# SCEC TPV13¶

TPV13 is similar to TPV12 except for that material properties are non-associative Drucker-Prager plastic. To run TPV13, it requires to recompile SeisSol with “-DPLASTICITY=ON” in the configuration python script.

The material is characterized by six constitutive parameters:

Bulk friction = 0.85

Fluid pressure = 1000 kg/m3 Based 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 (https://github.com/SeisSol/Examples/blob/master/tpv12_13/parameters.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.