Hyper-reduction framework for model calibration in plasticity-induced fatigue

Background: Many mechanical experiments in plasticity-induced fatigue are prepared by the recourse to finite element simulations. Usual simulation outputs, like local stress estimations or lifetime predictions, are useful to choose boundary conditions and the shape of a specimen. In practice, many other numerical data are also generated by these simulations. But unfortunately, these data are ignored, although they can facilitate the calibration procedure. The focus of this paper is to illustrate a new simulation protocol for finite-element model calibration. By the recourse to hyper-reduction of mechanical models, more data science is involved in the proposed protocol, in order to solve less nonlinear mechanical equations during the calibration of mechanical parameters. Usually, the location of the crack initiation is very sensitive to the heterogeneities in the material. The proposed protocol is versatile enough in order to focus the hyper-reduced predictions where the first crack is initiated during the fatigue test. Methods: In this paper, we restrict our attention to elastoplasticity or elastoviscoplasticity without damage nor crack propagation. We propose to take advantage of the duration of both the experiment design and the experimental protocol, to collect numerical data aiming to reduce the computational complexity of the calibration procedure. Until experimental data are available, we have time to prepare the calibration by substituting numerical data to nonlinear equations. This substitution is performed by the recourse to the hyper-reduction method (Ryckelynck in J Comput Phys 202(1):346–366, 2005, Int J Numer Method Eng 77(1):75–89, 2009). An hyper-reduced order model involves a reduced basis for the displacement approximation, a reduced basis for stress predictions and a reduced integration domain for the setting of reduced governing equations. The reduced integration domain incorporates a zone of interest that covers the location of the crack initiation. This zone of interest is updated according to experimental observations performed during the fatigue test. Results: Bending experiments have been performed to study the influence of a grain boundary on AM1 superalloy oligocyclic fatigue at high temperature. The proposed hyper-reduction framework is shown to be relevant for the modeling of these experiments. To account for the microstructure generated by a real industrial casting process, the specimen has been machined in a turbine blade. The model calibration aims to identify the loading condition applied on the specimen in order to estimate the stress at the point where the first crack is initiated, before the crack propagation. The model parameters are related to the load distribution on the specimen. The calibration