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An analytical framework is developed for investigating the effect of viscoelasticity on irregular hexagonal lattices. At room temperature many polymers are found to be near their glass temperature. Elastic moduli of honeycombs made of such materials are not constant, but changes in the time or frequency domain. Thus consideration of viscoelastic properties are essential for such honeycombs. Irregularity in lattice structures being inevitable from practical point of view, analysis of the compound effect considering both irregularity and viscoelasticty is crucial for such structural forms. On the basis of a mechanics based bottom-up approach, computationally efficient closed-form formulae are derived in frequency domain. The spatially correlated structural and material attributes are obtained based on Karhunen-Loève expansion, which is integrated with the developed analytical approach to quantify the viscoelastic effect for irregular lattices. Consideration of such spatially correlated behaviour can simulate the practical stochastic system more closely. The two effective complex Young’s moduli and shear modulus are found to be dependent on the viscoelastic parameters, while the two in-plane effective Poisson’s ratios are found to be independent of viscoelastic parameters and frequency. Results are presented in both deterministic and stochastic regime, wherein it is observed that the amplitude of Young’s moduli and shear modulus are significantly amplified in the frequency domain. The response bounds are quantified considering two different forms of irregularity, randomly inhomogeneous irregularity and randomly homogeneous irregularity. The computationally efficient analytical approach presented in this study can be quite attractive for practical purposes to analyse and design lattices with predominantly viscoelastic behaviour along with consideration of structural and material irregularity.
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σ(t) = Zt
−∞
g(tτ)∂(τ)
∂τ
tR+σ(t)(t)g(t)
¯σ(s) = s¯
G(s(s)
¯σ(s) ¯(s)¯
G(s)σ(t)(t)g(t)sC
g(t)
U(t)δ(t)
U(t) =
1t0,
0t < 0.
δ(t) =
0t6= 0,
R
−∞ δ(t)dt= 1
g(t) = µe(µ/η)tU(t)
g(t) = ηδ(t) + µU(t)
g(t) = ER1(1 τσ
τ
)et/τU(t)
g(t) = "n
X
j=1
µje(µjj)t#U(t)
¯
G(s) = ¯
G(iω) = G(ω)
ωR+G(ω)
G(ω) = G0(ω)+iG00(ω) = |G(ω)|eiφ(ω)
G0(ω)G00(ω)
G0(ω) = G+2
πZ
0
uG00(u)
ω2u2du
G00(ω) = 2ω
πZ
0
G0(u)
u2ω2du
G=G(ω→ ∞)R
G(ω)
ln |G0(ω)|= ln |G|+2
πZ
0
(u)
ω2u2du
φ(ω) = 2ω
πZ
0
ln |G(u)|
u2ω2du
G(ω)
E(ω) = ES1 + iω
µ+ iω
G(ω) = G0+Pn
k=1
akiω
iω+bk
G(ω) = G0+G(iωτ)β
1+(iωτ )β
G(ω) = G0h1 + Pkαkω2+2iξkωkω
ω2+2iξkωkω+ω2
ki
G(ω) = G0h1 + Pn
k=1 kω2+iωk
ω2+Ω2
ki
G(ω) = G0h1 + η1est0
st0i
G(ω) = G0h1 + η1+2(st0)2est0
1+2(st0)2i
G(ω) = G0h1 + η eω2/4µn1erf iω
2µoi
µ 
Es
|E(ω)|=ESsµ2+ω2(1 + )2
µ2+ω2
φ
φE(ω)= tan1µω
µ2+ω2(1 + )
|E(ω)| → ESµ→ ∞ |E(ω)| → ES(1 + )µ0ω > 0
|E(ω)| → ESω0|E(ω)| → ES(1 + )ω→ ∞ µ > 0
φE(ω)0µ→ ∞ φE(ω)0µ0ω > 0
