Coupled Bayesian Identification of Residual Stress and Fracture Strength in Thin-Film Fragmentation: A Physics-Informed Neural Network Framework with Synthetic Validation of Interface Adhesion Energy

Residual stress in brittle films on compliant substrates is routinely inferred from fragmentation experiments by combining an elastic stress-transfer model with a fracture strength criterion. This inversion is inherently coupled because the observed crack spacing depends jointly on the residual stress and the film fracture strength. Conventional closed-form estimators typically rely on a single feature, such as the cracking onset strain, and prescribe the fracture strength a priori, often at its bulk value. This practice discards most of the information encoded in the full crack-spacing evolution. It also obscures two sources of uncertainty: the intrinsic variability of thin-film fracture strength and the limited sensitivity of any single observable to individual parameters. Here, we recast the inversion as a Bayesian physics-informed neural network (B-PINN) in which the entire measured curve of the mean crack spacing versus applied strain is likely to occur. Stochastic gradient Langevin dynamics then sample the joint posterior of residual stress and fracture strength. A central finding is that crack-spacing data alone constrain only the difference between fracture strength and residual stress, confining the posterior to a one-dimensional manifold in parameter space and leaving each quantity individually unresolved. A single substrate curvature measurement, which, through the Stoney relation, depends on the residual stress but not on the fracture strength, provides the missing orthogonal constraint and collapses the posterior to a tight, well-resolved region. We further derive an identifiability condition under which buckle-wavelength observations serve as a third independent channel for recovering interface adhesion energy, and provide a synthetic proof-of-concept of this three-channel extension on DLC/Si and Mo/Si datasets; an experimental validation of the adhesion channel is identified as the natural next step but lies beyond the present scope. Requiring only standard fragmentation measurements and a single non-destructive curvature scan, the framework converts a point-estimate procedure into a posterior-quantified inverse method that makes explicit what can, and cannot, be learned from thin-film mechanics experiments.