# Drucker-Prager Constitutive Model (LAWS 10, 21 and 81)

## Drucker-Prager (LAW10 and LAW21)

For materials, like soils and rocks, the frictional and dilatational effects are significant. In these materials, the plastic behavior depends on the pressure as the internal friction is proportional to the normal force.

^{1}yield criterion uses a modified von Mises yield criteria to incorporate the effects of pressure for massive structures:

- $${J}_{2}$$
- Second invariant of deviatoric stress $${J}_{2}=\frac{1}{2}{s}_{ij}{s}_{ij}$$
- $$P$$
- Pressure
- $${A}_{0}$$, $${A}_{1}$$, $${A}_{2}$$
- Material coefficients

Figure 1 shows Equation 1 in the plane of $$\sqrt{{J}_{2}}$$ and $$P$$. The criterion expressed in the space of principal stresses represents a revolutionary surface with an axis parallel to the trisecting of the space as shown in Figure 2. This representation is in contrast with the von Mises criteria where yield criterion has a cylindrical shape. Drucker-Prager criterion is a simple approach to model the materials with internal friction because of the symmetry of the revolution surface and the continuity in variation of normal to the yield surface.

For LAW10 pressure evaluation for EOS is described with /EOS/COMPATION.

for loading $$d\mu >0$$

for unloading $$d\mu <0$$ and $$\mu <0$$

for unloading $$d\mu <0$$ and$$\mu >0$$

## Drucker-Prager Constitutive Model with Cap (LAW81)

### Yield Surface

- $$q$$
- von Mises stress
- $$p$$
- Pressure
- $${s}_{ij}$$
- Deviatoric stress
- $${}_{c}$$
- Cohesion
- $\beta $
- Friction angle
- $${p}_{0}$$
- Pressure value

### Plastic Flow

Plastic flow is governed by the non-associated flow potential $$G$$ defined as:

The plastic potential is continuous as you have $$\frac{\partial G}{\partial p}\left({p}_{0}\right)=\frac{\partial F}{\partial p}\left({p}_{0}\right)=0$$.

The scalar $$d\Lambda $$ will be determined in order to satisfy consistency and experimental hardening/softening.

### Hardening/Softening

- $${p}_{a0}$$ and $${p}_{b0}$$
- Initial value of $${p}_{a}$$ and $${p}_{b}$$

`fct_ID`

_{pb}.

- Shear yielding has an effect on $${p}_{b}$$, which depends on the possible dilantancy
imposed by the flow rule. An option to prevent this phenomenon is provided,
for example, for rocks (cap softening deactivation flag
`I`_{soft}). - $${p}_{a}$$ is derived from $${p}_{b}$$ via Equation 12
- If softening is allowed, the condition $${p}_{a}>0$$ is imposed, otherwise, $$d{\epsilon}_{v}^{p}\ge 0$$

### Derive Stress-Strain Relationships

and $$d{\epsilon}_{d}^{p}=d\Lambda \frac{\partial G}{\partial q}$$ with $$\frac{\partial G}{\partial q}=1$$.

With $$h=3\mu +K\frac{\partial F}{\partial p}\frac{\partial G}{\partial p}-\frac{\partial F}{\partial c}\frac{dc}{d{\epsilon}_{d}^{p}}-\frac{\partial G}{\partial p}\frac{\partial F}{\partial {p}_{b}}\frac{d{p}_{b}}{d{\epsilon}_{v}^{p}}$$

You can then compute all terms in Equation 18.

If $$p\le {p}_{a}$$, then $$\frac{\partial F}{\partial p}=-\mathrm{tan}\beta $$, $$\frac{\partial F}{\partial c}=-1$$, $$\frac{\partial F}{\partial {p}_{b}}=0$$.

$$\frac{\partial F}{\partial c}=-{r}_{c}\left(p\right)$$

and $$\frac{\partial F}{\partial {p}_{b}}=\frac{-p\left(p-{p}_{a}\right)}{{r}_{c}{p}_{b}{\left({p}_{b}-{p}_{a}\right)}^{2}}\left(p\mathrm{tan}\beta +c\right)$$

If $$p\le {p}_{a}$$, then $$\frac{\partial G}{\partial p}=-\mathrm{tan}\psi $$

If $$p\ge {p}_{0}$$, then $$\frac{\partial F}{\partial p}=\frac{\partial G}{\partial p}$$

When $$p<{p}_{0}$$ and $$\frac{\partial G}{\partial p}<0$$ leads to softening of the cap. If the no-softening cap flag is set, the last term in Equation 14 is irrelevant. To achieve this, set $$\frac{\partial F}{\partial {p}_{b}}=0$$ and impose on the hardening parameter $$d{\epsilon}_{v}^{p}$$ not decrease, although there is some volumetric plastic flow $$d{\epsilon}_{v}^{p}$$.

For $$p\to {p}_{b}$$, $$\frac{d{r}_{c}}{dp}\to \infty $$, $$d\Lambda \to \infty $$, so that $$d{\epsilon}_{v}^{p}$$ is undetermined in Equation 17.

### Elastic Properties

Yielding the cap actually models the compaction process. The elastic properties should thus increase when the porosity decreases, that is, $${\epsilon}_{v}^{p}$$ increases.

### Porosity Model

^{2}and assumes the soils are made of elastic grains with voids and is for low energies when the soil is not fully compacted. For a fully compacted soil at high energy, an equation of state should be used. In this material law, the variation of the volume of voids has an elastic part due to the elastic deformation of the skeleton and a plastic part which corresponds to the rearrangement of grains which induces compaction upon pressure loadings and dilatancy when undergoing shear loadings.

The above voids can be partly or totally filled with water.
In soil mechanics, when the soil is not saturated $S<1$ the only effect of water is its weight and mass so
the water pressure $u=0$; the mechanical properties are then the same as the
drained soil. When the soil is saturated $S\ge 1$, the water pressure $u$ is taken into account using Terzaghi’s assumption.
^{3} The total pressure is $p={p}^{\text{'}}+u$, where $p$ is the effective pressure in the structure that has
the voids. Also, assume the initial water pressure does not exceed the initial
pressure in the skeleton.

Where, ${\rho}_{w0}$ is the initial density of the water.

For stability reasons, a viscousity term is added.

If $${\mu}_{w}>-tol\text{then}{u}_{vis}=-{\alpha}_{v}\sqrt{{K}_{w}\rho}{\left(Vol\right)}^{\frac{1}{3}}{\epsilon}_{v}$$ and is added to $${u}^{*}$$.

^{1}Drücker D. and Prager W., “Soil mechanics and plastic analysis of limit design”, Quart. Appl. Math., Vol. 10, 157-165, 1952.

^{2}R. Kohler and G. Hofstetter, A cap model for partially saturated soils, Wiley & Sons, 2007

^{3}Karl Terzaghi, Theoretical Soil Mechanics, Wiley & Sons, 1943