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An analytical framework is developed for predicting the effective in-plane elastic moduli (longitudinal and transverse Young's modulus, Poisson's ratios and shear modulus) of irregular hexagonal lattices with generalized form of spatially random structural geometry. On the basis of a mechanics based bottom-up multi-step approach, computationally efficient closed-form formulae are derived in this article. As a special case when there is no irregularity, the derived analytical expressions reduce to the respective well known formulae of regular honeycombs available in literature. Previous analytical investigations include the derivation of effective in-plane elastic moduli for hexagonal lattices with spatially random variation of cell angles, which is a special case of the generalized form of irregularity in material and structural attributes considered in this paper. The present study also includes development of a highly generalized finite element code for obtaining equivalent elastic properties of random lattices, which is employed to validate the proposed analytical formulae. The statistical results of elastic moduli obtained using the developed analytical expressions and using direct finite element simulations are noticed to be in good agreement affirming the accuracy and validity of the proposed analytical framework. All the in-plane elastic moduli are found to be significantly influenced by spatially random irregularity resulting in a decrease of the mean values for the two Young's moduli and two Poisson's ratios, while an increase of the mean value for the shear modulus.
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ν12
ν12
ν12
ν21
ν21
ν21
l1l2l3α β
γ Es
ZU(ω)ω
Z U
σ1
E1ν12
E2ν21 σ2τ
G12
ZUZI
U
ˆ
Z Zeq
Z E1U
i j
i= 1,2,3, ..., m j = 1,2,3, ..., n
m
n
ij
ith jth Zij j
jth Zj
rd
rd<min h
2,l
2, l cos θ
r
r=πr2
d×100
2lcos θ(h+lsin θ)
r
r
σ1
E1U
E1U
P=σ1Lyb Lyb
M1M2
M1=1
2(P l1sin αCl1cos α)
M2=1
2(P l2sin βCl2cos β)
C
C=Pl1sin αl2sin β
l1cos αl2cos β
δh
AO
P C
δh
AO =P l3
1sin α
12EsICl3
1cos α
12EsIsin α
P C h
Es
I I =bt3/12 t
E1U
δh
BO =P l3
2sin β
12EsICl3
1cos β
12EsIsin β
δO=l2sin βl1cos αl1sin αl2cos β
l1cos αl2cos β
δh
1=δh
AO
δO
l1sin α+δh
BO
δO
l2sin β
=σ1Lyl2
1l2
2(l1+l2) (cos αsin βsin αcos β)2
Est3(l1cos αl2cos β)2
h
1=σ1Lyl2
1l2
2(l1+l2) (cos αsin βsin αcos β)2
Est3(l1cos αl2cos β)3
E1U=Est3(l1cos αl2cos β)3
Lyl2
1l2
2(l1+l2) (cos αsin βsin αcos β)2
E1U
Ly
EI
1ULjjth
EI
1U
P Bij
AN I E1U
=P Bij
AIEI
1U
ANI =Lyb AI=Ljb
EI
1U=E1U
Ly
Lj
E1eq σ1
jth
σ11j
1ij
jth
1j=
m
X
i=1
1ij
1jBj=
m
X
i=1
1ij Bij
1jBjjth
Bij = (l1ij cos αij l2ij cos βij )Bj=
Pm
i=1 Bij
σ1Bj
ˆ
E1j
=
m
X
i=1
σ1Bij
EI
1Uij
jth ˆ
E1j
ˆ
E1j=Bj
m
P
i=1
Bij
EI
1Uij
EI
1Uij
i j
E1eq
ˆ
E1j
σ1Lb =
n
X
j=1
σ1jLjb
Ljjth L=
n
P
j=1
Lj
b
n
E1eqL=
n
X
j=1
ˆ
E1jLj
E1eq
E1eq =1
L
n
X
j=1
BjLj
m
P
i=1
Bij
EI
1Uij
E1eq =Est3
L
n
X
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij )
m
P
i=1
l2
1ij l2
2ij (l1ij +l2ij ) (cos αij sin βij sin αij cos βij )2
(l1ij cos αij l2ij cos βij )2
σ2
E2U
δv
bCO =W l3
3cos2γ
12EsI
W=σ2b(l1cos αl2cos β)v
E2U
M0=W l3cos γ
2M0
M0
M0
OA =l2
l1+l2
M0M0
OB =l1
l1+l2
M0
l M
δ=Ml2
6EI φ=M0
OAl1
6EsI
M0
OA M0
δv
rCO =l1l2l2
3cos2γ
12EsI(l1+l2)W
M0
1=W1l1cos α
2+l2
l1+l2
M0
M00
1=W1l1cos α
2l2
l1+l2
M0
M0
2=W2l2cos β
2l1
l1+l2
M0
M00
2=W2l2cos β
2+l1
l1+l2
M0
W1=l2cos β
