ArticlePDF Available

Theoretical limits for negative elastic moduli in subacoustic lattice materials

Authors:

Abstract and Figures

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 fundamental 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 developed 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 accelerate the process of its potential exploitation in various engineered materials and structural systems under dynamic regime across the length-scales.
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=2(1 m)
EI (1 + ζk). 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 (ω0), 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,E25.598 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
mω
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= 30and 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)
∂x4t +m2V(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
dx42v+ iωbc2v= 0 (3)
or d4v
dx4b4v= 0 (4)
where
b4=2iωbc2
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=2(1 iζm)
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
λ4b4= 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 eibx, eibx, ebx and ebx so that in our notations s(x, ω) =
eibx, eibx , ebx, ebx 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 vC4is 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=R1T(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
cC11
2
1+sScC
cC11
2
cS+C s
cC1
1
2
cC+sS 1
cC1
1
2
cC+sS 1
b(cC1)
1
2
Cs+cS
b(cC1) 1
2
1+sScC
b(cC1) 1
2
Cs+cS
b(cC1)
1
2
S+s
cC1
1
2
Cc
cC1
1
2
S+s
cC11
2
Cc
cC1
1
2
Cc
b(cC1) 1
2
Ss
b(cC1) 1
2
Cc
b(cC1) 1
2
Ss
b(cC1)
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(Cc)
sbS Cs +cS b (Cc)S+s
b2(S+s)b(Cc)b2(cS +Cs)sbS
b(Cc)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=2
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
l313
35 m lω259
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 ω0, 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
l313
35 m lω259
161700
l5m2ω4
EI = 0
or ω
E1,E25.598 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(55461h9l191664h5l5+ 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)20
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 h4b42h2l3(2 h+l)0 (56)
Solving this equation for bone obtains
b230
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
mω
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= 30and 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. ω0) 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. ω0) 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
... Articially engineered lattice microstructures can oer physical properties that are rare in naturallyoccurring materials. They can be designed to exhibit extraordinary multi-functional properties like negative Poisson's ratio, extremely lightweight characteristics, negative elastic moduli, pentamode material characteristics (meta-uid), negative mass density, programmable constitutive laws, tunable wave propagation characteristics etc. [1,2,3,4,5,6,7,8,9,10]. Their properties span over varying ranges and degrees depending on application-specic requirements that have garnered attention of the scientic community signicantly over the last decade. ...
... For a beam with rectangular cross-section if b and t are the width and thickness, respectively, we have A = bt and I = bt 3 12 . Note that in the context of honeycomb lattice, the constituting beam elements have cell wall thickness t, while depth of the honeycomb panel in the direction perpendicular to 1-2 plane is b. ...
Article
Full-text available
Characterization of the effective elastic properties of lattice-type materials is essential for adopting such artificial microstructures in various multi-functional mechanical systems across varying length-scales with the requirement of adequate structural performances. Even though the recent advancements in manufacturing have enabled large-scale production of the complex lattice microstructures, it simultaneously brings along different aspects of manufacturing irregularity into the system. One of the most prevailing such effects is the presence of intrinsic residual stresses, which can significantly influence the effective elastic properties. Here we have proposed closed-form analytical expressions for the effective elastic moduli of lattice materials considering the influence of residual stresses. Besides characterization of the effect of manufacturing irregularities, the presence of such prestress could be viewed from a different perspective. From the materials innovation viewpoint, this essentially expands the design space for property modulation significantly. The proposed analytical framework is directly useful for both property characterization and materials development aspects. The numerical results reveal that the presence of residual stresses, along with the compound effect of other influencing factors, could influence the effective elastic moduli of lattices significantly, leading to the realization of its importance and prospective exploitation of the expanded design space for inclusive materials innovation.
... Of great contemporary interest in engineering science is a class of materials known as elastic metamaterials. Elastic metamaterials are new-age materials engineered to exhibit certain desired elastic properties which do not manifest in naturally occurring materials, including negative dynamic mass density [1], negative Poisson's ratio [2], negative shear modulus [3,4], negative stiffness [5][6][7], or combinations of the aforementioned properties [8][9][10][11]. Elastic metamaterials have garnered great interest from the research ...
