How this instrument works
Mark a small square on the side of a block and push its top sideways without letting it rise. The square becomes a parallelogram, and the corner that once measured ninety degrees closes by some angle θ. Engineering shear strain γ is the tangent of that closing angle — the travel Δx of the moving face divided by the height h between faces. Length over length, so no unit survives the division. The working range is enormous: structural steel yields in shear near γ = 0.0018, laminated elastomeric bearings under a bridge deck are routinely designed to swing through 0.5, dry sand starts rearranging its grains past roughly 0.0001, and rock across a plate boundary gathers a few hundred nanostrain in a year.
The bookkeeping goes back to Cauchy, who between 1822 and 1827 assembled the linear strain tensor that still underpins elasticity, and who defined its off-diagonal terms symmetrically as ε_xy = ½(∂u/∂y + ∂v/∂x). Engineers kept a different quantity — γ_xy = ∂u/∂y + ∂v/∂x, twice as large — because that is what a protractor laid on the deformed square would register, and because it pairs directly with stress as τ = Gγ. Both definitions are correct, both remain current, and a factor of two therefore lurks in every conversation between a hand calculation and a solver's output. Read the axis label before comparing figures.
Δx over h assumes simple shear: the two faces stay flat and parallel, displacement grows linearly with height, and nothing bulges or rotates on its own account. A slender specimen breaks that at once, since bending contributes travel that scales with the cube of height while genuine skew scales only with height, and the ratio reports the surplus as distortion that was never there. The linear picture also expires once deformation gets large — past roughly ten percent the gap between tan θ and θ stops being ignorable, volume no longer holds still, and finite-strain measures such as the Green–Lagrange or logarithmic families take over. Nor is γ a property of the material: it is one component of a tensor, so rotating your axes changes it, and along the principal directions it disappears altogether.
- Enter Sideways deflection — how far the moving face has travelled relative to the fixed one, measured parallel to both. Millimetres, centimetres, metres or inches.
- Enter Height of the element: the perpendicular distance between the fixed face and the moving one, not the block's length or width.
- Read Shear strain (dimensionless) as a bare ratio. Multiply by 100 for percent, or by a million for microstrain.
- Take Shear angle (degrees) when the geometry matters more than the ratio — it is the arctangent of the strain, so the two part company as the lean grows.
Worked example — a bridge bearing on a warm afternoon
A short-span highway bridge rests on laminated elastomeric bearings, each pad 100 mm tall between the girder soffit and the abutment seat. The deck warms through the afternoon, expands along its length, and drags the top of the pad 2 mm while the underside stays bonded to concrete. Enter 0.002 m into Sideways deflection and 0.1 m into Height of the element.
Shear strain (dimensionless) returns 0.002 ⁄ 0.1 = 0.02 exactly — two percent, or 20 000 microstrain. Shear angle (degrees) gives arctan(0.02) = 1.1458°, so the pad's vertical edges now stand a shade over one degree off plumb. Notice how close the two descriptions stay at this size: the ratio 0.02 and the same angle expressed in radians, 0.0199973, agree to better than 0.02 percent. That closeness is what makes the small-angle shortcut safe here and reckless at fifty percent.
Two percent is comfortable for rubber and alarming for metal. A pad of this type will work through 0.5 without anyone losing sleep, whereas 0.02 imposed on a structural steel plate is more than ten times its shear yield strain. That gap is the whole argument for elastomeric bearings: they absorb the movement a superstructure cannot.
Questions
What unit does shear strain use?
None at all. A length divided by a length leaves a bare number, so γ carries no unit and only convention decides how it is dressed: 0.02 may be written as 2 percent, as 20 000 microstrain, or as 20 000 µm/m. Degrees belong to Shear angle (degrees) and never to the ratio itself. Writing γ = 1.15 degrees quietly swaps the angle for its tangent, which is harmless at small leans and off by about one percent by the time the lean reaches ten degrees.
Why do two textbooks give different shear strains for the same block?
Because two conventions coexist. Engineering shear strain γ is the whole closing of the right angle and pairs with stress as τ = Gγ. Tensor shear strain ε_xy is half that, chosen so the components transform properly under rotation of axes. Hand calculations, strain-gauge handbooks and Mohr's circle for strain normally mean γ; continuum mechanics and much solver output means ε_xy. Sign works alike in both: positive when the right angle between the positive axes closes, negative when it opens. A stiffness that looks twice too large is almost always this.
How is shear strain measured in practice?
Rarely head-on, since a bonded foil gauge senses stretching along its own grid and is nearly blind to skew. The standard dodge is a rectangular rosette — three grids at 0, 45 and 90 degrees — giving γ = 2ε₄₅ − ε₀ − ε₉₀. Optical methods read distortion directly instead: digital image correlation tracks a sprayed speckle pattern and differentiates the displacement field, while photoelastic fringes map the difference of principal strains. At continental scale, permanent GPS networks resolve crustal shear strain rates of tens of nanostrain per year.
Can shear strain be zero even though the material is deformed?
Yes, and that is the point of principal directions. Any state of plane strain can be rotated onto a pair of perpendicular axes along which the material merely stretches or contracts with no skew whatsoever. Turn 45 degrees away from those axes and distortion peaks at γ_max = ε₁ − ε₂. This is why a ductile bar pulled in plain tension tears along planes tilted near 45 degrees to the pull, and why the figure returned here belongs to the pair of faces you measured rather than to the material itself.
Is this the same thing as shear rate?
No, although they share a letter. Shear strain γ is an amount of distortion a solid holds while it is loaded; shear rate γ̇ is how quickly distortion accumulates, measured per second, and it is what drives a fluid through τ = ηγ̇. A rheometer sweeps both — small-amplitude oscillatory tests hold strain amplitude near 0.01 to stay inside the linear region, while flow curves push rates from 0.01 up to 1000 per second. Treating an amount as a rate turns a viscosity into a modulus and vice versa.
How much shear strain can a material take?
It depends entirely on the material, and the spread is wide. Structural steel yields in shear near γ = 0.0018, which is simply a shear yield around 145 MPa divided by a rigidity of 79 GPa. Concrete cracks in diagonal tension well before reaching that. Elastomeric bridge bearings are specified to work through 0.5 and survive considerably more. Saturated loose sand can begin to liquefy under cyclic shear strain of about 0.0001. The two percent in the worked example is routine for rubber and far past yield for any structural metal.