NonlinearDruckerPrager

Drucker-Prager Material Model

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

$$F(\sigma,c)=\sqrt{J_2}+\eta_yp-\xi{}c$$

in which $J_2=\dfrac{1}{2}s:s$ is the second invariant of stress $\sigma$, $p=\dfrac{1}{3}( \sigma_1+\sigma_2+\sigma_3)$ is the hydrostatic stress, $c(\bar{\varepsilon_p})$ 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

$$\Phi(\sigma,c)=\sqrt{J_2}+\eta_fp$$

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 $\bar{\varepsilon_p}$ is defined as

$$\dot{\bar{\varepsilon_p}}=\xi\gamma,$$

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 $\phi$ and initial cohesion $c_0$ shall be determined by experiments.

Outer Mohr-Coulomb

$$\eta_y=\dfrac{6\sin\phi}{\sqrt{3}(3-\sin\phi)},\qquad\xi=\dfrac{6\cos\phi}{\sqrt{3}(3-\sin\phi)}$$

Inner Mohr-Coulomb

$$\eta_y=\dfrac{6\sin\phi}{\sqrt{3}(3+\sin\phi)},\qquad\xi=\dfrac{6\cos\phi}{\sqrt{3}(3+\sin\phi)}$$

Plane Strain Fitting

$$\eta_y=\dfrac{3\tan\phi}{\sqrt{9+12\tan^2\phi}},\qquad\xi=\dfrac{3}{\sqrt{9+12\tan^2\phi}}$$

Concrete, Rock, etc.

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

$$\phi=\sin^{-1}\dfrac{f_c-f_t}{f_c+f_t},\qquad{}c=\dfrac{f_cf_t}{f_c-f_t}\tan\phi$$

in which $f_t\ge0$ and $f_c\ge0$ are tension and compression strength respectively.

Uniaxial Tension/Compression

$$\eta_y=\dfrac{3\sin\phi}{\sqrt{3}},\qquad\xi=\dfrac{2\sin\phi}{\sqrt{3}}$$

Biaxial Tension/Compression

$$\eta_y=\dfrac{3\sin\phi}{2\sqrt{3}},\qquad\xi=\dfrac{2\sin\phi}{\sqrt{3}}$$