Content uploaded by Tanmoy Mukhopadhyay

Author content

All content in this area was uploaded by Tanmoy Mukhopadhyay on Jun 01, 2019

Content may be subject to copyright.

Theoretical limits for negative elastic moduli in sub-acoustic lattice materials

T. Mukhopadhyay∗

Department of Engineering Science, University of Oxford, Oxford, UK

S. Adhikari

College of Engineering, Swansea University, Bay Campus, Swansea, UK

A. Alu

Advanced Science Research Center, City University of New York, New York, USA

(Dated: March 4, 2019)

An insightful mechanics-based bottom-up framework is developed for probing the frequency-

dependence of lattice material microstructures. Under a vibrating condition, eﬀective elastic moduli

of such microstructured materials can become negative for certain frequency values, leading to an

unusual mechanical behaviour with a multitude of potential applications. We have derived the fun-

damental theoretical limits for the minimum frequency, beyond which the negative eﬀective moduli

of the materials could be obtained. An eﬃcient dynamic stiﬀness matrix based approach is devel-

oped to obtain the closed-form limits, which can exactly capture the sub-wavelength scale dynamics.

The limits turn out to be a fundamental property of the lattice materials and depend on certain

material and geometric parameters of the lattice in a unique manner. An explicit characterization of

the theoretical limits of negative elastic moduli along with adequate physical insights would accel-

erate the process of its potential exploitation in various engineered materials and structural systems

under dynamic regime across the length-scales.

Introduction. – The global mechanical properties can

be engineered in lattice materials by intelligently iden-

tifying the material microstructures as the properties in

these materials are often deﬁned by their structural con-

ﬁguration along with the intrinsic material properties of

the constituent members. This novel class of materials

with tailorable application-speciﬁc mechanical properties

(like equivalent elastic moduli, buckling, vibration and

wave propagation characteristics with modulation fea-

tures) have tremendous potential applications for future

aerospace, civil, mechanical, electronics and medical ap-

plications across the length-scales. Naturally occurring

materials cannot exhibit unprecedented and fascinating

properties such as extremely lightweight, negative elas-

tic moduli, negative mass density, pentamode material

characteristics (meta-ﬂuid), which can be achieved by an

intelligent microstructural design [1, 2]. For example, the

conventional positive value of Poisson’s ratio in a hexag-

onal lattice metamaterial can be converted to a negative

value [3] by making the cell angle θin ﬁgure 1(b) nega-

tive. Other unusual and exciting properties can be real-

ized in metamaterials under dynamic condition, such as

negative bulk modulus induced by monopolar resonance

[4], negative mass density induced by dipolar resonance

[5], and negative shear modulus induced by quadrupolar

resonance [6]. Elastic cloaks [7] and various other un-

precedented dynamic behaviour of such materials have

been widely reported in literature [8–14].

Lattice microstructures are often modelled as a contin-

uous solid medium with a set of eﬀective elastic moduli

throughout the entire domain based on an unit cell ap-

proach [15–17]. The basic mechanics of deformation for

the lattices being scale-independent, the formulations de-

veloped in this context are generally applicable for wide

range of materials and structural forms. Two dimen-

sional hexagonal lattices of natural and artiﬁcial nature

∗Electronic address: tanmoy.mukhopadhyay@eng.ox.ac.uk

FIG. 1: Bottom-up approach (involving an hierarchy of anal-

ysis with beam element, unit cell and lattice structure) for

analysing the frequency-dependent elastic moduli of lattice

materials (a) Typical representation of a hexagonal cellular

structure in a dynamic environment (such as the honeycomb

as part of a host structure experiencing wave propagation,

vibrating structural component etc.). The curved arrows are

symbolically used to indicate propagation of wave (b) One

hexagonal unit cell under dynamic environment (c) A dy-

namic beam element for the damped bending vibration with

two nodes and four degrees of freedom)

can be identiﬁed across diﬀerent length-scales (nano to

macro) in auxetic and non-auxetic forms [18, 19]. This

has led to our focus on hexagonal lattices in this article

while selecting a lattice conﬁguration to demonstrate the

concepts.

Honeycombs and other forms of lattice microstructures

are often intended to be utilized in vibrating structures

such as sandwich panels [20–22] used in aircraft struc-

tures [23]. Hexagonal lattice-like structural form being

a predominant material structure at nano-scale (such as

graphene, hBN etc [24–27]), analysis of vibrating nanos-

tructures are quite relevant to various applications at

nanoscale. Besides that, recent developments in the ﬁeld

2

of metamaterials have prompted its use as advanced ma-

terials in aircraft and other machineries that experience

vibration during operation. Dynamic homogenization of

metamaterials have been reported in various recent pa-

pers [28, 29]. For relatively low-frequency vibrations, the

length of each unit cell will be signiﬁcantly smaller than

the wave-lengths of the global vibration modes. As a

result, each unit cell would eﬀectively behave as a sub-

wavelength scale resonator. Several exciting and unusual

bulk properties of metamaterials have been reported ex-

ploiting sub-wavelength scale resonators [30]. These in-

clude negative stiﬀness [31], negative density (or mass)

[32], or both [33], anisotropy in the eﬀective mass or den-

sity [34, 35], and non-reciprocal response [36, 37].

Theoretically, lattice materials under the eﬀect of dy-

namic forces can also show similar unusual behaviour of

negative elastic moduli due to the sub-wavelength scale

resonator. However, this has not been widely reported

primarily due to the diﬃculties in modelling complex

lattice unit cells as sub-wavelength scale resonators. In

principle, this is possible using very ﬁne ﬁnite element

discretisations of the individual beam elements in an unit

cell. Such an approach will be purely numerical involving

infeasible computationally intensive simulations. Besides

that, a large-scale simulation based approach cannot pro-

vide an insightful physical framework for deriving the

theoretical limits of the frequencies to obtain negative

elastic moduli.

We aim to develop physically insightful theoretical lim-

its of natural frequency to obtain negative axial and shear

moduli in hexagonal lattice materials. We would exploit

the tremendous implicit capabilities of dynamic stiﬀness

method [38] at high frequencies coupled with the con-

cepts of structural mechanics to derive closed-form ana-

lytical limits, which are valid for steady-state dynamics

under harmonic excitations. Though we concentrate on

hexagonal lattices in this article, the basic concepts are

general and it would be applicable to other two and three

dimensional lattice geometries.

Negative elastic moduli of lattice materials and their

theoretical limits. – A bottom-up theoretical frame-

work is developed here (refer to ﬁgure 1) to investi-

gate the limits of natural frequency that would cause

negative axial or shear moduli. A lattice-like structure

can be analysed by considering a unit cell as shown in

ﬁgure 1(b), while the unit cell consists of beam ele-

ments. In a vibrating condition, the dynamic motion

of the overall lattice corresponds to vibration of individ-

ual beams, which would exhibit a diﬀerent frequency-

dependent deformation behaviour compared to the con-

ventional static analyses. Thus, we ﬁrst form the

frequency-dependent elastic stiﬀness matrix for a beam

element (D(ω) = [Dij],where i, j ∈[1,2, ...4] and ωis

the frequency of vibration) and thereby, the frequency-

dependent deformation characteristics of a unit cell are

developed. Here the dynamic stiﬀness matrix accounts

for the compound eﬀect of mass and stiﬀness matrices

as D(ω) = K(ω)−ω2M(ω), wherein the dynamic equi-

librium D(ω)b

v(ω) = bf(ω) is satisﬁed (refer to section

1.3 of the supplementary material for further details).

Eventually, frequency-dependent equivalent elastic mod-

uli of the overall lattice structure are derived based on

the deformation behaviour of a unit cell. A multitude

of critical analyses can be carried out based on the in-

sightful closed-form expressions of frequency-dependent

elastic moduli. The theoretical limits of frequencies to

obtain negative elastic moduli are derived using their re-

spective frequency-dependent expressions.

