# Equations

The second Newton’s law allows us to define the fundamental equation of the solid mechanic, as follows

 \rho^*_{s} \frac{\partial^2 \boldsymbol{\eta}_s}{\partial t^2} - \nabla \cdot \left(\boldsymbol{F}_s \boldsymbol{\Sigma}_s\right) = \boldsymbol{f}^t_s

## Linear elasticity

\begin{align} \boldsymbol{F}_s &= \text{Identity} \\ \boldsymbol{\Sigma}_s &=\lambda_s tr( \boldsymbol{\epsilon}_s)\boldsymbol{I} + 2\mu_s\boldsymbol{\epsilon}_s \end{align}

## Hyper-elasticity

### Saint-Venant-Kirchhoff

\boldsymbol{\Sigma}_s=\lambda_s tr( \boldsymbol{E}_s)\boldsymbol{I} + 2\mu_s\boldsymbol{E}_s

### Neo-Hookean

\boldsymbol{\Sigma}_s= \mu_s J^{-2/3}(\boldsymbol{I} - \frac{1}{3} \text{tr}(\boldsymbol{C}) \ \boldsymbol{C}^{-1})
\boldsymbol{\Sigma}_s^ = \boldsymbol{\Sigma}_s^\text{iso} + \boldsymbol{\Sigma}_s^\text{vol}

#### Isochoric part : \boldsymbol{\Sigma}_s^\text{iso}

Table 1. Isochoric law
Name \mathcal{W}_S(J_s) \boldsymbol{\Sigma}_s^{\text{iso}}

Neo-Hookean

\mu_s J^{-2/3}(\boldsymbol{I} - \frac{1}{3} \text{tr}(\boldsymbol{C}) \ \boldsymbol{C}^{-1})

#### Volumetric part : \boldsymbol{\Sigma}_s^\text{vol}

Table 2. Volumetric law
Name \mathcal{W}_S(J_s) \boldsymbol{\Sigma}_s^\text{vol}

classic

\frac{\kappa}{2} \left( J_s - 1 \right)^2

simo1985

\frac{\kappa}{2} \left( ln(J_s) \right)

## Axisymmetric reduced model

We interest us here to a 1D reduced model, named generalized string.

The axisymmetric form, which will interest us here, is a tube of length L and radius R_0. It is oriented following the axis z and r represent the radial axis. The reduced domain, named \Omega_s^* is represented by the dotted line. So the domain, where radial displacement \eta_s is calculated, is \Omega_s^*=\lbrack0,L\rbrack.

We introduce then \Omega_s^{'*}, where we also need to estimate a radial displacement as before. The unique variance is this displacement direction.

Figure 1 : Geometry of the reduce model

The mathematical problem associated to this reduce model can be describe as

 \rho^*_s h \frac{\partial^2 \eta_s}{\partial t^2} - k G_s h \frac{\partial^2 \eta_s}{\partial x^2} + \frac{E_s h}{1-\nu_s^2} \frac{\eta_s}{R_0^2} - \gamma_v \frac{\partial^3 \eta}{\partial x^2 \partial t} = f_s.

where \eta_s, the radial displacement that satisfy this equation, k is the Timoshenko’s correction factor, and \gamma_v is a viscoelasticity parameter. The material is defined by its density \rho_s^*, its Young’s modulus E_s, its Poisson’s ratio \nu_s and its shear modulus G_s

At the end, we take  \eta_s=0\text{ on }\partial\Omega_s^* as a boundary condition, which will fixed the wall to its extremities.

# CSM Toolbox

## Models

The solid mechanics model can be selected in json file :

Listing : select solid model
``"Model":"Hyper-Elasticity"``
Table 3. Table of Models for `model` option
Model Name in json

Linear Elasticity

`Elasticity`

Hyper Elasticity

`Hyper-Elasticity`

When materials are closed to incompressibility formulation in displacement/pressure are available.

Table 4. Table of Models for `material_law` with hyper elasticity model
Model Name Volumic law

Saint-Venant-Kirchhoff

`SaintVenantKirchhoff`

classic, simo1985

NeoHookean

`NeoHookean`

classic, simo1985

option: `mechanicalproperties.compressible.volumic_law`

## Materials

The Lamé coefficients are deducing from the Young’s modulus E_s and the Poisson’s ratio \nu_s of the material we work on and can be express

\lambda_s = \frac{E_s\nu_s}{(1+\nu_s)(1-2\nu_s)} \hspace{0.5 cm} , \hspace{0.5 cm} \mu_s = \frac{E_s}{2(1+\nu_s)}
Materials section
``````"Materials":
{
"<marker>":
{
"name":"solid",
"E":"1.4e6",
"nu":"0.4",
"rho":"1e3"
}
}``````

where `E` stands for the Young’s modulus in Pa, `nu` the Poisson’s ratio ( dimensionless ) and `rho` the density in kg\cdot m^{-3}.

## Boundary Conditions

Table 5. Boundary conditions
Name Options Type

Dirichlet

faces, edges and component-wise

"Dirichlet"

Neumann

scalar, vectorial

"Neumann_scalar" or "Neumann_vectorial"

Pressure follower ,

Nonlinear boundary condition set in deformed domain

TODO

Robin

TODO

TODO

## Body forces

Table 6. Volumic forces
Name Options Type

Expression

Vectorial

"VolumicForces"

## Post Process

### Exports for visualisation

The fields allowed to be exported in the `Fields` section are:

• displacement

• velocity

• acceleration

• stress or normal-stress

• pressure

• material-properties

• pid

• fsi

• Von-Mises

• Tresca

• principal-stresses

• all

### Measures

• Points

• Maximum

• Minimum

• VolumeVariation

#### Points

Same syntax as FluidMechanics with available Fields :

• displacement

• velocity

• acceleration

• pressure

• principal-stress-0

• principal-stress-1

• principal-stress-2

• sigma_xx, sigma_xy, …​

#### Maximum/Minimum

The Maximum and minimum can be evaluated and save on .csv file. User need to define (i) <Type> ("Maximum" or "Minimum"), (ii) "<tag>" representing this data in the .csv file, (iii) "<marker>" representing the name of marked entities and (iv) the field where extremum is computed.

``````"<Type>":
{
"<tag>":
{
"markers":"marker>",
"fields":["displacement","velocity"]
}
}``````

#### VolumeVariation

``"VolumeVariation":<marker>``

## Run simulations

programme avalaible :

• `feelpp_toolbox_solid_2d`

• `feelpp_toolbox_solid_3d`

`mpirun -np 4 feelpp_toolbox_solid_2d --config-file=<myfile.cfg>`