Cauchyspanningstensor

Cauchyspanningstensor is another term for the Cauchy stress tensor, a fundamental quantity in continuum mechanics that describes the internal forces acting at a point inside a continuous medium. Named after Augustin-Louis Cauchy, it relates the traction vector on any plane passing through the point to the plane’s unit normal.

Given a surface with unit normal n, the traction t exerted by the surrounding material on that surface is t = σ · n, where σ is the Cauchy stress tensor. In component form, t_i = σ_ij n_j. The tensor σ is symmetric (σ_ij = σ_ji) in the absence of body couples and it transforms as a second-order tensor under coordinate changes, reflecting its status as a true spatial quantity defined in the current configuration.

The Cauchy stress is defined in the current (spatial) configuration and is therefore distinguished from the material (reference) configuration stresses, such as the first and second Piola-Kirchhoff stresses. It plays a central role in the balance of linear momentum, expressed as ∇·σ + b = 0, where b is the body force per unit volume.

Physical interpretation includes the notion of principal stresses, which are the eigenvalues of σ and represent normal stresses along principal directions, and stress invariants that characterize the tensor independently of the chosen coordinate frame. In fluids, σ often takes the form σ = −pI + τ, with p the pressure and τ the viscous deviatoric stress; in solids, constitutive laws relate σ to deformation, as in Hooke’s law for linear elastic materials (σ = C : ε).

History and usage: the tensor is foundational in continuum mechanics, underpinning theories of solid mechanics, fluid dynamics, and numerical methods such as finite element analysis. See also stress tensor, traction, strain, and constitutive relations.