φE(ω)0ω0φE(ω)0ω→ ∞ µ > 0
µ→ ∞ ω0
µ0ω→ ∞
l1l2l3α β γ Es
E1eq =t3
L
n
X
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1
l2
1ijl2
2ij (l1ij +l2ij) (cos αij sin βij sin αij cos βij )2
Esij((l1ij cos αij l2ij cos βij )2)
E2eq =Lt3
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1
Esij l2
3ij cos2γij l3ij +l1ij l2ij
l1ij +l2ij +l2
1ij l2
2ij (l1ij +l2ij ) cos2αij cos2βij
(l1ij cos αij l2ij cos βij )21
ν12eq =1
L
n
X
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1
(cos αij sin βij sin αij cos βij)
cos αij cos βij
ν21eq =L
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1
l2
1ijl2
2ij (l1ij +l2ij) cos αij cos βij (cos αij sin βij sin αij cos βij )
(l1ij cos αij l2ij cos βij)2l2
3ij cos2γij l3ij +l1ij l2ij
l1ij +l2ij +l2
1ij l2
2ij (l1ij +l2ij ) cos2αij cos2βij
(l1ij cos αij l2ij cos βij )2
G12eq =Lt3
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1
Esij l2
3ij sin2γij l3ij +l1ij l2ij
l1ij +l2ij 1
Esij
t
L i j
i= 1,2,3, ..., m j = 1,2,3, ..., n
m n
ij
ith jth
Es
Esij Esij 1 + ij
iω
µij + iω
E1v(ω) = t3
L
n
X
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1
l2
1ijl2
2ij (l1ij +l2ij) (cos αij sin βij sin αij cos βij )2
Esij 1 + ij
iω
µij + iω((l1ij cos αij l2ij cos βij)2)
E2v(ω) = Lt3
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1
Esij 1 + ij
iω
µij + iωl2
3ij cos2γij l3ij +l1ij l2ij
l1ij +l2ij +l2
1ij l2
2ij (l1ij +l2ij ) cos2αij cos2βij
(l1ij cos αij l2ij cos βij )21
G12v(ω) = Lt3
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1
Esij 1 + ij
iω
µij + iωl2
3ij sin2γij l3ij +l1ij l2ij
l1ij +l2ij 1
L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180θ γij = 90
i j
E1v=κ1t
l3cos θ
(h
l+ sin θ) sin2θ
E2v=κ2t
l3(h
l+ sin θ)
cos3θ
G12v=κ2t
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θ
κ1κ2
κ1=m
n
n
X
j=1
1
m
P
i=1
1
Esij 1 + ij
iω
µij + iω
κ2=n
m
1
n
P
j=1
1
m
P
i=1
Esij 1 + ij
iω
µij + iω
κ1κ2
ω0
Esij =Es
µij =µ ij = i = 1,2,3, ..., m j = 1,2,3, ..., n κ1
κ2Es
Esij =Esµij =µ ij = i = 1,2,3, ..., m j = 1,2,3, ..., n
E1v=ES1 + iω
µ+ iωζ1
E2v=ES1 + iω
µ+ iωζ2
G12v=ES1 + iω
µ+ iωζ3
ζi(i= 1,2,3)
ζ1=t3
L
n
X
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1
l2
1ijl2
2ij (l1ij +l2ij) (cos αij sin βij sin αij cos βij )2
(l1ij cos αij l2ij cos βij)2
ζ2=Lt3
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1 l2
3ij cos2γij l3ij +l1ij l2ij
l1ij +l2ij +l2
1ij l2
2ij (l1ij +l2ij ) cos2αij cos2βij
(l1ij cos αij l2ij cos βij )21
ζ3=Lt3
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1 l2
3ij sin2γij l3ij +l1ij l2ij
l1ij +l2ij 1
|E1v|=Esζ1sµ2+ω2(1 + )2
µ2+ω2
|E2v|=Esζ2sµ2+ω2(1 + )2
µ2+ω2
|G12v|=Esζ3sµ2+ω2(1 + )2
µ2+ω2
φ
φE1v=φE2v=φG12v= tan1µω
µ2+ω2(1 + )
ω0
L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180θ γij = 90i j
µ0µ→ ∞ ω0ω→ ∞
|E1v| → ESζ1µ→ ∞ |E1v| → ES(1 + )ζ1µ0ω > 0
|E1v| → ESζ1ω0|E1v| → ES(1 + )ζ1ω→ ∞ µ > 0
|E2v| → ESζ2µ→ ∞ |E2v| → ES(1 + )ζ2µ0ω > 0
|E2v| → ESζ2ω0|E2v| → ES(1 + )ζ2ω→ ∞ µ > 0
