Connected Triangles Exhibiting Negative Poisson's Ratios and Negative Thermal Expansion

Abstract

Materials normally get thinner when stretched (positive Poisson’s ratio) and expand when heated (positive thermal expansion coefficients). However, not all systems behave in this way and systems (materials or structures) which defy common expectation and become wider when stretched (negative Poisson’s ratios, NPR, commonly referred to as auxetic) or contract when heated (negative thermal expansion, NTE) do exist and are now well documented. Materials exhibiting one of these unusual properties are known to be very useful in various practical applications. For example it has been shown that the presence of a negative Poisson’s ratio gives auxetic materials various enhanced physical characteristics over their conventional counterparts ranging from increased indentation resistance and a natural ability to form dome-shaped surfaces to improved acoustic damping properties. Similarly, negative thermal expansion materials can be extremely useful in various applications, such as in the manufacture of composites exhibiting some pre-determined thermal expansion coefficients since the presence of a component with negative thermal expansion lowers the overall thermal expansion of a composite. Although in recent years considerable progress and developments were made in the distinct fields of NPR and NTE, so far, research on systems which could exhibit both of these anomalous properties simultaneously is still in its infancy. Nevertheless, it is known that NTE and NPR are not mutually exclusive properties and in fact, single-crystalline polyacetylene networks (hypothetical carbon allotropes, which so far have not been synthesised) have been predicted to exhibit both properties. Here we will show how existing knowledge on systems exhibiting negative Poisson’s ratios and negative thermal expansion can be combined to produce a novel and easily constructible system which can exhibit both negative properties simultaneously. In particular, we note that as we have recently shown, the construct made from beams connected as illustrated in Fig. 1(a) can exhibit NPR since when it is uniaxially stretched, the beams will flex as illustrated in Fig. 1(b) with the net effect that the triangles rotate relative to each other to produce a more open structure (the ‘‘rotating triangles’’ effect, resulting in negative Poisson’s ratios of 1 in-plane if the triangles are equilateral. In such systems, the straight beams of length 2l can either be welded together or connected through the use of ‘‘pin joint’’-type connections at ends and centres of the beams (corresponding to the vertices of the triangles) where the axes of the ‘‘pin joints’’ are orthogonal to the plane of the structure. The use of such ‘‘pin joints’’ is possible since the formation of the triangles reduces the degrees of freedom of the system and give the structure its characteristic shear rigidity. The presence of these triangles also provide us with a route for turning this auxetic structure into one which can also exhibit negative thermal expansion since as recently proposed, equilateral triangles constructed in such a way that their base is made from a material with a thermal expansion coefficient being at least four times that of the other two sides will get shorter when heated (the ‘‘shortening of triangles’’ mechanism). In fact, it has been shown that dh, the change in height h when the equilateral triangle in Fig. 2 is subjected to a temperature change dT is given by: