Theory of Solid Mechanics

1. 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}

2. Equations

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

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

2.1. Linear elasticity

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

2.2. Hyperelasticity

2.2.1. Saint-Venant-Kirchhoff

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

2.2.2. 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

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

2.3. 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.

Reduced Model Geometry
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.