Two-dimensional regular theoretical units that give a negative Poisson's ratio (NPR) are well documented and well understood. Predicted mechanical properties resulting from these models are reasonably accurate in two dimensions but fall down when used for heterogeneous real-world materials. Manufacturing processes are seldom perfect and some measure of heterogeneity is therefore required to account for the deviations from the regular unit cells in this real-life situation. Analysis of heterogeneous materials in three dimensions is a formidable problem; we must first understand heterogeneity in two dimensions. This paper approaches the problem of finding a link between heterogeneous networks and its material properties from a new angle. Existing optimisation tools are used to create random two-dimensional topologies that display NPR, and the disorder in the structure and its relationship with NPR is investigated.
The auxetic foams first produced by Lakes have been modelled in a variety of ways, each model trying to reproduce some observed feature of the microscale of the foams. Such features include bent or broken ribs or inverted angles between ribs. These models can reproduce the Poisson's ratio or Poisson's function of auxetic foam if the model parameters are carefully chosen. However these model parameters may not actually reflect the internal structure of the foams. A big problem is that measurement of parameters such as lengths and angles is not straightforward within a 3-d sample. In this work a sample of auxetic foam has been imaged by 3-d X-ray computed tomography. The resulting image is translated to a form that emphasises the geometrical structure of connected ribs. This connected rib data are suitably analysed to describe both the microstructural construction of auxetic foams and the statistical spread of structure, that is, the heterogeneity of an auxetic foam. From the analysis of the microstructure, observations are made about the requirements for microstructural models and comparisons made to previous existing models. From the statistical data, measures of heterogeneity are made that will help with future modelling that includes the heterogeneous aspect of auxetic foams.
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