Band gap enhancement in periodic frames using hierarchical structures

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arXiv:2210.11063v1 [physics.app-ph] 20 Oct 2022

Band gap enhancement in periodic frames using

hierarchical structures

Vin´ıcius F. Dal Poggetto1, Federico Bosia2, Marco Miniaci3, Nicola M.

Pugno1,4

Abstract

The quest for novel designs for lightweight phononic crystals and elastic meta-

materials with wide low-frequency band gaps has proven to be a signiﬁcant

challenge in recent years. In this context, lattice-type materials represent a

promising solution, providing both lightweight properties and signiﬁcant pos-

sibilities of tailoring mechanical and dynamic properties. Additionally, lattice

structures also enable the generation of hierarchical architectures, in which

basic constitutive elements with diﬀerent characteristic length scales can be

combined. In this work, we propose 1D- and 2D-periodic phononic crystals

made of spatial frames inspired by a spider web-based architecture. Speciﬁ-

cally, hierarchical plane structures based on a combination of frames with a

variable cross-section are proposed and exploited to open and enhance band

gaps with respect to their non-hierarchical counterparts. Our results show

Email address: nicola.pugno@unitn.it (Nicola M. Pugno)

1Laboratory of Bio-inspired, Bionic, Nano, Meta Materials & Mechanics, Department

of Civil, Environmental and Mechanical Engineering, University of Trento, 38123 Trento,

Italy

2DISAT, Politecnico di Torino, 10129 Torino, Italy

3CNRS, Centrale Lille, ISEN, Univ. Lille, Univ. Valenciennes, UMR 8520 - IEMN,

F-59000 Lille, France

4School of Engineering and Materials Science, Queen Mary University of London, Mile

End Road, London E1 4NS, United Kingdom

Preprint submitted to International Journal of Solids and Structures October 21, 2022

that hierarchy is eﬀective in broadening existing band gaps as well as opening

new full band gaps in non-hierarchical periodic structures.

Keywords: Phononic crystals and elastic metamaterials, Hierarchy,

Bioinspired metamaterials, Band gaps, Wave propagation

1. Introduction

Phononic crystals (PCs) and elastic metamaterials (MMs) oﬀer inter-

esting opportunities for unprecedented wave manipulation (Brillouin, 1953;

Sigalas and Economou, 1993; Mart´ınez-Sala et al., 1995; Liu et al., 2000; Khelif et al.,

2006; Craster and Guenneau, 2012; Deymier, 2013; Laude, 2015; Ma and Sheng,

2016), and can be deﬁned as composite structures designed to yield speciﬁc

wave dispersion characteristics, exploiting Bragg scattering and/or local res-

onance eﬀects. Among the plethora of unconventional dynamic properties

that PCs present, their capability to open frequency band gaps (BGs) is one

of the most investigated. A phononic BG is a frequency region where wave

propagation is not allowed, since only evanescent waves are present in the cor-

responding dispersion diagram (Lee, 2009; Laude et al., 2009). These eﬀects

can be achieved by properly choosing speciﬁc material and geometrical con-

ﬁgurations that allow an impedance mismatch in the system, responsible for

constructive/destructive interferences associated with: (i) the lattice dimen-

sions (Bragg scattering) (Khelif et al., 2006; Deymier, 2013; Laude, 2015) or

(ii) local resonances (Liu et al., 2000; Craster and Guenneau, 2012; Deymier,

2013). This opened up new perspectives in many ﬁelds, ranging from mi-

croelectromechanical systems to nondestructive evaluation (Pennec et al.,

2010; Miniaci et al., 2017; Gliozzi et al., 2019), including but not limited to

wave ﬁlters and waveguiding (Miniaci et al., 2018b), beam and wave splitting

(Li et al., 2015; Miniaci et al., 2019), large scale building vibration shielding

(Miniaci et al., 2016a), and subwavelength imaging (Sukhovich et al., 2009;

Moler´on and Daraio, 2015).

So far, the main approach to obtain wave attenuation has involved com-

bining materials to form the necessary impedance mismatch (Miranda Jr. and Dos Santos,

2017) or adding local resonators (Xiao et al., 2013; Nobrega et al., 2016;

Dal Poggetto et al., 2019), but recent advances have shown that variations

in the cross-section of speciﬁc regions of the unit cell can also lead to eﬀec-

tive vibration reduction in one- (Sorokin, 2016; Pelat et al., 2019) and two-

dimensional (Tang and Cheng, 2019; Bibi et al., 2019; Dal Poggetto and Arruda,

2021) systems. In this context, designing simultaneously strong and lightweight

PCs/MMs with desirable dynamic properties has been the quest of many re-

searchers for decades. Among the various approaches (continuous single-

phase materials, composite structures, etc.), lattice-type materials, given

their porous structure and well-deﬁned unit cell geometries that allow de-

viating from the properties of bulk materials, are excellent candidates to

obtain lightweight structures with precisely tailored dynamic and mechani-

cal properties. In addition, lattice- or frame-like structures allow the easy

integration of hierarchical architectures in the design.

