Materials failure theory,strain energy,and the principal stresses eigenvalue problem: how these three historical areas are interrelated and mutually reinforcing,thus enabling new results for materials failure

The three sub-disciplines of principal stresses, strain energy, and failure theory are shown to depend upon four basic invariants. These four invariants are derived from the eigenvalue problem of the principal stress formulation. Thus, the most basic form of all three areas is that of the eigenvalue problem for the principal stresses. The proper forms for the strain energy and the failure problem follow from the eigenvalue problem, both being appropriate to isotropic materials. After further showing the specific relationship between the failure theory and the strain energy, some implications of the failure theory are examined.