Cauchy Stress Tensor

The Cauchy stress tensor provides a complete description of the internal forces acting within a deformable body. Imagine an infinitesimal cube of material at a point P. Forces are exerted on each face of this cube by the surrounding material. The stress tensor [latex]\boldsymbol{\sigma}[/latex] is a 3×3 matrix whose components [latex]\sigma_{ij}[/latex] represent the stress on the i-th face in the j-th direction. The diagonal components ([latex]\sigma_{11}, \sigma_{22}, \sigma_{33}[/latex]) are normal stresses, representing pulling (tension) or pushing (compression) perpendicular to the face. The off-diagonal components ([latex]\sigma_{12}, \sigma_{23},[/latex] etc.) are shear stresses, representing forces acting parallel to the face.

A key result, known as Cauchy’s stress theorem, states that knowledge of the stress vectors on three mutually perpendicular planes is sufficient to determine the stress vector on any other plane passing through that point. This is encapsulated in the formula [latex]\mathbf{T}^{(mathbf{n})} = \boldsymbol{\sigma}^T \mathbf{n}[/latex]. Furthermore, the conservation of angular momentum requires the stress tensor to be symmetric ([latex]\sigma_{ij} = \sigma_{ji}[/latex]), which reduces the number of independent components from nine to six. This tensor is fundamental because it allows engineers to analyze the stress state at any point within an object, regardless of its orientation, and to predict whether the material will yield or fracture under applied loads by comparing the stress state to the material’s strength properties.