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Poisson's Ratio

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Quick
Poisson's ratio (pronounced kind of like "pweh-so"), abbreviated by the symbol ν, is the ratio between the lateral and longitudinal strains in the elastic region. Poisson's ratio is positive for all engineering materials.


Equations

General equation, which is Poisson's ratio:

ν  =  −

lateral strain axial strain

=  −

εy εx

=  −

εz εx

The following equations fully describe the condition of strain under an axial load applied in a direction parallel to the x-axis:

εx  =  − σx E

εy  =  εz  =  − νσx E

Nomenclature

symbol	description
σ	stress
E	modulus of elasticity
ε	strain
εx	longitudinal strain in x direction
εy	lateral strain in y direction
εz	lateral strain in z direction

Explanation

From the lesson Deformation Due To Axial Loading, when a homogeneous slender bar is axially loaded, the resulting stress and strain satisfy Hooke's law, as long as the elastic limit of the material is not exceeded.



Assuming that load P is directed along the x-axis as shown in the above figure, then σ_x = P/A, where A is the cross-sectional area of the bar. So from Hooke's law:

εx  =

σx E

It may also be noted that the normal stresses on faces respectively perpendicular to the y and z axes are zero: σ_y = σ_z = 0, as shown in the figure below:



It would be tempting to conclude that the corresponding strains ε_y and ε_z are also zero. This, however, is not the case. It would also be tempting, but equally wrong, to assume that the volume of the rod remains unchanged as a result of the combined effect of the axial elongation and transverse contraction. In most engineering materials, the elongation produced by an axial tensile force P in the direction of the force is accompanied by a contraction in any transverse direction, as illustrated in the following figure:



The strain must have the same value for any transverse direction. Therefore, for the loading shown in the following figure, ε_y = ε_z. This common value is referred to as lateral strain. An important constant for a given material is the Poisson's ratio, denoted by ν and defined as:

ν  =  −

lateral strain axial strain

or

ν  =  −

εy εx

=  −

εz εx

for the loading condition represented in the following figure. Note that the use of a minus sign in the above equations to obtain a positive value for ν, the axial and lateral strains having opposite signs for most engineering materials. Solving Eq1 for ε_y and ε_z, and recalling Hooke's law, the following relations can be written, which fully describe the condition of strain under an axial load applied in a direction parallel to the x-axis:

εx  =  −

σx E

and:

εy  =  εz  =  −

νσx E