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

Integration is performed using an implicit backward Euler scheme with a return mapping normal to the flow surface g. This law can take into account the influence of:

• the first stress invariant, i.e. the yield surface is either parallel to the pressure axis in the p-q plane (see figure 1-A)
• the third stress invariant, i.e. the Lode angle : the trace in the deviatoric plane is either a circle or a smoothed irregular hexagon (see figure 1-B).

Fig. 1: A) Influence of the first stress invariant in the p-q plane; B) Influence of the third stress invariant in the deviatoric plane

The von Mises, the Drücker Prager and a smoothed Mohr Coulomb yield surfaces can be represented.

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

Stresses and stress invariants
\[ I_{\sigma} = \sigma_{ij}\delta_{ij} = \sigma_{ii}; \widehat{\sigma}_{ij} = \sigma_{ij} - \frac{I_\sigma}{3}\delta_{ij};\] \[ II_{\widehat{\sigma}} = \sqrt{\frac{1}{2}\widehat{\sigma}_{ij}\widehat{\sigma}_{ij}}; III_{\widehat{\sigma}} = \frac{1}{3}\widehat{\sigma}_{ij}\widehat{\sigma}_{jk}\widehat{\sigma}_{ki} ;\] \[\beta = -\frac{1}{3}\sin^{-1}\left( \frac{3\sqrt{3}}{2} \frac{III_{\widehat{\sigma}}}{II_{\widehat{\sigma}}^3} \right) \]

Criterion with friction angle different from 0 (Drücker Prager or Van Eekelen):
Regular criterion used if $I_{sigma} - m’ II_{\widehat{\sigma}} < (3c/\tan \phi_c)$ \[ f = II_{\widehat{\sigma}} + m\left( I_{\sigma} - \frac{3c}{\tan\phi_c} \right) = 0\] with

where $a$ and $b$ are function of $\phi_C$, $\phi_E$ and $n$.
Apex criterion used if $I_{\sigma} - m’ II_{\widehat{\sigma}} \geq (3c / \tan \phi_c)$ \[ f = I_{\sigma} - \frac{3c}{\tan\phi_c} = 0 \] $m’$ is the equivalent of m but for the flow surface (i.e. $\phi$ is replaced by $\psi$)

Criterion with friction angle egal tp 0 (Von Mises criterion): \[ f = II_{\widehat{\sigma}} - \frac{2c}{\sqrt{3}} = 0 \]

Hardening/softening is assumed to be represented by the evolution of friction angles and/or cohesion as a function of the Von Mises equivalent plastic strain \[ \varepsilon_{ep}^p = \sqrt{\frac{2}{3}\widehat{\varepsilon}_{ij}^p \widehat{\varepsilon}_{ij}^p }\] Hyperbolic functions are used:

1. If ILODE = 1 or 2,
   
   • if $\varepsilon_{eq}^p <$ decphi : $\phi_C = \phi_{C0}$
   • if $\varepsilon_{eq}^p >$ decphi : $\phi_C = \phi_{C0}+\frac{(\phi_{Cf} - \phi_{C0})(\varepsilon_{eq}^p - decphi)}{B_p + (\varepsilon_{eq}^p -decphi)}$
   • if $\varepsilon_{eq}^p <$ decphi : $c = c_0$
   • if $\varepsilon_{eq}^p >$ decphi : $c = c_0 + \frac{(c_f - c_0)(\varepsilon_{eq}^p - deccoh)}{B_c + (\varepsilon_{eq}^p -deccoh)} $
2. ONLY if ILODE = 2,
   • if $\varepsilon_{eq}^p <$ decphi : $\phi_E = \phi_{E0}$
   • if $\varepsilon_{eq}^p >$ decphi : $\phi_E = \phi_{E0}+\frac{(\phi_{Ef} - \phi_{E0})(\varepsilon_{eq}^p - decphi)}{B_p + (\varepsilon_{eq}^p -decphi)}$

Where coefficients $B_p$ and $B_c$ are respectively the values of equivalent plastic strain for which half of the hardening/softening on friction angles and cohesion is achieved (see figure 2). Fig. 2: Hardening softening hyperbolic relation for 2 values of coefficient Bp

