Análise dinâmica de estruturas periódicas utilizando uma abordagem de propagação de ondas e técnicas de sub-estruturação

Orientadores: José Roberto de França Arruda, Jean-Mathieu MencikTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia MecânicaResumo: Nesta tese de doutorado, o método dos elementos finitos ondulatórios é utilizado para cálculo da resposta harmônica de sistemas mecânicos envolvendo estruturas com periodicidade unidimensional, i.e., estruturas compostas por subestruturas idênticas arranjadas ao longo de uma direção. Tais sistemas mecânicos podem ser complexos e são comumente encontrados em aplicações de engenharia como, por exemplo, nas fuselagens de aviões. A primeira parte da tese é dedicada ao cálculo das ondas que se propagam ao longo dessas estruturas. Uma breve revisão da literatura sobre as formulações disponíveis para o problema de autovalor associado ao método dos elementos finitos ondulatórios é apresentada, assim como um estudo dos erros numéricos induzidos por estes problemas de autovalor no caso de um guia de ondas sólido. Na segunda parte desta tese, modelagens de superelementos para estruturas periódicas são propostas. Neste contexto, matrizes de rigidez dinâmica e de receptância ou flexibilidade de estruturas periódicas são expressas a partir dos modos de onda. Comparadas às matrizes de rigidez dinâmica e receptância obtidas pelo método dos elementos finitos convencional, as matrizes baseadas no método dos elementos finitos ondulatórios são calculadas de forma bastante rápida e sem perda de acuracidade. Ademais, uma estratégia eficiente de redução de ordem de modelo é apresentada. Comparada às formulações que utilizam a base completa de ondas, esta estratégia proporciona redução do tempo computacional requerido para cálculo da resposta forçada de estruturas periódicas. De fato, é mostrado que elementos espectrais numéricos de alta ordem podem ser construídos a partir do método dos elementos finitos ondulatórios. Isto constitui uma alternativa ao método dos elementos espectrais convencional, cuja utilização está limitada a estruturas simples para as quais soluções analíticas por ondas existam. A motivação por trás das formulações de matrizes de superelementos a partir do método dos elementos finitos ondulatórios está na utilização do conceito de ondas numéricas para calcular a resposta harmônica de sistemas mecânicos acoplados que envolvam estruturas com periodicidade unidimensional e junções elásticas a partir de procedimentos de montagem clássicos de elementos finitos ou técnicas de decomposição de domínio. Este assunto é tratado na terceira parte desta tese. Nesse caso, o método de Craig-Bampton é usado para expressar as matrizes de superelementos de junções por meio de modos estáticos e de interface fixa. Um critério baseado no método dos elementos finitos ondulatórios é considerado para a seleção dos modos da junção que mais contribuem para a resposta forçada do sistema. Isto também contribui para o aumento da eficiência da simulação numérica de sistemas acoplados. Finalmente, na quarta parte desta tese, o método dos elementos finitos ondulatórios é utilizado para mostrar que é possível projetar estruturas periódicas com potencial para funcionar como filtros de vibração em bandas de frequência específicas. Com o intuito de destacar a relevância dos desenvolvimentos propostos nessa tese, ensaios numéricos envolvendo guias de onda sólidos, pórticos planos e estruturas tridimensionais do tipo fuselagem aeronáutica são realizadosAbstract: In this thesis, the wave finite element (WFE) method is used for assessing the harmonic forced response of mechanical systems that involve structures with one-dimensional periodicity, i.e., structures which are made up of several identical substructures along one direction. Such mechanical systems can be quite complex and are commonly encountered in engineering applications, e.g., aircraft fuselages. The first part of the thesis is concerned with the computation of wave modes traveling along these structures. A brief literature review is presented regarding the available formulations for the WFE eigenproblem, which need to be solved for expressing the wave modes, as well as a study of the numerical errors induced by these eigenproblems in the case of a solid waveguide. In the second part of the thesis, the WFE-based superelement modeling of periodic structures is proposed. In this context, the dynamic stiffness matrices and receptance matrices of periodic structures are expressed in terms of wave modes. Compared to the conventional FE-based dynamic stiffness and receptance matrices, the WFE-based matrices can be computed in a very fast way without loss of accuracy. In addition, an accurate strategy for WFE-based model order reduction is presented. It provides significant computational time savings for the forced response analysis of periodic structures compared to WFE-based superelement modeling, which makes use of the full wave basis. Indeed, it is shown that higher-order numerical spectral elements can be built by means of the WFE method. This is an alternative to the conventional spectral element method, which is limited to simple structures for which closed-form wave solutions exist. The motivation behind the formulation of WFE-based superelement matrices is the use of the concept of numerical wave modes to assess the forced response of coupled mechanical systems that involve structures with one-dimensional periodicity and coupling elastic junctions through classic finite element assembly procedures or domain decomposition techniques. This issue is addressed in the third part of this thesis. In this case, the Craig-Bampton method is used to express superelement matrices of coupling junctions by means of static and fixed-interface modes. A WFE-based criterion is considered to select among junction modes those that contribute most to the system forced response. This also contributes to enhancing the efficiency of the numerical simulation of coupled systems. Finally, in the fourth part of this thesis, the WFE method is used to show the potential of designing periodic structures which work as vibration filters within specific frequency bands. In order to highlight the relevance of the developments proposed in this thesis, numerical experiments which involve solid waveguides, two-dimensional frame structures, and three-dimensional aircraft fuselage-like structures are carried outDoutoradoMecanica dos Sólidos e Projeto MecanicoDoutora em Engenharia Mecânica2010/17317-9FAPES