l1cos αl2cos βW
W2=l1cos α
l1cos αl2cos βW
δv
AO =W1l3
1cos2α
12EsI
δv
BO =W2l3
2cos2β
12EsI
δv
O=δAO(l2cos β) + δB O (l1cos α)
l1cos αl2cos β
δv
AO δv
BO
δv
O=l2
1l2
2cos2αcos2β(l1+l2)
12EsI(l1cos αl2cos β)2W
δv
2=δv
bCO +δv
rCO +δv
O
=W
12EsIl2
3cos2γl3+l1l2
l1+l2+l2
1l2
2(l1+l2) cos2αcos2β
(l1cos αl2cos β)2
v
2=σ2(l1cos αl2cos β)
Est3Lyl2
3cos2γl3+l1l2
l1+l2+l2
1l2
2(l1+l2) cos2αcos2β
(l1cos αl2cos β)2
E2U=Est3Ly
(l1cos αl2cos β)l2
3cos2γl3+l1l2
l1+l2+l2
1l2
2(l1+l2) cos2αcos2β
(l1cos αl2cos β)21
E2ULy
EI
2ULj
jth EI
2U
W Ly
AE2U
=W Lj
AEI
2U
A=Bij b
EI
2U=E2U
Lj
Ly
E2eq EI
2U
σ2
jth σ2
σ2Bb = m
X
i=1
σ2ij Bij !b
2ij
jth
ˆ
E2j2jBj=
m
X
i=1
EI
2U ij 2ij Bij
2ij =2ji= 1,2...m jth ˆ
E2j
jth
ˆ
E2j=
m
P
i=1
EI
2U ij Bij
Bj
2j
g
2L=
n
X
j=1
2jLj
g
22jjth
Ljjth
σ2L
E2eq
=
n
X
j=1
σ2Lj
ˆ
E2j
E2eq =L
n
P
j=1
LjBj
m
P
i=1
EI
2U ij Bij
E2eq =LEst3
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
G12Uτ
δs
OA =δs
OB
F1=cot α
cot αcot βF
F2=cot β
cot αcot βF
Vs=1
cot αcot βF
δs
OA =F1sin αl3
1
12EsIVscos αl3
1
12EsIsin α
s F1
G12U
Vsδs
OA =δs
OB = 0
τ
δs
bCO =F l3
3sin2γ
12EsI
F=τ b (l1cos αl2cos β)Ms=F l3sin γ
2Ms
MsMs
1=l2
l1+l2
MsMs
2=l1
l1+l2
Ms
l
M δ =Ml2
6EI
φ=l1l2Ms
6EsI(l1+l2)
δs
rCO =φl3sin γ=l1l2l2
3sin2γF
12EsI(l1+l2)
δs
CO =δs
bCO +δs
rCO =F l2
3sin2γ
12EsIl3+l1l2
l1+l2
F
γs=τ(l1cos αl2cos β)l2
3sin2γ
Est3Lyl3+l1l2
l1+l2
G12U=τ
γs
=Est3Ly
l2
3sin2γ(l1cos αl2cos β)l3+l1l2
l1+l2
G12ULy
GI
12U
Ljjth
GI
12U
τ
G12U
Ly=τ
GI
12U
Lj
GI
12U=G12U
Lj
Ly
τ
G12eq τ jth
τBj=
m
X
i=1
τij Bij
ˆ
G12jγjBj=
m
X
i=1
GI
12U ij γij Bij
ˆ
G12jjth γjγij
jth jth GI
12Uij
i j
γj=γij i= 1,2, ..., m jth
ˆ
G12j=
m
P
i=1
GI
12U ij Bij
Bj
τ
n
γgL=
n
X
j=1
γjLj
γgG12
τ
G12eq
L=
n
X
j=1
τj
ˆ
G12j
Lj
τ=τj
G12eq =L
n
P
j=1
LjBj
m
P
i=1
GI
12U ij Bij
GI
12Uij
G12eq =LEst3
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
ν12
ν12
ν12U
ν12U=0
2
0
1
0
10
2
0
1(= h
1)
δ0
AO =P l3
1sin α
12EsICl3
1cos α
12EsIcos α
P C
δ0
BO =P l3
2sin β
12EsICl3
1cos β
12EsIcos β
δ0
O=δ0
AO(l2cos β) + δ0
BO (l1cos α)
l1cos αl2cos β
=σ1l2
1l2
2Ly(l1+l2) cos αcos β(cos αsin βsin αcos β)
Est3(l1cos αl2cos β)2
0
2=σ1l2
1l2
2(l1+l2) cos αcos β(cos αsin βsin αcos β)
Est3(l1cos αl2cos β)2
ν12U=cos αcos β(l1cos αl2cos β)
(cos αsin βsin αcos β)Ly
ν12ULy
νI
12U
Ljjth
νI
12U0
1Bij =0I
1Bij
0
2Ly=0I
2Lj(.)I
νI
12U=ν12U
Ly
Lj
ν12
ν12eq νI
12U
ν12eq σ1
σ1jth
ν12
2j
ˆν12j
Bj=
m
X
i=1
2ij Bij
νI
U12ij
2j2ij jth jth
νI
U12ij
i j ˆν12jjth
2j=2ij i= 1,2, ..., m jth
ˆν12j=Bj
m
P
i=1
Bij
νI
12Uij
σ1
n
g12
2L=
n
X
j=1
2jLj
g12
2ν12
ν12eqg12
1L=
n
X
j=1
ν12j1jLj
g12
11jjth
g12
1=1jj= 1,2, ..., n
ν12eq =1
L
n
X
j=1
BLj
m
P
i=1
Bij
νI
12Uij
νI
12Uij
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
ν21
ν21
ν21U
ν21U=00
1
00
2
00
100
2
00
2(= v
2)
00
1
δ00
AO =W1l3
1cos αsin α
12EsI
δ00
BO =W2l3
2cos βsin β
12EsI
W1W2
δ00
1=δ00
AO
δO
l1sin α+δ00
BO
δO
l2sin β
=W l2
1l2
2(l1+l2) cos αcos β(cos αsin βsin αcos β)
12EsI(l1cos αl2cos β)2
δO
00
1=σ2l2
1l2
2(l1+l2) cos αcos β(cos αsin βsin αcos β)
Est3(l1cos αl2cos β)2
ν21U=Lyl2
1l2
2(l1+l2) cos αcos β(cos αsin βsin αcos β)
(l1cos αl2cos β)3l2
3cos2γl3+l1l2
l1+l2+l2
1l2
2(l1+l2) cos2αcos2β
(l1cos αl2cos β)2
ν21ULy
νI
21U
Ljjth
νI
21U00
1Bij =00I
1Bij
00
2Ly=00I
2Lj(.)I
νI
21U=ν21U
Lj
Ly
ν21
ν21eq νI
21U
σ2ν21eq
E2eq σ2jth
jth
1jBj=
m
X
i=1
1ij Bij
ν21
ˆν21j2jBj=
m
X
i=1
νI
21U ij 2ij Bij
ˆν21jjth 2j2ij
jth jth
νI
21Uij
i j 2j=2ij i= 1,2, ..., m jth
