A flat, rigid plate is turned by a small angle about an axis through the origin…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Strain and stress in a deformed solid

Problem

A flat, rigid plate is turned by a small angle ϕ=0.020 rad\phi=0.020\ \mathrm{rad} about an axis through the origin perpendicular to the plate, with no bending or stretching. Its displacement field is u1=−ϕx2u_1=-\phi x_2, u2=ϕx1u_2=\phi x_1. Someone argues: 'The point originally at (0.10,0,0) m(0.10,0,0)\ \mathrm{m} moves through u2=ϕ(0.10)=2.0×10−3 mu_2=\phi(0.10)=2.0\times10^{-3}\ \mathrm{m}, so ∂u2/∂x1=ϕ\partial u_2/\partial x_1=\phi; treating this alone as the shear strain gives ε12=∂u2/∂x1=ϕ=0.020\varepsilon_{12}=\partial u_2/\partial x_1=\phi=0.020, so the plate is shearing as it turns.' ((a)) Find the actual strain tensor ε11\varepsilon_{11}, ε22\varepsilon_{22}, ε12\varepsilon_{12} for this displacement field. ((b)) Say in words what is wrong with the argument, and name the quantity that its number 0.0200.020 actually equals.

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