Atenuação de vibrações e manipulação de ondas elásticas em estruturas periódicas utilizando bandas proibidas e não reciprocidade

Orientador: José Roberto de França ArrudaTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia MecânicaResumo: A propagação de ondas mecânicas em cristais fonônicos e metamateriais tem sido extensivamente investigada para atenuação de ruídos e vibrações, e para manipulação de ondas (e.g., focalização de energia, guia de ondas, colheta de energia e invisibilidade). Apoiado e inspirado por estas extraordinárias potencialidades, esta tese investiga a propagação de ondas e o comportamento dinâmico de estruturas periódicas rotativas, metamateriais com ressonadores interconectados, e metaestruturas com variabilidade espacialmente correlacionada. Primeiro, os métodos dos elementos espectrais (SE) e dos elementos finitos de onda (WFE) são aplicados em estruturas rotativas periódicas, onde o efeito giroscópico quebra a simetria do tempo reverso, e, consequentemente, a simetria da propagação de ondas. A aceleração de Coriolis produz um diagrama de bandas assimétrico e causa que os modos naturais de vibração bifurquem. Estas características foram usadas para propor um circulador mecânico para ondas elásticas. Do ponto de vista numérico, a estratégia de WFE é proposta, onde o problema de autovalor da estrutura rotativa é projetado em uma base de ondas simplética, bem condicionada e reduzida. Esta formulação também pode ser usada em análise paramétrica, como no cálculo do diagrama de Campbell, bem como em análise de incertezas de estruturas periódicas com um baixo custo computacional. Um metamaterial com ressonadores interconectados é então proposto. Nesta configuração, os movimentos de translação e rotação na cadeia de ressonadores estão acoplados, o que abre uma larga banda proibida em baixas frequências sem aumentar a massa dos ressonadores. Este mecanismo é validado com medidas experimentais feitas em amostras construídas em manufatura aditiva. Para o caso bidimensional, que é uma placa com ressonadores interconectados, a vibração de flexão pode ser focalizada em direções específicas usando bandas proibidas parciais. Finalmente, são analisados os efeitos físicos da variabilidade espacial das propriedades elásticas na resposta dinâmica de vigas de cristais fonônicos e metamateriais. É observado uma alta correlação entre a distribuição espacial das propriedades do material com a performance das bandas proibidas tanto no modelo númerico quanto nos experimentos. Além do mais, a disordem produz alargamento ou aniquilição da largura da banda de atenuação, e o fenômeno de aprisionamento de ondas. Os resultados e as análises apresentadas nesta tese podem ser extendidos para outros sistemas periódicos e podem ser usados: I) do ponto de vista físico para atenuação de vibrações (e.g., dinâmica de rotores e estruturas leves) e para manipulação de ondas (e.g., guias de onda não recíprocos, focalização de energia e aprisionamento de ondas); II) do ponto de visa numérico, para reduzir tempo computacional em análises paramétricas e de incertezas de estruturas periódicas; e III) do ponto de vista experimental, para investigar estruturas periódicas construídas em impressoras 3DAbstract: The mechanical wave propagation in phononic crystals and metamaterials has been extensively investigated for noise and vibration attenuation, and for wave manipulation (e.g., energy focalization, guiding, harvesting and cloaking). Based upon and inspired by these extraordinary potentialities, this thesis investigates the wave propagation and dynamic behavior of rotating periodic structures, metamaterials with interconnected resonators, and metastructures with spatially correlated variability. First, the spectral element (SE) and wave finite element (WFE) methods are applied to rotating periodic structures, where the gyroscopic effect breaks the time-reversal symmetry, and, hence, the wave propagation symmetry. The Coriolis acceleration makes the band structure asymmetric and cause the natural vibration modes to split. These features are used to propose a mechanical circulator for elastic waves. From the numerical point of view, a WFE strategy is proposed, where the eigenvalue problem of the rotating structure is projected on a reduced, symplectic and well-conditioned wave basis. This approach can also be used for parametric analyses, such as Campbell diagram computations, as well as for uncertainty analyses of periodic structures with low computational cost. A metamaterial with interconnected resonators is then proposed. In this configuration, the translational and rotational motion of the resonator chain are coupled, which leads to a wide band gap at low frequency, without increasing the resonator mass. This mechanism is validated with experimental measurements on samples printed out using additive manufacturing. For the two-dimensional configuration, which is a plate with interconnected resonators, the flexural vibration can be localized in specific directions by using partial band gaps. Finally, the physical effects of spatial variability of the elastic properties in the dynamic response of phononic and metamaterial beams are analyzed. A high correlation between the spatial material distributions and the band gap performance is observed in both numerical and experimental models. In addition, the disorder can promote the widening or annihilation of the attenuation bandwidth and the wave trapping phenomenon. The results and analyses presented in this thesis can be extended to other periodic systems and can be used: I) from the physical point of view for vibration attenuation (e.g., rotordynamics and light structures) and for wave manipulation (e.g., nonreciprocal waveguiding, energy focalization, and wave trapping); II) from the numerical point of view, to save computational time in parametric and uncertainty analysis of periodic systems; and III) from the experimental point of view, to investigate periodic structures constructed by 3D printingDoutoradoMecanica dos Sólidos e Projeto MecanicoDoutor em Engenharia Mecânica2014/19054-6 e 2015/1578-02013-2/156459FAPESPCNP

