theory.adoc

Theory of Solid Mechanics

Notations and units

Notation	Quantity	Unit
\$\boldsymbol{\eta}_s\$	displacement	\$m\$
\$\rho_s\$	density	\$kg.m^{-3}\$
\$\lambda_s\$	first Lamé coefficients	\$N.m^{-2}\$
\$\mu_s\$	second Lamé coefficients	\$N.m^{-2}\$
\$E_s\$	Young modulus	\$kg.m^{-1}.s^{-2}\$
\$\nu_s\$	Poisson’s ratio	dimensionless
\$\boldsymbol{F}_s\$	deformation gradient
\$\boldsymbol{\Sigma}_s\$	second Piola-Kirchhoff tensor
\$f_s^t\$	body force

• strain tensor \$\boldsymbol{F}_s = \boldsymbol{I} + \nabla \boldsymbol{\eta}_s\$

• Cauchy-Green tensor \$\boldsymbol{C}_s = \boldsymbol{F}_s^{T} \boldsymbol{F}_s\$

• Green-Lagrange tensor

\begin{align} \boldsymbol{E}_s &= \frac{1}{2} \left( \boldsymbol{C}_s - \boldsymbol{I} \right) \\ &= \underbrace{\frac{1}{2} \left( \nabla \boldsymbol{\eta}_s + \left(\nabla \boldsymbol{\eta}_s\right)^{T} \right)}_{\boldsymbol{\epsilon}_s} + \underbrace{\frac{1}{2} \left(\left(\nabla \boldsymbol{\eta}_s\right)^{T} \nabla \boldsymbol{\eta}_s \right)}_{\boldsymbol{\gamma}_s} \end{align}

Equations

Newton’s second law allows us to define the fundamental equation of solid mechanics, 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}\$

Hyperelasticity

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

Here, we are interested in 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 \$z\$ axis and \$r\$ represents the radial axis. The reduced domain, named \$\Omega_s^*\$ is represented by the dotted line. So, the radial displacement \$\eta_s\$ is calculated in the domain \$\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 reduced model

The mathematical problem associated to this reduced model can be described 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\$ is the radial displacement that satisfies 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\$

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