Dynamic Homogenisation of randomly irregular viscoelastic metamaterials

Dynamic homogenisation of randomly
irregular viscoelastic metamaterials
S. Adhikari1
, T. Mukhopadhyay2
, A. Batou3
1
Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea
University, Bay Campus, Swansea, Wales, UK, Email: S.Adhikari@swansea.ac.uk, Twitter:
@ProfAdhikari, Web: http://engweb.swan.ac.uk/~adhikaris
1
Department of Engineering Science, University of Oxford, Oxford, UK
3
Liverpool Institute for Risk and Uncertainty, University of Liverpool, Liverpool, UK
University of Texas at Austin: Swansea-Texas Strategic
Partnership
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 1

Swansea University

My research interests
• Development of fundamental computational methods for
structural dynamics and uncertainty quantiﬁcation
A. Dynamics of complex systems
B. Inverse problems for linear and non-linear dynamics
C. Uncertainty quantiﬁcation in computational mechanics
• Applications of computational mechanics to emerging
multidisciplinary research areas
D. Vibration energy harvesting / dynamics of wind turbines
E. Computational mechanics for mechanics and multi-scale
systems

Outline
1 Introduction
Regular lattices
Irregular lattices
2 Formulation for the viscoelastic analysis
3 Equivalent elastic properties of randomly irregular lattices
4 Effective properties of irregular lattices: uncorrelated
uncertainty
General results - closed-form expressions
Special case 1: Only spatial variation of the material properties
Special case 2: Only geometric irregularities
Special case 3: Regular hexagonal lattices
5 Effective properties of irregular lattices: correlated
uncertainty
6 Results and discussions
Spatially correlated irregular elastic lattices
Viscoelastic properties of regular lattices
Spatially correlated irregular viscoelastic lattices
7 Conclusions

Lattice based metamaterials
• Metamaterials are artiﬁcial materials designed to outperform
naturally occurring materials in various fronts. These include, but
are not limited to, electromagnetics, acoustics, optics, terahertz,
infrared, dynamics and mechanical properties.
• Lattice based metamaterials are abundant in man-made and
natural systems at various length scales
• Lattice based metamaterials are made of periodic
identical/near-identical geometric units

Hexagonal lattices in 2D
• Among various lattice geometries (triangle, square, rectangle,
pentagon, hexagon), hexagonal lattice is most common (note
that hexagon is the highest “space ﬁlling” pattern in 2D).
• This talk is about in-plane elastic properties of 2D hexagonal
lattice structures - commonly known as “honeycombs”

Lattice structures - nano scale
Illustrations of a single layer graphene sheet and a born nitride nano sheet

Lattice structures - nature
Top left: cork, top right: balsa, next down left: sponge, next down right: trabecular bone, next down left: coral, next down right: cuttleﬁsh bone, bottom le
leaf tissue, bottom right: plant stem, third column - epidermal cells (from web.mit.edu)

Some questions of general interest
• Shall we consider lattices as “structures” or “materials” from a
mechanics point of view?
• At what relative length-scale a lattice structure can be
considered as a material with equivalent elastic properties?
• In what ways structural irregularities “mess up” equivalent elastic
/ viscoelastic properties? Can we evaluate it in a quantitative as
well as in a qualitative manner?
• What is the consequence of random structural irregularities on
the homogenisation approach in general? Can we obtain
statistical measures?
• How can we efﬁciently compute equivalent elastic / viscoelastic
properties of random lattice structures?

Regular lattice structures
• Hexagonal lattice structures have been modelled as a continuous
solid with an equivalent elastic moduli throughout its domain.
• This approach eliminates the need of detail ﬁnite element
modelling of lattices in complex structural systems like sandwich
structures.
• Extensive amount of research has been carried out to predict the
equivalent elastic / viscoelastic properties of regular lattices
consisting of perfectly periodic hexagonal cells (El-Sayed et al.,
1979; Gibson and Ashby, 1999; Goswami, 2006; Masters and
Evans, 1996; Zhang and Ashby, 1992).
• Analysis of two dimensional hexagonal lattices dealing with
in-plane elastic properties are commonly based on an unit cell
approach, which is applicable only for perfectly periodic cellular
structures.
• For the dynamic analysis of perfectly periodic structures,
Floquet-Bloch theorem is normally employed to characterise
wave propagation.

