Library / Mechanical / MEC-83
Mechanical
Shear modulus from Young’s modulus and Poisson’s ratio
Link between the shear and tensile stiffness of an isotropic linear elastic material.
- Ref
- MEC-83
- Rev
- v1
- Field
- Mechanical
- Result
- G [GPa]
For an isotropic elastic material only two of the elastic constants are independent. The shear modulus follows from Young’s modulus E and Poisson’s ratio ν as G = E/[2(1 + ν)]. For steel this gives about 77 GPa from E = 200 GPa and ν = 0.3. It is used when the shear modulus is not tabulated but E and ν are known. It does not hold for composites or other anisotropic materials.
Link between the shear and tensile stiffness of an isotropic linear elastic material. If any two of E, ν and G are known, the third follows.
Calculator
- Ratio G/E0.3846
- Bulk modulus K from the same E and ν166.7 GPa
- Check: E from G and ν, 2G(1 + ν)200 GPa
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.
- With E = 200 GPa and ν = 0.3, the shear modulus is 76.92 GPa, 38.5 % of Young’s modulus.
- For any ν between 0 and 0.5, G lies between E/3 and E/2, so G is always between one third and one half of E for an isotropic material.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (GPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| E Young’s modulus | 1 to 500 GPa | 0.3846 | 192.3 | 191.9 | +10% | +10 % |
| ν Poisson’s ratio | 0 to 0.5 | 66.67 | 100 | 33.33 | +10% | −2.26 % |
Sweep of Poisson’s ratio
| Poisson’s ratio | Shear modulus (GPa) |
|---|---|
| 0 | 100 |
| 0.0625 | 94.12 |
| 0.125 | 88.89 |
| 0.1875 | 84.21 |
| 0.25 | 80 |
| 0.3125 | 76.19 |
| 0.375 | 72.73 |
| 0.4375 | 69.57 |
| 0.5 | 66.67 |
Sweep one variable
Variables and how each one contributes
Shear modulus
What the equation gives you. Everything else feeds it.
Young’s modulus
Young’s modulus, in GPa. Raises G in direct proportion. Raising it by 10 GPa (to 210 GPa) moves the result by +5 %, from 76.92 to 80.77 GPa.
Poisson’s ratio
Poisson’s ratio, dimensionless and between 0 and 0.5 for most materials. Lowers G: a higher Poisson’s ratio gives a smaller shear modulus for the same E. Raising it by 0.05 (to 0.35) moves the result by −3.7 %, from 76.92 to 74.07 GPa.
Worked example with the values above
- Denominator: 2(1 + ν) = 2 × (1 + 0.3) = 2.6.
- Divide the modulus: 200 / 2.6.
- G = 76.92 GPa.
Assumptions and limits
- An isotropic, linear elastic material.
- E and ν measured under the same conditions of temperature and rate.
Common mistakes
- Applying it to timber, composites or rolled sheet, which are not isotropic.
- Using the relation with ν outside −1 to 0.5.
Statistics and uncertainty
The relation is exact for an isotropic elastic material. Quoted values of E and ν vary by a few per cent between grades, so G computed this way inherits that variation; a measured G (torsion test) is better if torsion is critical.
Where it is used
- Finding G for torsion and shear calculations when only E and ν are given.
- Checking whether a set of quoted elastic constants is consistent.
Maths you need
- Stress and strain: force per area and relative change in length
- Elastic modulus: stress divided by strain in the elastic range