Stress/Strain Invariants
In FLAC3D, assuming \(s_{ij}\) is the deviatoric part of the stress tensor \(\sigma_{ij}\), and \(e_{ij}\) is the deviatoric part of the strain tensor \(\varepsilon_{ij}\), the stress/strain invariants in FLAC3D are defined and summarized in Table 1.
For example, in a triaxial compression test, the cell pressure in the \(xy\)-plane is 100, the vertical compression pressure in the \(z\)-direction is 400, the stress tensor in the sample is thus
\[\sigma_{ij} = \{\sigma_{xx}, \sigma_{yy}, \sigma_{zz}, \sigma_{xy}, \sigma_{yz}, \sigma_{xz} \} = \{-100,-100,-400,0,0,0\}\]
According to equations in Table 1, we have
\(I_1 = -600\),\(I_2 = 9e4\),\(I_3 = -4e6\)\(\sigma_v = -600\),\(\sigma_m = -200\),\(e_{ij} = \{s_{xx}, s_{yy}, s_{zz}, s_{xy}, s_{yz}, s_{xz} \} = \{100,100,-200,0,0,0\}\),\(J_1 = 0\),\(J_2 = 3e4\),\(J_3 = -2e6\),\(\theta_{\sigma} = \pi/6\),\(\sigma_1 = -400\),\(\sigma_2 = \sigma_3 = -100\),\(\sigma_{eq} = q = 300\),\(\sigma_{oct} = 100\sqrt{2}\),\(\tau_{max} = 100\),\(\sigma_{norm} = 300\sqrt{2}\), and\(\sigma_{tm} = 300\sqrt{2}\).
Stress Invariant | Definition | Strain Invariant | Definition |
---|---|---|---|
1st stress invariant | \(I_1 = \sigma_{kk}\) | 1st strain invariant | \(I'_1 = \varepsilon_{kk}\) |
2nd stress invariant | \(I_2 = (\sigma_{xx}\sigma_{yy}+\sigma_{yy}\sigma_{zz}+\sigma_{zz}\sigma_{xx}) \\- (\sigma_{xy}\sigma_{yz}+\sigma_{yz}\sigma_{zx}+\sigma_{zx}\sigma_{xy})\) | 2nd strain invariant | \(I'_2 = (\varepsilon_{xx}\varepsilon_{yy}+\varepsilon_{yy}\varepsilon_{zz}+\varepsilon_{zz}\varepsilon_{xx}) \\- (\varepsilon_{xy}\varepsilon_{yz}+\varepsilon_{yz}\varepsilon_{zx}+\varepsilon_{zx}\varepsilon_{xy})\) |
3rd stress invariant | \(I_3 = |\sigma_{ij}|\) | 3rd strain invariant | \(I'_3 = |\varepsilon_{ij}|\) |
volumetric stress | \(\sigma_v = \sigma_{kk}\) | volumetric strain | \(\varepsilon_{v} = \varepsilon_{kk}\) |
mean stress | \(\sigma_m = \sigma_{kk}/3\) | mean strain | \(\varepsilon_{m} = \varepsilon_{kk}/3\) |
1st deviatoric stress invariant | \(J_1 \equiv 0\) | 1st deviatoric strain invariant | \(J'_1 \equiv 0\) |
2nd deviatoric stress invariant | \(J_2 = (s_{ij}s_{ij})/2\) | 2nd deviatoric strain invariant | \(J'_2 = (e_{ij}e_{ij})/2\) |
3rd deviatoric stress invariant | \(J_3 = (s_{ij}s_{ik}{s_{ki}})/3\) | 3rd deviatoric strain invariant | \(J'_3 = (e_{ij}e_{ik}{e_{ki}})/3\) |
stress Loge’s angle | \(\theta_{\sigma} = {{1} \over {3}} \arcsin{\left( -{{3\sqrt{3}} \over {2}} {{J_3} \over {J^{1.5}_2}} \right)}\) | strain Loge’s angle | \(\theta_{\varepsilon} = {{1} \over {3}} \arcsin{\left(- {{3\sqrt{3}} \over {2}} {{J'_3} \over {J'^{1.5}_2}} \right)}\) |
minimum principal stress | \(\sigma_1 = {2\over{\sqrt{3}}}\sqrt{J_2}\sin{(\theta_{\sigma} - {2\over{3}}\pi)} + \sigma_m\) | minimum principal strain | \(\varepsilon_1 = {2\over{\sqrt{3}}}\sqrt{J'_2}\sin{(\theta_{\varepsilon} - {2\over{3}}\pi)} + \varepsilon_m\) |
intermediate principal stress | \(\sigma_2 = {2\over{\sqrt{3}}}\sqrt{J_2}\sin{(\theta_{\sigma})} + \sigma_m\) | intermediate principal strain | \(\varepsilon_2 = {2\over{\sqrt{3}}}\sqrt{J'_2}\sin{(\theta_{\varepsilon})} + \varepsilon_m\) |
maximum principal stress | \(\sigma_3 = {2\over{\sqrt{3}}}\sqrt{J_2}\sin{(\theta_{\sigma} + {2\over{3}}\pi)} + \sigma_m\) | maximum principal strain | \(\varepsilon_3 = {2\over{\sqrt{3}}}\sqrt{J'_2}\sin{(\theta_{\varepsilon} + {2\over{3}}\pi)} + \varepsilon_m\) |
von Mises equivalent stress | \(\sigma_{eq} = \sqrt{3 J_2}\) | von Mises equivalent strain | \(\varepsilon_{eq} = \sqrt{{{4}\over{3}}J'_2}\) |
equivalent deviatoric stress | \(q = \sqrt{3 J_2}\) | equivalent (engineering) shear strain | \(\gamma = \sqrt{4 J'_2}\) |
octahedral stress | \(\sigma_{oct} = \sqrt{{{2}\over{3}}J_2}\) | octahedral strain | \(\varepsilon_{oct} = \sqrt{{{8}\over{3}}J'_2}\) |
maximum shear stress | \(\tau_{max} = (\sigma_3 - \sigma_1)/2\) | maximum shear strain | \(\gamma_{max} = \varepsilon_3 - \epsilon_1\) |
stress norm | \(\sigma_{norm} = \sqrt{\sigma_{ij}\sigma_{ij}}\) | strain norm | \(\varepsilon_{norm} = \sqrt{\varepsilon_{ij}\varepsilon_{ij}}\) |
stress total measure | \(\sigma_{tm} = \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2} = \sqrt{I_1^2/3+2J_2}\) | strain total measure | \(\varepsilon_{tm} = \sqrt{\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2} = \sqrt{I_1^{'2}/3+2J'_2}\) |
Was this helpful? ... | PFC 6.0 © 2019, Itasca | Updated: Nov 19, 2021 |