The frequency dependent elastic stiﬀness matrix of a

beam element is obtained based on an eﬃcient dynamic

stiﬀness method [39, 40], which is a high ﬁdelity ap-

proach at low to high frequencies compared to the con-

ventional “static” ﬁnite element method. For character-

izing the frequency-dependent elastic moduli, the con-

ventional “static” ﬁnite element method could require

very ﬁne discretization for higher frequencies that may

be practically impossible to achieve in a complex lat-

tice metamaterial. The displacement ﬁeld within the ele-

ments can be expressed by complex frequency dependent

shape functions in dynamic stiﬀness method, leading to

a radically signiﬁcant computational eﬃciency at higher

frequencies. The major advantages of this method and

derivation of the frequency-dependent elastic moduli of

the hexagonal lattices is provided as supplementary ma-

terial. Expressions of the frequency-dependent Young’s

moduli and shear modulus [41]can be obtained based on

the concepts of structural mechanics using the elements

of [D(ω)] matrix as (refer to the supplementary material

for derivation)

E1(ω) = D33lcos θ

(h+lsin θ)¯

bsin2θ(1)

E2(ω) = D33(h+lsin θ)

l¯

bcos3θ(2)

G12(ω) = (h+lsin θ)

2l¯

bcos θ

1

−h2

4lDs

43 +2

Dv

33−

(Dv

34)2

Dv

44

(3)

For detailed description regarding the elements of dy-

namic stiﬀness matrix [D(ω)] involved in the above ex-

pressions, refer to the supplementary material. It can

be noted in the above expressions that the elements of

[D(ω)] matrix are functions of the frequency-dependent

parameter b, where b4=mω2(1 −iζm/ω)

EI (1 + iωζk). The quanti-

ties ζkand ζmare stiﬀness and mass proportional damp-

ing factors. Here Eis the intrinsic Young’s modulus of

the lattice material i.e. the Youngs modulus of the mate-

rial of the individual beam elements, while E1and E2are

the equivalent Youngs moduli of the entire lattice struc-

ture.The parameter mdenotes mass per unit length and

tis the thickness of lattice wall. The quantities h,land

θare the length of cell walls and cell angle as shown in

ﬁgure 1(b). Two in-plane Poisson’s ratios are found to

be independent of the frequency.

ν12 =1

ν21

=lcos2θ

(h+lsin θ) sin θ(4)

Primary scope of this work is to extend the well-known

Gibson and Ashby’s formulae [15] for static elastic mod-

uli of lattice structures to the dynamic domain. In most

of the engineering applications, the elastic properties re-

quired in design are presented in terms of the two princi-

ple axes, such as E1,E2,G12 etc. Thus we concentrated

3

on these quantities in the current paper to ﬁnd out the

eﬀect of vibration and deriving the expression for fre-

quencies to cause the onset of negative elastic moduli.

It can be noted in this context that the expressions of

E1,E2and G12 for the undamped case converge to the

closed-form solution provided in [15], when the frequency

parameter (ω) tends to zero, while the expressions of the

Poisson’s ratios are exactly same as that provided in [15].

The expressions of frequency dependent elastic moduli

also conform the reciprocal theorem, i.e. E1(ω)ν21 =

E2(ω)ν12. Regular lattice material (θ= 30◦) shows an

isotropic behaviour under dynamic condition

E1=E2=4

√3

D33

¯

b(5)

At the static limit (ω→0), the isotropic behaviour of a

regular lattice material (θ= 30◦) can be expressed as

E1=E2=4

√3Et

l3

(6)

The isotropic behaviour of a regular lattice depends on

two factors: the interaction between diﬀerent elements of

the [D(ω)] matrix (i.e. the dynamic stiﬀness matrix of

a single beam element) and the geometry of a unit cell.

It can be noted that the Youngs moduli E1and E2of a

hexagonal lattice depend on a single element D33 (refer

to equations 1 and 2), except the geometric parameters.

For a regular hexagonal lattice, the rest of the compo-

nents in the expression of E1and E2(i.e. the geometric

part) become same for h=land θ= 30o. This causes

the isotropy in a regular hexagonal system. For other

kind of regular lattices (e.g. triangular or square [42]),

the isotropic behaviour will depend on the above men-

tioned two factors, the crucial insights of which could be

obtained following a similar framework as proposed in

this paper.

The expressions of E1and E2are proportional to the

complex frequency-dependent element D33 of the [D(ω)]

matrix. Therefore, we study it’s behaviour in the un-

damped limit to understand the if the real part of E1and

E2can become negative. Assuming no damping in the

system, the critical value of frequency beyond which the

Young’s moduli become negative can be obtained based

on Taylor series expansion of D33 (refer to the supple-

mentary material for detailed derivation)

ω∗

E1,E2≈5.598 1

l2rEI

m(7)

Here, ω∗

E1,E2represents the fundamental inﬂection fre-

quency, where the Young’s moduli change sign from pos-

itive to negative. For lightly damped systems, beyond

this frequency value, the equivalent Young’s moduli E1

and E2will be negative for the ﬁrst time when viewed

on the frequency axis. As the frequency increases, the

Young’s moduli will become positive and negative again.

The signiﬁcance of the fundamental inﬂection frequency

derived in equation 7 is that it is the lowest frequency

value beyond which the eﬀective Young’s modulus can

become negative. Physically, negative Young’s modu-

lus means that when a force is applied at the inﬂection

frequency, the direction of the steady-state dynamic re-

sponse will be in the opposite direction to the applied

forcing at the same frequency.

Since the discovery of the Young’s modulus over three

centuries ago, it has been generally recognised as a pos-

itive quantity. This can be mathematically explained

in the light of equation 7. Since m6= 0, this implies

that ω∗

E1,E2>0 for any lattice with ﬁnite-length beams.

A static analysis normally used to obtain the classical

Young’s modulus can be viewed as a dynamic analysis

with ω= 0. Therefore, according to equation 7 it is

not possible to observe a negative Young’s modulus as

ω∗

E1,E2>0. Only when a dynamic equilibrium is con-

sidered, our results show that for cellular metamaterials

the Young’s moduli can be negative, apparently contra-

dicting notions established for centuries. It should be

noted that a similar observation has been made in the

context of acoustics metamaterials with sub-wavelength

scale oscillators (see the review paper [30] for more dis-

cussions). The result derived through equation 7 is the

ﬁrst explicit analysis towards establishing the existence

of negative Young’s modulus in the context of dynamics

of elastic cellular metamaterials.

0 2 4 6 8 10 12 14 16 18

Frequency (rad/s)

-400

-200

0

200

400

600

800

Normalised E1, E2

Real E1, E2

Imag E1, E2

Abs E1, E2

FIG. 2: The real and imaginary parts and the amplitude of

the normalised value of E1and E2as a function of frequency.

Here the ﬁrst frequency value when the Young’s moduli be-

come negative is marked by ‘+’.

Unlike the case of Young’s moduli, the frequency-

dependent closed-form expression 3 for shear modulus

shows a compound eﬀect of multiple elements of the

[D(ω)] matrix. It is possible to obtain the expression of

a tight bound for the frequency, beyond which the shear

modulus becomes negative. Expanding the closed-form

expression of G12 in a Taylor series, the following fun-

damental inequality regarding the frequency for negative

value of G12 can be derived (refer to the supplementary

material for detailed derivation)

120

q160 + 75 (h/l)4

1

l2rEI

m≤ω∗

G12

≤30.2715s1 + 2(h/l)

8 + 9(h/l)5

1

l2rEI

m

(8)

Here, ω∗

G12 represents the fundamental inﬂection fre-

quency for shear modulus, where the shear modulus

changes sign.

Adequate physical insights can be drawn from the

closed-form expressions for the elastic moduli in terms

4

of explicit characterization of the parameters involved

in the onset of negative Youngs moduli or shear modu-

lus. For example, if we notice equation 7, it is clear that

ω∗

E1,E2is inversely proportional to the parameters land

m, while proportional to the ﬂexural rigidity EI. Fur-

ther, based on the power(/ exponent) of the parameters,

it can be realized that the sensitivity of l(with a power of

2) is much higher than the other two parameters mand

EI (with a power of 0.5). Unlike the equivalent expres-

sion for the Young’s moduli E1and E2in equation 7, for

the minimum frequency above which G12 becomes neg-

ative depends on the h/l ratio in addition to the other

parameters (i.e. l,EI and m). Similar conclusions as

the Young’s moduli can be readily derived in case of the

shear modulus on the dependence of the onset of negative

shear modulus on diﬀerent system parameters.

