Mumbai University > Electronics Engineering > Sem 8 > MEMS Technology

Marks: 4M

• Poisson's ratio, named after Siméon Poisson, also known as the coefficient of expansion on the transverse axial, is the negative ratio of transverse to axial strain.
• When a material is compressed in one direction, it usually tends to expand in the other two directions perpendicular to the direction of compression.
• This phenomenon is called the Poisson effect.
• Poisson's ratio (nu) is a measure of this effect. The Poisson ratio is the fraction (or percent) of expansion divided by the fraction (or percent) of compression, for small values of these changes.
• Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching.
• That is a common observation when a rubber band is stretched, it becomes noticeably thinner.
• Again, the Poisson ratio will be the ratio of relative contraction to relative expansion and will have the same value as above.
• In certain rare cases, a material will actually shrink in the transverse direction when compressed (or expand when stretched) which will yield a negative value of the Poisson ratio.
• The Poisson's ratio of a stable, isotropic, linear elastic material cannot be less than −1.0 or greater than 0.5 because of the requirement for Young's modulus, theshear modulus and bulk modulus to have positive values.[1]
• Most materials have Poisson's ratio values ranging between 0.0 and 0.5.
• A perfectly incompressible material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Most steels and rigid polymers when used within their design limits (beforeyield) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation which occurs largely at constant volume.[2]
• Rubber has a Poisson ratio of nearly 0.5.
• Cork's Poisson ratio is close to 0: showing very little lateral expansion when compressed. Some materials, mostly polymer foams, and materials with special geometries such as zigzag-based materials[3] can have a negative Poisson's ratio;
• if these auxetic materials are stretched in one direction, they become thicker in perpendicular direction. Some anisotropic materials, such as zigzag-based folded sheet materials,[3] have one or more Poisson's ratios above 0.5 in some directions.

Assuming that the material is stretched or compressed along the axial direction (the x axis in the below diagram):

$$\nu = - \frac{d \epsilon_{trans}}{d \epsilon_{axial}} = - \frac{d \epsilon_y}{d \epsilon_x} = - \frac{d \epsilon_z}{d \epsilon_x} $$

where,

$\nu$ is the resulting Poisson's ratio,

$\epsilon_{trans}$ is transverse strain (negative for axial tension (stretching), positive for axial compression) $\epsilon_{axial}$ is axial strain (positive for axial tension, negative for axial compression).