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Dynamic homogenisation of randomlyirregular viscoelastic metamaterials
S. Adhikari1, T. Mukhopadhyay2, A. Batou3
1 Zienkiewicz Centre for Computational Engineering, College of Engineering, SwanseaUniversity, Bay Campus, Swansea, Wales, UK, Email: S.Adhikari@swansea.ac.uk, Twitter:
@ProfAdhikari, Web: http://engweb.swan.ac.uk/~adhikaris1 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 StrategicPartnership
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 1
Swansea University
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 2
Swansea University
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 3
My research interests
Development of fundamental computational methods forstructural dynamics and uncertainty quantificationA. Dynamics of complex systemsB. Inverse problems for linear and non-linear dynamicsC. Uncertainty quantification in computational mechanics
Applications of computational mechanics to emergingmultidisciplinary research areasD. Vibration energy harvesting / dynamics of wind turbinesE. Computational mechanics for mechanics and multi-scalesystems
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 4
Outline
1 IntroductionRegular latticesIrregular lattices
2 Formulation for the viscoelastic analysis3 Equivalent elastic properties of randomly irregular lattices4 Effective properties of irregular lattices: uncorrelated
uncertaintyGeneral results - closed-form expressionsSpecial case 1: Only spatial variation of the material propertiesSpecial case 2: Only geometric irregularitiesSpecial case 3: Regular hexagonal lattices
5 Effective properties of irregular lattices: correlateduncertainty
6 Results and discussionsSpatially correlated irregular elastic latticesViscoelastic properties of regular latticesSpatially correlated irregular viscoelastic lattices
7 ConclusionsAdhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 5
Lattice based metamaterials
Metamaterials are artificial materials designed to outperformnaturally occurring materials in various fronts. These include, butare not limited to, electromagnetics, acoustics, optics, terahertz,infrared, dynamics and mechanical properties.
Lattice based metamaterials are abundant in man-made andnatural systems at various length scales
Lattice based metamaterials are made of periodicidentical/near-identical geometric units
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 6
Hexagonal lattices in 2D
Among various lattice geometries (triangle, square, rectangle,pentagon, hexagon), hexagonal lattice is most common (notethat hexagon is the highest space filling pattern in 2D).
This talk is about in-plane elastic properties of 2D hexagonallattice structures - commonly known as honeycombs
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 7
Lattice structures - nano scale
Illustrations of a single layer graphene sheet and a born nitride nano sheet
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 8
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 left:leaf tissue, bottom right: plant stem, third column - epidermal cells (from web.mit.edu)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 9
Some questions of general interest
Shall we consider lattices as structures or materials from amechanics point of view?
At what relative length-scale a lattice structure can beconsidered 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 aswell as in a qualitative manner?
What is the consequence of random structural irregularities onthe homogenisation approach in general? Can we obtainstatistical measures?
How can we efficiently compute equivalent elastic / viscoelasticproperties of random lattice structures?
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 10
Regular lattice structures
Hexagonal lattice structures have been modelled as a continuoussolid with an equivalent elastic moduli throughout its domain.
This approach eliminates the need of detail finite elementmodelling of lattices in complex structural systems like sandwichstructures.
Extensive amount of research has been carried out to predict theequivalent elastic / viscoelastic properties of regular latticesconsisting of perfectly periodic hexagonal cells (El-Sayed et al.,1979; Gibson and Ashby, 1999; Goswami, 2006; Masters andEvans, 1996; Zhang and Ashby, 1992).
Analysis of two dimensional hexagonal lattices dealing within-plane elastic properties are commonly based on an unit cellapproach, which is applicable only for perfectly periodic cellularstructures.
For the dynamic analysis of perfectly periodic structures,Floquet-Bloch theorem is normally employed to characterisewave propagation.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 11
Equivalent elastic properties of regular hexagonal lattices
Unit cell approach - Gibson and Ashby (1999)
(a) Regular hexagon ( = 30) (b) Unit cell
We are interested in homogenised equivalent in-plane elasticproperties
This way, we can avoid a detailed structural analysis consideringall the beams and treat it as a material
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 12
Equivalent elastic properties of regular hexagonal lattices
The cell walls are treated as beams of thickness t , depth b andYoungs modulus Es. l and h are the lengths of inclined cell wallshaving inclination angle and the vertical cell walls respectively.
