INGENIA
Index/Materials/MAT-32

MAT-32

Poisson ratio

ν = −εlat / εlong. Lateral contraction over axial stretch.

Reading speed
ElasticityPoisson 1827

Governing equation

ν=−εlat/εlong\nu=-\varepsilon_{\mathrm{lat}}/\varepsilon_{\mathrm{long}}

εlat∈[−0.01,0.01]\varepsilon_{\mathrm{lat}} \in \left[-0.01, 0.01\right]

where

\varepsilon_{\mathrm{lat}}
Lateral strain (—)
\varepsilon_{\mathrm{long}}
Axial strain (—)
\nu
Poisson ratio (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

MAT-32 · beam
00:0 / 00:08
①

Governing law — red

ν=−εlat/εlong\nu=-\varepsilon_{\mathrm{lat}}/\varepsilon_{\mathrm{long}}

εlat∈[−0.01,0.01]\varepsilon_{\mathrm{lat}} \in \left[-0.01, 0.01\right]

εlat=−0.01\varepsilon_{\mathrm{lat}} = -0.01ν=10\nu = 10

②

Domain of the variable — blue

\varepsilon_{\mathrm{lat}}_{\min} = -0.01,\quad \varepsilon_{\mathrm{lat}}_{\max} = 0.01

εlat∈[−0.01,0.01]\varepsilon_{\mathrm{lat}} \in \left[-0.01, 0.01\right]

εlat=−0.01\varepsilon_{\mathrm{lat}} = -0.01εlat=−0.01\varepsilon_{\mathrm{lat}} = -0.01

③

Response curve — green

ν=−εlat/εlong\nu=-\varepsilon_{\mathrm{lat}}/\varepsilon_{\mathrm{long}}

εlat∈[−0.01,0.01]\varepsilon_{\mathrm{lat}} \in \left[-0.01, 0.01\right]

εlat=−0.01\varepsilon_{\mathrm{lat}} = -0.01ν=10\nu = 10

④

Rate of change — amber

dνdεlat\frac{\mathrm{d}\nu}{\mathrm{d}\varepsilon_{\mathrm{lat}}}

εlat∈[−0.01,0.01]\varepsilon_{\mathrm{lat}} \in \left[-0.01, 0.01\right]

εlat=−0.01\varepsilon_{\mathrm{lat}} = -0.01ν′=−1000\nu^{\prime} = -1000

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-32 — Poisson ratio) is the form associated with Poisson 1827. Working symbols: εlat\varepsilon_{\mathrm{lat}}, εlong\varepsilon_{\mathrm{long}} →\rightarrow ν\nu. Poisson's ratio is a thermodynamic elastic constant. For isotropic solids −1 < ν < 0.5, with 0.5 the incompressible limit.

Purpose

Purpose: compute ν\nu from εlat\varepsilon_{\mathrm{lat}}, εlong\varepsilon_{\mathrm{long}} in Materials via ν=−εlat/εlong\nu=-\varepsilon_{\mathrm{lat}}/\varepsilon_{\mathrm{long}} ν = −εlat / εlong. Lateral contraction over axial stretch. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given εlat=−3.000e−4 —\varepsilon_{\mathrm{lat}} = -3.000e-4\,\mathrm{—}, εlong=0.001 —\varepsilon_{\mathrm{long}} = 0.001\,\mathrm{—}, the governing relation ν=−εlat/εlong\nu=-\varepsilon_{\mathrm{lat}}/\varepsilon_{\mathrm{long}} yields ν=0.300 —\nu = 0.300\,\mathrm{—}. A stretched bar, a thinning, a ν ratio. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Poisson ratio \nu0.300 —
Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

MAT-32 · beam
00:0 / 00:08

Narration of this film

A stretched bar, a thinning, a ν ratio.

Poisson's ratio is a thermodynamic elastic constant. For isotropic solids −1 < ν < 0.5, with 0.5 the incompressible limit.

Reading speed

Watch on YouTube