NonlinearDruckerPrager

Drucker-Prager Material Model

The Drucker-Prager model use the following function as the yield surface.

$$F(\sigma ,c)=\sqrt{J_2}+{\eta }_{y}p-\xi c$$

in which $J_2={\displaystyle \frac{1}{2}}s:s$ is the second invariant of stress $\sigma$, $p={\displaystyle \frac{1}{3}}({\sigma }_1+{\sigma }_2+{\sigma }_3)$ is the hydrostatic stress, $c(\overline{{\epsilon }_p})$ is cohesion, ${\eta }_y$ and $\xi$ are material constants.

Either associated or non-associated flow rule can be applied. The flow potential is defined as

$$\mathrm{\Phi }(\sigma ,c)=\sqrt{J_2}+{\eta }_{f}p$$

with ${\eta }_f$ is another material constant. If ${\eta }_f={\eta }_y$, the associative plasticity is defined so that the symmetry of stiffness matrix is recovered.

The accumulated plastic strain $\overline{{\epsilon }_p}$ is defined as

$$\dot{\overline{{\epsilon }_p}}=\xi \gamma ,$$

where $\gamma$ is the plasticity multiplier.

History Variable Layout

location	parameter
initial_history(0)	accumulated plastic strain

Determination of Material Constants

There are quite a lot of schemes to determine the material constants used in Drucker-Prager model. Here a few are presented.

Geomaterials

The friction angle $\varphi$ and initial cohesion $c_0$ shall be determined by experiments.

Outer Mohr-Coulomb

$${\eta }_y={\displaystyle \frac{6\sin \varphi }{\sqrt{3}(3-\sin \varphi )}},\xi ={\displaystyle \frac{6\cos \varphi }{\sqrt{3}(3-\sin \varphi )}}$$

Inner Mohr-Coulomb

$${\eta }_y={\displaystyle \frac{6\sin \varphi }{\sqrt{3}(3+\sin \varphi )}},\xi ={\displaystyle \frac{6\cos \varphi }{\sqrt{3}(3+\sin \varphi )}}$$

Plane Strain Fitting

$${\eta }_y={\displaystyle \frac{3\tan \varphi }{\sqrt{9+12{\tan }^2\varphi }}},\xi ={\displaystyle \frac{3}{\sqrt{9+12{\tan }^2\varphi }}}$$

Concrete, Rock, etc.

To fit uniaxial tension and compression strength, the friction angle and cohesion shall be computed as

$$\varphi ={\sin }^{-1}{\displaystyle \frac{f_c-f_t}{f_c+f_t}},c={\displaystyle \frac{f_c{f}_t}{f_c-f_t}}\tan \varphi$$

in which $f_t\ge 0$ and $f_c\ge 0$ are tension and compression strength respectively.

Uniaxial Tension/Compression

$${\eta }_y={\displaystyle \frac{3\sin \varphi }{\sqrt{3}}},\xi ={\displaystyle \frac{2\sin \varphi }{\sqrt{3}}}$$

Biaxial Tension/Compression

$${\eta }_y={\displaystyle \frac{3\sin \varphi }{2\sqrt{3}}},\xi ={\displaystyle \frac{2\sin \varphi }{\sqrt{3}}}$$