CauchySpannungstensors

The CauchySpannungstensors, or Cauchy stress tensors, are second-order tensors that describe the distribution of internal forces within a continuum in the current (spatial) configuration. For any smooth surface with unit normal n, the traction vector t acting on that surface is t = σ n, where σ is the Cauchy stress tensor. Thus σ maps surface normals to the corresponding traction and encodes normal and shear stresses on all planes through a material point.

In classical continua without couple stresses, the Cauchy stress tensor is symmetric, σ = σ^T, which follows from the balance of angular momentum. The components of σ depend on the current state of deformation, temperature, and material behavior. The mechanical equilibrium equation is ∇·σ + b = 0, where b is the body-force density.

Relation to other stress measures: In finite deformation, the Cauchy stress is related to the first Piola-Kirchhoff stress P by P = σ F^{-T}, with F the deformation gradient. The left and right Cauchy–Green tensors, B = F F^T and C = F^T F, connect σ to constitutive models. In small-strain linear elasticity, σ = λ tr(ε) I + 2 μ ε, where ε is the small strain tensor and λ, μ are Lamé parameters. In Newtonian fluids, σ = -p I + 2 μ d, with p the pressure and d = (∇v + ∇v^T)/2 the rate-of-deformation tensor.

Notes: The Cauchy stress is defined with respect to the current configuration and is objective under rigid body motions. Constitutive models must respect frame-indifference and comply with the governing balance laws and boundary conditions for accurate predictions.