# 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
$$\sigma_{tm} = 300\sqrt{2}$$.

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