Dynamic Mode Decomposition of Deformation Fields in Elastic and Elastic-Plastic Solids

Abstract

Without recourse to constitutive assumptions, without knowledge of material properties, and without solving any of the conservation equations, we construct deformation fields for linear solids using Dynamic Mode Decomposition (DMD). Originally developed for analyzing experimental or computational field variables in fluid mechanics, it takes as input vectors of flow variables assembled as columns of a data matrix, and requires a remarkably few lines of code. It is a natural fit for solid mechanics wherein surface displacements can be used for the analysis. Vectors of surface displacements are arrayed in columns to created a data matrix. Singular Value Decomposition of the time-shifted data matrix affords selection of dominant modes (rank) in the deformation field. The DMD algorithm operates on time-shifted data matrices, obtains a reduced-order model that can reconstruct the deformation field, and also provides the dominant temporal and spatial modes of the deformation. In the case of linear elastic solids, DMD can be used to: reconstruct or predict the displacement states. In elastic-plastic solids, the transition from elastic to plastic results in the eigenvalues of the low-rank data matrix going out of the unit circle (implying an unstable growth mode). Using a combination of finite element analyses and displacement measurements obtained using Digital Image Correlation (DIC), we make the case for using DMD for state-estimation and state prediction in elastic solids and identifying onset of plasticity in elastic-plastic solids.