The equivalent elastic properties are:
E1 = Es
(tl
)3 cos ( hl + sin ) sin
2 (1)
E2 = Es
(tl
)3 ( hl + sin )cos3
(2)
12 =cos2
( hl + sin ) sin (3)
21 =( hl + sin ) sin
cos2 (4)
G12 = Es
(tl
)3 ( hl + sin
)( hl
)2(1 + 2 hl ) cos
(5)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 13
Finite element modelling and verification
A finite element code has been developed to obtain the in-planeelastic moduli numerically for hexagonal lattices.
Each cell wall has been modelled as an Euler-Bernoulli beamelement having three degrees of freedom at each node.
For E1 and 12: two opposite edges parallel to direction-2 of theentire hexagonal lattice structure are considered. Along one ofthese two edges, uniform stress parallel to direction-1 is appliedwhile the opposite edge is restrained against translation indirection-1. Remaining two edges (parallel to direction-1) arekept free.
For E2 and 21: two opposite edges parallel to direction-1 of theentire hexagonal lattice structure are considered. Along one ofthese two edges, uniform stress parallel to direction-2 is appliedwhile the opposite edge is restrained against translation indirection-2. Remaining two edges (parallel to direction-2) arekept free.
For G12: uniform shear stress is applied along one edge keepingthe opposite edge restrained against translation in direction-1and 2, while the remaining two edges are kept free.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 14
Finite element modelling and verification
0 500 1000 1500 20000.9
0.95
1
1.05
1.1
1.15
1.2
Number of unit cells
Rati
o o
f ela
sti
c m
od
ulu
s
E1
E2
12
21
G12
= 30, h/l = 1.5. FE results converge to analytical predictions after1681 cells.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 15
Irregular lattice structures
(c) Cedar wood (d) Manufactured honeycomb core
(e) Graphene image (f) Fabricated CNT surfaceAdhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 16
Irregular lattice structures
A significant limitation of the aforementioned unit cell approach isthat it cannot account for the spatial irregularity, which ispractically inevitable.
Spatial irregularity may occur due to manufacturing uncertainty,structural defects, variation in temperature, pre-stressing andmicro-structural variabilities.
To include the effect of irregularity, voronoi honeycombs havebeen considered in several studies.
The effect of different forms of irregularity on elastic propertiesand structural responses of hexagonal lattices are generallybased on direct finite element (FE) simulation.
In the FE approach, a small change in geometry of a single cellmay require completely new geometry and meshing of the entirestructure. In general this makes the entire process timeconsuming and tedious.
The problem becomes worse for uncertainty quantification of theresponses, where the expensive finite element model is neededto be simulated for a large number of samples while using aMonte Carlo based approach.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 17
Examples of some viscoelastic materials
(g) Viscoelastic foam (h) Viscoelastic membrane
(i) Viscoelastic sheet (j) Viscoelastic sheetAdhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 18
Overview of the viscoelastic behaviour
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 19
Fundamental equation for the viscoelastic behaviour
When a linear viscoelastic model is employed, the stress atsome point of a structure can be expressed as a convolutionintegral over a kernel function as
(t) = t
g(t )()
(6)
t R+ is the time, (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 ofdifferent subjects.
In practice, the kernel function is often defined in the frequencydomain (or Laplace domain). Taking the Laplace transform ofEquation (6), we have
(s) = sG(s)(s) (7)
Here (s), (s) and G(s) are Laplace transforms of (t), (t) andg(t) respectively and s C is the (complex) Laplace domainparameter.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 20
Mathematical representation of the kernel function
The kernel function in Equation (7) is a complex function in thefrequency domain. For notational convenience we denote
G(s) = G(i) = G() (8)
where R+ is the frequency. The complex modulus G() can be expressed in terms of its real
and imaginary parts or in terms of its amplitude and phase asfollows
G() = G() + iG() = |G()|ei() (9)The real and imaginary parts of the complex modulus, that is,G() and G() are also known as the storage and loss modulirespectively.
