Does anyone know how Bloch-Floquet boundary conditions work? 1

[tugni925]

I am working with microstructures and buckling, and often this term comes up:

"Subsequently, under a given macroscopic stress state, material buckling strength can be evaluated using linear buckling analysis with
Bloch-Floquet boundary conditions to cover all the possible buckling modes in the material microstructure. For a
prescribed macroscopic stress state, σ0, the effective elasticity matrix is used to transform the macroscopic stress to
the macroscopic strain, written as..." (Extreme 3D architected isotropic materials with tunable stiffness and buckling strength)

Is it possible to explain how Bloch-Floquet boundary conditions work?

 

[rb1957]

I did a quick google for "Bloch-Floquet boundary conditions" ... not sure if you've done this (you don't say so I assume you didn't).

got several good looking hits, but seemed like gibberish to me ('cause I haven't come across the term before, and the paper presented Way too much math).

I'd've thought that any FEA program that could handle this analysis would be able to help (ie can your FEA help desk, if you don't have an FEA call around ... all the usual suspects).

another day in paradise, or is paradise one day closer ?

 

[tugni925]

Googling it is my first go-to, and like you said it was a bit difficult to understand unfortunately.

 

[onatirec]

Here's what I posted here last time this question was asked...:

"floquet-bloch b.c.s are used in quantum mechanics to solve schrodingers equation with a periodic potential. You can think of the crystalline structure of a metal as having periodic boundary conditions. Effectively, this allows you to extrapolate out from a single crystal (or periodic element), the behavior on a -macro scale. By looking at a single element, you can see the effects of modes that are larger in span than a single element. Practically, this means you can exploit the periodicity to simplify the calculations (and reduce the meshed region to a single period)."

 

[NING GA]

Actually, the Floquet-Bloch boundary condition is similar to the periodic boundary condition of homogenization method. In the homogenization method, the initial unit strain can transform to the pre-defined displacement boundary. Then, u can solve the KU=F to obtain the displacement of each node. Finally, u can calculate the strain energy to obtain all the components in the elastic tensor. Similarly, the Floquet-Bloch boundary condition should satisfy the equation you posted. That means the eigenvector of some nodes on the boundary have already defined and the two symmetrical boundaries should satisfy the equation. Then you can solve the Equation to get the eigenvalues and the eigenvalues is the buckling load factor. Finally, you can change the value of the k-vector t get the band gap.