## 3 Results
### Compressive Behaviour
#### 3.2.2 Poisson's Ratio  
The longitudinal and transverse strain were calculated by using the virtual extensometer as illustrated in Figure 16. Poisson's ratio of the whole structure may not be the same as the unit cell in  
Figure 14: Elastic modulus and yield strength for auxetic structures under uniaxial compression.  
Figure 15: DIC images. (**a**) and (**b**): deformation of samples under compression in the x direction; (**c**) and (**d**): deformation of samples under compression in the y direction.  
the middle of the structure due to presence of boundary effects through compression in both the x and y directions. The boundary effects during compression in the y direction are due to an imbalance of force and moment created due to the termination of structure, which causes the vertical struts to deform more. Similar boundary effects were observed in the work carried out by [PERSON] [46], who also stated that the deformation of the vertical strut due to boundary effect increases with the increase in the length of the vertical strut. Therefore, it was necessary to analyse Poisson's ratio at various regions of the structure in order to determine the degree to which Poisson's ratio of the unit cell at the centre represent the whole structure. Measurement of transverse and longitudinal strain was made in four regions of structure 6, transverse strain being measured in top, middle and bottom layers in regions A and B. Longitudinal strain was measured in the left, middle and right layers of region A and in the left and right layers in region B, as illustrated in Figure 17a,b (x and y direction, respectively). From the results (Table 3), it can be seen that \(\
u\) of the unit cell in the centre (C and D) is higher than other regions. The error in \(\
u\) in Table 3 is largely due to only single samples being taken. Furthermore, \(\
u\) decreases as it proceeds towards outside, which shows that the \(\
u\) of the unit cell in the centre cannot represent the whole re-entrant structure under compression in the x direction. The influence of end effects in \(\
u\) can be reduced by fabricating large samples by increasing the number of unit cells. The calculated Poisson's ratios, using the averaging method, for all samples, are given in Figure 18.  
Figure 16: Illustration of the measurement of transverse and longitudinal strain from the DIC image.  
Figure 17: Poisson’s ratio DIC strain analysis. (**a**) Location of unit cells for compression in x, (**d**) location of unit cells for compression in y, (**b**) measurement of transverse strain in the x direction, (**c**) measurement of longitudinal strain in the x direction, (**e**) measurement of transverse strain in the y direction, (**f**) measurement of longitudinal strain in the y direction.  
It can be seen that there is little variation in [PERSON]'s ratio for compression in the y direction, but large deviation in values is observed for compression in the x direction. Although samples 2 and 3 have similar relative density, there is a significant variation in E and \(\sigma\) in both directions and of [PERSON]'s ratio in the x direction. Similar variation can also be seen for samples 6 and 8 with similar relative density. This indicates that there is no correlation between relative density and [PERSON]'s ratio. [PERSON] stated that [PERSON]'s ratio would contribute to the difference in E and \(\sigma\)[46], but as observed here, this is true only for compression in x. The number of unit cells in the x and y directions were equal for all samples, and, thus, the overall dimensions of the samples differed, However, comparing samples 3 and 7 in Figure 5, there does not appear to be any correlation between variation in overall size and Poisson's ratio for compression in x.