Results and discussion. – Numerical results based on

the derived expressions of analytical limits of negative

elastic moduli are presented in the following paragraphs.

However, before discussing the results concerning nega-

tive axial and shear moduli, the dynamic stiﬀness based

framework needs to be validated ﬁrst. We have presented

representative results for validation of the analytical ex-

pression for frequency dependent Young’s modulus in the

supplementary material. Unless otherwise mentioned,

numerical results are presented for a structural conﬁg-

uration of θ= 30◦and h/l = 1, with ζk= 0.002 and

ζm= 0.05. The geometric parameters of the honey-

comb and intrinsic material properties are assumed as:

l= 3.67 mm, h=l,E= 69.5×103N/mm2,d= 0.8

mm, t= 0.0635 mm and m= 0.137 kg/mm.

The Young’s moduli E1and E2are functions of only

the frequency dependent coeﬃcient D33 (refer to equa-

tions 1 and 2). When E1and E2are normalised with

respect to their equivalent static values, they both essen-

tially become same mathematical function

E1

E1s

=E2

E2s

=D33

12EI /l3(9)

For any positive values of the damping coeﬃcients, D33

becomes complex. This in turn makes the Young’s mod-

uli E1and E2as complex quantities. The real and imagi-

nary parts and also the amplitude of the normalised value

of E1and E2(see equation 9) are shown in ﬁgure 2. It can

be observed that the real part of E1and E2becomes neg-

ative and then changes to positive again with the change

of frequency. This conﬁrms that the value of the elastic

moduli E1and E2(and subsequently the axial stiﬀness in

the two directions) will be negative at certain frequencies.

In ﬁgure 2, the frequency axis is zoomed to observe the

ﬁrst frequency point when D33 becomes negative. This

frequency point is predicted by equation 7 as ω= 1.2231.

This matches exactly with what is observed (marked by

‘+’) in ﬁgure 2 conﬁrming the validity of equation equa-

tion 7. The frequency at which the Young’s moduli E1

and E2of a hexagonal lattice becomes negative is a fun-

damental property of the lattice and it depends only on

the length of the inclined beams (l), the bending rigidity

(EI ) and mass density per unit length (m). The imagi-

nary parts of E1and E2remain positive at all frequency

for any positive value of damping.

The normalised shear modulus is presented in ﬁgure 3

for two diﬀerent values of h/l ratios. The real and imag-

inary parts along with the absolute values are shown in

the ﬁgure. The upper and lower bounds of the values of

ω∗

G12 , the frequency at which G12 becomes negative are

shown by ‘x’ and ‘+’ in the ﬁgure. It is found that the

actual value of ω∗

G12 lies within the bounds given by equa-

tion 8. The value of ω∗

G12 reduces with the increase in h/l

ratio, which is also evident from the derived inequality.

It can be noted here that the real part becomes negative

for all the three elastic moduli beyond the fundamen-

tal inﬂection frequency. Amplitude is always a positive

quantity by deﬁnition. The imaginary part can not be

negative for a positive value of damping in a stable dy-

namic system.

Summary and perspective. – We have developed a ro-

bust analytical framework to explain the negative elastic

moduli (the real parts of E1,E2and G12 ) of lattice ma-

terials under vibrating condition. In the steady-state dy-

namic environment, a metamaterial could subsequently

be developed with both negative elastic moduli and nega-

tive Poisson’s ratio when the cell angle becomes negative

(refer to ﬁgure 1(b)). Similar observation of negative

stiﬀness was made for acoustic metamaterials [31] and

through destabilizations of (meta)stable equilibria of the

constituents [43, 44]. Here we demonstrate such a pos-

sibility for lattice materials in the sub-acoustic range.

Theoretical limits of frequencies are reported for the ﬁrst

time to achieve such negative axial and shear moduli.

The main approach to establish the negative eﬀective

elastic moduli hinges upon exploitation of the dynamic

stiﬀness matrix. In contrast to the conventional static

analysis, the dynamic stiﬀness approach accurately mod-

els the sub-wavelength scale dynamics of the unit cells.

Assuming the undamped limit, an explicit closed-form

expression of the minimum frequency value, referred as

fundamental inﬂection frequency (refer to equation 7) be-

yond which the eﬀective elastic moduli E1and E2be-

come negative has been obtained. This is achieved using

a Taylor series expansion of a relevant dynamic stiﬀness

coeﬃcient. For the shear modulus, a closed-form solu-

tion for the frequency (fundamental inﬂection frequency)

when it becomes negative was not found. However, a

tight bound has been derived (refer to equation 8) . The

frequencies ω∗

E1,E2and ω∗

G12 are fundamental properties

of a lattice metamaterial and they depend only on the

length of the inclined and vertical beams, the bending

rigidity and the mass density per unit length. The imag-

inary part of the elastic moduli remain positive for all fre-

quency values indicating that the material would result in

dynamically stable responses. The expressions of ω∗

E1,E2

and ω∗

G12 clearly show the relative mass (m) and stiﬀness

(EI ) contributions on the critical frequencies. A higher

value of the stiﬀness contribution increases the critical

frequencies and vice versa, while the mass contribution

has an opposite eﬀect. The values of the fundamental

inﬂection frequencies are proportional to the square root

of the ratio EI

m. In addition to this ratio, ω∗

E1,E2de-

pends only on l, while ω∗

G12 depends on both land h/l

ratio. The expressions reveal another interesting fact in

terms of static limits. In the static limit, the contribution

of mass (eﬀect of inertia) tends to zero. This leads to the

value of ω∗

E1,E2and ω∗

G12 as inﬁnity. In other words, there

cannot be a negative value of the Young’s modulus and

the shear modulus in the static case. Thus besides char-

acterizing the negative elastic moduli, our analysis gives

5

FIG. 3: The real and imaginary parts and the amplitude of the normalised value of G12 as a function of frequency for two

diﬀerent values of h/l. Bounds of ω∗

G12 , the frequency at the which the G12 becomes negative, are calculated from equation

8. Here the upper and lower bounds of frequency where G12 becomes negative for the ﬁrst time are shown by ‘x’ and ‘+’

respectively.

a new and alternative explanation of the classical posi-

tive elastic moduli of lattice metamaterials. Although we

have focused here on hexagonal two-dimensional lattices

to present numerical results, the disseminated concepts

can be extended to other forms of lattices and metama-

terials in two and three dimensions, the complexity of

which will depend on the nature of microstructure.

Realization of negative elastic moduli in metamateri-

als is not new, as discussed in the introduction section.

However, the contribution of this paper is to develop

the fundamental limits for the minimum frequency, be-

yond which the negative elastic moduli (Youngs modu-

lus and shear modulus) can be realized. These are de-

rived in closed-form for the ﬁrst time. The limits turn

out to be intrinsic properties of the lattice material and

certain geometric parameters. Exact characterization of

the inﬂuencing intrinsic mechanical properties at the on-

set of negative elastic properties is an important aspect

for mechanical metamaterials. These closed-form lim-

its will have tremendous impact in eﬃcient development

of future microstructured materials within a dynamic

paradigm exploiting the accurate onset of negative elastic

moduli.

In summary, this article sheds light on the negative

axial and shear moduli of lattice materials under sub-

acoustic conditions based on a physics-based insightful

framework. Theoretical limits of the minimum frequency

beyond which the elastic moduli change sign, referred as

the fundamental inﬂection frequencies, have been derived

in closed-form. These frequency values are intrinsic prop-

erty of the lattice and are unique to a given geometrical

pattern and material properties. These expressions and

the disseminated generic concepts can be used to pin-

point the onset of negative elastic moduli and help to

design and develop next-generation of lattice materials

in diﬀerent length-scales.

Acknowledgements. – TM acknowledges the ﬁnancial

support from Swansea University through the award of

the Zienkiewicz Scholarship.

[1] Fleck, N. A., Deshpande, V. S., Ashby, M. F., 2010.

