# SCEC TPV13¶

TPV13 is similar to TPV12 except for that material properties are **non-associative Drucker-Prager plastic**. TPV13 is using non-associative Drucker-Prager plasticity with yielding in shear. The material is characterized by six constitutive parameters:

Bulk friction = 0.85

Fluid pressure = 1000 kg/m3Based on the SCEC benchmark, the stress components\(\sigma_{ij}\) are defined as :

Mean stress:\(\sigma_m = (\sigma_{11}+\sigma_{22}+\sigma_{33})/3\)

Stress deviator: \(s_{ij} = \sigma_{ij} - \sigma_m \delta_{ij}\) Second invariant of the stress deviator:

\(J_2(\sigma) = 1/2 *\sum_{ij} s_{ij} s_{ji}\)

Drucker-Prager yield stress:

\(Y(\sigma) =\max(0,c\cos \phi - (\sigma_m +P_f)\sin \phi)\)

Drucker-Prager yield function:

\(F(\sigma)=\sqrt{J_s(\sigma)-Y(\sigma)}\)

The Drucker-Prager material is required to satisfy the yield equation:

\(F(\sigma)\leq 0\)

When \(F(\sigma) < 0\), the material behaves like a linear isotropic elastic material, with Lame parameters \(\lambda\) and \(\mu\).

Wen \(F(\sigma) = 0\), if the material is subjected to a strain that tends to cause an increase in \(F(\sigma)\), then the material yields. For TPV13, we assume that the material yields in shear. Yielding in shear means that when the material yields, the stress tensor \(\sigma_{ij}\) changes by an amount proportional to the stress deviator \(s_{ij}\), so as to preserve the condition \(F(\sigma)\) with no change in mean stress \(\sigma_m\) .

## Nucleation¶

TPV13 uses the same nucleation method as TPV12

## Plasticity parameters¶

To turn on plasticity in SeisSol, add the following lines in
*parameter.par*:

&SourceType Plasticity = 1 ! default = 0 Tv = 0.03 ! Plastic relaxation /

In the **material.yaml**, add plasticity parameters:

```
!Switch
[rho, mu, lambda, plastCo, bulkFriction]: !ConstantMap
map:
rho: 2700
mu: 2.9403e+010
lambda: 2.941e+010
plastCo: 5.0e+06
bulkFriction: 0.85
[s_xx, s_yy, s_zz, s_xy, s_yz, s_xz]: !Include tpv12_13_initial_stress.yaml
```

## Results¶

Figure [fig:tpv13compare] shows the comparison between TPV12 (elastic) and TPV13 (plastic). The peak slip rate in TPV12 is higher than TPV13. This difference attributes to the response of the off-fault plasticity. Refer to Wollherr et al. [2018] for detailed discussions.