A hierarchical architecture is here understood as a structure characterized

by multiple nested levels of unit cells repeated at diﬀerent size scales. The

structuring of these materials can be obtained by using self-similarity between

diﬀerent hierarchical levels (Mousanezhad et al., 2015; Meza et al., 2015) or

exploiting diﬀerent multi-scale structuring (Chen et al., 2016). The use of

hierarchical structures in the quasi-static regime, e.g., in architectural design

and civil engineering has been largely explored, leading to considerable weight

reduction without loss of mechanical performance, whereas in the dynamic

regime the properties of such materials have been somewhat less investigated.

In terms of wave dynamics and attenuation, Movchan and Guenneau

(2004) have shown that localized modes can be used to tune and create

low-frequency BGs, which was further reinforced by Huang and Sun (2010),

showing that BGs can be shifted by the proper selection of internal stiﬀ-

ness and masses of the system. Hierarchical structures have been proposed

by Chen and Wang (2016) using honeycomb architectures to design stiﬀ and

lightweight PCs, and by Lim et al. (2015) using self-similar beam structures

to improve wave propagation characteristics in hexagonal lattices. Among

the various types of hierarchical metamaterials, bioinspired hierarchical PCs

have shown interesting characteristics in terms of broadband wave ﬁlter-

ing, as demonstrated by Zhang and To (2013) using multiscale periodic-

ity in one-dimensional PCs, and Chen and Wang (2014, 2015a,b), who also

demonstrated how bioinspired composites can have their geometry tailored

for broadband vibration ﬁltering. Miniaci et al. (2016b) showed how a spe-

ciﬁc hierarchical arrangement of lattice-type structures inspired by a spider

web organization (i.e., lattice-type structure with radial and circular threads

linked by nodes of variable stiﬀness) are able to control wave propagation.

The eﬀect of bioinspired hierarchical organization on wave attenuation prop-

erties has also been experimentally investigated for the case of continuous

elastic metamaterials made of single-phase continuous structures formed by

self-similar unit cells with diﬀerent hierarchical levels and types of hierarchy

(Miniaci et al., 2018a). However, practically oriented metamaterials with

multi-scale wave attenuation are yet to be fully explored. One of these possi-

bilities is the use of frame-like bioinspired structures to harness the concep-

tual advantages of structures with varying cross-sections.

In this paper, we propose the use of a hierarchical spider web-based

lightweight solution to broaden the low- and mid-frequency BGs in a periodic

metamaterial frame. The proposed hierarchical structures are constructed by

replacing the regular frame elements of two-dimensional lattices by frame el-

ements forming a 1D periodic PC. The paper is organized as follows: Section

2 presents the models and methods, Section 3 illustrates the obtained results,

and Section 4 presents some concluding remarks.

2. Models and methods

2.1. Hierarchical structure

The hierarchical order associated with a structure is usually deﬁned as

the number nof length scales at which a recognizable sub-structure occurs

(Lakes, 1993): n= 0 corresponds to a continuous (1D, 2D or 3D) element,

n= 1 (ﬁrst order) represents a structure formed by elements occurring at

a single length scale, n= 2 (second-order) an arrangement of ﬁrst-order

structures, and so on.

2.1.1. First hierarchical level

We begin by deﬁning a structure (n= 1) obtained by arranging one-

dimensional solid frame elements (n= 0), so that it can be used as a one-

dimensional PC, controlling waves in its longitudinal direction. The rationale

for the development of this structure is depicted in Figure 1.

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arXiv:2210.11063v1[physics.app-ph]20Oct2022BandgapenhancementinperiodicframesusinghierarchicalstructuresVin´ıciusF.DalPoggetto1,FedericoBosia2,MarcoMiniaci3,NicolaM.Pugno1,4AbstractThequestfornoveldesignsforlightweightphononiccrystalsandelasticmeta-materialswithwidelow-frequencybandgapshasproventobe...

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Band gap enhancement in periodic frames using hierarchical structures

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