Micro-architectured materials: past, present and future.

Proceedings of the Royal Society of London A: Mathe-

matical, Physical and Engineering Sciences 466 (2121),

2495–2516.

[2] Zadpoor, A. A., 2016. Mechanical meta-materials. Mate-

rials Horizons 3 (5), 371–381.

[3] Mukhopadhyay, T., Adhikari, S., 2016. Eﬀective in-plane

elastic properties of auxetic honeycombs with spatial ir-

regularity. Mechanics of Materials 95, 204 – 222.

[4] Li, J., Chan, C. T., 2004. Double-negative acoustic meta-

material. Phys. Rev. E 70, 055602.

[5] Liu, Z., Zhang, X., Mao, Y., Zhu, Y. Y., Yang, Z., Chan,

C. T., Sheng, P., 2000. Locally resonant sonic materials.

Science 289 (5485), 1734–1736.

[6] Wu, Y., Lai, Y., Zhang, Z.-Q., 2011. Elastic metamateri-

als with simultaneously negative eﬀective shear modulus

and mass density. Phys. Rev. Lett. 107, 105506.

[7] Milton, G. W., Briane, M., Willis, J. R., 2006. On cloak-

ing for elasticity and physical equations with a transfor-

mation invariant form. New Journal of Physics 8 (10),

248.

[8] Cummer, S. A., Christensen, J., Al`u, A., 2016. Control-

ling sound with acoustic metamaterials. Nature Reviews

Materials 1 (3), 16001.

[9] Lai, Y., Wu, Y., Sheng, P., Zhang, Z.-Q., 2011. Hybrid

elastic solids. Nature materials 10 (8), 620.

[10] Liu, Z., Chan, C. T., Sheng, P., 2005. Analytic model of

phononic crystals with local resonances. Phys. Rev. B 71,

014103.

[11] Ma, G., Sheng, P., 2016. Acoustic metamaterials: From

local resonances to broad horizons. Science Advances

6

2 (2) e1501595.

[12] Milton, G. W., Willis, J. R., 2007. On modiﬁcations of

newton’s second law and linear continuum elastodynam-

ics. Proceedings of the Royal Society of London A: Math-

ematical, Physical and Engineering Sciences 463 (2079),

855–880.

[13] Wu, Ying and Lai, Yun and Zhang, Zhao-Qing, 2007.

Eﬀective medium theory for elastic metamaterials in two

dimensions. Physical Review B 76 (20), 205313.

[14] Mei, Jun and Liu, Zhengyou and Wen, Weijia and Sheng,

Ping, 2007. Eﬀective dynamic mass density of compos-

ites. Physical Review B 76 (13), 134205.

[15] Gibson, L., Ashby, M. F., 1999. Cellular Solids Structure

and Properties. Cambridge University Press, Cambridge,

UK.

[16] Mukhopadhyay, T., Adhikari, S., 2017. Stochastic me-

chanics of metamaterials. Composite Structures 162

8597.

[17] Mukhopadhyay, T., Adhikari, S., Batou A., 2019. Fre-

quency domain homogenization for the viscoelastic prop-

erties of spatially correlated quasi-periodic lattices. In-

ternational Journal of Mechanical Sciences 150 784 806.

[18] Mukhopadhyay, T., Adhikari, S., 2016. Equivalent in-

plane elastic properties of irregular honeycombs: An an-

alytical approach. International Journal of Solids and

Structures 91, 169 – 184.

[19] Mukhopadhyay, T., Adhikari, S., 2017. Eﬀective in-plane

elastic moduli of quasi-random spatially irregular hexag-

onal lattices. International Journal of Engineering Sci-

ence 119 142179.

[20] Kumar R. R., Mukhopadhyay T., Pandey K. M., Dey

S., 2019. Stochastic buckling analysis of sandwich plates:

The importance of higher order modes. International

Journal of Mechanical Sciences 152 630-643.

[21] Dey S., Mukhopadhyay T., Naskar S., Dey T. K., Chalak

H. D., Adhikari S., 2019. Probabilistic characterization

for dynamics and stability of laminated soft core sand-

wich plates. Journal of Sandwich Structures & Materials

21(1) 366 397.

[22] Mukhopadhyay T., Adhikari S., 2016. Free vibration

analysis of sandwich panels with randomly irregular hon-

eycomb core. Journal of Engineering Mechanics 142(11)

06016008.

[23] Dey S., Mukhopadhyay T., Adhikari S., 2018. Uncer-

tainty quantiﬁcation in laminated composites: A meta-

model based approach. CRC Press, Taylor & Francis

Group.

[24] Mukhopadhyay, T., Mahata, A., Adhikari, S., Zaeem,

M. A., 2017. Eﬀective elastic properties of two dimen-

sional multiplanar hexagonal nanostructures. 2D Materi-

als 4 (2), 025006.

[25] Mahata A., Mukhopadhyay T., 2018. Probing the

chirality-dependent elastic properties and crack propa-

gation behavior of single and bilayer stanene. Physical

Chemistry Chemical Physics 20 2276822782.

[26] Mukhopadhyay T., Mahata A., Adhikari S., Asle Za-

eem M., 2018. Probing the shear modulus of two-

dimensional multiplanar nanostructures and heterostruc-

tures. Nanoscale 10 5280 5294.

[27] Mukhopadhyay T., Mahata A., Adhikari S., Asle Zaeem

M., 2017. Eﬀective mechanical properties of multilayer

nano-heterostructures. Scientiﬁc Reports 7 15818.

[28] Muhlestein, M. B., Haberman, M. R., 2016. A microme-

chanical approach for homogenization of elastic metama-

terials with dynamic microstructure. Proceedings of the

Royal Society of London A: Mathematical, Physical and

Engineering Sciences 472 (2192).

[29] Srivastava, A., 2015. Elastic metamaterials and dy-

namic homogenization: a review. International Journal

of Smart and Nano Materials 6 (1), 41–60.

[30] Hussein, M. I., Leamy, M. J., Ruzzene, M., 05 2014. Dy-

namics of phononic materials and structures: historical

origins, recent progress, and future outlook. Applied Me-

chanics Reviews 66 (4), 040802–38.

[31] Fang, N., Xi, D., Xu, J., Ambati, M., Srituravanich, W.,

Sun, C., Zhang, X., 06 2006. Ultrasonic metamaterials

with negative modulus. Nature Materials 5 (6), 452–456.

[32] Yang, Z., Mei, J., Yang, M., Chan, N. H., Sheng, P., Nov

2008. Membrane-type acoustic metamaterial with nega-

tive dynamic mass. Physical Review Letters 101, 204301.

[33] Ding, Y., Liu, Z., Qiu, C., Shi, J., 2007. Metamate-

rial with simultaneously negative bulk modulus and mass

density. Physical Review Letters 99, 093904.

[34] Huang, H., Sun, C., 2011. Locally resonant acoustic

metamaterials with 2D anisotropic eﬀective mass density.

Philosophical Magazine 91 (6), 981–996.

[35] Torrent, D., Sanchez-Dehesa, J., 2008. Anisotropic mass

density by two-dimensional acoustic metamaterials. New

Journal of Physics 10 (2), 023004.

[36] Fleury, R., Sounas, D. L., Sieck, C. F., Haberman,

M. R., Al`u, A., 2014. Sound isolation and giant linear

nonreciprocity in a compact acoustic circulator. Science

343 (6170), 516–519.

[37] Miri, M.-A., Verhagen, E., Al`u, A., May 2017. Op-

tomechanically induced spontaneous symmetry breaking.

Physical Review A 95, 053822.

[38] Banerjee, J. R., 1997. Dynamic stiﬀness formulation for

structural elements: A general approach. Computer and

Structures 63 (1), 101–103.

[39] Doyle, J. F., 1989. Wave Propagation in Structures.

Springer Verlag, New York.

[40] Manohar, C., Adhikari, S., 1998. Dynamic stiﬀness of

randomly parametered beams. Probabilistic Engineering

Mechanics 13 (1), 39 – 51.

[41] Mukhopadhyay T., Adhikari S., Alu A, 2019. Probing the

frequency-dependent elastic moduli of lattice materials.

