Cauchy stress tensor

The Cauchy stress tensor maps a surface’s unit normal to the traction acting across that surface at a point in a continuum. Its components describe normal and shear stresses in a chosen coordinate system.

Cauchy's traction theorem: Cauchy's traction theorem states that traction on a plane through a point depends linearly on its unit normal. It gives the relation that defines the tensor’s action on surface orientation.
Continuum mechanics: Continuum mechanics models matter as continuously distributed, deformable material rather than as discrete particles. The tensor describes internal forces within this continuous material model.
Von Mises yield criterion: The von Mises yield criterion predicts yielding in ductile materials from the distortional part of the stress state. It reduces the stress tensor to an equivalent scalar used to assess yielding.
First Piola–Kirchhoff stress tensor: The first Piola–Kirchhoff stress tensor maps reference-area normals to forces per reference area in finite-deformation mechanics. Unlike Cauchy stress, it uses reference areas and is generally not symmetric.
Traction vector: The traction vector is force per unit area acting across an oriented surface in a continuum. The tensor maps each surface normal to this force-per-area vector.
Cauchy stress principle: The Cauchy stress principle states that internal force across a small surface element is proportional to its area as the element shrinks. It supports representing local surface forces by a finite traction vector.
Mohr's circle: Mohr's circle is a graphical method for representing stress transformation in two dimensions. It visualizes how the tensor’s normal and shear components vary across plane orientations.
Second Piola–Kirchhoff stress tensor: The second Piola–Kirchhoff stress tensor measures stress in the reference configuration and is conjugate to Green–Lagrange strain. It describes material-frame stress rather than force per current area.
Stress transformation: Stress transformation describes how stress components change when coordinate axes or plane orientation changes. Changing the basis alters the tensor’s components without changing the physical stress state.
Tensor: A tensor is a mathematical object whose components transform consistently under changes of coordinates. The stress components depend on axes, while the represented physical quantity does not.

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