# Joe Bishop

## Research student, School of Mathematics

bishoprj@cardiff.ac.uk
Room M/1.08, 21-23 Senghennydd Road, Cathays, Cardiff, CF24 4AG

## Research

### Thesis

#### Homogenisation of linear and non-linear problems

In this project I will derive the $\Gamma$-limit of a thin elastic periodically perforated plate, via simultaneous homogenisation and dimension reduction, starting from the non-linear elasticity setting. The plate has a thickness of $h$ and the period of the perforations is $\varepsilon$, where $h$ and $\varepsilon$ are both small parameters. I will use techniques from linearisation, homogenisation, dimension reduction and non-linear elasticity in my work. I will using ideas from the works by M. Cherdantsev, K. Cherednichenko, S. Neukammm, P.Hornung, I Vel\u{c}i\'{c}, G. Friesecke, R. James and S. M{\"u}ller. I will look at the cases where the period is of the same order as the thickness of the plate $(\varepsilon = O(h))$, where the thickness is greater than the period of the plate ($h\gg \varepsilon$) and when the thickness is less than the period of the plate ($\varepsilon^2 \gg h \gg \varepsilon$ and $h \ll\varepsilon^2)$. The form of the limit homogenised functional will depend on the particular choice of the regime. Upon successful completion of this task, I will study a similar problem were the perforations are filled with soft material.

Senior Lecturer