Table 6 Parameter determination of extended binary medium model.

Bonded elements

\({K}^{b}\) (kPa)

\(187.4{P}_{a}{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{0.5032}\)

\({G}^{b}\) (kPa)

\(112.4{P}_{a}{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{0.5032}\)

Frictional elements

\({K}^{f}\) (kPa)

\(73{P}_{a}{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{0.3105}\)

\({G}^{f}\) (kPa)

\(21.3{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{0.2223}\)

\({\alpha }_{m}\)

\(0.9743{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{-0.3184}\)

\({c}_{1}\)

\(-0.3741\mathrm{ln}\left(0.1589\left(\frac{{\sigma }_{3}}{{P}_{a}}-0.2426\right)\right)-1.681\)

\({c}_{2}\)

\(-0.07499{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{-3.055}+0.3954\)

\({p}_{0}\)

200

n

5

n₁

5

\(\psi\)

0.0585

κ

0.011

\({e}_{0}\)

1.22

Root

\({K}^{r}\)(kPa)

135,000

\({G}^{r}\)(kPa)

20,250

Breakage ratio parameter

\({\alpha }_{v}\)

25

\(-0.01194\mathrm{ln}\left(0.8742N+0.000092\right)+0.139-0.005r\)

50

\(0.1206\mathrm{ln}\left(0.7369N+1.95\right)+0.1632-0.015r\)

100

\(0.0499\mathrm{ln}\left(0.1268\mathrm{N}+0.66\right)+0.2707-0.015r\)

200

\(0.0539\mathrm{ln}\left(0.03369N+0.0542\right)+0.4075-0.015r\)

\({\beta }_{v}\)

\(0.06841\mathrm{ln}\left(0.4938\frac{{\sigma }_{3}}{{P}_{a}}-0.0689\right)+0.2102\)

\({\theta }_{v}\)

25

\(0.4321\mathrm{ln}\left(1.513N+0.7682\right)+2.119\)

50

\(0.04871\mathrm{ln}\left(1.726N+1.334\right)+1.99\)

100

\(0.09783\mathrm{ln}\left(1.136N+0.8878\right)+2.019\)

200

\(0.5199\mathrm{ln}\left(1.523N+4.1578\right)+1.265\)

\({\omega }_{v}\)

0.25

\({\alpha }_{s}\)

2

\({\beta }_{s}\)

3

\({\theta }_{s}\)

− 0.002

\({\omega }_{s}\)

0.1

Stress concentration parameter

\({m}_{1}\)

− 0.4

\({m}_{2}\)

− 0.3

\({C}_{m}^{r}\)

\(0.7981{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{-0.1619}\)

\({C}_{s}^{r}\)

\(0.8978{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{-0.07796}\)