One of the main restriction on the form of the kernel functioncomes from the fact that the response of the structure must notstart before the application of the forces.
This causality condition imposes a mathematical relationshipbetween real and imaginary parts of the complex modulus,known as Kramers-Kronig relations
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 21
Mathematical representation of the kernel function
Kramers-Kronig relations specifies that the real and imaginaryparts should be related by a Hilbert transform pair, but can begeneral otherwise. Mathematically this can be expressed as
G() = G +2
0
uG(u)2 u2
du
G() =2
0
G(u)u2 2
du(10)
where the unrelaxed modulus G = G( ) R. Equivalent relationships linking the modulus and the phase of
G() can expressed as
ln |G()| = ln |G|+2
0
u(u)2 u2
du
() =2
0
ln |G(u)|u2 2
du(11)
Complex modulus derived using a physics based principleautomatically satisfy these conditions. However, there can bemany other function which would also satisfy these condition.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 22
Viscoelastic models
(k) Maxwell model (l) Voigt model
(m) Standard linear model (n) Generalised Maxwell model
Figure: Springs and dashpots based models viscoelastic materials.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 23
Viscoelastic models
The viscoelastic kernel function can be expressed for the four modelsas Maxwell model:
g(t) = e(/)tU(t) (12)
Voigt model:g(t) = (t) + U(t) (13)
Standard linear model:
g(t) = ER
[1 (1
)et/
]U(t) (14)
Generalised Maxwell model:
g(t) =
nj=1
je(j/j )t
U(t) (15)Models similar to this is also known as the Pony series model.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 24
Viscoelastic models
Viscoelasticmodel
Complex modulus
Biot model G() = G0 +n
k=1ak ii+bk
Fractional deriva-tive
G() = G0+G(i)
1+(i)
GHM G() = G0[1 +
k k
2+2ikk2+2ikk+2k
]ADF G() = G0
[1 +
nk=1 k
2+ik2+2k
]Step-function G() = G0
[1 + 1e
st0
st0
]Half cosine model G() = G0
[1 + 1+2(st0/)
2est01+2(st0/)2
]Gaussian model G() = G0
[1 + e
2/4{
1 erf(
i2
)}]Complex modulus for some viscoelastic models in the frequency
domain
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 25
The Biot Model
We consider that each constitutive element of a hexagonal unitwithin the lattice structure is modelled using viscoelasticproperties. For simplicity, we use Biot model with only one term.Frequency dependent complex elastic modulus for an element isexpresses as
E() = ES
(1 +
i
+ i
)(16)
where and are the relaxation parameter and a constantdefining the strength of viscosity, respectively. Es is the intrinsicYoungs modulus.
The amplitude of this complex elastic modulus is given by
|E()| = ES
2 + 2 (1 + )2
2 + 2(17)
The phase () of this complex elastic modulus is given by
(E()
)= tan1
(
2 + 2(1 + )
)(18)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 26
Mathematical idealisation of irregularity in lattice structures
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 27
Irregular honeycombs
Random spatial irregularity in cell angle is considered in thisstudy.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 28
Irregular lattice structures
Direct numerical simulation to deal with irregularity in latticestructures may not necessarily provide proper understanding ofthe underlying physics of the system. An analytical approachcould be a simple, insightful, yet an efficient way to obtaineffective elastic properties of lattice structures.
This work develops a structural mechanics based analyticalframework for predicting equivalent in-plane elastic properties ofirregular lattices having spatially random variations in cell angles.
Closed-form analytical expressions will be derived for equivalentin-plane elastic properties.
An approach based on the frequency-domain representation ofthe viscoelastic property of the constituent elements in the cellsis used.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 29
The philosophy of the analytical approach for irregular lattices
(a)Typical representation of an irregular lattice (b) Representative unitcell element (RUCE) (c) Illustration to define degree of irregularity (d)Unit cell considered for regular hexagonal lattice by Gibson andAshby (1999).
