In materials science and solid mechanics, Poisson's ratio (symbol: ν (nu)) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. For small values of these changes, ν is the amount of transversal elongation divided by the amount of axial compression. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2 to 0.3. The ratio is named after the French mathematician and physicist Siméon Poisson.
Definition Assuming that the material is stretched or compressed in only one direction (the x or y axis in the diagram):
ν = − d ε t r a n s d ε a x i a l = − d ε y d ε x = − d ε z d ε x {\displaystyle \nu =-{\frac {d\varepsilon _{\mathrm {trans} }}{d\varepsilon _{\mathrm {axial} }}}=-{\frac {d\varepsilon _{\mathrm {y} }}{d\varepsilon _{\mathrm {x} }}}=-{\frac {d\varepsilon _{\mathrm {z} }}{d\varepsilon _{\mathrm {x} }}}}
where
ν is the resulting Poisson's ratio, εtrans is transverse strain εaxial is axial strain and positive strain indicates extension and negative strain indicates contraction.
Origin Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression. Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching. It 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 must be between −1.0 and +0.5 because of the requirement for Young's modulus, the shear modulus and bulk modulus to have positive values. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. 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. Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0, showing very little lateral expansion when compressed. Glass is between 0.18 and 0.30. Some materials, e.g. some polymer foams, origami folds, and certain cells can exhibit negative Poisson's ratio, and are referred to as auxetic materials. If these auxetic materials are stretched in one direction, they become thicker in the perpendicular direction. In contrast, some anisotropic materials, such as carbon nanotubes, zigzag-based folded sheet materials, and honeycomb auxetic metamaterials to name a few, can exhibit one or more Poisson's ratios above 0.5 in certain directions.
Changing geometry
Length change
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
d ε x = d x x , d ε y = d y y , d ε z = d z z . {\displaystyle d\varepsilon _{x}={\frac {dx}{x}},\qquad d\varepsilon _{y}={\frac {dy}{y}},\qquad d\varepsilon _{z}={\frac {dz}{z}}.}
If Poisson's ratio is constant through deformation, integrating these expressions and using the definition of Poisson's ratio gives
− ν ∫ L L + Δ L d x x = ∫ L L + Δ L ′ d y y = ∫ L L + Δ L ′ d z z . {\displaystyle -\nu \int _{L}^{L+\Delta L}{\frac {dx}{x}}=\int _{L}^{L+\Delta L'}{\frac {dy}{y}}=\int _{L}^{L+\Delta L'}{\frac {dz}{z}}.}
Solving and exponentiating, the relationship between ΔL and ΔL′ is then
( 1 + Δ L L ) − ν = 1 + Δ L ′ L . {\displaystyle \left(1+{\frac {\Delta L}{L}}\right)^{-\nu }=1+{\frac {\Delta L'}{L}}.}
For very small values of ΔL and ΔL′, the first-order approximation yields:
ν ≈ − Δ L ′ Δ L . {\displaystyle \nu \approx -{\frac {\Delta L'}{\Delta L}}.}
Volumetric change The relative change of volume ΔV/V of a cube due to the stretch of the material can now be calculated. Since V = L3 and
V + Δ V = ( L + Δ L ) ( L + Δ L ′ ) 2 {\displaystyle V+\Delta V=(L+\Delta L)\left(L+\Delta L'\right)^{2}}
one can derive
Δ V V = ( 1 + Δ L L ) ( 1 + Δ L ′ L ) 2 − 1 {\displaystyle {\frac {\Delta V}{V}}=\left(1+{\frac {\Delta L}{L}}\right)\left(1+{\frac {\Delta L'}{L}}\right)^{2}-1}
Using the above derived relationship between ΔL and ΔL′:
Δ V V = ( 1 + Δ L L ) 1 − 2 ν − 1 {\displaystyle {\frac {\Delta V}{V}}=\left(1+{\frac {\Delta L}{L}}\right)^{1-2\nu }-1}
and for very small values of ΔL and ΔL′, the first-order approximation yields:
Δ V V ≈ ( 1 − 2 ν ) Δ L L {\displaystyle {\frac {\Delta V}{V}}\approx (1-2\nu ){\frac {\Delta L}{L}}}
For isotropic materials we can use Lamé's relation
ν ≈ 1 2 − E 6 K {\displaystyle \nu \approx {\frac {1}{2}}-{\frac {E}{6K}}}
where K is bulk modulus and E is Young's modulus.
