Evolution of the Mesoscopic Parameters and Mechanical Properties of Granular Materials upon Loading

<div>Two particles interact at contact. The branch vector <svg height="15.8442pt" id="M5" style="vertical-align:-4.2006pt" version="1.1" viewbox="-0.0498162 -11.6436 14.1188 15.8442" width="14.1188pt" 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="M238 681C243 705 239 712 230 712C217 712 156 682 75 674L70 648H105C148 648 153 641 144 598L39 110C18 11 35 -12 55 -12C90 -12 166 36 221 103L205 125C174 93 130 65 118 65C112 65 108 68 114 96L238 681Z" id="g113-109"></path></g><g transform="matrix(.0091,0,0,-0.0091,3.498,-6.844)"><path d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z" id="g50-113"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.932,-6.844)"><path d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z" id="g50-114"></path></g><g transform="matrix(.0091,0,0,-0.0091,3.172,4.041)"><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></g></svg> is the vector connecting the centers of two particles that are in contact. The corresponding unit normal vector at the contact is <span class="nowrap"><svg height="15.8442pt" id="M6" style="vertical-align:-4.2006pt" version="1.1" viewbox="-0.0498162 -11.6436 17.1592 15.8442" width="17.1592pt" 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="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z" id="g113-111"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.526,-6.844)"><path d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z" id="g50-113"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.96,-6.844)"><path d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z" id="g50-114"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.461,4.041)"><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></g></svg>.</span></div>

Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: Evolution of the Mesoscopic Parameters and Mechanical Properties of Granular Materials upon Loading 