\[\require{physics} \renewcommand{\vec}[1]{\underline{#1}} \def\mat#1{\vec{\vec{#1}}} \def\ten#1{\vec{\vec{\vec{\vec{#1}}}}}\]

About

Toppic

The reason for me is to solve classical problems in Solid Mechanics. This section, and those below will introduce the typical equations that are encountered in solid mechanics. This section is not exhaustive and it might be extended in the future to discuss more details.

Bram Lagerweij
11 Feb 2020

The examples will become gradually more complex. It starts with the simplest problem, the Laplace equation:

\[{\grad}^2 u(\vec{m}) = 0 \qquad \forall \vec{m}\in\Omega\]

In here one can imagine various levels of complication:

With a simple geometry, no sharp corners, and a combination of Neuman and Diriclet boundary conditions.
With a more complex geometry, sharp corners, cracks and inclusions.
With a ‘non-linear’ stiffness, \({\grad}\vdot\qty(\vec{C}\,{\grad}u(\vec{m}))\) adding a non-constant variable \(\vec{C}\) which is a function depending somehowe on \(\grad u\).
Where \(\vec{C}\) is non-linear and history dependent, aka \(\vec{C}^{(n+1)}\) is a function af all previous timesteps.
With softening in the non-linear stiffness \(C\), that is the tangent of \(\vec{C}\,\grad u\) will become negative at some point.
Versions in 3D

Moving on to solids where we solve elasticity and plasticity equations:

\[\begin{split}\grad\vdot\,\mat{\sigma} + \vec{b} &= 0 \qquad \forall \vec{m}\in\Omega\\ \qq{where} & \mat{\sigma} = \ten{C}:\mat{\varepsilon} \\ & \mat{\varepsilon} = \frac{1}{2}\big(\grad \vec{u} + (\grad \vec{u})^T\big)\end{split}\]

The simplest problem would be linear elasticity, but more complicated versions can be build as well.

With a simple geometry, no sharp corners, and a combination of Neuman and Diriclet boundary conditions.
With a more complex geometry, sharp corners, cracks and inclusions.
Large displacements (geometrically non-linear) and deformations (this might require a different strain measure).
Softening and possbily fracture.
Self Contact.