Shear Modulus

Imagine pushing the top of a stack of playing cards sideways while the bottom stays fixed. Each card slides a little relative to its neighbor, and the whole deck distorts into a slanted shape without changing its volume. This type of deformation — where parallel layers slide past each other — is called shear deformation, and the shear modulus (also called the modulus of rigidity, symbol G) measures how strongly a material resists it. The shear modulus is one of the three primary elastic constants used to characterise solid materials, alongside Young's modulus (resistance to stretching/compression) and the bulk modulus (resistance to uniform pressure). While Young's modulus describes what happens when you pull a rod along its length, the shear modulus describes what happens when you twist a shaft, cut with scissors, or wobble a block of jelly sideways. Liquids and gases have a shear modulus of zero — they cannot resist shear at all, which is precisely why they flow. Quantitatively, the shear modulus is defined as the ratio of shear stress to shear strain. A high value (e.g., steel at ~80 GPa) means the material barely deforms under a given sideways force; a low value (e.g., rubber at ~0.001 GPa) means it deforms easily. Understanding G is essential for designing drive shafts, earthquake-resistant buildings, surgical instruments, and even the strings of musical instruments.

Key takeaways
  • What is Shear Stress?
  • What is Shear Strain?
  • Hooke's Law for Shear
  • Relation to Other Elastic Constants
  • Why Fluids Have G = 0

What is Shear Stress?

Shear stress (τ) is a force applied tangentially (parallel) to a surface, divided by the area of that surface: τ = F/A. Unlike normal stress, which pushes perpendicular to a face, shear stress tries to make one face slide relative to the opposite face. Units are Pascals (Pa = N/m²). Think of it as the 'sliding pressure' on a surface.

What is Shear Strain?

Shear strain (φ, sometimes written γ) is the tangential displacement of the top face divided by the height of the block: φ = Δx / h. This is equal to tan θ, where θ is the angle by which a vertical edge tilts. For small deformations (the usual case in elastic behaviour), tan θ ≈ θ in radians, so shear strain is dimensionless and equal to the tilt angle. A shear strain of 0.01 means the top face has shifted 1 cm for every metre of height.

Hooke's Law for Shear

Just as a spring obeys F = kx, an elastic solid under shear obeys a linear relationship between shear stress and shear strain, provided the deformation is small. The proportionality constant is the shear modulus G: τ = G·φ. This is the shear analogue of Hooke's law. The linearity holds only in the elastic regime — push too hard and the material yields (plastically deforms) or fractures.

Relation to Other Elastic Constants

For an isotropic material (same properties in all directions), G is not independent of Young's modulus E and Poisson's ratio ν. They are linked by G = E / [2(1 + ν)]. For most metals ν ≈ 0.3, giving G ≈ E/2.6 — so the shear modulus is always less than Young's modulus for common materials. This relationship is extremely useful: measure any two of {E, G, ν} and you can calculate the third.

Why Fluids Have G = 0

A fluid cannot sustain a static shear stress — any shear force, no matter how tiny, causes continuous flow rather than a fixed deformation. Mathematically, for a fluid φ → ∞ for any finite τ, so G = τ/φ → 0. This is the fundamental physical distinction between solids and fluids: solids have G > 0, fluids have G = 0. Highly viscous fluids like pitch or glass on very long timescales do flow under shear, illustrating that this distinction can be time-dependent.

The Key Equation

G=τϕ=F/AΔx/hG = \frac{\tau}{\phi} = \frac{F/A}{\Delta x / h}
  • G — Shear Modulus (Modulus of Rigidity) (Pascal (Pa)): The material constant measuring resistance to shear deformation; higher G means stiffer in shear.
  • \tau — Shear Stress (Pascal (Pa)): Tangential (parallel) force F applied over area A: τ = F/A.
  • \phi — Shear Strain (Dimensionless (radians)): Lateral displacement Δx divided by original height h; equals the tilt angle θ for small deformations.
  • F — Applied Tangential Force (Newton (N)): The sideways force causing the shear deformation.
  • A — Cross-sectional Area (Square metre (m²)): The area of the face over which the tangential force is distributed.
  • \Delta x — Lateral Displacement (Metre (m)): How far the top face moves sideways relative to the fixed bottom face.
  • h — Height of Block (Metre (m)): The perpendicular distance between the two sheared faces.

Worked Example

Problem. A rubber block has a square base of side 8 cm and a height of 5 cm. A horizontal force of 24 N is applied to its top face while the bottom is fixed. The top face shifts sideways by 3 mm. (a) Calculate the shear stress, (b) the shear strain, and (c) the shear modulus of the rubber.

τ=3.75 kPa,ϕ=0.06,G=62.5 kPa\tau = 3.75\,\text{kPa},\quad \phi = 0.06,\quad G = 62.5\,\text{kPa}

Answer. Shear stress = 3.75 kPa; Shear strain = 0.06; Shear modulus G = 62.5 kPa

Figure
Diagram
Shear Deformation of a Solid BlockFixed base (A = L²)OriginalShearedBlockhΔxFθShear ModulusG = τ / φτ = F/Aφ = Δx/h = tan θ
Origins · 1 minClaude-Louis Navier and how this was discovered
CN
Claude-Louis Navier
1785–1836 · French

Navier was a pioneer of continuum mechanics and structural engineering. He developed the mathematical framework for elasticity theory, correctly describing how internal stresses and strains are related in solid bodies — foundational work that underpins the very definition of elastic moduli including the shear modulus. He also derived (independently of Stokes) the Navier–Stokes equations governing fluid flow. Navier designed a suspension bridge over the Seine in Paris in the 1820s, but it was demolished before opening due to engineering controversies — a painful reminder that even great theorists face practical challenges!

  • Young's Modulus — Young's modulus measures resistance to tensile/compressive stress along an axis and is mathematically related to the shear modulus via Poisson's ratio: G = E/[2(1+ν)].
  • Bulk Modulus — The bulk modulus measures resistance to uniform (volumetric) compression and completes the trio of elastic constants alongside Young's modulus and the shear modulus.
  • Poisson's Ratio — Poisson's ratio ν links lateral and axial strains and connects all three elastic constants; you cannot relate G to E without knowing ν.
  • Hooke's Law — The shear modulus is the shear analogue of Hooke's spring constant — both express a linear (proportional) relationship between a deforming force and the resulting deformation.
  • Stress and Strain — Shear stress and shear strain are the two fundamental quantities whose ratio defines G; a firm grasp of normal stress/strain is needed before extending the ideas to shear.