The Young’s Modulus and Poisson’s Ratio of Hard Coals in Laboratory Tests
Abstract
:1. Introduction
2. Materials and Methods
2.1. Specimens
Chronostratigraphic Division | Lithostratigraphic Units/Beds | No. of Seams | ||
---|---|---|---|---|
Carboniferous | Westphalian | Krakow Sandstone Series | Libiąż | 110–119 |
Łaziska | 201–215 | |||
Mudstone Series | Orzesze | 301–326 | ||
Załęże * | 327–406 | |||
Namurian | Upper Silesian Sandstone Series | Ruda | 407–419 | |
Saddle * | 501–510 | |||
Jejkowice | - | |||
Paralic Series | Poruba | 601–630 | ||
Jaklovec | 701–723 | |||
Hrušov | 801–848 | |||
Petřkovice | 901–915 |
2.2. Experimental Program
Parameter | Methodology |
---|---|
Applied stress condition: σ1 > 0, σ2 = σ3 Differential stress | Young’s modulus and stress parameters were determined in conventional triaxial compression tests, using a Karman pressure chamber and confining pressures of σ2 = σ3 = 10, 20, 30, and 50 MPa, strain rates of 5 × 10−5 to 10−4 s−1 and 10−1 s−1, and cylindrical specimens with a diameter of 30 mm and a slenderness of 3 [27]. |
Applied stress condition: σ1 > 0, σ2 = σ3 = 0 Uniaxial compressive strength Static Young’s modulus Static Poisson’s ratio | The specimens were tested in air-dry conditions. Specimen shape: cube with a base edge of 50 mm or cylinder with a diameter of 50 mm. Slenderness of the specimens: 1.0. An empirical factor of 0.89 was used to account for a specimen slenderness of over 2.0, as recommended by the ISRM. Load direction: perpendicular to the lamination. Piston rate: about 0.008 mm/s, i.e., the strain rate of rock in the area of mining excavations. Young’s modulus was determined over the entire height of the compressed rock specimen as a tangent of the inclination angle of the tangent to the x-axis, which is a linear approximation of the stress and strain characteristic. Poisson’s ratio was determined within the longitudinal elastic strains and using a roller chain with a sensor for recording and measuring circumferential strains [3,7,22,25,27]. |
Dynamic elastic moduli and conditions of testing | Confining pressure, 21.1 MPa; pore pressure, 8.8 MPa; effective pressure, 12.3 MPa; temperature, 37 °C. Core specimens with a diameter of 1.5 inches; a length depending on the condition of individual specimens was cut out perpendicular to the lamination. Before testing, specimens were saturated with a 2% potassium chloride solution in a vacuum chamber, at a pressure of 1 bar, for at least 12 h, before being additionally saturated with the same solution at a backpressure of at least 500 psi, into the core holder of the ultrasonic device [26,28]. |
3. Results and Discussion
- Maximum vertical stress, σ1 (uniaxial compressive strength, UCS), and differential stress, σ1–σ3;
- Young’s modulus: static, Est, and dynamic, Edyn;
- Poisson’s ratio: static νst and dynamic νdyn;
- Bulk density, ρo;
- Total porosity, P.
4. Conclusions
- The geologic age of coal, as well as its petrographic structure, had a significant influence on the dependence between uniaxial compressive strength and the static Young’s modulus determined on the basis of the stress and strain curve (R2 = 0.76–0.96).
- The functional dependences determined on the basis of tests conducted under a complex state of stress and at various confining pressures and high strain rates were characterized by a very high correlation between the differential stress and the static Young’s modulus.
- A strong and very strong correlation between the S-wave velocity and the elastic parameters (R2 = 0.76–0.96) were obtained on the basis of the conducted laboratory ultrasonic tests. However, no correlation was observed between the P-wave velocity and the elastic parameters, which also confirms the results of experiments conducted by other researchers. The determined values of VP/VS for carboniferous coals revealed a strong correlation with the dynamic Young’s modulus (R2 = 0.65) and a very strong linear dependence with the dynamic Poisson’s ratio (R2 = 0.89).
