# Partially Auxetic Structure of Rotating Rectangles
## 4 Discussion  
The present work effectively investigates the occurrence of mechanical phase transitions in auxetic structures. In this way, the concept of phase transition has also been extended to mechanics to distinguish the behavior of structures at different stages of the deformation process. Considering the mathematical aspects of phase transition, it can be stated, similarly to [PERSON]'s concept, that mechanical phase transitions occur when a discontinuity of Poisson's ratio changes is obtained with a continuous change of dimensions, e.g., elongation. Assuming that the metamaterial phase transition concerns changes in the sign of Poisson's ratio (from negative to positive) with a continuous change in the linear dimensions of the structure, it was discovered that this predicted phenomenon occurs in modified structures of rotating rectangles. These structures are, therefore, partially auxetic. It has been known so far that metamaterial structures in the form of a system of rectangular unit cells can obtain large negative values of Poisson's ratio. This property has been emphasized in well-known publications [15, 22, 23, 24]. It was also predicted that Poisson's ratio could change sign [28, 29, 30], i.e., change from negative to positive values, although, as noted, this requires large dimensional changes in the structures. It has been investigated theoretically and practically that compliant structures subjected to loading can only undergo slight dimensional changes, which are limited by elastic properties. With larger deformations, they are destroyed. One of the known proposals to remove this shortcoming was the introduction of articulated gratings [31] but without practical implementation. It should be emphasized that although the occurrence of a metamaterial phase transition has been demonstrated in theory, it has not been realized in a realistic way for metamaterial structures. In order to ensure the proper functioning of structures subjected to tension and compression, a solution was proposed for a stable connection of rigid structural elements, ensuring high values of linear expansion in the longitudinal or transverse direction. By using a structural modification that connects elementary cells with rotational axes, auxetic properties that depend on geometric parameters (a/b and x) and the degree of deformation are achieved. In particular, it is necessary to consider the relationship between the degree of deformation and Poisson's ratio in these structures. It should be noted that the structure of the rotating squares does not show this effect. The movement of the structure from the initial to the final position and the associated change in dimensions determine the value of this complex parameter, Poisson's ratio. Based on the study of rotating rectangular structures, it has been confirmed that for fixed parameters a/b and x, Poisson's ratio does not depend on the number of unit cells. Increasing the number of unit cells keeps Poisson's ratio constant. For modified structures, the change of one of the geometric parameters (a/b or x) can be selected in such a way as to obtain a structure with the intended auxetic properties. For example, increasing the quotient a/b from 1.6 to 2 (with a constant geometric parameter x) results in a smaller range of linear dimension changes in the auxetic area, while the negative Poisson's ratio increases sharply. However, for the parameters a/b = 2 and x = 0.1, negative and positive Poisson's ratios were obtained, thus creating a partially auxetic structure with a phase transition. The phase transition can be controlled by selecting structure parameters (a/b and x). Like most phase transitions, mechanical phase transitions are also rapid. At the moment of transition, the system has discontinuities in Poisson's ratio. For a certain elongation value, an infinitely large negative and infinitely large positive value of Poisson's ratio is reached, which also means that one of the linear dimensions of the stretched structure undergoes zero dimensional change.