Dynamic Homogenisation of randomly irregular viscoelastic metamaterials

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: [email protected], 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


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


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


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


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 left: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 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?


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.


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


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)


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.


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.


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


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.


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


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.


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


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.


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 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.


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


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)


Mathematical idealisation of irregularity in lattice structures


Irregular honeycombs

Random spatial irregularity in cell angle is considered in thisstudy.



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.


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).


The idealisation of RUCE and the bottom-up homogenisation approach


Unit cell geometry

(a) Classical unit cell for regular lattices (b) Representative unit cellelement (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 () =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


)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)


Equivalent shear Modulus G12

Equivalent G12

G12v () =Lt3

nj=1

mi=1


)m

i=1Esij

(1 + ij

i

ij + i

)(l23ij sin

2 ij

(l3ij +

l1ij l2ijl1ij +l2ij

))1(21)


Poissons ratios 12, 21

Equivalent 12

12eq = 1L

nj=1

mi=1


)m

i=1

(cosij sinij sinij cosij

)cosij cosij

(22)

Equivalent 21

21eq = L

nj=1

mi=1


)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)


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


2 (24)

E2v = 2

(tl

)3 ( hl + sin )cos3

(25)

and G12v = 2

(tl

)3 ( hl + sin

)( hl

)2(1 + 2 hl ) cos

(26)


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.


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)


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


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


mi=1

(l23ij sin

2 ij(

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 ); 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


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.


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)


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.


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


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)



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)


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 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.


Samples of the random fields

Movement of the top vertices of a tessellating hexagonal unit cell withrespect 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 = 30 andh/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 irregularhexagonal lattice for multiple random realisations considering = 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 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).


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


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


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


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


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.


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 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).


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


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.


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 () =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


)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


)m

i=1Esij

1 + ij iij + i

(l23ij sin2 ij(

l3ij +l1ij l2ij

l1ij +l2ij

))1(55)


Closed-form expressions: Poissons ratios

12eq = 1

L

nj=1

mi=1


)m

i=1

(cosij sin ij sinij cos ij

)cosij cos ij

(56)

21eq = L

nj=1

mi=1


)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)


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, 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.


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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