The Princess and the Pea: Mathematical Design of Neutral Inclusions and their Fabrication

Lead Research Organisation: University of Manchester
Department Name: Mathematics


Hans Christian Andersen's fairytale of the ``Princess and the Pea'' describes how a girl (subsequently a princess) cannot sleep in her bed due to the location of a very hard pea under her numerous mattresses. Despite its distance from the top mattress the influence of the pea extends upwards, ensuring that the girl in question is uncomfortable due to its presence. The presence of the pea here is analogous to that of inclusions, which are ubiquitous in structures, composite materials and metamaterials, and like the pea, their presence disturbs the local stress and strain fields in the background or matrix medium in which they reside. Inclusions are introduced to enhance material properties, e.g. stiffness, toughness, thermal or electrical conductivity, etc. but the resulting stress concentrations can often lead to crack initiation and subsequent global material failure. This can have catastrophic consequences in applications, including ships sinking and planes falling out of the sky. A classic example is the Comet aircraft's square window, which contributed to the structural failure of the plane in flight, the window itself here being the ``inclusion'' in the aircraft skin. Numerous mechanisms to mitigate these problems have been proposed and introduced. These include careful design of the shape of inclusions (cf. windows on the Comet) and indeed whole textbooks have been written detailing stress concentrations around specific structural artefacts. Reinforcements or coatings are also used on inclusions, although typically the aim of this is to improve the bonding between the matrix and inclusion (stress concentrations remain relatively high) or to obtain only modest improvements in composite toughness.

In this project we seek to develop a concept that has never been fully exploited: that of the Neutral Inclusion (NI). These are coated inclusions, where the coatings are designed to ensure that stress fields exterior to the NIs are unperturbed upon loading, as if the NI was absent. The concept is that coatings can ``cloak'' the presence of the inclusion under a range of applied loadings, so e.g. a ``neutral pea'' would not perturb the mattresses around it, thus yielding a more comfortable bed. In principle therefore, NIs enable a redistribution of stress, a decrease in stress concentration (reduced failure likelihood) and an enhanced materials design space, including lighter, stronger materials.

Although advances have been made in thermal and electrical applications, the vectorial nature of the equations of elasticity means that it has not been possible to design complete NIs, i.e. NIs that yield neutrality to more than one loading state. We thus call NIs that are neutral only for one loading state "incomplete", given that this property significantly reduces their attractiveness for advanced materials applications. Our new, very recent results however, illustrated that this bottleneck was due to the previous overly simplistic restriction of isotropic NI coatings. Relaxing this assumption and incorporating anisotropic coatings opens up a host of new opportunities, including neutrality under multiple loading states, giving rise to the potential for complete NIs.

In this project we will use mathematical techniques to design complete NIs and then fabricate them for the first time, with a view to deployment in advanced materials applications, enabling the design of lighter, stiffer and more durable materials. The project will open up new avenues in metamaterials research associated with the control of material properties for structural applications and in the manipulation of elastic waves.


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Description The aim of this project was to use mathematics to design and fabricate neutral inclusions. These are coated inclusions with coatings designed to reduce the stress concentration that builds up around inclusions when used in composite materials.
In this project the maths employed in the paper that motivated the work (Norris and Parnell, 2020) was employed as a building block to design samples including long fibres. The idea was to measure the strain inside and around such microscale fibrous inclusions (and thus to infer the stress) and then to design coatings to reduce the resulting concentrations. Samples were fabricated and the challenge was then to measure the strain using Digital Volume Correlation (DVC). DVC requires small tracer particles to be embedded uniformly in the matrix medium hosting the fibre. A key finding was the development of a repeatable, stable technique to fabricate samples with such particles uniformly mixed in the matrix.
Although early results were promising issues with DVC software meant that strain fields could not be imaged accurately. Investigations are ongoing as to why this was the case. Due to these issues work as started on macro-scale inclusions embedded in a thin plate. These are photo-elastic materials that can be 3d printed. Their photo-elasticity combined with their larger scale means that the resulting strain and stress distributions are more easily visualised than the microscale fibres described above. Early stage experiments show promise and work will continue this year via alternative routes now that the project has finished.
In terms of mathematics, significant progress has been made on the three-dimensional neutral inclusion problem, which is the natural extension of the Norris & Parnell (2020) paper that motivated this work. We anticipate that this work will be completed over the next 6-12 months and will thus act as a broader tool in neutral inclusion design.
Exploitation Route We have generated interest around the concept of neutral inclusions and their design, particularly with regard to talks by Parnell (the PI) on the concepts of neutral inclusions and their link to metamaterials and composites. An important facet of this work was to establish a collaboration between the mathematicians and materials scientists involved in the project. This team has now developed a range of future projects and this initial pilot project was an important part of that development. We anticipate that the project will lead to interesting and important work in the future. More time is required to fully exploit the initial work undertaken but with time we envisage that the work has the potential to be important in the areas of composite materials development.
Sectors Aerospace, Defence and Marine,Manufacturing, including Industrial Biotechology,Transport