At a point in a moving liquid the stress is

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Fluid Dynamics, Ch The stress tensor completed: constitutive law and fluid properties

Problem

\sigma_{xx}=\sigma_{yy}=\sigma_{zz}=-p

\sigma_{xy}=\sigma_{yz}=\tau

and

\sigma_{xz}=0

, with

p=2.0\times10^{5}\,\mathrm{Pa}

and

\tau=3.0\,\mathrm{Pa}

. ((a)) Find the three principal stresses and their principal directions. Is any principal axis one of the coordinate axes? ((b)) Find the largest shear traction on any plane, and a unit normal of a plane that carries it. ((c)) Check your principal stresses against the trace and the determinant of

\sigma_{ij}+p\,\delta_{ij}

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