ˆν21j=
m
P
i=1
νI
21U ij Bij
Bj
σ2
n
g21
2L=
n
X
j=1
2jLj
g21
2ν21
g21
1
ν21eq
L=
n
X
j=1
1j
ˆν21j
Lj
g21
1
g21
1=1jj= 1,2, ..., n
ν21eq =L
n
P
j=1
Bj
m
P
i=1
νI
21U ij Bij
Ly
νI
21Uij
ν21eq =L
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij )
m
P
i=1
l2
1ij l2
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
βij >90
GA
L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180θ γij = 90
i j
θ= 30
E
1
Es
=E
2
Es
= 2.3t
l3
E
1E
2
E1E2G12 ν12 ν21
E1GA =
Est
l3cos θ
(h
l+sin θ) sin2θ
E2GA =Est
l3(h
l+sin θ)
cos3θ
G12GA =
Est
l3(h
l+sin θ)
(h
l)2(1+2 h
l) cos θ
ν12GA =cos2θ
(h
l+sin θ) sin θ
ν21GA =(h
l+sin θ) sin θ
cos2θ
E1eq =Est3
L
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij )
m
P
i=1
l2
1ij l2
2ij (l1ij +l2ij ) (cos αij sin βij sin αij cos βij )2
(l1ij cos αij l2ij cos βij )2
E2eq =LEst3
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1 l2
3ij cos2γij l3ij +l1ijl2ij
l1ij+l2ij +l2
1ijl2
2ij(l1ij +l2ij ) cos2αij cos2βij
(l1ij cos αijl2ij cos βij )21
G12eq =LEst3
n
P
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij)
m
P
i=1 l2
3ij sin2γij l3ij +l1ijl2ij
l1ij+l2ij 1
ν12eq =1
L
n
P
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 +l1ijl2ij
l1ij+l2ij +l2
1ijl2
2ij(l1ij +l2ij ) cos2αij cos2βij
(l1ij cos αijl2ij cos βij )2
θ= 30
G
12
Es
= 0.57 t
l3
G
12
E
2ν
12 =E
1ν
21 ν
12 ν
21
ν
12 =ν
21 = 1 G=E/2(1 + ν)
E G ν
ith jth
E1Uij =Esij t3(l1ij cos αij l2ij cos βij )3
Lyl2
1ij l2
2ij (l1ij +l2ij ) (cos αij sin βij sin αij cos βij )2
Esij ith jth
E1eq =t3
L
n
X
j=1
m
P
i=1
(l1ij cos αij l2ij cos βij )
m
P
i=1
l2
1ij l2
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
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
L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180θ γij = 90i j
E1eq =κ1t
l3cos θ
(h
l+ sin θ) sin2θ
E2eq =κ2t
l3(h
l+ sin θ)
cos3θ
G12eq =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
κ2=n
m
1
n
P
j=1
1
m
P
i=1
Esij
ν12eq =ν12GA ν21eq =ν21GA
Esij Esij =Es
i= 1,2,3, ..., m j = 1,2,3, ..., n κ1κ2Es
30h/l = 1
l1l2l3α β γ t
L
Es
r
l1l2l3α β γ
r
r
h=l θ = 30rd= 0.257l r = 8
h l
rd= 0.257l
h/l t/l 102
h/l
θ◦ ◦
Esρ3
E1E2ν12 ν21 G12
¯
E1=E1eq
Esρ3¯
E2=E2eq
Esρ3¯ν12 =ν12eq ¯ν21 =ν21eq ¯
G12 =G12eq
Esρ3(¯)
ρ
θ= 30;h/l = 1
θ
1.3 1.35 1.4 1.45 1.5 1.55 1.6
0
2
4
6
8
10
12
14
¯
E1
Density
Analytical
F EM
E1
θ= 30h/l = 1 r= 8
θ
E1
E1
E1
r
h/l = 1 h/l = 1.5
h/l = 2
E1θ= 30
h/l = 1 h/l = 1.5
h/l = 2
E1θ= 45
h/l = 1 h/l = 1.5
h/l = 2
E1θ= 60
E1
E2ν12 ν21
G12
E1G12
θ E1E2
ν12 ν21 G12
h/l
% % E1E2%
h/l = 1 h/l = 1.5
h/l = 2
E2θ= 30
h/l = 1 h/l = 1.5
h/l = 2
E2θ= 45
h/l = 1 h/l = 1.5
h/l = 2
E2θ= 60
G12 % % ν12 ν21
m
mr
h/l = 1 h/l = 1.5
h/l = 2
G12 θ= 30
h/l = 1 h/l = 1.5
h/l = 2
G12 θ= 45
h/l = 1 h/l = 1.5
h/l = 2
G12 θ= 60
h/l = 1 h/l = 1.5
h/l = 2
ν12 θ= 30
h/l = 1 h/l = 1.5
h/l = 2
ν12 θ= 45
h/l = 1 h/l = 1.5
h/l = 2
ν12 θ= 60
θ
E1E2
G12
h/l = 1 h/l = 1.5
h/l = 2
ν21 θ= 30
h/l = 1 h/l = 1.5
h/l = 2
ν21 θ= 45
h/l = 1 h/l = 1.5
h/l = 2
ν21 θ= 60
E2ν21 G12
E1ν12
E1E2
ν12 ν21 G12
γ
%
&
... Various methods have been presented in the literature to determine the elastic properties 6-10 of the lattice structure. Variation in the effective elastic moduli due to the presence of the irregularities in the passive lattice structures 11 have been investigated extensively. A few experimental studies 12,13 have also been conducted to understand the behaviour of the lattice structures. ...
