I've been trying to create a simple finite element analysis script for determining the strains and stresses in 3-D structures composed of a linearly elastic isotropic material.
I've found a couple of examples of something similar to what I'd like to do here for the 3D structures, and also here in a simplified 2D version (Go to "Deformation of a Beam under load" section).
My current issue is that I don't understand the equation input as the PDE. It looks like it might be related to the Navier-Cauchy Equation, i.e.
$$\left( \lambda+\mu\right)*\nabla(\nabla \cdot u)+\mu *\nabla^2u+F=0$$ Where:
$\ F$ is the force vector
$\mu$ is the second Lame parameter
$\lambda$ is the first Lame parameter/ Shear Modulus
$\ u$ is the displacement vector.
I'm not totally sure this is the equation used, and if it is I can't figure out how or why it was manipulated into the form used in these examples.
Any help would be greatly appreciated, thank you.
It seems that they have simply written down the operator $$ \mathrm{div}\,\sigma(u) $$ explicitly in the corresponding dimensions. In linear elasticity, $$ \sigma(u) = 2 \mu\,\varepsilon(u) + \lambda\,\mathrm{tr}\,\varepsilon(u) I $$ and $$ \varepsilon(u) = \frac{1}{2}(\nabla u + \nabla u^T). $$ You can find a relationship between the Young's modulus, Poisson ratio $(Y,\nu)$ and the Lame parameters $(\mu,\lambda)$ from Wikipedia and work out the details by hand or by a CAS.