Concept

Cauchy stress tensor

Summary
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_{ij} that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector e to the traction vector T(e) across an imaginary surface perpendicular to e: :\mathbf{T}^{(\mathbf e)} = \mathbf e \cdot\boldsymbol{\sigma}\quad \text{or} \quad T_{j}^{(e)}= \sigma_{ij}e_i, or, :\left[{\begin{matrix} T^{(\mathbf e)}_1 & T^{(\mathbf e)}_2 & T^{(\mathbf e)}_3\end{matrix}}\right]=\left[{\begin{matrix} e_1 & e_2 & e_3 \end{matrix}}\right]\cdot \left[{\begin{matrix} \sigma _{11} & \sigma _{12} & \sigma _{13} \ \sigma _{21} & \sigma _{22} & \sigma _{23} \ \sigma _{31} & \sigma _{32} & \sigma _{33} \ \en
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