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