# Physical Models

## Overview

SeisSol includes various physical models to simulate realistic earthquake scenarios.

### Elastic

This is the standard model in SeisSol and it implements isotropic elastic materials. The constitutive behaviour is $$\sigma_{ij} = \lambda \delta_{ij} \epsilon_{kk} + 2\mu \epsilon_{ij}$$ with stress $$\sigma$$ and strain $$\epsilon$$. Elastic materials can be extended to elastoplastic materials (see SCEC TPV13).

### Anisotropic

This is an extension of the elastic material, where direction-dependent effects also play a role. The stress strain relation is given by the $$\sigma_{ij} = c_{ijkl} \epsilon_{kl}$$. Whereas isotropic materials are described by two material parameters, the tensor $$c$$ has $$81$$ entries. Due to symmetry considerations there are only $$21$$ independent parameters, which have to be specified in the material file: c11, c12, c13, c14, c15, c16, c22, c23, c24, c25, c26, c33, c34, c35, c36, c44, c45, c46, c55, c56, c66. For more details about the anisotropic stiffness tensor, see: https://en.wikipedia.org/wiki/Hooke%27s_law#Anisotropic_materials. All parameters have to be set, even if they are zero. If only the two Lamé parameters are provided, SeisSol assumes isotropic behaviour.

Example of an material file for a homogeneous anisotropic fullspace. The material describes the tilted transversally isotropic medium from chapter 5.4 in https://link.springer.com/chapter/10.1007/978-3-030-50420-5_3.

!ConstantMap
map:
rho:  2590
c11:  6.66000000000000e10
c12:  2.46250000000000e10
c13:  3.44750000000000e10
c14: -8.53035022727672e9
c15:  0.0
c16:  0.0
c22:  6.29062500000000e10
c23:  3.64187500000000e10
c24:  4.05949408023955e9
c25:  0.0
c26:  0.0
c33:  4.95562500000000e10
c34:  7.50194506028270e9
c35:  0.0
c36:  0.0
c44:  7.91875000000000e9
c45:  0.0
c46:  0.0
c55:  1.40375000000000e10
c56:  5.43430940874735e9
c66:  2.03125000000000e10


Anisotropy together with plasticity and dynamic rupture is not tested yet. You can define a dynamic rupture fault embedded in an isotropic material and have anisotropic regions elsewhere in the domain.

### Poroelastic

In poroelastic materials a fluid and a solid phase interact with each other. The material model introduces the pressure $$p$$ and the relative fluid velocities $$u_f, v_f, w_f$$ to the model, so that we observe 13 quantities in total. A poroelastic material is characterised by the following material parameters:

Parameter

SeisSol name

Abbreviation

Unit

Solid Bulk modulus

bulk_solid

$$K_S$$

$$Pa$$

Solid density

rho

$$\rho_S$$

$$kg \cdot m^{-3}$$

Matrix $$1^{st}$$ Lamé parameter

lambda

$$\lambda_M$$

$$Pa$$

Matrix $$2^{nd}$$ Lamé parameter

mu

$$\mu_M$$

$$Pa$$

Matrix permeability

permeability

$$\kappa$$

$$m^2$$

Matrix porosity

porosity

$$\phi$$

Matrix tortuosity

tortuosity

$$T$$

Fluid bulk modulus

bulk_fluid

$$K_F$$

$$Pa$$

Fluid density

rho_fluid

$$\rho_F$$

$$kg \cdot m^{-3}$$

Fluid viscosity

viscosity

$$\nu$$

$$Pa \cdot s$$

The implementation of poroelasticity is tested for point sources, material interfaces and free-surfaces. Plasticity and dynamic rupture together with poroelasticity are not tested.

### Viscoelastic

Viscoelasticity is used to model the dissipation of wave energy over time. A full documentation can be found in Attenuation.