When a material is flattened, it tends to bulge out at right angles to the compression direction. If it's stretched, it tends to constrict. We would not expect a thin rod to bulge or constrict as much as a thick cylinder. Such materials are called dilatant . Many unconsolidated materials are dilatant.
Rocks can become dilatant just before failure because microcracks increase the volume of the rock.
There are a few weird synthetic foams with negative Poisson's Ratios. These materials are light froths whose bubble walls collapse inward under compression. Shear Modulus Poisson's Ratio describes transverse strain, so it obviously has a connection with shear. The Shear Modulus , usually abbreviated G, plays the same role in describing shear as Young's Modulus does in describing longitudinal strain. There are really only two independent quantities, so if we know any two quantities E, v, G and K, we can calculate any others.