Cauchystresstensorn

Cauchystresstensorn, commonly referred to in standard literature as the Cauchy stress tensor, is a fundamental quantity in continuum mechanics that describes internal forces within a material. It is a second-order tensor defined at each point of a deformable body and relates the traction (the force per unit area) on an imagined plane by the plane’s normal vector.

For a plane with unit normal n, the traction t acting on that plane is t = σ · n, where σ is the Cauchy stress tensor. In a Cartesian basis, the components are σij, with i indexing the direction of the traction and j indexing the orientation of the plane. The tensor encodes normal stresses (diagonal components) and shear stresses (off-diagonal components). In classical continuum mechanics without body couples, σ is symmetric: σij = σji.

The tensor is the primary bridge between microscopic interactions and observable mechanical behavior. Its invariants, such as the hydrostatic pressure p = -1/3 tr(σ), and its deviatoric part, play central roles in material models and failure criteria. Under a change of coordinates, σ transforms as a second-order tensor: σ′ = Q σ Q^T.

Governing equations couple σ to motion and equilibrium. In static or quasi-static problems, balance of linear momentum gives ∇·σ + f = 0, where f is the body force per unit volume. In dynamics, ρa = ∇·σ + b, with ρ the density and a the acceleration. The Cauchy stress tensor is named after Augustin-Louis Cauchy, who formalized its use in the early 19th century. Notes: the term Cauchystresstensorn is uncommon; standard nomenclature is the Cauchy stress tensor.