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Mechanical
Shear strain from shear stress
Engineering shear strain (the change in angle between two originally perpendicular lines) produced by a shear stress in a linear elastic material.
- Ref
- MEC-82
- Rev
- v1
- Field
- Mechanical
- Result
- γxy [µrad]
In the elastic range, shear stress and shear strain are linked by the shear modulus G: γ = τ/G. The engineering shear strain γ is the change of angle of an originally right-angled corner, in radians. With τ in MPa and G in GPa, multiplying by 1000 gives γ in microradians (µrad). It has the same form as Hooke’s law for tension, but with G in place of E.
Engineering shear strain (the change in angle between two originally perpendicular lines) produced by a shear stress in a linear elastic material. It is twice the tensor shear strain.
Calculator
- Shear strain in milliradians0.5202 mrad
- Shear strain in degrees0.0298 °
- Shear stress that gives a strain of 1000 µrad76.9 MPa
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 shear stress of 40 MPa in a material with G = 76.9 GPa distorts a right angle by 520.2 µrad (0.5202 mrad), a very small angle.
- The stress needed to reach a strain of 0.1 % is G/1000, which is 76.9 MPa, so ordinary shear stresses in elastic metals give strains of a few hundred microradians.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (µrad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| G Shear modulus | 1 to 200 GPa | 200 | 40,000 | 39,800 | +10% | −9.09 % |
| τxy Shear stress | −300 to 300 MPa | −3,901 | 3,901 | 7,802 | +12 MPa | +30 % |
Sweep of Shear stress
| Shear stress (MPa) | Engineering shear strain (µrad) |
|---|---|
| −300 | −3,901 |
| −225 | −2,926 |
| −150 | −1,951 |
| −75 | −975.3 |
| 0 | 0 |
| 75 | 975.3 |
| 150 | 1,951 |
| 225 | 2,926 |
| 300 | 3,901 |
Sweep one variable
Variables and how each one contributes
Engineering shear strain
What the equation gives you. Everything else feeds it.
Shear stress
The shear stress, in MPa. Raises the shear strain in direct proportion. Raising it by 5 MPa (to 45 MPa) moves the result by +12.5 %, from 520.2 to 585.2 µrad.
Shear modulus
The shear modulus, in GPa (about 77 for steel, 26 for aluminium). Lowers the shear strain in inverse proportion: a stiffer material distorts less for the same stress. Raising it by 5 GPa (to 81.9 GPa) moves the result by −6.11 %, from 520.2 to 488.4 µrad.
Worked example with the values above
- Stress over modulus, with consistent units: τ/G = 40 MPa / 76,900 MPa = 5.202×10−4 rad.
- Convert to microradians: × 10⁶.
- γ = 520.2 µrad.
Assumptions and limits
- A linear elastic, homogeneous and isotropic material, with strains small.
- Stresses in MPa with moduli in GPa, giving strain in microstrain.
- Pure shear in a single plane, within the elastic limit.
Common mistakes
- Using Young’s modulus in place of the shear modulus.
- Confusing the engineering shear strain γ with the tensor shear strain γ/2.
Statistics and uncertainty
Shear strains are small and hard to measure directly; they are usually inferred from two normal strains on a rosette. An error of 1 % in G gives 1 % in γ.
Where it is used
- The shear distortion of bolts, adhesive layers and shafts.
- Converting a shear stress to a strain for gauge comparisons.
Maths you need
- Stress and strain: force per area and relative change in length
- Elastic modulus: stress divided by strain in the elastic range