... The stiffness matrix given in Equation (11) has been obtained by considering 4 DOF, as we have neglected the axial deformation. The deformation of the hybrid beam can be obtained by using the stiffness matrix given by Equation (11). ...
... The stiffness matrix given in Equation (11) has been obtained by considering 4 DOF, as we have neglected the axial deformation. The deformation of the hybrid beam can be obtained by using the stiffness matrix given by Equation (11). The obtained results have been compared with the existing results from the literature. ...
Article
2-D lattice structures have gained significant attention in the last few decades. Extensive analytical and experimental studies have been conducted to determine the elastic properties of the lattice structures. Further, the variation in the elastic properties of the passive lattice structures by changing various dimensional parameters and geometry have also been studied. However, once manufactured, it is impossible to vary the elastic properties of these lattice structures. A few studies have been conducted to understand the modulation of the elastic properties in symmetric hybrid lattice structures. This article proposes a geometrically asymmetric hybrid lattice structure having piezoelectric material on the opposite faces (top and bottom) of the consecutive inclined cell walls, respectively. The closed-form expressions have been derived by considering a bottom-up approach neglecting the axial deformation of the cell walls. Young's modulus has emerged to be a function of externally applied voltage, warranting control of the elastic properties of the structure even after manufacturing. In contrast, Poisson's ratio is independent of externally applied voltage. The transition from negative to positive values for Young's modulus has also been observed at specific cell angle values and externally applied voltage to stress ratio. This study intends to provide the basic framework for voltage-dependent elastic properties in asymmetric lattice structures for potential use in various futuristic multi-functional structural systems and devices across different length scales.
... Another broad aspect of research on metamaterials is the quantification of spatial irregularities [32] and estimation of effective physical properties including its effect. The practically inevitable spatial irregularities exist in lattices due to manufacturing uncertainty, microstructural defects, pre-stressing, etc. [33]. Voronoi geometries are one of the popular configurations studied in literature to model spatial irregularity in honeycombs. ...
... Most of such studies are experimental and purely numerical (finite element), which suffer from a lack of physical insights in addition to high computational time and cost. A few analytical studies providing an in-depth understanding of the physics of disorder and irregularities are found in recent literature [32,33]. ...
Article
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Over the last decade, lattice-based artificial materials have demonstrated the possibility of tailoring multifunctional capabilities that are not achievable in traditional materials. While a large set of mechanical properties can be simultaneously modulated by adopting an appropriate network architecture in the conventional periodic lattices, the prospect of enhancing global specific stiffness and failure strength 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 and failure strength that can be obtained only by lattice-level geometries. Here we propose a novel concept of anti-curvature in the design of lattice materials, which reveals a dramatic capability in terms of enhancing the elastic failure strength in the nonlinear regime while keeping the relative density unaltered. A semi-analytical bottom-up framework is developed for estimating the onset of failure in honeycomb lattices with the anti-curvature effect in cell walls considering geometric nonlinearity under large deformation. The physically insightful semi-analytical model captures nonlinearity in elastic failure strength of anti-curvature lattices as a function of the degree of curvature and applied stress together with conventional microstructural and intrinsic material properties.
... Literature exemplifies a number of works concerning the mechanics of lattice materials with periodic (crystalline) geometries. However, only a limited number of studies examined disordered or non-periodic architectures [28][29][30][31][32][33][34]. Hence, the present study focuses on lattices with constant node connectivity, with a large range of beam aspect ratios and levels of geometrical disorder. ...
... In bending-dominated lattice (honeycomb-based lattice), increasing disorder stiffens the material and increases E/(s/ 0 ) 3 by ∼ 40% for the maximum disorder (u = 0.45 0 ), see Fig. 12(a). This variation is qualitatively consistent but quantitatively more pronounced than what is reported in the literature [29,32,44,45]. The root cause of these differences lies with how disordering occurs, by displacing randomly the points, the Voronoi tessellation of which provides the initial periodic honeycomb lattice. ...
Article
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We examine how disordering joint position influences the linear elastic behavior of lattice materials via numerical simulations in two-dimensional beam networks. Three distinct initial crystalline geometries are selected as representative of mechanically isotropic materials with low connectivity, mechanically isotropic materials with high connectivity, and mechanically anisotropic materials with intermediate connectivity. Introducing disorder generates spatial fluctuations in the elasticity tensor at the local (joint) scale. Proper coarse-graining reveals a well-defined continuum-level scale elasticity tensor. Increasing disorder aids in making initially anisotropic materials more isotropic. The disorder impact on the material stiffness depends on the lattice connectivity: Increasing the disorder softens lattices with high connectivity and stiffens those with low connectivity, without modifying the scaling between elastic modulus and density (linear scaling for high connectivity and cubic scaling for low connectivity). Introducing disorder in lattices with intermediate fixed connectivity reveals both scaling: the linear scaling occurs for low density, the cubic one at high density, and the crossover density increases with disorder. Contrary to classical formulations, this work demonstrates that connectivity is not the sole parameter governing elastic modulus scaling. It offers a promising route to access novel mechanical properties in lattice materials via disordering the architectures.
