# Differential Energy Criterion for Brittle Fracture: Conceptualization and Application to the Analysis of Axial and Lateral Deformation in Uniaxial Compression of Rocks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Conceptual Aspects, Methodology and Results

#### 2.1. Preliminary Remarks

#### 2.2. Research Hypothesis

#### 2.3. Estimates of the Undamaged and Damaged Area Depending on the Strain

#### 2.4. Degradation of the Cross-Sectional Area

#### 2.5. Justification of the Load–Displacement Relation in Uniaxial Compression

#### 2.6. Justification of the Stress–Strain Relation in Uniaxial Compression

#### 2.7. Previous Models of the Class under Discussion

#### 2.8. Pre- and Post-Peak Modulus of Elasticity

#### 2.9. Comparison of Simulation Results and Experimental Data

#### 2.10. The Point of Maximum Modulus of Elasticity on the Ascending Branch of the Stress–Strain Curve as the Point of Highest Density of a Brittle Material

#### 2.11. The Concept of Virtual Material Transformation

#### 2.12. Relationship between $d{W}_{e}$ and $d{W}_{d}$, Strength Condition and Fracture Point Coordinates

#### 2.13. Graphic Definition of the Brittle Fracture Point

- On the pre-peak branch, determine the point that corresponds to the largest tangential modulus of elasticity (the point 1 in Figure 8).
- Draw a tangent through point 1 and define point 2.
- From any point (for example, point 3) on the tangent, draw a perpendicular to the abscissa axis. Find point 4.
- Define point 5 as the midpoint of segment 3–4.
- Draw a line through points 2 and 5. The point of intersection of this line with the post-peak branch of the stress–strain curve simulates the point of failure; this is point 6 in Figure 8.

## 3. Discussion

## 4. Conclusions

- The hypothesis of the study was formulated (1): the damage to the cross-sectional area is proportional to the undamaged part of the area and to displacement (strain).
- In a logical connection with the research hypothesis (1), the residual resource function of the cross-sectional area (3) is justified.
- The question of analytical determination of the highest (pre-peak) and lowest (post-peak) values of the modulus of elasticity is considered. Equations (20) and (21) for calculating the coordinates and values (18) of these moduli are obtained. Examples of determining the pre-peak and post-peak modulus of elasticity in uniaxial compression of marble are given (Table 3).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$A$ | cross-sectional area of the specimen. |

$B$ | model parameter. |

$D$ | damage variable. |

$E$ | Young’s modulus. |

EMC | concept of equivalent material [12]. |

$F$ | Force (or Load). |

FMC | concept of fictitious material [13]. |

$n$ | model parameter. |

$u$ | displacement. |

${u}_{peak}$ | peak displacement. |

${W}_{d}$ | dissipated energy. |

${W}_{e}$ | stored energy. |

$\epsilon $ | strain. |

${\epsilon}_{fracture}$ | strain in fracture point. |

${\epsilon}_{peak}$ | peak strain. |

$\sigma $ | stress. |

${\sigma}_{fracture}$ | stress in fracture point. |

${\sigma}_{peak}$ | peak stress. |

θ | undamaged part of the original cross-sectional area. |

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**Figure 1.**Typical plot of cross-sectional area $A$ versus displacement $u$ under uniaxial compression.

**Figure 3.**Force $F\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\u2206H=u$.

**Figure 4.**Plots of axial stress–axial strain and axial stress–lateral strain during uniaxial compression of marble. The semi-transparent color shows experimental data from [33] obtained with test machines of different stiffness (4.7, 5.6, 6.4 and 7.4 GN/m). Colored circles 1, 2, 3 and 4 correspond to the points at which the condition (19) is satisfied; tangents (thin slanted lines) pass through these points. The tangent angle of these lines determines the pre- and post-peak moduli (Table 3).

**Figure 5.**Axial stress–axial strain relationships for an ideal material (thin straight line) and a real material (curve).

**Figure 6.**Change in dissipated ($d{W}_{d}$) and stored ($d{W}_{e}$) energy at infinitesimal increment of deformation ($d\epsilon $). The modulus of elasticity is greatest at point 1. Point 1 can be determined numerically (by processing measurements during material tests) or analytically (by differentiating the function (17)); see also Section 2.8. The stress–strain relation ($\sigma =\epsilon E$) for the ideal material is represented by the tangent to the stress–strain curve (17) for the real material; point 1 is the common point of these dependencies, i.e., as noted above, the elastic modulus of the ideal material at this point is equal to that of the real material.

