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A Global Finite Element Reverse Approach for Identifying the Material Elasticity and Current State of Stress


The mechanical response of solids exhibiting complex material behavior has traditionally been determined by fitting constitutive models of specified functional form to experimentally derived force-displacement (stress-strain) data. However, characterizing the nonlinear mechanical behavior of complex materials requires a method of quantifying material behavior that is not restricted by a specific constitutive relation. To this end, a new method, termed \textit{the reverse updated Lagrangian finite element method} (RULFEM), which is based on the three-dimensional displacement field of the deformed solid and finite element method, is developed for incrementally linear materials. In RULEFM, the body is discretized by finite elements and its material properties are determined element-wise, i.e., the properties are assumed to be uniform at the element level and may vary from one element to another. The validity of RULFEM is demonstrated by three noise-free numerical examples and three numerical examples with various input noise levels. Two methods to assess the global and local errors of the results due to error in the measured input data (noisy data) are also discussed.

The life expectancy of a solid is traditionally predicted by assessing its expected stress cycle and comparing it to the experimentally determined stress state at failure. The accuracy of this procedure is often compromised by unforeseen extremes in the loading cycle or progressive material degradation. Often residually stressed part can either produce longer or shorter lifespans than predicted. Thus determination of the current state of stress (i.e., the residual stress in the absence of external loading) and material properties is particularly important. Typically, the material properties of a solid are determined by fitting the experimental data collected from the measured response to deformations of a stress-free configuration. However, the characterization of the mechanical behavior of a residually stressed body requires, in principle, a method that is not restricted to specific constitutive models. Complementing RULFEM \cite{tartibi2014reverse}, a new method, called \textit{estimating the current-state-of-stress} (ECSS) is developed in this work. Similar to RULFEM, ECSS also uses as input three-dimensional full-field displacement and force data of the body, which is perturbed by small displacements. ECSS complements the first step of the incremental RULFEM method. It generates the nodal current-state-of-stress (or residual stress in the absence of external tractions) as well as the incremental elasticity tensor pertaining to each of the finite elements used to discretize the body. The ECSS method is used to simulate two noise-free examples.

The linear dynamic response of a solid body has been used for material identification and even defect detection in low-damping (negligible viscoelastic behavior) materials. RULFEM and ECSS input full-field data, obtained from the statically perturbed body, are used to determine the initial state of stress and the current material elasticity tensors. A new formulation of ECSS, called \textit{dynamically estimating residual stress} (DERS) is developed in the fifth chapter of this dissertation. This is based on three-dimensional full-field displacement and force data of the dynamically perturbed body. In addition to nodal residual stress tensors and element-wise incremental elasticity tensor, DRES generates element-wise material density of the discretized body.

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