Periodic Elastic Structures Research Articles

Surface Wave Bloch Mode Synthesis for Accelerating the Calculations of Elastic Periodic Structures

Surface Wave Bloch Mode Synthesis for Accelerating the Calculations of Elastic Periodic Structures

An analysis of the propagation of surface acoustic waves in a substrate covered by a metal T-plate with the Carrera unified formulation

In this work, the wave propagation of Rayleigh type through a periodic elastic element covered with a T-plate is analyzed. Viscous-spring artificial boundaries are used to satisfy boundary conditions of a periodic structure. By using the Carrera Unified Formulation (CUF), the various kinematics of the three-dimensional structure are consistently expressed and the propagation of waves within the model is obtained. Lagrange expansion is employed to generate the mathematical model of the structure. The numerical evaluation and the comparison with the results obtained by the analytical method and COMSOL involving different structures reveal that the results of this study are reliable and show that the model developed provides accurate results for the analysis of the periodic elastic structure of T-plates.

A simple example of the tunnelling effect in periodic elastic structures

The phenomenon of tunnelling manifests itself in many branches of physics. It is generated by the presence of turning points, where the wave undergoes a transition from cut-on to cut-off and backwards. Despite the enormous interest in wave propagation in periodic media in general, its existence in continuous periodic elastic structures remains obscured. This paper is concerned with demonstrating this effect in, probably, the simplest possible model. The closed-form analytical solutions obtained in the Floquet theory framework and by means of modal analysis are shown to be in perfect agreement with conventional calculations of insertion losses.

Homogenization of a Periodic Elastic Structure Saturated with a Maxwell Fluid

Homogenization of a Periodic Elastic Structure Saturated with a Maxwell Fluid

Homogenization of an elastic material reinforced by very strong fibres arranged along a periodic lattice

We provide in this paper homogenization results for the L 2 -topology leading to complete strain-gradient models and generalized continua. Actually, we extend to the L 2 -topology the results obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5 , 259–288) using a topology adapted to minimization problems set in varying domains. Contrary to (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5 , 259–288) we consider elastic lattices embedded in a soft elastic matrix. Thus our study is placed in the usual framework of homogenization. The contrast between the elastic stiffnesses of the matrix and the reinforcement zone is assumed to be very large. We prove that a suitable choice of the stiffness on the weak part ensures the compactness of minimizing sequences while the energy contained in the matrix disappears at the limit: the Γ-limit energies we obtain are identical to those obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5 , 259–288).

On periodic homogenization of highly contrasted elastic structures

While homogenization of periodic linear elastic structures is now a well-known procedure when the stiffness of the material varies inside fixed bounds, no homogenization formula is known which enables to compute the effective properties of highly contrasted structures. Examples have been given in which the effective energy involves the strain-gradient but no general formula provides this strain-gradient dependence. Some formulas have been proposed which involve such terms and provide a small correction to the classical effective energy still when the stiffness of the material varies inside fixed bounds. The goal of this paper is to check the applicability of these formulas for highly contrasted structures. To that aim we focus on structures whose limit energy is already known and we compare the energies given by (i) the convergence results, (ii) the corrective formulas and (iii) by a direct numerical simulation of the complete structure.

Polarization-dependent and valley-protected Lamb waves in asymmetric pillared phononic crystals

We present the realization of the topological valley-protected zero-order antisymmetric (A0) or symmetric (S0) and zero-order shear-horizontal (SH0) Lamb waves at different domain walls based on topologically distinct asymmetric double-sided pillared phononic crystals. The elastic periodic structures have either the triangular or the honeycomb symmetry and give rise to a double-negative branch in the dispersion curves. By artificially folding the doubly negative branch, a degenerate Dirac cone is achieved. Different polarization-dependent propagation along the same primary direction along the constituent branches are presented. Moreover, divergent polarization-dependent phenomena along different primary directions along a given branch are also reported. By imposing two large space-inversion symmetry (SIS) breaking perturbations the topological phase transition is obtained. We show that the Berry curvature becomes strongly anisotropic when the wave vector gets away from the valleys. Further, we demonstrate the unidirectional transport of A0, S0, and SH0 Lamb waves at different domain walls in straight or Z-shaped wave guides. In the large SIS breaking case, we show negligible reflection at the zigzag outlet of the straight wave guide and occurrence of weak inter-valley scattering at the bending corners of the Z-shaped wave guide. For a larger strength of SIS breaking, the edge states are gapped and strong reflection at the zigzag outlet and bending corners is observed. The topological protection cannot be guaranteed anymore in that case.

Narrow Band Solid-Liquid Composite Arrangements: Alternative Solutions for Phononic Crystal-Based Liquid Sensors.

