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

\(\varepsilon_{tm} = \sqrt{\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2} = \sqrt{I_1^{'2}/3+2J'_2}\)