Equations of Elasticity in Transient Case

In page Equations of Elasticity in Static Case, we see the formulation of elastic equation in static case in axisymmetric. In this section we see the elastic problem in transient case in axisymmetric.

1. In General Case

1.1. Differential Formulation

We search the displacement \(\mathbf{u} \in \mathbb{R}^3\) induced by a transformation. The domain of resolution is \(\Omega_c\) with bounds \(\Gamma_c\), \(\Gamma_{D \hspace{0.05cm} elas}\) the bound of Dirichlet conditions, \(\Gamma_{N \hspace{0.05cm} elas}\) the bound of Neumann conditions and \(\Gamma_{R \hspace{0.05cm} elas}\) the bound of Neumann conditions such that \(\Gamma_{D \hspace{0.05cm} elas} \cup \Gamma_{N \hspace{0.05cm} elas} \cup \Gamma_{R \hspace{0.05cm} elas} = \Gamma_c\).

We introduce the Lamé’s coefficients :

\[\lambda = \frac{E \, \nu}{(1-2 \nu) \, (1+\nu)}, \ \mu = \frac{E}{2 \, (1+\nu)}\]

With :

Young modulus \(E\), it represents the rigidity of materials (\(10^5 \leq \lambda \leq 10^{10}\))
Poisson’s coefficient \(\nu\), it represents the incompressibility of materials (\(0 \leq \mu \leq 0.5\)). If \(\mu = 0.5\), the materials is perfectly incompressible.

Conversely :

\[ \nu = \frac{\lambda}{2 (\lambda + \mu)}, E = \frac{\mu (3\lambda + 2\mu)}{\lambda + \mu}\]

We introduce the tensor of small deformations :

\[ \bar{\bar{\epsilon}}_{ij} = \frac{1}{2} \, \left( \frac{\partial u_{j}}{\partial j} + \frac{\partial u_{i}}{\partial i} \right) \ \text{ for } i,j = x, y, z\]

And, the stress tensor :

\[ \bar{\bar{\sigma}} = C \, \bar{\bar{\epsilon}}\]

With :

\(C\) …

To add the transient part, we have the Momentum : \(\mathbf{p} = \frac{\partial \left( \rho \, \mathbf {u} \right)}{\partial t}\) with \(\rho\) the density

The elastic equations with boundary conditions of Dirichlet, Neumann and Robin is written like :

Differential Formulation of Elasticity Equation

\[\text{(Elasticity)} \left\{ \begin{matrix} \frac{\partial \mathbf{p}}{\partial t} - \nabla \cdot \bar{\bar{\sigma}} = F & \text{on } \Omega_c & \text{(Elas-1)} \\ \mathbf{u} = \mathbf{u_0} & \text{on } \Gamma_{D \hspace{0.05cm} elas} & \text{(D elas)} \\ \bar{\bar{\sigma}} \cdot \mathbf{n} = g_N & \text{on } \Gamma_{N \hspace{0.05cm} elas} & \text{(N elas)} \\ \bar{\bar{\sigma}} \cdot \mathbf{n} + k \, \mathbf{u} = g_R & \text{on } \Gamma_{R \hspace{0.05cm} elas} & \text{(R elas)} \end{matrix} \right.\]

With :

Displacement \(\mathbf{u} = \begin{pmatrix} u_x \\ u_y \\ u_z \end{pmatrix}\)
Momentum \(\mathbf{p} = \frac{\partial \left( \rho \, \mathbf{u} \right)}{\partial t} = \begin{pmatrix} p_x \\ p_y \\ p_z \end{pmatrix}\) with \(\rho\) the density
Stress Tensor : \(\bar{\bar{\sigma}}\)
Stiffness tensor : \(k\)
Volumic Forces : \(\mathbf{F}\)

And notations :

Divergence of tensor : \(\nabla \cdot \bar{\bar{\sigma}} = \begin{pmatrix} \nabla \cdot \bar{\bar{\sigma}}_{x,:} \\ \nabla \cdot \bar{\bar{\sigma}}_{y,:} \\ \nabla \cdot \bar{\bar{\sigma}}_{z,:} \end{pmatrix} = \begin{pmatrix} \frac{\partial \bar{\bar{\sigma}}_{xx}}{\partial x} + \frac{\partial \bar{\bar{\sigma}}_{xy}}{\partial y} + \frac{\partial \bar{\bar{\sigma}}_{xz}}{\partial z} \\ \frac{\partial \bar{\bar{\sigma}}_{yx}}{\partial x} + \frac{\partial \bar{\bar{\sigma}}_{yy}}{\partial y} + \frac{\partial \bar{\bar{\sigma}}_{yz}}{\partial z} \\ \frac{\partial \bar{\bar{\sigma}}_{zx}}{\partial x} + \frac{\partial \bar{\bar{\sigma}}_{zy}}{\partial y} + \frac{\partial \bar{\bar{\sigma}}_{zz}}{\partial z} \end{pmatrix}\)
Scalar produce of tensor : \(\bar{\bar{\sigma}} \cdot \mathbf{n} = \begin{pmatrix} \bar{\bar{\sigma}}_{x,:} \cdot \mathbf{n} \\ \bar{\bar{\sigma}}_{y,:} \cdot \mathbf{n} \\ \bar{\bar{\sigma}}_{z,:} \cdot \mathbf{n} \end{pmatrix} = \begin{pmatrix} \bar{\bar{\sigma}}_{xx} \, n_x + \bar{\bar{\sigma}}_{xy} \, n_y + \bar{\bar{\sigma}}_{xz} \, n_z \\ \bar{\bar{\sigma}}_{yx} \, n_x + \bar{\bar{\sigma}}_{yy} \, n_y + \bar{\bar{\sigma}}_{yz} \, n_z \\ \bar{\bar{\sigma}}_{zx} \, n_x + \bar{\bar{\sigma}}_{zy} \, n_y + \bar{\bar{\sigma}}_{zz} \, n_z \end{pmatrix}\)

In the rest of this section, we use the Hooke law :

\[ \bar{\bar{\sigma}}^E = \lambda \, Tr(\bar{\bar{\epsilon}}) \, Id + 2 \, \mu \, \bar{\bar{\epsilon}}\]

It can rewrite like :

\[ \bar{\bar{\sigma}}_{ij}^E = \lambda \, \delta_{ij} \, \nabla \cdot \mathbf{u} + 2 \, \mu \, \bar{\bar{\epsilon}}_{ij}\]

Plus the Thermal Dilatation :

\[ \bar{\bar{\sigma}}^T = - (3 \lambda + 2 \mu) \, \alpha_T \, \left( T - T_0 \right) \, Id\]

With linear dilatation coefficient \(\alpha_T\), \(T\) the temperature and \(T_0\) the temperature of reference (or rest temperature).

