Frequency domain homogenization for the viscoelastic properties of spatially correlated quasi-periodic lattices

Abstract

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.

Full text

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∗

σ(t) = Zt
−∞
g(t−τ)∂(τ)
∂τ dτ
t∈R+σ(t)(t)g(t)
¯σ(s) = s¯
G(s)¯(s)
¯σ(s) ¯(s)¯
G(s)σ(t)(t)g(t)s∈C
g(t)
U(t)δ(t)
U(t) = 




1t≥0,
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) = ER1−(1 −τσ
τ
)e−t/τU(t)
•
g(t) = "n
X
j=1
µje−(µj/ηj)t#U(t)
¯
G(s) = ¯
G(iω) = G(ω)

ω∈R+G(ω)
G(ω) = G0(ω)+iG00(ω) = |G(ω)|eiφ(ω)
G0(ω)G00(ω)
G0(ω) = G∞+2
πZ∞
0
uG00(u)
ω2−u2du
G00(ω) = 2ω
πZ∞
0
G0(u)
u2−ω2du
G∞=G(ω→ ∞)∈R
G(ω)
ln |G0(ω)|= ln |G∞|+2
πZ∞
0
uφ(u)
ω2−u2du
φ(ω) = 2ω
πZ∞
0
ln |G(u)|
u2−ω2du
G(ω)
E(ω) = ES1 + iω
µ+ iω

G(ω) = G0+Pn
k=1
akiω
iω+bk
G(ω) = G0+G∞(iωτ)β
1+(iωτ )β
G(ω) = G0h1 + Pkαk−ω2+2iξkωkω
−ω2+2iξkωkω+ω2
ki
G(ω) = G0h1 + Pn
k=1 ∆kω2+iωΩk
ω2+Ω2
ki
G(ω) = G0h1 + η1−e−st0
st0i
G(ω) = G0h1 + η1+2(st0/π)2−e−st0
1+2(st0/π)2i
G(ω) = G0h1 + η eω2/4µn1−erf iω
2√µoi
µ 
Es
|E(ω)|=ESsµ2+ω2(1 + )2
µ2+ω2
φ
φE(ω)= tan−1µω
µ2+ω2(1 + )
|E(ω)| → ESµ→ ∞ |E(ω)| → ES(1 + )µ→0∀ω > 0
|E(ω)| → ESω→0|E(ω)| → ES(1 + )ω→ ∞ ∀µ > 0
φE(ω)→0µ→ ∞ φE(ω)→0µ→0∀ω > 0
φE(ω)→0ω→0φE(ω)→0ω→ ∞ ∀µ > 0
µ→ ∞ ω→0
µ→0ω→ ∞

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 )2−1
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)2l2
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 + iωl2
3ij cos2γij l3ij +l1ij l2ij
l1ij +l2ij +l2
1ij l2
2ij (l1ij +l2ij ) cos2αij cos2βij
(l1ij cos αij −l2ij cos βij )2−1
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 + iωl2
3ij sin2γij l3ij +l1ij l2ij
l1ij +l2ij −1
L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180◦−θ γij = 90◦
i j

E1v=κ1t
l3cos θ
(h
l+ sin θ) sin2θ
E2v=κ2t
l3(h
l+ sin θ)
cos3θ
G12v=κ2t
l3h
l+ sin θ
h
l2(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
ω→0
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=ES1 + iω
µ+ iωζ1
E2v=ES1 + iω
µ+ iωζ2

G12v=ES1 + iω
µ+ iωζ3
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 )2−1
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
φE1v=φE2v=φG12v= tan−1µω
µ2+ω2(1 + )

ω→0
L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180◦−θ γij = 90◦i j
µ→0µ→ ∞ ω→0ω→ ∞
|E1v| → ESζ1µ→ ∞ |E1v| → ES(1 + )ζ1µ→0∀ω > 0
|E1v| → ESζ1ω→0|E1v| → ES(1 + )ζ1ω→ ∞ ∀µ > 0
|E2v| → ESζ2µ→ ∞ |E2v| → ES(1 + )ζ2µ→0∀ω > 0
|E2v| → ESζ2ω→0|E2v| → ES(1 + )ζ2ω→ ∞ ∀µ > 0
|G12v| → ESζ3µ→ ∞ |G12v| → ES(1 + )ζ3µ→0∀ω > 0
|G12v| → ESζ3ω→0|G12v| → ES(1 + )ζ3ω→ ∞ ∀µ > 0
φE1v, φE2v, φG12v→0µ→ ∞
φE1v, φE2v, φG12v→0µ→0∀ω > 0
φE1v, φE2v, φG12v→0ω→0
φE1v, φE2v, φG12v→0ω→ ∞ ∀µ > 0
L=n(h+lsin θ)l1ij =l2ij =l3ij =l
αij =θ βij = 180◦−θ γij = 90◦i j
Esij =Esµij =µ ij =
i= 1,2,3, ..., m j = 1,2,3, ..., n
E1v=Es1 + iω
µ+ iωt
l3cos θ
(h
l+ sin θ) sin2θ

