At a point in a loaded steel plate the stress tensor is _ ij = pmatrix 50 & 30 & 0 30…

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture · Solid continuum mechanics and complex fluids

Fluids and Continuum Physics, Ch The continuum hypothesis, pressure and the stress tensor

Problem

At a point in a loaded steel plate the stress tensor is

\sigma_{ij} = \begin{pmatrix} 50 & 30 & 0\\ 30 & -30 & 0\\ 0 & 0 & 0\end{pmatrix}\ \mathrm{MPa}.

A small cut through the point has unit normal

\hat{\mathbf{n}} = (\cos\theta,\ \sin\theta,\ 0)

, and

\hat{\mathbf{s}} = (-\sin\theta,\ \cos\theta,\ 0)

is the unit vector that lies in the cut and in the

xy

plane. (a) For any symmetric stress tensor with entries

\sigma_{xx}

\sigma_{yy}

and

\sigma_{xy} = \sigma_{yx} = \tau

in the

xy

plane and

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

, show that the traction on the cut has normal component

t_n = \tfrac12(\sigma_{xx}+\sigma_{yy}) + \tfrac12(\sigma_{xx}-\sigma_{yy})\cos2\theta + \tau\sin2\theta

and in-plane component

t_s = \tau\cos2\theta - \tfrac12(\sigma_{xx}-\sigma_{yy})\sin2\theta

. (b) For the steel plate, find the two orientations with

0 \le \theta < 180^{\circ}

at which the cut carries no shear traction, and the normal traction on each. (c) A sheared glycerine film has

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

with

p = 1.00\times10^{5}\ \mathrm{Pa}

, and

\tau = 14\ \mathrm{Pa}

. Use part (a) to find its zero-shear orientations and the normal traction on each. A classmate argues that because the film is flowing, every cut through it must carry some shear traction. Use your results to say whether this is right.

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