Identification of Effective Coefficients of Thermal Contraction of Structural States in Cast Bronze by Solving Inverse Problems of Thermodeformation Modeling

The temperature-dependent formation of free linear shrinkage in cast CuAl7Si2Sn1.5Mn0.3 aluminum bronze was investigated using a cascade inverse-identification approach. Experimental data for 17 bronze compositions (Kᵣ = 0.41–0.85), including differential thermal analysis, quantitative metallography, and free-linear-shrinkage measurements, were used to reconstruct three genetic structure-formation pathways. Unlike conventional phase-averaging approaches, the proposed methodology identifies effective integral-average coefficients of linear thermal contraction (IA-CLTCs) of successive structural states directly from integral shrinkage data without requiring independently prescribed thermophysical properties of individual phases. The identified IA-CLTC sequences were 20.10 → 15.44 → 18.99 × 10−6 K−1 for pathway I, 21.88 → 29.40 → 11.57 × 10−6 K−1 for pathway II, and 25.90 → 36.65 → approximately 11.67 × 10−6 K−1 for pathway III. The first coefficient of pathway I was adopted as a fixed reference value. The model reproduced the experimental shrinkage with R2 = 0.998 and RMSE = 0.00562 percentage points. Conditional cascade leave-one-composition-out cross-validation yielded RMSE = 0.00995 percentage points, compared with 0.00915 percentage points for a simple linear-regression benchmark. Near the boundary between pathways II and III, the reconstructed thermal and structural-transformation contributions exhibited nearly compensating changes, although their quantitative separation remained model-dependent. Numerical sensitivity analysis revealed strong parameter correlation and limited practical identifiability for pathway III. The proposed approach therefore provides a framework for reconstructing effective temperature–deformation characteristics of successive structural states, while independent experimental validation remains necessary for their broader predictive and physical interpretation.