When viscosity is taken into account (IVISCO=1), one can assume that the plastic strain is composed of a time independent instantaneous strain $\varepsilon_{ij}^p$, but also of a time-dependent creep strain $\varepsilon_{ij}^{vp}$. The total strain (elastic, plastic and viscoplastic) reads : \[\varepsilon_{ij} = \varepsilon_{ij}^e +\varepsilon_{ij}^p + \varepsilon_{ij}^{vp}\] The material viscosity implies a time-dependent strain $\varepsilon_{ij}^{vp}$ which is a delayed plastic deformation. Under soil mechanic convention (compressive stress is taken as positive), the loading surface of the viscoplastic flow $f_{vp}$ and the viscoplastic potential $Q_{vp}$ are controlled by a delayed viscoplastic hardening function $\alpha_{vp}$ and read : \[ f_{vp} = q - \alpha_{vp} g(\theta) \bar{R}\sqrt{A\left(C_s + \frac{p'}{\bar{R}}\right)} \geq 0 \] \[ Q_{vp} = q - g(\theta)\left( \alpha_{vp} - \beta_p \right)\left( p' + C_s \bar{R} \right)\] \[ \alpha_{vp} = \alpha_{vp,0} + \left( 1 - \alpha_{vp,0} \right) \frac{\gamma_{vp}}{B_{vp} + \gamma_{vp}}\] where:

• $q=\sqrt{3}II_{\widehat{\sigma}}$ is the deviatoric stress
• $p' = \frac{\sigma_{ii}'}{3}$ is the mean effective stress
• $\gamma_{vp}$ is the equivalent plastic shear strain, i.e. the generalized plastic distorsion
• $g(\theta)$ is a function allowing to take into account the influence of the Lode angle $\theta$, i.e. the third deviatoric stress invariant, on yield surface in the deviatoric plane, $g(\theta) = 1$ for the sake of simplicity
• $\bar{R}$ is a normalising parameter, for convenience it is taken as equal to the uniaxial compression strength $\bar{R} = R_c$
• $\alpha_{vp,0}$ is the initial threshold for the viscoplastic flow (a value of $\alpha_{vp,0} = 0$ could be taken in agreement with the instantaneous plastic mechanism)
• $B_{vp}$ is a parameter controlling the evolution of $\alpha_{vp}$ and therefore of $f_{vp}$
• $A$ is an internal friction coefficient defining the curvature of the failure surface and $C_s$ is a cohesion coefficient, it denotes the material cohesion in saturated condition. Finally, $\beta_p$ is a parameter which defines the transition from compressibility $(\alpha_{vp} < \beta_p)$ to dilatancy $(\alpha_{vp} > \beta_p)$. $A$, $C_s$ and $\beta_p$ are parameters linked to the instantaneous plastic deformation modelling.
  

The equivalent plastic shear strain increment is given by : \[ \dot{\gamma_{vp}} = \sqrt{\frac{2}{3}\dot{e}_{ij}^{vp}\dot{e}_{ij}^{vp}} \] where $\dot{e}_{ij}^{vp}$ is the viscoplastic deviatoric strain : \[ \dot{e}_{ij}^{vp} = \dot{\varepsilon}_{ij}^{vp} - \dot{\varepsilon}_{kk}^{vp}\delta_{ij} \] where $\delta_{ij}$ is the Kronecker symbol. The viscoplastic flow rule is determined as follows : \[ \dot{\varepsilon}_{ij}^{vp} = A(T) \langle \frac{f_{vp}}{\bar{R}} \rangle^n \frac{\partial Q_{vp}}{\partial \sigma_{ij}} \] where $\langle\rangle$ is the Macauley bracket, $\langle x\rangle = x$ if $x\geq 0$ and $\langle x\rangle = 0$ if $x<0$, $A(T)$ is thefluidity coefficient generally dependent of the temperature $T$ and $n$ is a parameter which describes the shape of the creep curve. The following function is used for the fluidity : \[ A(T) = A_0 \exp\left( -\frac{\zeta}{RT} \right) \] where $A_0$ is the fluidity value at a reference temperature, $R$ is the perfect gas universal constant, $T$ is the absolute temperature and $\zeta$ is a parameter controlling the influence of temperature on the material viscosity. However, the temperature is generally assumed constant.

The viscoplastic law definition and typical values of parameters for sandstone and hard clay can be found in Zhou et al. (2008)¹⁾, Jia et al. (2008)²⁾. To modify parameters go to the file “LMLVP.F” in LAGAMINE code.

Prepro: LPLA.F
Lagamine: PLA2EA.F, PLA3D.F

= 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 :

= 43 : for 2D plane strain analysis with bifurcation criterion (ICBIF=1)
= 31 : in all the other cases