Library / Mechanical / MEC-67
Mechanical
Poisson’s ratio from measured strains
Ratio of the sideways contraction to the lengthwise stretch in a bar loaded in one direction.
- Ref
- MEC-67
- Rev
- v1
- Field
- Mechanical
- Result
- ν
Pull a bar and it gets longer and thinner. Poisson’s ratio is the size of the thinning relative to the stretching, ν = −εlat/εaxial, where the minus sign makes ν positive for the usual case of a sideways contraction under axial extension. Strains are in microstrain (10⁻⁶); the ratio itself has no units, as long as both strains use the same one. For a stable isotropic elastic material −1 < ν < 0.5, with 0.5 the incompressible limit; most metals lie near 0.3. Obtaining stress from strain also needs a constitutive model such as Hooke’s law.
Ratio of the sideways contraction to the lengthwise stretch in a bar loaded in one direction. For a stable isotropic elastic material it lies between −1 and 0.5, with 0.5 meaning no change in volume.
Calculator
- Volumetric strain, εaxial + 2εlat400 µε
- Ratio of Young’s modulus to shear modulus, E/G = 2(1 + ν)2.6
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A tension specimen that stretches by 1,000 µε and contracts sideways by 300 µε has ν = 0.3, typical of steel. The volume rises by only 400 µε, much less than the axial strain, because most of the stretch is balanced by thinning.
- At ν = 0.5 the lateral strain is half the axial strain and the volume does not change; at ν = 0 (cork) there is no lateral movement at all. Auxetic materials have negative ν and get wider when stretched.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| εlat Lateral strain (negative for contraction) | −1,500 to 1,500 µε | −1.5 | 1.5 | 3 | +60 µε | −20 % |
| εaxial Axial strain | 100 to 3,000 µε | 0.1 | 3 | 2.9 | +10% | −9.09 % |
Sweep of Lateral strain (negative for contraction)
| Lateral strain (negative for contraction) (µε) | Poisson’s ratio |
|---|---|
| −1,500 | 1.5 |
| −1,125 | 1.125 |
| −750 | 0.75 |
| −375 | 0.375 |
| 0 | 0 |
| 375 | −0.375 |
| 750 | −0.75 |
| 1,125 | −1.125 |
| 1,500 | −1.5 |
Sweep one variable
Variables and how each one contributes
Poisson’s ratio
What the equation gives you. Everything else feeds it.
Lateral strain (negative for contraction)
The strain across the bar, in microstrain. It is negative for a contraction when the bar is stretched, so the slider runs from −1500 to +1500 µε. A larger contraction (a more negative strain) gives a larger ν; ν is proportional to the size of the contraction. Raising it by 50 µε (to −250 µε) moves the result by −16.7 %, from 0.3 to 0.25.
Axial strain
The strain along the bar, in microstrain. A positive value is an extension. The slider runs from 100 to 3000 µε. Lowers ν as the extension increases for the same sideways contraction, because the contraction becomes smaller in proportion. Raising it by 10 % (to 1,100) moves the result by −9.09 %, from 0.3 to 0.2727.
Worked example with the values above
- Ratio of strains: εlat/εaxial = −300 / 1,000 = −0.3.
- Change the sign: ν = −(−0.3).
- ν = 0.3.
Assumptions and limits
- A uniaxial stress state in a linear elastic, isotropic material.
- Lateral and axial strains measured at the same point at the same time, in the same units.
Common mistakes
- Forgetting the minus sign, which gives a negative ν for an ordinary material.
- Using it far into plastic deformation, where the apparent ratio tends towards 0.5.
Statistics and uncertainty
ν is a ratio of two measured strains, so relative errors add in quadrature: with ±2 % on each strain, ν has about ±2.8 %. Transverse sensitivity of the gauges and any bending in the specimen add to this, so ν from a single pair of gauges is rarely quoted to better than ±5 %.
Where it is used
- Getting ν from a pair of strain gauges on a tension specimen.
- Completing the elastic constants (E, G, ν) needed to turn strains into stresses.
Maths you need
- Strain: change in length divided by original length
- Hooke’s law: stress proportional to strain