An implicit self-consistent clustering analysis with total formulation for heterogeneous materials and its application for multiscale analysis

The traditional self-consistent clustering analysis (SCA) methods encounter convergence difficulties in damage and failure simulations, while the approximate algorithm that artificially decouple damage from the equilibrium iteration has deficiencies such as accuracy, local damage effect capture, and physical consistency. In this paper, a rigorous convergence analysis of traditional SCA reveals the origin of its convergence breakdown in the presence of material softening and damage evolution, and a novel implicit SCA method based on total formulation is developed, where the algorithmic tangent operator is replaced by a secant stiffness tensor through consistent linearization, thereby ensuring algorithmic stability and robust convergence for strongly nonlinear heterogeneous materials. Based on the implicit SCA method with total formulation, a class of implicit concurrent multiscale methods with consistent multiscale coupling across multiple hierarchical levels is developed for damage and failure analysis of heterogeneous materials and structures, enabling robust convergence of the global Newton scheme. Finally, through typical examples, it is demonstrated that the proposed implicit SCA method is significantly superior to the traditional SCA method (post-processing approximate algorithm) in terms of accuracy, capture of local damage effects, and physical consistency in damage and failure simulations. In multiscale damage and failure simulations, the proposed implicit concurrent multiscale framework preserves these key advantages while reducing computational cost by approximately two orders of magnitude compared with the multiscale method based on traditional SCA. Furthermore, the framework avoids the need for expensive experimental calibration and enables the direct representation of underlying mechanisms across scales, thereby ensuring general applicability.