# Shear modulus
The shear modulus, usually written G and also called the modulus of rigidity or torsional modulus, is the ratio of shear stress to the corresponding shear strain in a material, measured below the proportional limit where the response is linear[1](https://www.iteh.us/catalog/standards/astm/5e714b7c-f0f7-4691-a55c-f48753406d40/astm-e143-20). It measures a solid's resistance to a change in shape under conditions where the volume stays constant, and it has the units of pressure, pascals in SI[2](https://msecore.northwestern.edu/406/406text.pdf).
| Key fact | Detail |
|---|---|
| Definition | Ratio of shear stress to shear strain below the proportional limit; also called torsional modulus and modulus of rigidity[1](https://www.iteh.us/catalog/standards/astm/5e714b7c-f0f7-4691-a55c-f48753406d40/astm-e143-20) |
| Governing relation | \( \tau = G \gamma \), valid for pure shear; in tensor form \( G = \tfrac{1}{2} \cdot S_{13}/\varepsilon_{13} \)[3](https://ocw.mit.edu/courses/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/02ba48770baeafa2478e4d212be75d0d_zm17_20.pdf) • [4](https://johnfoster.pge.utexas.edu/PGE334-ResGeomechanics/slides/Lecture11-ConstitutiveModeling-Elasticity.slides.html) |
| Isotropic relations | \( E = 2G(1+\nu) \) and \( \nu = (3K - 2G)/(6K + 2G) \), linking G to Young's modulus E, bulk modulus K, and Poisson's ratio ν[5](https://lakeslab.ep.wisc.edu/PoissonIntro2.html) |
| Standard test | ASTM E143-20, static torsion testing at room temperature, \( G = TL/(J\theta) \)[1](https://www.iteh.us/catalog/standards/astm/5e714b7c-f0f7-4691-a55c-f48753406d40/astm-e143-20) |
| Typical Poisson's ratio | About 0.33 for most materials, giving \( G \approx 3E/8 \)[3](https://ocw.mit.edu/courses/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/02ba48770baeafa2478e4d212be75d0d_zm17_20.pdf) |
| Applications | Design of rotating shafts and helical compression springs[6](https://store.astm.org/standards/e143) |
| Anisotropic case | G is an independent constant, not derivable from E and ν; the ratio \( A = 2c_{44}/(c_{11}-c_{12}) \) measures anisotropy, with A = 1 for isotropic materials[7](https://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php) |
## Definition and the strain convention
An applied shear stress produces a shear strain. In engineering notation the shear strain γ equals the total change in angle of a right angle, and the shear modulus is defined consistently with [Young's modulus](https://www.edgepedia.org/youngs-modulus) by \( \tau = G\gamma \)[3](https://ocw.mit.edu/courses/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/02ba48770baeafa2478e4d212be75d0d_zm17_20.pdf). This relation holds only when a pure shear state is applied to the specimen[3](https://ocw.mit.edu/courses/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/02ba48770baeafa2478e4d212be75d0d_zm17_20.pdf).
**Two strain conventions.** The engineering shear strain γ_xy and the tensorial shear strain ε_xy differ by a factor of 2: \( \epsilon_{xy} = \tfrac{1}{2}\gamma_{xy} \)[8](https://academic.oup.com/book/4888/chapter/147281580). This factor of two is a common source of error in strain transformation and in listing material constants[8](https://academic.oup.com/book/4888/chapter/147281580).
## Measurement by torsion
The standard laboratory determination is ASTM E143-20, which covers static torsion testing of structural materials at room temperature, limited to materials and stress levels where creep is negligible compared with the immediate elastic strain[1](https://www.iteh.us/catalog/standards/astm/5e714b7c-f0f7-4691-a55c-f48753406d40/astm-e143-20) • [6](https://store.astm.org/standards/e143). For a specimen of circular cross-section the modulus follows from the torque–twist relation
\[ G = \frac{TL}{J\theta} \]
where T is the applied torque, L the gauge length, J the polar moment of inertia of the section, and θ the angle of twist in radians[1](https://www.iteh.us/catalog/standards/astm/5e714b7c-f0f7-4691-a55c-f48753406d40/astm-e143-20). For a solid cylinder \( J = \pi D^{4}/32 \), and for a tube \( J = \pi(D_{o}^{4} - D_{i}^{4})/32 \), with D the diameter and D_o and D_i the outside and inside diameters[1](https://www.iteh.us/catalog/standards/astm/5e714b7c-f0f7-4691-a55c-f48753406d40/astm-e143-20).
**Measurement pitfalls.** Precise determination depends on specimen characteristics such as residual stress, concentricity, wall thickness in tubes, previous strain history, and specimen dimensions, and on testing conditions including temperature and apparatus maintenance[1](https://www.iteh.us/catalog/standards/astm/5e714b7c-f0f7-4691-a55c-f48753406d40/astm-e143-20). Because structural shear displacement before yielding is very small, rotation measured at the machine drive can capture the wind-up and compliance of the fixtures and drive train rather than the deformation of the specimen, producing artificially low modulus readings; direct-contact twist gauges on the gauge length address this[9](https://mechanicaltest.org/technical-app-notes/astm-e143/).