... In this section, the properties of the surface waves are investigated, with particular interest given to the dispersion relation of the waves. The dispersion relation can be normalized by multiplying equation (38) by 1=k 4 and making use of the definition of c, as well as b 1 given by equation (23). The following expression for wavespeed c is obtained: ...
Article
Elastic metamaterials are man-made materials engineered with the purpose of inducing atypical bulk elastic properties. To model a type of elastic metamaterial with local rotational effects, a new two-dimensional continuum model which incorporates local rotation as a scalar field superposed on the translational displacement fields is utilized. This paper provides a comprehensive study of surface wave propagation in this new continuous medium. Unlike the classical Rayleigh wave, the surface wave in this new medium is dispersive. An explicit dispersion relation is obtained, and a closed-form solution of the dispersion curve is derived. The dispersion relation is then used to evaluate the general behaviour of the surface wave. It is found that even for the cases where the effect of local rotation is relatively weak, the surface wave still clearly shows dispersion. The phase velocity of the surface wave falls mostly between the classical Rayleigh wave and the shear wave, especially for cases where the effect of local rotation is weak. In addition to the classical Rayleigh wave, there exists another surface wave which possesses a wavespeed depending only on local rotational parameters. It was also found that particles residing on the free surface of the material move in an elliptical fashion similar to that of classical Rayleigh wave propagation.
... Over the past few years, various shapes and topologies (triangular, Kagome, hexagonal, N-Kagome, foam structures, origami, star-shaped, chiral, square and other tailor-made geometries) of such lattice structures have been studied to understand the variation in the effective mechanical properties as the microstructure of the lattice changes (Zhang et al., 2008;Shan et al., 2015;Wei et al., 2020;Bückmann et al., 2014;Grima et al., 2000;Song et al., 2008;Lakes, 1987;Schenk and Guest, 2013;Bacigalupo and Gambarotta, 2020;Xu et al., 2021;Li et al., 2019;Qi et al., 2021;Huang et al., 2021). These artificially engineered metastructures (often referred to as metamaterials) have a wide range of applications in the field of vibration and wave propagation, multi-functional modulation of static deformations, impact resistance, indentation, stabilty control and programmable shape modulation (Fleck et al., 2010;Mukhopadhyay et al., 2019;Du et al., 2021). In this paper, we deal with the contactless active modulation of elastic properties that in turn influence such applications. ...
Article
Full-text available
2D lattices are widely popular in micro-architected metamaterial design as they are easy to manufacture and provide lightweight multifunctional properties. The mechanical properties of such lattice structures are predominantly an intrinsic geometric function of the microstructural topology, which are generally referred to as passive metamaterials since there is no possibility to alter the properties after manufacturing if the application requirement changes. A few studies have been conducted recently to show that the active modulation of elastic properties is possible in piezoelectric hybrid lattice structures, wherein the major drawback is that complicated electrical circuits are required to be physically attached to the micro-beams. This paper proposes a novel hybrid lattice structure by incorporating magnetostrictive patches that allow contactless active modulation of Young’s modulus and Poisson’s ratio as per real-time demands. We have presented closed-form expressions of the elastic properties based on a bottom-up approach considering both axial and bending deformations at the unit cell level. The generic expressions can be used for different configurations (both unimorph or bimorph) and unit cell topologies under variable vertical or horizontal magnetic field intensity. The study reveals that extreme on-demand contactless modulation including sign reversal of Young’s modulus and Poisson’s ratio (such as auxetic behavior in a structurally non-auxetic configuration, or vice-versa) is achievable by controlling the magnetic field remotely. Orders of difference in the magnitude of Young’s modulus can be realized actively in the metamaterial, which necessarily means that the same material can behave both like a soft polymer or a stiff metal depending on the functional demands. The new class of active mechanical metamaterials proposed in this article will bring about a wide variety of design and application paradigms in the field of functional materials and structures.
