Constitutive modeling and reliability evaluation of interfacial slip damage in three-dimensional stacked micro/nano-scale through-silicon vias

Micro/nano-interconnect structures are critical components in three-dimensional high-density packaging, yet their reliability is critically constrained by the copper pumping effect induced by the thermal mismatch between copper and silicon. This study investigates interfacial failure at the copper/silicon interface in through-silicon via (TSV) composite structures. Based on continuum damage mechanics, a constitutive relation for interfacial slip is established using energy dissipation as a damage indicator. The reliability under thermoelectrical coupling is evaluated, and the model is further applied to analyze thermal residual stress and strain in common liner materials. A Bayesian support vector regression (BSVR) model is integrated into a cross-entropy-based framework for failure probability assessment. Within this framework, importance sampling is employed to optimize BSVR hyperparameter selection, while BSVR serves as a surrogate to reduce computational cost in constitutive simulations. To address the analytical intractability of the marginal likelihood integral in BSVR, a simulation-based sampling method is proposed as an alternative to Laplace approximation. The proposed computational framework effectively combines the advantages of cross-entropy methods and BSVR surrogate modeling in handling high-dimensional problems with small failure probabilities, thereby mitigating efficiency degradation caused by data sparsity and complex failure domains.