E2v=Es1 + iω
µ+ iωt
l3(h
l+ sin θ)
cos3θ
G12v=Es1 + iω
µ+ iωt
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θ
|E1v|=Essµ2+ω2(1 + )2
µ2+ω2t
l3cos θ
(h
l+ sin θ) sin2θ
|E2v|=Essµ2+ω2(1 + )2
µ2+ω2t
l3(h
l+ sin θ)
cos3θ
|G12v|=Essµ2+ω2(1 + )2
µ2+ω2t
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θ
φ
φE1v=φE2v=φG12v= tan−1µω
µ2+ω2(1 + )
ω→0µ→0µ→ ∞ ω→0ω→ ∞

|E1v| → ESt
l3cos θ
(h
l+ sin θ) sin2θµ→ ∞
|E1v| → ES(1 + )t
l3cos θ
(h
l+ sin θ) sin2θµ→0∀ω > 0
|E1v| → ESt
l3cos θ
(h
l+ sin θ) sin2θω→0
|E1v| → ES(1 + )t
l3cos θ
(h
l+ sin θ) sin2θω→ ∞ ∀µ > 0
|E2v| → ESt
l3(h
l+ sin θ)
cos3θµ→ ∞
|E2v| → ES(1 + )t
l3(h
l+ sin θ)
cos3θµ→0∀ω > 0
|E2v| → ESt
l3(h
l+ sin θ)
cos3θω→0
|E2v| → ES(1 + )t
l3(h
l+ sin θ)
cos3θω→ ∞ ∀µ > 0
|G12v| → ESt
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θµ→ ∞
|G12v| → ES(1 + )t
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θµ→0∀ω > 0
|G12v| → ESt
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θω→0
|G12v| → ES(1 + )t
l3h
l+ sin θ
h
l2(1 + 2h
l) cos θω→ ∞ ∀µ > 0
φE1v, φE2v, φG12v→0µ→ ∞
φE1v, φE2v, φG12v→0µ→0∀ω > 0
φE1v, φE2v, φG12v→0ω→0
φE1v, φE2v, φG12v→0ω→ ∞ ∀µ > 0
θ= 30◦
E1v=E2v= 2.3ES1 + iω
µ+ iωt
l3
θ= 30◦
G12v= 0.57ES1 + iω
µ+ iωt
l3

|E1v|=|E2v|= 2.3Essµ2+ω2(1 + )2
µ2+ω2t
l3
|G12v|= 0.57Essµ2+ω2(1 + )2
µ2+ω2t
l3
φ
φE1v=φE2v=φG12v= tan−1µω
µ2+ω2(1 + )
θ= 30◦
E2vν12v=E1vν21v=ES1 + iω
µ+ iωt
l31
sin θcos θ
ν12 =
ν21 = 1
G=E/2(1 + ν)E G ν
(Θ,F,P) Θ F P
σ− H (x, θ)
(Θ,F,P)θ∈ΘxH(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(−|y1−y2|/by)+(−|z1−z2|/bz)

bybz
σ2
H
λiψi(y2, z2) = Za1
−a1Za2
−a2
ΓH(y1, z1;y2, z2)ψi(y1, z1)dy1dz1
−a16y6a1−a26z6a2
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(−|y1−y2|/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)
qa−sin(2ω∗
ia)
2ωi∗
λ∗
i=2σ2
Hb
ω∗2
i+b2i
b= 1/by1/bza=a1a2ζ ωiωi∗
b−ωitan(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= 30◦h
l= 1 θ= 30◦h
l= 1.5
θ= 45◦h
l= 1 θ= 45◦h
l= 1.5
E1
θ= 30◦h
l= 1 θ= 30◦h
l= 1.5
θ= 45◦h
l= 1 θ= 45◦h
l= 1.5
E2

h/l
θ= 30◦θ= 45◦t/l ∼10−2
r
θ= 30◦h/l = 1
h/l = 1
h/l = 1.5θ

θ= 30◦h
l= 1 θ= 30◦h
l= 1.5
θ= 45◦h
l= 1 θ= 45◦h
l= 1.5
G12
µ=ωmax/5ωmax
= 2
ω→0

θ= 30◦h
l= 1 θ= 30◦h
l= 1.5
θ= 45◦h
l= 1 θ= 45◦h
l= 1.5
ν12
θ= 30◦h
l= 1 θ= 30◦h
l= 1.5
θ= 45◦h
l= 1 θ= 45◦h
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 Es∆m= 0.002

θ= 30◦h/l = 1 r= 0 r= 2 r= 4 r= 6
θ= 30◦h/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= 30◦h/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= 45◦h/l = 1

&