... To address this issue, here numerical models for the wave propagation analysis would be developed based on SEM [27,28], which is capable of obtaining accurate results in an efficient framework. SEM is formulated from the wave equation's analytical solution that reduces the number of elements in the simulation significantly [30,31]. The spectral transfer matrix method is employed to estimate dispersion diagrams of the unimorph beam. ...
Article
Full-text available
Randomness in the media breaks its periodicity affecting the vibration and wave propagation performance. Such disorder caused by the variability may lead to interesting physical phenomena such as trapping and scattering waves, wave reflection, and energy localisation. While the randomness may be attributed to manufacturing irregularities and quantifying its effect is crucial for ensuring adequate performance of a range of smart systems, these effects can also be exploited for manipulating the wave properties. Here we investigate a smart metastructure in the form of a beam integrated with piezoelectric transducers coupled to a resonant shunt circuit. The piezoelectric shunt in a periodical arrangement can induce locally resonant bandgaps that can be employed in wave and vibration manipulation (and control). This paper quantifies the uncertainty associated with electrical circuit components that affect the circuit impedance. Such uncertainty essentially propagates to the beam smart metamaterial, influencing its wave and vibration control feature. Numerical results of the unimorph meta-beam with single and multi-frequency shunt configuration show that the bandgap behaviour is sensitive to the random disorder associated with circuit impedance parameters, which can, in turn, be exploited for enhanced functionalities based on optimal RL shunt circuits for controlling structural vibration response along with wave propagation and attenuation.
... The point of transition for unit-cells in a column or row can be gradually varied by designing meta-sheets with a graded microstructure. For lattice-based cellular metamaterials [69,70], a lack of mixed-mode deformation with a transition point directly correlates to the unlikeliness of a programmable stiness having sudden jumps in the value. Scientic studies [4] have been performed to obtain programmable stiness through contact and structural deformation. ...
Article
Full-text available
This paper develops kirigami-inspired modular materials with programmable deformation-dependent stiffness and multidirectional auxeticity. Mixed-mode deformation behaviour of the proposed metastructure involving both rigid origami motion and structural deformation has been realized through analytical and computational analyses, supported by elementary-level qualitative physical experiments. It is revealed that the metamaterial can transition from a phase of low stiffness to a contact-induced phase that brings forth an extensive rise in stiffness with programmable features during the deformation process. Transition to the contact phase as a function of far-field global deformation can be designed through the material's microstructure. A deformation-dependent mixed-mode Poisson’s ratio can be achieved with the capability of transition from positive to negative values in both in-plane and out-of-plane directions, wherein it can further be programmed to have a wide-ranging auxeticity as a function of the microstructural geometry. We have demonstrated that uniform and graded configurations of multi-layer tessellated material can be developed to modulate the constitutive law of the metastructure with augmented programmability as per application-specific demands. Since the fundamental mechanics of the proposed kirigami-based metamaterial is scale-independent, it can be directly utilized for application in multi-scale systems, ranging from meter-scale transformable architectures and energy storage systems to micrometer-scale electro-mechanical systems.
... In a periodic structure, one unit cell (i.e. repeating units) can be analyzed with appropriate periodic boundary conditions to obtain the global behaviour of the entire lattice [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81]. Figure 1(A) shows the schematic diagram of the parent 3DCDL unit cell. It has two planar rings (loops) in two orthogonal planes with four connected beams. ...
Article
Full-text available
If we compress a conventional material in one direction, it will try to expand in the other two perpendicular directions and vice‐versa, indicating a positive Poisson’s ratio. Recently auxetic materials with negative Poisson’s ratios, which can be realized through artificial microstructuring, are attracting increasing attention due to enhanced mechanical performances in multiple applications. Most of the proposed auxetic materials show different degrees of in‐plane auxeticity depending on their microstructural configurations. However, this restricts harnessing the advantages of auxeticity in 3D systems and devices where multi‐directional functionalities are warranted. Thus, there exists a strong rationale to develop microstructures that can exhibit auxeticity both in the in‐plane and out‐of‐plane directions. Here we propose generic 3D connected double loop (3DCDL) type periodic microstructures for multi‐directional modulation of Poisson’s ratios. Based on the bending dominated behaviour of elementary beams with variable curvature, we demonstrate mixed‐mode auxeticity following the framework of multi‐material unit cells. The proposed 3DCDL unit cell and expanded unit cells formed based on their clusters are capable of achieving partially auxetic, purely auxetic, purely non‐auxetic and null‐auxetic behaviour. Comprehensive numerical results are presented for the entire spectrum of combinations concerning the auxetic behaviour in the in‐plane and out‐of‐plane directions including their relative degrees. This article is protected by copyright. All rights reserved.
