next up previous contents
Next: Damage Up: Material deformation and flow Previous: Elasticity   Contents

Plasticity

Plastic strain

In plastic analysis, the materi_strain_elasti rate follows by subtracting from the materi_strain_total rate the materi_strain_plasti rate


\begin{displaymath}
\dot{\epsilon_{ij}}^{\rm elas} = \dot{\epsilon_{ij}} -
\dot{\epsilon_{ij}}^{\rm plas}
\end{displaymath}

where the materi_strain_total rate is


\begin{displaymath}
\dot{\epsilon_{ij}} = 0.5 ( \frac{\partial v_i}{\partial x_j} +
\frac{\partial v_j}{\partial x_i} )
\end{displaymath}

The materi_strain_plasti rate follows from the condition that the stress cannot exceed the yield surface. This condition is specified by a yield function $f^{\rm yield}(\sigma_{ij})=0$. The direction of the plastic strain rate is specified by the stress gradient of a flow function $\frac{\partial f^{\rm flow}}{\partial \sigma_{ij}}$. If the yield function and flow function are chosen to be the same, the plasticity is called associative, else it is non-associative.

Von-Mises is typically used for metal plasticity. Mohr-Coulomb and Drucker-Prager are typically used for soils and other frictional materials. The plasticity models can freely be combined; the combination of the plasticity surfaces defines the total plasticity surface.

Von-Mises stress, mean stress and deviatoric stress

First some stress quantities which are used in most of the plasticity models are listed.

Equivalent Von-Mises stress:

\begin{displaymath}
\bar{\sigma} = \sqrt{ \frac{ s_{ij}s_{ij} } {2} }
\end{displaymath}

Mean stress:

\begin{displaymath}
\sigma_m = \frac{ \sigma_{11} + \sigma_{22} + \sigma_{33} } {3}
\end{displaymath}

Deviatoric stress:

\begin{displaymath}
s_{ij} = \sigma_{ij} - \sigma_m \delta_{ij}
\end{displaymath}

Compression limiting plasticity model

This group_materi_plasti_compression model uses a special definition for the equivalent stress


\begin{displaymath}
\bar{\sigma} =
\sqrt{ {\sigma_1}^2 + {\sigma_2}^2 + {\sigma_3}^2 }
\end{displaymath}

where $\sigma_1$, $\sigma_2$ and $\sigma_3$ are the first, second and third principal stress respectively. Each of these is only incorporated if it is a compression stress. The model now reads


\begin{displaymath}
\bar{\sigma} - \sigma_y = 0
\end{displaymath}

This plasticity surface limits the allowed compression stresses.

Drucker-Prager plasticity model

The group_materi_plasti_druckprag model reads


\begin{displaymath}
3 \alpha \sigma_m + \bar{\sigma} - K = 0
\end{displaymath}


\begin{displaymath}
\alpha = \frac{2 \sin( \phi )}{\sqrt{3} ( 3 - \sin(\phi) )}
\end{displaymath}


\begin{displaymath}
K = \frac{ 6 c \cos( \phi )}{\sqrt{3} ( 3 - \sin(\phi) )}
\end{displaymath}

Here c is the cohesion, which needs to be specified both for the yield function and the flow rule; by choosing different values non-associative plasticity is obtained.

Gurson plasticity model

The group_materi_plasti_gurson model reads

\begin{displaymath}
\frac{3 \bar{\sigma}^2}{\sigma_y^2} +
2 q_1 f^* \cosh ( q_2 \frac{3 \sigma_m}{2 \sigma_y} ) -
(1 + ( q_3 f^* ) ^2 ) = 0
\end{displaymath}

Here f* is the volume fraction of voids. The rate equation

\begin{displaymath}
\dot{f^*} = ( 1 - f^*) f^* \epsilon_{kk}^{\rm plas}
\end{displaymath}

defines the evolution of f* if the start value for f* is specified. Furthermore, q1, q2 and q3 are model parameters.

Tension limiting plasticity model

This group_materi_plasti_tension model uses a special definition for the equivalent stress


\begin{displaymath}
\bar{\sigma} =
\sqrt{ {\sigma_1}^2 + {\sigma_2}^2 + {\sigma_3}^2 }
\end{displaymath}

where $\sigma_1$, $\sigma_2$ and $\sigma_3$ are the first, second and third principal stress respectively. Each of these is only incorporated if it is a tension stress. The model now reads


\begin{displaymath}
\bar{\sigma} - \sigma_y = 0
\end{displaymath}

This plasticity surface limits the allowed tension stresses.

A simple model for concrete can be obtained as follows. Ue group_materi_plasti_tension to limit the tension strength ft. Use group_materi_plasti_compression to limit the compressive strength fc. The tension strength could be softened to zero over a plastic strain of, say, 1 percent. The compressive strength could be softened to zero over a plastic strain of, say, 10 percent.

Von-Mises plasticity model

The group_materi_plasti_vonmis model reads


\begin{displaymath}
\sqrt{3} ~ \bar{\sigma} - \sigma_y = 0
\end{displaymath}

Mohr-Coulomb plasticity model

The group_materi_plasti_mohrcoul model reads


\begin{displaymath}
0.5 ( \sigma_1 - \sigma_3 ) + 0.5 ( \sigma_1 + \sigma_3 ) \sin ( \phi )
- c ~ \cos ( \phi ) = 0
\end{displaymath}

Here c is the cohesion, $\sigma_1$ is the maximal principal stress and $\sigma_3$ is the minimal principal stress. The angle $\phi$ needs to be specified for both the yield condition and the flow rule; by choosing different values, non-associative plasticity is obtained.

Isotropic Hardening

The size of the plastic strains rate is measured by the materi_plasti_kappa parameter

\begin{displaymath}
\dot{\kappa} = \sqrt{ 0.5 \dot{\epsilon}_{ij}^{\rm plas} \dot{\epsilon}_{ij}^{\rm plas} }
\end{displaymath}

This parameter can be used for isotropic hardening. Use the dependency_diagram for this.

Kinematic Hardening

The materi_plasti_rho matrix $\rho_{ij}$, governs the kinematic hardening in the plasticity models. It is used in the yield rule and flow rule to get a new origin by using the argument $\sigma_{ij} - \rho_{ij}$:


\begin{displaymath}
f^{\rm yield} = f^{\rm yield}(\sigma_{ij} - \rho_{ij})
\end{displaymath}


\begin{displaymath}
f^{\rm flow} = f^{\rm flow}(\sigma_{ij} - \rho_{ij})
\end{displaymath}

where the rate of the matrix $\rho_{ij}$ is taken to be


\begin{displaymath}
\dot { \rho_{ij} } = a \;\; \dot{\epsilon_{ij}}^{\rm plas}
\end{displaymath}

where a is a user specified factor (see group_materi_plasti_kinematic_hardening).

Plastic heat generation

The plastic energy loss can be partially turned into heat rate per unit volume q:


\begin{displaymath}
q = \eta \: \sigma_{ij} \: \dot{\epsilon_{ij}}^{\rm plas}
\end{displaymath}

where $\eta$ is a user specified parameter (between 0 and 1) specifying which part of the plastic energy loss is turned into heat (see group_materi_plasti_heat_generation).


next up previous contents
Next: Damage Up: Material deformation and flow Previous: Elasticity   Contents
root
1998-11-16