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 30
The idealisation of RUCE and the bottom-up homogenisation approach
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 31
Unit cell geometry
(a) Classical unit cell for regular lattices (b) Representative unit cellelement (RUCE) geometry for irregular lattices
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 32
RUCE and free-body diagram for the derivation of E1
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 33
RUCE and free-body diagram for the derivation of E2
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 34
RUCE and free-body diagram for the derivation of G12
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 35
Equivalent E1,E2
Equivalent E1
E1v () =t3
L
nj=1
mi=1
(l1ij cosij l2ij cosij
)m
i=1
l21ij l22ij
(l1ij + l2ij
) (cosij sinij sinij cosij
)2Esij
(1 + ij
i
ij + i
)((l1ij cosij l2ij cosij
)2)
(19)
Equivalent Youngs moduli E2
E2v () =Lt3
nj=1
mi=1
(l1ij cosij l2ij cosij
)m
i=1Esij
(1 + ij
i
ij + i
)(l23ij cos2 ij
(l3ij +
l1ij l2ijl1ij +l2ij
)+
l21ij l22ij(l1ij +l2ij) cos2 ij cos2 ij(l1ij cosijl2ij cos ij)
2
)1(20)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 36
Equivalent shear Modulus G12
Equivalent G12
G12v () =Lt3
nj=1
mi=1
(l1ij cosij l2ij cosij
)m
i=1Esij
(1 + ij
i
ij + i
)(l23ij sin
2 ij
(l3ij +
l1ij l2ijl1ij +l2ij
))1(21)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 37
Poissons ratios 12, 21
Equivalent 12
12eq = 1L
nj=1
mi=1
(l1ij cosij l2ij cosij
)m
i=1
(cosij sinij sinij cosij
)cosij cosij
(22)
Equivalent 21
21eq = L
nj=1
mi=1
(l1ij cosij l2ij cosij
)m
i=1
l21ij l22ij
(l1ij + l2ij
)cosij cosij
(cosij sinij sinij cosij
)(l1ij cosij l2ij cosij
)2(l23ij cos2 ij (l3ij + l1ij l2ijl1ij +l2ij )+ l21ij l22ij(l1ij +l2ij) cos2 ij cos2 ij(l1ij cosijl2ij cos ij)2)
(23)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 38
Only spatial variation of the material properties
According to the notations used for a regular lattice by Gibsonand Ashby (1999), the notations for lattices without any structuralirregularity can be expressed as: L = n(h + l sin );l1ij = l2ij = l3ij = l ; ij = ; ij = 180 ; ij = 90, for all i and j .
Using these transformations in case of the spatial variation ofonly material properties, the closed-form formulae for compoundvariation of material and geometric properties (equations 1921)can be reduced to:
E1v = 1
(tl
)3 cos ( hl + sin ) sin
2 (24)
E2v = 2
(tl
)3 ( hl + sin )cos3
(25)
and G12v = 2
(tl
)3 ( hl + sin
)( hl
)2(1 + 2 hl ) cos
(26)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 39
Only spatial variation of the material properties
The multiplication factors 1 and 2 arising due to theconsideration of spatially random variation of intrinsic materialproperties can be expressed as
1 =mn
nj=1
1m
i=1
1
Esij
(1 + ij
i
ij + i
) (27)
and 2 =nm
1n
j=1
1m
i=1Esij
(1 + ij
i
ij + i
) (28)
In the special case when 0 and there is no spatial variabilities in thematerial properties of the lattice, all viscoelastic material propertiesbecome identical (i.e. Esij = Es, ij = and ij = for i = 1, 2, 3, ...,mand j = 1, 2, 3, ..., n) and subsequently the amplitude of 1 and 2becomes exactly 1. This confirms that the expressions in 27 and 28 givethe necessary generalisations of the classical expressions of Gibsonand Ashby (1999) through 2426.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 40
Only geometric irregularities
In case of only spatially random variation of structural geometrybut constant viscoelastic material properties (i.e. Esij = Es,ij = and ij = for i = 1,2,3, ...,m and j = 1,2,3, ...,n) the1921 lead to
