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.

History Variable Layout

location	parameter
initial_history(0)	accumulated plastic strain

Decision 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

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