... Literature exemplifies a number of works concerning the mechanics of lattice materials with periodic (crystalline) geometries. However, only a limited number of studies examined disordered or non-periodic architectures [28][29][30][31][32][33][34]. Hence, the present study focuses on lattices with constant node connectivity, with a large range of beam aspect ratios and levels of geometrical disorder. ...
... In bending-dominated lattice (honeycomb-based lattice), increasing disorder stiffens the material and increases E/(s/ 0 ) 3 by ∼ 40% for the maximum disorder (u = 0.45 0 ), see Fig. 12(a). This variation is qualitatively consistent but quantitatively more pronounced than what is reported in the literature [29,32,44,45]. The root cause of these differences lies with how disordering occurs, by displacing randomly the points, the Voronoi tessellation of which provides the initial periodic honeycomb lattice. ...
Preprint
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We examine how disordering joint position influences the linear elastic behavior of lattice materials via numerical simulations in two-dimensional beam networks. Three distinct initial crystalline geometries are selected as representative of mechanically isotropic materials low connectivity, mechanically isotropic materials with high connectivity, and mechanically anisotropic materials with intermediate connectivity. Introducing disorder generates spatial fluctuations in the elasticity tensor at the local (joint) scale. Proper coarse-graining reveals a well-defined continuum-level scale elasticity tensor. Increasing disorder aids in making initially anisotropic materials more isotropic. The disorder impact on the material stiffness depends on the lattice connectivity: Increasing the disorder softens lattices with high connectivity and stiffens those with low connectivity, without modifying the scaling between elastic modulus and density (linear scaling for high connectivity and cubic scaling for low connectivity). Introducing disorder in lattices with intermediate fixed connectivity reveals both scaling: the linear scaling occurs for low density, the cubic one at high density, and the crossover density increases with disorder. Contrary to classical formulations, this work demonstrates that connectivity is not the sole parameter governing elastic modulus scaling. It offers a promising route to access novel mechanical properties in lattice materials via disordering the architectures.
... Quantication of spatial irregularities and estimation of eective physical properties, including their eect, is another signicant area of research on metamaterials [44]. Due to manufacturing uncertainty, microstructural aws, pre-stressing, and other factors, spatial irregularities in lattices are practically unavoidable [45]. One of the most often studied congurations for modeling spatial irregularity in honeycombs is Voronoi geometries. ...
... The majority of this research is experimental and solely numerical (nite element) and thus suers from a lack of physical insight and high computational times and costs. Recent literature has a few analytical investigations that provide an in-depth understanding of the physics of disorder and irregularity [44,45]. Because mechanical metamaterials include scale-free mechanics of periodic forms over a wide range of length scales (nano, micro, and macro), the research area of eective mechanical property estimation is relevant to a wide range of structures, from macro-level (such as honeycomb cores in sandwich structures) to nanomaterials with regular honeycomb-like congurations (such as graphene and hBN) and microstructures of various woods and bones [54,55,56]. ...
Article
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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.
... Physics based numerical modelling approaches such as finite element method are also quite cumbersome since each new configuration of the composite requires a fresh modelling, meshing and analysis. However, the assistance of focused computational solutions that learn the provided data and modify the configurations accordingly can help in accommodating solutions to complex analytical J o u r n a l P r e -p r o o f problems [13]. Artificial Intelligence (AI) has been already used in past for optimizing the composite designs based on the calculations of Finite Element Method (FEM) for different loading conditions and various forms of static and dynamic analyses [14][15][16]. ...
Research
Microstructural image based convolutional neural networks for efficient prediction of full-field stress maps in short fiber polymer composites Abstract The increased demand for superior materials has highlighted the need of investigating the mechanical properties of composites to achieve enhanced constitutive relationships. Fiber-reinforced polymer composites have emerged as an integral part of materials development with tailored mechanical properties. However, the complexity and heterogeneity of such composites make it considerably more challenging to have precise quantification of properties and attain an optimal design of structures through experimental and computational approaches. In order to avoid the complex, cumbersome, and labor-intensive experimental and numerical modeling approaches, a machine learning (ML) model is proposed here such that it takes the microstructural image as input with a different range of Young's modulus of carbon fibers and neat epoxy, and obtains output as visualization of the stress component S 11 (principal stress in the x-direction). For obtaining the training data of the ML model, a short carbon fiber-filled specimen under quasi-static tension is modeled based on 2D Representative Area Element (RAE) using finite element analysis. The composite is inclusive of short carbon fibers with an aspect ratio of 7.5 that are infilled in the epoxy systems at various random orientations and positions generated using the Simple Sequential Inhibition (SSI) process. The study reveals that the pix2pix deep learning Convolutional Neural Network (CNN) model is robust enough to predict the stress fields in the composite for a given arrangement of short fibers filled in epoxy over the specified range of Young's modulus with high accuracy. The CNN model achieves a correlation score of about 0.999 and L2 norm of less than 0.005 for a majority of the samples in the design spectrum, indicating excellent prediction capability. In this paper, we have focused on the stage-wise chronological development of the CNN model with optimized performance for predicting the full-field stress maps of the fiber-reinforced composite specimens. The development of such a robust and efficient algorithm would significantly reduce the amount of time and cost required to study and design new composite materials through the elimination of numerical inputs by direct microstructural images.
