## Abstract

In this article, using a two-dimensional numerical model based on the finite element method, the fracture pressure of rock samples with a circular hole in the center is obtained. In this model, a mixed criterion is used to determine the fracture pressure, which is a combination of the tensile strength criterion and the fracture toughness criterion. The superiority of this model over the tensile strength criterion or the fracture toughness criterion is that it shows well the effect of the size of the hole in the middle of the sample on the fracture stress. The main and innovative finding of this article is that if the radius of the hole in the center of the sample is more than 70 mm, the fracture strength of the sample is dependent on the tensile strength of the rock material, and if the radius of the hole is less than 70 mm, the fracture strength is dependent on the fracture toughness. According to other results, the fracture strength decreases with the increase of the radius of the hole in the center of the sample, the decrease rate is high at first and then decreases with the increase of the radius of the hole.

You have full access to this open access chapter, Download conference paper PDF

## Keywords

## 1 Introduction

In rock mechanics laboratories, for various reasons, fracture resistance tests are performed on rock samples with a hole in the center. For example, in hydraulic fracturing tests, to determine the fracture pressure, rock samples with a circular cavity are subjected to triaxial compressive stresses. But it has been observed that the fracture pressure in rock samples, although it depends on the mechanical properties of the rock and the fluid injection rate, but it also depends on the hole radius. In hydraulic fracturing tests, the fracture pressure is also dependent on the length of the initial crack in the cavity wall. The dependence of the fracture pressure on the size of the borehole is also very important in the field, where the breakdown pressure of the hydraulic fracture is supposed to be used in the estimation of the in situ stresses.

In rock and rock masses, failure is mainly caused by shear and tensile stresses caused by compressive stresses [1,2,3,4,5,6,7], in other words, when the tensile stresses exceed the tensile strength of the rock, tensile failure occurs [8,9,10,11,12,13]. This is the traditional classic view of tensile failure, against which there is the view of energy instability. The theory of energy instability, which today has led to the linear elastic fracture mechanics (LEFM) method, states that when the stress intensity factor at the crack tip exceeds the fracture toughness, a crack starts to propagate [14, 15]. On the other hand Leguillon (2002) showed that both the energy criterion and the stress criterion are necessary conditions for failure, but neither of them alone is sufficient [15]. This is while the energy criterion provides the lower bound for crack length and the stress criterion leads to the upper bound for admissible crack length.

If there is a hole in the center of the sample, due to the concentration of tensile stress, tensile failure occurs in the cavity wall and along the maximum in situ stress. Although, due to the use of linear elastic theory, the hole size is not included in the classical criterion based on tensile strength, but the fracture mechanics criterion is able to consider the effect of the initial crack length on the fracture pressure.

Some laboratory studies showed that the fracture pressure is dependent on the tensile strength and the fracture toughness of the material [7, 15], in such conditions, the effect of the size of the hole in the center of the sample can be seen in the fracture pressure. Recently, the authors obtained the breakdown pressure of hydraulic fracture by combining the tensile strength criterion and the fracture mechanics criterion and investigated the effect of borehole radius and initial crack length on the breakdown pressure [16].

This study investigates fracture pressure of brittle rock samples with a circular cavity using a plane strain two-dimensional finite element model. The mixed failure criterion is based on a combination of tensile strength (stress criterion) and fracture toughness (energetic criterion) of the material. The superiority of this model over the previous models is that it examines both the tensile strength of the material and the fracture mechanics at the same time, and by using it, the effect of the radius of the hole in the center of the sample can be seen on the fracture resistance. Basically, the mixed criterion in this paper covers the gap between the tensile strength criterion and the toughness criterion. This model, while being simple, is very effective in determining the fracture stress of brittle materials. In the next section, the geometry of the problem and the solution method are explained, and in the Sect. 3, the results of the numerical analysis are given.