**Figure 7.**The coordinates of the fracture point ${\epsilon}_{fracture}$ and ${\sigma}_{fracture}$ rupture are determined from Equations (34) and (33), respectively. Lines 1 and 2 intersect at the point with coordinates ${\epsilon}_{x}={\epsilon}_{E}-\frac{{\sigma}_{E}}{E}$; $\sigma =0$. Elastic modulus $E$ calculated from Equations (18) and (20). The point with coordinates ${\epsilon}_{E},{\sigma}_{E}$ corresponds to the highest value of the modulus of elasticity.

**Figure 8.**Point 1 corresponds to the highest value of the modulus of elasticity. Point 6 corresponds to brittle fracture. Points 2–4 are auxiliary points.

**Figure 9.**The ellipse limits the area of predicted fracture points on the curves by Figure 4.

**Figure 10.**The experimental curves are adapted from [40]; points 1–5 on these curves predict the failure of granite specimens with different modulus of elasticity in uniaxial compression. Points 1–5 obtained as a solution of Equation (34) using the graphical algorithm (Section 2.13).

Specimen Number | Input Data | Calibration Parameters | ||
---|---|---|---|---|

${\mathit{\epsilon}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$ (%) | ${\mathit{\sigma}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$ (MPa) | $\mathit{n}$ | $\mathit{B}$ | |

1 | 0.268 | 95.6 | 9.5 | 1.3 |

2 | 0.270 | 95.8 | 8.0 | 1.3 |

3 | 0.274 | 96.3 | 7.0 | 1.3 |

4 | 0.279 | 96.4 | 6.0 | 1.3 |

Specimen Number | Input Data | Calibration Parameters | |||
---|---|---|---|---|---|

Poisson’s Ratio ν | ${\mathit{\epsilon}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$ (%) | ${\mathit{\sigma}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$ (MPa) | $\mathit{n}$ | $\mathit{B}$ | |

1 | 0.2 | −0.0536 | 95.6 | 1.00 | 0.10 |

2 | 0.2 | −0.0540 | 95.8 | 1.05 | 0.20 |

3 | 0.2 | −0.0548 | 96.3 | 1.08 | 0.20 |

4 | 0.2 | −0.0578 | 96.4 | 1.15 | 0.20 |

Specimen Number | Pre-Peak Module | Post-Peak Module | ||||
---|---|---|---|---|---|---|

E (GPa) | $\mathit{\epsilon}$ (%) | $\mathit{\sigma}$ (MPa) | $\mathit{M}$ (GPa) | $\mathit{\epsilon}$ (%) | $\mathit{\sigma}$ (MPa) | |

1 | 46.1 (46.8) ^{1} | 0.183 | 66.67 | −133.34 | 0.336 | 45.85 |

2 | 45.9 (44.7) | 0.175 | 63.89 | −112.48 | 0.346 | 47.18 |

3 | 45.7 (44.6) | 0.169 | 61.83 | −98.05 | 0.359 | 48.47 |

4 | 45.2 (45.5) | 0.163 | 59.01 | −83.21 | 0.374 | 49.81 |

^{1}The experimental values of the elastic modulus from [33] are given in parentheses.

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**MDPI and ACS Style**

Shekov, V.; Kolesnikov, G.
Differential Energy Criterion for Brittle Fracture: Conceptualization and Application to the Analysis of Axial and Lateral Deformation in Uniaxial Compression of Rocks. *Materials* **2023**, *16*, 4875.
https://doi.org/10.3390/ma16134875

**AMA Style**

Shekov V, Kolesnikov G.
Differential Energy Criterion for Brittle Fracture: Conceptualization and Application to the Analysis of Axial and Lateral Deformation in Uniaxial Compression of Rocks. *Materials*. 2023; 16(13):4875.
https://doi.org/10.3390/ma16134875

**Chicago/Turabian Style**

Shekov, Vitali, and Gennady Kolesnikov.
2023. "Differential Energy Criterion for Brittle Fracture: Conceptualization and Application to the Analysis of Axial and Lateral Deformation in Uniaxial Compression of Rocks" *Materials* 16, no. 13: 4875.
https://doi.org/10.3390/ma16134875