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17: Appendix F- The Cauchy Stress Tensor

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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:

    \[\tau_{i j}=f_{j}^{(i)}.\]

    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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