Wave attenuation in viscoelastic hierarchical plates

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Wave attenuation in viscoelastic hierarchical plates
Vin´ıcius F. Dal Poggetto1,∗, Edson J. P. Miranda Jr.2,3,4,∗, Jos´e Maria C. Dos
Santos4, Nicola M. Pugno1,5
Abstract
Phononic crystals (PCs) are periodic structures obtained by the spatial arrangement
of materials with contrasting properties, which can be designed to efficiently manip-
ulate mechanical waves. Plate structures can be modeled using the Mindlin-Reissner
plate theory and have been extensively used to analyze the dispersion relations of PCs.
Although the analysis of the propagating characteristics of PCs may be sufficient for
simple elastic structures, analyzing the evanescent wave behavior becomes fundamental
if the PC contains viscoelastic components. Another complication is that increasingly
intricate material distributions in the unit cell of PCs with hierarchical configuration
may render the calculation of the complex band structure (i.e., considering both prop-
agating and evanescent waves) prohibitive due to excessive computational workload.
In this work, we propose a new extended plane wave expansion formulation to com-
pute the complex band structure of thick PC plates with arbitrary material distribution
using the Mindlin-Reissner plate theory containing constituents with a viscoelastic be-
havior approximated by a Kelvin-Voigt model. We apply the method to investigate
the evanescent behavior of periodic hierarchically structured plates for either (i) a
hard purely elastic matrix with soft viscoelastic inclusions or (ii) a soft viscoelastic
matrix with hard purely elastic inclusions. Our results show that for (i), an increase
in the hierarchical order leads to a weight reduction with relatively preserved attenu-
ation characteristics, including attenuation peaks due to locally resonant modes that
present a decrease in attenuation upon increasing viscosity levels. For (ii), changing
the hierarchical order implies in opening band gaps in distinct frequency ranges, with
an overall attenuation improved by an increase in the viscosity levels.
∗Corresponding authors
Email addresses: v.fonsecadalpoggetto@unitn.it (Vin´ıcius F. Dal Poggetto),
edson.jansen@ifma.edu.br (Edson J. P. Miranda Jr.)
1Laboratory for Bio-inspired, Bionic, Nano, Meta Materials & Mechanics, Department of Civil,
Environmental and Mechanical Engineering, University of Trento, 38123 Trento, Italy
2Federal Institute of Maranh˜ao, IFMA-EIB-DE, Rua Afonso Pena, 174, CEP 65010-030, S˜ao
Lu´ıs, MA, Brazil
3Federal Institute of Maranh˜ao, IFMA-PPGEM, Avenida Get´ulio Vargas, 4, CEP 65030-005, S˜ao
Lu´ıs, MA, Brazil
4University of Campinas, UNICAMP-FEM-DMC, Rua Mendeleyev, 200, CEP 13083-970, Camp-
inas, SP, Brazil
5School of Engineering and Materials Science, Queen Mary University of London, Mile End Road,
London E1 4NS, United Kingdom
Preprint submitted to Journal October 26, 2022
arXiv:2210.14068v1 [physics.app-ph] 25 Oct 2022
Keywords: Plane wave expansion, Hierarchical structure, Mindlin-Reissner plate,
Evanescent wave, Viscoelastic material.
1. Introduction
Hierarchical materials present a structured composition across multiple length
scales and have long been a subject of study [1] since their occurrence in nature is
associated with excellent static [2] and dynamic characteristics [3]. Such properties
may be harnessed to design novel materials through the selection of constituents in
a proper multi-level structuring [4]. Although the static properties of hierarchical
structures have been thoroughly exposed in different contexts [5–7], their application
to obtain interesting dynamic properties remains to be fully explored. In particular,
hierarchical periodic structures can be used to attenuate waves in a particularly broad
manner [8–10], which has been demonstrated both theoretically and experimentally
with the use of dissipative elastic metamaterials [11].
Periodic structures, which can be obtained by the repetition of a representative
unit cell, are known for their ability to manipulate waves [12], leading to applications
in mechanical systems such as vibration attenuation, imaging, and cloaking [13, 14].
A remarkable feature that can be found in a specific class of periodic structures named
phononic crystals (PCs) [15] is that impedance mismatches achieved, for instance, by
using spatial modulations in single-phase materials [16–19] or by combining materials
with contrasting elastic properties [20–22] can lead to the occurrence of frequency
ranges named band gaps (BGs). Such frequency ranges are typically created in PCs
by the destructive interference of waves (Bragg scattering) [23], thus prohibiting free
wave propagation due to the resulting purely evanescent behavior of waves [24, 25].