Thus, the stress tensor is writen like :

\[\begin{eqnarray*} \bar{\bar{\sigma}} &=& \bar{\bar{\sigma}}^E + \bar{\bar{\sigma}}^T \\ &=& \lambda \, Tr(\bar{\bar{\epsilon}}) \, Id + 2 \, \mu \, \bar{\bar{\epsilon}} - (3 \lambda + 2 \mu) \, \alpha_T \, \left( T - T_0 \right) \, Id \end{eqnarray*}\]

Or :

\[ \bar{\bar{\sigma}}_{ij} = \lambda \, \delta_{ij} \, \nabla \cdot \mathbf{u} + 2 \, \mu \, \bar{\bar{\epsilon}}_{ij} - (3 \lambda + 2 \mu) \, \delta_{ij} \, \alpha_T \, \left( T - T_0 \right)\]

1.2. Weak Formulation

We express the weak formulation of (Elasticity) component by component.

We will here deal only with strong Dirichlet conditions (on \(\Gamma_{D \hspace{0.05cm} elas}\)) that is why the function space is set to \(H^1_{u_0}(\Omega) = \{ u \in H^1(\Omega), \ u = u_0 \text{ on } \Gamma_{D \hspace{0.05cm} elas} \}\).

By multiplying the x component of (Elas-1) by \(\xi_x\) of \(\mathbf{ξ} = \begin{pmatrix} \xi_x \\ \xi_y \\ \xi_z \end{pmatrix} \in H^1_{u_0}(\Omega)\) and integrating over \(\Omega_c\), we obtain :

\[ \int_{\Omega_c}{ \frac{\partial p_x}{\partial t} \xi_x } - \int_{\Omega_c}{\left( \frac{\partial \bar{\bar{\sigma}}_{xx}}{\partial x} + \frac{\partial \bar{\bar{\sigma}}_{xy}}{\partial y} + \frac{\partial \bar{\bar{\sigma}}_{xz}}{\partial z} \right) \, \xi_x } = \int_{\Omega_c}{ F_x \, \xi_x }\]

By the Formula of Green (First Form) :

\[ \int_{\Omega_c}{ \frac{\partial p_x}{\partial t} \xi_x } + \int_{\Omega_c}{ \bar{\bar{\sigma}}_{xx} \, \frac{\partial \xi_x}{\partial x} + \bar{\bar{\sigma}}_{xy} \, \frac{\partial \xi_x}{\partial y} + \bar{\bar{\sigma}}_{xz} \, \frac{\partial \xi_x}{\partial z} } - \int_{\Gamma_c}{ \bar{\bar{\sigma}}_{xx} \, \xi_x \, n_x + \bar{\bar{\sigma}}_{xy} \, \xi_x \, n_y + \bar{\bar{\sigma}}_{xz} \, \xi_x \, n_z } = \int_{\Omega_c}{ F_x \, \xi_x }\]

With notations :

\[ \int_{\Omega_c}{ \frac{\partial p_x}{\partial t} \xi_x } + \int_{\Omega_c}{ \bar{\bar{\sigma}}_{xx} \, \frac{\partial \xi_x}{\partial x} + \bar{\bar{\sigma}}_{xy} \, \frac{\partial \xi_x}{\partial y} + \bar{\bar{\sigma}}_{xz} \, \frac{\partial \xi_x}{\partial z} } - \int_{\Gamma_c}{ \left( \bar{\bar{\sigma}}_{x,:} \cdot \mathbf{n} \right) \, \xi_x } = \int_{\Omega_c}{ F_x \, \xi_x }\]

We apply the boundary conditions :

\[ \int_{\Omega_c}{ \frac{\partial p_x}{\partial t} \xi_x } + \int_{\Omega_c}{ \bar{\bar{\sigma}}_{xx} \, \frac{\partial \xi_x}{\partial x} + \bar{\bar{\sigma}}_{xy} \, \frac{\partial \xi_x}{\partial y} + \bar{\bar{\sigma}}_{xz} \, \frac{\partial \xi_x}{\partial z} } - \int_{\Gamma_{N \hspace{0.05cm} elas}}{ g_{Nx} \, \xi_x } - \int_{\Gamma_{R \hspace{0.05cm} elas}}{ \left( g_{Rx} - \left( k \, u \right)_x \right) \, \xi_x } = \int_{\Omega_c}{ F_x \, \xi_x }\]

\[ \int_{\Omega_c}{ \frac{\partial p_x}{\partial t} \xi_x } + \int_{\Omega_c}{ \bar{\bar{\sigma}}_{xx} \, \frac{\partial \xi_x}{\partial x} + \bar{\bar{\sigma}}_{xy} \, \frac{\partial \xi_x}{\partial y} + \bar{\bar{\sigma}}_{xz} \, \frac{\partial \xi_x}{\partial z} } = \int_{\Omega_c}{ F_x \, \xi_x } + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ g_{Nx} \, \xi_x } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ \left( g_{Rx} - \left( k \, u \right)_x \right) \, \xi_x }\]

We do the same with \(y\) and \(z\) components and we obtain the weak formulation of (Elasticity) component by component :

Weak Formulation of Elasticity Equation Case Component by Component

\[\text{(Weak Elasticity)} \\ \left\{ \begin{matrix} \int_{\Omega_c}{ \frac{\partial p_x}{\partial t} \xi_x } + \int_{\Omega_c}{ \bar{\bar{\sigma}}_{xx} \, \frac{\partial \xi_x}{\partial x} + \bar{\bar{\sigma}}_{xy} \, \frac{\partial \xi_x}{\partial y} + \bar{\bar{\sigma}}_{xz} \, \frac{\partial \xi_x}{\partial z} } = \int_{\Omega_c}{ F_x \, \xi_x } + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ g_{Nx} \, \xi_x } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ \left( g_{Rx} - \left( k \, u \right)_x \right) \, \xi_x } \\ \int_{\Omega_c}{ \frac{\partial p_y}{\partial t} \xi_y } + \int_{\Omega_c}{ \bar{\bar{\sigma}}_{yx} \, \frac{\partial \xi_y}{\partial x} + \bar{\bar{\sigma}}_{yy} \, \frac{\partial \xi_y}{\partial y} + \bar{\bar{\sigma}}_{yz} \, \frac{\partial \xi_y}{\partial z} } = \int_{\Omega_c}{ F_y \, \xi_y } + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ g_{Ny} \, \xi_y } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ \left( g_{Ry} - \left( k \, u \right)_y \right) \, \xi_y } \\ \int_{\Omega_c}{ \frac{\partial p_z}{\partial t} \xi_z } + \int_{\Omega_c}{ \bar{\bar{\sigma}}_{zx} \, \frac{\partial \xi_z}{\partial x} + \bar{\bar{\sigma}}_{zy} \, \frac{\partial \xi_z}{\partial y} + \bar{\bar{\sigma}}_{zz} \, \frac{\partial \xi_z}{\partial z} } = \int_{\Omega_c}{ F_z \, \xi_z } + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ g_{Nz} \, \xi_z } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ \left( g_{Rz} - \left( k \, u \right)_z \right) \, \xi_z } \\ \text{for } \mathbf{ξ} = \begin{pmatrix} \xi_x \\ \xi_y \\ \xi_z \end{pmatrix} \in H^1_{u_0}(\Omega_c) \end{matrix} \right.\]

