Uniaxial loading
 Tensile/compressive stress in the xdirection produces tensile/compressive strain in the xdirection, with the stress and strain related through Young's modulus E:
 Strain in the xdirection produces strains the y and zdirections. The strains in the y and zdirections are proportional to the strain in the xdirection through the negative of the Poisson's ratio.
Positive Poisson's ratio
If the Poisson's ratio of the material if positive, then tensile/compressive strains in the xdirection produce compressive/tensile strains in the y and zdirections, as shown below.
Negative Poisson's ratio
If the Poisson's ratio of the material if positive, then tensile/compressive strains in the xdirection produce tensile/compressive strains in the y and zdirections, just the opposite of that for a positive Poisson's ratio.
QUESTION: Can a material possess a negative Poisson’s ratio? The answer is “yes”. Such materials are known as “auxetic materials”. From above, we see than an auxetic material will expand in the transverse directions for a tensile axial load. Consider the animation below of such a material, and study how a negative value for Poisson's ratio is possible with this material. As can be seen here, a compressive axial load produces contraction in the transverse direction, and a tensile axial load produces expansion in the transverse direction.
Some examples of auxetic materials include:

 Auxetic polyurethane foam.
 Kevlar woven composite materials used for body armor.
 Nuclei of mouse embryonic stem cells in exiting pluripotent state.
 Certain states of crystalline materials: Li, Na, K, Cu, Rb, Ag, Fe, Ni, Co, Cs, Au, Be, Ca, Zn, Sr, Sb, MoS_{2}, BAsO_{4}, and others.
 Certain rocks and minerals.
 Graphene, which can be made auxetic through the introduction of vacancy defects.
 Carbon diamondlike phases.
 Noncarbon nanotubes.
 Living bone tissue (although this is only suspected).
 Tendons within their normal range of motion.
 Specific variants of polytetrafluorethylene polymers such as GoreTex.
 Several types of origami folds like the DiamondFoldingStructure (RFS), the herringbonefoldstructure (FFS) or the miura fold, and other periodic patterns derived from it.
 Tailored structures designed to exhibit special designed Poisson's ratios.
 Chain organic molecules. Recent researches revealed that organic crystals like nparaffins and similar to them may demonstrate an auxetic behavior.