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Table 3 Estimated \(\mu_{\text{d}}\), assumed \(\mu_{\text{s}}\), and calculated \(S\) and \(L_{\text{c}}\)

From: 3-D dynamic rupture simulations of the 2016 Kumamoto, Japan, earthquake

Case name \(\sigma_{1}\) \(\sigma_{3}\) \(\sigma_{\text{n}}^{\text{r}}\) (MPa) \(\tau^{\text{r}}\) (MPa) \(\mu_{\text{d}}\) \(\mu_{\text{s}}\) \(S\) \({{L_{\text{c}}^{\text{II}} } \mathord{\left/ {\vphantom {{L_{\text{c}}^{\text{II}} } {D_{\text{c}} }}} \right. \kern-0pt} {D_{\text{c}} }}\) d
Hinagu Futagawa Hinagu Futagawa Hinagu (\(S_{\text{H}}\)) Futagawa Hinagu Futagawa
Case A 100 50 77.262 63.077 23.956 19.632 0.261 0.4a
        0.349 0.8b 0.758 \(6.3 \times 10^{3}\) \(7.4 \times 10^{3}\)
        0.369 1.2c 1.148 \(7.8 \times 10^{3}\) \(9.1 \times 10^{3}\)
        0.389 1.6 1.539 \(9.2 \times 10^{3}\) \(1.1 \times 10^{4}\)
        0.408 2.0 1.930 \(1.1 \times 10^{4}\) \(1.2 \times 10^{4}\)
Case B 100 70 86.357 77.846 14.373 11.779 0.116 0.187 0.4 1.000 \(4.3 \times 10^{3}\) \(9.7 \times 10^{3}\)
        0.207 0.8 1.571 \(5.5 \times 10^{3}\) \(1.2 \times 10^{4}\)
        0.227 1.2 2.143 \(6.7 \times 10^{3}\) \(1.5 \times 10^{4}\)
        0.247 1.6 2.714 \(8.0 \times 10^{3}\) \(1.8 \times 10^{4}\)
        0.267 2.0 3.285 \(9.2 \times 10^{3}\) \(2.1 \times 10^{4}\)
Case C 300 160 236.334 196.616 67.076 54.969 0.266 0.4a
        0.8a
        1.2a
        0.312 1.6 2.413 \(8.2 \times 10^{3}\) \(1.7 \times 10^{4}\)
        0.319 2.0 2.938 \(9.5 \times 10^{3}\) \(2.0 \times 10^{4}\)
Case D 300 260 281.810 270.462 19.165 15.705 0.050 0.0752 0.4 2.12407 \(3.7 \times 10^{3}\) \(1.9 \times 10^{4}\)
        0.0824 0.8 3.016662 \(4.7 \times 10^{3}\) \(2.5 \times 10^{4}\)
        0.0896 1.2 3.909253 \(5.8 \times 10^{3}\) \(3.0 \times 10^{4}\)
        0.0968 1.6 4.801845 \(6.8 \times 10^{3}\) \(3.5 \times 10^{4}\)
        0.104 2.0 5.694436 \(7.9 \times 10^{3}\) \(4.1 \times 10^{4}\)
Case E 500 290 404.502 344.924 100.614 82.454 0.235 0.4a
        0.8a
        0.265 1.2 6.464 \(5.3 \times 10^{3}\) \(7.1 \times 10^{4}\)
        0.271 1.6 7.822 \(6.2 \times 10^{3}\) \(8.4 \times 10^{4}\)
        0.276 2.0 9.179 \(7.2 \times 10^{3}\) \(9.7 \times 10^{4}\)
Case F 500 400 454.525 426.154 47.912 39.264 0.090 0.112 0.4 9.106 \(2.7 \times 10^{3}\) \(1.5 \times 10^{5}\)
        0.118 0.8 11.994 \(3.4 \times 10^{3}\) \(1.9 \times 10^{5}\)
        0.124 1.2 14.881 \(4.2 \times 10^{3}\) \(2.3 \times 10^{5}\)
        0.130 1.6 17.768 \(5.0 \times 10^{3}\) \(2.8 \times 10^{5}\)
        0.136 2.0 20.656 \(5.7 \times 10^{3}\) \(3.2 \times 10^{5}\)
Case G 500 460 481.810 470.462 19.165 15.705 0.029 0.0441 0.4 2.442 \(3.6 \times 10^{3}\) \(2.2 \times 10^{4}\)
        0.0484 0.8 3.426 \(4.6 \times 10^{3}\) \(2.9 \times 10^{4}\)
        0.0527 1.2 4.409 \(5.7 \times 10^{3}\) \(3.5 \times 10^{4}\)
        0.0570 1.6 5.393 \(6.7 \times 10^{3}\) \(4.1 \times 10^{4}\)
        0.0613 2.0 6.376 \(7.7 \times 10^{3}\) \(4.8 \times 10^{4}\)
  1. aWe did not conduct simulations of these cases because \(\mu_{\text{s}}\) is smaller than \(\mu^{\text{r}} = {{\tau^{\text{r}} } \mathord{\left/ {\vphantom {{\tau^{\text{r}} } {\sigma_{\text{n}}^{\text{r}} }}} \right. \kern-0pt} {\sigma_{\text{n}}^{\text{r}} }}\) in the junction elements
  2. bThe case with \(D_{\text{c}}\) of 0.35 m is referred to as case A2 in the text and figures
  3. cThe case with \(D_{\text{c}}\) of 0.75 m is referred to as case A1 in the text and figures
  4. d \(L_{\text{c}}^{\text{II}}\) is \(L_{\text{c}}\) for mode II
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