In the case of locally resonant PCs [26], Fano-like interference mechanisms can also
occur in the sub-wavelength scale [27], thus typically leading to low-frequency BGs.
Although the opening of BGs is evident in the case of purely elastic materials, the
inclusion of damping leads to complications in the determination of BGs, since spatial
attenuation becomes inherent in such cases [28–30].
Band diagrams can be used to conveniently analyze the dispersion relation (i.e.,
the relation between the wavenumber and frequency) of periodic structures and can
be obtained through a variety of techniques. Although finite element (FE)-based
techniques are widely employed [31–34], these methods usually suffer from disadvan-
tages in terms of computational burden, which may render their use prohibitive when
distinct orders of hierarchy are considered due to the inherently detailed modeling
which is required. Dispersion relations can also be computed using the plane wave
expansion (PWE) method, which typically results in a reduced computational effort
[35–37]. On the other hand, the PWE method usually requires the use of analytical
expressions for the shape of the scatterers included in the PC matrix material, which
limits the applicability of the method. Also, the conventional PWE method does not
offer information about the evanescent behavior of waves [25, 38], which is necessary
to characterize the complex band structure of damped systems. A solution to this
2
limitation is proposed by the extended plane wave expansion (EPWE) method [39],
which yields both propagating and evanescent parts of the wave vectors for a given
frequency of interest at the expense of a greater computational cost.
Recent advances in the experimental observations of guided waves in biological
structures such as the human skull [40] have revealed the propagation of Lamb waves
[41, 42], which motivates the analysis of the wave propagation in structured media
using plate theories [43]. Plates have been thoroughly explored as versatile structures
in the field of metamaterials and PCs using the Kirchhoff plate theory [44] with
periodic arrays of embedded resonators [45, 46], periodic arrays of local resonators
[47, 48], the inclusion of point defects [49], or the Mindlin plate theory [50] with
embedded [37, 51] or attached resonators [52, 53]. The use of plate theories for the
computation of dispersion relations seems thus to be the most common solution when
compared to the use of solid models with stress-free boundaries [54] or equivalent
low impedance surrounding media [55]. Although the Kirchhoff-Love plate theory
can be considered under the assumption of negligible shear strain and rotational
inertia in the low-frequency range, its use may require additional refinements of the
kinematic model or adjustments to properly include inertial terms [56, 57]. On the
other hand, the Mindlin-Reissner plate theory already accounts for shear strain and
rotational inertia terms, although requiring larger computational models, being also
more suited to analyze structures which operate in higher frequency ranges, which is
the case of PCs. Previous works have computed the dispersion relation considering
the viscoelastic material behavior for the SH-wave of two-dimensional PCs [58] and
quasi-periodic lattices [59]. The investigation of the effects of hierarchical structuring
on plates, however, especially when considering the complex band structure necessary
to fully understand the implications of components that present damping, remains
largely unexplored.
In this work, we propose the numerical investigation of the evanescent behavior
of viscoelastic hierarchical plate PCs with the use of the EPWE method applied
using the Mindlin-Reissner plate theory. This paper is organized as follows: Section 2
presents the considered plate theory, the material behavior, and the application of the
EPWE method to plates with discrete geometries. Section 3 presents the obtained
results, and Section 4 presents our concluding remarks.
2. Models and methods
In this section, we present the analytical derivations regarding the calculation of
dispersion curves for periodic PC structures using the Mindlin-Reissner plate theory,
which considers non-negligible rotational inertia and transverse shear strain [60]. The
related dynamic equations are expressed considering periodic solutions for both dis-
placements and rotations, and also periodic material properties. Then, a Kelvin-Voigt
viscoelastic model is included to represent the dissipative behavior of constituents,
which is considered when formulating the equations that allow to compute the com-
plex band structure of the unit cell.
3
2.1. Wave propagation in periodic plates using the Mindlin-Reissner plate theory
Plate theories can be employed for the analysis of structural elements with one
dimension (thickness) considerably smaller than the other two ones [61]. Considering
the Mindlin-Reissner plate theory, the equation that describes the dynamic behavior
in the time domain (t) of an isotropic plate lying in the xy plane without applied
loads is given by
∂
∂x κµh∂uz
∂x −ψx+∂
∂y κµh∂uz
∂y −ψy=ρh∂2uz
∂t2,
(1a)
∂
∂x D∂ψx
∂x +ν∂ψy
∂y +∂
∂y D(1−ν)
2∂ψx
∂y +∂ψy
∂x +κµh∂uz
∂x −ψx=ρh3
12
∂2ψx
∂t2,
(1b)
∂
∂x D(1−ν)
2∂ψx
∂y +∂ψy
∂x +∂
∂y Dν∂ψx
∂x +∂ψy
∂y +κµh∂uz
∂y −ψy=ρh3
12
∂2ψy
∂t2,
(1c)
where uz=uz(x, y, t)is the plate out-of-plane displacement, ψx=ψx(x, y, t)and
ψy=ψx(x, y, t)represent the rotations of the plate midplane normal direction, κ
is the plate shear correction factor [62], µ=µ(x, y)is the material shear modulus,
ν=ν(x, y)is the material Poisson’s ratio, ρ=ρ(x, y)is the material mass density, h
is the plate thickness, and D=D(x, y)is the plate flexural stiffness, given by
D(x, y)=E(x, y)h3
12(1−ν2(x, y)),(2)
where E=E(x, y)is the material Young’s modulus. If the plate material proper-
ties are periodic, the resulting displacement and rotation solutions present the same
periodicity [63], which can be used to obtain the dispersion curves of the periodic
medium.