Periodic elastic composite structures attract great attention. They offer the ability to design artificial properties to advance the control over the propagation of elastic/acoustic waves. In previous work, we drew attention to composite periodic structures comprising liquids. It was shown that the transmission spectrum of the structure, specifically a well-isolated peak, follows the material properties of liquid constituent in a distinct manner. This idea was realized in several liquid sensor concepts that launched the field of phononic crystal liquid sensors. In this work we introduce a novel concept—narrow band solid-liquid composite arrangements. We demonstrate two different concepts to design narrow band structures, and show the results of theoretical studies and results of experimental investigations that confirm the theoretical predictions. This work extends prior studies in the field of phononic crystal liquid sensors with novel concepts and results that have a high potential in a field of volumetric liquid properties evaluation.

A topology optimisation of periodic elastic structures with negative refraction indices

A topology optimisation of periodic elastic structures with negative refraction indices

A topology optimisation of multi-layered elastic periodic structures based on the scattering matrix and boundary element method

A topology optimisation of multi-layered elastic periodic structures based on the scattering matrix and boundary element method

Discrete and continuous aspects of some metamaterial elastic structures with band gaps

We study three different 1D continuous models (extensional rods, Euler and Timoshenko beams) for addressing the dynamic properties of those microstructural materials containing a density of resonators. These models correspond to metamaterials which show interesting properties: In particular, the property that is the objective of this paper is the capacity of eliminating the vibration amplitude in a specific frequency range, which is called hereinafter band gap. The simplicity of these models emphasizes those microstructural properties having a relation with the band gap. We show that the rigidity of the hosting structure does not affect the values of the frequency band gap; it affects only the distance between the load-source of vibration and those points where the amplitude attenuation is visible. We also study, from a numerical point of view and using the Euler beam as the hosting structure, the case of a finite number of resonators. In particular, we study the minimum number of resonators which provides the same band gap as in the case of the presence of a density of resonators. We finally perform a numerical study on a periodic 2D elastic structure, which behaves like the Timoshenko beam model and for which an identification procedure is given.

Extension/Compression-Controlled Complete Band Gaps in 2D Chiral Square-Lattice-Like Structures

Achieving tunable band gaps in a structure by external stimuli is of great importance in acoustic applications. This paper aims to investigate the tunability of band gaps in square-lattice-like elastic periodic structures that are usually not featured with notable band gaps. Endowed with chirality, the periodic structures here are able to undergo imperfection-insensitive large deformation under extension or compression. The influences of geometric parameters on band gaps are discussed via the nonlinear finite element method. It is shown that the band gaps in such structures with curved beams can be very rich and, more importantly, can be efficiently and robustly tuned by applying appropriate mechanical loadings without inducing buckling. As expected, geometry plays a more significant role than material nonlinearity does in the evolution of band gaps. The dynamic tunability of band gaps through mechanical loading is further studied. Results show that closing, opening, and shifting of band gaps can be realized by exerting real-time global extension or compression on the structure. The proposed periodic structure with well-designed chiral symmetry can be useful in the design of particular acoustic devices.

Homogenization of periodic graph-based elastic structures

In the framework of Γ-convergence and periodic homogenization of highly contrasted materials, we study cylindrical structures made of one material and voids. Interest in high contrast homogenization is growing rapidly but assumptions are generally made in order to remain in the framework of classical elasticity. On the contrary, we obtain homogenized energies taking into account second gradient (i.e. strain gradient) effects. We first show that we can reduce the study of the considered structures to discrete systems corresponding to frame lattices. Our study of such lattices differs from the literature in the fact that we must take into account the different orders of magnitude of the extensional and flexural stiffnesses. This allows us to consider structures which would have been floppy when considering only extensional stiffness and completely rigid when considering flexural stiffnesses of the same order of magnitude than the extensional ones. At our knowledge, this paper provides the first rigorous homogenization result with a complete second gradient limit energy.

Using PWE/FE Method to Calculate the Band Structures of the Semi-Infinite PCs: Periodic in x–y Plane and Finite in z -direction

Abstract This paper introduces the concept of semi-infinite phononic crystal (PC) on account of the infinite periodicity in x-y plane and finiteness in z-direction. The plane wave expansion and finite element methods are coupled and formulized to calculate the band structures of the proposed periodic elastic composite structures based on the typical geometric properties. First, the coupled plane wave expansion and finite element (PWE/FE) method is applied to calculate the band structures of the Pb/rubber, steel/epoxy and steel/aluminum semi-infinite PCs with cylindrical scatters. Then, it is used to calculate the band structure of the Pb/rubber semi-infinite PC with cubic scatter. Last, the band structure of the rubbercoated Pb/epoxy three-component semi-infinite PC is calculated by the proposed method. Besides, all the results are compared with those calculated by the finite element (FE) method implemented by adopting COMSOL Multiphysics. Numerical results and further analysis demonstrate that the proposed PWE/FE method has strong applicability and high accuracy.

