*In this series our group members pick five images that each illustrate an aspect of their PhD.*

**Will Parnell**, Professor of Applied Mathematics. PhD in Applied Mathematics, University of Manchester, 2004

**My thesis title:**Homogenization techniques for wave propagation in composite materials

**What does this mean?**

*Composite material*. A composite material is a material that is produced from two or more constituents, which usually have quite distinct properties. This usually gives rise to a material that has enhanced properties, e.g. unusually stiff or conductive. Composites typically have a microstructure which can be as small as hundreds of nanometres.*Wave propagation*. By “waves” here we mean any wave that propagates through a composite material, so it could be an acoustic, elastic or electromagnetic wave. Although the physics of these waves are quite different, the mathematics governing how we describe them is remarkably similar and so it means that methods developed for one type of wave can be used to describe another type with remarkable ease.*Homogenization technique*. Mathematically, homogenization is the study of differential equations with rapidly oscillating coefficients. This is important because given that composites consist of at least two different constituents, the properties of the material change with space and often over a very small length scale as noted above. Mathematically, this manifests itself as a differential equation that governs some physical property (e.g. acoustic pressure) with coefficients that vary rapidly in space, corresponding to e.g. density or bulk modulus.

The homogenisation technique makes sense of what the waves “see” or “feel” when they propagate through the composite material in question, normally working best in the case when the wave’s wavelength is much longer than the variation in microstructure. The techniques allow one to determine the “effective material properties” of the composite medium. I focussed mainly on elastic properties in my PhD.

Hover over the images to read the captions, or see below for plain text version.

*elastic waves*propagate inside

*fibre-reinforced*composites where the fibres are very long. Here their length is along the x3 axis. In such fibrous materials the microstructure (the fibres) can be considered to be characterised by a repeating periodic cell in the x1-x2 plane. The effective elastic properties of the material can then determined by applying homogenisation techniques. (At least) three different types of elastic waves can be considered to propagate – here illustrated as SH, SV and P waves.

*isotropic*, so that the wave speed is independent of direction of propagation. Here we illustrate this by determining the associated effective “antiplane” shear modulus of the material as a function of the size (radius r) of the fibre. The fibre is ten times as soft as the matrix, so that the more fibre added, the softer the composite becomes (the effective shear modulus decreases). I considered a new approach to the implementation of the so-called

*asymptotic homogenization scheme,*which is plotted here in orange. It can be seen that the results agree well with existing methods and energy bounds.

*integral equation method*of homogenisation. This is an extremely efficient new technique, which can be used straightforwardly to model composites where the inclusions have rather arbitrary cross-section and also to geometries where fibres are placed on unusual lattices. Here we illustrate results when fibres are located on a

*rectangular lattice.*This induces

*anisotropy*, so that the elastic waves propagate with different speeds in different directions. This is illustrated here by plotting the associated effective shear modulus when waves propagate in the x1 and x2 directions. Our new integral equation method is plotted in comparison to the asymptotic method here, showing good agreement in both cases.