Development of Transversely Isotropic Elastoplastic Constitutive Model in FLAC<sup>3D</sup> and Its Application in Tunnel Engineering

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr class="thead"><td class="align_left">AN-MC model</td><td class="align_center">Functions</td></tr><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr><td class="align_left">Elastic relationship</td><td class="align_center"><span style="width: 84.894ptpx;"><svg height="18.3157pt" id="M1" style="vertical-align:-5.8304pt" version="1.1" viewbox="-0.0498162 -12.4853 84.894 18.3157" width="84.894pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,2.0537e-05)"><path d="M363 -410V-380C268 -368 256 -319 256 -264C256 -189 277 -5 277 130C277 195 252 252 189 269V273C253 288 277 346 277 410C277 542 256 679 256 809C256 861 268 908 363 920V950C295 950 197 933 197 793C197 667 208 520 208 391C208 349 204 294 87 287V254C204 246 208 192 208 150C208 26 197 -118 197 -257S295 -410 363 -410Z"></path></g><g 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d="M351 254V287C234 294 230 349 230 391C230 520 241 667 241 793C241 933 143 950 75 950V920C170 908 182 861 182 809C182 679 161 542 161 410C161 346 185 288 249 273V269C186 252 161 195 161 130C161 -5 182 -189 182 -264C182 -319 170 -368 75 -380V-410C143 -410 241 -396 241 -257S230 26 230 150C230 192 234 246 351 254Z"></path></g></svg></span></td></tr><tr><td class="align_left">Yield criteria</td><td class="align_center">Shear yield criterion: <svg height="15.4854pt" id="M2" style="vertical-align:-5.52901pt" version="1.1" viewbox="-0.0498162 -9.95639 131.032 15.4854" width="131.032pt" 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="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.352,-5.741)"><path d="M357 391C357 416 325 451 270 451C241 451 179 431 146 400C109 365 97 334 97 307C97 248 143 213 196 180C250 146 261 123 261 101C261 75 237 40 190 40C153 40 110 72 86 109C81 116 68 119 55 112C37 102 24 87 24 66C24 30 85 -12 137 -12C229 -12 331 62 331 140C331 189 304 218 236 260C197 284 164 309 164 346C164 381 194 400 220 400C259 400 282 380 304 350C310 342 321 339 330 344C347 353 357 372 357 391Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.962,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.225,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 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204C24 97 93 14 213 -8L188 -131C170 -220 197 -250 217 -257L224 -254L278 -7C427 -3 552 98 552 243ZM478 228C478 124 408 37 287 40L363 405C401 395 478 336 478 228Z"></path></g><g transform="matrix(.013,0,0,-0.013,82.807,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,93.343,0)"><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,-0.013,99.583,0)"><path d="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z"></path></g><g transform="matrix(.013,0,0,-0.013,104.865,.013)"><path d="M782 731H740L476 -79H474L280 352L133 274L151 239L242 286L473 -219L782 731Z"></path></g><g transform="matrix(.013,0,0,-0.013,114.835,0)"><path d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,124.897,3.132)"><path d="M552 243C552 372 474 431 385 468C378 471 378 492 382 510L419 675L401 683L345 637L224 54C173 71 118 128 118 218C118 323 174 382 270 423L255 458C130 420 24 339 24 204C24 97 93 14 213 -8L188 -131C170 -220 197 -250 217 -257L224 -254L278 -7C427 -3 552 98 552 243ZM478 228C478 124 408 37 287 40L363 405C401 395 478 336 478 228Z"></path></g></svg><br/>Tensile yield criterion: <svg height="14.6007pt" id="M3" style="vertical-align:-3.429501pt" version="1.1" viewbox="-0.0498162 -11.1712 62.7233 14.6007" width="62.7233pt" 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="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.352,-5.741)"><path d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z"></path></g><g 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375V377Z"></path></g><g transform="matrix(.013,0,0,-0.013,41.526,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,52.063,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,58.823,3.132)"><path d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z"></path></g></svg></td></tr><tr><td class="align_left">Plastic potential functions</td><td class="align_center">Shear plastic potential function: <svg height="15.4854pt" id="M4" style="vertical-align:-5.52901pt" version="1.1" viewbox="-0.0498162 -9.95639 80.0869 15.4854" width="80.0869pt" 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="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.398,-5.741)"><path d="M357 391C357 416 325 451 270 451C241 451 179 431 146 400C109 365 97 334 97 307C97 248 143 213 196 180C250 146 261 123 261 101C261 75 237 40 190 40C153 40 110 72 86 109C81 116 68 119 55 112C37 102 24 87 24 66C24 30 85 -12 137 -12C229 -12 331 62 331 140C331 189 304 218 236 260C197 284 164 309 164 346C164 381 194 400 220 400C259 400 282 380 304 350C310 342 321 339 330 344C347 353 357 372 357 391Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.008,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,26.271,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,33.031,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g><g transform="matrix(.013,0,0,-0.013,40.883,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,51.42,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,58.18,3.132)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"></path></g><g transform="matrix(.013,0,0,-0.013,63.126,0)"><path d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,73.188,3.132)"><path d="M659 416L649 451C600 438 564 414 546 396C527 377 518 354 506 308C495 265 474 200 456 152C437 102 400 45 326 34L424 510L415 514L351 494L266 44C213 55 182 94 182 174C182 200 187 243 191 277C194 304 196 331 196 357C196 415 178 451 138 451C108 451 66 431 24 386L32 355C49 370 66 377 81 377C91 377 110 370 110 328C110 300 107 269 104 240C101 214 99 188 99 162C99 37 166 -5 257 -13L220 -202C215 -228 229 -251 236 -257L274 -218L318 -11C349 -4 378 8 400 21C444 48 489 89 518 157L586 327C601 369 619 393 659 416Z"></path></g></svg><br/>Tensile plasticity potential function: <svg height="14.6007pt" id="M5" style="vertical-align:-3.429501pt" version="1.1" viewbox="-0.0498162 -11.1712 45.5067 14.6007" width="45.5067pt" 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="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z"></path></g><g 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<div>Basic equation of the AN-MC constitutive model.</div>

Geofluids

tab1

Table 1

Table 1: Development of Transversely Isotropic Elastoplastic Constitutive Model in FLAC<sup>3D</sup> and Its Application in Tunnel Engineering 

Geofluids

Development of Transversely Isotropic Elastoplastic Constitutive Model in FLAC3D and Its Application in Tunnel Engineering

Table 1

Development of Transversely Isotropic Elastoplastic Constitutive Model in FLAC^3D and Its Application in Tunnel Engineering