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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:

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

    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.

    This page titled 17: Appendix F- The Cauchy Stress Tensor is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Bill Smyth via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.