Abstract
Auxetic materials, characterised by a negative Poisson's ratio, have properties that are different from most conventional materials. These are a result of the constraints on the kinematics of the material's basic structural components, and have important technological implications. Models of these materials have been studied extensively, but theoretical descriptions have remained largely limited to materials with an ordered microstructure. Here we investigate whether negative Poisson's ratios can arise spontaneously in disordered systems. To this end, we develop a quantitative description of the structure in systems of connected basic elements, which enables us to analyse the local and global responses to small external tensile forces. We find that the Poisson's ratios in these disordered systems are equally likely to be positive or negative on both the element and system scales. Separating the strain into translational, rotational and expansive components, we find that the translational strains of neighbouring basic structural elements are positively correlated, while their rotations are negatively correlated. There is no correlation in this type of system between the local auxeticity and local structural characteristics. Our results suggest that auxeticity is more common in disordered structures than the ubiquity of positive Poisson's ratios in macroscopic materials would suggest.
Highlights
Materials with a negative Poisson’s ratio (PR), called auxetics, have positively correlated horizontal and vertical strains, and expand in both directions when stretched.[1]
Auxeticity of disordered structures is still far from fully understood. We address this problem by modelling auxeticity in isostatic structures that consist of minimally connected constituent units that can freely fold, expand and contract.[18,19]
Provides a local definition of the PR and makes it possible to study the spatial distribution of this property across the system
Summary
Materials with a negative Poisson’s ratio (PR), called auxetics, have positively correlated horizontal and vertical strains, and expand (contract) in both directions when stretched (compressed).[1]. Identifying the Nb boundary triangles in this structure, it is convenient for the purpose of our analysis to enclose this graph within a frame to which the boundary triangles are connected by one of their vertices. For such a graph, Euler’s topological relation for two-dimensional graphs in the plane is,[32]. The vectors rct and Rct are the diagonals of a quadrilateral volume element, which is called quadron.[29] Inter-triangle forces (red) are calculated as the difference between the loop forces l c (blue) of neighbouring cells. By calculating the fabric tensor, the quantitative description of the local structure of our isostatic systems can be coarse-grained to the overall system.[26,27,30] the fabric tensor is related to local rotational strains,[31] and can be used to identify and isolate rotational strains in auxetic materials.[18,19] In this paper, we extend this latter development and apply it to random systems
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