When is the combined load identifiable from a stress-intensity profile? A coupled forward-inverse study on SIFBench finite-element data

This work studies the inverse problem of recovering the relative magnitudes of the tension, bending, and bearing loads acting on a crack from its stress-intensity-factor profile along the crack front, using the public SIFBench finite-element data. The central claim is not forensic load recovery on field cases, but a rigorous characterization of when the combined load is identifiable at all, together with an estimator that returns calibrated uncertainty precisely in the regimes where it is not. For a known geometry the forward map from loads to profile is exactly linear, and identifiability reduces to a single geometric question: whether the three elementary load profiles are linearly independent as functions along the front. When they are nearly dependent, many different load combinations produce almost the same profile and the inverse problem is illposed; the analysis shows that the degree of ill-posedness is controlled by an intrinsic stability margin, not by the conditioning number alone. A single crack-front operator serves both as a structured forward surrogate and as the differentiable map required by a simplex-constrained, set-valued inverse estimator. On the SIFBench corner-crack scenario the empirical behaviour matches the theory: the typical geometry is well posed while a sizable minority is genuinely ill-posed, so a point estimate is reliable on the majority and provably uninformative on the rest. Validation is on controlled synthetic noise; no real fracture cases are used or claimed.