Equivalent elastic properties of regular hexagonal lattices
• Unit cell approach - Gibson and Ashby (1999)
(a) Regular hexagon (θ = 30◦) (b) Unit cell
• We are interested in homogenised equivalent in-plane elastic
properties
• This way, we can avoid a detailed structural analysis considering
all the beams and treat it as a material

Equivalent elastic properties of regular hexagonal lattices
• The cell walls are treated as beams of thickness t, depth b and
Young’s modulus Es. l and h are the lengths of inclined cell walls
having inclination angle θ and the vertical cell walls respectively.
• The equivalent elastic properties are:
E1 = Es
t
l
3
cos θ
(h
l + sin θ) sin2
θ
(1)
E2 = Es
t
l
3
(h
l + sin θ)
cos3 θ
(2)
ν12 =
cos2
θ
(h
l + sin θ) sin θ
(3)
ν21 =
(h
l + sin θ) sin θ
cos2 θ
(4)
G12 = Es
t
l
3 h
l + sin θ
h
l
2
(1 + 2h
l ) cos θ
(5)

Finite element modelling and veriﬁcation
• A ﬁnite element code has been developed to obtain the in-plane
elastic moduli numerically for hexagonal lattices.
• Each cell wall has been modelled as an Euler-Bernoulli beam
element having three degrees of freedom at each node.
• For E1 and ν12: two opposite edges parallel to direction-2 of the
entire hexagonal lattice structure are considered. Along one of
these two edges, uniform stress parallel to direction-1 is applied
while the opposite edge is restrained against translation in
direction-1. Remaining two edges (parallel to direction-1) are
kept free.
• For E2 and ν21: two opposite edges parallel to direction-1 of the
entire hexagonal lattice structure are considered. Along one of
these two edges, uniform stress parallel to direction-2 is applied
while the opposite edge is restrained against translation in
direction-2. Remaining two edges (parallel to direction-2) are
kept free.
• For G12: uniform shear stress is applied along one edge keeping
the opposite edge restrained against translation in direction-1
and 2, while the remaining two edges are kept free.

Finite element modelling and veriﬁcation
0 500 1000 1500 2000
0.9
0.95
1
1.05
1.1
1.15
1.2
Number of unit cells
Ratioofelasticmodulus E
1
E
2
ν
12
ν
21
G
12
θ = 30◦
, h/l = 1.5. FE results converge to analytical predictions after
1681 cells.

Irregular lattice structures
(c) Cedar wood (d) Manufactured honeycomb core
(e) Graphene image (f) Fabricated CNT surface

• A significant limitation of the aforementioned unit cell approach is
that it cannot account for the spatial irregularity, which is
practically inevitable.
• Spatial irregularity may occur due to manufacturing uncertainty,
structural defects, variation in temperature, pre-stressing and
micro-structural variabilities.
• To include the effect of irregularity, voronoi honeycombs have
been considered in several studies.
• The effect of different forms of irregularity on elastic properties
and structural responses of hexagonal lattices are generally
based on direct finite element (FE) simulation.
• In the FE approach, a small change in geometry of a single cell
may require completely new geometry and meshing of the entire
structure. In general this makes the entire process time
consuming and tedious.
• The problem becomes worse for uncertainty quantification of the
responses, where the expensive finite element model is needed
to be simulated for a large number of samples while using a
Monte Carlo based approach.

Examples of some viscoelastic materials
(g) Viscoelastic foam (h) Viscoelastic membrane
(i) Viscoelastic sheet (j) Viscoelastic sheet

Overview of the viscoelastic behaviour

Fundamental equation for the viscoelastic behaviour
• When a linear viscoelastic model is employed, the stress at
some point of a structure can be expressed as a convolution
integral over a kernel function as
σ(t) =
t
−∞
g(t − τ)
∂ (τ)
∂τ
τ (6)
• t ∈ R+
is the time, σ(t) is stress and (t) is strain.
• The kernel function g(t) also known as ‘hereditary function’,
‘relaxation function’ or ‘after-effect function’ in the context of
different subjects.
• In practice, the kernel function is often deﬁned in the frequency
domain (or Laplace domain). Taking the Laplace transform of
Equation (6), we have
¯σ(s) = s ¯G(s)¯(s) (7)
Here ¯σ(s), ¯(s) and ¯G(s) are Laplace transforms of σ(t), (t) and
g(t) respectively and s ∈ C is the (complex) Laplace domain
parameter.