|G12v| → ESζ3µ→ ∞ |G12v| → ES(1 + )ζ3µ0ω > 0
|G12v| → ESζ3ω0|G12v| → ES(1 + )ζ3ω→ ∞ µ > 0
φE1v, φE2v, φG12v0µ→ ∞
φE1v, φE2v, φG12v0µ0ω > 0
φE1v, φE2v, φG12v0ω0
φE1v, φE2v, φG12v0ω→ ∞ µ > 0
L=n(h+lsin θ)l1ij =l2ij =l3ij =l
αij =θ βij = 180θ γij = 90i j
Esij =Esµij =µ ij =
i= 1,2,3, ..., m j = 1,2,3, ..., n
E1v=Es1 + iω
µ+ iωt
l3cos θ
(h
l+ sin θ) sin2θ
E2v=Es1 + iω
µ+ iωt
l3(h
l+ sin θ)
cos3θ
G12v=Es1 + iω
µ+ iωt
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θ
|E1v|=Essµ2+ω2(1 + )2
µ2+ω2t
l3cos θ
(h
l+ sin θ) sin2θ
|E2v|=Essµ2+ω2(1 + )2
µ2+ω2t
l3(h
l+ sin θ)
cos3θ
|G12v|=Essµ2+ω2(1 + )2
µ2+ω2t
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θ
φ
φE1v=φE2v=φG12v= tan1µω
µ2+ω2(1 + )
ω0µ0µ→ ∞ ω0ω→ ∞
|E1v| → ESt
l3cos θ
(h
l+ sin θ) sin2θµ→ ∞
|E1v| → ES(1 + )t
l3cos θ
(h
l+ sin θ) sin2θµ0ω > 0
|E1v| → ESt
l3cos θ
(h
l+ sin θ) sin2θω0
|E1v| → ES(1 + )t
l3cos θ
(h
l+ sin θ) sin2θω→ ∞ µ > 0
|E2v| → ESt
l3(h
l+ sin θ)
cos3θµ→ ∞
|E2v| → ES(1 + )t
l3(h
l+ sin θ)
cos3θµ0ω > 0
|E2v| → ESt
l3(h
l+ sin θ)
cos3θω0
|E2v| → ES(1 + )t
l3(h
l+ sin θ)
cos3θω→ ∞ µ > 0
|G12v| → ESt
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θµ→ ∞
|G12v| → ES(1 + )t
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θµ0ω > 0
|G12v| → ESt
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θω0
|G12v| → ES(1 + )t
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θω→ ∞ µ > 0
φE1v, φE2v, φG12v0µ→ ∞
φE1v, φE2v, φG12v0µ0ω > 0
φE1v, φE2v, φG12v0ω0
φE1v, φE2v, φG12v0ω→ ∞ µ > 0
θ= 30
E1v=E2v= 2.3ES1 + iω
µ+ iωt
l3
θ= 30
G12v= 0.57ES1 + iω
µ+ iωt
l3
|E1v|=|E2v|= 2.3Essµ2+ω2(1 + )2
µ2+ω2t
l3
|G12v|= 0.57Essµ2+ω2(1 + )2
µ2+ω2t
l3
φ
φE1v=φE2v=φG12v= tan1µω
µ2+ω2(1 + )
θ= 30
E2vν12v=E1vν21v=ES1 + iω
µ+ iωt
l31
sin θcos θ
ν12 =
ν21 = 1
G=E/2(1 + ν)E G ν
,F,P) Θ F P
σ H (x, θ)
,F,P)θΘxH(x, θ)
H(x, θ)
H(x, θ)
H(x, θ) ΓH(x1,x2)
H(x, θ)
H(x, θ) = ¯
H(x) +
X
i=1 pλiξi(θ)ψi(x)
{ξi(θ)} {λi} {ψi(x)}
ΓH(x1,x2)
Z
<N
ΓH(x1,x2)ψi(x1)dx1=λiψi(x2)
˜
H(x, θ)
=¯
H(x) +
M
X
i=1 pλiξi(θ)ψi(x)
H(x, θ)M→ ∞
ΓH(x1,x2)˜
H(x, θ)
˜
H(x, θ)
=G"¯
H(x) +
M
X
i=1 pλiξi(θ)ψi(x)#
H(x, θ)
H(x, θ)
ΓH(y1, z1;y2, z2) = σ2
He(−|y1y2|/by)+(−|z1z2|/bz)
bybz
σ2
H
λiψi(y2, z2) = Za1
a1Za2
a2
ΓH(y1, z1;y2, z2)ψi(y1, z1)dy1dz1
a16y6a1a26z6a2
ψi(y2, z2) = ψ(y)
i(y2)ψ(z)
i(z2)
λi(y2, z2) = λ(y)
i(y2)λ(z)
i(z2)
λ(y)
iψ(y)
i(y1) = Za1
a1
e(−|y1y2|/by)ψ(y)
i(y2)dy2
ψi(ζ) = cos(ωiζ)
qa+sin(2ωia)
2ωi
λi=2σ2
Hb
ω2
i+b2i
ψi(ζ) = sin(ω
iζ)
qasin(2ω
ia)
2ωi
λ
i=2σ2
Hb
ω2
i+b2i
b= 1/by1/bza=a1a2ζ ωiωi
bωitan(ωia)=0 ωi+btan(ωia)=0
r
m
Esµ
 r
m
r
Esρ3
E1E2ν12 ν21 G12
¯
E1=E1eq
Esρ3¯
E2=E2eq
Esρ3¯ν12 =ν12eq ¯ν21 =ν21eq ¯
G12 =G12eq
Esρ3
(¯) ρ
θ= 30h
l= 1 θ= 30h
l= 1.5
θ= 45h
l= 1 θ= 45h
l= 1.5
E1
θ= 30h
l= 1 θ= 30h
l= 1.5
θ= 45h
l= 1 θ= 45h
l= 1.5
E2
h/l
θ= 30θ= 45t/l 102
r
θ= 30h/l = 1
h/l = 1
h/l = 1.5θ
θ= 30h
l= 1 θ= 30h
l= 1.5
θ= 45h
l= 1 θ= 45h
l= 1.5
G12
µ=ωmax/5ωmax
= 2
ω0
θ= 30h
l= 1 θ= 30h
l= 1.5
θ= 45h
l= 1 θ= 45h
l= 1.5
ν12
θ= 30h
l= 1 θ= 30h
l= 1.5
θ= 45h
l= 1 θ= 45h
l= 1.5