- Testing of the studied carboniferous coals demonstrated a weak linear correlation between porosity and the Young’s modulus and Poisson’s ratio.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
UCS | uniaxial compressive strength |
USCB | Upper Silesian Coal Basin |
ISRM | International Society for Rock Mechanics |
Est | static Young’s modulus |
νst | static Poisson’s ratio |
σ1 | axial stress, vertical stress |
σ2, σ3 | radial stress, horizontal stress |
ρo | bulk density |
P | total porosity |
VP | P-wave velocity |
VS | S-wave velocity |
Edyn | dynamic Young’s modulus |
νdyn | dynamic Poisson’s ratio |
BI | brittleness index |
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Number of Samples | UCS | Est | νst | |||
---|---|---|---|---|---|---|
Mean Values MPa | SD MPa | Mean Values MPa | SD MPa | Mean Values - | SD - | |
4 | 11.2 | 2.2 | 1069 | 327 | 0.25 | 0.06 |
4 | 5.9 | 1.3 | 613 | 175 | 0.28 | 0.04 |
4 | 6.3 | 2.1 | 638 | 303 | * | - |
3 | 6.6 | 0.1 | 836 | 16 | * | - |
4 | 14.2 | 1.7 | 1585 | 151 | 0.27 | 0.05 |
3 | 11.1 | 2.9 | 1120 | 227 | 0.24 | 0.03 |
1 | 7.8 | - | 768 | - | * | - |
1 | 5.5 | - | 536 | - | * | - |
3 | 10.5 | 1.5 | 1239 | 246 | * | - |
4 | 6.8 | 2.4 | 606 | 331 | * | - |
3 | 7.1 | 5.8 | 814 | 766 | 0.22 | 0.06 |
3 | 8.9 | 0.7 | 601 | 159 | 0.26 | 0.06 |
3 | 9.0 | 2.1 | 942 | 309 | 0.32 | 0.08 |
2 | 4.7 | 1.4 | 573 | 276 | 0.34 | 0.08 |
2 | 4.6 | 0.6 | 389 | 75 | 0.29 | 0.09 |
4 | 8.0 | 2.9 | 945 | 372 | 0.22 | 0.10 |
4 | 7.1 | 2.6 | 784 | 421 | 0.32 | 0.05 |
4 | 8.5 | 1.2 | 1031 | 230 | 0.29 | 0.02 |
2 | 8.1 | 2.1 | 900 | 324 | 0.38 | 0.01 |
2 | 5.5 | 2.8 | 654 | 542 | 0.29 | 0.04 |
4 | 6.3 | 0.8 | 619 | 248 | 0.28 | 0.05 |
4 | 4.5 | 2.1 | 503 | 127 | 0.31 | 0.08 |
4 | 4.5 | 3.0 | 673 | 479 | 0.27 | 0.10 |
4 | 7.4 | 2.8 | 948 | 308 | 0.25 | 0.08 |
3 | 5.7 | 0.6 | 656 | 68 | 0.16 | 0.02 |
2 | 4.8 | 1.4 | 523 | 146 | 0.16 | 0.06 |
2 | 11.0 | 2.4 | 1033 | 394 | 0.28 | 0.06 |
2 | 9.2 | 0.2 | 1066 | 155 | 0.39 | 0.06 |
4 | 7.5 | 2.9 | 734 | 470 | 0.27 | 0.11 |
4 | 26.6 | 4.4 | 2399 | 382 | 0.30 | 0.05 |
3 | 19.1 | 1.9 | 1751 | 151 | 0.37 | 0.02 |
2 | 23.2 | 0.8 | 1903 | 96 | 0.24 | 0.12 |
3 | 15.2 | 1.1 | 1233 | 178 | 0.23 | 0.04 |
3 | 8.8 | 2.1 | 765 | 251 | 0.37 | 0.02 |
2 | 10.9 | 0.4 | 1061 | 404 | 0.24 | 0.12 |
3 | 26.3 | 5.7 | 2073 | 402 | 0.36 | 0.05 |
3 | 27.8 | 7.7 | 2371 | 340 | 0.31 | 0.05 |
4 | 28.2 | 3.0 | 2312 | 348 | 0.30 | 0.08 |
3 | 24.4 | 2.8 | 2372 | 149 | 0.33 | 0.02 |
3 | 21.1 | 3.5 | 2026 | 367 | 0.34 | 0.08 |
2 | 23.7 | 11.1 | 2234 | 612 | 0.30 | 0.08 |
4 | 16.8 | 2.7 | 1794 | 220 | 0.20 | 0.08 |
3 | 17.6 | 0.9 | 1872 | 125 | 0.24 | 0.12 |
Number of Samples | VP m/s | VS m/s | VP/VS - | Edyn GPa | νdyn - |
---|---|---|---|---|---|
1 | 2409 | 1200 | 2.01 | 5.21 | 0.34 |
1 | 2399 | 1259 | 1.91 | 5.63 | 0.31 |
1 | 2435 | 1250 | 1.95 | 5.41 | 0.32 |
1 | 2392 | 959 | 2.49 | 3.37 | 0.40 |
1 | 2488 | 1299 | 1.92 | 6.15 | 0.31 |
1 | 2421 | 1249 | 1.94 | 5.71 | 0.32 |
1 | 2441 | 1104 | 2.21 | 4.36 | 0.37 |
1 | 2441 | 1271 | 1.92 | 5.50 | 0.31 |
1 | 2481 | 1283 | 1.93 | 5.61 | 0.32 |
1 | 2425 | 1154 | 2.10 | 4.75 | 0.35 |
1 | 2098 | 1034 | 2.03 | 3.74 | 0.34 |
1 | 2389 | 1273 | 1.88 | 5.56 | 0.30 |
1 | 2441 | 1509 | 1.62 | 7.09 | 0.19 |
1 | 2504 | 1377 | 1.82 | 6.33 | 0.28 |
