Shear modulus from stress, strain, and geometry
Shear modulus, also called modulus of rigidity, relates shear stress to shear strain in a material’s linear elastic range.
The geometry mode derives stress from tangential force divided by area and strain from lateral displacement divided by specimen height.
How to use the shear modulus calculator
- Select an input method: Use direct stress and strain or force and geometry.
- Choose the unknown: In direct mode, select modulus, stress, or strain.
- Enter positive values: Keep quantities within consistent units and the material’s elastic range.
- Calculate: Review modulus in GPa, stress in MPa, and dimensionless strain.
Formula and variables
Shear stress and modulus use pressure units; shear strain is a dimensionless ratio.
G = τ/γ; τ = F/A; γ = Δx/L₀- G — Shear modulus
- Ratio of shear stress to shear strain (Pa)
- τ — Shear stress
- Tangential force per area (Pa)
- γ — Shear strain
- Lateral displacement divided by height
- F — Tangential force
- Force parallel to the sheared face (N)
Direct shear response
A specimen experiences 80 MPa shear stress at 0.001 shear strain.
- Stress
- 80 MPa
- Strain
- 0.001
- G = 80 MPa / 0.001
Result: G = 80 GPa.
The linear stress-to-strain ratio is 80 gigapascals.
Understanding your results
Use the elastic-range slope
A modulus result represents the linear elastic stress-strain ratio.
- Higher G means greater resistance to shear deformation.
- Stress and modulus share pressure dimensions.
- Shear strain is dimensionless.
- Outside the linear range, one constant modulus may not describe behavior.
Assumptions
- Material response is linear elastic.
- Stress and strain are representative of the specimen.
- Geometry mode uses small deformation and uniform shear.
Limitations
- Does not model plasticity, anisotropy, viscoelasticity, or stress concentration.
- Geometry mode is an idealized uniform-shear calculation.
- Engineering design requires verified material data and safety factors.
Common mistakes
- Treating strain as having pressure units.
- Mixing Pa, MPa, and GPa without conversion.
- Using total force instead of the tangential component.
- Applying the linear formula beyond the elastic range.
Practical use cases
Mechanics coursework
Rearrange the shear stress-strain equation.
Specimen checks
Estimate an apparent modulus from force and displacement measurements.
Frequently asked questions
Is shear strain measured in radians?
It is dimensionless; for small angles its numerical value corresponds to the shear angle in radians.
Is shear modulus the same as Young’s modulus?
No. They describe different deformation modes and are related only with additional material assumptions.
Sources and review
- Stress, Strain, and Elastic Modulus — OpenStax University Physics Volume 1. Accessed 2026-07-14.
Reviewed 2026-07-14.