Elasto‑plastic constitutive law for solid elements at constant temperature (non-associated) with linear anisotropic elasticity. Isotropic hardening/softening of friction angle and cohesion is possible.

This law is a combination of ORTHO3D (for the elastic part of the law) and PLA3D (for the plastic behaviour).

This law is only used for mechanical analysis of elasto-plastic anisotropic porous media undergoing large strains.

Prepro: LORTHOPLA.F

See PLASOL law

See PLASOL law (above)

The material cohesion depends on the angle $\alpha_{\sigma_1}$ between the major compressive principal stress $\vec{\sigma_1}$ and the normal to the bedding plane $\vec{e_3}$ : \[ \alpha_{\sigma_1} = \arccos\left( \frac{\vec{e_3}\vec{\sigma_{1}}'}{||\vec{e_3}||\ ||\vec{\sigma_1}'||} \right) \]

Fig. 1: Schematic view of the angle between the normal to bedding plane and the direction of major principal stress

Three cohesion values are defined ($c_{0^{\circ}}, c_{min}, c_{90^{\circ}}$), for major principal stress parallel $\alpha_{\sigma_1} = 0^{\circ}$ (perpendicular), perpendicular $\alpha_{\sigma_1} = 90^{\circ}$ (parallel) and with an angle of $\alpha_{\sigma_1, min}$ with respect to the normal to bedding plane (with respect to the bedding plane) (Salehnia, 2015)¹⁾. Between those values, cohesion varies linearly with $\alpha_{\sigma_1}$. The mathematical expression of the cohesion is as follows: \[ c = \max \left[\left( \frac{c_{min} - c_{0^{\circ}}}{\alpha_{\sigma_1, min}} \right)\alpha_{\sigma_1} + c_{0^{\circ}} ; \left( \frac{c_{90^{\circ}} - c_{min}}{90^{\circ} - \alpha_{\sigma_1, min}} \right)\left( \alpha_{\sigma_1} - \alpha_{\sigma_1, min} \right)+ c_{0^{\circ}} \right] \]

Fig. 2: Schematic view of the cohesion evolution as a function of the angle between the normal to bedding plane and the direction of major principal stress

The material cohesion anisotropy is defined with a microstructure tensor $a_{ij}$, which is a measure of the material fabric. The eigenvectors of this tensor correspond to the preferred or principal material axes (orthotropic axes) $e_1, e_2, e_3$. The cohesion corresponds to the projection of this tensor on a generalized loading vector l , therefore the cohesion specifies the effect of load orientation relative to the material axes. \[c = a_{ij}l_il_j\] \[l_i = \sqrt{\frac{\sigma_{l1}^2 + \sigma_{l2}^2 +\sigma_{l3}^2}{\sigma_{ij}\sigma_{ij}}}\]

Where $\sigma_{ij}$ expressed in reference to the material axes. The cohesion can be expressed as : \[c = c_0\left( 1 + A_{ij}l_il_j\right)\] \[A_{ij} = \frac{dev(a_{ij})}{c_0} =\frac{a_{ij}}{c_0} - \delta_{ij}\] \[c_0 = \frac{a_{kk}}{3}\]

where $A_{ij}$ is a symmetric traceless operator, $A_{kk}=0$. The above expression can be generalized by considering higher order tensors : \[c= c_0 \left( 1+A_{ij}l_il_j + b_1(A_{ij}l_il_j)^2 + b_2(A_{ij}l_il_j)^3 + … \right)\] where $B_1,b_2,…$ are constants.

Considering cross-anisotropy, i.e. transverse isotropy, and refering the problem to the principal material axes implies $A_{ij} = 0$ for $i \neq j$, $A_{ii} = A_{11}+A_{22}+A_{33} = 0$, $A_{11} = A_{33}$ if the bedding plane is in ($e_1, e_3$) anisotropic plane, $A_{22} = -2A_{11}$, implying : \[A_{ij}l_il_j = A_{l1}(1-3l_2^2)\] where $A_{11}$ is the component of the microstructure operator $A_{ij}$ in the isotropic (bedding) plane. The late expression for cohesion becomes (Pardoen, 2015)²⁾: \[c= c_0 \left( 1+A_{l1}(1-3l_2^2) + b_1A_{l1}^2(1-3l_2^2)^2 + b_2A_{l1}^3(1-3l_2^2)^3 + … \right)\]

The constants $c_0, A_{11}, b_1, b_2, …$ can be obtained from experimental data and laboratory tests. For uniaxial compression : $l_2 = \cos(\alpha)$ , $\alpha$ being the angle between the compression direction and the normal to the bedding plane (figure 1), $\alpha=0^{\circ}$ if the loading is perpendicular to the bedding plane and $\alpha = 90^{\circ}$ if parallel. The cohesion and the uniaxial compressive straight $f_c$ are linked by $f_c = 2c\cos\phi /(1-\sin\phi)$. The constants can be obtained from results coming from tests performed for differents orientation $\alpha$ (figure 3). For axial compression $\sigma_1$ with confinement $p_0$ (triaxial) : $l_2^2 = \frac{p_0^2\sin^2(\alpha)+\sigma_1^2\cos^2(\alpha)}{2p_0^2+\sigma_1^2}$.

Fig. 3: Schematic view of the cohesion and uniaxial compressive strength evolution as a function of the angle between the normal to bedding plane and the direction of axial loading

See PLASOL
Remark : For anisotropic Biot’s coeffcient, the deviatoric stress is calculated from the effective stresses (more details about this anisotropy are available in the definition of element CSOL2 and ISOL=9 in Appendix 7).

= 4 : for 2D analysis
= 6 : for 3D analysis

The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates.
For 2D analysis :

For the 3-D analysis :

= 48 : for 2D plane strain analysis with bifurcation criterion (ICBIF=1)
= 36 : in all the other cases