Elastoplastic Analysis of Circular Opening Based on a New Strain-Softening Constitutive Model and Its Engineering Application in Hydraulic Fracturing

<div>Stress distribution law for borehole expansion (note: because the maximum critical inner hydraulic pressure <svg height="12.0836pt" id="M153" style="vertical-align:-5.30063pt" version="1.1" viewbox="-0.0498162 -6.78297 38.8918 12.0836" width="38.8918pt" 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="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.384,3.132)"><path d="M384 109C349 76 310 59 268 59C197 59 123 113 120 245C240 248 367 259 384 262C403 266 407 275 407 296C407 376 338 451 254 451H253C201 451 148 425 105 378C65 334 39 272 39 203C39 88 112 -12 235 -12C266 -12 339 5 402 84L384 109ZM230 411C285 411 319 365 319 313C319 298 313 293 294 293C236 291 180 290 125 290C141 370 185 411 230 411Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.333,3.132)"><path d="M482 0V30C420 38 407 48 370 103L273 248C308 299 331 332 352 357C387 397 401 404 463 409V439H277V409C318 405 324 400 307 369C290 337 270 307 251 279L194 367C175 396 180 402 220 409V439H18V409C74 401 87 395 117 349L204 213C173 168 147 130 119 95C78 46 67 38 5 30V0H195V30C144 38 143 47 162 79C181 112 202 149 225 181L297 70C314 43 307 35 264 30V0H482Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,15.72,3.132)"><path d="M173 382V461C125 442 68 425 25 418V390C88 382 92 382 92 317V-127C92 -191 84 -199 18 -204V-235H259V-204C180 -196 173 -192 173 -120V7C188 -1 212 -11 243 -12C375 12 495 109 495 261C495 361 427 451 316 451C303 451 283 446 266 435L173 382ZM173 345C200 365 243 388 274 388C346 388 409 329 409 222C409 111 353 40 266 40C231 40 196 56 173 79V345Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,22.172,3.132)"><path d="M810 0V30C751 35 744 40 744 106V296C744 396 694 451 614 451C583 451 558 438 537 425C512 409 485 391 454 368C434 420 390 451 342 451C309 451 285 439 260 424C228 405 206 387 185 372V453C141 434 90 421 43 413V386C102 378 105 373 105 309V106C105 41 96 35 29 30V0H243V30C194 35 185 41 185 106V335C216 360 256 389 297 389C357 389 384 348 384 275V106C384 41 374 35 312 30V0H530V30C475 35 465 41 465 104V294C465 313 464 324 463 336C502 367 539 389 576 389C637 389 663 346 663 274V109C663 41 653 35 593 30V0H810Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.734,3.132)"><path d="M442 39L431 68C420 63 406 57 394 57C377 57 359 71 359 118V301C359 353 349 393 314 424C291 443 261 451 230 451C171 438 107 401 79 381C59 366 46 355 46 340C46 316 72 295 91 295C106 295 117 303 122 319C131 350 139 371 152 384C163 397 178 404 197 404C246 404 281 365 281 293V278C260 262 186 232 124 213C69 194 41 161 41 112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,33.82,3.132)"><path d="M482 0V30C420 38 407 48 370 103L273 248C308 299 331 332 352 357C387 397 401 404 463 409V439H277V409C318 405 324 400 307 369C290 337 270 307 251 279L194 367C175 396 180 402 220 409V439H18V409C74 401 87 395 117 349L204 213C173 168 147 130 119 95C78 46 67 38 5 30V0H195V30C144 38 143 47 162 79C181 112 202 149 225 181L297 70C314 43 307 35 264 30V0H482Z"></path></g></svg> is taken as the calculated inner pressure, the fracture zone does not appear).</div>

Advances in Civil Engineering

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Figure 4

Figure 4: Elastoplastic Analysis of Circular Opening Based on a New Strain-Softening Constitutive Model and Its Engineering Application in Hydraulic Fracturing 

Figure 4 | Elastoplastic Analysis of Circular Opening Based on a New Strain-Softening Constitutive Model and Its Engineering Application in Hydraulic Fracturing 