shear-modulus-uses-sideways-strain - Elasticity and Nonlinear Dynamics Step by Step

The shear modulus G ties shear stress to shear strain, tau = G * gamma, where the strain is the sideways displacement over the height of the block, the analog of Young's modulus for a shape change. The arithmetic is exact only for the stated shear modulus, area, and sideways strain in the ideal linear elastic model. Real materials add nonlinear response, geometry tolerance, temperature/loading history, fixture compliance, calibration, and measurement uncertainty.

A shear modulus ledger uses sideways displacement over height. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Shear uses sideways displacement over height

The sideways displacement and height form a dimensionless strain. The shear force and area form shear stress.

x=1\ \text{m}\qquad h=3\ \text{m}

With shear strain fixed, modulus follows stress

The displacement-over-height ratio is fixed in every row. Larger shear force gives larger shear stress and modulus.

\begin{array}{c|c|c|c|c}F&A&x/h&\tau&G\\3\ \text{N}&3\ \text{m}^{2}&\frac{1}{3}&1\ \text{Pa}&3\ \text{Pa}\\6\ \text{N}&3\ \text{m}^{2}&\frac{1}{3}&2\ \text{Pa}&6\ \text{Pa}\\9\ \text{N}&3\ \text{m}^{2}&\frac{1}{3}&3\ \text{Pa}&9\ \text{Pa}\\\end{array}

Shear modulus uses sideways strain

The shear ledger separates shear stress from sideways strain before computing the modulus.

\tau=\frac{F}{A}=3\ \text{Pa};\quad \gamma=\frac{x}{h}=\frac{1}{3};\quad G=\frac{\tau}{\gamma}=9\ \text{Pa}