Finally, we have the vectorial form :

Weak Formulation of Elasticity Equation Case Vectorial

\[\text{(Weak Elasticity)} \\ \left\{ \begin{matrix} \int_{\Omega_c}{ \frac{\partial \mathbf{p}}{\partial t} \mathbf{ξ} } + \int_{\Omega_c}{ \bar{\bar{\sigma}}_{xx} \, \frac{\partial \xi_x}{\partial x} + \bar{\bar{\sigma}}_{xy} \, \frac{\partial \xi_x}{\partial y} + \bar{\bar{\sigma}}_{xz} \, \frac{\partial \xi_x}{\partial z} + \bar{\bar{\sigma}}_{yx} \, \frac{\partial \xi_y}{\partial x} + \bar{\bar{\sigma}}_{yy} \, \frac{\partial \xi_y}{\partial y} + \bar{\bar{\sigma}}_{yz} \, \frac{\partial \xi_y}{\partial z} + \bar{\bar{\sigma}}_{zx} \, \frac{\partial \xi_z}{\partial x} + \bar{\bar{\sigma}}_{zy} \, \frac{\partial \xi_z}{\partial y} + \bar{\bar{\sigma}}_{zz} \, \frac{\partial \xi_z}{\partial z} } = \int_{\Omega_c}{ \mathbf{F} \cdot \mathbf{ξ} } + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ \mathbf{g_N} \, \mathbf{ξ} } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ \left( \mathbf{g_R} - \left( k \, \mathbf{u} \right) \right) \, \mathbf{ξ} } \\ \end{matrix} \right.\]

2. In axisymmetric coordinates

2.1. Differential Formulation

On equations (Elasticity), we suppose the geometry and the parameters are independent by \(\theta\) (of cylindric coordinates \((r,\theta,z)\)).

We note \(\Omega^{axis}_c\) (respectively \(\Gamma_c^{axis}\), \(\Gamma_{D \hspace{0.05cm} elas}^{axis}\), \(\Gamma_{N \hspace{0.05cm} elas}^{axis}\) and \(\Gamma_{R \hspace{0.05cm} elas}^{axis}\)) the representation of \(\Omega_c\) (respectively \(\Gamma_c\), \(\Gamma_{D \hspace{0.05cm} elas}\), \(\Gamma_{N \hspace{0.05cm} elas}\)) and \(\Gamma_{R \hspace{0.05cm} elas}\) in axisymmetric coordinates.

We note \(\mathbf{u} = \begin{pmatrix} u_r \\ u_{\theta} \\ u_z \end{pmatrix}_{cyl}\) the coordinates of \(u \in \mathbb{R}^3\) in cylindrical base.

We note \(\mathbf{n}^{axis} = \begin{pmatrix} n^{axis}_r \\ n^{axis}_z \end{pmatrix}_{cyl}\) the exterior normal of \(\Gamma^{axis}\) on \(\Omega^{axis}\).

The tensor of small deformations becomes :

\[\begin{align*} \bar{\bar{\epsilon}}_{rr} &= \frac{\partial u_r}{\partial r} , & \bar{\bar{\epsilon}}_{rz} &= \frac{1}{2} \left( \frac{\partial u_z}{\partial r} + \frac{\partial u_r}{\partial z} \right) \\ \bar{\bar{\epsilon}}_{zz} &= \frac{\partial u_z}{\partial z} , & \bar{\bar{\epsilon}}_{r\theta} &= 0 \\ \bar{\bar{\epsilon}}_{\theta \theta} &= \frac{1}{r} \, u_r , & \bar{\bar{\epsilon}}_{\theta z} &= 0 \end{align*}\]

And the stress tensor :

\[\begin{align*} \bar{\bar{\sigma}}_{rr} &= \frac{\lambda}{r} \, u_r + \left( \lambda + 2\mu \right) \, \frac{\partial u_r}{\partial r} + \lambda \, \frac{\partial u_z}{\partial z} - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) , & \bar{\bar{\sigma}}_{rz} &= \mu \, \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \\ \bar{\bar{\sigma}}_{zz} &= \frac{\lambda}{r} \, u_r + \lambda \, \frac{\partial u_r}{\partial r} + \left( \lambda + 2\mu \right) \, \frac{\partial u_z}{\partial z} - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) , & \bar{\bar{\sigma}}_{r \theta} &= 0 \\ \bar{\bar{\sigma}}_{\theta \theta} &= \frac{\lambda + 2\mu}{r} \, u_r + \lambda \, \frac{\partial u_r}{\partial r} + \lambda \, \frac{\partial u_z}{\partial z} - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right), & \bar{\bar{\sigma}}_{z \theta} &= 0 \end{align*}\]

We have :

\[ \nabla \cdot \bar{\bar{\sigma}} = \begin{pmatrix} \nabla \cdot \bar{\bar{\sigma}}_{r,:} + \frac{\bar{\bar{\sigma}}_{rr} - \bar{\bar{\sigma}}_{\theta \theta}}{r} \\ \nabla \cdot \bar{\bar{\sigma}}_{z,:} + \frac{\bar{\bar{\sigma}}_{rz}}{r} \end{pmatrix}\]

With \(\bar{\bar{\sigma}}_{r,:} = \begin{pmatrix} \bar{\bar{\sigma}}_{rr} & \bar{\bar{\sigma}}_{rz} \end{pmatrix}\) and \(\bar{\bar{\sigma}}_{z,:} = \begin{pmatrix} \bar{\bar{\sigma}}_{rz} & \bar{\bar{\sigma}}_{zz} \end{pmatrix}\)

Differential Formulation of Elasticity Equation Case in Axisymmetric

\[\text{(Elasticity Axis)} \left\{ \begin{matrix} \frac{\partial \mathbf{p}}{\partial t} - \nabla \cdot \bar{\bar{\sigma}} = F & \text{on } \Omega_c^{axis} & \\ \mathbf{u} = 0 & \text{on } \Gamma_{D \hspace{0.05cm} elas}^{axis} & \text{(D axis elas)} \\ \bar{\bar{\sigma}} \cdot \mathbf{n}^{axis} = g_N & \text{on } \Gamma_{N \hspace{0.05cm} elas}^{axis} & \text{(N axis elas)} \\ \bar{\bar{\sigma}} \cdot \mathbf{n}^{axis} + k \, \mathbf{u} = g_R & \text{on } \Gamma_{R \hspace{0.05cm} elas}^{axis} & \text{(R axis elas)} \end{matrix} \right.\]

With : \(\mathbf{p} = \frac{\partial \rho \, \mathbf{u}}{\partial t}\) and \(\nabla \cdot \bar{\bar{\sigma}} = \begin{pmatrix} \nabla \cdot \bar{\bar{\sigma}}_{r,:} + \frac{\bar{\bar{\sigma}}_{rr} - \bar{\bar{\sigma}}_{\theta \theta}}{r} \\ \nabla \cdot \bar{\bar{\sigma}}_{z,:} + \frac{\bar{\bar{\sigma}}_{rz}}{r} \end{pmatrix}\)

2.2. Weak Formulation

This section are based on calculus of book Theorical Physics Reference.