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Wave propagation in periodic buckled beams

Folding of the earth's crust, wrinkling of the skin, rippling of fruits, vegetables and leaves are all examples of natural structures that can have periodic buckling. Periodic buckling is also present in engineering structures such as compressed lattices, cylinders, thin films, stretchable electronics, tissues, etc., and the question is to understand how wave propagation is affected by such media. These structures possess geometrical nonlinearities and intrinsic dispersive sources, two conditions which are necessary to the formation of stable, nonlinear waves called solitary waves. These waves are particular since dispersive effects are balanced by nonlinear ones, such that the wave characteristics remain constant during the propagation, without any decay or modification in the shape. It is the goal of this thesis to demonstrate that solitary waves can propagate in periodic buckled structures. This manuscript focuses specifically on periodically buckled beams that require either guided or pinned supports for stability purposes. Buckling is initially considered statically and investigations are made on stability, role played by imperfections, shape of the deflection, etc. Linear dispersion is analyzed employing the semi-analytical dispersion equation, a new method that relates the frequency explicitly to the propagation constant of the acoustic branch. This allows the quantification of the different dispersive sources and it is found that in addition to periodicity, transverse inertial and coupling effects are playing a dominant role. Modeling the system by a mass-spring chain that accounts for additional dispersive sources, homogenization and asymptotic procedures lead to the double-dispersion Boussinesq equation. Varying the pre-compression level and the support type, the main result of this thesis is to show that four different waves are possible, namely compressive supersonic, rarefaction (tension) supersonic, compressive subsonic and rarefaction subsonic solitary waves. For high-amplitude waves, models based on strongly-nonlinear PDEs as the one modeling wave propagation in granular media (Hertz power law) are more appropriate and adaptation of existing work is done. Analytical model results are then compared to finite-element simulations of the structure and experiments, and are found in excellent agreement. In this thesis, in addition to the semi-analytical dispersion equation, two other new methods are proposed. For periodic structures by translation with additional glide symmetries (e.g. buckled beams), Bloch theorem is revisited and allows the use of a smaller unit cell. Advantages are dispersion curves easier to interpret and computational cost reduced. Finally, the last contribution of this thesis is the use of NURBS-based isogeometric analysis (IGA) to solve the extensible-elastica problem requiring at least C1-continuous basis functions, which was not possible before with classical finite-element methods. The formulation is found efficient to solve dynamic problems involving slender beams as buckling