ν21
 π/2
µ
ω= 0
µ 
E1E2G12
µ
µ 
E1
E2
G12
µ
= 2
µ=ωmax/5Z E1E2G12 Z0
ω= 0
µ
µ=ωmax/5
µ=ωmax/5Z E1E2G12
r= 6 Esm= 0.002
θ= 30h/l = 1 r= 0 r= 2 r= 4 r= 6
θ= 30h/l = 1 r= 6
E1
r= 3 r= 6
E2
r= 3 r= 6
G12
r= 3 r= 6
r= 3
r= 6
m
Esµ 
θ= 30h/l = 1 r= 6
m= 0.002
r
E1
r= 3 r= 6
E2
r= 3 r= 6
G12
r= 3 r= 6
r= 6
mθ= 45
h/l = 1
m
r= 1000×
h l θ
Esµ  r
θ= 45h/l = 1
&
... Now to implement viscoelasticity of material in the analysis, instantaneous stresses are assumed as a function of strains history by employing linear viscoelastic model. In a simplified linear viscoelastic mathematical model, the stress σ(t) at any point within the structure is a function of time and can be written in the form of convolution integral on the kernel function [75] as σ t (t) = g t (t) ⊛ ε(t) (11) In above equation, t ∈ R + is the dimensionless time parameter, σ(t) and ε(t) are representing timedependent stress and strain, respectively. Here, it is assumed that the strain is zero for negative times. ...
... Dirac's delta δ(t) distribution represents the unit impulse function which is a generalized function or distribution over the real numbers [78,79]. Based on these assumptions, the viscoelastic kernel function (g t (t)) for the four models can be expressed [75][76][77] as Maxwell viscoelastic model: ...
... The Biot model (mentioned in Table 1) with only one term has been used for computational simplicity. Hence, complex elastic and shear modulus in frequency domain [75] can be expressed as ...
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... Micromechanics and homogenization method are efficient approaches to predict the effective viscoelastic behaviors of the composite systems (generally including the porous materials) [6][7][8]. Viscoelastic properties such as the relaxation modulus, creep compliance, and complex modulus of the porous materials have been investigated tremendously by adopting the micromechanics and homogenization approaches [7][8][9][10][11][12]. As for micromechanical constitutive modeling, one of the widely used ways is the Laplace transform originally proposed by Hashin [13] to investigate the effective viscoelastic properties of heterogeneous media. ...
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Phononic crystals (PCs) are periodic structures obtained by the spatial arrangement of materials with contrasting properties, which can be designed to efficiently manipulate 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 propagating and evanescent waves) prohibitive due to excessive computational workload. In this work, we propose a new extended plane wave expansion formulation to compute the complex band structure of thick PC plates with arbitrary material distribution using the Mindlin-Reissner plate theory containing constituents with a viscoelastic behavior 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 attenuation 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.