... Physics based numerical modelling approaches such as finite element method are also quite cumbersome since each new configuration of the composite requires a fresh modelling, meshing and analysis. However, the assistance of focused computational solutions that learn the provided data and modify the configurations accordingly can help in accommodating solutions to complex analytical problems [13]. Artificial Intelligence (AI) has been already used in past for optimizing the composite designs based on the calculations of Finite Element Method (FEM) for different loading conditions and various forms of static and dynamic analyses [14][15][16]. ...
Article
Full-text available
The increased demand for superior materials has highlighted the need of investigating the mechanical properties of composites to achieve enhanced constitutive relationships. Fiber-reinforced polymer composites have emerged as an integral part of materials development with tailored mechanical properties. However, the complexity and heterogeneity of such composites make it considerably more challenging to have precise quantification of properties and attain an optimal design of structures through experimental and computational approaches. In order to avoid the complex, cumbersome, and labor-intensive experimental and numerical modeling approaches, a machine learning (ML) model is proposed here such that it takes the microstructural image as input with a different range of Young’s modulus of carbon fibers and neat epoxy, and obtains output as visualization of the stress component S11 (principal stress in the x-direction). For obtaining the training data of the ML model, a short carbon fiber-filled specimen under quasi-static tension is modeled based on 2D Representative Area Element (RAE) using finite element analysis. The composite is inclusive of short carbon fibers with an aspect ratio of 7.5 that are infilled in the epoxy systems at various random orientations and positions generated using the Simple Sequential Inhibition (SSI) process. The study reveals that the pix2pix deep learning Convolutional Neural Network (CNN) model is robust enough to predict the stress fields in the composite for a given arrangement of short fibers filled in epoxy over the specified range of Young’s modulus with high accuracy. The CNN model achieves a correlation score of about 0.999 and L2 norm of less than 0.005 for a majority of the samples in the design spectrum, indicating excellent prediction capability. In this paper, we have focused on the stage-wise chronological development of the CNN model with optimized performance for predicting the full-field stress maps of the fiber-reinforced composite specimens. The development of such a robust and efficient algorithm would significantly reduce the amount of time and cost required to study and design new composite materials through the elimination of numerical inputs by direct microstructural images.
... A suitable computational model must be integrated with statistical uncertainty quantification and propagation methods to account for the inherent uncertainty in honeycombs [35,36]. As demonstrated in Fig. 1, the honeycombs have a multiscale hierarchical structure. ...
Article
The honeycomb manufacturing process involves complex and inter-related procedures spanning multiple physics domains and scales. The geometry heterogeneity in honeycomb cells inevitably exists and propagates to the uncertainty of the honeycomb’s mechanical performance. Quantitatively characterizing the effect of cell geometry variability on the effective mechanical properties of honeycomb hence becomes critically important. In this study, a Fast Fourier Transform (FFT) based method was developed to predict the effective in-plane elastic properties of irregular hexagonal honeycomb. The honeycomb geometry heterogeneity was identified with a digital image technology by examining a large number of cells. Correlation analysis on the resulting statistical data enables determining the interdependence of cell wall lengths and angles. Using a Representative Unit Cell (RUC) with varying cell wall lengths and angles, an analytical function for the effective in-plane elastic parameters was established by incorporating the dependencies. The statistical quantification of cell geometry and the propagation of RUC’s elastic characteristics were further integrated into the FFT-based multiscale framework for predicting the in-plane elastic moduli of irregular honeycomb. The tensile and shear tests on the irregular honeycombs, together with the corresponding finite element analysis, were used to verify the proposed method. Both experimental and numerical results confirmed that the uncertainty of elastic moduli achieved by the FFT-based method provides a reasonable estimation.
... Based on this study we have decided the converged number of unit cells for any further analysis (as discussed later in this paper). [43], (b) honeycomb microstructure of fluorinated Si surface [44], (c) top SEM view of honeycomb structured lithiated silicon [45], (d) lightweight heat insulation bricks for fire resistance (e) the architecture of Hex tower, (f) microstructural view of natural cork [8], (g) hexagonal configurations of 2D materials in nano-scale [12,92,94], (h) SEM image of TiO2 surface after photoelectrochemical etching under intense (+1.0 V) anodic polarization [46], (i) typical silica micro honeycomb with channel structure [47], (j) honeycomb sandwiched panels for lightweight structural appliocations, (k) honeycomb mirror for James Webb telescope, (l) honeycomb network of anthraquinone molecules with open pores [48], (m) typical microstructural view of osteoporosis bone [49], (n) schematic illustrations depicting the accordion-like honeycomb form made up of two overlappings [50], (o) topologies of sintered and extruded ceramic honeycombs [51], (p) airless tires having a honeycomb core designed for obstacles like hot spots, curbs, rocks and other terrains, (q) SEM images of poly-dimethylsiloxane (PDMS) star polymer sheet with honeycomb architecture [52], (r) honeycomb structured viable building unit [52], (s) honeycomb seals for turbine, (t) modular hexagonal furniture. ...