E1v = ES
(1 +
i
+ i
)1 (29)
E2v = ES
(1 +
i
+ i
)2 (30)
G12v = ES
(1 +
i
+ i
)3 (31)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 41
Only geometric irregularities
The random coefficients i (i = 1, 2, 3) are
1 =t3
L
nj=1
mi=1
(l1ij cosij l2ij cosij)
mi=1
l21ij l22ij (l1ij + l2ij) (cosij sinij sinij cosij)
2
(l1ij cosij l2ij cosij)2
(32)
2 =Lt3
nj=1
mi=1
(l1ij cosij l2ij cosij)
mi=1
(l23ij cos2 ij
(l3ij +
l1ij l2ijl1ij +l2ij
)+
l21ij l22ij(l1ij +l2ij) cos2 ij cos2 ij(l1ij cosijl2ij cos ij)
2
)1(33)
3 =Lt3
nj=1
mi=1
(l1ij cosij l2ij cosij)
mi=1
(l23ij sin
2 ij(
l3ij +l1ij l2ij
l1ij +l2ij
))1(34)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 42
Regular hexagonal lattices
The geometric notations for regular lattices can be expressed as:L = n(h + l sin ); l1ij = l2ij = l3ij = l ; ij = ; ij = 180 ; ij = 90, forall i and j . Using these transformations, the expressions of in-planeelastic moduli for regular hexagonal lattices (without the viscoelasticeffect) can be obtained.
The in-plane Youngs moduli and shear modulus (viscosity dependentin-plane elastic properties) can be expressed as
E1v = Es(
1 + i
+ i
)(tl
)3 cos ( hl + sin ) sin
2 (35)
E2v = Es(
1 + i
+ i
)(tl
)3 ( hl + sin )cos3
(36)
G12v = Es(
1 + i
+ i
)(tl
)3 ( hl + sin
)( hl
)2(1 + 2 hl ) cos
(37)
The amplitude of the elastic moduli obtained based on the aboveexpressions converge to the closed-form equation provided by Gibsonand Ashby (1999) in the limiting case of 0.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 43
Regular uniform hexagonal lattices
In the case of regular uniform lattices with = 30, we have
E1v = E2v = 2.3ES
(1 +
i
+ i
)(tl
)3(38)
Similarly, in the case of shear modulus for regular uniform lattices( = 30)
G12v = 0.57ES
(1 +
i
+ i
)(tl
)3(39)
Regular viscoelastic lattices satisfy the reciprocal theorem
E2v12v = E1v21v = ES
(1 +
i
+ i
)(tl
)3 1sin cos
(40)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 44
Random field model for material and geometric properties
Correlated structural and material attributes can be modelledrandom fields H (x, ).
The traditional way of dealing with random field is to discretisethe random field into finite number of random variables. Theavailable schemes for discretising random fields can be broadlydivided into three groups: (1) point discretisation (e.g., midpointmethod, shape function method, integration point method,optimal linear estimate method); (2) average discretisationmethod (e.g., spatial average, weighted integral method), and (3)series expansion method (e.g., orthogonal series expansion).
An advantageous alternative for discretising H (x, ) is torepresent it in a generalised Fourier type of series as, oftentermed as Karhunen-Loeve (KL) expansion.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 45
Karhunen-Loeve (KL) expansion
Suppose, H (x, ) is a random field with covariance functionH(x1,x2) defined in the probability space (,F ,P). The KLexpansion for H (x, ) takes the following form
H (x, ) = H (x) +i=1
ii ()i (x) (41)
where {i ()} is a set of uncorrelated random variables. {i} and {i (x)} are the eigenvalues and eigenfunctions of the
covariance kernel H(x1,x2), satisfying the integral equation
Karhunen-Loeve (KL) expansion
Gaussian and lognormal random fields have been considered.The covariance function is represented as:
Z = 2Z
e(|y1y2|/by )+(|z1z2|/bz ) (44)
where by and bz are the correlation parameters at y and zdirections (that corresponds to direction - 1 and direction - 2respectively). These quantities control the rate at which thecovariance decays.
