NonlinearDruckerPrager

Drucker-Prager Material Model

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

F(σ,c)=J2+ηypξcF(\sigma,c)=\sqrt{J_2}+\eta_yp-\xi{}c

in which J2=12s:sJ_2=\dfrac{1}{2}s:s is the second invariant of stress σ\sigma, p=13(σ1+σ2+σ3)p=\dfrac{1}{3}( \sigma_1+\sigma_2+\sigma_3) is the hydrostatic stress, c(εpˉ)c(\bar{\varepsilon_p}) is cohesion, ηy\eta_y and ξ\xi are material constants.

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

Φ(σ,c)=J2+ηfp\Phi(\sigma,c)=\sqrt{J_2}+\eta_fp

with ηf\eta_f is another material constant. If ηf=ηy\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 ϕ\phi and initial cohesion c0c_0 shall be determined by experiments.

Outer Mohr-Coulomb

ηy=6sinϕ3(3sinϕ),ξ=6cosϕ3(3sinϕ)\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

ηy=6sinϕ3(3+sinϕ),ξ=6cosϕ3(3+sinϕ)\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

ηy=3tanϕ9+12tan2ϕ,ξ=39+12tan2ϕ\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

ϕ=sin1fcftfc+ft,c=fcftfcfttanϕ\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 ft0f_t\ge0 and fc0f_c\ge0 are tension and compression strength respectively.

Uniaxial Tension/Compression

ηy=3sinϕ3,ξ=2sinϕ3\eta_y=\dfrac{3\sin\phi}{\sqrt{3}},\qquad\xi=\dfrac{2\sin\phi}{\sqrt{3}}

Biaxial Tension/Compression

ηy=3sinϕ23,ξ=2sinϕ3\eta_y=\dfrac{3\sin\phi}{2\sqrt{3}},\qquad\xi=\dfrac{2\sin\phi}{\sqrt{3}}

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