Cauchy stress tensor

Representation of mechanical stress at every point within a deformed 3D object

In continuum mechanics, the Cauchy stress tensor (symbol ⁠ σ {\displaystyle {\boldsymbol {\sigma }}} ⁠, named after Augustin-Louis Cauchy), also called true stress tensor or simply stress tensor, completely defines the state of stress at a point inside a material in the deformed state, placement, or configuration. The second order tensor consists of nine components σ i j {\displaystyle \sigma _{ij}} and relates a unit-length direction vector e to the traction vector T(e) across a surface perpendicular to e: T ( e) = e ⋅ σ or T j ( e) = ∑ i σ i j e i . {\displaystyle \mathbf {T} ^{(\mathbf {e})}=\mathbf {e} \cdot {\boldsymbol {\sigma }}\quad {\text{or}}\quad T_{j}^{(\mathbf {e})}=\sum _{i}\sigma _{ij}e_{i}.} The SI unit of both stress tensor and traction vector is the newton per square metre (N/m²) or pascal (Pa), corresponding to the stress scalar.

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