... Yang et al. [13] presented experimentally and theoretically that membrane-type acoustic meta-materials show negative dynamic mass at a frequency around the total reection frequency. Recently Mukhopadhyay and Adhikari have presented analytical and experimental investigations for 2D lattice-type metamaterials showing extreme modulation of eective elastic moduli under dynamic condition, including attainment of the negative values at certain critical frequency ranges [14,15,16,17]. ...
Article
Full-text available
Metastructures and phononic crystals could have several unique physical properties, such as effective negative parameters, tunable band gaps, negative refraction, and so on, which allow them to improve multi-physical performances at the materials level. Motivated by the elastic negative mass metastructures, this work reports the enhancement of bandwidth and vibration suppression, while achieving better energy harvesting via non-linear attachments. We propose to consider the effect of spring softening and spring hardening simultaneously along with exploiting the coupled influence of multiple variables like spring stiffness, damping, number of unit cells, electro-mechanical coupling coefficient and masses. A mathematical model of the metastructure having linear spring with nonlinear attachments is developed and analyzed numerically including the effect of functional gradation. Dimensionless parametric study is performed to tune two-cell and multi-cell models in order to enhance vibration suppression and energy harvesting performances. In an eight-cell model, the non-linear characteristic parameter is functionally graded from softening to hardening using exponential and power law to explore the dual functionality further. It is revealed that the resonant peak can be reduced by non-linear softening characteristics. For enhanced energy harvesting, a smaller value of mass ratio is preferred, while a larger value of damping characteristic is suitable for vibration suppression. Under certain configurations, band structure of the phononic metastructure is capable of achieving absolute band gaps, resulting in frequency ranges where waves cannot propagate. The comprehensive analysis presented here on the effect of various system parameters would lead to the design of non-linear multi-resonator metamaterials for the dual functionality of vibration attenuation and energy harvesting that can be applied in a wide range of automated systems and self-powered devices including the capabilities of real-time monitoring and active behaviour.
... Metamaterials: Next generation of architected composites. Metamaterials are a unique class of materials that are designed around different scales and patterns of microstructure in a way that the light/energy interaction of these materials results in novel properties which are not attainable naturally [446][447][448][449][450][451][452]. The core concept of metamaterials is the customization ability of the structural units to achieve the desired (multi-) functionality. ...
Article
Full-text available
The superior multi-functional properties of polymer composites have made them an ideal choice for aerospace, automobile, marine, civil, and many other technologically demanding industries. The increasing demand of these composites calls for an extensive investigation of their physical, chemical and mechanical behavior under different exposure conditions. Machine learning (ML) has been recognized as a powerful predictive tool for data-driven multi-physical modeling, leading to unprecedented insights and exploration of the system properties beyond the capability of traditional computational and experimental analyses. Here we aim to abridge the findings of the large volume of relevant literature and highlight the broad spectrum potential of ML in applications like prediction, optimization, feature identification, uncertainty quantification, reliability and sensitivity analysis along with the framework of different ML algorithms concerning polymer composites. Challenges like the curse of dimensionality, overfitting, noise and mixed variable problems are discussed, including the latest advancements in ML that have the potential to be integrated in the field of polymer composites. Based on the extensive literature survey, a few recommendations on the exploitation of various ML algorithms for addressing different critical problems concerning polymer composites are provided along with insightful perspectives on the potential directions of future research.
