# Energy Criterion for Fracture of Rocks and Rock-like Materials on the Descending Branch of the Load–Displacement Curve

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Research Problem

#### 1.2. Two Classes of Fracture Criteria for Brittle Materials

#### 1.2.1. Micro- and Meso-Level Models

#### 1.2.2. Macro-Level Models

#### 1.3. Working Hypothesis and Purpose of the Study

## 2. Methodology

#### 2.1. Complete Stress–Strain Curve of a Brittle Material

#### 2.2. Justification of the Energy Differential Fracture Criterion for Brittle Materials

## 3. Examples and Comparison with Experiments Known in the Literature

#### 3.1. Example 1. Sandstone

#### 3.2. Example 2. Medium Coarse Sand (−10 °C)

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Wawersik, W.R.; Fairhurst, C. A Study of brittle Rock Fracture in Laboratory Compression Experiments. Int. J. Rock Mech. Min. Sci. Geomech.
**1970**, 7, 561–575. [Google Scholar] [CrossRef] - Liu, G.; Chen, Y.; Du, X.; Wang, S.; Fernández-Steeger, T.M. Evolutionary Analysis of Heterogeneous Granite Microcracks Based on Digital Image Processing in Grain-Block Model. Materials
**2022**, 15, 1941. [Google Scholar] [CrossRef] - Xu, T.; Fu, T.F.; Heap, M.J.; Meredith, P.G.; Mitchell, T.M.; Baud, P. Mesoscopic Damage and Fracturing of Heterogeneous Brittle Rocks Based on Three-Dimensional Polycrystalline Discrete Element Method. Rock Mech. Rock Eng.
**2020**, 53, 5389–5409. [Google Scholar] [CrossRef] - Parbhakar-Fox, A.; Fox, N.; Jackson, L.; Cornelius, R. Forecasting Geoenvironmental Risks: Integrated Applications of Mineralogical and Chemical Data. Minerals
**2018**, 8, 541. [Google Scholar] [CrossRef][Green Version] - Đurđevac Ignjatović, L.; Krstić, V.; Radonjanin, V.; Jovanović, V.; Malešev, M.; Ignjatović, D.; Đurđevac, V. Application of Cement Paste in Mining Works, Environmental Protection, and the Sustainable Development Goals in the Mining Industry. Sustainability
**2022**, 14, 7902. [Google Scholar] [CrossRef] - Zhang, J.Z.; Zhou, X.P. Forecasting catastrophic rupture in brittle rocks using precursory AE time series. J. Geophys. Res. Solid Earth
**2020**, 125, e2019JB019276. [Google Scholar] [CrossRef] - Zhou, X.P.; Zhang, J.Z.; Qian, Q.H.; Niu, Y. Experimental investigation of progressive cracking processes in granite under uniaxial loading using digital imaging and ae techniques. J. Struct. Geol.
**2019**, 126, 129–145. [Google Scholar] [CrossRef] - Li, L.P.; Shang, C.S.; Chu, K.W.; Zhou, Z.Q.; Song, S.G.; Liu, Z.H.; Chen, Y.H. Large-scale geo-mechanical model tests for stability assessment of super-large cross-section tunnel. Tunn. Undergr. Space Technol.
**2021**, 109, 103756. [Google Scholar] [CrossRef] - Contreras Inga, C.E.; Walton, G.; Holley, E. Statistical Assessment of the Effects of Grain-Structure Representation and Micro-Properties on the Behavior of Bonded Block Models for Brittle Rock Damage Prediction. Sustainability
**2021**, 13, 7889. [Google Scholar] [CrossRef] - Mishra, S.; Slabunov, A.I.; Svetov, S.A.; Kervinen, A.V.; Nesterova, N.S. Zircons from Collisional Granites, Garhwal Himalaya, NW India: U–Th–Pb Age, Geochemistry and Protolith Constraints. Minerals
**2021**, 11, 1071. [Google Scholar] [CrossRef] - Liu, Y.; Dai, F. A review of experimental and theoretical research on the deformation and failure behavior of rocks subjected to cyclic loading. J. Rock Mech. Geotech. Eng.
**2021**, 13, 1203–1230. [Google Scholar] [CrossRef] - Christophersen, A.; Behr, Y.; Miller, C. Automated Eruption Forecasting at Frequently Active Volcanoes Using Bayesian Networks Learned From Monitoring Data and Expert Elicitation: Application to Mt Ruapehu, Aotearoa, New Zealand. Front. Earth Sci.