To obtain the weak formulation of axisymetric elastic equation, we pass by the weak formulation of elastic problem in cartesian (Weak Elasticity).

We have the formula of changement to cartesian to cylindric :

\[\begin{eqnarray*} u_x &=& cos \theta \, u_r \\ u_y &=& sin \theta \, u_r \\ u_z &=& u_z \end{eqnarray*}\]

\[\begin{eqnarray*} \frac{\partial u_x}{\partial x} &=& \frac{\partial u_r}{\partial r} cos^2 \theta + \frac{1}{r} u_r sin^2 \theta \\ \frac{\partial u_x}{\partial y} &=& \frac{\partial u_r}{\partial r} cos \theta sin \theta + \frac{1}{r} u_r cos \theta sin \theta \\ \frac{\partial u_x}{\partial z} &=& \frac{\partial u_z}{\partial z} cos \theta \end{eqnarray*}\]

\[\begin{eqnarray*} \frac{\partial u_y}{\partial x} &=& \frac{\partial u_r}{\partial r} cos \theta sin \theta + \frac{1}{r} u_r cos \theta sin \theta \\ \frac{\partial u_y}{\partial y} &=& \frac{\partial u_r}{\partial r} sin^2 \theta + \frac{1}{r} u_r cos^2 \theta \\ \frac{\partial u_y}{\partial z} &=& \frac{\partial u_z}{\partial z} sin \theta \end{eqnarray*}\]

\[\begin{eqnarray*} \frac{\partial u_z}{\partial x} &=& \frac{\partial u_z}{\partial z} cos \theta \\ \frac{\partial u_z}{\partial y} &=& \frac{\partial u_z}{\partial z} sin \theta \\ \frac{\partial u_z}{\partial z} &=& \frac{\partial u_x}{\partial z} \end{eqnarray*}\]

The x component of (Weak Elasticity) becomes :

\[\scriptsize{ \begin{eqnarray*} \int_{\Omega}{ r \, \frac{\partial p_r}{\partial t} \xi_r cos^2 \theta \, dr d\theta dz } + \int_{\Omega_c} \left( r \left[ \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) + 2\mu \left( \frac{\partial u_r}{\partial r} cos^2 \theta + \frac{1}{r} u_r sin^2 \theta \right) - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right] \left( \frac{\partial \xi_r}{\partial r} cos^2 \theta + \frac{1}{r} \xi_r sin^2 \theta \right) \right. \\ + r 2\mu \left( \frac{\partial u_r}{\partial r} cos \theta sin \theta - \frac{1}{r} u_r cos \theta sin \theta \right) \left( \frac{\partial \xi_r}{\partial r} cos \theta sin \theta - \frac{1}{r} \xi_r cos \theta sin \theta \right) \\ \left. + r\mu \left( \frac{\partial u_r}{\partial z} cos \theta + \frac{\partial u_z}{\partial r} cos \theta \right) \frac{\partial \xi_r}{\partial z} cos \theta \right) \ dr d\theta dz \\ = \int_{\Omega_c}{r \, F_r \, \xi_r \, cos^2 \theta \ dr d\theta dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ r \, g_r \, \xi_r \, cos^2 \theta \ dr d\theta dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ r \left( g_r - (k \mathbf{u})_r \right) \, \xi_r \, cos^2 \theta \ dr d\theta dz } \end{eqnarray*} }\]

And the y :

\[\scriptsize{ \begin{eqnarray*} \int_{\Omega}{ r \, \frac{\partial p_r}{\partial t} \xi_r sin^2 \theta \, dr d\theta dz } + \int_{\Omega_c} \left( r 2\mu \left( \frac{\partial u_r}{\partial r} cos \theta sin \theta - \frac{1}{r} u_r cos \theta sin \theta \right) \left( \frac{\partial \xi_r}{\partial r} cos \theta sin \theta - \frac{1}{r} \xi_r cos \theta sin \theta \right) \right. \\ + r \left[ \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) + 2\mu \left( \frac{\partial u_r}{\partial r} sin^2 \theta + \frac{1}{r} u_r cos^2 \theta \right) - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right] \left( \frac{\partial \xi_r}{\partial r} sin^2 \theta + \frac{1}{r} \xi_r cos^2 \theta \right) \\ \left. + r\mu \left( \frac{\partial u_r}{\partial z} sin \theta + \frac{\partial u_z}{\partial r} sin \theta \right) \frac{\partial \xi_r}{\partial z} sin \theta \right) \ dr d\theta dz \\ = \int_{\Omega_c}{r \, F_r \, \xi_z \, sin^2 \theta \ dr d\theta dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ r g_r sin^2 \theta \ dr d\theta dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ r \left( g_r - (k \mathbf{u})_r \right) \xi_z \, sin^2 \theta \, dr d\theta dz } \end{eqnarray*} }\]

Adding these two equations :

\[\scriptsize{ \begin{eqnarray*} \int_{\Omega_c}{ r \, \frac{d p_r}{d t} \, \xi_r \, dr d\theta dz} + \int_{\Omega_c} r \left( \left( \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right) \left( \frac{\partial \xi_r}{\partial r} + \frac{1}{r} \xi_r \right) + r \mu \left[ 2 \left( \frac{\partial u_r}{\partial r} \frac{\partial \xi_r}{\partial r} + \frac{1}{r^2} u_r \xi_r \right) + \left( \frac{\partial u_r}{\partial z} \frac{\partial \xi_r}{\partial z} + \frac{\partial u_z}{\partial r} \frac{\partial \xi_r}{\partial z} \right) \right] \right) dr d\theta dz\\ = \int_{\Omega_c}{r \, F_r \, \xi_r \ dr d\theta dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ r \, g_r \, \xi_r \ dr d\theta dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ r \left( g_r - (k \mathbf{u})_r \right) \xi_r \ dr d\theta dz } \end{eqnarray*} }\]

