Physics-Regularized Low-Rank–Sparse Decomposition for Structural Damage Localization and Severity-Sensitive Characterization Using Full-Field Displacement Responses

Under complex environmental disturbances and visual measurement noise, globally coherent components, damage-induced local anomalies, and random disturbances are coupled in full-field vertical displacement responses. This coupling limits conventional low-rank–sparse decomposition because of its lack of mechanics-based constraints and can obscure weak damage-induced anomalies. To address this issue, this study proposes a physics-regularized low-rank–sparse damage identification method incorporating a physics prior derived from curvature-strain-energy perturbation. The method first extracts deflection curvature from the full-field displacement responses of the healthy and damaged states. A normalized physical evidence field is then constructed from the curvature-energy difference through Gaussian spatial regularization and mapped into spatially varying sparsity weights to modulate anomaly separation. Subsequently, the Physics-Regularized Differential Damage Index (PRDDI) is constructed from the difference in physics-regularized sparse anomaly intensity between the two states for damage localization and severity-sensitive characterization. The proposed method is primarily intended for beam-like structures satisfying the small-deformation bending assumption. For more complex structures, such as continuous beams, frames, plates, and shells, the corresponding mechanics-based physical evidence and spatial neighborhood relationships can be extended according to their load-transfer mechanisms and spatial geometries. Experimental and numerical results show that the peak-to-background ratio of the physics-regularized sparse anomaly field reaches approximately 2.77 times that of conventional robust principal component analysis (RPCA), while the background level is reduced by approximately 60%, and spurious peaks in non-damaged regions are markedly suppressed. For local stiffness reductions of 5–30%, the PRDDI localization error remains within 0–1 spatial measurement points. Both the peak value and local integrated area within the damaged region increase consistently with the degree of stiffness reduction, with coefficients of determination R2 exceeding 0.99 and Spearman rank correlation coefficients of 1.00. For representative dual-damage cases, the proposed method maintains good dual-peak resolution. Under 10 dB noise, the complete dual-damage detection rate is approximately 87%, while the missed-detection rate for weak damage is approximately 10%. The physics prior derived from curvature-strain-energy perturbation improves consistency with structural mechanics, spatial separability, and the identification reliability of local damage anomaly extraction under complex measurement conditions. By exploiting spatially continuous, vision-based full-field displacement measurements, the proposed method can identify local damage regions in bridges and characterize variations in damage severity, providing a basis for subsequent detailed inspection and condition assessment.