In a two dimensional physical space the eigensolutions of thecovariance function are obtained by solving the integral equationanalytically
ii (y2, z2) = a1a1
a2a2
(y1, z1; y2, z2)i (y1, z1)dy1dz1 (45)
where a1 6 y 6 a1 and a2 6 z 6 a2. Assume the eigen-solutions are separable in y and z directions,
i.e.i (y2, z2) =
(y)i (y2)
(z)i (z2) (46)
i (y2, z2) = (y)i (y2)
(z)i (z2) (47)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 47
Karhunen-Loeve (KL) expansion
The solution of the integral equation reduces to the product of thesolutions of two equations of the form
(y)i
(y)i (y1) =
a1a1
e(|y1y2|/by )(y)i (y2)dy2 (48)
The solution of this equation, which is the eigensolution (eigenvaluesand eigenfunctions) of an exponential covariance kernel for aone-dimensional random field is obtained as
i() =cos(i)a + sin(2i a)2i
i =22z b2i + b2
for i odd
i() =sin(i )a sin(2
i a)
2i
i =22z b2i + b2
for i even(49)
where b = 1/by or 1/bz and a = a1 or a2. can be either y or z and ipresents the period of the random field.
The final eigenfunctions are given by
k (y , z) = (y)i (y)
(z)i (z) (50)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 48
Samples of the random fields
Spatial variability of the intrinsic elastic modulus (Es) with m = 0.002
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 49
The degree of geometric irregularity
To define the degree of irregularity, it is assumed that eachconnecting node of the lattice moves randomly within a certainradius (rd ) around the respective node corresponding to theregular deterministic configuration. For physically realisticvariabilities, it is considered that a given node do not cross aneighbouring node, that is
rd < min(
h2,
l2, l cos
)(51)
In each realization of the Monte Carlo simulation, all the nodes ofthe lattice move simultaneously to new random locations withinthe specified circular bounds. Thus, the degree of irregularity (r )is defined as a non-dimensional ratio of the area of the circle andthe area of one regular hexagonal unit as
r =r2d 100
2l cos (h + l sin )(52)
The degree of irregularity (r ) has been expressed as percentagevalues for presenting the results.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 50
Samples of the random fields
Movement of the top vertices of a tessellating hexagonal unit cell withrespect to the corresponding deterministic locations (r = 6)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 51
Random geometric configurations
Structural configurations for a single random realisation of an irregular hexagonal lattice considering deterministic cell angle = 30 andh/l = 1: (a) r = 0 (b) r = 2 (c) r = 4 (d) r = 6
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 52
Samples of random geometric configurations
Figure: Simulation bound of the structural configuration of an irregularhexagonal lattice for multiple random realisations considering = 30,h/l = 1 and r = 6. The regular configuration is presented using red colour.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 53
Samples of random geometric configurations
In randomly inhomogeneous correlated system, spatial variabilityof the stochastic structural attributes are accounted, whereineach sample of the Monte Carlo simulation includes the spatiallyrandom distribution of structural and materials attributes with arule of correlation.
The spatial variability in structural and material properties (Es, and ) are physically attributed by degree of structural irregularity(r ) and degree of material property variation (m) respectively.
As the two Youngs moduli and shear modulus for low densitylattices are proportional to Es3 (Zhu et al., 2001), thenon-dimensional results for in-plane elastic moduli E1, E2, andG12, unless otherwise mentioned, are presented as:
E1 =E1eqEs3
, E2 =E2eqEs3
G12 =G12eqEs3
is the relative density of the lattice (defined as a ratio of theplanar area of solid to the total planar area of the lattice).
