A Mathematical Model for Calculating the “Brittleness-Ductility” Drop Coefficient of Sandstone in Mining Zones

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="8"><hr/></td></tr><tr class="thead"><td class="align_left">Pressures (MPa)</td><td class="align_center"><i>S</i><sub>BAC</sub>-<i>S</i><sub>EDF</sub> (10<sup>3</sup> J/m<sup>3</sup>)</td><td class="align_center"><svg height="12.7178pt" id="M59" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 71.3582 12.7178" width="71.3582pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M538 96L524 119C492 88 454 63 446 63C439 63 434 70 440 101C463 223 491 341 518 448H508L433 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329 341 329C355 329 387 352 387 375Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,22.959,3.132)"><path d="M43 650V622C120 616 128 612 128 526V124C128 39 120 33 34 27V0H270C392 0 492 25 567 83C643 141 690 230 690 350C690 444 655 517 605 565C543 625 450 650 323 650H43ZM213 547C213 587 217 598 226 604C236 612 262 617 304 617C371 617 429 604 474 576C554 529 592 439 592 336C592 176 505 36 319 36C246 36 213 55 213 131V547Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.076,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,43.613,0)"><path d="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 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404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z"></path></g></svg> (10<sup>3</sup> J/m<sup>3</sup>)</td><td class="align_center"><svg height="20.427pt" id="M60" style="vertical-align:-7.7173pt" version="1.1" viewbox="-0.0498162 -12.7097 45.8576 20.427" width="45.8576pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,.007)"><path d="M480 853C480 879 456 900 424 900C303 900 237 760 239 491L245 -163C246 -299 234 -327 208 -327C187 -327 171 -289 137 -289C112 -289 90 -309 90 -335C90 -363 116 -389 155 -389C277 -389 342 -245 338 99L332 673C330 820 346 844 370 844C396 844 396 808 432 808C460 808 480 826 480 853Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.438,-8.495)"><path d="M396 374C396 399 365 451 262 451C173 451 84 405 84 327C84 295 107 257 157 243V241C91 226 24 184 24 119C24 41 92 -12 189 -12C272 -12 344 31 386 91L367 118C325 76 274 50 222 50C163 50 120 84 120 138C120 191 169 218 223 218C247 218 263 217 278 216L312 260L308 267H259C200 267 167 297 167 338S202 414 250 414C297 414 326 387 336 341C338 331 341 328 349 328C361 328 396 349 396 374Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,11.032,-6.394)"><path d="M43 650V622C120 616 128 612 128 526V124C128 39 120 33 34 27V0H270C392 0 492 25 567 83C643 141 690 230 690 350C690 444 655 517 605 565C543 625 450 650 323 650H43ZM213 547C213 587 217 598 226 604C236 612 262 617 304 617C371 617 429 604 474 576C554 529 592 439 592 336C592 176 505 36 319 36C246 36 213 55 213 131V547Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.438,5.567)"><path d="M396 374C396 399 365 451 262 451C173 451 84 405 84 327C84 295 107 257 157 243V241C91 226 24 184 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274Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,23.597,3.132)"><path d="M481 445L456 468C426 437 403 427 388 427C372 427 356 428 302 438C256 447 221 451 187 451C159 451 130 428 105 384C93 363 80 340 65 305L96 290C122 342 149 381 193 381C216 381 267 377 294 374S346 372 365 376L75 105C37 70 24 35 24 16C24 2 30 -5 35 -8C42 -8 46 -6 48 -2C58 14 75 35 90 41C102 46 116 46 147 32C200 8 267 -21 298 -21C347 -21 390 12 448 126L420 151C369 77 338 56 311 56C287 56 249 68 212 81C180 92 149 100 127 98C212 172 353 308 413 372L481 445Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.708,0)"><path d="M517 51L485 54C448 58 441 63 441 115V712C404 700 337 684 285 678V653C357 648 362 645 362 580V437C339 446 309 449 295 449C159 449 38 340 38 201C38 61 143 -12 223 -12C234 -12 261 -6 301 17L362 53V-12C420 9 495 22 517 26V51ZM362 85C338 67 301 51 266 51C201 51 128 109 128 228C128 373 212 411 259 411C296 411 338 395 362 360V85Z"></path></g><g transform="matrix(.013,0,0,-0.013,35.598,0)"><path d="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 375Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,40.603,3.132)"><path d="M481 445L456 468C426 437 403 427 388 427C372 427 356 428 302 438C256 447 221 451 187 451C159 451 130 428 105 384C93 363 80 340 65 305L96 290C122 342 149 381 193 381C216 381 267 377 294 374S346 372 365 376L75 105C37 70 24 35 24 16C24 2 30 -5 35 -8C42 -8 46 -6 48 -2C58 14 75 35 90 41C102 46 116 46 147 32C200 8 267 -21 298 -21C347 -21 390 12 448 126L420 151C369 77 338 56 311 56C287 56 249 68 212 81C180 92 149 100 127 98C212 172 353 308 413 372L481 445Z"></path></g></svg> (10<sup>3</sup> J/m<sup>3</sup>)</td><td class="align_center"><i>S</i><sub>CADF</sub> (10<sup>3</sup> J/m)</td><td class="align_center"><i>H</i></td><td class="align_center"><span style="width: 16.2018ptpx;"><svg height="8.8423pt" id="M61" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 16.2018 8.8423" width="16.2018pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.662,0)"><path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z"></path></g></svg></span></td><td class="align_center"><i>θ</i></td></tr><tr><td class="thead-hr" colspan="8"><hr/></td></tr><tr><td class="align_left">4</td><td class="align_center">29.95</td><td class="align_center">6.19</td><td class="align_center">100.8</td><td class="align_center">60.82</td><td class="align_center">0.314</td><td class="align_center">1.82</td><td class="align_center">—</td></tr><tr><td class="align_left">6</td><td class="align_center">24.78</td><td class="align_center">9.29</td><td class="align_center">110.9</td><td class="align_center">70.06</td><td class="align_center">0.377</td><td class="align_center">2.45</td><td class="align_center">56°</td></tr><tr><td class="align_left">8</td><td class="align_center">10.07</td><td class="align_center">12.38</td><td class="align_center">118.8</td><td class="align_center">71.30</td><td class="align_center">0.381</td><td class="align_center">5.85</td><td class="align_center">75°</td></tr><tr><td class="align_left">10</td><td class="align_center">15.56</td><td class="align_center">15.48</td><td class="align_center">165.5</td><td class="align_center">82.36</td><td class="align_center">0.393</td><td class="align_center">4.30</td><td class="align_center">82°</td></tr><tr class="table-tr"><td colspan="8"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Calculation of sandstone energy drop coefficients.</div>

Discrete Dynamics in Nature and Society

tab1

Table 1

Table 1: A Mathematical Model for Calculating the “Brittleness-Ductility” Drop Coefficient of Sandstone in Mining Zones 