Discontinuous Deformation Analysis Enriched by the Bonding Block Model

<div>Contact springs between the elements <svg height="12.6569pt" id="M303" style="vertical-align:-3.42776pt" version="1.1" viewbox="-0.0498162 -9.22914 15.4614 12.6569" width="15.4614pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path d="M63 95C70 84 85 83 93 83C121 83 136 106 136 131C136 160 113 179 85 179C52 179 26 150 26 107C26 41 92 -15 199 -15C302 -15 399 18 484 106C535 159 576 245 617 333C684 348 739 389 774 457L758 467C722 411 681 381 633 368C679 468 726 563 779 603C807 603 888 604 888 655C888 667 875 675 859 675C848 675 818 671 783 648C758 651 718 656 665 656C605 656 537 649 482 616C434 585 396 536 396 471C396 392 448 345 520 330C511 311 502 292 493 271C400 61 306 16 181 16C110 16 61 68 61 95H63ZM729 601C648 547 595 474 536 362C465 370 437 423 437 462C437 518 477 560 522 580S625 604 676 604C696 604 714 604 729 603V601Z" id="g198-10"></path><glyph.data ascent="687" descent="-226" horiz-adv-x="897" vert-adv-y="897"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,12.153,3.264)"><path d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z" id="g50-106"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="280" vert-adv-y="280"></glyph.data></g></svg> and <svg height="14.9891pt" id="M304" style="vertical-align:-5.75996pt" version="1.1" viewbox="-0.0498162 -9.22914 16.8893 14.9891" width="16.8893pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path d="M63 95C70 84 85 83 93 83C121 83 136 106 136 131C136 160 113 179 85 179C52 179 26 150 26 107C26 41 92 -15 199 -15C302 -15 399 18 484 106C535 159 576 245 617 333C684 348 739 389 774 457L758 467C722 411 681 381 633 368C679 468 726 563 779 603C807 603 888 604 888 655C888 667 875 675 859 675C848 675 818 671 783 648C758 651 718 656 665 656C605 656 537 649 482 616C434 585 396 536 396 471C396 392 448 345 520 330C511 311 502 292 493 271C400 61 306 16 181 16C110 16 61 68 61 95H63ZM729 601C648 547 595 474 536 362C465 370 437 423 437 462C437 518 477 560 522 580S625 604 676 604C696 604 714 604 729 603V601Z" id="g198-10"></path><glyph.data ascent="687" descent="-226" horiz-adv-x="897" vert-adv-y="897"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,12.153,3.264)"><path d="M400 606C400 634 383 656 353 656C316 656 294 620 294 593C294 564 317 545 343 545C375 545 400 573 400 606ZM366 351C379 413 381 451 356 451C323 451 251 408 183 341L199 313C223 335 267 365 277 365C285 365 284 354 277 312C245 132 222 27 193 -100C182 -148 160 -188 131 -188C113 -188 90 -178 75 -170C64 -164 55 -168 48 -175C38 -185 24 -203 24 -222S48 -257 71 -257C89 -257 131 -241 186 -192C243 -141 286 -46 310 74L366 351Z" id="g50-107"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="430" vert-adv-y="430"></glyph.data></g></svg>. (a) The normal contact spring with the stiffness <svg height="9.52372pt" id="M305" style="vertical-align:-3.31356pt" version="1.1" viewbox="-0.0498162 -6.21016 13.8064 9.52372" width="13.8064pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z" id="g113-232"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="507" vert-adv-y="507"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.53,3.264)"><path d="M645 0V53H381V555H319V53H55V0H645Z" id="g54-183"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="699" vert-adv-y="699"></glyph.data></g></svg> and the shear contact spring with the stiffness <svg height="11.0849pt" id="M306" style="vertical-align:-4.87474pt" version="1.1" viewbox="-0.0498162 -6.21016 11.1506 11.0849" width="11.1506pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z" id="g113-232"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="507" vert-adv-y="507"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.53,3.264)"><path d="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="420" vert-adv-y="420"></glyph.data></g></svg>. (b) The normal contact distance <svg height="12.9915pt" id="M307" style="vertical-align:-3.3136pt" version="1.1" viewbox="-0.0498162 -9.6779 14.2008 12.9915" width="14.2008pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z" id="g113-105"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="516" vert-adv-y="516"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.923,3.264)"><path d="M645 0V53H381V555H319V53H55V0H645Z" id="g54-183"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="699" vert-adv-y="699"></glyph.data></g></svg> and the shear contact distance <svg height="14.5526pt" id="M308" style="vertical-align:-4.8747pt" version="1.1" viewbox="-0.0498162 -9.6779 11.5449 14.5526" width="11.5449pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z" id="g113-105"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="516" vert-adv-y="516"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.923,3.264)"><path d="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="420" vert-adv-y="420"></glyph.data></g></svg>.</div>

Mathematical Problems in Engineering

fig6

Figure 6

Figure 6: Discontinuous Deformation Analysis Enriched by the Bonding Block Model 