Article
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Honeycomb lattices exhibit remarkable structural properties and novel functionalities, such as high specific energy absorption, excellent vibroacoustic properties, and tailorable specific strength and stiffness. A range of modern structural applications demands for maximizing the failure strength and energy absorption capacity simultaneously with the minimum additional weight of material to the structure. To this end, conventional approaches of designing the periodic microstructural geometry have possibly reached to a saturation point. This creates a strong rationale in this field to exploit the recent advances in artificial intelligence and machine learning for further enhancement in the mechanical performance of artificially engineered lattice structures. Here we propose to strengthen the lattice structure locally by identifying the failure pattern through the emerging capabilities of machine learning. We have developed a Gaussian Process Regression (GPR) assisted surrogate modelling algorithm, supported by finite element simulations, for the prediction of failure bands in lattice structures. Subsequently, we strengthen the identified failure bands locally instead of adopting a global strengthing approach to optimize the material utilization and lattice density. A range of sequential local strengthening schemes is explored logically, among which the schemes having localized gradation by increasing the elastoplastic properties and lowering Young's modulus of the intrinsic material lead to an increase up to 37.54% in the failure stress of the lattice structure along with 32.12% increase in energy absorption. The comprehensive numerical results presented in this paper convincingly demonstrate the abilities of machine learning in material microstructure design for enhancing failure strength and energy absorption capacity simultaneously when it is coupled with the physics-based understanding of material and structural behavior.
... The point of transition for unit-cells in a column or row can be gradually varied by designing meta-sheets with a graded microstructure. For lattice-based cellular metamaterials [69,70], a lack of mixed-mode deformation with a transition point directly correlates to the unlikeliness of a programmable stiness having sudden jumps in the value. Scientic studies [4] have been performed to obtain programmable stiness through contact and structural deformation. ...
Article
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This paper develops kirigami-inspired modular materials with programmable deformation-dependent stiffness and multidirectional auxeticity. Mixed-mode deformation behaviour of the proposed metastructure involving both rigid origami motion and structural deformation has been realized through analytical and computational analyses, supported by elementary-level qualitative physical experiments. It is revealed that the metamaterial can transition from a phase of low stiffness to a contact-induced phase that brings forth an extensive rise in stiffness with programmable features during the deformation process. Transition to the contact phase as a function of far-field global deformation can be designed through the material's microstructure. A deformation-dependent mixed-mode Poisson’s ratio can be achieved with the capability of transition from positive to negative values in both in-plane and out-of-plane directions, wherein it can further be programmed to have a wide-ranging auxeticity as a function of the microstructural geometry. We have demonstrated that uniform and graded configurations of multi-layer tessellated material can be developed to modulate the constitutive law of the metastructure with augmented programmability as per application-specific demands. Since the fundamental mechanics of the proposed kirigami-based metamaterial is scale-independent, it can be directly utilized for application in multi-scale systems, ranging from meter-scale transformable architectures and energy storage systems to micrometer-scale electro-mechanical systems.
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A generalized analytical approach is presented to derive closed-form formulae for the elastic moduli of hexagonal multiplanar nano-structures. Hexagonal nano-structural forms are common for various materials. Four different classes of materials (single layer) from a structural point of view are proposed to demonstrate the validity and prospective application of the developed formulae. For example, graphene, an allotrope of carbon, consists of only carbon atoms to form a honeycomb like hexagonal lattice in a single plane, while hexagonal boron nitride (hBN) consists of boron and nitrogen atoms to form the hexagonal lattice in a single plane. Unlike graphene and hBN, there are plenty of other materials with hexagonal nano-structures that have the atoms placed in multiple planes such as stanene (consists of only Sn atoms) and molybdenum disulfide (consists of two different atoms: Mo and S). The physics based high-fidelity analytical model developed in this article are capable of obtaining the elastic properties in a computationally efficient manner for wide range of such materials with hexagonal nano-structures that are broadly classified in four classes from structural viewpoint. Results are provided for materials belonging to all the four classes, wherein a good agreement between the elastic moduli obtained using the proposed formulae and available scientific literature is observed.
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The effect of stochasticity in mechanical behaviour of metamaterials is quantified in a probabilistic framework. The stochasticity has been accounted in the form of random material distribution and structural irregularity, which are often encountered due to manufacturing and operational uncertainties. An analytical framework has been developed for analysing the effective stochastic in-plane elastic properties of irregular hexagonal structural forms with spatially random variations of cell angles and intrinsic material properties. Probabilistic distributions of the in-plane elastic moduli have been presented considering both randomly homogeneous and randomly inhomogeneous stochasticity in the system, followed by an insightful comparative discussion. The ergodic behaviour in spatially irregular lattices is investigated as a part of this study. It is found that the effect of random micro-structural variability in structural and material distribution has considerable influence on mechanical behaviour of metamaterials.
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We investigate the dependency of strain rate, temperature and size on yield strength of hexagonal close packed (HCP) nanowires based on large-scale molecular dynamics (MD) simulation. A variance-based analysis has been proposed to quantify relative sensitivity of the three controlling factors on the yield strength of the material. One of the major drawbacks of conventional MD simulation based studies is that the simulations are computationally very intensive and economically expensive. Large scale molecular dynamics simulation needs supercomputing access and the larger the number of atoms, the longer it takes time and computational resources. For this reason it becomes practically impossible to perform a robust and comprehensive analysis that requires multiple simulations such as sensitivity analysis, uncertainty quantification and optimization. We propose a novel surrogate based molecular dynamics (SBMD) simulation approach that enables us to carry out thousands of virtual simulations for different combinations of the controlling factors in a computationally efficient way by performing only few MD simulations. Following the SBMD simulation approach an efficient optimum design scheme has been developed to predict optimized size of the nanowire to maximize the yield strength. Subsequently the effect of inevitable uncertainty associated with the controlling factors has been quantified using Monte Carlo simulation. Though we have confined our analyses in this article for Magnesium nanowires only, the proposed approach can be extended to other materials for computationally intensive nano-scale investigation involving multiple factors of influence.
