Improved inverse methods in the characterisation of mechanical behaviour of materials

Abstract

The accuracy of finite element simulations of material-dependent processes depend heavily on applying properly determined material behaviour characteristics in the simulation process. Due to practical constraints in using the analytical direct methods in some applications, inverse identification methods are widely employed to identify the material behaviour model. In this study we develop and study improved inverse methods to characterise mechanical behaviour of materials, namely model class selection and model parameter identification. First we propose a progressive inverse identification algorithm to characterize flow stress from the material response, independent of choosing an a priori hardening constitutive model. In contrast to the conventional forward flow stress identification methods, the flow stress is characterized by a multi-linear curve rather than a limited number of a hardening model parameters. The proposed algorithm optimises the slopes and lengths of the curve increments simultaneously. We employ the algorithm to identify flow stress of a 304 stainless steel tube in a tube bulge test as an example to illustrate application of the algorithm. Since there is no need for a priori choosing the hardening model, there is no risk for choosing an improper hardening model, which in turn facilitates solving the inverse problem. The progressive inverse identification approach is classified as the forward inverse identification approach, and it is limited to characterise flow stress. An alternative but lesser-known approach to solve an inverse material behaviour identification problem, called a direct inverse map, directly maps the measured response to the parameters of a material model. Therefore, there is no need for optimisation, which reduces the computational burden of solving an inverse problem. In this study we investigate the potential pitfalls of the well-known stochastic noise and lesser-known model errors when constructing direct inverse maps. We show how to address these problems, explaining in particular the importance of projecting the measured response onto the domain of the simulated responses before mapping it to the material parameters. The study concludes by proposing partial least squares regression as an elegant and computationally efficient approach to address stochastic and systematic (model) errors. This study also gives insight into the nature of the inverse problem under consideration. x The above-mentioned approaches to solve an inverse problem are classified as deterministic approaches. In order to incorporate the effects of uncertainties into the measured material response and the response of the inverse problem, probabilistic inverse problems are introduced. Therefore, we set out and justify a unified framework for Bayesian inference in Mechanical properties of Materials Characterisation (BMMC), with the aim of model class comparison and model parameter distribution inference. We integrate Bayes_ rule, nested sampling, Galilean Monte Carlo sampling, artificial neural networks and principal component analysis to construct a unified framework for BMMC. The specific design of the constructed framework justifies its application for materialcharacterisation-related problems in a computationally efficient way. We demonstrate the application of the developed framework in three cases. First we compare different error correlation models related to different material responses of a tube subjected to a tube punch test in a Bayesian framework. We then compare the material model classes, which comprise combination of a Ludwik and a bilinear hardening model, along with a Hill48 anisotropy and an isotropy yield model, in a Bayesian framework. Selecting the most supportable material model class, the corresponding model parameter distributions are inferred simultaneously with no need for additional effort. The proposed method can be applied for any model class comparison and parameter distribution inference. However, its application is in particular justified for material characterisation where the problem comprises high dimensional material responses obtained from different sensors. The unified framework for BMMC avoids choosing insupportable material model classes. Moreover, the uncertainty of the material models are identified which in turn determines the uncertainty of the finite element analysis of a system made of the characterised material. The study presented offers at least two main contributions. Firstly the design of the constructed unified framework for BMMC is novel. In addition, its practical application in material characterisation problems, namely hardening and yield identification in multi-axial state of stress is justified and demonstrated. Throughout this thesis we mostly illustrated application of the methods on mechanical behaviour of tubular materials characterisation. However, one can generalise them for other applications.