Deviatoric

Deviatoric refers to the part of a tensor that remains after removing its isotropic, or volumetric, component. In three-dimensional space, for a second-order tensor A, the deviatoric part A' is defined as A' = A − (tr A)/3 · I, where tr(A) is the trace (sum of diagonal entries) and I is the identity tensor. This operation projects A onto the subspace of traceless, or zero-trace, tensors.

Properties and interpretation: The deviatoric part satisfies tr(A') = 0, and it transforms as a tensor under basis changes. The eigenvalues of A' are the original eigenvalues shifted by subtracting one-third of tr(A); the sum of the eigenvalues of A' is zero. The deviatoric part captures distortional or shape-changing components of A, while the isotropic part (tr(A)/3 · I) corresponds to uniform expansion or compression.

Applications in continuum mechanics: Any second-order tensor A can be decomposed into a volumetric part and a deviatoric part: A = (tr A)/3 · I + A'. In stress analysis, the hydrostatic pressure is p = (1/3) tr(σ), and the deviatoric stress is σ' = σ − p I. The deviatoric component is associated with distortion without volume change; many constitutive models, such as the von Mises yield criterion, depend on the magnitude of the deviatoric stress. In fluid mechanics, the rate-of-deformation tensor D can similarly be decomposed into a deviatoric part that governs shear-like deformations.

Generalization: The concept extends to n dimensions, with A' = A − (tr A)/n · I. Deviatoric operators are linear projections onto the trace-free subspace of tensors.