Acta Materialia 165 654-665.

[42] Wu, Ying and Zhang, Zhao-Qing, 2009. Dispersion rela-

tions and their symmetry properties of electromagnetic

and elastic metamaterials in two dimensions. Physical

Review B 79 (19), 195111.

[43] Grima, J. N., Caruana-Gauci, R., 2012. Mechanical

metamaterials: materials that push back. Nature mate-

rials 11 (7), 565.

[44] Nicolaou, Z. G., Motter, A. E., 2012. Mechanical meta-

materials with negative compressibility transitions. Na-

ture materials 11 (7), 608.

Supplementary material

Theoretical limits for negative elastic moduli in sub-acoustic lattice materials

T. Mukhopadhyaya,∗, S. Adhikarib, A. Aluc

aDepartment of Engineering Science, University of Oxford, Oxford, UK

bCollege of Engineering, Swansea University, Swansea, UK

cAdvanced Science Research Center, City University of New York, New York, USA

To obtain the closed-form theoretical limits of natural frequency, beyond which negative elastic moduli

can be realized in lattice materials, the frequency-dependent expressions for elastic moduli of lattices

are needed. On the basis of a unit cell based approach, closed-form expressions for the complex elastic

moduli are derived as a function of frequency by employing the dynamic stiﬀness matrix of a damped

beam element. In this supplementary material, for the ready-reference of readers, we have provided the

derivation of the dynamic stiﬀness matrix for a single beam element ﬁrst [19]. Thereby, the derivation of

closed-form expressions of the frequency-dependent elastic moduli of lattice materials are presented [19].

The theoretical limits of frequency to obtain negative elastic moduli are obtained based on their respective

closed-form expressions. It can be noted that the contribution of this article is in deriving the limits for

negative elastic moduli of lattice materials based on the closed-form expressions of the frequency-dependent

elastic moduli. For the completeness of this article, here we also show results for validation of the proposed

dynamic stiﬀness based framework, based on which the theoretical limits are derived.

Contents

1 Dynamic stiﬀness approach 2

1.1 Equationofmotion........................................ 2

1.2 Frequency dependent shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Element dynamic stiﬀness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 The derivation of frequency-dependent elastic moduli 6

2.1 Derivation of Young’s modulus E1................................ 7

2.2 Derivation of Young’s modulus E2................................ 8

∗Corresponding author: Tanmoy Mukhopadhyay

Email address: tanmoy.mukhopadhyay@eng.ox.ac.uk (T. Mukhopadhyay )

Preprint submitted to Physical Review Letters March 4, 2019

2.3 Derivation of shear modulus G12 ................................. 9

2.4 Derivation of Poisson’s ratios ν12 and ν21 ............................ 10

3 Theoretical limits of frequency for negative elastic moduli 11

4 Numerical validation of the dynamic stiﬀness based framework 13

1. Dynamic stiﬀness approach

1.1. Equation of motion

Dynamic motion of the overall cellular structure corresponds to vibration of individual beams which

constitute each hexagonal unit cells. A pictorial depiction of the beam is shown in ﬁgure 1(c) of the main

manuscript. One honeycomb unit cell under dynamic environment is shown in ﬁgure 1(b) of the main

manuscript, wherein a vibrating mode of each constituent members is symbolically shown. If external

forces are applied to such vibrating honeycomb, the members will deform following a diﬀerent rule. Thus

the eﬀective elastic moduli of the entire lattice will be diﬀerent from conventional static elastic moduli.

The eﬀect of vibration in the eﬀective elastic moduli of hexagonal lattices can be captured based on

dynamic stiﬀness method [9, 16, 19]. The dynamic stiﬀness matrix of a single beam element is derived ﬁrst

(section 1); thereby the expressions of frequency dependent elastic moduli of the lattice metamaterial are

developed based on the elements of the dynamic stiﬀness matrix of a single beam (section 2).

The equation of motion of free vibration of a damped beam can be expressed as

EI ∂4V(x, t)

∂x4+bc1

∂5V(x, t)

∂x4∂t +m∂2V(x, t)

∂t2+bc2

∂V (x, t)

∂t = 0 (1)

It is assumed that the behaviour of the beam follows the Euler-Bernoulli hypotheses. In the above equation

EI is the bending rigidity, mis mass per unit length, bc1is the strain-rate-dependent viscous damping coef-

ﬁcient, bc2is the velocity-dependent viscous damping coeﬃcient and V(x, t) is the transverse displacement.

The length of the beam is assumed to be L. Considering a harmonic motion with frequency ωwe have

V(x, t) = v(x) exp [iωt] (2)

where i = √−1. Substituting this in the beam equation (1) one obtains

EI d4v

dx4+ iωbc1

d4v

dx4−mω2v+ iωbc2v= 0 (3)

or d4v

dx4−b4v= 0 (4)

where

b4=mω2−iωbc2

EI + iωbc1

(5)

2

Following the damping convention in dynamic analysis as in [18], we consider stiﬀness and mass proportional

damping. Therefore, we express the damping constants as

bc1=ζk(EI) and bc2=ζm(m) (6)

where ζkand ζmare stiﬀness and mass proportional damping factors. Substituting these, from Eq. (5) we

have

b4=mω2(1 −iζm/ω)

EI (1 + iωζk)(7)

The constant bis in general a complex number for any physically realistic damping values. The eﬀect of

mass proportional damping factor ζmlinearly decreases with higher frequency whereas the eﬀect of stiﬀness

proportional damping factor ζklinearly increases with higher frequency. To obtain the characteristic

equation, we consider

v(x) = exp [λx] (8)

Substituting this in Eq. (4) one obtains

λ4−b4= 0 (9)

or λ= ib, −ib, b, −b(10)

Next we use these solutions to obtain the dynamic shape functions of the beam.

1.2. Frequency dependent shape functions

For classical (static) ﬁnite element analysis of beams, cubic polynomials are used as shape functions

(see for example [21]). Here we aim to incorporate frequency dependent dynamic shape functions, as used

with the framework of the dynamic ﬁnite element method. The dynamic ﬁnite element method belongs

to the general class of spectral methods for linear dynamical systems [9]. This approach, or approaches

very similar to this, is known by various names such as the dynamic stiﬀness method [1–7, 10, 11, 17, 20],

spectral ﬁnite element method [9, 13] and dynamic ﬁnite element method [14, 15].

The dynamic shape functions are obtained such that the equation of dynamic equilibrium is satisﬁed

exactly at all points within the element. Similar to the classical ﬁnite element method, assume that the

frequency-dependent displacement within an element is interpolated from the nodal displacements as

v(x, ω) = NT(x, ω)b

v(ω) (11)

Here b

v(ω)∈Cnis the nodal displacement vector N(x, ω)∈Cnis the vector of frequency-dependent shape

functions and n= 4 is the number of the nodal degrees-of-freedom. Suppose the sj(x, ω)∈C, j = 1,··· ,4

3

are the basis functions which exactly satisfy Eq. (4). It can be shown that the shape function vector can

be expressed as

N(x, ω) = Γ(ω)s(x, ω) (12)

where the vector s(x, ω) = {sj(x, ω)}T,∀j= 1,··· ,4 and the complex matrix Γ(ω)∈C4×4depends on

the boundary conditions. The elements of s(x, ω) constitutes exp[λjx] where the values of λjare obtained

from the solution of the characteristics equation as given in Eq. (10). An element for the damped beam

under bending vibration is shown in ﬁgure 1(c) of the main manuscript. The degrees-of-freedom for each

nodal point include a vertical and a rotational degrees-of-freedom.

In view of the solutions in Eq. (10), the displacement ﬁeld with the element can be expressed by

a linear combination of the basic functions e−ibx, eibx, ebx and e−bx so that in our notations s(x, ω) =

e−ibx, eibx , ebx, e−bx T. We can also express s(x, ω) in terms of trigonometric functions. Considering e±ibx =

cos(bx)±i sin(bx) and e±bx = cosh(bx)±i sinh(bx), the vector s(x, ω) can be alternatively expressed as

s(x, ω) =

sin(bx)

cos(bx)

sinh(bx)

cosh(bx)

∈C4(13)

For steady-state dynamic response, the displacement ﬁeld within the element can be expressed as

v(x) = s(x, ω)Tv(14)

where v∈C4is the vector of constants to be determined from the boundary conditions.