The interior of integrals are independant by \(\theta\) :

\[\scriptsize{ \begin{eqnarray*} \int_{\Omega_c^{axis}}{ r \, \frac{d p_r}{d t} \, \xi_r \, dr dz} + \int_{\Omega_c^{axis}} r \left( \left( \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right) \left( \frac{\partial \xi_r}{\partial r} + \frac{1}{r} \xi_r \right) + r \mu \left[ 2 \left( \frac{\partial u_r}{\partial r} \frac{\partial \xi_r}{\partial r} + \frac{1}{r^2} u_r \xi_r \right) + \left( \frac{\partial u_r}{\partial z} \frac{\partial \xi_r}{\partial z} + \frac{\partial u_z}{\partial r} \frac{\partial \xi_r}{\partial z} \right) \right] \right) dr dz \\ = \int_{\Omega_c^{axis}}{r \, F_r \, \xi_r \ dr dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}^{axis}}{ r \, g_r \, \xi_r \ dr dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}^{axis}}{ r \left( g_r - (k \mathbf{u})_r \right) \xi_r \ dr dz } \end{eqnarray*} }\]

In the other hand, the z component becomes :

\[\scriptsize{ \begin{eqnarray*} \int_{\Omega_c}{ r \, \frac{d p_z}{dt} \, \xi_z \, dr d\theta dz} + \int_{\Omega_c} \left( r \mu \left( \frac{\partial u_r}{\partial z} cos \theta + \frac{\partial u_z}{\partial r} cos \theta \right) \frac{\partial \xi_z}{\partial r} cos \theta + r \mu \left( \frac{\partial u_r}{\partial z} sin \theta + \frac{\partial u_z}{\partial r} sin \theta \right) \frac{\partial \xi_z}{\partial r} sin \theta + r \left[ \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) + 2\mu \frac{\partial u_z}{\partial z} - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right] \frac{\partial \xi_z}{\partial z} \right) dr d\theta dz \\ = \int_{\Omega_c}{ r \, F_z \, \xi_z} + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ r \, g_z \, \xi_z } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ r \left( g_z - (k \mathbf{u})_z \right) \, \xi_z } \end{eqnarray*} }\]

This gives us :

\[\scriptsize{ \begin{eqnarray*} \int_{\Omega_c}{ r \, \frac{d p_z}{dt} \, \xi_z \, dr d\theta dr } + \int_{\Omega_c} \left( r \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \frac{\partial \xi_z}{\partial r} + r \left[ \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) + 2\mu \frac{\partial u_z}{\partial z} - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right] \frac{\partial \xi_z}{\partial z} \right) \ dr d\theta dz \\ = \int_{\Omega_c}{ r \, F_z \, \xi_z \ dr d\theta dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ r \, g_z \, \xi_z \ dr d\theta dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}}{ \left( g_z - (k \mathbf{u})_z \right) \xi_z \ dr d\theta dz } \end{eqnarray*} }\]

The interior of integrals are independant by \(\theta\) :

\[\scriptsize{ \begin{eqnarray*} \int_{\Omega_c^{axis}}{ r \, \frac{d p_z}{dt} \, \xi_z \, dr dz } + \int_{\Omega_c^{axis}} \left( r \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \frac{\partial \xi_z}{\partial r} + r \left[ \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) + 2\mu \frac{\partial u_z}{\partial z} - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right] \frac{\partial \xi_z}{\partial z} \right) \ dr dz \\ = \int_{\Omega_c^{axis}}{ r \, F_z \, \xi_z \ dr dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}^{axis}}{ r \, g_z \, \xi_z \ dr dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}^{axis}}{ \left( g_z - (k \mathbf{u})_z \right) \xi_z \ dr dz } \end{eqnarray*} }\]

We have the weak formulation for axisymetric problem component by component :

Weak Formulation of Elasticity Equation in Axisymmetric Component by Component

\[\tiny{ \text{(Weak Elasticity Axis)} \\ \left\{ \begin{align*} \int_{\Omega_c^{axis}}{ r \, \frac{d p_r}{dt} \, \xi_r \, dr dr } + \int_{\Omega_c^{axis}} \left( r \left( \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right) \left( \frac{\partial \xi_r}{\partial r} + \frac{1}{r} \xi_r \right) + r \mu \left[ 2 \left( \frac{\partial u_r}{\partial r} \frac{\partial \xi_r}{\partial r} + \frac{1}{r^2} u_r \xi_r \right) + \left( \frac{\partial u_r}{\partial z} \frac{\partial \xi_r}{\partial z} + \frac{\partial u_z}{\partial r} \frac{\partial \xi_r}{\partial z} \right) \right] \right) dr dz \\ = \int_{\Omega_c^{axis}}{r \, F_r \, \xi_r \ dr dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}^{axis}}{ r \, g_r \, \xi_r \ dr dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}^{axis}}{ r \left( g_r - (k \mathbf{u})_r \right) \xi_r \ dr dz } & \hspace{0.5cm} \text{r-(Weak Static Elasticity Axis)} \\ \int_{\Omega_c^{axis}}{ r \, \frac{d p_z}{dt} \, \xi_z \, dr dr } + \int_{\Omega_c^{axis}} \left( r \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \frac{\partial \xi_z}{\partial r} + r \left[ \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) + 2\mu \frac{\partial u_z}{\partial z} - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right] \frac{\partial \xi_z}{\partial z} \right) \ dr dz \\ = \int_{\Omega_c^{axis}}{ r \, F_z \, \xi_z \ dr dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}^{axis}}{ r \, g_z \, \xi_z \ dr dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}^{axis}}{ \left( g_z - (k \mathbf{u})_z \right) \xi_z \ dr dz } & \hspace{0.5cm} \text{z-(Weak Static Elasticity Axis)} \end{align*} \right. }\]

Finally, we have the vectorial form :

Weak Formulation of Elasticity Equation in Axisymmetric Vectorial

\[\tiny{ \text{(Weak Elasticity Axis)} \\ \left\{ \begin{align*} \int_{\Omega_c^{axis}}{ r \, \frac{d \mathbf{p}}{dt} \, \mathbf{ξ} \, dr dr } + \int_{\Omega_c^{axis}} \left( r \left( \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right) \left( \frac{\partial \xi_r}{\partial r} + \frac{1}{r} \xi_r \right) + r \mu \left[ 2 \left( \frac{\partial u_r}{\partial r} \frac{\partial \xi_r}{\partial r} + \frac{1}{r^2} u_r \xi_r \right) + \left( \frac{\partial u_r}{\partial z} \frac{\partial \xi_r}{\partial z} + \frac{\partial u_z}{\partial r} \frac{\partial \xi_r}{\partial z} \right) \right] \right) + \left( r \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \frac{\partial \xi_z}{\partial r} + r \left[ \lambda \left( \frac{\partial u_r}{\partial r} + \frac{1}{r} u_r + \frac{\partial u_z}{\partial z} \right) + 2\mu \frac{\partial u_z}{\partial z} - (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right] \frac{\partial \xi_z}{\partial z} \right) dr dz \\ = \int_{\Omega_c^{axis}}{r \, \mathbf{F} \cdot \mathbf{ξ} \ dr dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}^{axis}}{ r \, \mathbf{g} \cdot \mathbf{ξ} \ dr dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}^{axis}}{ r \left( \mathbf{g} - (k \mathbf{u}) \right) \cdot \mathbf{ξ} \ dr dz } & \hspace{0.5cm} \text{(Weak Static Elasticity Axis)} \\ \end{align*} \right. }\]

2.3. Adaptation to CFPDEs

To adapt (Elasticity Axis) on CFPDEs, we have two way Scalar Form and Vectorial Form.