1 | 2530 | 1537 | 1.65 | 7.77 | 0.21 |
1 | 2491 | 1501 | 1.66 | 7.24 | 0.21 |
1 | 1982 | 1150 | 1.72 | 4.34 | 0.25 |
1 | 2500 | 1175 | 2.09 | 5.50 | 0.35 |
1 | 2446 | 1289 | 1.90 | 5.89 | 0.31 |
1 | 2457 | 1416 | 1.73 | 6.52 | 0.25 |
1 | 2503 | 1332 | 1.88 | 6.12 | 0.30 |
1 | 2533 | 1364 | 1.86 | 6.41 | 0.30 |
1 | 2472 | 1130 | 2.19 | 4.52 | 0.37 |
Lithostratigraphic Series | Coal Lithotype | UCS stress Condition σ1 > 0, σ2 = σ3 = 0 MPa | Static Young’s Modulus MPa | Static Poisson’s Ratio - | Bulk Density kg/m3 |
---|---|---|---|---|---|
Mudstone Series | Vitrain | 4.5–11.2 | 389–1069 | 0.16–0.34 | 1112–1376 |
Clarain | 5.5–11.1 | 601–1239 | 0.24–0.39 | 1220–1302 | |
Durain/vitrain | 8.0–14.2 | 942–1585 | 0.22–0.32 | 1229–1366 | |
Upper Silesian Sandstone Series | Vitrain | 8.8–19.1 | 765–1751 | 0.23–0.37 | 1268–1490 |
Clarain | 21.1–28.2 | 1903–2399 | 0.24–0.36 | 1262–1384 |
Coals | Mudstone Series | Upper Silesian Sandstone Series |
---|---|---|
Vitrain | Very low and low strength | Very low and low strength |
Clarain | High strength | |
Durain/vitrain | Not present in the rock mass in the sampling regions |
Relationship | R2 Coefficient | Estimated SSE Standard Error |
---|---|---|
Mudstone Series Est = 94.824UCS + 107.18 | 0.7623 | 128.79 |
Upper Silesian Sandstone Series Est = 79.592UCS + 216.47 | 0.9278 | 159.44 |
Vitrain Est = 80.56UCS + 179.52 | 0.8842 | 106.55 |
Clarain and durain Est = 78.905UCS + 255.48 | 0.9386 | 170.55 |
Relationship | R2 Coefficient | Estimated SSE Standard Error |
---|---|---|
Est = 11,471(σ1–σ3) + 2417.9 strain rates = 10−5 to 10−4 s−1; confining pressures of 10, 20, 30, and 50 MPa | 0.9543 | 83.78 |
Est = 5932(σ1–σ3) + 2878.1 strain rates = 10−1 s−1; confining pressures of 10, 20, 30, and 50 MPa | 0.8455 | 106.62 |
Parameter | SI Unit | Min. Value | Max. Value |
---|---|---|---|
Compressional (P) wave velocity | m/s | 1982 | 2533 |
Shear (S) wave velocity | m/s | 959 | 1537 |
VP/VS | - | 1.62 | 2.49 |
Dynamic Young’s modulus | GPa | 3.370 | 7.770 |
Dynamic Poisson’s ratio | - | 0.19 | 0.40 |
Porosity | % | 1.96 | 8.72 |
Bulk density | kg/m3 | 1290 | 1390 |
Relationship | R2 Coefficient | Estimated SSE Standard Error |
---|---|---|
Edyn = 0.0073VS − 3.637 | 0.9643 | 0.2100 |
Edyn = −4.3609(VP/VS) + 14.017 | 0.6485 | 0.6590 |
νdyn = 0.0003VS + 0.7072 | 0.7579 | 0.0270 |
νdyn = 0.2524 (VP/VS) − 0.1825 | 0.8990 | 0.0170 |
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Bukowska, M.; Kasza, P.; Moska, R.; Jureczka, J. The Young’s Modulus and Poisson’s Ratio of Hard Coals in Laboratory Tests. Energies 2022, 15, 2477. https://doi.org/10.3390/en15072477
Bukowska M, Kasza P, Moska R, Jureczka J. The Young’s Modulus and Poisson’s Ratio of Hard Coals in Laboratory Tests. Energies. 2022; 15(7):2477. https://doi.org/10.3390/en15072477
Chicago/Turabian StyleBukowska, Mirosława, Piotr Kasza, Rafał Moska, and Janusz Jureczka. 2022. "The Young’s Modulus and Poisson’s Ratio of Hard Coals in Laboratory Tests" Energies 15, no. 7: 2477. https://doi.org/10.3390/en15072477
APA StyleBukowska, M., Kasza, P., Moska, R., & Jureczka, J. (2022). The Young’s Modulus and Poisson’s Ratio of Hard Coals in Laboratory Tests. Energies, 15(7), 2477. https://doi.org/10.3390/en15072477