Mathematical representation of the kernel function
• The kernel function in Equation (7) is a complex function in the
frequency domain. For notational convenience we denote
¯G(s) = ¯G(iω) = G(ω) (8)
where ω ∈ R+
is the frequency.
• The complex modulus G(ω) can be expressed in terms of its real
and imaginary parts or in terms of its amplitude and phase as
follows
G(ω) = G (ω) + iG (ω) = |G(ω)|eiφ(ω)
(9)
The real and imaginary parts of the complex modulus, that is,
G (ω) and G (ω) are also known as the storage and loss moduli
respectively.
• One of the main restriction on the form of the kernel function
comes from the fact that the response of the structure must not
start before the application of the forces.
• This causality condition imposes a mathematical relationship
between real and imaginary parts of the complex modulus,
known as Kramers-Kronig relations

Mathematical representation of the kernel function
• Kramers-Kronig relations speciﬁes that the real and imaginary
parts should be related by a Hilbert transform pair, but can be
general otherwise. Mathematically this can be expressed as
G (ω) = G∞ +
2
π
∞
0
uG (u)
ω2 − u2
du
G (ω) =
2ω
π
∞
0
G (u)
u2 − ω2
du
(10)
where the unrelaxed modulus G∞ = G(ω → ∞) ∈ R.
• Equivalent relationships linking the modulus and the phase of
G(ω) can expressed as
ln |G (ω)| = ln |G∞| +
2
π
∞
0
uφ(u)
ω2 − u2
du
φ(ω) =
2ω
π
∞
0
ln |G(u)|
u2 − ω2
du
(11)
• Complex modulus derived using a physics based principle
automatically satisfy these conditions. However, there can be
many other function which would also satisfy these condition.

Viscoelastic models
(k) Maxwell model (l) Voigt model
(m) Standard linear model (n) Generalised Maxwell model
Figure: Springs and dashpots based models viscoelastic materials.

Viscoelastic models
The viscoelastic kernel function can be expressed for the four models
as
• Maxwell model:
g(t) = µe−(µ/η)t
U(t) (12)
• Voigt model:
g(t) = ηδ(t) + µU(t) (13)
• Standard linear model:
g(t) = ER 1 − (1 −
τσ
τ
)e−t/τ
U(t) (14)
• Generalised Maxwell model:
g(t) =


n
j=1
µj e−(µj /ηj )t

 U(t) (15)
Models similar to this is also known as the Pony series model.

Viscoelastic models
Viscoelastic
model
Complex modulus
Biot model G(ω) = G0 +
n
k=1
ak iω
iω+bk
Fractional deriva-
tive
G(ω) = G0+G∞(iωτ)β
1+(iωτ)β
GHM G(ω) = G0 1 + k αk
−ω2
+2iξk ωk ω
−ω2+2iξk ωk ω+ω2
k
ADF G(ω) = G0 1 +
n
k=1 ∆k
ω2
+iωΩk
ω2+Ω2
k
Step-function G(ω) = G0 1 + η 1−e−st0
st0
Half cosine model G(ω) = G0 1 + η 1+2(st0/π)2
−e−st0
1+2(st0/π)2
Gaussian model G(ω) = G0 1 + η eω2
/4µ
1 − erf iω
2
√
µ
Complex modulus for some viscoelastic models in the frequency
domain

The Biot Model
• We consider that each constitutive element of a hexagonal unit
within the lattice structure is modelled using viscoelastic
properties. For simplicity, we use Biot model with only one term.
Frequency dependent complex elastic modulus for an element is
expresses as
E(ω) = ES 1 +
iω
µ + iω
(16)
where µ and are the relaxation parameter and a constant
deﬁning the ‘strength’ of viscosity, respectively. Es is the intrinsic
Young’s modulus.
• The amplitude of this complex elastic modulus is given by
|E(ω)| = ES
µ2 + ω2 (1 + )
2
µ2 + ω2
(17)
• The phase (φ) of this complex elastic modulus is given by
φ E(ω) = tan−1 µω
µ2 + ω2(1 + )
(18)

Mathematical idealisation of irregularity in lattice structures

Irregular honeycombs
• Random spatial irregularity in cell angle is considered in this
study.