**2022**, 10, 905965. [Google Scholar] [CrossRef] - Hu, X.; Li, Q.; Wu, Z.; Yang, S. Modelling fracture process zone width and length for quasi-brittle fracture of rock, concrete and ceramics. Eng. Fract. Mech.
**2022**, 259, 108158. [Google Scholar] [CrossRef] - Parsajoo, M.; Armaghani, D.J.; Mohammed, A.S.; Khari, M.; Jahandari, S. Tensile strength prediction of rock material using non-destructive tests: A comparative intelligent study. Transp. Geotech.
**2021**, 31, 100652. [Google Scholar] [CrossRef] - Zhao, P.; Masoumi, Z.; Kalantari, M.; Aflaki, M.; Mansourian, A. A GIS-Based Landslide Susceptibility Mapping and Variable Importance Analysis Using Artificial Intelligent Training-Based Methods. Remote Sens.
**2022**, 14, 211. [Google Scholar] [CrossRef] - Jędrzejczyk, A.; Firek, K.; Rusek, J. Convolutional Neural Network and Support Vector Machine for Prediction of Damage Intensity to Multi-Storey Prefabricated RC Buildings. Energies
**2022**, 15, 4736. [Google Scholar] [CrossRef] - Morgenroth, J.; Khan, U.T.; Perras, M.A. An Overview of Opportunities for Machine Learning Methods in Underground Rock Engineering Design. Geosciences
**2019**, 9, 504. [Google Scholar] [CrossRef][Green Version] - Abdelmaboud, A.; Abaker, M.; Osman, M.; Alghobiri, M.; Abdelmotlab, A.; Dafaalla, H. Hybrid Early Warning System for Rock-Fall Risks Reduction. Appl. Sci.
**2021**, 11, 9506. [Google Scholar] [CrossRef] - Bai, E.; Guo, W.; Tan, Y.; Guo, M.; Wen, P.; Liu, Z.; Ma, Z.; Yang, W. Regional Division and Its Criteria of Mining Fractures Based on Overburden Critical Failure. Sustainability
**2022**, 14, 5161. [Google Scholar] [CrossRef] - Yang, B.; Liu, Y. Application of Fractals to Evaluate Fractures of Rock Due to Mining. Fractal Fract.
**2022**, 6, 96. [Google Scholar] [CrossRef] - Wang, S.; Cai, X.; Zhou, J.; Song, Z.; Li, X. Analytical, Numerical and Big-Data-Based Methods in Deep Rock Mechanics. Mathematics
**2022**, 10, 3403. [Google Scholar] [CrossRef] - Zhang, Y.; Chu, W.H.; Ahmad, M. The establishment of prediction model for soil liquefaction based on the seismic energy using the neural network. Environ. Earth Sci.
**2022**, 81, 11. [Google Scholar] [CrossRef] - Cornetti, P.; Pugno, N.; Carpinteri, A.; Taylor, D. Finite fracture mechanics: A coupled stress and energy failure criterion. Eng. Fract. Mech.
**2006**, 73, 2021–2033. [Google Scholar] [CrossRef] - Ayatollahi, M.R.; Moghaddam, M.R.; Berto, F. A generalized strain energy density criterion for mixed mode fracture analysis in brittle and quasi-brittle materials. Theor. Appl. Fract. Mech.
**2015**, 79, 70–76. [Google Scholar] [CrossRef] - Khaji, Z.; Fakoor, M.; Farid, H.M.; Alderliesten, R. Applying the new experimental midpoint concept on strain energy density for fracture assessment of composite materials. Theor. Appl. Fract. Mech.
**2022**, 121, 103522. [Google Scholar] [CrossRef] - Guo, Y.; Chen, X.; Wang, Z.; Ning, Y.; Bai, L. Identification of mixed mode damage types on rock-concrete interface under cyclic loading. Int. J. Fatigue
**2022**, 166, 107273. [Google Scholar] [CrossRef] - Weißgraeber, P.; Becker, W. Finite fracture mechanics model for mixed mode fracture in adhesive joints. Int. J. Solids Struct.
**2013**, 50, 2383–2394. [Google Scholar] [CrossRef][Green Version] - Meng, Q.; Zhang, M.; Han, L.; Pu, H.; Chen, Y. Acoustic emission characteristics of red sandstone specimens under uniaxial cyclic loading and unloading compression. Rock Mech. Rock Eng.
**2018**, 51, 969–988. [Google Scholar] [CrossRef] - Lin, Q.; Wan, B.; Wang, Y.; Lu, Y.; Labuz, J.F. Unifying acoustic emission and digital imaging observations of quasi-brittle fracture. Theor. Appl. Fract. Mech.
