Honeycombs with negative values of θ have negative values of Poisson's ratio for loading in the plane of the cells.
To understand why honeycombs with negative values of θ (angle between the loading direction and the normal to the honeycomb cells) have negative values of Poisson's ratio, let's first clarify what Poisson's ratio represents.
Poisson's ratio (ν) is a dimensionless material property that describes the lateral contraction or expansion of a material when stretched or compressed in one direction. It is defined as the ratio of the transverse strain (lateral contraction or expansion) to the longitudinal strain (axial deformation).
In the case of honeycombs, when they are subjected to loading in the plane of the cells, the cells tend to deform due to compression or expansion in the loading direction. This deformation leads to a change in the lateral dimensions of the honeycomb.
Now, let's consider the effect of the angle θ on the deformation behavior of the honeycomb cells. Negative values of θ indicate that the loading direction is opposite to the normal direction of the honeycomb cells. In other words, the honeycomb is compressed in the opposite direction of the cell walls.
When a honeycomb with negative θ value is compressed, the cell walls tend to come closer to each other in the loading direction. Consequently, the cell walls experience lateral expansion in perpendicular directions. This behavior is counterintuitive to the conventional definition of Poisson's ratio, where the lateral contraction is expected when the material is compressed.
Thus, in honeycombs with negative θ values, the lateral expansion is opposite to the transverse strain, resulting in a negative Poisson's ratio. It indicates that the material expands laterally when subjected to compression and contracts laterally when subjected to tension.
To validate this observation and find actual numerical values of Poisson's ratio for honeycombs with specific negative θ values, further analysis and experimentation would be required. Theoretical calculations or experimental tests involving different loading conditions and material properties would be needed to determine the exact values of Poisson's ratio in such cases.