# Partially Auxetic Structure of Rotating Rectangles
## 1 Introduction  
Auxetic structures are primarily mathematical concepts that have been given graphical form. Thanks to simulation studies, it has been discovered that they may have unusual mechanical properties that one would like to exploit. The technique awaits the applications of auxetics, and the community awaits the benefits of it. Numerous review papers present various aspects of auxetics and theoretically possible potential applications [1, 2, 3, 4]. Auxetic structures can be used to produce advanced lightweight cellular materials for the aerospace, automotive, and construction industries [5], including in sandwich panels [6] and similar sandwich structures [7], for creating efficient structures for energy absorption [8, 9], and for applications in biomedical engineering [10], especially in bone tissue and blood vessel engineering and as a stent [11]. Since 2000, there have been more than 105 review-type publications on auxetics, where the mentioned potential applications are a natural motivation for the analysis of auxetic structures. The idea is to exploit the effect of multidirectional expansion when structures are stretched and the effect of vertical and horizontal contraction when they are compressed. A special group of mechanical metamaterials are so-called structures of rotating geometric figures, with squares, rectangles, triangles, or rhombuses appearing as unit cells. These structures in the form of computer graphics were first proposed by [PERSON] [12, 13] in 1965. However, it is only since 2000 that these structures have been popularized byGrima groups, where in a very large number of publications, they have been subjected to careful analysis, e.g., [14; 15; 16], and it has been shown that, rotating structures have a negative Poisson's ratio. A particular advantage of such structures is that their auxetic behavior was also illuminated in many other publications devoted to structures of rotating squares [17; 18; 19; 20; 21], or rotating rectangles [22; 23; 24; 25]. Structures of rotating geometric figures are created by regularly arranging two-dimensional unit cells connected by corners. The idea is to use the effect of multidirectional expansion when structures are stretched and the effect of vertical and horizontal contraction when they are compressed. In publications, subjecting a representative number of unity cells in the structure to theoretical analysis is customary. Such an ideal image of auxetic structures has no relation to the practiced construction of these structures. Especially in all previous constructions, the connections of unit cells are the weak points of the structures because the damaging stresses pile up. In addition, the changes in the linear dimensions of such structures under tension or compression are small. These changes are those that the elastic properties of the materials of elementary cell connections allow. The subject of the study is the known structures of rotating rectangles, which are characterized by the possibility of obtaining large negative values of Poisson's ratio. For these structures, a simple and proven modification [26] was used, which consists of connecting rectangular elementary cells with axes of rotation. In this solution, the corners of the unit cells are overlapped, with the position of the axis of rotation passing through the overlapped corners being an important geometric parameter of the modified structures. The distance of the axis of rotation from the edge of the unit cell was defined as the x parameter. When such structures are stretched or compressed, the structural units not only rotate and the angles between adjacent sides change, but the units also move, which leads to planar expansion or contraction, which is an auxetic behavior. The structure deforms not only by the relative rotation of the squares but also by their translation. This paper presents a mathematical analysis and experimental design of rotating rectangular structures with a large number of unit cells.