Necessary Corrections of the Mathematical Foundation of Fracture Mechanics

Abstract

It is shown that the applied fracture mechanics textbook crack tip boundary value problem solution is identical to the sum of the exact solutions of pure normal stress loading alone, and of pure shear stress loading alone, although these two solutions exclude each other and cannot apply at the same time. Evidently, this is against the existing “mixed mode” failure
criterion in all acting stresses, which necessarily follows as solution of the crack boundary value problem. Further, by an improper small variables transformation, the pure shear loading solution is shown to be not correct. This all delivers, a wrong, incompatible, equilibrium system, which does not satisfy the crack boundary conditions and the failure criterion, thus does not represent linear elastic fracture mechanics. As correction, the right exact limit analysis solution is given, which, as such, provides the derivation of “mixed I-IImode” fracture criterion, which is precisely verified by empirical research. It is further shown that the postulated textbook stress function is based on integration of the copied mono-mode stress solutions, without regarding right crack-boundary values. The copied mono-mode stress solutions are shown to be based on the Stevenson potentials. The necessary limit analysis approach for failure, as exact calculation method, provides the necessary linear elastic analysis up to yield. Linear elastic stress and displacement terms also represent the non-vanishing first order expansion terms of virtual work behavior of any non-linear stress division. This has consequences for assumed non-linear elastic fracture mechanics, which is shown, to be replaceable by the linear approach of limit analysis. As example, the critical stress intensity is given of a test series with different initial crack lengths, which, in terms of nonlinear fracture mechanics, was subject to undeterminable critical J-integral behavior.