• Direct numerical simulation to deal with irregularity in lattice
structures may not necessarily provide proper understanding of
the underlying physics of the system. An analytical approach
could be a simple, insightful, yet an efﬁcient way to obtain
effective elastic properties of lattice structures.
• This work develops a structural mechanics based analytical
framework for predicting equivalent in-plane elastic properties of
irregular lattices having spatially random variations in cell angles.
• Closed-form analytical expressions will be derived for equivalent
in-plane elastic properties.
• An approach based on the frequency-domain representation of
the viscoelastic property of the constituent elements in the cells
is used.

The philosophy of the analytical approach for irregular lattices
(a)
Typical representation of an irregular lattice (b) Representative unit
cell element (RUCE) (c) Illustration to deﬁne degree of irregularity (d)
Unit cell considered for regular hexagonal lattice by Gibson and
Ashby (1999).

The idealisation of RUCE and the bottom-up homogenisation approach

Unit cell geometry
(a) Classical unit cell for regular lattices (b) Representative unit cell
element (RUCE) geometry for irregular lattices

RUCE and free-body diagram for the derivation of E1

RUCE and free-body diagram for the derivation of E2

RUCE and free-body diagram for the derivation of G12

Equivalent E1, E2
Equivalent E1
E1v (ω) =
t3
L
n
j=1
m
i=1
l1ij cos αij − l2ij cos βij
m
i=1
l2
1ij l2
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
)
(19)
Equivalent Young’s moduli E2
E2v (ω) =
Lt3
n
j=1
m
i=1
m
i=1
Esij 1 + ij
iω
µij + iω
l2
3ij cos2 γij l3ij +
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
2ij (l1ij +l2ij )cos2 αij cos2 βij
(l1ij cos αij −l2ij cos βij )
2
−1
(20)

Equivalent shear Modulus G12
Equivalent G12
G12v (ω) =
Lt3
n
j=1
m
i=1
m
i=1
Esij 1 + ij
iω
µij + iω
l2
3ij sin2
γij l3ij +
l1ij l2ij
l1ij +l2ij
−1
(21)

Poisson’s ratios ν12, ν21
Equivalent ν12
ν12eq = −
1
L
n
j=1
m
i=1
m
i=1
cos αij sin βij − sin αij cos βij
cos αij cos βij
(22)
Equivalent ν21
ν21eq = −
L
n
j=1
m
i=1
m
i=1
l2
1ij l2
2ij l1ij + l2ij cos αij cos βij cos αij sin βij − sin αij cos βij
2
l2
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
(l1ij cos αij −l2ij cos βij )
2
(23)

Only spatial variation of the material properties
• According to the notations used for a regular lattice by Gibson
and Ashby (1999), the notations for lattices without any structural
irregularity can be expressed as: L = n(h + l sin θ);
l1ij = l2ij = l3ij = l; αij = θ; βij = 180◦
− θ; γij = 90◦
, for all i and j.
• Using these transformations in case of the spatial variation of
only material properties, the closed-form formulae for compound
variation of material and geometric properties (equations 19–21)
can be reduced to:
E1v = κ1
t
l
3
cos θ
(h
l + sin θ) sin2
θ
(24)
E2v = κ2
t
l
3
(h
l + sin θ)
cos3 θ
(25)
and G12v = κ2
t
l
3 h
l + sin θ
h
l
2
(1 + 2h
l ) cos θ
(26)