2.3.1. Scalar Form

CFPDEs is an application of software Feel++. It solve by finite element the equations of form :

Differential Formulation of cfpdes : Scalar Form

\[d \frac{\partial u}{\partial t} + \nabla \cdot \left( -c \nabla u - \alpha u + \mathbf{γ} \right) + \mathbf{β} \cdot \nabla u + a u = f \text{ on domain } \Omega \in \mathbb{R}^d\]

With :

\(u: \Omega \rightarrow \mathbb{R}\) the unknown

\(\mathbf{γ}\) conservative flux source term

\(d\) damping or mass coefficient

\(\mathbf{β}\) convection coefficient

\(c\) diffusion coefficient

\(a\) absorption or reaction coefficient

\(\alpha\) conservative flux convection coefficient

\(f\) source term

To adapt (Weak Elasticity Axis), we introduce the equations of definition of momentum \(\mathbf{p}\) :

\[\left\{ \begin{eqnarray*} p_r &=& \rho \frac{\partial u_r}{\partial t} \\ p_z &=& \rho \frac{\partial u_z}{\partial t} \end{eqnarray*}\right.\]

In weak formulation, it give :

\[\left\{ \begin{eqnarray*} \int_{\Omega_c^{axis}}{ r \rho \, \frac{\partial u_r }{\partial t} \, \eta_r \ dr dz } &=& \int_{\Omega_c^{axis}}{ r \, p_r \, \eta_r \ dr dz } & \text{(ur-(Weak Elasticity Axis))} \\ \int_{\Omega_c^{axis}}{ r \rho \, \frac{\partial u_z}{\partial t} \, \eta_z \ dr dz } &=& \int_{\Omega_c^{axis}}{ r \, p_z \, \eta_z \ dr dz } & \text{(uz-(Weak Elasticity Axis))} \\ && \text{with } \mathbf{η} = \begin{pmatrix} \eta_r \\ \eta_z \end{pmatrix} \in \left( H^1(\Omega_c^{axis}) \right)^2 \end{eqnarray*}\right.\]

And we reorder the equations r-(Weak Elasticity Axis) and z-(Weak Elasticity Axis) renamed by r-(Weak Elasticity Axis) and z-(Weak Elasticity Axis) plus ur-(Weak Elasticity Axis) and uz-(Weak Elasticity Axis) :

\[\scriptsize{ \text{(Weak Static Elasticity Axis)} \\ \left\{ \begin{align*} \int_{\Omega_c^{axis}}{ r \frac{d p_r}{dt} \, \xi_r \, dr dz } + \int_{\Omega_c^{axis}} \left( r (\lambda+2\mu) \frac{\partial u_r}{\partial r} + \lambda u_r + r \lambda \frac{\partial u_z}{\partial z} - r (3\lambda+2\mu) \alpha_T \left( T - T_0 \right) \right) \frac{\partial \xi_r}{\partial r} + r \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \frac{\partial \xi_r}{\partial z} \ dr dz \\ = \int_{\Omega_c^{axis}}{ \left( r F_r - \lambda \frac{\partial u_r}{\partial r} - \frac{\lambda + 2\mu}{r} u_r - \lambda \frac{\partial u_z}{\partial z} + (3\lambda+2\mu) \alpha_T \left( T - T_0 \right) \right) \xi_r \, dr dz } + \int_{\Gamma_{N \hspace{0.05cm} elas}}{ g_r r \xi_r \ dr dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}^{axis}}{ r \left( g_r - (k \mathbf{u})_r \right) dr dz } & \hspace{0.5cm} \text{pr-(Weak Static Elasticity Axis)} \\ \int_{\Omega_c^{axis}}{ r \rho \, \frac{\partial u_r}{\partial t} \, \eta_r \ dr dz } = \int_{\Omega_c^{axis}}{ r \, p_r \, \eta_r \ dr dz } & \hspace{0.5cm} \text{ur-(Weak Elasticity Axis)} \\ \int_{\Omega_c^{axis}}{ r \frac{d p_z}{dt} \, \xi_z \, dr dz } + \int_{\Omega_c^{axis}} r \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \frac{\partial \xi_z}{\partial r} + \left[ r (\lambda+2\mu) \frac{\partial u_r}{\partial r} + \lambda u_r + r \lambda \frac{\partial u_z}{\partial z} - r (3\lambda + 2 \mu) \, \alpha_T \, \left( T-T_0 \right) \right] \frac{\partial \xi_z}{\partial z} \ dr dz \\ = \int_{\Omega_c^{axis}}{ r \, F_z \, \xi_z \ dr dz} + \int_{\Gamma_{N \hspace{0.05cm} elas}^{axis}}{ r \, g_z \, \xi_z \ dr dz } + \int_{\Gamma_{R \hspace{0.05cm} elas}^{axis}}{ \left( g_z - (k \mathbf{u})_z \right) \xi_z \ dr dz } & \hspace{0.5cm} \text{pz-(Weak Static Elasticity Axis)} \\ \int_{\Omega_c^{axis}}{ r \rho \, \frac{\partial u_z}{\partial t} \, \eta_z \ dr dz } = \int_{\Omega_c^{axis}}{ r \, p_z \, \eta_z \ dr dz } & \hspace{0.5cm} \text{uz-(Weak Elasticity Axis)} \\ \end{align*} \right. }\]

We have four equations :

\(\text{ur-(Weak Elasticity Axis)}\) of unknow \(u_r\)
\(\text{pr-(Weak Elasticity Axis)}\) of unknow \(p_r\)
\(\text{uz-(Weak Elasticity Axis)}\) of unknow \(u_z\)
\(\text{pz-(Weak Elasticity Axis)}\) of unknow \(p_z\)

It give the coefficients :

For \(\text{pr-(Weak Elasticity Axis)}\) :

Coefficient

Description

Expression

\(d\)

damping or mass coefficient

\(r\)

