# Computational Solid Mechanics

Computational solid mechanics ( or CSM ) is a part of mechanics that studies solid deformations or motions under applied forces or other parameters.

## Second Newton’s law

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 (\boldsymbol{F}_s\boldsymbol{\Sigma}_s) = \boldsymbol{f}^t_s

It’s define here into a Lagrangian frame.

### Variables, symbols and units

 Notation Quantity Unit \rho_s^* strucure density kg/m^3 \boldsymbol{\eta}_s displacement m \boldsymbol{F}_s deformation gradient \boldsymbol{\Sigma}_s second Piola-Kirchhoff tensor N/m^2 f_s^t body force N/m^2

## Lamé coefficients

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)}

## 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.