Limit Analysis of Collapse States in Cellular Solids

蜂窝固体是被分成多个细胞的二维或三维物体，其壁由固体材料制成，能够进行(大的)弹性变形而不发生塑性破坏或断裂。由于其特殊的机械效率，它们在自然界和工业中无处不在，但与几乎任何其他类别的材料相比，人们对它们的了解较少。这些结构最广为人知的机械特性之一是它们的高强度与重量比和能量吸收能力，这是由于其几何结构和底层固体材料的本构性质之间密不可分的关系。这个项目的目的是通过提供一个新的框架来理解非线性弹性材料的蜂窝结构的力学特性，并提供严格的数学方法来分析它们在外部载荷作用下的大弹性变形，从而深入了解非线性弹性材料的蜂窝结构的特性和它们在载荷下的整体响应。考虑到弹性蜂窝壁的几何和力学性质之间的相互作用，将设计新的数值方法来计算各种蜂窝结构中由蜂窝闭合引起致密化的临界载荷的有效上下界，并将界限间隙作为计算误差的指标。在这种背景下，所提出的研究和非标准数值方法是新颖的，具有许多潜在的应用。例如，生物医学应用中用于软组织再生长的新型柔性支架和支架的开发是生物材料和组织工程的一个快速发展的多学科领域，许多为缓冲和重复使用而设计的泡沫和海绵也可以在日常生活以及几个工业领域中找到，例如微电子、航空航天、制药和食品加工。为了理解和优化这些复杂材料的机械响应，需要有严格的数学机械分析支持的可靠的计算机模型，并可能为新的应用开辟道路。

Cellular solids are two or three dimensional bodies divided into cells, the walls of which are made of a solid material capable of undertaking (large) elastic deformations without plastic failure or fracture. Due to their exceptional mechanical efficiency, they are ubiquitous in nature and industry, yet they are less well understood than almost any other class of materials. Among the best known mechanical qualities of these structures are their high strength-to-weight ratio and their energy absorption capacity, which are due the inextricable relation between the geometric architecture and the constitutive properties of the underlying solid mater. The aim of this project is to gain insight into the properties of cellular structures of nonlinear elastic material and their overall response under loads by providing both a new framework to understand the mechanical properties of these structures and rigorous mathematical techniques for the analysis of their large elastic deformations caused by the application of external loads. Taking into account the interplay between the geometry and the mechanical qualities of the elastic cell walls, novel numerical methods will be devised to compute effective lower and upper bounds for the critical load causing densification by cell closure in various cellular structures, and the bounds gap will be used as an indicator for the computational error. In this context, the proposed investigation and non-standard numerical procedures are novel and have many potential applications. For example, the development of new flexible stents and scaffolds for soft tissue re-growth in biomedical applications is a rapidly growing multidisciplinary area of biomaterials and tissue engineering, and many foams and sponges designed for cushioning and re-usability can also be found in everyday life as well as in several industrial areas, e.g. microelectronics, aerospace, pharmaceutical and food processes. For these complex materials to be understood and optimised with respect to their mechanical response, reliable computer models supported by rigorous mathematical mechanical analysis are needed, and may also open the way for new applications.