A thin steel plate carries a plane stress state

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

A thin steel plate carries a plane stress state, referred to axes

x,y

fixed in the plate, given by

\sigma_{xx} = -1.0\times10^{5}\,\mathrm{Pa}

\sigma_{yy} = -0.4\times10^{5}\,\mathrm{Pa}

\sigma_{xy} = 0.2\times10^{5}\,\mathrm{Pa}

, with

\sigma_{zz}=\sigma_{xz}=\sigma_{yz}=0

(a two-dimensional state). ((a)) Using

\sigma'_{ij} = R_{ik}R_{jl}\sigma_{kl}

for axes

x',y'

rotated by

\theta = 30^\circ

anticlockwise about

z

from

x,y

, find

\sigma'_{x'x'}

and

\sigma'_{x'y'}

. ((b)) Compute

\sigma'_{y'y'}

from the same transformation law, and check your results by confirming that

\sigma'_{x'x'}+\sigma'_{y'y'}=\sigma_{xx}+\sigma_{yy}

and that the determinant of the in-plane block is unchanged.

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