Feel++

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.

Reduced Model Geometry
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.