Only spatial variation of the material properties
• The multiplication factors κ1 and κ2 arising due to the
consideration of spatially random variation of intrinsic material
properties can be expressed as
κ1 =
m
n
n
j=1
1
m
i=1
1
Esij 1 + ij
iω
µij + iω
(27)
and κ2 =
n
m
1
n
j=1
1
m
i=1
Esij 1 + ij
iω
µij + iω
(28)
• In the special case when ω → 0 and there is no spatial variabilities in the
material properties of the lattice, all viscoelastic material properties
become identical (i.e. Esij = Es, µij = µ and ij = for i = 1, 2, 3, ..., m
and j = 1, 2, 3, ..., n) and subsequently the amplitude of κ1 and κ2
becomes exactly 1. This conﬁrms that the expressions in 27 and 28 give
the necessary generalisations of the classical expressions of Gibson
and Ashby (1999) through 24–26.

Only geometric irregularities
• In case of only spatially random variation of structural geometry
but constant viscoelastic material properties (i.e. Esij = Es,
µij = µ and ij = for i = 1, 2, 3, ..., m and j = 1, 2, 3, ..., n) the
19–21 lead to
E1v = ES 1 +
iω
µ + iω
ζ1 (29)
E2v = ES 1 +
iω
µ + iω
ζ2 (30)
G12v = ES 1 +
iω
µ + iω
ζ3 (31)

Only geometric irregularities
• The random coefﬁcients ζi (i = 1, 2, 3) are
ζ1 =
t3
L
n
j=1
m
i=1
(l1ij cos αij − l2ij cos βij )
m
i=1
l2
1ij l2
2ij (l1ij + l2ij ) (cos αij sin βij − sin αij cos βij )2
(l1ij cos αij − l2ij cos βij )2
(32)
ζ2 =
Lt3
n
j=1
m
i=1
m
i=1
l2
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
(l1ij cos αij −l2ij cos βij )2
−1
(33)
ζ3 =
Lt3
n
j=1
m
i=1
m
i=1
l2
3ij sin2
γij l3ij +
l1ij l2ij
l1ij +l2ij
−1
(34)

Regular hexagonal lattices
• The geometric notations for regular lattices can be expressed as:
L = n(h + l sin θ); l1ij = l2ij = l3ij = l; αij = θ; βij = 180◦
− θ; γij = 90◦
, for
all i and j. Using these transformations, the expressions of in-plane
elastic moduli for regular hexagonal lattices (without the viscoelastic
effect) can be obtained.
• The in-plane Young’s moduli and shear modulus (viscosity dependent
in-plane elastic properties) can be expressed as
E1v = Es 1 +
iω
µ + iω
t
l
3
cos θ
(h
l
+ sin θ) sin2
θ
(35)
E2v = Es 1 +
iω
µ + iω
t
l
3
(h
l
+ sin θ)
cos3 θ
(36)
G12v = Es 1 +
iω
µ + iω
t
l
3 h
l
+ sin θ
h
l
2
(1 + 2h
l
) cos θ
(37)
• The amplitude of the elastic moduli obtained based on the above
expressions converge to the closed-form equation provided by Gibson
and Ashby (1999) in the limiting case of ω → 0.

Regular uniform hexagonal lattices
• In the case of regular uniform lattices with θ = 30◦
, we have
E1v = E2v = 2.3ES 1 +
iω
µ + iω
t
l
3
(38)
• Similarly, in the case of shear modulus for regular uniform lattices
(θ = 30◦
)
G12v = 0.57ES 1 +
iω
µ + iω
t
l
3
(39)
• Regular viscoelastic lattices satisfy the reciprocal theorem
E2v ν12v = E1v ν21v = ES 1 +
iω
µ + iω
t
l
3
1
sin θ cos θ
(40)

Random field model for material and geometric properties
• Correlated structural and material attributes can be modelled
random fields H (x, θ).
• The traditional way of dealing with random field is to discretise
the random field into finite number of random variables. The
available schemes for discretising random fields can be broadly
divided into three groups: (1) point discretisation (e.g., midpoint
method, shape function method, integration point method,
optimal linear estimate method); (2) average discretisation
method (e.g., spatial average, weighted integral method), and (3)
series expansion method (e.g., orthogonal series expansion).
• An advantageous alternative for discretising H (x, θ) is to
represent it in a generalised Fourier type of series as, often
termed as Karhunen-Loève (KL) expansion.