... 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. ...
Article
Phononic crystals (PCs) are periodic structures obtained by the spatial arrangement of materials with contrasting properties, which can be designed to efficiently manipulate 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 propagating and evanescent waves) prohibitive due to excessive computational workload. In this work, we propose a new extended plane wave expansion formulation to compute the complex band structure of thick PC plates with arbitrary material distribution using the Mindlin-Reissner plate theory containing constituents with a viscoelastic behavior 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 attenuation 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.
... The mechanical properties of these articially engineered materials depend not only on intrinsic material characteristics but also on their microstructural conguration, giving rise to rare mechanical properties that could be exploited to create advanced materials with novel functionality [1, 2, 3, 4,5,6]. With the revolutionary advances in additive manufacturing in fabricating materials with arbitrarily complex micro/nano-architecture, the research area on design and development of various application-specic metamaterials have received signicant attention from material scientists and engineers over the last few years [7,8,9,10,11,12,13]. It has been successfully demonstrated that several multifunctional material properties required in modern-day advanced engineering applications can be achieved through intelligent microstructural design [14,15,16]. ...
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Shear modulus assumes an important role in characterizing the applicability of different materials in various multi-functional systems and devices such as deformation under shear and torsional modes, and vibrational behaviour involving torsion, wrinkling and rippling effects. Lattice-based artificial microstructures have been receiving significant attention from the scientific community over the past decade due to the possibility of developing materials with tailored multifunctional capabilities that are not achievable in naturally occurring materials. In general, the lattice materials can be conceptualized as a network of beams with different periodic architectures, wherein the common practice is to adopt initially straight beams. While shear modulus and multiple other mechanical properties can be simultaneously modulated by adopting an appropriate network architecture in the conventional periodic lattices, the prospect of on-demand global specific stiffness and flexibility modulation has become rather saturated lately due to intense investigation in this field. Thus there exists a strong rationale for innovative design at a more elementary level in order to break the conventional bounds of specific stiffness that can be obtained only by lattice-level geometries. In this article, we propose a novel concept of anti-curvature in the design of lattice materials, which reveals a dramatic capability in terms of enhancing shear modulus in the nonlinear regime while keeping the relative density unaltered. A semi-analytical bottom-up framework is developed for estimating effective shear modulus of honeycomb lattices with the anti-curvature effect in cell walls considering geometric nonlinearity under large deformation. We propose to consider the complementary deformed shapes of cell walls of honeycomb lattices under anti-clockwise or clockwise modes of shear stress as the initial beam-level elementary configuration. A substantially increased resistance against deformation can be realized when such a lattice is subjected to the opposite mode of shear stress, leading to increased effective shear modulus. Within the framework of a unit cell based approach, initially curved lattice cell walls are modeled as programmed curved beams under large deformation. The combined effect of bending, stretching and shear deformation is considered in the framework of Reddy’s third order shear deformation theory in a body embedded curvilinear frame. Governing equation of the elementary beam problem is derived using variational energy principle based Ritz method. In addition to application-specific design and enhancement of shear modulus, unlike conventional materials, we demonstrate through numerical results that it is possible to achieve non-invariant shear modulus under anti-clockwise and clockwise modes of shear stress. The developed physically insightful semi-analytical model captures nonlinearity in shear modulus as a function of the degree of anti-curvature and applied shear stress along with conventional parameters related to unit cell geometry and intrinsic material property. The concept of anti-curvature in lattices would introduce novel exploitable dimensions in mode-dependent effective shear modulus modulation, leading to an expanded design space including more generic scopes of nonlinear large deformation analysis.
... In this context, high-order dynamic homogenisation theories could provide effective models for pre-stressed viscoelastic structures valid at moderate frequencies (Hu and Oskay, 2017). Another potential direction of research is the study of manufactured phononic crystals with an irregular lattice (Mukhopadhyay et al., 2019). ...
