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Auxetic and other mechanical metamaterials are typically studied in situations where they are subjected solely to mechanical forces or displacements even though they may be designed to exhibit additional anomalous behaviour or tunability when subjected to other disturbances such as changes in temperature or magnetic fields. It is shown that externally applied magnetic fields can tune the geometry and macroscopic properties of known auxetics that incorporate magnetic component/s, thus resulting in a change of their macroscopic properties. Anomalous properties which are observed in such novel magneto-mechanical systems include tunable Poisson's ratios, bi-stability or multi-stability, depending on the applied magnetic fields, and other electromagnetic-mechanical effects such as strain dependent induced electric currents and magnetic fields. The properties exhibited depend, amongst other things, on the relative position and orientation of the magnetic insertion/s within the structure, the geometry of the system and the magnetic strength of the magnetic components, including that of the external magnetic field.

A three-dimensional cellular system that may be made to exhibit some very unusual but highly useful mechanical properties, including negative Poisson's ratio (auxetic), zero Poisson's ratio, negative linear and negative area compressibility, is proposed and discussed. It is shown that such behaviour is scale-independent and may be obtained from particular conformations of this highly versatile system. This model may be used to explain the auxetic behaviour in auxetic foams and in other related cellular systems; such materials are widely known for their superior performance in various practical applications. It may also be used as a blueprint for the design and manufacture of new man-made multifunctional systems, including auxetic and negative compressibility systems, which can be made to have tailor-made mechanical properties.

Unimode metamaterials made from rotating rigid triangles are analysed mathematically for their mechanical and thermal expansion properties. It is shown that these unimode systems exhibit positive Poisson's ratios irrespective of size, shape and angle of aperture, with the Poisson's ratio exhibiting giant values for certain conformations. When the Poisson's ratio in one loading direction is larger than +1, the systems were found to exhibit the anomalous property of negative linear compressibility along this direction, that is, the systems expand in this direction when hydrostatically compressed. Also discussed are the thermal expansion properties of these systems under the assumption that the units exhibit increased rotational agitation once subjected to an increase in temperature. The effect of the geometric parameters on the aforementioned thermo-mechanical properties of the system, are discussed, with the aim of identifying negative behaviour.

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