Karhunen-Loève (KL) expansion
• Suppose, H (x, θ) is a random field with covariance function
ΓH(x1, x2) defined in the probability space (Θ, F, P). The KL
expansion for H (x, θ) takes the following form
H (x, θ) = ¯H (x) +
∞
i=1
λi ξi (θ) ψi (x) (41)
where {ξi (θ)} is a set of uncorrelated random variables.
• {λi } and {ψi (x)} are the eigenvalues and eigenfunctions of the
covariance kernel ΓH(x1, x2), satisfying the integral equation
N
ΓH(x1, x2)ψi (x1) dx1 = λi ψi (x2) (42)
• In practise, the infinite series of 41 must be truncated, yielding a
truncated KL approximation
˜H (x, θ) ∼= ¯H (x) +
M
i=1
λi ξi (θ) ψi (x) (43)

• Gaussian and lognormal random ﬁelds have been considered.
The covariance function is represented as:
ΓαZ
= σ2
αZ
e(−|y1−y2|/by )+(−|z1−z2|/bz )
(44)
where by and bz are the correlation parameters at y and z
directions (that corresponds to direction - 1 and direction - 2
respectively). These quantities control the rate at which the
covariance decays.
• In a two dimensional physical space the eigensolutions of the
covariance function are obtained by solving the integral equation
analytically
λi ψi (y2, z2) =
a1
−a1
a2
−a2
Γ(y1, z1; y2, z2)ψi (y1, z1)dy1dz1 (45)
where −a1 y a1 and −a2 z a2.
• Assume the eigen-solutions are separable in y and z directions,
i.e.
ψi (y2, z2) = ψ
(y)
i (y2)ψ
(z)
i (z2) (46)
λi (y2, z2) = λ
(y)
i (y2)λ
(z)
i (z2) (47)

• The solution of the integral equation reduces to the product of the
solutions of two equations of the form
λ
(y)
i ψ
(y)
i (y1) =
a1
−a1
e(−|y1−y2|/by )
ψ
(y)
i (y2)dy2 (48)
• The solution of this equation, which is the eigensolution (eigenvalues
and eigenfunctions) of an exponential covariance kernel for a
one-dimensional random field is obtained as



ψi (ζ) =
cos(ωi ζ)
a + sin(2ωi a)
2ωi
λi =
2σ2
αz
b
ω2
i + b2
for i odd
ψi (ζ) =
sin(ω∗
i ζ)
a −
sin(2ω∗
i
a)
2ωi ∗
λ∗
i =
2σ2
αz
b
ω∗2
i + b2
for i even
(49)
where b = 1/by or 1/bz and a = a1 or a2. ζ can be either y or z and ωi
presents the period of the random field.
• The final eigenfunctions are given by
ψk (y, z) = ψ
(y)
i (y)ψ
(z)
i (z) (50)

Samples of the random ﬁelds
Spatial variability of the intrinsic elastic modulus (Es) with ∆m = 0.002

The degree of geometric irregularity
• To define the degree of irregularity, it is assumed that each
connecting node of the lattice moves randomly within a certain
radius (rd ) around the respective node corresponding to the
regular deterministic configuration. For physically realistic
variabilities, it is considered that a given node do not cross a
neighbouring node, that is
rd < min
h
2
,
l
2
, l cos θ (51)
• In each realization of the Monte Carlo simulation, all the nodes of
the lattice move simultaneously to new random locations within
the specified circular bounds. Thus, the degree of irregularity (r)
is defined as a non-dimensional ratio of the area of the circle and
the area of one regular hexagonal unit as
r =
πr2
d × 100
2l cos θ(h + l sin θ)
(52)
• The degree of irregularity (r) has been expressed as percentage
values for presenting the results.

