Wave dispersion in periodic post-buckled structures

Wave propagation in pinned-supported, post-buckled beams can be described with the Korteweg de Vries (KdV) equation. Finite-element simulations however show that the KdV is applicable only to post-buckled beams with strong pre-compression. For weak and moderate pre-stress, a dispersive front is present and it is the aim of the current paper to analyze sources of dispersion beyond periodicity given three support types: guided, pinned, and free. Bloch theorem and a transfer-matrix method are employed to obtain numerical dispersion relations and characteristic wave modes, which are used to analyze the effects of pre-stress, initial curvature, and the influence of support types. Additionally, a new method is proposed to obtain a semi-analytical dispersion equation for the acoustic branch. Powers of frequency and the propagation constant are explicitly expressed and their coefficients are based on stiffness and mass-matrix components obtained from finite elements. This allows a physical interpretation of the dispersion sources, based on which, equivalent mass-spring models of post-buckled beam are proposed. It is found that mass and stiffness coupling are significant dispersion sources. In the present paper, a reduced form of Bloch theorem is presented exploiting glide-reflection symmetries, reducing the size of the unit cell and allowing an easier representation and interpretation of results.

Wave propagation in periodic buckled beams

Florian Paul Robert Maurin

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.

EPFL2015

Low-frequency wave propagation in post-buckled structures

Florian Paul Robert Maurin, Alessandro Spadoni

Nonlinear wave propagation in solids and material structures provides a physical basis to derive nonlinear canonical equations which govern disparate phenomena such as vortex filaments, plasma waves, and traveling loops. Nonlinear waves in solids however remain a challenging proposition since nonlinearity is often associated with irreversible processes, such as plastic deformations. Finite deformations, also a source of nonlinearity, may be reversible as for hyperelastic materials. In this work, we consider geometric bucking as a source of reversible nonlinear behavior. Namely, we investigate wave propagation in initially compressed and post-buckled structures with linear-elastic material behavior. Such structures present both intrinsic dispersion, due to buckling wavelengths, and nonlinear behavior. We find that dispersion is strongly dependent on pre-compression and we compute waves with a dispersive front or tail. In the case of post-buckled structures with large initial pre-compression, we find that wave propagation is well described by the KdV equation. We employ finite-element, difference-differential, and analytical models to support our conclusions.

Elsevier2014

Sur la méthode de Buser-Silhol pour l'uniformisation des surfaces de Riemann hyperelliptiques

Grégoire Aubry

The Uniformization Theorem due to Koebe and Poincaré implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

EPFL2010

Wave propagation in periodic buckled beams

Florian Paul Robert Maurin

EPFL2015