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17: Appendix F- The Cauchy Stress Tensor

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Jul 23, 2022
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- 16: Appendix E- Vector Identities
- 17.1: F.1 Local Equilibrium

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In section 6.3.2 we defined the Cauchy array, whose elements are the components of the stress vector $\vec{f}$ acting on each of the three coordinate planes:

$\tau_{i j}=f_{j}^{(i)}. \nonumber$

In this appendix we will demonstrate three additional properties of this array:

The stress vector acting on any plane is given by $f_j=\tau_{ij}n_i$ , where $\hat{n}$ is the unit normal to the plane in question.
The array $\underset{\sim}{\tau}$ transforms as a 2nd-order tensor.
$\underset{\sim}{\tau}$ is symmetric.

We will do this by applying Newton’s second law to carefully chosen fluid parcels and imagining the result if we take the size of the parcel to zero.

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