## 2 Finite Element Numerical Model, Mixed Criterion

As seen in Fig. 1, a sample with a circular cavity of radius *R*_{o} = 5 cm is considered. This sample is under compressive stress *p*_{o}. Here, the dimensions of the model are considered to be \(2 \times 2\;m^{2}\). The purpose of this article is to determine the pressure *p*_{o} so that tensile failure occurs in the wall of the cavity and along *p*_{o}. In this article, the mixed criterion is used to find the fracture pressure, which means that material tensile strength criterion and the fracture mechanics criterion must both be satisfied, i.e.:

In this relationship, *p*_{f} is the fracture pressure and *c*_{f} is the fracture initiation length. \(\sigma_{\theta }\) is the maximum tangential stress along the imaginary crack with length *c*. \(\sigma_{t}\) is the tensile strength of the rock material and *K*_{Ic} is the fracture toughness. *K*_{I} is the stress intensity factor at the crack tip with a length of *c*. To find these values, we draw the graphical form of the first and second criterion in one plot. The point of intersection of two criteria indicates the fracture pressure and the length of fracture initiation. Figure 2 shows an example of a mixed criterion.

After discretizing the geometry of the problem and applying the boundary conditions, the problem is solved by the finite element method and the fracture pressure is obtained. For this, a code is written in the MATLAB program.

## 3 Numerical Analysis Results

For the characteristics of the materials presented in Table 1 and using the mixed criteria explained above, various numerical analysis are performed, the results of which are given below.

Figure 3 shows the fracture pressure (*p*_{f}) obtained from the mixed criterion against the hole radius for different fracture toughnesses. In this figure, it can be seen that the rate of change of fracture pressure against the hole radius is high at first and then decreases. Therefore, the dependence of the fracture pressure on the cavity radius can be clearly seen. If the fracture pressure is normalized by the tensile strength of the material and plotted against the radius of the hole, Fig. 4 is obtained. The important point that can be obtained from this Fig. 4 is that for *R*_{o} > 70 mm, \(\frac{{p_{f} }}{{\sigma_{t} }} \simeq 1\), that is, for holes with a large radius, the tensile strength of the rock determines the fracture pressure, while for holes with a small radius, the fracture mechanics criterion plays an essential role in determining the fracture pressure of the samples. Thus, the mixed criterion captures the size effect well. Figures 3 and 4 show that with increasing toughness, the fracture pressure increases. But for *R*_{o} > 70 mm, the fracture pressure for different fracture toughnesses is almost the same, and therefore toughness has no effect on the fracture pressure of the rock samples, this is because for *R*_{o} > 70 mm as mentioned, it is the tensile strength of the rock that is effective in the fracture of the model (Fig. 3). As the radius of the hole decreases, the distance between the curves with different toughness increases, because for* R*_{o} < 70 mm, the fracture toughness is effective in the fracture of the model.

Figure 5 shows the fracture pressure against the hole radius for different values of tensile strength. In this figure, it can be seen that for *R*_{o} < 70 mm, the sensitivity of the model to the tensile strength of the rock is low, but for *R*_{o} > 70 mm, the fracture pressure increases slightly with the increase of tensile strength.

Figures 6 and 7 also show the fracture initiation length (*c*_{f}) versus the hole radius. It can be seen in Fig. 6 that for samples with the same hole radius, the length of fracture initiation increases with the increase of fracture toughness. But Fig. 7 illustrates that, for samples with the same hole radius, as the tensile strength of the sample increases, the fracture pressure decreases. Also, Figs. 6 and 7 illustrate that as the radius of the hole increases, the initiation fracture length increases, the rate of increase of the initiation length of the fracture is high at first, but then it decreases.

Figure 8 shows the comparison between the results of the numerical model presented in this paper and the experiments conducted by Carter et al. (1992) on sandstone [7]. This Fig. Shows that there is a good agreement between the two sets of results.

## 4 Conclusion

In this article, using a numerical model based on the finite element method and mixed tensile fracture criterion technique, the fracture pressure of rock samples with a hole in its centre was obtained. Using the technique of mixed fracture criterion, the effect of the size of the hole in the middle of the sample was observed on the fracture pressure. The obtained results show that when *R*_{o} > 70 mm, the tensile strength criterion plays a role in determining the fracture pressure of rock samples, while for *R*_{o} < 70 mm, the fracture mechanics criterion determines the fracture pressure of the model. Also, as the radius of the hole increases, the fracture pressure decreases, the fracture pressure reduction rate is high at first and then decreases, and the fracture pressure converges to the tensile strength of the rock.

