17: Appendix F- The Cauchy Stress Tensor
- Page ID
- 18044
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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.