version 3.10.0
Loading...
Searching...
No Matches

Deformation of a solid body using the theory of linear elasticity (small deformations) More...

Go to the source code of this file.

Description

This model describes the deformation of a solid body using the theory of linear elasticity. The momentum balance equation of the solid body can be expressed by

\[\nabla\cdot\boldsymbol{\sigma} + \rho \mathbf{g} + \mathbf{f} = \rho\ddot{\mathbf{u}}, \]

where \( \boldsymbol{\sigma} \) is the stress tensor, \( \rho \) is the density of the solid, \( \mathbf{f} \) in \( \mathrm{N/m^3} \) is the external force acting on the body per unit volume (e.g. magnetism), and \( \mathbf{u} = \mathbf{x} - \mathbf{x}_{\mathrm{initial}} \) is the displacement, defined as the difference in material points \( \mathbf{x} \) and \( \mathbf{x}_{\mathrm{initial}} \) in the deformed and undeformed (initial) state, respectively. The model assumes quasi-static conditions, that is, the above momentum balance equation is solved under the assumption that the acceleration term \( \rho\ddot{\mathbf{u}} \approx 0\).

Per default, Hookes' Law is used for expressing the stress tensor \( \boldsymbol{\sigma} \) as a function of the displacement:

\[\boldsymbol{\sigma} = \lambda\mathrm{tr}(\boldsymbol{\varepsilon}) \mathbf{I} + 2G \boldsymbol{\varepsilon}, \]

with

\[\boldsymbol{\varepsilon} = \frac{1}{2} \left[ \nabla\mathbf{u} + (\nabla\mathbf{u})^{\mathrm{T}} \right]. \]

Primary variables are the displacements in each direction \( \mathbf{u} \). Gravity can be enabled or disabled via a runtime parameter.