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

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## 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

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**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**).

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**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