Novel superior materials based on aperiodic tilings

我们的世界包含了大量的秩序表现，无论是在人类文明的产品（如艺术，音乐或建筑）和自然世界（与水晶代表最终有序的结构）。准晶的发现令人惊讶，并获得了2011年诺贝尔化学奖，这激发了人们对这些迷人材料的性质和应用的持续研究，这些材料表现出无周期的有序性，并在非周期有序结构的数学中得到了形式化。非周期有序的空间排列为特定结构（包括智能材料）提供了迷人的机会。增材制造使廉价可靠地生产此类材料成为可能，并对广泛的应用领域产生潜在的巨大影响，例如定制骨科植入物，一种由宝贵原材料生产的空间部件和航空航天部件。非周期性超材料的研究才刚刚开始，重点是光子材料;还没有尝试探索基于非周期性超材料的具有上级机械性能的材料。例如，细胞结构被用作定制骨科植入物的刚度以匹配接收骨的刚度的手段。现有技术采用立方或六边形晶格，这导致不期望的机械各向异性，通常需要过度设计以确保足够的机械性能。使用数字制造以可重复的方式重建骨中存在的伪随机结构是实现“按订单打印”骨科植入物的关键限制。非周期结构有一个关键的优势：它们可以实现与晶格周期性不相容的更高的对称性，从而可以避免各向异性，同时保持定义明确的，确定性的，算法可重复的结构。我们将创建，建立，评估和验证数学模型，以确定基于非周期性平铺的近各向同性蜂窝材料的弹性特性。

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.