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The effective dynamic properties of specific periodic structures involving rubber-like materials can be adjusted by pre-strain, thus facilitating the design of custom acoustic filters. While nonlinear viscoelastic behaviour is one of the main features of soft solids, it has been rarely incorporated in the study of such phononic media. Here, we study the dynamic response of nonlinear viscoelastic solids within a ‘small-on-large’ acoustoelasticity framework, that is we consider the propagation of small amplitude waves superimposed on a large static deformation. Incompressible soft solids whose behaviour is described by the Fung–Simo quasi-linear viscoelasticity theory (QLV) are considered. We derive the incremental equations using stress-like memory variables governed by linear evolution equations. Thus, we show that wave dispersion follows a strain-dependent generalised Maxwell rheology. Illustrations cover the propagation of plane waves under homogeneous tensile strain in a QLV Mooney–Rivlin solid. The acoustoelasticity theory is then applied to phononic crystals involving a lattice of hollow cylinders, by making use of a dedicated perturbation approach. In particular, results highlight the influence of viscoelastic dissipation on the location of the first band gap. We show that dissipation shifts the band gap frequencies, simultaneously increasing the band gap width. These results are relevant to practical applications of soft viscoelastic solids subject to static pre-stress.
... In a periodic structure, one unit cell (i.e. repeating units) can be analyzed with appropriate periodic boundary conditions to obtain the global behaviour of the entire lattice [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81]. Figure 1(A) shows the schematic diagram of the parent 3DCDL unit cell. It has two planar rings (loops) in two orthogonal planes with four connected beams. ...
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If we compress a conventional material in one direction, it will try to expand in the other two perpendicular directions and vice‐versa, indicating a positive Poisson’s ratio. Recently auxetic materials with negative Poisson’s ratios, which can be realized through artificial microstructuring, are attracting increasing attention due to enhanced mechanical performances in multiple applications. Most of the proposed auxetic materials show different degrees of in‐plane auxeticity depending on their microstructural configurations. However, this restricts harnessing the advantages of auxeticity in 3D systems and devices where multi‐directional functionalities are warranted. Thus, there exists a strong rationale to develop microstructures that can exhibit auxeticity both in the in‐plane and out‐of‐plane directions. Here we propose generic 3D connected double loop (3DCDL) type periodic microstructures for multi‐directional modulation of Poisson’s ratios. Based on the bending dominated behaviour of elementary beams with variable curvature, we demonstrate mixed‐mode auxeticity following the framework of multi‐material unit cells. The proposed 3DCDL unit cell and expanded unit cells formed based on their clusters are capable of achieving partially auxetic, purely auxetic, purely non‐auxetic and null‐auxetic behaviour. Comprehensive numerical results are presented for the entire spectrum of combinations concerning the auxetic behaviour in the in‐plane and out‐of‐plane directions including their relative degrees. This article is protected by copyright. All rights reserved.
... Lattice material is a kind of artificial periodic material with good lightweight, porosity, and designability [1,2]. By adjusting the topology configuration, cell sizes and cell arrangement rules, the mechanical properties of lattice materials can be improved to adapt to different engineering applications [3][4][5]. With the development of the study, it is found that lattice materials include many other important functional potentials, such as cushioning, vibration attenuation, energy absorption, heat dissipation, noise reduction, and electromagnetic shielding, which are expected to be widely used in aerospace, transportation, machinery, and many others fields [6][7][8][9]. ...
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A heterogeneous lattice material composed of different cells is proposed to improve the energy absorption capacity. The heterogeneous structure is formed by setting layers of body-centered XY rods (BCCxy) cells as the reinforcement in the body-centered cubic (GBCC) uniform lattice material. The heterogeneous lattice samples are designed and processed by additive manufacturing technology. The stress wave propagation and energy absorption properties of heterogeneous lattice materials under impact load are analyzed by finite element simulation (FES) and Hopkinson pressure bar (SHPB) experiments. The results show that, compared with the GBCC uniform lattice material, the spreading velocity of the stress of the (GBCC)3(BCCxy)2 heterogeneous lattice material is reduced by 18.1%, the impact time is prolonged 27.9%, the stress peak of the transmitted bar is reduced by 34.8%, and the strain energy peak is reduced by 29.7%. It indicates that the heterogeneous lattice materials are able to reduce the spreading velocity of stress and improve the energy absorption capacity. In addition, the number of layers of reinforcement is an important factor affecting the stress wave propagation and energy absorption properties.
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Despite the common use of the standard linear solid model (SLSM) in viscoelasticity, the physical significance as well as the difference between the Maxwell and Kelvin forms of SLSM are still not clear. This paper demonstrates that each parameter of those two models has its specific physical meaning, and introduces the relationships allowing the transformations between those parameters. Regardless of their physical significance, those two models are equivalent in terms of their mathematical properties. Hence, no matter which model is chosen, consistent analysis results can always be obtained as long as the physical meaning of each parameter is accurately interpreted.
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