Stress/Strain Invariants

In FLAC, 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 FLAC 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

\(\sigma_{tm} = 300\sqrt{2}\).

Table 1: Stress/Strain Invariants

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 Lode’s angle	\(\theta_{\sigma} = {{1} \over {3}} \arcsin{\left( -{{3\sqrt{3}} \over {2}} {{J_3} \over {J^{1.5}_2}} \right)}\)	strain Lode’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	\(\varepsilon_{tm} = \sqrt{\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2} = \sqrt{I_1^{'2}/3+2J'_2}\)