NEMESIS: NEw Mathematics for Materials Modelling in the Engineering Sciences and Industrial Sectors

For millennia, mankind has recognized the importance of inhomogeneous media: a combination of two or more individual materials. Such materials are far better than the sum of their parts, e.g. leading to a huge increase in stiffness or strength. Today a plethora of so-called composite or smart materials exist enabling wondrous scientific and engineering advances in a many sectors including aerospace, automobile, structural, communications and acoustic engineering, biotechnology and health, leisure and the nuclear sector to name but a few. Inhomogeneous materials also arise naturally in a number of contexts, e.g. biological tissues where there are often several scales of inhomogeneity.

A natural, often difficult research challenge is to predict the effective behaviour of inhomogeneous materials from knowledge of the properties of the constituent phases and their distribution. Such materials often possess rather surprising and counter-intuitive properties, for example the speed of sound in bubbly water is faster than that in either water or air! Amongst other benefits, models of inhomogeneous media are important for design optimization strategies for composites, for the replacement of prohibitively costly experiments in engineering applications and for understanding structure-function relationships in biomechanics.

In addition to their effective behaviour, the way that waves propagate through such materials is of great importance. In recent times metamaterials have been devised which allow incredible non-intuitive properties such as strong absorption and filtering properties, waveguiding and localization capabilities and the exciting notions of negative refraction, focussing behaviour and even cloaking!

This project focuses on the development and application of new mathematical methods and models associated with complex inhomogeneous, generally nonlinear, materials. Three themes focus on (A) Industrial composites, (B) metamaterials and phononics, (C) Soft biomaterials. Despite there being three distinct themes, there exists a great deal of overlap between these topics meaning that methods developed in one area can also apply to other, apparently unconnected topics. This is the beauty of applied mathematics!

In theme (A) the team will work with project partner Thales Underwater Systems Ltd in order to understand the way that sound propagates through complex composite materials when they are subject to high pressures. The load significantly modifies the microstructure of the material and subsequent response to propagating waves and as such the prediction of the reflected and transmitted sound field from such materials is a non-trivial task.

Theme (B) will further research into hyperelastic cloaking theory, a technique recently developed by the PI, which uses pre-stressed materials in order to guide waves around specific regions of space. They will also understand further the way that special materials with periodic microstructure can act as wave filters by permitting or restricting wave propagation at given frequencies. In particular the interest is tunable materials so that we can modify the material response at will be applying a pre-stress, or magnetic field for example.

In theme (C) the team will develop models for the behaviour of soft tissues: tendon and skin, using information from the microstructure in order to ``upscale'' to macroscopic models. Soft tissues are highly deformable and in particular are viscoelastic meaning that energy is lost during deformation. The prediction of the loading and unloading of such materials is a notoriously difficult task, made even harder in skin due to its complex structural organization. A full understanding of the way that such materials behave has a multitude of applications in medicine and pharmaceutical industries.

Models developed are continuously informed and iterated by input from experimental collaborators, whose work is of great importance to this project.

We shall describe the beneficiaries of the research under each theme of the proposed project.

Theme A is focused on industrial composites and specifically composite materials of interest to the project partner Thales Underwater Systems Ltd, a British company with several offices, including the main contact base at Cheadle Heath, close to Manchester. As such they stand to directly benefit economically from the research undertaken in this theme. In particular, we quote from their letter of support:

"In terms of impact, TUSL would anticipate that this research leads to a significant advance in the capability of theoretical models that can be applied in the early design phase of major military systems such as submarines. The advanced models would be used to investigate the impact of the application of new composite materials on for example the radiated noise signature of such platforms and how by exploring different design options its acoustic signature can be minimised in the most cost effective way."

Additionally, in the letter Thales say

"As well as a benefit to TUSL in maintaining its leading industrial position is this aspect of platform characterisation the techniques when implemented should also be regarded as a UK strategic asset in maintaining a national capability at the forefront of future submarine design. "

As a result of this we expect long term economic benefit to UK PLC but also societal benefit of having strong defence mechanisms in place.

In theme B the topic is the development of new metamaterials and phononic materials. Major breakthroughs in this area can lead to the development of new smart materials for manipulating elastodynamic and acoustic waves over an extremely broad range of frequencies, from very low frequency (seismic) waves through to sonic and ultrasonic waves (medical sensing, aerospace and defence, sound systems, traffic and other noise pollution). Success will enable exciting new developments in both theoretical and practical aspects of smart materials and device design for use in a multitude of novel applications, for example in underwater acoustics, sound reduction in buildings, use of composite materials in the aerospace, defence and automobile industries, structural applications such as seismic isolation and vibration reduction and nano-electronics. Developments in these areas are of great interest to a number of companies, one of which, the British company Dyson have just initiated a collaboration with the PI on a separate project in this line of research. Impact in these areas and subsequent material and device improvement has direct economic benefit through the input to such companies and also societal benefit through the improved products and associated impact on peoples' lifestyles.

In theme C, we focus on an improved understanding of soft tissues and specifically tendon and skin. A current EPSRC Doctoral prize project with which the PI is involved works on similar developments in the cruciate ligament with potential influence on surgical developments by engaging with a local consultant surgeon at Stepping Hill Hospital, UK. We expect that similar developments can be forthcoming with the work on tendon proposed in this project. Long term societal benefits are the possible improvement to surgical techniques. Specific to skin, microstructural models can give important insights into the structure-function relationships of living tissues. For example, the structural arrangement of collagen is often associated with pathological disorders, such as the Ehlers-Danlos syndrome, a connective tissue disorder caused by a defect in the synthesis of collagen. In this context therefore, full biomechanical models of the skin would allow the investigation of the individual dermal components on the overall tissue response. Such models therefore have long-term societal benefit in the sense that a better understanding can lead to more effective treatment strategies.