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 54
Spatially correlated irregular elastic lattices: E1
(a) = 30; hl = 1 (b) = 30; hl = 1.5
(c) = 45; hl = 1 (d) = 45; hl = 1.5
Figure: Effective Youngs modulus (E1) of irregular lattices
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 55
Spatially correlated irregular elastic lattices: E2
(a) = 30; hl = 1 (b) = 30; hl = 1.5
(c) = 45; hl = 1 (d) = 45; hl = 1.5
Figure: Effective Youngs modulus (E2) of irregular lattices with differentstructural configurations considering correlated attributes
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 56
Spatially correlated irregular elastic lattices: G12
(a) = 30; hl = 1 (b) = 30; hl = 1.5
(c) = 45; hl = 1 (d) = 45; hl = 1.5
Figure: Effective shear modulus (G12) of irregular lattices with differentstructural configurations considering correlated attributes
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 57
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 themagnitude 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
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 58
Viscoelastic properties of regular lattices
(a) Effect of variation of on the viscoelastic modulus of regular hexagonal lattices (considering a constant value of = 2) (b) Effect ofvariation of on the viscoelastic modulus of regular hexagonal lattices (considering a constant value of = max/5). Here Z represents
the viscoelastic moduli (i.e. E1, E2 and G12) and Z0 is the corresponding elastic modulus value for = 0.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 59
Spatially correlated irregular viscoelastic lattices: E1
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 60
Spatially correlated irregular viscoelastic lattices: E2
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 61
Spatially correlated irregular elastic lattices: G12
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 62
Probability density function: random geometry
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 63
Probability density function: random material property
Probability density function plots for the amplitude of the elastic moduli considering randomly inhomogeneous form of stochasticity fordifferent values of m (i.e. coefficient of variation for spatially random correlated material properties, such as Es , and ). Results are
presented as a ratio of the values corresponding to irregular configurations and respective deterministic values (for a frequency of 800 Hz).
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 64
Combined material and geometric uncertianty
Probabilistic descriptions for the amplitudes of three effective viscoelastic properties corresponding to a frequency of 800 Hz consideringindividual and compound effect of stochasticity in material and structural attributes with cov = 0.006
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 65
Conclusions
The effect of viscoelasticity on irregular hexagonal lattices isinvestigated in frequency domain considering two different forms ofirregularity in structural and material parameters (spatially uncorrelatedand correlated).
Spatially correlated structural and material attributes are considered toaccount for the effect of randomly inhomogeneous form of irregularitybased on Karhunen-Loeve expansion.
The two Youngs moduli and shear modulus are dependent on theviscoelastic parameters. Two in-plane Poissons ratios depend only onstructural geometry of the lattice structure.
The classical closed-form expressions for equivalent in-plane and out ofplane elastic properties of regular hexagonal lattice structures havebeen generalised to consider geometric and material irregularity andviscoelasticity.
Using the principle of basic structural mechanics on a newly defined unitcell with a homogenisation technique, closed-form expressions havebeen obtained for E1, E2, 12, 21 and G12.
The new results reduce to classical formulae of Gibson and Ashby forthe special case of no irregularities and no viscoelastic effect.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 66
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
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 67
Closed-form expressions: Elastic Moduli
E1v () =t3
L
nj=1
mi=1
(l1ij cosij l2ij cos ij
)m
i=1
l21ij l22ij
(l1ij + l2ij
) (cosij sin ij sinij cos ij
)2Esij
1 + ij iij + i
((l1ij cosij l2ij cos ij)2)(53)
E2v () =Lt3
nj=1
mi=1
(l1ij cosij l2ij cos ij
)m
i=1Esij
1 + ij iij + i
l23ij cos2 ij
(l3ij +
l1ij l2ijl1ij +l2ij
)+
l21ij l22ij
(l1ij +l2ij
)cos2 ij cos
2 ij(l1ij cosijl2ij cos ij
)21
(54)
G12v () =Lt3
nj=1
mi=1
(l1ij cosij l2ij cos ij
)m
i=1Esij
1 + ij iij + i
(l23ij sin2 ij(
l3ij +l1ij l2ij
l1ij +l2ij
))1(55)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 68
Closed-form expressions: Poissons ratios
12eq = 1
L
nj=1
mi=1
(l1ij cosij l2ij cos ij
)m
i=1
(cosij sin ij sinij cos ij
)cosij cos ij
(56)
21eq = L
nj=1
mi=1
(l1ij cosij l2ij cos ij
)m
i=1
l21ij l22ij
(l1ij + l2ij
)cosij cos ij
(cosij sin ij sinij cos ij
)(
l1ij cosij l2ij cos ij)2 l23ij cos2 ij
(l3ij +
l1ij l2ijl1ij +l2ij
)+
l21ij l22ij
(l1ij +l2ij
)cos2 ij cos
2 ij(l1ij cosijl2ij cos ij
)2
(57)
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 69
Some of our papers on this topic
1 Mukhopadhyay, T. and Adhikari, S., Effective in-plane elastic propertiesof quasi-random spatially irregular hexagonal lattices, InternationalJournal of Engineering Science, (revised version submitted).
