Novel superior materials based on aperiodic tilings
Lead Research Organisation:
The Open University
Department Name: Faculty of Sci, Tech, Eng & Maths (STEM)
Abstract
Our world contains ample manifestations of order, both in products of human civilisation (such as art, music or architecture) and in the natural world (with crystals representing the ultimate ordered structure). The surprise discovery of quasicrystals, honoured with the 2011 Nobel Prize in Chemistry, inspires continuing research into properties and applications of these fascinating materials, which exhibit order without periodicity, and are formalised in the mathematics of aperiodically ordered structures.
Aperiodically ordered spatial arrangements open up fascinating opportunities of purpose-made structures, including smart materials. Additive Manufacturing makes it possible to produce such materials cheaply and reliably, with potentially huge impact across a vast area of applications, such as bespoke orthopaedic implants, one of a kind space components and aerospace components produced from valuable raw materials. The investigation of aperiodic metamaterials has only just begun, with an emphasis on photonic materials; there have been no attempts as yet to explore materials with superior mechanical properties based on aperiodic arrangements.
For instance, cellular structures are used as a means of tailoring the stiffness of orthopaedic implants to match that of the receiving bone. The current state of the art employs cubic or hexagonal lattices, which result in undesirable mechanical anisotropy, often requiring over-engineering to ensure sufficient mechanical performance. To recreate the pseudo-random structures present in bone in a repeatable fashion using digital manufacturing is a key limitation in the realisation of 'print-to-order' orthopaedic implants. Aperiodic structures have a key advantage: they can realise higher symmetries that are incompatible with lattice periodicity, making it possible to avoid anisotropies while maintaining a well-defined, deterministic, algorithmically reproducible structure.
The proposed research will lay the mathematical groundwork for applications. We will create, establish, evaluate and verify mathematical models to determine the elastic properties of nearly isotropic cellular materials based on aperiodic tilings.
Aperiodically ordered spatial arrangements open up fascinating opportunities of purpose-made structures, including smart materials. Additive Manufacturing makes it possible to produce such materials cheaply and reliably, with potentially huge impact across a vast area of applications, such as bespoke orthopaedic implants, one of a kind space components and aerospace components produced from valuable raw materials. The investigation of aperiodic metamaterials has only just begun, with an emphasis on photonic materials; there have been no attempts as yet to explore materials with superior mechanical properties based on aperiodic arrangements.
For instance, cellular structures are used as a means of tailoring the stiffness of orthopaedic implants to match that of the receiving bone. The current state of the art employs cubic or hexagonal lattices, which result in undesirable mechanical anisotropy, often requiring over-engineering to ensure sufficient mechanical performance. To recreate the pseudo-random structures present in bone in a repeatable fashion using digital manufacturing is a key limitation in the realisation of 'print-to-order' orthopaedic implants. Aperiodic structures have a key advantage: they can realise higher symmetries that are incompatible with lattice periodicity, making it possible to avoid anisotropies while maintaining a well-defined, deterministic, algorithmically reproducible structure.
The proposed research will lay the mathematical groundwork for applications. We will create, establish, evaluate and verify mathematical models to determine the elastic properties of nearly isotropic cellular materials based on aperiodic tilings.
Organisations
Publications
Clarke D
(2023)
A systematic numerical and experimental study into the mechanical properties of five honeycombs
in Composites Part B: Engineering
Clarke D
(2023)
An isotropic zero Poisson's ratio metamaterial based on the aperiodic 'hat' monotile
in Applied Materials Today
Clarke D
(2024)
Identification of mechanically representative samples for aperiodic honeycombs
in Materials Today Communications
Imediegwu C
(2023)
Mechanical characterisation of novel aperiodic lattice structures
in Materials & Design
Imediegwu C
(2023)
A computational method for determining the linear elastic properties of 2D aperiodic lattice structures
in The Journal of Strain Analysis for Engineering Design
Moat R
(2024)
A class of aperiodic honeycombs with tuneable mechanical properties
in Applied Materials Today
Moat R
(2022)
Compressive behaviour of cellular structures with aperiodic order
in Results in Materials
| Description | The Novel Superior Materials Based on Aperiodic Tilings project has for the first time applied the geometry of aperiodic patterns to the design of mechanical meta-materials. Mechanical meta-materials are engineered structures, the behaviour of which is derived from internal geometry as well as the properties of the base-material and much research has gone into different design methodologies to vary the resulting properties. In the project the meta-material designs investigated range from the famous Penrose tilings to the recently discovered "Hat" monotile. The resulting structures have shown remarkable isotropic mechanical properties with otherwise unprecedented variability and with scope to tune different elastic mechanical properties independently. To achieve these results the project developed a new methodology for simulating the orientational dependent mechanical properties of non-periodic meta-materials. The first challenge when studying aperiodic metamaterials is the number of tests or simulations needed to quantify the orientation dependent performance. For known patterns assumptions can be made based on symmetry, however for aperiodic tilings little was known about the level of anisotropy expected or if similar links between anisotropy and rotational symmetry found in periodic patterns also exist. Therefore