity can be seen as an alternative to the flat punch test. However, it should be mentioned in this context that this will also lead to a situation where plastic (irreversible) deformation is due to the self-similarity introduced immediately at contact.

(22)

The theoretical model based on the Weibull failure criteria predicts this slope to be

σST =

−0.138 F 1 1 ∝σ ∝ ∝ =a 1 3 πa2 (Veff )1/ m θmax (a ) 21.8

(23)

where the Weibull parameter is taken from the three point bending tests on Bohus granite.20 A possible reason behind the mismatch between the model prediction and the experimental data could be due to the error in the cubic relation between the punch radius and the effective volume. This is a consequence of the major simplification of the stress distribution as being linear elastic. However, the rock directly under the punch experiences very high compressive stresses and is accordingly crushed and even expanded due to dilation. It is reasonable to expect that this to some degree influence both the contact problem and the stress field in the medium. Furthermore, the compressive behavior in this region may also be size dependent similar to the tensile behavior. Finally, the possibility of the Weibull theory itself not being able to capture accurately the tensile failure phenomena at hand cannot be discarded. The length scale of these indentation tests, in the sense of size of the loaded volume, range from about 1 mm and upwards. At the

5. Conclusions In this work, the focus is towards an investigation of the load capacity of selected hard rocks subjected to circular flat punch indentation. The compacted zone underneath the indenter is assumed to be limited and only responsible for the load transition to the rest of the material. Therefore, the theory of elasticity is used to define the stress state in a semi-infinite medium loaded by a flat punch indenter. A tensile failure criterion is used and it is assumed that the final load capacity is related to the formation of a sub-surface median crack at the symmetry line. The median crack initiates due to tensile hoop stresses. Therefore the final failure should occur at a force level in which the hoop stress is greater than the tensile strength of the rock. For the tensile strength, the Weibull size effect is adopted and the strength of the rock is scaled using an effective volume. Therefore, the initiation of

Fig. 5. Contact pressure at failure (which is equal to stamp strength) shown as function of punch radius. The solid line represents the size dependency of the model which obviously is underestimated for norite and Bohus granite.

175

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the median crack, responsible for the final load capacity, is treated as a probabilistic phenomenon. It should be mentioned that the material properties for Bohus granite obtained from three point bending tests is used as reference values in the Weibull size effect.20 It is likely that the presently proposed methodology is applicable for other types of semibrittle materials and indenter shapes. The effective volume is calculated based on the part of the material that is subjected to a positive hoop stress. Since the rock is considered as a semi-infinite medium, the effective volume should be calculated based on an infinite volume, which is problematic. However, arbitrary outer boundaries are imposed on the integrals and the effective volume is calculated numerically. It is shown that the effective volume converges to a constant value, as the arbitrary outer boundary gets larger than a certain level. The experimental results for the stamp load capacity of three selected hard rocks are taken from the literature. They are considered as similar to the reference material in this paper, which is Bohus granite. The model describes the observed load capacity with a very good accuracy for all three rocks. Finally, it should be mentioned that a possible application of the present results concerns percussive drilling. As a first approximation then, dynamic effects could be addressed by changing the explicit values on the tensile strength accounting for rate effects. However, considering the small values on the effective volume during indentation testing obtained in this paper, the rate dependency of Bohus granite based on previous studies performed by the authors and also the range of strain rates occurring at percussive drilling,23 the dynamic loading should not have a significant effect on stamp test results.

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Conflict of interests The authors declare that there is no conflict of interest regarding the publication of this paper. References 1. Wagner H, Schümann EHR. The stamp-load bearing strength of rock an experimental and theoretical investigation. Rock Mech. 1971;3(4):185–207. http://dx.doi.org/10. 1007/BF01238179. 2. Lawn B, Wilshaw R. Indentation fracture: principles and applications. J Mater Sci.

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