Modeling Crack Propagation in Polycrystalline Microstructure Using Variational Multiscale Method

<div>Representation of crack opening <svg height="11.439pt" id="M15" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 19.9207 11.439" width="19.9207pt" 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="M427 -163V-135C320 -126 318 -122 318 -44V583C318 662 321 666 427 675V703H120V-163H427ZM268 -126C183 -117 181 -99 181 -44V583C181 638 184 657 268 666V-126Z" id="g113-35"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="482" vert-adv-y="482"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.266,0)"><path d="M542 56L520 58C483 62 478 72 478 124V449C455 445 422 441 387 438C351 435 316 432 287 431V402L318 395C345 389 350 383 350 336V93C328 71 306 65 279 65C245 65 211 88 211 153V298C211 356 211 407 214 449C187 445 151 441 118 438C84 435 53 432 28 431V402L51 395C77 388 83 383 83 337V130C83 42 131 -12 214 -12C245 -12 272 -1 300 18C322 31 333 41 350 55V-5L359 -12C389 -5 420 2 452 8C485 15 513 20 542 24V56Z" id="g13-115"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="555" vert-adv-y="555"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,13.481,0)"><path d="M363 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H363ZM302 -43C302 -98 299 -117 215 -126V667C299 657 302 639 302 584V-43Z" id="g113-37"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="482" vert-adv-y="482"></glyph.data></g></svg>, domain <svg height="11.927pt" id="M16" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 15.1523 11.927" width="15.1523pt" 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="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z" id="g113-153"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="757" vert-adv-y="757"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,9.841,3.132)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z" id="g50-111"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="512" vert-adv-y="512"></glyph.data></g></svg>, and crack surface <svg height="13.8999pt" id="M17" style="vertical-align:-3.94351pt" version="1.1" viewbox="-0.0498162 -9.95639 10.9676 13.8999" width="10.9676pt" 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="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z" id="g113-132"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="507" vert-adv-y="507"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.595,-5.741)"><path d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z" id="g50-100"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="411" vert-adv-y="411"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,5.486,3.784)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z" id="g50-111"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="512" vert-adv-y="512"></glyph.data></g></svg>. Crack surface has normal direction <svg height="6.089pt" id="M18" style="vertical-align:-0.04982042pt" version="1.1" viewbox="-0.0498162 -6.03918 7.48522 6.089" width="7.48522pt" 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="M549 0V32C497 40 491 42 491 105V294C491 394 441 454 360 454C331 454 312 445 282 426C260 412 240 396 220 383V456C161 439 95 425 39 418V390C87 380 92 377 92 317V109C92 42 88 41 26 32V0H267V32C225 38 220 42 220 104V346C246 366 268 377 294 377C330 377 363 355 363 283V113C363 42 358 38 316 32V0H549Z" id="g13-108"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="566" vert-adv-y="566"></glyph.data></g></svg> and tangent direction <svg height="6.089pt" id="M19" style="vertical-align:-0.04982042pt" version="1.1" viewbox="-0.0498162 -6.03918 11.0997 6.089" width="11.0997pt" 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="M823 0V32C771 40 767 43 767 100V293C767 395 719 454 632 454C603 454 576 441 554 426C530 409 507 394 482 376C464 421 424 454 368 454C339 454 312 442 281 422C259 409 240 395 218 379V456C157 439 89 425 42 418V389C89 382 93 373 93 315V105C93 42 88 39 30 32V0H265V32C227 38 220 43 220 100V344C248 366 273 376 297 376C334 376 366 354 366 282V100C366 41 362 38 315 32V0H546V32C501 38 494 43 494 100V302C494 317 493 331 492 342C524 369 549 376 569 376C601 376 639 359 639 277V105C639 42 632 38 588 32V0H823Z" id="g13-107"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="843" vert-adv-y="843"></glyph.data></g></svg>. Nominal traction on the crack surface is denoted as <svg height="10.1889pt" id="M20" style="vertical-align:-0.2325096pt" version="1.1" viewbox="-0.0498162 -9.95639 11.7505 10.1889" width="11.7505pt" 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="M329 37L319 72C301 64 285 60 268 60C248 60 212 69 212 147V394H304C320 402 321 431 310 442H212V586L198 593L84 515V442H44L16 406L21 394H84V114C84 24 130 -12 207 -12C214 -12 230 -8 247 -1L329 37Z" id="g13-114"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="339" vert-adv-y="339"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,4.407,-5.741)"><path d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z" id="g50-100"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="411" vert-adv-y="411"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,8.659,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g113-47"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g></svg></div>

Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: Modeling Crack Propagation in Polycrystalline Microstructure Using Variational Multiscale Method 