In this work, we introduce a computational framework for capturing the evolution of phenomena across different scales within the nonlinear quasi-static solid mechanics context. This framework encompasses three key components: - Adaptivity. Our approach lies on a multilevel Adaptive Mesh Refinement (AMR) method, namely Local Defect Correction (LDC) approach. We showcase its innate ability to generate a hierarchy of levels with more and more refined meshes, following dynamically the evolution of studied phenomena over time. Solutions computed on different levels are interlinked through an iterative multigrid-like process based on prolongation and restriction operators. This approach is versatile (allows different solvers, material models, and meshes on different levels), eliminates the necessity for a complete remeshing of the domain, and benefits from a non-intrusive implementation. - Automation. Important feature of the algorithm allowing an on-the-fly refinement, is the automatic and smart selection of regions to be refined, facilitated by an a posteriori error estimator. - Error control. Our AMR mechanism is driven by user-prescribed tolerances on errors, ensuring that simulation results meet a predefined accuracy. We introduce a reliable remeshing algorithm designed to minimize the frequency of mesh regeneration over time, ensuring compliance with prescribed tolerances. The proposed framework manages the transfer of fields between time steps and guarantees error control over time through an efficient equilibration strategy. This algorithm enables us to optimize computations in both memory space and computational time. Its effectiveness and robustness are validated on several numerical experiments involving diverse material behaviors and variable multiscale loadings.