The relationship between the shape functions and the boundary conditions can be represented as in

Table 1, where boundary conditions in each column give rise to the corresponding shape function. Writing

Table 1: The relationship between the boundary conditions and the shape functions for the bending vibration of beams.

N1(x, ω)N2(x, ω)N3(x, ω)N4(x, ω)

y(0) 1 0 0 0

dy

dx(0) 0 1 0 0

y(L)0010

dy

dx(L)0001

4

Eq. (14) for the above four sets of boundary conditions, one obtains

[R]y1,y2,y3,y4=I(15)

where

R=

s1(0) s2(0) s3(0) s4(0)

ds1

dx(0) ds2

dx(0) ds3

dx(0) ds4

dx(0)

s1(L)s2(L)s3(L)s4(L)

ds1

dx(L)ds2

dx(L)ds3

dx(L)ds4

dx(L)

(16)

and ykis the vector of constants giving rise to the kth shape function. In view of the boundary conditions

represented in Table 1 and equation (15), the shape functions for bending vibration can be shown to be

given by Eq. (12) where

Γ(ω) = y1,y2,y3,y4T=R−1T(17)

By obtaining the matrix Γ(ω) from the above equation, the shape function vector can be obtained from Eq.

(12). After some algebraic simpliﬁcations, we have represented the frequency dependent complex shape

functions as

N1(x, ω)

N2(x, ω)

N3(x, ω)

N4(x, ω)

=

1

2

cS+C s

cC−1−1

2

1+sS−cC

cC−1−1

2

cS+C s

cC−1

1

2

cC+sS −1

cC−1

1

2

cC+sS −1

b(cC−1)

1

2

−Cs+cS

b(cC−1) −1

2

1+sS−cC

b(cC−1) −1

2

−Cs+cS

b(cC−1)

−1

2

S+s

cC−1

1

2

C−c

cC−1

1

2

S+s

cC−1−1

2

C−c

cC−1

1

2

C−c

b(cC−1) −1

2

S−s

b(cC−1) −1

2

C−c

b(cC−1) −1

2

S−s

b(cC−1)

sin bx

cos bx

sinh bx

cosh bx

(18)

where

C= cosh(bL), c = cos(bL), S = sinh(bL) and s= sin(bL) (19)

and bis deﬁned in (7).

1.3. Element dynamic stiﬀness matrix

The stiﬀness and mass matrices can be obtained following the conventional variational formulation [8].

The only diﬀerence is instead of classical cubic polynomials as the shape functions, frequency dependent

shape functions in (18) should be used. It is convenient to deﬁne the dynamic stiﬀness matrix as

D(ω) = K(ω)−ω2M(ω) (20)

so that the equation of dynamic equilibrium is

D(ω)b

v(ω) = b

f(ω) (21)

5

In Eq. (20), the frequency-dependent stiﬀness and mass matrices can be obtained from

K(ω) = EI ZL

0

d2N(x, ω)

dx2

d2NT(x, ω)

dx2dx(22)

and M(ω) = mZL

0

N(x, ω)NT(x, ω)dx(23)

After some algebraic simpliﬁcations it can be shown that the dynamic stiﬀness matrix is given by the

following closed-form expression

D(ω) = EIb

(cC −1)

−b2(cS +Cs)−sbS b2(S+s)−b(C−c)

−sbS −Cs +cS b (C−c)−S+s

b2(S+s)b(C−c)−b2(cS +Cs)sbS

−b(C−c)−S+s sbS −Cs +cS

(24)

The elements of this matrix are frequency dependent complex quantities because bis a function of ωand

the damping factors.

2. The derivation of frequency-dependent elastic moduli

Considering only the static deformation of a unit cell, [12] obtained the equivalent elastic moduli of the

hexagonal cellular materials as

E1GA =Et

l3cos θ

(h

l+ sin θ) sin2θ(25)

E2GA =Et

l3(h

l+ sin θ)

cos3θ(26)

ν12GA =cos2θ

(h

l+ sin θ) sin θ(27)

ν21GA =(h

l+ sin θ) sin θ

cos2θ(28)

and G12GA =Et

l3h

l+ sin θ

h

l2(1 + 2h

l) cos θ(29)

where (.)GA represents the expressions of elastic moduli of regular hexagonal honeycombs. The cell walls

are treated as beams of thickness tand Young’s modulus E. The quantities land hare the lengths of

inclined cell walls having inclination angle θand the vertical cell walls respectively. A key interest in this

section is to obtain equivalent expressions when harmonic forcing is considered. The central idea behind

the proposed derivation is to exploit the physical interpretation of the elements of the dynamic stiﬀness

matrix obtained in the previous section.

6

Using equation (24), the analytical expressions of the frequency dependent in-plane elastic moduli will

be obtained. For the purpose of deriving the expressions, the dynamic stiﬀness matrix is written in the

following form for notational convenience

D(ω) =

D11 D12 D13 D14

D21 D22 D23 D24

D31 D32 D33 D34

D41 D42 D43 D44

(30)

where Dij (i, j = 1,2,3,4) has the expressions corresponding to the terms of equation (24).

2.1. Derivation of Young’s modulus E1

One cell wall is considered for deriving the expression of the Young’s modulus E1under the application

of stress in direction - 1 as shown in ﬁgure 1(a) [12]. In the free body diagram of the slant member in ﬁgure

1(a), the rotational displacements of both ends and the bending displacement of one end is considered as

zero. To satisfy the equilibrium of forces in direction -2, the force Cis needed to be zero. Thus from the

dynamic stiﬀness matrix presented in equation (30), the bending deﬂection of one end of the slant member

with respect to the other end can be written as

δ=Psin θ

D33

(31)

where P=σ1(h+lsin θ)¯

b(geometric dimensions of a single honeycomb cell is shown in ﬁgure 1(b) of the

main manuscript. ¯

bis the width of the beam i.e. thickness of the honeycomb sheet). The component of

δin direction - 1 is δsin θ. Thus the strain component in direction - 1 due to applied stress in the same

direction can be expressed as

11 =δsin θ

lcos θ

=σ1(h+lsin θ)¯

bsin2θ

D33lcos θ

(32)

The expression of D33 is given in equation (24) and (30). Replacing the expression for D33 and I=¯

bt3

12 ,

the Young’s modulus E1can be obtained as

E1(ω) = σ1

11

=D33lcos θ

(h+lsin θ)¯

bsin2θ

=Et3lcos θb3(cos(bl) sinh(bl) + cosh(bl) sin(bl))

12(h+lsin θ) sin2θ(1 −cos(bl) cosh(bl))

(33)

7

Figure 1: Deformed shapes and free body diagrams under the application of direct stresses and shear stress. The undeformed

shapes of the hexagonal cell are indicated using blue colour for each of the loading conditions.

The expression of bis provided in equation (7). Eis the intrinsic elastic modulus of the honeycomb material

and tis the thickness of honeycomb wall.

2.2. Derivation of Young’s modulus E2

Similar to the derivation of E1, the bending deformation of one end of the slant beam under the

application of σ2(as shown in ﬁgure 1(b)) can be expressed as

δ=Wcos θ

D33

(34)

where W=σ2l¯

bcos θ. The expression for strain component in direction - 2 due to application of stress in

the same direction can be obtained as

22 =δcos θ

(h+lsin θ)

=σ2l¯

bcos3θ

D33(h+lsin θ)

(35)

8

Replacing the expression for D33 and I=¯

bt3

12 , the Young’s modulus E2can be obtained as

E2(ω) = σ2

22

=D33(h+lsin θ)

l¯

bcos3θ

=Et3(h+lsin θ)b3(cos(bl) sinh(bl) + cosh(bl) sin(bl))

12lcos3θ(1 −cos(bl) cosh(bl))

(36)

The expression of bis provided in equation (7), Eis the intrinsic elastic modulus of the honeycomb material

and tis the thickness of honeycomb wall as before.

