Physics Model

From the general theory of linear elasticity, the following relation holds:

\[ C \epsilon =\sigma \] where $C$ is the hook tensor, $\epsilon$ is the strain tensor, and $\sigma$ is the stress tensor.

With Einstein notation:

\[ C_{ijkl} \epsilon_{kl}= \sigma_{ij} \]

The $\epsilon$ tensor from definition:

\[ \epsilon_{ij} = \frac{1}{2} (\frac{\partial u_i}{\partial r_j} +\frac{\partial u_j}{\partial r_i}) \] where $\vec{u}$ is the displacement field.

For one volume element the Newton II is the following:

\[ div(\sigma) = \rho \frac{\partial^2 \vec{u}}{\partial t^2} + S_{RL} \frac{\partial \epsilon }{\partial t} + K_{RL} \frac{\partial \vec{u}}{\partial t} + \vec{f} \] Where, the $\rho$ is the density of the material, and $K_{RL}$, $S_{RL}$ are the Rayleigh matrices. The $\vec{f}$ is the external force density. \[ div(C \epsilon) = \rho \frac{\partial^2 \vec{u}}{\partial t^2} + S_{RL} \frac{\partial \epsilon }{\partial t} + K_{RL} \frac{\partial \vec{u}}{\partial t} + \vec{f} \]

In this equation, when $\epsilon$ is replaced with its definition, the only unknown is the $\vec{u}$ displacement.

Stationary

The $\frac{\partial}{\partial t}$ derivatives can be neglected. The equation above simplifies to the following:

\[ div(C \epsilon) = \vec{f} \]

Periodic and Eigenvalues

Conditions: $\epsilon = \epsilon_A \cdot e^{i \omega t}$, $\vec{u} = \vec{u_A} \cdot e^{i \omega t}$, $\vec{f} = \vec{f_A} \cdot e^{i \omega t}$

\[ div(C \epsilon_A) = - \rho \omega^2 \vec{u_A} + i \cdot \omega S_{RL} \epsilon_A + i \cdot \omega K_{RL} \vec{u_A} + \vec{f_A} \]

The eigenvalue solver finds the $\omega$ values, which fulfill the equation above.

Crystal Stress

For crystal simulation a new material parameter could be added, the lattice constants. When two joining crystal face have different lattice constants, it builds up an additional strain. It alters the equations above with the following term: \[ \vec{f} = \vec{f_{ext}} +\vec{f_{latticestrain}} = \vec{f_{ext}} + div(C\epsilon_{latticestrain}) \]

The $\epsilon_{latticestrain}$ definied in the principal directions:

\[ \epsilon_{latticestrain} = \begin{bmatrix}\frac{a_1}{a_2}-1 & 0& 0\\0 & \frac{b_1}{b_2}-1& 0\\0 & 0& \frac{c_1}{c_2}-1\end{bmatrix} \]

Where $a_i$, $b_i$, $c_i$ are the lattice contacts of the two joining crystals.

The total strain is the difference of the built up strain and the strain from the displacement: \[ \epsilon_{ij} = \frac{1}{2}(\frac{\partial u_i}{\partial r_j}+\frac{\partial u_j}{\partial r_i}) - \epsilon_{latticestrain_{ij}} \]

Hook tensor

With different material structures the structure of the Hook tensor is also different. With Voigt notation presented.

Isotropic material

\[ C_{isotropic} = \begin{bmatrix} 2 \mu +\lambda & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & 2\mu+ \lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & 2\mu+ \lambda & 0 & 0 & 0 \\ 0&0&0&\mu&0&0\\ 0&0&0&0&\mu&0\\ 0&0&0&0&0&\mu\\ \end{bmatrix} \] with $\lambda$, $\mu$ Lamé constants.

Examples

2D Simulations

Stationary study 2D

Periodic Study 2D

Crystal Stress 2D

3D Simulations

Nut under volume force

Nanowire Bending