Cauchystresstensor

The Cauchy stress tensor, denoted sigma, is a second-order tensor that relates the traction vector on a plane to the plane’s unit normal in the current (deformed) configuration of a continuum. For a plane with unit normal n, the traction (force per unit area) acting on that plane is t = sigma · n. Its components sigma_ij describe the state of stress at a material point.

In standard (non-polar) continuum mechanics, the Cauchy stress tensor is symmetric: sigma = sigma^T. This symmetry follows from the balance of angular momentum in the absence of distributed couple stresses. The eigenvalues of sigma are the principal stresses, and the corresponding eigenvectors give the principal directions, i.e., the orientations where the traction is normal to the plane. The invariants I1 = trace(sigma), I2 = 1/2[(trace sigma)^2 − trace(sigma^2)], and I3 = det(sigma) characterize the tensor independently of the coordinate system.

The Cauchy stress is defined in the current configuration. In finite deformation, it is related to other stress measures defined in the reference configuration, such as the second Piola-Kirchhoff stress S, by the relation sigma = (1/J) F S F^T, where F is the deformation gradient and J = det F. In linear isotropic elasticity, sigma = lambda tr(eps) I + 2 mu eps, where eps is the small-strain tensor and lambda, mu are Lamé constants.

Practically, the Cauchy stress is determined from traction measurements and constitutive models, and it is central to continuum mechanics, structural analysis, and materials science. See also stress tensor, deformation gradient, and Piola-Kirchhoff stresses.