## Relation to the other elastic constants
For an isotropic solid, one whose properties are independent of direction, the shear modulus is tied to Young's modulus E, the bulk modulus K, and [Poisson's ratio](https://www.edgepedia.org/poissons-ratio) ν by
\[ E = 2G(1+\nu), \qquad \nu = \frac{3K - 2G}{6K + 2G} \][5](https://lakeslab.ep.wisc.edu/PoissonIntro2.html)
Two moduli are sufficient to completely specify the elastic behavior of a homogeneous isotropic solid, and tables of interrelations allow any two known constants to be converted into the other three[10](https://link.springer.com/chapter/10.1007/978-3-030-44787-8_4). For most materials Poisson's ratio is approximately 0.33, so \( K \approx E \) and \( G \approx 3E/8 \); the shear modulus is usually somewhat smaller than E[3](https://ocw.mit.edu/courses/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/02ba48770baeafa2478e4d212be75d0d_zm17_20.pdf) • [7](https://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php).
**Stability bounds.** Requiring the bulk modulus and the shear modulus to be positive restricts Poisson's ratio to \( -1 < \nu < \tfrac{1}{2} \)[10](https://link.springer.com/chapter/10.1007/978-3-030-44787-8_4). Most materials fall between zero (cork) and one half (rubber); auxetic materials with negative Poisson's ratio are uncommon and usually anisotropic, with low mass density and complicated internal structure such as solid foams[10](https://link.springer.com/chapter/10.1007/978-3-030-44787-8_4).
## Anisotropic materials
In anisotropic solids, such as wood or fiber-reinforced composites, the shear modulus is an independent material constant and is not related to the Young's moduli or Poisson's ratios through the isotropic formula[7](https://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php) • [11](https://structures.dhu.edu.cn/_upload/article/files/f6/62/f5c6159f4c86ae7a86fbd6b48811/1f577c4d-2e0a-405f-bf2e-5930185280c8.pdf). The value depends on the shear plane and direction, and for crystals it can be expressed through components of the stiffness tensor; the ratio \( A = 2c_{44}/(c_{11}-c_{12}) \) serves as a convenient measure of anisotropy, with A = 1 corresponding to isotropic behavior[7](https://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php). Theoretical work has established necessary and sufficient conditions for an anisotropic material to exhibit a direction-independent shear modulus, and found that such a material can only be transversely isotropic or orthotropic[12](https://www.sciencedirect.com/science/article/pii/S0020722504001892).
## Physical meaning and limits
The shear modulus quantifies a material's resistance to volume-preserving shear deformation[7](https://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php). Most materials resist a change in volume, governed by the bulk modulus, more than they resist a change in shape, governed by the shear modulus, which is why stretched solids usually narrow in cross-section[5](https://lakeslab.ep.wisc.edu/PoissonIntro2.html). Fluids at rest cannot sustain static shear, and their shear modulus is zero, in contrast with crystals[13](https://arxiv.org/abs/0905.0764).
In engineering practice the modulus enters the calculation of torsional compliance of structural members, provided they follow [Hooke's law](https://www.edgepedia.org/hookes-law), meaning the angle of twist is proportional to the applied torque; rotating shafts and helical compression springs are the standard examples[6](https://store.astm.org/standards/e143). For materials with nonlinear elastic stress-strain behavior, tangent or chord shear moduli can be used to estimate the change in torsional strain over a specified stress range, though such determinations fall outside the scope of the ASTM method[1](https://www.iteh.us/catalog/standards/astm/5e714b7c-f0f7-4691-a55c-f48753406d40/astm-e143-20).
## References
1. [ASTM E143-20, Standard Test Method for Shear Modulus at Room Temperature](https://www.iteh.us/catalog/standards/astm/5e714b7c-f0f7-4691-a55c-f48753406d40/astm-e143-20)
2. [Stress and Strain: Deformation Modes, Northwestern University MSE 406 course text](https://msecore.northwestern.edu/406/406text.pdf)
3. [MIT OCW 16.01 Unified Engineering, Stress and Strain notes](https://ocw.mit.edu/courses/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/02ba48770baeafa2478e4d212be75d0d_zm17_20.pdf)
4. [Lecture 11: Constitutive Modeling, Elasticity, University of Texas at Austin](https://johnfoster.pge.utexas.edu/PGE334-ResGeomechanics/slides/Lecture11-ConstitutiveModeling-Elasticity.slides.html)
5. [What is Poisson's ratio? page 2, R.S. Lakes, University of Wisconsin](https://lakeslab.ep.wisc.edu/PoissonIntro2.html)
6. [ASTM E143, Standard Test Method for Shear Modulus at Room Temperature, ASTM International store](https://store.astm.org/standards/e143)
7. [A.F. Bower, Applied Mechanics of Solids, Section 3.2: Linear Elastic Materials](https://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php)
8. [Strain, Engineering Mechanics of Deformable Solids, Oxford Academic](https://academic.oup.com/book/4888/chapter/147281580)
9. [Testing Products for ASTM E143, Newton Characterization technical app note](https://mechanicaltest.org/technical-app-notes/astm-e143/)
10. [Elasticity of Solids, Springer Nature Link](https://link.springer.com/chapter/10.1007/978-3-030-44787-8_4)
11. [Engineering Mechanics, Donghua University course notes](https://structures.dhu.edu.cn/_upload/article/files/f6/62/f5c6159f4c86ae7a86fbd6b48811/1f577c4d-2e0a-405f-bf2e-5930185280c8.pdf)
12. [Q.-C. He, Characterization of the anisotropic materials capable of exhibiting an isotropic Young or shear or area modulus, International Journal of Engineering Science 42 (2004)](https://www.sciencedirect.com/science/article/pii/S0020722504001892)
13. [Viscoelastic Properties of Crystals, arXiv:0905.0764](https://arxiv.org/abs/0905.0764)
---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Continuum, solid, and fluid mechanics › Solid mechanics › Elasticity › Elastic moduli and constants*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*
License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license