To properly analyze the propagation of elastic waves in periodic plates, the PWE
method requires the expression of displacements, rotations, and material properties
considering their respective spatial periodicity. Let us denote the position vector rin
terms of its Cartesian components, i.e., r=xˆ
i+yˆ
j. Considering a Bloch solution [63]
for the displacements uz(x, y, t)=uz(r, t), one has
uz(r, t)=uzk(r)e−iωt ,(3)
where uzk(r)is a spatial function, and ωis the considered angular frequency. Ac-
cording to Bloch’s theorem, uzk(r)must obey
uzk(r)=eik⋅ruz0(r),(4)
where kis the two-dimensional wave vector, which can be written in terms of its
Cartesian components as k=kxˆ
i+kyˆ
j, and uz0(r)is a periodic function with the
4
same periodicity as the medium. Thus, uz0(r)can be represented as a Fourier series
using
uz0(r)=
G
ˆuz(G)eiG⋅r,(5)
where ˆuz(G)denotes a Fourier coefficient for the representation of the out-of-plane
displacements, which must be summed for infinite reciprocal lattice vectors of the form
G=Gxˆ
i+Gyˆ
j. For a square lattice unit cell of side length a,Gx=nx2π
aand Gy=ny2π
a
for {nx, ny}∈Z. Thus, for −N≤{nx, ny}≤N,N∈N, a total of nG=(1+2N)2plane
waves is considered.
The expression of displacements in the periodic medium can thus be obtained by
combining Eqs. (3)-(5) in the form
uz(r, t)=e−iωt
G
ˆuz(G)ei(k+G)⋅r,(6)
which presents a form appropriate for its application in the PWE method.
An analogous procedure can be applied to the rotations ψx(r, t)and ψy(r, t),
allowing to write
ψx(r, t)=e−iωt
G
ˆ
ψx(G)ei(k+G)⋅r,
ψy(r, t)=e−iωt
G
ˆ
ψy(G)ei(k+G)⋅r,(7)
where ˆ
ψx(G)and ˆ
ψy(G)denote the Fourier coefficients for the representation of the
midplane rotations ψxand ψy, respectively.
The wave vector kand its components can be classified according to its real and
imaginary parts, assumed to be co-linear [64]: purely real wave vectors yield prop-
agating waves, purely imaginary wave vectors yield evanescent waves, and complex
conjugate solutions yield decaying propagating waves.
2.2. Material properties
Let us consider a square unit cell with a side length of a, divided in a set of
square elements (pixels) used to describe its spatial configuration. Each pixel can be
described by the x- and y-coordinates of its center, denoted as xcand yc, respectively,
its side length lc, and a corresponding material property pc(Figure 1). Although the
pixels do not necessarily form a regular grid, they cover the entire area Aof the unit
cell, i.e., ∑Al2
c=a2. Furthermore, a four-fold symmetry is assumed so that the band
structure of the medium can be investigated by analyzing a reduced region of the unit
cell [65].
The PWE method also requires that the material properties be written in terms
of their Fourier series. Thus, a general expression can be stated for a given material
property pusing
p(r)=
G
ˆp(G)eiG⋅r,(8)
where ˆp(G)denotes a Fourier series term for the representation of the corresponding
material property, which theoretically must be summed for an infinite number of
reciprocal lattice vectors G.
5
摘要：

WaveattenuationinviscoelastichierarchicalplatesVinciusF.DalPoggetto1,‡,EdsonJ.P.MirandaJr.2,3,4,‡,JoseMariaC.DosSantos4,NicolaM.Pugno1,5AbstractPhononiccrystals(PCs)areperiodicstructuresobtainedbythespatialarrangementofmaterialswithcontrastingproperties,whichcanbedesignedtoecientlymanip-ulatemec...

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