# SCEC TPV13

TPV13 is similar to TPV12 except for the non-associative Drucker-Prager visco-plastic rheology.

The Drucker-Prager yield function is given by:

$$F(\sigma)=\sqrt{J_2(\sigma)-Y(\sigma)}$$

$$Y(\sigma)$$ is the Drucker-Prager yield stress, given as:

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

with $$\sigma_m = (\sigma_{11}+\sigma_{22}+\sigma_{33})/3$$ the mean stress,

$$c$$ the bulk cohesion, $$\phi$$ the bulk friction and $$P_f$$ the fluid pressure (1000 kg/m $$^3$$). In TPV13 benchmark, $$c=$$ 5.0e+06 Pa and $$\phi$$ =0.85.

$$J_2$$ is the second invariant of the stress deviator:

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

with $$s_{ij} = \sigma_{ij} - \sigma_m \delta_{ij}$$ the deviator stress components.

The yield equation has to be satisfied:

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

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 and plastic strains accumulates.

## Nucleation

TPV13 uses the same nucleation strategy as TPV12.

## Plasticity parameters

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

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


Plasticity related parameters are defined in material.yaml:

!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 1 compares the slip-rates along strike and dip in TPV12 (elastic) and TPV13 (visco-plastic). The peak slip rate in TPV12 is higher than in TPV13. This difference can be attributed to the inelastic response of the off-fault material. See Wollherr et al. (2018) for detailed discussions.