A new mixed model based on the enhanced-Refined Zigzag Theory for the analysis of thick multilayered composite plates

The Refined Zigzag Theory (RZT) has been widely used in the numerical analysis of multilayered and sandwich plates in the last decay. It has been demonstrated its high accuracy in predicting global quantities, such as maximum displacement, frequencies and buckling loads, and local quantities such as through-the-thickness distribution of displacements and in-plane stresses [1,2]. Moreover, the C0 continuity conditions make this theory appealing to finite element formulations [3]. The standard RZT, due to the derivation of the zigzag functions, cannot be used to investigate the structural behaviour of angle-ply laminated plates. This drawback has been recently solved by introducing a new set of generalized zigzag functions that allow the coupling effect between the local contribution of the zigzag displacements [4]. The newly developed theory has been named enhanced Refined Zigzag Theory (en- RZT) and has been demonstrated to be very accurate in the prediction of displacements, frequencies, buckling loads and stresses. The predictive capabilities of standard RZT for transverse shear stress distributions can be improved using the Reissner’s Mixed Variational Theorem (RMVT). In the mixed RZT, named RZT(m) [5], the assumed transverse shear stresses are derived from the integration of local three-dimensional equilibrium equations. Following the variational statement described by Auricchio and Sacco [6], the purpose of this work is to implement a mixed variational formulation for the en-RZT, in order to improve the accuracy of the predicted transverse stress distributions. The assumed kinematic field is cubic for the in-plane displacements and parabolic for the transverse one. Using an appropriate procedure enforcing the transverse shear stresses null on both the top and bottom surface, a new set of enhanced piecewise cubic zigzag functions are obtained. The transverse normal stress is assumed as a smeared cubic function along the laminate thickness. The assumed transverse shear stresses profile is derived from the integration of local three-dimensional equilibrium equations. The variational functional is the sum of three contributions: (1) one related to the membrane-bending deformation with a full displacement formulation, (2) the Hellinger-Reissner functional for the transverse normal and shear terms and (3) a penalty functional adopted to enforce the compatibility between the strains coming from the displacement field and new “strain” independent variables. The entire formulation is developed and the governing equations are derived for cases with existing analytical solutions. Finally, to assess the proposed model’s predictive capabilities, results are compared with an exact three-dimensional solution, when available, or high-fidelity finite elements 3D models. References: [1] Tessler A, Di Sciuva M, Gherlone M. Refined Zigzag Theory for Laminated Composite and Sandwich Plates. NASA/TP- 2009-215561 2009:1–53. [2] Iurlaro L, Gherlone M, Di Sciuva M, Tessler A. Assessment of the Refined Zigzag Theory for bending, vibration, and buckling of sandwich plates: a comparative study of different theories. Composite Structures 2013;106:777–92. https://doi.org/10.1016/j.compstruct.2013.07.019. [3] Di Sciuva M, Gherlone M, Iurlaro L, Tessler A. A class of higher-order C0 composite and sandwich beam elements based on the Refined Zigzag Theory. Composite Structures 2015;132:784–803. https://doi.org/10.1016/j.compstruct.2015.06.071. [4] Sorrenti M, Di Sciuva M. An enhancement of the warping shear functions of Refined Zigzag Theory. Journal of Applied Mechanics 2021;88:7. https://doi.org/10.1115/1.4050908. [5] Iurlaro L, Gherlone M, Di Sciuva M, Tessler A. A Multi-scale Refined Zigzag Theory for Multilayered Composite and Sandwich Plates with Improved Transverse Shear Stresses, Ibiza, Spain: 2013. [6] Auricchio F, Sacco E. Refined First-Order Shear Deformation Theory Models for Composite Laminates. J Appl Mech 2003;70:381–90. https://doi.org/10.1115/1.1572901

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”