2.3. Derivation of shear modulus G12

For deriving the expression of G12, two members of the honeycomb cell are needed to be considered

(vertical member with length h

2and a slant member with length l) as shown in ﬁgure 1(c). The points

A, B and C will not have any relative movement due to symmetrical structure. The total shear deﬂection

usconsists of two components, bending deﬂection of the member BD and its deﬂection due to rotation of

joint B.

It can be noted here that the elements of the dynamic stiﬀness matrix (refer to equation(30)) will be

diﬀerent for the vertical member and the slant member due to their diﬀerent lengths. Using the stiﬀness

components of the dynamic stiﬀness matrix (refer to equation (30)), the bending deformation of point D

with respect to point B in direction - 1 can be obtained as

δb=F

Dv

33 −Dv

34Dv

43

Dv

44 =F

Dv

33 −(Dv

34)2

Dv

44 (37)

Here F= 2τl¯

bcos θand we make use of the symmetry of the elements of the dynamic stiﬀness matrix.

The superscript vin the elements of the dynamic stiﬀness matrix is used to indicate the stiﬀness element

corresponding to the vertical member.

From the free body diagram presented in ﬁgure 1(c),

M=F h

4(38)

On the basis of equation (30), deﬂection of the end B with respect to the end C due to application of

moment Mat the end B is given as

δr=M

−Ds

43

(39)

Here the superscript sin D43 is used to indicate the stiﬀness element corresponding to the slant member

and the negative arise due to the direction of the rotation as given in ﬁgure 1(c) of the main manuscript.

9

Thus the rotation of joint B can be expressed as

φ=δr

l

=−F h

4lDs

43

(40)

Total shear deformation under the application of shear stress τcan be expressed as

us=1

2φh +δb

=−F h2

8lDs

43

+F

Dv

33 −(Dv

34)2

Dv

44 (41)

The shear strain is given by

γ=2us

(h+lsin θ)

=F

(h+lsin θ)

−h2

4lDs

43

+2

Dv

33 −(Dv

34)2

Dv

44

=2τl¯

bcos θ

(h+lsin θ)

−h2

4lDs

43

+2

Dv

33 −(Dv

34)2

Dv

44

(42)

Replacing the expressions for the stiﬀness components from equation (24) and (30), the shear modulus can

be obtained as

G12(ω) = τ

γ=(h+lsin θ)

2l¯

bcos θ

1

−h2

4lDs

43 +2

Dv

33−(Dv

34)2

Dv

44 !

=(h+lsin θ)

2l¯

bcos θ

4EI b3sin(bl) sinh(bl) (1 + cos(bh/2) cosh(bh/2))

h2b(1 −cos(bl) cosh(bl)) (1 + cos(bh/2) cosh(bh/2))

+ 8lsin(bl) sinh(bl) (cosh(bh/2) sin(bh/2) −sinh(bh/2) cos(bh/2))

=Et3(h+lsin θ)b3sin(bl) sinh(bl) (1 + cos(bh/2) cosh(bh/2))

6lcos θ[h2b(1 −cos(bl) cosh(bl)) (1 + cos(bh/2) cosh(bh/2))

+8lsin(bl) sinh(bl) (cosh(bh/2) sin(bh/2) −sinh(bh/2) cos(bh/2))]

(43)

The expression of the complex variable bis provided in equation (7).

2.4. Derivation of Poisson’s ratios ν12 and ν21

The strain components in direction - 1 and direction - 2 under the application of stress σ1are given by

(refer to ﬁgure 1(a))

11 =δsin θ

lcos θ(44)

10

21 =−δcos θ

h+lsin θ(45)

Thus the Poisson’s ratio for loading direction - 1 can be obtained as

ν12 =−21

11

=lcos2θ

(h+lsin θ) sin θ

(46)

Similarly the Poisson’s ratio for loading direction - 2 can be obtained as

ν21 =−12

22

=(h+lsin θ) sin θ

lcos2θ

(47)

It can be noted that the in-plane Poisson’s ratios (Equation 46 and 47) are not dependent on frequency

and the expressions are same as the case of static deformation provided by [12].

3. Theoretical limits of frequency for negative elastic moduli

Expressions of the frequency-dependent elastic moduli, as derived in the preceding section, can be

summarized as [19]:

E1(ω) = D33lcos θ

(h+lsin θ)¯

bsin2θ=Et3lcos θb3(cos(bl) sinh(bl) + cosh(bl) sin(bl))

12(h+lsin θ) sin2θ(1 −cos(bl) cosh(bl)) (48)

E2(ω) = D33(h+lsin θ)

l¯

bcos3θ=Et3(h+lsin θ)b3(cos(bl) sinh(bl) + cosh(bl) sin(bl))

12lcos3θ(1 −cos(bl) cosh(bl)) (49)

G12(ω) = (h+lsin θ)

2l¯

bcos θ

1

−h2

4lDs

43 +2

Dv

33−(Dv

34)2

Dv

44 !

=Et3(h+lsin θ)b3sin(bl) sinh(bl) (1 + cos(bh/2) cosh(bh/2))

6lcos θ[h2b(1 −cos(bl) cosh(bl)) (1 + cos(bh/2) cosh(bh/2))

+8lsin(bl) sinh(bl) (cosh(bh/2) sin(bh/2) −sinh(bh/2) cos(bh/2))]

(50)

It can be noted here that the expressions of E1and E2are proportional to the complex frequency-dependent

element D33 of the [D(ω)] matrix. Therefore, we study it’s behaviour in the undamped limit to understand

the if the real part of E1and E2can become negative. Assuming no damping in the system, the parameter

bbecomes

b4=mω2

EI (51)

11

Substituting this in the expression of D33 and expanding the expression by a Taylor series in the frequency

parameter ωwe have

D33 = 12 EI

l3−13

35 m lω2−59

161700

l5m2ω4

EI −551

794593800

l9m3ω6

EI 2+··· (52)

Note that coeﬃcients of some higher order terms of ωare negative. We observe that D33 appears as a

multiplicative term in the expressions of E1(ω) and E2(ω) in equations (48) and (49) and the other terms

are positive. Therefore, near the vicinity of ω≈0, there exists a frequency beyond where the eﬀective

elastic moduli of honeycomb will be negative. Retaining up to terms of order ω4in equation (52), the

critical value of ωcan be obtained by setting D33 = 0 as

D33 ≈12 EI

l3−13

35 m lω2−59

161700

l5m2ω4

EI = 0

or ω∗

E1,E2≈5.598 1

l2rEI

m

(53)

Here, ω∗

E1,E2represents the fundamental inﬂection frequency, where the Young’s moduli change sign from

positive to negative. For lightly damped systems, beyond this frequency value, the equivalent Young’s

moduli E1and E2will be negative for the ﬁrst time when viewed on the frequency axis. As the frequency

increases, the Young’s moduli will become positive and negative again. The signiﬁcance of the fundamental

inﬂection frequency derived in equation (53) is that it is the lowest frequency value beyond which the

eﬀective Young’s modulus can become negative. Physically, negative Young’s modulus means that when

a force is applied at the inﬂection frequency, the direction of the steady-state dynamic response will be in

the opposite direction to the applied forcing at the same frequency.

Unlike the case of Young’s moduli, the frequency-dependent closed-form expression 50 for shear modulus

shows a compound eﬀect of multiple elements of the [D(ω)] matrix. It is possible to obtain the expression

of a tight bound for the frequency, beyond which the shear modulus becomes negative. For the shear

modulus, the frequency dependent expression (50) can be extended in a Taylor series in ωabout ω= 0 as

G12(ω) = (h+lsin θ)

2l¯

bcos θ24 EI

h2(2h+l)−11

420

m(9h5+ 8l5)ω2

h2(2h+l)2

−1

46569600

m2(55461h9l−191664h5l5+ 198912l9h+ 3111h10 + 14272l10)ω4

EI h2(2h+l)3+···(54)

Considering only up to the second-order terms we can obtain the upper-bound of the frequency, beyond

12

which the G12 will be negative as

24 EI

h2(2h+l)−11

420

m(9h5+ 8l5)ω2

h2(2h+l)2≈0

or ω∗

G12 /30.2715s1 + 2(h/l)

8 + 9(h/l)5

1

l2rEI

m

(55)

The lower bound is obtained by considering the numerator of G12 in equation (50) and setting it to zero.