2 Mukhopadhyay, T., Mahata, A., Asle Zaeem, M. and Adhikari, S.,Effective elastic properties of two dimensional multiplanar hexagonalnano-structures, 2D Materials, 4[2] (2017), pp. 025006:1-15.
3 Mukhopadhyay, T. and Adhikari, S., Stochastic mechanics ofmetamaterials, Composite Structures, 162[2] (2017), pp. 85-97.
4 Mukhopadhyay, T. and Adhikari, S., Free vibration of sandwich panelswith randomly irregular honeycomb core, ASCE Journal of EngineeringMechanics,141[6] (2016), pp. 06016008:1-5..
5 Mukhopadhyay, T. and Adhikari, S., Equivalent in-plane elasticproperties 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 propertiesof auxetic honeycombs with spatial irregularity, Mechanics of Materials,95[2] (2016), pp. 204-222.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 70
Adhikari, S., May 1998. Energy dissipation in vibrating structures. Masters thesis, Cambridge University Engineering Department,Cambridge, UK, first Year Report.
Adhikari, S., Woodhouse, J., May 2001. Identification of damping: part 1, viscous damping. Journal of Sound and Vibration 243 (1), 4361.El-Sayed, F. K. A., Jones, R., Burgess, I. W., 1979. A theoretical approach to the deformation of honeycomb based composite materials.
Composites 10 (4), 209214.Gibson, L., Ashby, M. F., 1999. Cellular Solids Structure and Properties. Cambridge University Press, Cambridge, UK.Goswami, S., 2006. On the prediction of effective material properties of cellular hexagonal honeycomb core. Journal of Reinforced Plastics
and Composites 25 (4), 393405.Li, K., Gao, X. L., Subhash, G., 2005. Effects of cell shape and cell wall thickness variations on the elastic properties of two-dimensional
cellular solids. International Journal of Solids and Structures 42 (5-6), 17771795.Masters, I. G., Evans, K. E., 1996. Models for the elastic deformation of honeycombs. Composite Structures 35 (4), 403422.Zhang, J., Ashby, M. F., 1992. The out-of-plane properties of honeycombs. International Journal of Mechanical Sciences 34 (6), 475 489.Zhu, H. X., Hobdell, J. R., Miller, W., Windle, A. H., 2001. Effects of cell irregularity on the elastic properties of 2d voronoi honeycombs.
Journal of the Mechanics and Physics of Solids 49 (4), 857870.Zhu, H. X., Thorpe, S. M., Windle, A. H., 2006. The effect of cell irregularity on the high strain compression of 2d voronoi honeycombs.
International Journal of Solids and Structures 43 (5), 1061 1078.
Adhikari (Swansea) Homogenisation of randomly irregular metamaterials May 24, 2017 70
IntroductionRegular latticesIrregular lattices
Formulation for the viscoelastic analysisEquivalent elastic properties of randomly irregular latticesEffective properties of irregular lattices: uncorrelated uncertaintyGeneral results - closed-form expressionsSpecial case 1: Only spatial variation of the material propertiesSpecial case 2: Only geometric irregularitiesSpecial case 3: Regular hexagonal lattices
Effective properties of irregular lattices: correlated uncertaintyResults and discussionsSpatially correlated irregular elastic latticesViscoelastic properties of regular latticesSpatially correlated irregular viscoelastic lattices
Conclusions
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