**2019**, 103, 102301. [Google Scholar] [CrossRef] - Logoń, D.; Schabowicz, K. The Recognition of the Micro-Events in Cement Composites and the Identification of the Destruction Process Using Acoustic Emission and Sound Spectrum. Materials
**2020**, 13, 2988. [Google Scholar] [CrossRef] - Friedrich, L.F.; Tanzi, B.N.R.; Colpo, A.B.; Sobczyk, M.; Lacidogna, G.; Niccolini, G.; Iturrioz, I. Analysis of Acoustic Emission Activity during Progressive Failure in Heterogeneous Materials: Experimental and Numerical Investigation. Appl. Sci.
**2022**, 12, 3918. [Google Scholar] [CrossRef] - Meneghetti, G.; Ricotta, M. Evaluating the heat energy dissipated in a small volume surrounding the tip of a fatigue crack. Int. J. Fatigue
**2016**, 92, 605–615. [Google Scholar] [CrossRef] - Li, T.; Pei, X.; Wang, D.; Huang, R.; Tang, H. Nonlinear behavior and damage model for fractured rock under cyclic loading based on energy dissipation principle. Eng. Fract. Mech.
**2019**, 206, 330–341. [Google Scholar] [CrossRef] - Liang, Z.; Yu, Z.; Guo, L.; Huang, S.; Qin, N.; Wen, Z. Evaluation of white sandstone mechanical behaviour and the energy evolution of prepeak unloading damage. Sci. Rep.
**2022**, 12, 2793. [Google Scholar] [CrossRef] [PubMed] - Xu, X.; Yue, C.; Xu, L. Thermal Damage Constitutive Model and Brittleness Index Based on Energy Dissipation for Deep Rock. Mathematics
**2022**, 10, 410. [Google Scholar] [CrossRef] - Cai, M.; Hou, P.Y.; Zhang, X.W.; Feng, X.T. Post-peak stress–strain curves of brittle hard rocks under axial-strain-controlled loading. Int. J. Rock Mech. Min. Sci.
**2021**, 147, 104921. [Google Scholar] [CrossRef] - Liu, X.S.; Ning, J.G.; Tan, Y.L.; Gu, Q.H. Damage constitutive model based on energy dissipation for intact rock subjected to cyclic loading. Int. J. Rock Mech. Min.
**2016**, 85, 27–32. [Google Scholar] [CrossRef] - Zheng, H.; Ma, Z.; Zhou, L.; Zhang, D.; Liang, X. Effect of Loading Rate and Confining Pressure on Strength and Energy Characteristics of Mudstone under Pre-Cracking Damage. Energies
**2022**, 15, 3545. [Google Scholar] [CrossRef] - Tan, Y.; Gu, Q.; Ning, J.; Liu, X.; Jia, Z.; Huang, D. Uniaxial Compression Behavior of Cement Mortar and Its Damage-Constitutive Model Based on Energy Theory. Materials
**2019**, 12, 1309. [Google Scholar] [CrossRef][Green Version] - Wu, Y.; Huang, L.; Li, X.; Guo, Y.; Liu, H.; Wang, J. Effects of Strain Rate and Temperature on Physical Mechanical Properties and Energy Dissipation Features of Granite. Mathematics
**2022**, 10, 1521. [Google Scholar] [CrossRef] - Zhang, Y.; Feng, X.-T.; Yang, C.; Han, Q.; Wang, Z.; Kong, R. Evaluation Method of Rock Brittleness under True Triaxial Stress States Based on Pre-peak Deformation Characteristic and Post-peak Energy Evolution. Rock Mech. Rock Eng.
**2021**, 54, 1277–1291. [Google Scholar] [CrossRef] - Wen, T.; Tang, H.; Ma, J.; Liu, Y. Energy Analysis of the Deformation and Failure Process of Sandstone and Damage Constitutive Model. KSCE J. Civ. Eng.
**2018**, 23, 513–524. [Google Scholar] [CrossRef] - Gong, F.; Yan, J.; Luo, S.; Li, X. Investigation on the Linear Energy Storage and Dissipation Laws of Rock Materials Under Uniaxial Compression. Rock Mech. Rock Eng.
**2019**, 52, 4237–4255. [Google Scholar] [CrossRef] - Li, E.; Gao, L.; Jiang, X.; Duan, J.; Pu, S.; Wang, J. Analysis of dynamic compression property and energy dissipation of salt rock under three-dimensional pressure. Environ. Earth Sci.
**2019**, 78, 388. [Google Scholar] [CrossRef] - Zhou, T.B.; Qin, Y.P.; Ma, Q.F.; Liu, J. A constitutive model for rock based on energy dissipation and transformation principles. Arab. J. Geosci.
**2019**, 12, 492. [Google Scholar] [CrossRef] - Zhang, L.; Cheng, H.; Wang, X.; Liu, J.; Guo, L. Statistical Damage Constitutive Model for High-Strength Concrete Based on Dissipation Energy Density. Crystals
**2021**, 11, 800. [Google Scholar] [CrossRef] - Mohammadnejad, M.; Liu, H.; Chan, A.; Dehkhoda, S.; Fukuda, D. An overview on advances in computational fracture mechanics of rock. Geosyst. Eng.
