Poisson's ratio

In certain rare cases,[2] 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.

A perfectly incompressible isotropic 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 (before yield) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation which occurs largely at constant volume.

For a cube stretched in the x-direction (see Figure 1) with a length increase of ΔL in the x-direction, and a length decrease of ΔL′ in the y- and z-directions, the infinitesimal diagonal strains are given by If Poisson's ratio is constant through deformation, integrating these expressions and using the definition of Poisson's ratio gives Solving and exponentiating, the relationship between ΔL and ΔL′ is then For very small values of ΔL and ΔL′, the first-order approximation yields: The relative change of volume ⁠ΔV/V⁠ of a cube due to the stretch of the material can now be calculated.

Then Hooke's law can be expressed in matrix form as[13][14] where The Poisson ratio of an orthotropic material is different in each direction (x, y and z).

However, the symmetry of the stress and strain tensors implies that not all the six Poisson's ratios in the equation are independent.

If we assume that this plane of isotropy is the yz-plane, then Hooke's law takes the form[15] where we have used the yz-plane of isotropy to reduce the number of constants, that is, The symmetry of the stress and strain tensors implies that This leaves us with six independent constants Ex, Ey, Gxy, Gyz, νxy, νyz.

However, transverse isotropy gives rise to a further constraint between Gyz and Ey, νyz which is Therefore, there are five independent elastic material properties two of which are Poisson's ratios.

Studies have shown that certain solid wood types display negative Poisson's ratio exclusively during a compression creep test.

[20][21] Initially, the compression creep test shows positive Poisson's ratios, but gradually decreases until it reaches negative values.

Consequently, this also shows that Poisson's ratio for wood is time-dependent during constant loading, meaning that the strain in the axial and transverse direction do not increase in the same rate.

In a simple case auxeticity is obtained removing material and creating a periodic porous media.

[25][26][27] For example, Li, Na, K, Cu, Rb, Ag, Fe, Ni, Co, Cs, Au, Be, Ca, Zn Sr, Sb, MoS2 and others.

Due to Poisson's effect, this hoop stress will cause the pipe to increase in diameter and slightly decrease in length.

In a geological timescale, excessive erosion or sedimentation of Earth's crust can either create or remove large vertical stresses upon the underlying rock.

This change in strain in the horizontal direction can affect or form joints and dormant stresses in the rock.

Poisson's ratio of a material defines the ratio of transverse strain ( x direction) to the axial strain ( y direction)
Figure 1 : A cube with sides of length L of an isotropic linearly elastic material subject to tension along the x axis, with a Poisson's ratio of 0.5. The green cube is unstrained, the red is expanded in the x -direction by Δ L due to tension, and contracted in the y - and z -directions by Δ L ′ .
Figure 2: The blue slope represents a simplified formula (the top one in the legend) that works well for modest deformations, ∆ L , up to about ±3. The green curve represents a formula better suited for larger deformations.
Influences of selected glass component additions on Poisson's ratio of a specific base glass. [ 16 ]