Chapter 6 Chapter 1 On the Detection and Quantification of Nonlinearity via Statistics of the Gradients of a Black-Box Model Georgios Tsialiamanis and Charles R. Farrar Abstrac t Detection and identification of nonlinearity is a task of high importance for structural dynamics. On the one hand, identifying nonlinearity in a structure would allow one to build more accurate models of the structure. On the other hand, detecting nonlinearity in a structure, which has been designed to operate in its linear region, might indicate the existence of damage within the structure. Common damage cases which cause nonlinear behaviour are breathing cracks and points where some material may have reached its plastic region. Therefore, it is important, even for safety reasons, to detect when a structure exhibits nonlinear behaviour. In the current work, a method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest. The data-driven model selected for the current application is a neural network. The selection of such a type of model was done in order to not allow the user to decide how linear or nonlinear the model shall be, but to let the training algorithm of the neural network shape the level of nonlinearity according to the training data. The neural network is trained to predict the accelerations of the structure for a time-instant using as input accelerations of previous time-instants, i.e. one-step-ahead predictions. Afterwards, the gradients of the output of the neural network with respect to its inputs are calculated. Given that the structure is linear, the distribution of the aforementioned gradients should be unimodal and quite peaked, while in the case of a structure with nonlinearities, the distribution of the gradients shall be more spread and, potentially, multimodal. To test the above assumption, data from an experimental structure are considered. The structure is tested under different scenarios, some of which are linear and some of which are nonlinear. More specifically, the nonlinearity is introduced as a column-bumper nonlinearity, aimed at simulating the effects of a breathing crack and at different levels, i.e. different values of the initial gap between the bumper and the column. Following the proposed method, the statistics of the distributions of the gradients for the different scenarios can indeed be used to identify cases where nonlinearity is present. Moreover, via the proposed method one is able to quantify the nonlinearity by observing higher values of standard deviation of the distribution of the gradients for lower values of the initial column-bumper gap, i.e. for “more nonlinear” scenarios. Keyword s Structural health monitoring (SHM) · Structural dynamics · Nonlinear dynamics · Machine learning · Neural networks 1.1 Introduction In the pursuit of making everyday life safer, humans have extensively tried to model the environment around them. Structures are an important part of the environment, in which humans live. They are man-made and should be safe throughout their lifetime. Structures are exposed to numerous environmental factors, which may cause them to fail. Moreover, during operation, structures are subjected to dynamic loads, which, in time, may cause failure. Such failures will most probably result in economic damage to society and may even result in loss of human lives. Therefore, for the purpose of maintaining structures safe, the field of structural health monitoring (SHM) [1] has emerged. G. Tsialiamanis ( ) Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: g.tsialiamanis@sheffield.ac.uk C. R. Farrar Engineering Institute, MS T-001, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: farrar@lanl.gov © The Society for Experimental Mechanics, Inc. 2024 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-36999-5_1 1 Design and Optimization of Shock Absorbers Made of Grade Density Foams C. Sabbatini, G. Zandri, and M. Sasso Abstract In this paper, the adoption of density graded polymeric foams is evaluated for energy absorption purposes. The use of functionally graded materials is becoming increasingly widespread in the field of personal protection and industrial applications, thanks in part to recent advances that have simplified their production. To properly design an energy absorber, it is first necessary to identify the material used and characterize its behavior at different densities and strain rates. The necessary experimental tests were conducted using a pneumatic compression machine for low and intermediate strain rates, while for dynamic tests, the Hopkinson bar was employed. Tests were initially conducted on uniform samples of different densities; this allowed the analysis and calibration of a visco-elasto-plastic constitutive model borrowed from the literature. Then, the mentioned constitutive models have been implemented in numerical simulations with Abaqus/explicit finite element software, where the energy absorption capability of a density graded absorber was evaluated reproducing a puncture test. The density distribution in the absorber was designed and optimized to simultaneously reduce the peak of acceleration of the impacting mass and the maximum stress experienced by the absorber during the impact event. Keywords Polymeric foams · Ogden hyperfoam model · Viscoelasticity· Functionally graded materials Introduction Polyvinyl chloride (PVC) foam is widely used in various industries due to its lightweight structure, energy-absorbing capacity, and excellent mechanical properties. Understanding its behavior under different loading conditions is essential for optimizing its performance in applications such as impact protection [8], cushioning systems, and structural components [12]. As new technologies make it possible to produce increasingly complex cellular structures at lower costs [5], the use of functionally graded materials whose mechanical and physical properties vary spatially is gaining traction, depending on the specific application requirements [3, 9]. For foamed materials, one way to achieve such structures is to create a graded density distribution. However, for proper design and optimization of these materials, accurate characterization is necessary to enable reliable finite element simulations. In this study, compression tests were conducted on PVC foam across a range of strain rates, from quasi-static to dynamic, and considering different nominal densities. This approach allowed for the evaluation of both strain rate effects and densitydependent behavior [7]. The experimental results were then used to calibrate the Ogden hyperelastic model [10], the Prony series for viscoelasticity, and the Mullins effect to account for foam damage and stress softening. The goal of this work is to develop a user-defined constitutive model (VUMAT) capable of implementing all three models, thus accurately describing the behavior of PVC foam. Finally, an impact test simulation will be carried out on an absorber with a variable density distribution to verify the effectiveness of the VUMAT and assess the ability of graded-density structures to enhance energy absorption efficiency compared to an absorber with the same average density. C. Sabbatini · G. Zandri · M. Sasso Department of Mechanical Engineering, Universita` Politecnica delle Marche, Ancona, AN, 29208 Email: c.sabbatini@pm.univpm.it; g.zandri@staff.univpm.it; m.sasso@staff.univpm.it © The Author(s), under exclusive license to River Publishers 2025 43 Emily Retzlaff et al. (eds.), Mechanics of Additive & Advanced Manufacturing, Inverse Methods and Machine Learning, Vol. 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0831-2 6
RkJQdWJsaXNoZXIy MTMzNzEzMQ==