**2021**, 24, 206–229. [Google Scholar] [CrossRef] - Makarov, P.V. Evolutionary nature of structure formation in lithospheric material: Universal principle for fractality of solids. Russ. Geol. Geophys.
**2007**, 48, 558–574. [Google Scholar] [CrossRef] - Walley, S.M.; Rogers, S.J. Is Wood a Material? Taking the Size Effect Seriously. Materials
**2022**, 15, 5403. [Google Scholar] [CrossRef] - Kolesnikov, G. Analysis of Concrete Failure on the Descending Branch of the Load-Displacement Curve. Crystals
**2020**, 10, 921. [Google Scholar] [CrossRef] - Katarov, V.; Syunev, V.; Kolesnikov, G. Analytical Model for the Load-Bearing Capacity Analysis of Winter Forest Roads: Experiment and Estimation. Forests
**2022**, 13, 1538. [Google Scholar] [CrossRef] - Blagojević, M.; Pešić, D.; Mijalković, M.; Glišović, S. Jedinstvena funkcija za opisivanje naprezanja i deformacije betona u požaru. Građevinar
**2011**, 63, 19–24. Available online: https://hrcak.srce.hr/clanak/96329 (accessed on 14 July 2022). - Stojković, N.; Perić, D.; Stojić, D.; Marković, N. New stress–strain model for concrete at high temperatures. Teh. Vjesn.
**2017**, 24, 863–868. [Google Scholar] - Pereira, L.R.S.; Penna, S.S. Nonlinear analysis method of concrete structures under cyclic loading based on the generalized secant modulus. Rev. IBRACON Estrut. Mater.
**2022**, 15, e15406. [Google Scholar] [CrossRef] - Kolesnikov, G.; Meltser, R. A Damage Model to Trabecular Bone and Similar Materials: Residual Resource, Effective Elasticity Modulus, and Effective Stress under Uniaxial Compression. Symmetry
**2021**, 13, 1051. [Google Scholar] [CrossRef] - Chen, J.; Wang, L.; Yao, Z. Physical and mechanical performance of frozen rocks and soil in different regions. Adv. Civ. Eng.
**2020**, 2020, 8867414. [Google Scholar] [CrossRef] - Kolesnikov, G.; Zaitseva, M.; Petrov, A. Analytical Model with Independent Control of Load–Displacement Curve Branches for Brittle Material Strength Prediction Using Pre-Peak Test Loads. Symmetry
**2022**, 14, 2089. [Google Scholar] [CrossRef] - Cai, M.; Hou, P.Y. Post-peak stress–strain curves of brittle hard rocks under different loading environment system stiffness. Rock Mech. Rock Eng.
**2022**, 55, 3837–3857. [Google Scholar] [CrossRef] - Yin, Y.; Zheng, W.; Tang, X.; Xing, M.; Zhang, Y.; Zhu, Y. Test study on failure and energy supply characteristics of rock under different loading stiffness. Eng. Fail. Anal.
**2022**, 142, 106796. [Google Scholar] [CrossRef] - Guidotti, R.; Monreale, A.; Ruggieri, S.; Turini, F.; Giannotti, F.; Pedreschi, D. A survey of methods for explaining black box models. ACM Comput. Surv. (CSUR)
**2018**, 51, 1–42. [Google Scholar] [CrossRef][Green Version] - Lu, G.; He, X.; Wang, Q.; Shao, F.; Wang, J.; Jiang, Q. Bridge crack detection based on improved single shot multi-box detector. PLoS ONE
**2022**, 17, e0275538. [Google Scholar] [CrossRef] [PubMed] - Ziying, M.; Shaolin, H.; Xiaomin, H.; Ye, K. Fine Crack Detection Algorithm Based on Improved SSD. Sci. Technol.
**2022**, 8, 43–47. [Google Scholar] - Noii, N.; Khodadadian, A.; Wick, T. Bayesian Inversion Using Global-Local Forward Models Applied to Fracture Propagation in Porous Media. Int. J. Multiscale Comput. Eng.
**2022**, 20, 57–79. [Google Scholar] [CrossRef] - Noii, N.; Khodadadian, A.; Wick, T. Bayesian inversion for anisotropic hydraulic phase-field fracture. Comput. Methods Appl. Mech. Eng.
**2021**, 386, 114118. [Google Scholar] [CrossRef] - Noii, N.; Khodadadian, A.; Ulloa, J.; Aldakheel, F.; Wick, T.; François, S.; Wriggers, P. Bayesian Inversion with Open-Source Codes for Various One-Dimensional Model Problems in Computational Mechanics. Arch. Comput. Methods Eng.
**2022**, 29, 4285–4318. [Google Scholar] [CrossRef]