Width change If a rod with diameter (or width, or thickness) d and length L is subject to tension so that its length will change by ΔL then its diameter d will change by:
Δ d d = − ν Δ L L {\displaystyle {\frac {\Delta d}{d}}=-\nu {\frac {\Delta L}{L}}}
The above formula is true only in the case of small deformations; if deformations are large then the following (more precise) formula can be used:
Δ d = − d ( 1 − ( 1 + Δ L L ) − ν ) {\displaystyle \Delta d=-d\left(1-{\left(1+{\frac {\Delta L}{L}}\right)}^{-\nu }\right)}
where
d is original diameter Δd is rod diameter change ν is Poisson's ratio L is original length, before stretch ΔL is the change of length. The value is negative because it decreases with increase of length
Characteristic materials
Isotropic For a linear isotropic material subjected only to compressive (i.e. normal) forces, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axis in three dimensions. Thus it is possible to generalize Hooke's law (for compressive forces) into three dimensions:
ε x x = 1 E [ σ x x − ν ( σ y y + σ z z ) ] ε y y = 1 E [ σ y y − ν ( σ z z + σ x x ) ] ε z z = 1 E [ σ z z − ν ( σ x x + σ y y ) ] {\displaystyle {\begin{aligned}\varepsilon _{xx}&={\frac {1}{E}}\left[\sigma _{xx}-\nu \left(\sigma _{yy}+\sigma _{zz}\right)\right]\\[6px]\varepsilon _{yy}&={\frac {1}{E}}\left[\sigma _{yy}-\nu \left(\sigma _{zz}+\sigma _{xx}\right)\right]\\[6px]\varepsilon _{zz}&={\frac {1}{E}}\left[\sigma _{zz}-\nu \left(\sigma _{xx}+\sigma _{yy}\right)\right]\end{aligned}}}
where:
εxx, εyy, and εzz are strain in the direction of x, y and z σxx, σyy, and σzz are stress in the direction of x, y and z E is Young's modulus (the same in all directions for isotropic materials) ν is Poisson's ratio (the same in all directions for isotropic materials) these equations can be all synthesized in the following:
ε i i = 1 E [ σ i i ( 1 + ν ) − ν ∑ k σ k k ] {\displaystyle \varepsilon _{ii}={\frac {1}{E}}\left[\sigma _{ii}(1+\nu )-\nu \sum _{k}\sigma _{kk}\right]}
In the most general case, also shear stresses will hold as well as normal stresses, and the full generalization of Hooke's law is given by:
ε i j = 1 E [ σ i j ( 1 + ν ) − ν δ i j ∑ k σ k k ] {\displaystyle \varepsilon _{ij}={\frac {1}{E}}\left[\sigma _{ij}(1+\nu )-\nu \delta _{ij}\sum _{k}\sigma _{kk}\right]}
where δij is the Kronecker delta. The Einstein notation is usually adopted:
σ k k ≡ ∑ l δ k l σ k l {\displaystyle \sigma _{kk}\equiv \sum _{l}\delta _{kl}\sigma _{kl}}
to write the equation simply as:
ε i j = 1 E [ σ i j ( 1 + ν ) − ν δ i j σ k k ] {\displaystyle \varepsilon _{ij}={\frac {1}{E}}\left[\sigma _{ij}(1+\nu )-\nu \delta _{ij}\sigma _{kk}\right]}
Anisotropic For anisotropic materials, the Poisson ratio depends on the direction of extension and transverse deformation
ν ( n , m ) = − E ( n ) s i j α β n i n j m α m β E − 1 ( n ) = s i j α β n i n j n α n β {\displaystyle {\begin{aligned}\nu (\mathbf {n} ,\mathbf {m} )&=-E\left(\mathbf {n} \right)s_{ij\alpha \beta }n_{i}n_{j}m_{\alpha }m_{\beta }\\[4px]E^{-1}(\mathbf {n} )&=s_{ij\alpha \beta }n_{i}n_{j}n_{\alpha }n_{\beta }\end{aligned}}}
Here ν is Poisson's ratio, E is Young's modulus, n is a unit vector directed along the direction of extension, m is a unit vector directed perpendicular to the direction of extension. Poisson's ratio has a different number of special directions depending on the type of anisotropy.
Orthotropic
Orthotropic materials have three mutually perpendicular planes of symmetry in their material properties. An example is wood, which is most stiff (and strong) along the grain, and less so in the other directions. Then Hooke's law can be expressed in matrix form as
[ ϵ x x ϵ y y ϵ z z 2 ϵ y z 2 ϵ z x 2 ϵ x y ] = [ 1 E x − ν y x E y − ν z x E z 0 0 0 − ν x y E x 1 E y − ν z y E z 0 0 0 − ν x z E x − ν y z E y 1 E z 0 0 0 0 0 0 1 G y z 0 0 0 0 0 0 1 G z x 0 0 0 0 0 0 1 G x y ] [ σ x x σ y y σ z z σ y z σ z x σ x y ] {\displaystyle {\begin{bmatrix}\epsilon _{xx}\\\epsilon _{yy}\\\epsilon _{zz}\\2\epsilon _{yz}\\2\epsilon _{zx}\\2\epsilon _{xy}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{x}}}&-{\tfrac {\nu _{yx}}{E_{y}}}&-{\tfrac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xy}}{E_{x}}}&{\tfrac {1}{E_{y}}}&-{\tfrac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xz}}{E_{x}}}&-{\tfrac {\nu _{yz}}{E_{y}}}&{\tfrac {1}{E_{z}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{yz}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{zx}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}
where
Ei is the Young's modulus along axis i Gij is the shear modulus in direction j on the plane whose normal is in direction i νij is the Poisson ratio that corresponds to a contraction in direction j when an extension is applied in direction i. 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. There are only nine independent material properties: three elastic moduli, three shear moduli, and three Poisson's ratios. The remaining three Poisson's ratios can be obtained from the relations
ν y x E y = ν x y E x , ν z x E z = ν x z E x , ν y z E y = ν z y E z {\displaystyle {\frac {\nu _{yx}}{E_{y}}}={\frac {\nu _{xy}}{E_{x}}}\,,\qquad {\frac {\nu _{zx}}{E_{z}}}={\frac {\nu _{xz}}{E_{x}}}\,,\qquad {\frac {\nu _{yz}}{E_{y}}}={\frac {\nu _{zy}}{E_{z}}}}
From the above relations we can see that if Ex > Ey then νxy > νyx. The larger ratio (in this case νxy) is called the major Poisson ratio while the smaller one (in this case νyx) is called the minor Poisson ratio. We can find similar relations between the other Poisson ratios.
Transversely isotropic Transversely isotropic materials have a plane of isotropy in which the elastic properties are isotropic. If we assume that this plane of isotropy is the yz-plane, then Hooke's law takes the form
[ ϵ x x ϵ y y ϵ z z 2 ϵ y z 2 ϵ z x 2 ϵ x y ] = [ 1 E x − ν y x E y