Samples of the random ﬁelds
Movement of the top vertices of a tessellating hexagonal unit cell with
respect to the corresponding deterministic locations (r = 6)

Random geometric conﬁgurations
Structural conﬁgurations for a single random realisation of an irregular hexagonal lattice considering deterministic cell angle θ = 30◦ and
h/l = 1: (a) r = 0 (b) r = 2 (c) r = 4 (d) r = 6

Samples of random geometric configurations
Figure: Simulation bound of the structural configuration of an irregular
hexagonal lattice for multiple random realisations considering θ = 30◦
,
h/l = 1 and r = 6. The regular configuration is presented using red colour.

Samples of random geometric conﬁgurations
• In randomly inhomogeneous correlated system, spatial variability
of the stochastic structural attributes are accounted, wherein
each sample of the Monte Carlo simulation includes the spatially
random distribution of structural and materials attributes with a
rule of correlation.
• The spatial variability in structural and material properties (Es, µ
and ) are physically attributed by degree of structural irregularity
(r) and degree of material property variation (∆m) respectively.
• As the two Young’s moduli and shear modulus for low density
lattices are proportional to Esρ3
(Zhu et al., 2001), the
non-dimensional results for in-plane elastic moduli E1, E2, and
G12, unless otherwise mentioned, are presented as:
¯E1 =
E1eq
Esρ3
, ¯E2 =
E2eq
Esρ3
¯G12 =
G12eq
Esρ3
• ρ is the relative density of the lattice (deﬁned as a ratio of the
planar area of solid to the total planar area of the lattice).

Spatially correlated irregular elastic lattices: E1
(a) θ = 30◦; h
l
= 1 (b) θ = 30◦; h
l
= 1.5
(c) θ = 45◦; h
l
= 1 (d) θ = 45◦; h
l
= 1.5
Figure: Effective Young’s modulus (E1) of irregular lattices

Spatially correlated irregular elastic lattices: E2
(a) θ = 30◦; h
l
= 1 (b) θ = 30◦; h
l
= 1.5
(c) θ = 45◦; h
l
= 1 (d) θ = 45◦; h
l
= 1.5
Figure: Effective Young’s modulus (E2) of irregular lattices with different
structural conﬁgurations considering correlated attributes

Spatially correlated irregular elastic lattices: G12
(a) θ = 30◦; h
l
= 1 (b) θ = 30◦; h
l
= 1.5
(c) θ = 45◦; h
l
= 1 (d) θ = 45◦; h
l
= 1.5
Figure: Effective shear modulus (G12) of irregular lattices with different
structural conﬁgurations considering correlated attributes

Viscoelastic properties of regular lattices: E1, E2, G12
(a) Effect of viscoelasticity on the magnitude and phase angle of E1 for regular hexagonal lattices (b) Effect of viscoelasticity on the
magnitude and phase angle of E2 for regular hexagonal lattices (c) Effect of viscoelasticity on the magnitude and phase angle of G12 for
regular hexagonal lattices

Viscoelastic properties of regular lattices
(a) Effect of variation of µ on the viscoelastic modulus of regular hexagonal lattices (considering a constant value of = 2) (b) Effect of
variation of on the viscoelastic modulus of regular hexagonal lattices (considering a constant value of µ = ωmax /5). Here Z represents
the viscoelastic moduli (i.e. E1, E2 and G12) and Z0 is the corresponding elastic modulus value for ω = 0.

Spatially correlated irregular viscoelastic lattices: E1

Spatially correlated irregular viscoelastic lattices: E2

Spatially correlated irregular elastic lattices: G12

Probability density function: random geometry

Probability density function: random material property
Probability density function plots for the amplitude of the elastic moduli considering randomly inhomogeneous form of stochasticity for
different values of ∆m (i.e. coefﬁcient of variation for spatially random correlated material properties, such as Es, µ and ). Results are
presented as a ratio of the values corresponding to irregular conﬁgurations and respective deterministic values (for a frequency of 800 Hz).