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A novel computationally efficient damage identification methodology for web core fiber-reinforced polymer composite bridges has been developed in this article based on multivariate adaptive regression splines in conjunction with a multi-objective goal-attainment optimization algorithm. The proposed damage identification methodology has been validated for several single and multiple damage cases. The performance of the efficient multivariate adaptive regression splines-based approach for the inverse system identification process is found to be quite satisfactory. An iterative scheme in conjunction with the multi-objective optimization algorithm coupled with multivariate adaptive regression splines is proposed to increase damage identification accuracy. The effect of noise on the proposed damage identification algorithm has also been addressed subsequently using a probabilistic framework. The multivariate adaptive regression splines-based damage identification algorithm is general in nature; therefore, in future it can be implemented to other structures.
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Materials that become thicker when stretched and thinner when compressed are the subject of this review. The theory behind the counterintuitive behavior of these so-called auxetic materials is discussed, and examples and applications are examined. For example, blood vessels made from an auxetic material will tend to increase in wall thickness (rather than decrease) in response to a pulse of blood, thus preventing rupture of the vessel (see Figure).
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This paper presents a concise state-of-the-art review along with an exhaustive comparative investigation on surrogate models for critical comparative assessment of uncertainty in natural frequencies of composite plates on the basis of computational efficiency and accuracy. Both individual and combined variations of input parameters have been considered to account for the effect of low and high dimensional input parameter spaces in the surrogate based uncertainty quantification algorithms including the rate of convergence. Probabilistic characterization of the first three stochastic natural frequencies is carried out by using a finite element model that includes the effects of transverse shear deformation based on Mindlin’s theory in conjunction with a layer-wise random variable approach. The results obtained by different metamodels have been compared with the results of traditional Monte Carlo simulation (MCS) method for high fidelity uncertainty quantification. The crucial issue regarding influence of sampling techniques on the performance of metamodel based uncertainty quantification has been addressed as an integral part of this article.
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This paper presents a stochastic approach to study the natural frequencies of thin-walled laminated composite beams with spatially varying matrix cracking damage in a multi-scale framework. A novel concept of stochastic representative volume element (SRVE) is introduced for this purpose. An efficient radial basis function (RBF) based uncertainty quantification algorithm is developed to quantify the probabilistic variability in free vibration responses of the structure due to spatially random stochasticity in the micro-mechanical and geometric properties. The convergence of the proposed algorithm for stochastic natural frequency analysis of damaged thin-walled composite beam is verified and validated with original finite element method (FEM) along with traditional Monte Carlo simulation (MCS). Sensitivity analysis is carried out to ascertain the relative influence of different stochastic input parameters on the natural frequencies. Subsequently the influence of noise is investigated on radial basis function based uncertainty quantification algorithm to account for the inevitable variability for practical field applications. The study reveals that stochasticity/ system irregularity in structural and material attributes affects the system performance significantly. To ensure robustness, safety and sustainability of the structure, it is very crucial to consider such forms of uncertainties during the analysis.
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The present computational study investigates on stochastic natural frequency analyses of laminated composite curved panels with cutout based on support vector regression (SVR) model. The SVR based uncertainty quantification (UQ) algorithm in conjunction with Latin hypercube sampling is developed to achieve computational efficiency. The convergence of the present algorithm for laminated composite curved panels with cutout is validated with original finite element (FE) analysis along with traditional Monte Carlo simulation (MCS). The variations of input parameters (both individual and combined cases) are studied to portray their relative effect on the output quantity of interest. The performance of the SVR based uncertainty quantification is found to be satisfactory in the domain of input variables in dealing low and high dimensional spaces. The layer-wise variability of geometric and material properties are included considering the effect of twist angle, cutout sizes and geometries (such as cylindrical, spherical, hyperbolic paraboloid and plate). The sensitivities of input parameters in terms of coefficient of variation are enumerated to project the relative importance of different random inputs on natural frequencies. Subsequently, the noise induced effects on SVR based computational algorithm are presented to map the inevitable variability in practical field of applications.
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In this paper we show that a 1-D phononic crystal (laminate) can exhibit metamaterial wave phenomena which are traditionally associated with 2- and 3-D crystals. Moreover, due to the absence of a length scale in 2 of its dimensions, it can outperform higher dimensional crystals on some measures. This includes allowing only negative refraction over large frequency ranges and serving as a near-omnidirectional high-pass filter up to a large frequency value. First we provide a theoretical discussion on the salient characteristics of the dispersion relation of a laminate and formulate the solution of an interface problem by the application of the normal mode decomposition technique. We present a methodology with which to induce a pure negative refraction in the laminate. As a corollary to our approach of negative refraction, we show how the laminate can be used to steer beams over large angles for small changes in the incident angles (beam steering). Furthermore, we clarify how the transmitted modes in the laminate can be switched on and off by varying the angle of the incident wave by a small amount. Finally, we show that the laminate can be used as a remarkably efficient high-pass frequency filter. An appropriately designed laminate will reflect all plane waves from quasi-static to a large frequency, incident at it from all angles except for a small set of near-normal incidences. This will be true even if the homogeneous medium is impedance matched with the laminate. Due to the similarities between SH waves and electromagnetic (EM) waves it is expected that some or all of these results may also apply to EM waves in a layered periodic dielectric.