# The Influence of Specimen Geometry and Loading Conditions on the Mechanical Properties of Porous Brittle Media

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Method

^{−1}to 6000 s

^{−1}.

^{i}(t) of the required amplitude and duration (determined by the speed and length of the striker), propagating along the incident bar with the speed of sound C, reaches the specimen and loads it; while part of the wave is reflected back by the reflected pulse ε

^{r}(t), and part passes into the supported bar by the transmitted pulse ε

^{t}(t). Based on these strain pulses recorded by strain gauges in measuring bars, the parametric dependences of the development of the axial stress σ

_{s}(t), strain ε

_{s}(t) and strain rate ${\dot{\mathsf{\epsilon}}}_{s}(t)$ components of the specimen over time by the Kolsky method formulas [8,9] were determined:

_{0}is the initial length of the specimen, ${A}_{s}^{0}$ is the specimen initial cross-sectional area.

_{s}(ε

_{s}) with a dependence ${\dot{\mathsf{\epsilon}}}_{s}({\mathsf{\epsilon}}_{s})$.

- -
- Considerable difference in the acoustic impedances of the sample material and the material of the pressure bars;
- -
- Low speed of wave propagation in a porous medium;
- -
- The quality of the processing of the end surfaces of the specimen;
- -
- Significant porosity of the material.

^{i}(t) + ε

^{r}(t) = ε

^{t}(t)

## 3. Tested Specimens

_{2}O

_{3}in a molar fraction of 11–12%. Manufacturing technology was close to factory. To obtain the required fractional composition of ceramics (Table 1), the technology included grinding, chemical and magnetic cleaning, the addition of a temporary plasticizing binder, pressing (pressure 100 MPa) and annealing (2000 K for 13 h).

^{3}, porosity 20%, static compressive strength 39 MPa.

_{s}/2; where L and D are the length and diameter of the cylindrical specimen, respectively, and v

_{s}is the Poisson’s ratio of the material under test. With this L/D ratio, the components of axial and radial inertia are mutually compensated, therefore, the calculated stress in the sample is considered reliable. This ratio is valid for a variety of materials, including brittle media.

## 4. Results and Discussion

_{i}(t) with a flat top, the reflected pulse ε

_{r}(t) (the sample strain rate) first increases and then decreases due to an increase in the resistance of the sample during its deformation. After the end of affecting of the incident pulse on the sample, the transmitted pulse ε

_{t}(t) also begins to decrease, while the strain rate determined by the reflected pulse ε

_{r}(t) becomes negative. Thus, in the section of the sample active loading, both stress and deformation increase, then, with the beginning of the incident pulse decrease, the stress in the specimen decreases to almost zero, while the achieved deformation decreases by a certain amount, determined by the unloading capacity of the material and it is calculated by using the negative part of the reflected pulse.

_{t}(t)) and strain rate (pulse ε

_{r}(t)) takes place at the initial stage of specimen loading, however, after the point of maximum stress, the avalanche-like fracture process begins in the specimen. The stress after this point decreases, whereas the strain rate increases. So, although the amplitude of the incident pulse remains almost constant, the collapsing specimen does not completely pass the compression wave, its resistance to deformation is steadily decreasing.

_{s}(t) and ${\dot{\mathsf{\epsilon}}}_{s}(t)$, as well as the diagrams σ

_{s}(ε

_{s}) and ${\dot{\mathsf{\epsilon}}}_{s}({\mathsf{\epsilon}}_{s})$ themselves. The functions ${\dot{\mathsf{\epsilon}}}_{s}(t)$ and ${\dot{\mathsf{\epsilon}}}_{s}({\mathsf{\epsilon}}_{s})$ are represented on the graphs by dotted lines in the lower half of the figure field. The corresponding axis is located on the right side of the graphs. Digital markers on the lines are used to identify the curves and their mutual reference.

_{i}(t) and σ

_{s}(t) are shown.

_{s}(t) curve. These parameters for each curve are given in Table 2. When processing the experimental information for each diagram, in addition to the average strain rate ${\dot{\mathsf{\epsilon}}}_{s}$ of the sample, we determined the maximum values of the stress growth rate ${\dot{\sigma}}_{s}$ in the sample (Figure 8), which are also shown in Table 2.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**An example of strain pulses in measuring bars during registration (

**a**) and during synchronization (

**b**) when testing a ceramic specimen maintaining its visible integrity.

**Figure 3.**An example of strain pulses in measuring bars during registration (

**a**) and during synchronization (

**b**) when testing a ceramic specimen with its complete destruction.

**Figure 4.**Examples of functions σ

_{s}(t) and ${\dot{\mathsf{\epsilon}}}_{s}(t)$ (

**a**) and σ

_{s}(ε

_{s}) and ${\dot{\mathsf{\epsilon}}}_{s}({\mathsf{\epsilon}}_{s})$ (

**b**) for cases of maintaining integrity (curves 2) and complete destruction of a specimen (curves 1).

**Figure 5.**The parametric process of stress development in a specimen σ

_{s}in case of its destruction in comparison with a loading pulse σ

^{i}.

**Figure 11.**Definition of modules of load branches of ceramic diagrams for specimens of different lengths.

Sizes of Fractions, mm | |||||
---|---|---|---|---|---|

<0.05 | <0.2 | 0.2–0.315 | 0.315–0.4 | 0.63–1 | 1–2 |

40 | - | 20 | - | 40 | - |

Curve Number in the Diagram | The Module of the Load Branch, MPa | Average Strain Rate, 1/s | Stress Growth Rate, MPa/μs | Destruction Start Point | |||
---|---|---|---|---|---|---|---|

Strength, MPa | Strain, % | Time, μs | Energy Capacity, MJ/m^{3} | ||||

1 | 8000 | 220 | 1.9 | 71 | 1.5 | 82 | 0.93 |

2 | 10,625 | 400 | 6.5 | 80 | 1.4 | 40 | 2.42 |

3 | 14,000 | 1040 | 10.0 | 113 | 1.4 | 27 | 6.18 |

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**MDPI and ACS Style**

Bragov, A.M.; Lomunov, A.K.; Igumnov, L.A.; Belov, A.A.; Eremeyev, V.A.
The Influence of Specimen Geometry and Loading Conditions on the Mechanical Properties of Porous Brittle Media. *Materials* **2021**, *14*, 7144.
https://doi.org/10.3390/ma14237144

**AMA Style**

Bragov AM, Lomunov AK, Igumnov LA, Belov AA, Eremeyev VA.
The Influence of Specimen Geometry and Loading Conditions on the Mechanical Properties of Porous Brittle Media. *Materials*. 2021; 14(23):7144.
https://doi.org/10.3390/ma14237144

**Chicago/Turabian Style**

Bragov, Anatoly M., Andrey K. Lomunov, Leonid A. Igumnov, Aleksandr A. Belov, and Victor A. Eremeyev.
2021. "The Influence of Specimen Geometry and Loading Conditions on the Mechanical Properties of Porous Brittle Media" *Materials* 14, no. 23: 7144.
https://doi.org/10.3390/ma14237144