One of the assumptions of Linear Elastic Fracture Mechanics is that the crack faces are traction-free or, at most, loaded by bounded tractions. The standard Irwin's crack closure integral, widely used for the computation of the Energy Release Rate, also relies upon this assumption. However, there are practical situations where the load acting on the crack boundaries is singular. This is the case, for instance, in hydraulic fracturing, where the fluid inside the crack exerts singular tangential tractions at its front. Another example of unbounded tractions is the case of a rigid line inclusion (anticrack) embedded into an elastic body. In such situations, the classical Irwin's crack closure integral fails to provide the correct value of the Energy Release Rate. In this paper, we address the effects occurring when square-root singular tractions act at the boundary of a line defect in an elastic solid and provide a generalisation of Irwin's crack closure integral. The latter yields the correct Energy Release Rate and allows broad applications, including, among others, hydraulic fracturing, soft materials containing stiff inclusions, rigid inclusions, shear bands and cracks characterized by the Gurtin-Murdoch surface stress elasticity. We present the results in the most general form, where six Stress Intensity Factors are present: three of them are classical SIFs corresponding to the modes I-II-III and computed under the assumption of homogeneous boundary conditions at the defect surfaces, while the other three SIFs are associated with singular admissible tractions (those that lead to a finite ERR value). It is demonstrated that this approach resolves an ambiguity in using the same SIF's terminology in the cases of open cracks and rigid inclusions, among other benefits.