Effect of formulation method for plastic deformation rate on topology optimization considering elastic-plastic behavior

In continuum mechanics the deformation rate is a crucial tensor that drives deformation evolution. It plays an important role in the formulation of elastic and plastic deformations, and makes a distinction between Lagrangian and Eulerian formulations. Furthermore, because plastic deformation measure is not uniquely determined, the modeling method adopted has a bearing on topology optimization and structural design. However, research on this aspect has been scarce. The objective of this study was to investigate the effect of the formulation method chosen for the plastic deformation rate on topology optimization by considering both Lagrangian and Eulerian formulations. In this work, the Lagrangian framework adopted both phenomenological and crystal plasticity, while the Eulerian framework used microstructural vector theory. For the sake of brevity, the phenomenological plasticity in the Lagrangian framework has been denoted as classical plasticity in this work. The Eulerian formulation was the first to be implemented for topology optimization in this study. The plastic deformation rates from the three models were applied to the topology optimization with the finite element (FE) method. The Messerschmitt–Bölkow–Blohm beam problem, a widely known example in this regard, was considered in this study for comparison purposes. While classical and crystal plasticity models exhibited similar overall trends in terms of optimization, the Eulerian deformation rate presented distinctly different optimization results compared with the other two approaches. All the three models aimed to minimize free energy in the topology optimization loop. However, ultimately, differences in the constitutive equations affected the resultant values of the free energy leading to different optimization outcomes. The results of this study were validated through comparisons between experimental and FE analysis results for each design. The results clearly demonstrate that the plastic deformation rate significantly affects the topology optimization, and hence needs to be appropriately adapted based on the design objectives.