Combined material and geometric uncertianty
Probabilistic descriptions for the amplitudes of three effective viscoelastic properties corresponding to a frequency of 800 Hz considering
individual and compound effect of stochasticity in material and structural attributes with ∆cov = 0.006

Conclusions
• The effect of viscoelasticity on irregular hexagonal lattices is
investigated in frequency domain considering two different forms of
irregularity in structural and material parameters (spatially uncorrelated
and correlated).
• Spatially correlated structural and material attributes are considered to
account for the effect of randomly inhomogeneous form of irregularity
based on Karhunen-Lo`eve expansion.
• The two Young’s moduli and shear modulus are dependent on the
viscoelastic parameters. Two in-plane Poisson’s ratios depend only on
structural geometry of the lattice structure.
• The classical closed-form expressions for equivalent in-plane and out of
plane elastic properties of regular hexagonal lattice structures have
been generalised to consider geometric and material irregularity and
viscoelasticity.
• Using the principle of basic structural mechanics on a newly deﬁned unit
cell with a homogenisation technique, closed-form expressions have
been obtained for E1, E2, ν12, ν21 and G12.
• The new results reduce to classical formulae of Gibson and Ashby for
the special case of no irregularities and no viscoelastic effect.

Future works and collaborations
• Explicit dynamic analysis (inertia effect).
• Optimally designed variability (perfectly imperfect system)
• Band-gap analysis of viscoelastic metamaterials
• More general metamaterials with complex geometry
• Investigation of possible unusual properties arising due to randomness
and viscoelasticity

Closed-form expressions: Elastic Moduli
E1v (ω) =
t3
L
n
j=1
m
i=1
m
i=1
l2
1ij
l2
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
)
(53)
E2v (ω) =
Lt3
n
j=1
m
i=1
m
i=1
Esij

1 + ij
iω
µij + iω




l2
3ij
cos2 γij l3ij +
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
2ij
l1ij +l2ij cos2 αij cos2 βij
l1ij cos αij −l2ij cos βij
2



−1
(54)
G12v (ω) =
Lt3
n
j=1
m
i=1
m
i=1
Esij

1 + ij
iω
µij + iω

 l2
3ij
sin2 γij l3ij +
l1ij l2ij
l1ij +l2ij
−1
(55)

Closed-form expressions: Poisson’s ratios
ν12eq = −
1
L
n
j=1
m
i=1
m
i=1
cos αij sin βij − sin αij cos βij
cos αij cos βij
(56)
ν21eq = −
L
n
j=1
m
i=1
m
i=1
l2
1ij
l2
2ij
l1ij + l2ij cos αij cos βij cos αij sin βij − sin αij cos βij
2


l2
3ij
cos2 γij l3ij +
l1ij l2ij
l1ij +l2ij
+
l2
1ij
l2
2ij
l1ij +l2ij cos2 αij cos2 βij
l1ij cos αij −l2ij cos βij
2



(57)

Some of our papers on this topic
1 Mukhopadhyay, T. and Adhikari, S., “Effective in-plane elastic properties
of quasi-random spatially irregular hexagonal lattices”, International
Journal of Engineering Science, (revised version submitted).
2 Mukhopadhyay, T., Mahata, A., Asle Zaeem, M. and Adhikari, S.,
“Effective elastic properties of two dimensional multiplanar hexagonal
nano-structures”, 2D Materials, 4[2] (2017), pp. 025006:1-15.
3 Mukhopadhyay, T. and Adhikari, S., “Stochastic mechanics of
metamaterials”, Composite Structures, 162[2] (2017), pp. 85-97.
4 Mukhopadhyay, T. and Adhikari, S., “Free vibration of sandwich panels
with randomly irregular honeycomb core”, ASCE Journal of Engineering
Mechanics,141[6] (2016), pp. 06016008:1-5..
5 Mukhopadhyay, T. and Adhikari, S., “Equivalent in-plane elastic
properties of irregular honeycombs: An analytical approach”,
International Journal of Solids and Structures, 91[8] (2016), pp.
169-184.
6 Mukhopadhyay, T. and Adhikari, S., “Effective in-plane elastic properties
of auxetic honeycombs with spatial irregularity”, Mechanics of Materials,
95[2] (2016), pp. 204-222.

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El-Sayed, F. K. A., Jones, R., Burgess, I. W., 1979. A theoretical approach to the deformation of honeycomb based composite materials.
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Journal of the Mechanics and Physics of Solids 49 (4), 857–870.
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International Journal of Solids and Structures 43 (5), 1061 – 1078.

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