Elasticity — Question 6

Question 6

A shear pad separates lateral stiffness from normal compression

A rectangular elastic pad has bonded face area $A=.020\,\mathrm{m^2}$, thickness $h=.050\,\mathrm m$ and shear modulus $G=.80\,\mathrm{MPa}$. Opposite tangential forces of $160\,\mathrm N$ act on its faces. Use uniform small shear: $\tau=G\gamma$, $\gamma=\Delta x/h$.

1. Find shear stress, strain and relative face displacement.

2. Find stored energy and the effect of doubling $h$ at fixed force and area.

Question 6 - Solution

A shear pad separates lateral stiffness from normal compression

(a) Derivation and calculation

Uniform shear gives $$\boxed{\tau=F/A=160/.020=8.00\ \mathrm{kPa}},$$ $$\boxed{\gamma=\tau/G=8000/(.80\times10^6)=.0100},$$ $$\boxed{\Delta x=\gamma h=.0100(.050)=.500\ \mathrm{mm}}.$$

(b) Result and interpretation

The equivalent shear stiffness and stored energy are $$k_s=GA/h=3.20\times10^5\ \mathrm{N/m},$$ $$\boxed{U=\tfrac12F\Delta x=\tfrac12(160)(.000500)=.0400\ \mathrm J}.$$ At fixed $F,A,G$, doubling $h$ gives $k_s'=k_s/2$, $\Delta x'=2\Delta x=1.00$ mm and $U'=2U=.0800$ J. Stress and shear strain are unchanged.

Physical check

The computed shear strain is small, consistent with the stated linear small-shear model.