\(\mathbf{γ}\)

conservative flux source term

\(\begin{pmatrix} - r (\lambda + 2\mu) \frac{\partial u_r}{\partial r} - \lambda u_r - r\lambda \frac{\partial u_z}{\partial z} + r (3\lambda+2\mu) \alpha_T (T-T_0) \\ - r \mu \frac{\partial u_r}{\partial z} - r\mu \frac{\partial u_z}{\partial z} \end{pmatrix}\)

\(f\)

source term

\(r F_r - \lambda \, \frac{\partial u_r}{\partial r} - \lambda \, \frac{\partial u_z}{\partial z} - \frac{\lambda+2\mu}{r} u_r + (3\lambda+2\mu) \alpha_T (T-T_0)\)

For \(\text{ur-(Weak Elasticity Axis)}\) :

Coefficient

Description

Expression

\(d\)

damping or mass coefficient

\(r \, \rho\)

\(f\)

source term

\(r p_r\)

For \(\text{z-(Weak Elasticity Axis)}\) :

Coefficient

Description

Expression

\(d\)

damping or mass coefficient

\(r\)

\(\mathbf{γ}\)

conservative flux source term

\(\begin{pmatrix} - r \mu \frac{\partial u_r}{\partial z} - r\mu \frac{\partial u_z}{\partial z} \\ - r\lambda \frac{\partial u_r}{\partial r} - \lambda u_r - r (\lambda+2\mu) \frac{\partial u_z}{\partial z} + r (3\lambda+2\mu) \alpha_T (T-T_0) \end{pmatrix}\)

\(f\)

source term

\(r F_z\)

For \(\text{pz-(Weak Elasticity Axis)}\) :

Coefficient

Description

Expression

\(d\)

damping or mass coefficient

\(r \, \rho\)

\(f\)

source term

\(r p_z\)

For the boundary conditions :

We can see, the boundaries integrals are on pr-(Weak Elasticity Axis) and pz-(Weak Elasticity Axis) of unknow \(p_r\) and \(p_z\), thus the boundaries conditions are implement on those two equations. But, we use strong Dirichlet, thus the Dirichlet conditions are implement on ur-(Weak Elasticity Axis) and uz-(Weak Elasticity Axis) of unknow \(u_r\) and \(u_z\).

Dirichlet : we use strong Dirichlet condition, thus we put on JSON file :

For ur-(Weak Elasticity Axis) :

"Dirichlet":
{
    "Gamma_Delas"
    {
        "expr":"u0r:u0r"
    }
}

For uz-(Weak Elasticity Axis) :

"Dirichlet":
{
    "Gamma_Delas"
    {
        "expr":"u0z:u0z"
    }
}

Neumann : in the calcul, it appears the Jacobian \(r\), thus we put on JSON file :

For pr-(Weak Elasticity Axis) :

"Neumann":
{
    "Gamma_Nelas"
    {
        "expr":"r*g_Nr:r:g_Nr"
    }
}

For pz-(Weak Elasticity Axis) :

"Neumann":
{
    "Gamma_Nelas"
    {
        "expr":"r*g_Nz:r:g_Nz"
    }
}

Robin : in the calcul, it appears the Jacobian \(r\), thus we put on JSON file :

If the stifness matrix \(k\) isn’t diagonal, it appears a coupling between the two equations. But actually, we can’t implement coupling on boundary conditions on CFPDEs. Thus, we suppose that the stifness matrix \(k\) is diagonal (\(k=\begin{pmatrix} k_{00} & 0 \\ 0 && k_{11} \end{pmatrix}\)), the Robin condition is rewrited : \(\left\{ \begin{pmatrix} k_{00} u_r + \bar{\bar{\sigma}}_{r,:} \cdot \mathbf{n}^{axis} = g_r \\ k_{11} u_z + \bar{\bar{\sigma}}_{z,:} \cdot \mathbf{n}^{axis} = g_z \end{pmatrix} \right.\).

For pr-(Weak Elasticity Axis) :

"Neumann":
{
    "Gamma_Nelas"
    {
        "expr1":"r*k00:r:k00"
        "expr2":"r*g_Rr:r:g_Rr"
    }
}

For pz-(Weak Elasticity Axis) :

"Neumann":
{
    "Gamma_Nelas"
    {
        "expr1":"r*k11:r:k11"
        "expr2":"r*g_Rz:r:g_Rz"
    }
}

2.3.2. Vectorial Form

CFPDEs is an application of software Feel++. It solve by finite element the equations of form :

Set the domain of resolution \(\Omega \subset \mathbb{R}^d\) and unknow \(\mathbf{u} : \Omega \rightarrow \mathbf{R}^n\)

\[d \frac{\partial \mathbf{u}}{\partial t} + \nabla \cdot \left( -c \nabla \mathbf{u} + \mathbf{γ} \right) + \mathbf{β} \cdot \nabla \mathbf{u} + a \mathbf{u} = \mathbf{f} \text{ on domain } \Omega \in \mathbb{R}^d\]

With :

Symbol

Description

Dimension

\(\mathbf{u}\)

scalar unknown

\(2\)

\(d\)

damping or mass coefficient

\(1\)

\(c\)

diffusion coefficient

\(1\) or \(2 \times 2\)

\(\mathbf{γ}\)

conservative flux source term

\(2 \times 2\)

\(\mathbf{β}\)

convection coefficient

\(2\)

\(a\)

absorption or reaction coefficient

\(1\)

\(f\)

source term

\(2\)

To obtain the coefficients of (Static Elasticity Axis), we introduce a second equation of definition of momentum \(\mathbf{p}\) :

\[ \mathbf{p} = \rho \, \frac{\partial \mathbf{u}}{\partial t}\]

Its weak formulation is :

\[ \int_{\Omega^{axis}}{ \mathbf{p} \, r \, dr dz } = \int_{\Omega^{axis}}{ \rho \, \frac{\partial \mathbf{u}}{\partial t} \, r \, dr dz }\]

