Dispersive Waves in Microstructured Solids

The wave motion in micromorphic microstructured solids is studied. The mathematical model is based on ideas of Mindlin and governing equations are derived by making use of the Euler–Lagrange formalism. The same result is obtained by means of the internal variables approach. Actually such a model describes internal fields in microstructured solids under external loading and the interaction of these fields results in various physical effects. The emphasis of the paper is on dispersion analysis and wave profiles generated by initial or boundary conditions in a one-dimensional case

Deformation Waves in Microstructured Materials: Theory and Numerics

A linear model of the microstructured continuum based on Mindlin theory is adopted which can be represented in the framework of the internal variable theory. Fully coupled systems of equations for macro-motion and microstructure evolution are represented in the form of conservation laws. A modification of wave propagation algorithm is used for numerical calculations. Results of direct numerical simulations of wave propagation in periodic medium are compared with similar results for the continuous media with the modelled microstructure. It is shown that the proper choice of material constants should be made to match the results obtained by both approache

Continuum models for the dynamic behavior of 1D nonlinear structured solids

Mención Internacional en el título de doctorThe main objective of this thesis is to propose and validate generalized continuum models for analyzing and predicting the free vibrations of 1D linear elastic structured solids subjected to finite deformations. With this aim, two different generalized continuum models that account for geometric nonlinearities have been proposed: a nonlocal nonlinear model and a gradient velocity nonlinear one, here referred to as the inertia gradient nonlinear model. These models contain one or more microstructural parameters. The formulation of the cited theories has been devised for general 3D solids, but in this thesis they have been applied to the study of nonlinear axial and nonlinear axial-transverse coupled vibrations of a kind of 1D structured solids. The 1D structured solids chosen as reference are such that, under certain conditions, the classical nonlinear continuum model adequately reproduces its main features. However, it is well known that the classical theory is not able to capture the size effect presented in this kind of solids. Moreover, a non-standard continualization technique has been applied to the discrete system in order to get appropriate continuous equations of motion. Unlike the postulated generalized continuum models, this continualization technique permits to derive, from the characteristics of the solid structure, the additional parameters appearing in the formulation. The results for the two nonlinear analyzed problems (axial vibrations and axialtransverse coupled vibrations) obtained from the classical, generalized nonlinear continuum, and continualized models have been compared to those derived from the discrete solution taken as reference. From this critical comparison, the advantages and shortcomings of the different theories have been pointed out clearly showing the ability of the generalized continuum models to adequately address the size effects in structured solids, observed in many practical applications.El objetivo principal de esta tesis es proponer y validar modelos del continuo generalizado para analizar y predecir el comportamiento de vibraciones libres de sólidos estructurados unidimensionales elásticos y lineales sometidos a deformaciones finitas. Con este objetivo, se han propuesto dos modelos del continuo generalizado diferentes que consideran las no linealidades geométricas: un modelo no lineal no local y otro no lineal de gradiente de la velocidad, aquí referido como modelo con gradiente de inercia. Estos modelos contienen uno o más parámetros microestructurales. Las formulaciones de dichas teorías se han planteado para sólidos tridimensionales, pero en esta tesis se han aplicado al estudio de las vibraciones no lineales longitudinales y las vibraciones no lineales acopladas axiales-transversales en un tipo de sólidos 1D estructurados. El sólido 1D estructurado elegido como referencia es tal que, bajo ciertas condiciones, el modelo continuo no lineal clásico reproduce adecuadamente sus características principales. Sin embargo, es bien sabido que la teoría clásica no es capaz de capturar los efectos de tamaño que presentan este tipo de sólidos. Por otra parte, se ha aplicado una técnica de continualización no estándar al sistema discreto con el fin de obtener las ecuaciones continuas del movimiento. A diferencia de los modelos del continuo generalizado postulados, esta técnica de continualización permite derivar, a partir de las características del sólido estructurado, los parámetros adicionales que aparecen en la formulación del problema. Los resultados para los dos problemas no lineales analizados (vibraciones axiales y axiales-transversales acopladas) obtenidos mediante los modelos continuo clásico, continuos no lineales generalizados, y los continualizados se han comparado con los derivados de la solución discreta que se toma como referencia. A partir de esta comparación crítica, se han señalado las ventajas y los inconvenientes de las diferentes teorías, mostrando claramente la capacidad de los modelos del continuo generalizado para tratar adecuadamente los efectos de tamaño en los sólidos estructurados, observados ´estos en muchas aplicaciones prácticas.The financial support of the Ministerio de Ciencia e Innovación de España (Project DPI/2014-57989-P) is greatly acknowledged. The subject of this Thesis is a core part of the research project.Programa Oficial de Doctorado en Ingeniería Mecánica y de Organización IndustrialPresidente: Enrique Barbero Pozuelo.- Secretario: Eugenio Giner Maravilla.- Vocal: Julian J. Rimol

Low-order continualization of an anisotropic membrane lattice with next-nearest interactions. Enhanced prediction of its dynamic behaviour

In this paper, a novel anisotropic membrane lattice with nearest and next-nearest interactions (long-range forces) has been continualized through different standard and non-standard continualization procedures, which enables the development of new non-classical continuum models capable of accurately capturing the scale effects, present in the matter due to its discrete nature. The performance of these continuum models is assessed by means of both dispersion and natural frequencies analyses, where the discrete model is considered as a reference. In addition, the appearance of certain physical inconsistencies in some of the developed models is analysed, concluding that these only appear for those developed with continualizations based on Taylor expansion. Interestingly, the non-standard models suitably capture the dispersive behaviour of the discrete one, without both physical inconsistencies and higher-order spatial derivatives, thus avoiding the need for extra boundary conditions when finite solids are involved.The authors acknowledge support from MCIN/AEI /10.13039/501100011033 under Grants numbers PGC2018-098218-B-I00 and PRE2019-088002. FEDER: A way to make Europe. ESF invests in your future

Soliton-like solutions based on geometrically nonlinear Cosserat micropolar elasticity

The Cosserat model generalises an elastic material taking into account the possible microstructure of the elements of the material continuum. In particular, within the Cosserat model the structured material point is rigid and can only experience microrotations, which is also known as micropolar elasticity. We present the geometrically nonlinear theory taking into account all possible interaction terms between the elastic and microelastic structures. This is achieved by considering the irreducible pieces of the deformation gradient and of the dislocation curvature tensor. In addition we also consider the so-called Cosserat coupling term. In this setting we seek soliton type solutions assuming small elastic displacements, however, we allow the material points to experience full rotations which are not assumed to be small. By choosing a particular ansatz we are able to reduce the system of equations to a sine–Gordon type equation which is known to have soliton solutions

Solitary wave collisions in the regularized long wave equation

Abstract. The regularized long-wave equation admits families of positive and negative solitary waves. Interactions of these waves are studied, and it is found that interactions of pairs of positive and pairs of negative solitary waves feature the same phase shift asymptotically as the wave velocities grow large as long as the same amplitude ratio is maintained. The collision of a positive with a negative wave leads to a host of phenomena, including resonance, annihilation and creation of secondary waves. A sharp criterion on the resonance for positive-negative interactions is found

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