# 7 Drucker Prager

The Drucker-Prager material is very similar to the Mohr-Coulomb material but uses slightly different expressions for the yield and plastic potential function.

# 7.1 Summary of material parameters

The key material parameters are summarized below. More specialized parameters are elaborated on in later sections.

# Stiffness

E: Young's modulus
\nu: Poisson's ratio

# Strength

k: cohesion
M: friction coefficient

# 7.2 Governing equations

# 7.2.1 Elasticity

Isotropic elasticity defined by E and \nu is used (see Elasticity).

# 7.2.2 Failure surface

The Drucker-Prager failure surface is given by

F = q-Mp'-k \tag{7.1}

where k and M are the cohesion and friction coefficient respectively.

The Drucker-Prager failure surface depicts a regular cone in three-dimensional stress space as shown in Figure 7.1.

# 7.2.3 Flow rule

The flow potential is given by

G = q-Np' \tag{7.2}

where M is the dilation coefficient.

Figure 7.1: Drucker-Prager failure surface (left) and inner Drucker-Prager approximation to Mohr-Coulomb (right).

For Flow Rule = Nonassociated, the same choices regarding dilation cap as for the Mohr-Coulomb model are available.

# 7.2.4 Drucker-Prager versus Mohr-Coulomb

An inner Drucker-Prager approximation of the Mohr-Coulomb failure surface (see Figure 7.1) is achieved by setting:

M= \frac{3 \sin\phi'}{\sqrt{3+\sin^2\phi'}},~~~
k = \frac{3 c'\cos\phi'}{\sqrt{3+\sin^2\phi'}} \tag{7.3}

This approximation matches the Mohr-Coulomb criterion exactly for the special case of plane strain with \psi=\phi.