FLAC3D Theory and Background • Constitutive Models

Power-Mohr Model

This viscoplastic model cpower combines the behavior of the viscoelastic two-component power model and the elasto-plastic Mohr-Coulomb model. In the model formulation, the total strain rate \(\dot{\epsilon_{ij}}\) is decomposed into elastic (\(\dot{\epsilon}_{ij}^e\)), viscous (\(\dot{\epsilon}_{ij}^c\)), and plastic (\(\dot{\epsilon}_{ij}^p\)) components:

(1)\[\dot{\epsilon}_{ij} = \dot{\epsilon}_{ij}^e + \dot{\epsilon}_{ij}^c + \dot{\epsilon}_{ij}^p\]

The elastic strain rate \(\dot{\epsilon}_{ij}^e\) is the only component contributing to the stress rate; the deviatoric behavior is visco-elasto-plastic and is expressed as

(2)\[\dot{S}_{ij} = 2G \left( \dot{e}_{ij} - \dot{e}_{ij}^c - \dot{e}_{ij}^p\right)\]

where \(\dot{S}_{ij}\) and \(\dot{e}_{ij}\) are the deviatoric parts of the stress and strain rate tensors, \(\dot{\sigma}_{ij}\) and \(\dot{\epsilon}_{ij}\), and \(G\) is tangent shear modulus (see the Burgers model for a definition of the notation convention used in this section).

The volumetric behavior is elasto-plastic and has the form

(3)\[\dot{\sigma}_0 = K (\dot{e}_{vol}^e - \dot{e}_{vol}^p)\]

where \(\dot{\sigma}_0 = (\dot{\sigma}_{11} + \dot{\sigma}_{22} + \dot{\sigma}_{33} ) / 3\), \(\dot{e}_{vol} = \dot{e}_{11} + \dot{e}_{22} + \dot{e}_{33}\), and \(K\) is tangent bulk modulus.

Creep is activated by the von Mises stress \(q=\sqrt{3J_2}\) in accordance with the Norton Power Law (\(J_2 = 1/2 S_{ij} S_{ij}\) is the second invariant of stress deviatoric tensor), and the creep rate is

(4)\[\dot{e}_{ij}^c = \dot{e}_{cr} {{\partial q} \over {\partial S_{ij}}}\]

The direction of creep flow is derived from the definition of \(q\):

(5)\[{{\partial q} \over {\partial S_{ij}}} = {{3} \over {2}} {{S_{ij}} \over {q}}\]

By definition, the creep intensity has two components (see the power model):

(6)\[\dot{e}_{cr} = \dot{e}_{cr}^1 + \dot{e}_{cr}^2\]

where

\[\begin{split}\dot{e}^1_{cr} = \begin{cases} A_1 q^{n_1} & q \ge \sigma^{ref}_1 \\ 0 & q < \sigma^{ref}_1 \\ \end{cases}\end{split}\]

\[\begin{split}\dot{e}^2_{cr} = \begin{cases} A_2 q^{n_2} & q \le \sigma^{ref}_2 \\ 0 & q > \sigma^{ref}_2 \\ \end{cases}\end{split}\]

and \(\sigma_1^{ref}\) and \(\sigma_2^{ref}\) are two model parameters.

The plastic strain rate is defined using the Mohr-Coulomb flow rule,

(7)\[\dot{e}_{ij}^p = \dot{e}_{p} {{\partial g} \over {\partial {\sigma}_{ij}}} - {{1} \over {3}} \dot{e}_{vol}^p {\delta}_{ij}\]

where

(8)\[\dot{e}_{vol}^p = \dot{e}_{p} \left[ {{\partial g} \over {\partial {\sigma}_{11}}} + {{\partial g} \over {\partial {\sigma}_{22}}} + {{\partial g} \over {\partial {\sigma}_{33}}} \right]\]

The direction of plastic flow, \({\partial g} / {\partial {\sigma}_{ij}}\), is expressed using the definition of the Mohr-Coulomb potential function, \(g\), and the plastic flow rate intensity, \(\dot{e}_{p}\), is derived from the Mohr-Coulomb yield criterion \(f\) = 0 (see the Mohr-Coulomb model). In the principal axes formulation, the yield and potential functions for shear yielding are

(9)\[f=\sigma_1-\sigma_3N_{\phi} +2C\sqrt{N_{\phi} }\]

(10)\[g=\sigma_1-\sigma_3N_{\psi}\]

and for tension yielding, the functions are

(11)\[f=\sigma ^t-\sigma_3\]

(12)\[g=-\sigma_3\]

where \(\sigma_1\) and \(\sigma_3\) are the minimum and maximum principal stresses (compression negative), \(C\) is the material cohesion, \(\phi\) is the friction, \(\psi\) is the material dilation, \(\sigma ^t\) is the tensile strength, \(N_{\phi}=(1+\sin \phi )/(1-\sin \phi )\), and \(N_{\psi} =(1+\sin \psi )/(1-\sin \psi )\).

The model implementation closely follows the procedures for the power model and the Mohr-Coulomb model described in the manual. First, the viscoelastic response is calculated for the timestep \(\Delta t\). Principal stresses and principal directions are computed, and the yield criterion is checked. If the criterion is not met, plastic strain increments are added for the step, and the increment intensity \(\lambda = \dot{e}_p \Delta t\) is calculated to fulfil the yield condition \(f\) = 0. The procedure follows the lines presented in the Mohr-Coulomb model implementation, with the viscoelastic response replacing the “elastic guess” for the step.

power-mohr Model Properties

Use the following keywords with the zone property (FLAC3D) or zone property (3DEC) command to set these properties of the power-Mohr model.

power-mohr

bulk f

elastic bulk modulus, \(K\)

cohesion f

cohesion, \(c\)

constant-1 f

power-law constant, \(A\)₁

constant-2 f

power-law constant, \(A\)₂

dilation f

dilation angle, \(ψ\)

exponent-1 f

power-law exponent, \(n\)₁

exponent-2 f

power-law exponent, \(n\)₂

friction f

angle of internal friction, \(ϕ\)

poisson f

Poisson’s ratio, \(v\)

shear f

elastic shear modulus, \(G\)

stress-reference-1 f

reference stress, \(\sigma^{ref}_1\)

stress-reference-2 f

reference stress, \(\sigma^{ref}_2\)

tension f

tension limit, \(σ^t\). The default is 0.0.

young f

Young’s modulus, \(E\)

flag-brittle b (a)

If true, the tension limit is set to 0 in the event of tensile failure. The default is false.

Key

(a) Advanced property.

This property has a default value; simpler applications of the model do not need to provide a value for it.

Notes

• Only one of the two options is required to define the elasticity: bulk modulus \(K\) and shear modulus \(G\), or Young’s modulus \(E\) and Poisson’s ratio \(v\).
• The creep behavior is triggered by deviatoric stress, while the volumetric behavior does not consider creep.
• The tension cut-off is \(σ^t\) = min (\(σ^t\), \(c\)/tan\(ϕ\)).