Principal stresses

Symmetry of the stress tensor has the important consequence that, at each point x, there exist three mutually perpendicular directions along which there are no shear stresses. These directions are called the principal stress directions, and the corresponding normal stresses are called the principal stresses. If the principal stresses are ordered algebraically as σ_I, σ_II, and σ_III (Figure 4), then the normal stress on any face (given as σ_n = n · T) satisfies σ_I ≤ σ_n ≤ σ_III. The principal stresses are the eigenvalues (or characteristic values) s, and the principal directions the eigenvectors n, of the problem T = sn, or [σ]{n} = s{n} in matrix notation with the 3-column {n} representing n. It has solutions when det ([σ] − s[I ]) = −s³ + I₁s² + I₂s + I₃ = 0, with I₁ = tr[σ], I₂ = −(1/2)I 2/1 + (1/2)tr([σ][σ]), and I₃ = det [σ]. Here “det” denotes determinant and “tr” denotes trace, or sum of diagonal elements, of a matrix. Since the principal stresses are determined by I₁, I₂, and I₃ and can have no dependence on how one chooses the coordinate system with respect to which the components of stress are referred, I₁, I₂, and I₃ must be independent of that choice and are therefore called stress invariants. One may readily verify that they have the same values when evaluated in terms of σ_ij′ above as in terms of σ_ij by using the tensor transformation law and properties noted for the orthogonal transformation matrix.

Very often, in both nature and technology, there is interest in structural elements in forms that might be identified as strings, wires, rods, bars, beams, or columns, or as membranes, plates, or shells. These are usually idealized as, respectively, one- or two-dimensional continua. One possible approach is then to develop the consequences of the linear and angular momentum principles entirely within that idealization, working in terms of net axial and shear forces and bending and twisting torques at each point along a one-dimensional continuum, or in terms of forces and torques per unit length of surface in a two-dimensional continuum.