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