Mechanical low-frequency filter via modes separation in 3D periodic structures

This work presents a strategy to design three-dimensional elastic periodic structures endowed with complete bandgaps, the first of which is ultra-wide, where the top limits of the first two bandgaps are overstepped in terms of wave transmission in the finite structure. Thus, subsequent bandgaps are merged, approaching the behaviour of a three-dimensional low-pass mechanical filter. This result relies on a proper organization of the modal characteristics, and it is validated by performing numerical and analytical calculations over the unit cell. A prototype of the analysed layout, made of Nylon by means of additive manufacturing, is experimentally tested to assess the transmission spectrum of the finite structure, obtaining good agreement with numerical predictions. The presented strategy paves the way for the development of a class of periodic structures to be used in robust and reliable wave attenuation over a wide frequency band.

Study of liquid resonances in solid-liquid composite periodic structures (phononic crystals) – theoretical investigations and practical application for in-line analysis of conventional petroleum products

Periodic elastic structures attracted a great attention because of their ability to be designed with artificial acoustic properties. The ability to control the wave propagation and the spectral properties of transmitting or reflecting elastic waves allow to design the sensor structures that are able to exhibit a significant improvement in comparison to currently existing approaches. This contribution is specifically focused on the study of liquid induced resonances of solid-liquid periodic composite arrangements at the frequency range corresponding to the structure bandgap. In this work we bring together several structure designs and study their behavior with specific attention to resonances of liquid constituents. Developed sensor structures are investigated in terms of a frequency spectra variation depending on the speed of sound of a liquid analyte that constitutes the periodic arrangement. From the application side, the work is focused at an in-line analysis of conventional petroleum products and their additives. The sensor measuring circuit is only acoustically coupled to the petroleum under investigation that ensures the safety of proposed approach and allows minimizing the explosion risk in an emergency case. Both, numerical simulations and experimental results clearly disclose the ways and conceptual advantages of the phononic crystal based approach for the specified liquid sensor application.

Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect

We investigate elastic periodic structures characterized by topologically nontrivial bandgaps supporting backscattering suppressed edge waves. These edge waves are topologically protected and are obtained by breaking inversion symmetry within the unit cell. Examples for discrete one and two-dimensional lattices elucidate the concept and illustrate parallels with the quantum valley Hall effect. The concept is implemented on an elastic plate featuring an array of resonators arranged according to a hexagonal topology. The resulting continuous structures have non-trivial bandgaps supporting edge waves at the interface between two media with different topological invariants. The topological properties of the considered configurations are predicted by unit cell and finite strip dispersion analyses. Numerical simulations demonstrate edge wave propagation for excitation at frequencies belonging to the bulk bandgaps. The considered plate configurations define a framework for the implementation of topological concepts on continuous elastic structures of potential engineering relevance.

Periodic Homogenization and Material Symmetry in Linear Elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demonstrates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.

Multiscale Nonconforming Finite Element Computation to Small Periodic Composite Materials of Elastic Structures on Anisotropic Meshes

The small periodic elastic structures of composite materials with the multiscale asymptotic expansion and homogenized method are discussed. A nonconforming Crouzeix-Raviart finite element is applied to calculate every term of the asymptotic expansion on anisotropic meshes. The approximation scheme to the higher derivatives of the homogenized solution is also derived. Finally, the optimal error estimate in·hfor displacement vector is obtained.

Harnessing Multiple Folding Mechanisms in Soft Periodic Structures for Tunable Control of Elastic Waves

Mechanical instabilities in periodic porous elastic structures may lead to the formation of homogeneous patterns, opening avenues for a wide range of applications that are related to the geometry of the system. This study focuses on an elastomeric porous structure comprising a triangular array of circular holes, and shows that by controlling the loading direction, multiple pattern transformations can be induced by buckling. Interestingly, these different pattern transformations can be exploited to design materials with highly tunable properties. In particular, these results indicate that they can be effectively used to tune the propagation of elastic waves in phononic crystals, enhancing the tunability of the dynamic response of the system. Using a combination of finite element simulations and experiments, a proof‐of‐concept of the novel material is demonstrated. Since the proposed mechanism is induced by elastic instability, it is reversible, repeatable, and scale‐independent, opening avenues for the design of highly tunable materials and devices over a wide range of length scales.