Derivation of Universal Curves for Nonlinear Soil Consolidation with Potential Constitutive Dependences

<table class="fixed-width table-group" id="tab4"><tr><td><table class="table"><colgroup><col style="width:7.95em"/><col style="width:4.27em"/><col style="width:7.27em"/><col style="width:4.27em"/><col style="width:6.05em"/></colgroup><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr class="thead"><td class="align_left">Universal curves</td><td class="align_center" colspan="2">J-B most general model</td><td class="align_center" colspan="2">Tabulated model</td></tr><tr class="thead"><td class="align_left"><svg height="10.3116pt" id="M306" style="vertical-align:-4.35069pt" version="1.1" viewbox="-0.0498162 -5.96091 16.3919 10.3116" width="16.3919pt" 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="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.434,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.157,3.132)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.314,3.132)"><path d="M327 325C324 350 312 408 304 433C283 442 256 451 213 451C120 451 59 391 59 320C59 243 125 210 188 181C237 158 266 136 266 93C266 52 231 24 195 24C134 24 93 81 74 148L42 143C42 106 51 37 59 23C78 8 125 -12 177 -12C257 -12 345 35 345 127C345 198 292 234 216 264C171 283 131 305 131 348C131 387 156 416 195 416C243 416 277 378 297 317L327 325Z"></path></g></svg> (yr)</td><td class="align_center"><svg height="10.3116pt" id="M307" style="vertical-align:-4.35069pt" version="1.1" viewbox="-0.0498162 -5.96091 16.3919 10.3116" width="16.3919pt" 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="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.434,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.157,3.132)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.314,3.132)"><path d="M327 325C324 350 312 408 304 433C283 442 256 451 213 451C120 451 59 391 59 320C59 243 125 210 188 181C237 158 266 136 266 93C266 52 231 24 195 24C134 24 93 81 74 148L42 143C42 106 51 37 59 23C78 8 125 -12 177 -12C257 -12 345 35 345 127C345 198 292 234 216 264C171 283 131 305 131 348C131 387 156 416 195 416C243 416 277 378 297 317L327 325Z"></path></g></svg> (yr)</td><td class="align_center">Rel. error %</td><td class="align_center"><svg height="10.3116pt" id="M308" style="vertical-align:-4.35069pt" version="1.1" viewbox="-0.0498162 -5.96091 16.3919 10.3116" width="16.3919pt" 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="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.434,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.157,3.132)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.314,3.132)"><path d="M327 325C324 350 312 408 304 433C283 442 256 451 213 451C120 451 59 391 59 320C59 243 125 210 188 181C237 158 266 136 266 93C266 52 231 24 195 24C134 24 93 81 74 148L42 143C42 106 51 37 59 23C78 8 125 -12 177 -12C257 -12 345 35 345 127C345 198 292 234 216 264C171 283 131 305 131 348C131 387 156 416 195 416C243 416 277 378 297 317L327 325Z"></path></g></svg> (yr)</td><td class="align_center">Rel. error %</td></tr><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr><td class="align_left">0.748</td><td class="align_center">0.750</td><td class="align_center">0.26</td><td class="align_center">0.817</td><td class="align_center">8.20</td></tr><tr class="table-tr"><td colspan="5"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Case study solution for the characteristic time <span class="nowrap"><svg height="10.3116pt" id="M305" style="vertical-align:-4.35069pt" version="1.1" viewbox="-0.0498162 -5.96091 16.3919 10.3116" width="16.3919pt" 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="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.434,3.132)"><path d="M262 451C169 451 38 375 38 210C38 98 121 -12 261 -12C360 -12 481 64 481 227C481 351 386 451 262 451ZM249 414C337 414 385 320 385 204C385 70 333 25 272 25C189 25 134 117 134 241C134 355 189 414 249 414Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.157,3.132)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.314,3.132)"><path d="M327 325C324 350 312 408 304 433C283 442 256 451 213 451C120 451 59 391 59 320C59 243 125 210 188 181C237 158 266 136 266 93C266 52 231 24 195 24C134 24 93 81 74 148L42 143C42 106 51 37 59 23C78 8 125 -12 177 -12C257 -12 345 35 345 127C345 198 292 234 216 264C171 283 131 305 131 348C131 387 156 416 195 416C243 416 277 378 297 317L327 325Z"></path></g></svg>.</span> Comparison with numerical models.</div>

Mathematical Problems in Engineering

tab4

Table 4

Table 4: Derivation of Universal Curves for Nonlinear Soil Consolidation with Potential Constitutive Dependences 