multiple samples constructed or simulated in multiple orientations would be required to quantify the anisotropy. We adapted the computational analysis method of asymptotic expansions homogenisation to extract elastic two dimensional mechanical performance from single simulations and enable the assessment of many aperiodic tilings. This enabled the investigation of many aperiodic tilings in multiple orientations and densities and facilitated targeted testing using state of the art spatially resolved strain measurement to be conducted across a range of tilings. Applying these methods to the recently discovered "Hat" monotile family of tilings has resulted in the discovery of a structure with zero Poisson ratio. When loaded in one direction, this structures does not deform in the perpendicular direction. This behaviour is rare in synthetic materials but is observed in natural materials such as cork and in several human tissues and is desirable in many engineering application. Most significantly, unlike other published research into synthetic zero Poisson's ratio meta-materials, the "Hat" based meta-material exhibits this property at a wide range of relative densities and irrespective of the orientation of applied load. |
| Exploitation Route | This new class of meta-materials based on the mathematics of aperiodic tilings give rise to a wide range of mechanical properties not available from traditional periodic structures. As a result, they offer the opportunity to manipulate the mechanical performance of meta-material to the needs of the end user without the need for complex or proprietary optimisation software. Because of the wide variety of properties found the potential use cases are also varied. For instance, the "Hat" monotile family of tilings offers such a wide variety of performances, with Poisson's ratio equal to zero at one extreme, well suited to morphing wing aerospace applications, to 0.45 at the other extreme, well suited to matching the properties of cortical bone for medical implants. Most exciting however, is the possibility to achieve all properties in-between these extremes, opening up opportunities to tailor the mechanical properties where required, such as in bespoke medical devices. |
| Sectors | Aerospace Defence and Marine Creative Economy Energy Manufacturing including Industrial Biotechology Pharmaceuticals and Medical Biotechnology |
| Description | Aperiodic tilings have a general appeal and have often been used to encourage the public to engage with otherwise inaccessible mathematical concepts. This research has created an additional dimension to this showing practical engineering potential for these aesthetic patterns. Our work has been included in a series of mathematics outreach events including, British Science Festival, HatFest and Festival of Communities. These have attracted audiences in excess of 1000 attendees, including school children. |
| First Year Of Impact | 2022 |
| Sector | Education |
| Impact Types | Societal |
| Description | Aperiodic tilings exhibition |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | Regional |
| Primary Audience | Public/other audiences |
| Results and Impact | The event consisted of a formal interdisciplinary academic conference, Art exhibition and hands on engagement activities in memory of the passing of Prof Uwe Grimm. This brought together engineers, mathematicians, artists and educators with a common interest in tilings and patterns. The event was extremely successful with over 700 visitors to the exhibition over two days, including 450 school students. Comments from those who attended included: 'I was greatly impressed by the range and depth of art displayed. For me, this was the largest collection of aperiodic art I have ever seen' and 'The exhibition was a unique experience bringing together mathematicians, engineers, artists, educators, school children and members of the local community.' Teachers of students who attended further commented "They certainly were given the opportunity to see maths in a very different way" and "This has had such an effect on one pupil that they have set up a geometric art club in the library at lunchtime back in school." Researchers from the NOVMAT team exhibited work from the project, presented in the formal academic session and led groups during the student sessions. The event resulted in internal collaboration and interest from new external partners. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://www.open.ac.uk/stem/mathematics-and-statistics/aperiodic-tilings-2022 |
| Description | British Science festival exhibition table |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Public/other audiences |
| Results and Impact | Exhibited meta-materials based on aperiodic order which had been tuned to have differences in stiffness detectible by compression by the hand. This initiated the engagement of 100's of members of the public with the mathematics of aperiodic order. |
| Year(s) Of Engagement Activity | 2023 |
| Description | Community lecture (Royal Latin school, Buckingham) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Schools |
| Results and Impact | A community lecture at the Royal Latin School in Buckinghamshire funded by the Lillingstone Trust. The title of the talk was "Elusive Mechanical Properties Hiding in More Elusive Mathematics" and was given to an audience of approximately 60, including structures, parents and members of the local community. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://www.royallatin.org/lillingstone-lectures/ |
| Description | Exhibitors at Hat Fest 2023 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Hatfest was a two-day celebration of The Hat, an aperiodic monotile where our team presented an exhibition space showing the mechanical properties of aperiodic tilings which led to coverage in the Media, social media and sparked new collaborations |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://sites.google.com/view/thegrimmnetwork/hatfest |
| Description | Invited talk at University California Berkeley |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | Talk at the department of Engineering, primarily to the Gu Group which has lead to discussions and plans for future collaboration. |
| Year(s) Of Engagement Activity | 2023 |