Expanding the numerator of G12 in a Taylor series in bwe have

1

1440 h2l3(2 h+l)32 l4+ 15 h4b4−2h2l3(2 h+l)≈0 (56)

Solving this equation for bone obtains

b≈2√30

4

√160 l4+ 75 h4(57)

Using the relationship of ωfor the undamped case in equation (51) results in the following relationship

ω∗

G12 '120

q160 + 75 (h/l)4

1

l2rEI

m(58)

Combining equations (55) and (58) we obtain the following fundamental inequality regarding the frequency

for negative value of G12

120

q160 + 75 (h/l)4

1

l2rEI

m≤ω∗

G12 ≤30.2715s1 + 2(h/l)

8 + 9(h/l)5

1

l2rEI

m(59)

Here, ω∗

G12 represents the fundamental inﬂection frequency for shear modulus, where the shear modulus

changes sign. Unlike the equivalent expression for the Young’s moduli E1and E2in equation (53), for the

minimum frequency above which G12 becomes negative depends on the h/l ratio.

It can be noted that the derivation of the eﬃcient closed-form limits (equations 53 and 59) have only

been possible due to the proposed dynamic stiﬀness based approach to obtain the expressions for the elastic

moduli in a vibrating environment.

4. Numerical validation of the dynamic stiﬀness based framework

For discussing the results concerning negative elastic moduli with a high degree of conﬁdence, the

dynamic stiﬀness based framework needs to be validated ﬁrst. We have presented representative results

for validation of the analytical expression for frequency dependent Young’s modulus here [19].

Two diﬀerent validations are presented in this section. To verify the validity of the derived expression of

the dynamic stiﬀness matrix we compare the results with the conventional ﬁnite element method considering

13

Figure 2: (a) Frequency-dependent responses of a pinned-pinned beam under the application of a unit moment at the right

edge (b) Frequency dependent Young’s modulus E1(with h/l = 1) of hexagonal lattices with θ= 30◦and ζm= 0.05 and

ζk= 0.002 (obtained using the analytical expressions and ﬁnite element method). A validation of the ﬁnite element (denoted

by FE) model for static case (i.e. ω→0) is shown in the inset along with a convergence study for the number of unit cells.

The validation is presented with respect to the static analytical expression provided by [12] (denoted by GA).

a single beam element ﬁrst. In ﬁgure 2(a) the responses of a pinned-pinned beam under the application of

unit moment at the right edge are shown, wherein the rotational responses (radian) at the left and right

edges are compared. The conventional ﬁnite element results are obtained by discretizing the beam into 100

elements and taking ﬁrst 20 modes in the response calculations. The dynamic stiﬀness results are obtained

using the closed-form expression obtained from only one element. The results match very well, conﬁrming

the validity and eﬃciency of applying the dynamic stiﬀness method.

After establishing that a single beam element using the dynamic stiﬀness matrix is capable to capture

the dynamic behavior, we have validated the derived closed-form formulae for frequency-dependent elastic

moduli with respect to the ﬁnite element approach. We have written a bespoke ﬁnite element code for

the honeycomb lattice structure, where the stiﬀness matrix of each of the beam elements is used as the

dynamic stiﬀness matrix. The ﬁnite element approach here involves transforming the element dynamic

stiﬀness matrices for all beam elements into the global coordinate system, assembling them and applying

the boundary conditions. The ﬁnite element model itself is validated with literature [12] in case of the static

deformations (i.e. ω→0) as shown in the inset of ﬁgure 2(b). The geometry, node numbers and nodal

connectivity of the static case remains the same for the dynamic case. Therefore, the current validation

14

along with the dynamic validation for a single beam ensures the validation for the dynamic responses

of the entire lattice. Figure 2(b) shows representative results obtained from the proposed closed-form

expressions for the frequency dependent Young’s modulus along with the results generated using ﬁnite

element simulations. Numerical results obtained from the ﬁnite element approach for every frequency

value is compared with the closed-form analytical expressions derived in the paper. Minor diﬀerence in the

numerical values of the two results corresponding to a wide range of frequency corroborates the validity of

the proposed expressions.

References

[1] Adhikari, S., Manohar, C. S., November 2000. Transient dynamics of stochastically parametered

beams. ASCE Journal of Engineering Mechanics 126 (11), 1131–1140.

[2] Banerjee, J. R., 1989. Coupled bending torsional dynamic stiﬀness matrix for beam elements. Inter-

national Journal for Numerical Methods in Engineering 28 (6), 1283–1298.

[3] Banerjee, J. R., 1997. Dynamic stiﬀness formulation for structural elements: A general approach.

Computer and Structures 63 (1), 101–103.

[4] Banerjee, J. R., Fisher, S. A., 1992. Coupled bending torsional dynamic stiﬀness matrix for axially

loaded beam elements. International Journal for Numerical Methods in Engineering 33 (4), 739–751.

[5] Banerjee, J. R., Williams, F. W., 1985. Exact bernoulli-euler dynamic stiﬀness matrix for a range of

tapered beams. International Journal for Numerical Methods in Engineering 21 (12), 2289–2302.

[6] Banerjee, J. R., Williams, F. W., 1992. Coupled bending-torsional dynamic stiﬀness matrix for timo-

shenko beam elements. Computer and Structures 42 (3), 301–310.

[7] Banerjee, J. R., Williams, F. W., 1995. Free-vibration of composite beams - an exact method using

symbolic computation. Journal of Aircraft 32 (3), 636–642.

[8] Dawe, D., 1984. Matrix and Finite Element Displacement Analysis of Structures. Oxford University

Press, Oxford, UK.

[9] Doyle, J. F., 1989. Wave Propagation in Structures. Springer Verlag, New York.

[10] Ferguson, N. J., Pilkey, W. D., 1993. Literature review of variants of dynamic stiﬀness method, Part

1: The dynamic element method. The Shock and Vibration Digest 25 (2), 3–12.

[11] Ferguson, N. J., Pilkey, W. D., 1993. Literature review of variants of dynamic stiﬀness method, Part

2: Frequency-dependent matrix and other. The Shock and Vibration Digest 25 (4), 3–10.

15

[12] Gibson, L., Ashby, M. F., 1999. Cellular Solids Structure and Properties. Cambridge University Press,

Cambridge, UK.

[13] Gopalakrishnan, S., Chakraborty, A., Mahapatra, D. R., 2007. Spectral Finite Element Method.

Springer Verlag, New York.

[14] Hashemi, S. M., Richard, M. J., 2000. Free vibrational analysis of axially loaded bending-torsion

coupled beams: a dynamic ﬁnite element. Computer and Structures 77 (6), 711–724.

[15] Hashemi, S. M., Richard, M. J., Dhatt, G., 1999. A new Dynamic Finite Element (DFE) formulation

for lateral free vibrations of Euler-Bernoulli spinning beams using trigonometric shape functions.

Journal of Sound and Vibration 220 (4), 601–624.

[16] Manohar, C., Adhikari, S., 1998. Dynamic stiﬀness of randomly parametered beams. Probabilistic

Engineering Mechanics 13 (1), 39 – 51.

[17] Manohar, C. S., Adhikari, S., January 1998. Dynamic stiﬀness of randomly parametered beams. Prob-

abilistic Engineering Mechanics 13 (1), 39–51.

[18] Meirovitch, L., 1997. Principles and Techniques of Vibrations. Prentice-Hall International, Inc., New

Jersey.

[19] Mukhopadhyay, T., Adhikari, S., Alu, A., 2019. Probing the frequency-dependent elastic moduli of

lattice materials. Acta Materialia 165, 654–665.

[20] Paz, M., 1980. Structural Dynamics: Theory and Computation, 2nd Edition. Van Nostrand, Reinhold.

[21] Petyt, M., 1998. Introduction to Finite Element Vibration Analysis. Cambridge University Press,

Cambridge, UK.

16