## References

Hoek, E.: A photoelastic technique for the determination of potential fracture zones in rock structures. In: ARMA US Rock Mechanics/Geomechanics Symposium, pp. ARMA-66. ARMA (1966)

Bahrehdar, M., Lakirouhani, A.: Evaluation of the depth and width of progressive failure of breakout based on different failure criteria, using a finite element numerical model. Arab. J. Sci. Eng.

**47**(9), 11825–11839 (2022)Bahrehdar, M., Lakirouhani, A.: Assessment of interplay of mud cake and failure criteria on the lower limit of safe borehole pressure. Indian Geotech. J. 1–13 (2024). https://doi.org/10.1007/s40098-023-00856-8

Lakirouhani, A., Hasanzadehshooiili, H.: Review of rock strength criteria. In: Proceedings of the 22nd World Mining Congress & Expo, pp. 473–482 (2011)

Lakirouhani, A., Jolfaei, S.: Hydraulic fracturing breakdown pressure and prediction of maximum horizontal in situ stress. Adv. Civil Eng.

**2023**, 1–14 (2023)Lakirouhani, A., Bahrehdar, M., Medzvieckas, J., Kliukas, R.: Comparison of predicted failure area around the boreholes in the strike-slip faulting stress regime with Hoek-brown and Fairhurst generalized criteria. J. Civ. Eng. Manag.

**27**(5), 346–354 (2021)Carter, B.J., Lajtai, E.Z., Yuan, Y.: Tensile fracture from circular cavities loaded in compression. Int. J. Fract.

**57**, 221–236 (1992)Lakirouhani, A., Ghorbannezhad, S., Medzvieckas, J., Kliukas, R.: Failure analysis around oriented boreholes using an analytical model in different faulting stress regimes. J. Civ. Eng. Manag.

**29**(4), 360–371 (2023)Jolfaei, S., Lakirouhani, A.: Sensitivity analysis of effective parameters in borehole failure, using neural network. Adv. Civil Eng.

**2022**, 1–16 2022Hubbert, M.K., Willis, D.G.: Mechanics of hydraulic fracturing. Trans. AIME

**210**(01), 153–168 (1957)Zhang, X., Lu, Y., Tang, J., Zhou, Z., Liao, Y.: Experimental study on fracture initiation and propagation in shale using supercritical carbon dioxide fracturing. Fuel

**190**, 370–378 (2017)Bahrehdar, M., Lakirouhani, A.: Effect of eccentricity on breakout propagation around noncircular boreholes. Advances in Civil Engineering,

**2023**, 6962648 (2023)Jolfaei, S., Lakirouhani, A.: Initiation pressure and location of fracture initiation in elliptical wellbores. Geotech. Geol. Eng.

**41**, 1–20 (2023)Atkinson, C., Thiercelin, M.: The interaction between the wellbore and pressure-induced fractures. Int. J. Fract.

**59**, 23–40 (1993)Leguillon, D.: Strength or toughness? A criterion for crack onset at a notch. Eur. J. Mech.-A/Solids

**21**(1), 61–72 (2002)Lakirouhani, A., Jolfaei, S.: Assessment of hydraulic fracture initiation pressure using fracture mechanics criterion and coupled criterion with emphasis on the size effect. Arab. J. Sci. Eng.

**49**, 1–12 (2023)

## Author information

### Authors and Affiliations

### Corresponding author

## Editor information

### Editors and Affiliations

## Rights and permissions

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## Copyright information

© 2024 The Author(s)

## About this paper

### Cite this paper

Lakirouhani, A., Jolfaei, S. (2024). Conflict Between Fracture Toughness and Tensile Strength in Determining Fracture Strength of Rock Samples with a Circular Hole. In: Feng, G. (eds) Proceedings of the 10th International Conference on Civil Engineering. ICCE 2023. Lecture Notes in Civil Engineering, vol 526. Springer, Singapore. https://doi.org/10.1007/978-981-97-4355-1_46

### Download citation

DOI: https://doi.org/10.1007/978-981-97-4355-1_46

Published:

Publisher Name: Springer, Singapore

Print ISBN: 978-981-97-4354-4

Online ISBN: 978-981-97-4355-1

eBook Packages: EngineeringEngineering (R0)