Question

Show theoretically that poisons ratio is greater than 0.5

Answer 1

Answer:

This can be understood intitutively. When a material is stretched it can behave in two extreme ways as regard to change in volume. These extreme ways are - (1) no increase in volume , and (2) maximum possible increase in volume. Now consider a cube of a homogeneous and isotropic material subjected to stretching in x direction. For the case of no increase in volume, the strain in y and z directions must be half and negative that of positive strain in x direction, as increase in volume due to positive strain in x direction has to be compensated by decrease in volume due to negative strain in y and z directions. So the Poisson's ratio, which is defined as negative ratio of lateral strain to longitudinal strain, will be 0.5. Physically a material can not contract in volume when stretched, the maximum value of Poisson's ratio can't therefore be more than 0.5. Now let's consider the case of maximum possible increase in volume when material is stretched. This will happen when there is no nagative lateral strain - no contraction in y and z directions ( auxetic materials like foam polymers can have positive lateral strain when stretched - such materials are excluded from this discussion). This means the Poisson's ratio in this situation will be zero. Most materials undergo some increase in volume when stretched. Poisson's ratio for most metals is found to be between 0.25 and 0.35. We can also intitutively understand that softer materials like rubber, fresh concrete or molten metal will have Poisson's ratio close to 0.5 as the molecules in these materials can easily move in lateral direction when material is stretched in longitudinal direction. For stronger or harder materials like metals or hardened concrete the molecules can't easily move in laterally, therefore when such materials are stretched longitudinally, there will always be increase in volume, hence Poisson's ratio will be much lower than 0.5 for such materials.