After, we reorganize the equation (Static Elasticity Axis)

\[\begin{eqnarray*} \frac{\partial \mathbf{p}}{\partial t} - \nabla \bar{\bar{\sigma}} &=& \mathbf{F} \\ \frac{\partial \mathbf{p}}{\partial t} - \begin{pmatrix} \frac{\partial \bar{\bar{\sigma}}_{rr}}{\partial r} + \frac{\partial \bar{\bar{\sigma}}_{rz}}{\partial z} + \frac{\bar{\bar{\sigma}}_{rr} - \bar{\bar{\sigma}}_{\theta \theta}}{r} \\ \frac{\partial \bar{\bar{\sigma}}_{rz}}{\partial r} + \frac{\partial \bar{\bar{\sigma}}_{zz}}{\partial z} + \frac{\bar{\bar{\sigma}}_{rz}}{r} \end{pmatrix} &=& \mathbf{F} \\ \frac{\partial \mathbf{p}}{\partial t} + \tilde{\nabla} \cdot \begin{pmatrix} - \bar{\bar{\sigma}}_{rr} & - \bar{\bar{\sigma}}_{rz} \\ - \bar{\bar{\sigma}}_{rz} & - \bar{\bar{\sigma}}_{zz} \end{pmatrix} &=& \mathbf{F} + \begin{pmatrix} \frac{\bar{\bar{\sigma}}_{rr} - \bar{\bar{\sigma}}_{\theta \theta}}{r} \\ \frac{\bar{\bar{\sigma}}_{rz}}{r} \end{pmatrix} \\ \frac{\partial \mathbf{p}}{\partial t} + \tilde{\nabla} \cdot \begin{pmatrix} - \lambda \left( \frac{\partial u_r}{\partial r} + \frac{u_r}{r} + \frac{\partial u_z}{\partial z} \right) - 2\mu \frac{\partial u_r}{\partial r} - \sigma_T & - \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \\ - \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) & - \lambda \left( \frac{\partial u_r}{\partial r} + \frac{u_r}{r} + \frac{\partial u_z}{\partial z} \right) - 2\mu \frac{\partial u_z}{\partial z} - \sigma_T \end{pmatrix} &=& \begin{pmatrix} F_r + \frac{2\mu}{r} \left( \frac{\partial u_r}{\partial r} - \frac{u_r}{r} \right) \\ F_z + \frac{\mu}{r} \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \end{pmatrix} \end{eqnarray*}\]

We can write thise equation to the form :

\[ \hat{d} \frac{\partial \mathbf{p}}{\partial t} + \tilde{\nabla} \cdot \hat{\gamma} = \hat{f}\]

With :

\(\hat{d} = 1\)

\(\hat{\gamma} = \begin{pmatrix} - \lambda \left( \frac{\partial u_r}{\partial r} + \frac{u_r}{r} + \frac{\partial u_z}{\partial z} \right) - 2\mu \frac{\partial u_r}{\partial r} - \sigma_T & - \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \\ - \mu \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) & - \lambda \left( \frac{\partial u_r}{\partial r} + \frac{u_r}{r} + \frac{\partial u_z}{\partial z} \right) - 2\mu \frac{\partial u_z}{\partial z} - \sigma_T \end{pmatrix}\)

\(\hat{f} = \begin{pmatrix} F_r + \frac{2\mu}{r} \left( \frac{\partial u_r}{\partial r} - \frac{u_r}{r} \right) \\ F_z + \frac{\mu}{r} \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \end{pmatrix}\)

\(\hat{c}\), \(\hat{\gamma}\) and \(\hat{f}\) aren’t the CFPDEs coefficients of (Static Elasticity Axis).

Now, we pass to the weak formulation :

\[\begin{eqnarray*} \int_{\Omega^{axis}}{ \left( \hat{d} \frac{\partial \mathbf{p}}{\partial t} + \tilde{\nabla} \cdot \hat{\gamma} \right) \cdot \xi \, r \, dr dz } &=& \int_{\Omega^{axis}}{ \hat{f} \cdot \xi \, r \, dr dz} \\ \int_{\Omega^{axis}}{ \left( \hat{d} \frac{\partial \mathbf{p}}{\partial t} \right) \cdot \xi \, r \, dr dz } + \int_{\Omega^{axis}}{ -\hat{\gamma} \cdot \tilde{\nabla} (r \xi) dr dz } - \int_{\Gamma^{axis}}{ ... } &=& \int_{\Omega^{axis}}{ \hat{f} \cdot \xi \, r \, dr dz} \\ \int_{\Omega^{axis}}{ \left( \hat{d} \frac{\partial \mathbf{p}}{\partial t} \right) \cdot \xi \, r \, dr dz } + \int_{\Omega^{axis}}{ \left( -\hat{\gamma} \cdot \tilde{\nabla} \xi + \begin{pmatrix} -\hat{\gamma}_{00} \\ - \hat{\gamma}_{10} \end{pmatrix} \xi \right) dr dz } - \int_{\Gamma^{axis}}{ ... } &=& \int_{\Omega^{axis}}{ \hat{f} \cdot \xi \, r \, dr dz} \end{eqnarray*}\]

We have the system of two equations :

\[\left\{ \begin{eqnarray*} \int_{\Omega^{axis}}{ d \frac{\partial \mathbf{p}}{\partial t} + \tilde{\nabla} \cdot \gamma \ dr dz } &=& \int_{\Omega^{axis}}{ f \ dr dz } \\ \int_{\Omega^{axis}}{ d' \frac{\partial \mathbf{u}}{\partial t} \, dr dz } &=& \int_{\Omega^{axis}}{ f' \ dr dz } \end{eqnarray*} \right.\]

Thus the coefficients are :

For first equation (of unknow the momentum \(\mathbf{p}\))

Coefficient

Description

Expression

\(d\)

\(r \hat{d} = r \)

\(\gamma\)

conservative flux source term

\(r \hat{\gamma} = \begin{pmatrix} - \lambda \left( r \frac{\partial u_r}{\partial r} + u_r + r \frac{\partial u_z}{\partial z} \right) - \mu \, r \, \frac{\partial u_r}{\partial r} - r \sigma_T & - \mu \, r \, \frac{\partial u_z}{\partial r} \\ - \mu \, r \, \frac{\partial u_r}{\partial z} & - \lambda \left( r \, \frac{\partial u_r}{\partial r} + u_r + r \, \frac{\partial u_z}{\partial z} \right) - \mu \, r \, \frac{\partial u_z}{\partial z} - r \sigma_T \end{pmatrix}\)

\(f\)

source term

\(r \hat{f} + \begin{pmatrix} \hat{\gamma}_{00} \\ \hat{\gamma}_{10} \end{pmatrix} = \begin{pmatrix} r F_r - \lambda \frac{\partial u_r}{\partial r} - (\lambda+2\mu) \frac{u_r}{r} - \lambda \frac{\partial u_z}{\partial z} - \sigma_T \\ r F_z \end{pmatrix}\)

For the second equation (of unknow the displacement \(\mathbf{u}\)) :

Coefficient

Description

Expression

\(d'\)

\(r \rho \)

\(f'\)

source term

\(r \mathbf{p}\)

3. Documentation

Elasticity adn toolbox Coefficient Form PDEs, Céline Van Landeghem, 2021, unpublished, Download the PDF

Coupled modelling and simulation of electromagnets in stationnary and instationnary modes (axisymmetric and 3D), J. Nadarasa, June 2 2015, unpublished, page 20-24, Download the PDF

Theorical Physics Reference, Ondřej Čertík, November 29 2011, page 85-90 Download the PDF