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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