**Figure 3.**The line $\sigma =\epsilon {E}_{tangential}$ passes through the inflection point a, the coordinates of which can be found from the equation ${d}^{2}\sigma /d{\epsilon}^{2}=0$. The line $\sigma =\epsilon {E}_{secant}$_secant passes through the origin and point $b$, where $\epsilon ={\epsilon}_{peak}$, $\sigma ={\sigma}_{peak}$ and $d\sigma /d\epsilon =0$.

**Figure 4.**Strain energy $d{W}_{e}=\sigma d\epsilon $ and dissipation energy $d{W}_{d}=\left(\widehat{\sigma}-\sigma \right)d\epsilon $. The stress in an ideal material without dissipation is $\widehat{\sigma}=\epsilon {E}_{tangential}$. The voltage σ in a material with energy dissipation is determined from Equation (5).

**Figure 5.**Fracture point on curve (5) (big red circle): using tangent (

**left**) and secant modulus of elasticity (

**right**).

**Figure 6.**The stress–strain curve (5) for sandstone in uniaxial compression. A tangent (black dashed line) passes through point $k$, the slope angle of which determines the tangential modulus of elasticity. At point $t$, we predict failure according to criterion (12), at this point the line $\sigma =0.5\epsilon {E}_{tangential}$ intersects the curve (5) (see also Figure 5). A secant (red dotted line) passes through point $b$, the slope angle of which determines the secant modulus of elasticity. At point $s$ the failure is predicted by criterion (14), at this point the line $\sigma =0.5\epsilon {E}_{secant}$ intersects the curve (5). The red curve simulates the experimental curve from [36]. The thin red and black lines correspond to Figure 5.

**Figure 7.**Stress–strain curve (5) for frozen sand under uniaxial compression. A secant (red dotted line) passes through point b, the slope angle of which determines the secant modulus of elasticity. At point $s$, failure is predicted by criterion (14), at this point the line $\sigma =0.5\epsilon {E}_{secant}$ intersects the curve (5). The red curve simulates the experimental curve from [56].

**Figure 8.**Effect of deviations in ${\sigma}_{peak}$, ${\epsilon}_{peak}$ on uniaxial compression behavior of samples: (

**a**) Sandstone from example 1; (

**b**) Frozen sand from example 2. The red line corresponds to the parameters ${\sigma}_{peak}$, ${\epsilon}_{peak}$. Thin lines correspond to parameters with deviations:${\sigma}_{peak}\xb7\left(1\pm 0.05\right)$, ${\epsilon}_{peak}\xb7\left(1\pm 0.05\right)$. The red curve simulates the experimental curve from [36] (

**a**) and [56] (

**b**).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kolesnikov, G.; Shekov, V.
Energy Criterion for Fracture of Rocks and Rock-like Materials on the Descending Branch of the Load–Displacement Curve. *Materials* **2022**, *15*, 7907.
https://doi.org/10.3390/ma15227907

**AMA Style**

Kolesnikov G, Shekov V.
Energy Criterion for Fracture of Rocks and Rock-like Materials on the Descending Branch of the Load–Displacement Curve. *Materials*. 2022; 15(22):7907.
https://doi.org/10.3390/ma15227907

**Chicago/Turabian Style**

Kolesnikov, Gennady, and Vitali Shekov.
2022. "Energy Criterion for Fracture of Rocks and Rock-like Materials on the Descending Branch of the Load–Displacement Curve" *Materials* 15, no. 22: 7907.
https://doi.org/10.3390/ma15227907