Principalstresses

Principal stresses are the eigenvalues of the Cauchy stress tensor at a point in a continuous material. The corresponding eigenvectors are called principal directions and form an orthonormal basis in which the stress tensor is diagonal, meaning the shear components vanish in that coordinate system. In three dimensions there are three principal stresses, typically denoted σ1, σ2, σ3, often ordered so that σ1 ≥ σ2 ≥ σ3. In two dimensions there are two principal stresses.

Because the Cauchy stress tensor is symmetric, its eigenvalues are real and its eigenvectors are mutually orthogonal. The principal stresses can be found by solving the characteristic equation det(σ − λI) = 0; the roots λ are the principal stresses. The set of principal stresses is invariant under rotation of the coordinate system, and the values are related to the tensor invariants I1 = σ1 + σ2 + σ3, I2 = σ1σ2 + σ2σ3 + σ3σ1, and I3 = σ1σ2σ3.

The principal stresses provide a convenient description of the state of stress, since many material failure criteria depend on normal stresses in principal directions. For example, the maximum principal stress criterion (Rankine criterion) uses the largest principal stress to assess failure, while Mohr’s circle graphics illustrate the transition between stresses in any given plane and the corresponding principal stresses in two dimensions.

Special cases include hydrostatic stress, where all principal stresses are equal, and plane stress, where only two distinct principal stresses generally occur with the third being the out-of-plane component. The sign convention for tension and compression can affect the numerical values of the principal stresses.