Article
Lattice structures with Plateau borders (LSPB) have attracted increasing interests recently due to the improved stiffness, strength, energy absorption properties. Undoubtedly, vertexes with Plateau borders (VPB) play a significant role on the dynamic elastic moduli. This paper proposes an analytical framework of the frequency-dependent equivalent in-plane dynamic elastic moduli (Young’s moduli E1(ω), E2(ω), Poisson’s ratio ν12(ω), ν21(ω) and shear modulus G12(ω)) of LSPB. First, dynamic stiffness (DS) matrix of a lattice cell edge (based on rod and Timoshenko theories) connected to VPBs (modelled as rigid bodies) at both ends is formulated. Then, based on the above DS matrix and the unit cell method, closed-form expressions of equivalent in-plane dynamic elastic moduli are proposed, which are sufficient general to be applied to four types of lattices. The effects of mass, inertia moment and size of VPB on the equivalent dynamic elastic moduli are studied, with both physical and mathematical interpretations. Furthermore, the proposed expressions are applied to honeycomb, rectangular, auxetic and rhombus LSPB and some interesting and important observations are made. This research provides analytical expressions for broadband dynamic elastic moduli of LSPB, which can be directly used in the design and optimization of composite structures with lattice cores.
Article
Full-text available
Honeycomb lattices exhibit remarkable structural properties and novel functionalities, such as high specific energy absorption, excellent vibroacoustic properties, and tailorable specific strength and stiffness. A range of modern structural applications demands for maximizing the failure strength and energy absorption capacity simultaneously with the minimum additional weight of material to the structure. To this end, conventional approaches of designing the periodic microstructural geometry have possibly reached to a saturation point. This creates a strong rationale in this field to exploit the recent advances in artificial intelligence and machine learning for further enhancement in the mechanical performance of artificially engineered lattice structures. Here we propose to strengthen the lattice structure locally by identifying the failure pattern through the emerging capabilities of machine learning. We have developed a Gaussian Process Regression (GPR) assisted surrogate modelling algorithm, supported by finite element simulations, for the prediction of failure bands in lattice structures. Subsequently, we strengthen the identified failure bands locally instead of adopting a global strengthing approach to optimize the material utilization and lattice density. A range of sequential local strengthening schemes is explored logically, among which the schemes having localized gradation by increasing the elastoplastic properties and lowering Young's modulus of the intrinsic material lead to an increase up to 37.54% in the failure stress of the lattice structure along with 32.12% increase in energy absorption. The comprehensive numerical results presented in this paper convincingly demonstrate the abilities of machine learning in material microstructure design for enhancing failure strength and energy absorption capacity simultaneously when it is coupled with the physics-based understanding of material and structural behavior.
Article
Full-text available
The stochastic buckling behaviour of sandwich plates is presented considering uncertain system parameters (material and geometric uncertainty). The higher-order-zigzag theory (HOZT) coupled with stochastic finite element model is employed to evaluate the random first three buckling loads. A cubic in-plane displacement variation is considered for both face sheets and core while quadratic transverse displacement is considered within the core and assumed constant in the faces beyond the core. The global stiffness matrix is stored in a single array by using skyline technique and stochastic buckling equation is solved by simultaneous iteration technique. The individual as well as compound stochastic effect of ply-orientation angle, core thickness, face sheets thickness and material properties (both core and laminate) of sandwich plates are considered in this study. A significant level of computational efficiency is achieved by using artificial neural network (ANN) based surrogate model coupled with the finite element approach. Statistical analyses are carried out to illustrate the results of stochastic buckling behaviour. Normally in case of various engineering applications, the critical buckling load with the least Eigen value is deemed to be useful. However, the results presented in this paper demonstrate the importance of considering higher order buckling modes in case of a realistic stochastic analysis. Besides that, the probabilistic results for global stability behaviour of sandwich structures show that a significant level of variation with respect to the deterministic values could occur due to the presence of inevitable source-uncertainty in the input parameters demonstrating the requirement of an inclusive design paradigm considering stochastic effects.
Article
Full-text available
An insightful mechanics-based concept is developed for probing the frequency-dependence in in-plane elastic moduli of microstructured lattice materials. Closed-form expressions for the complex elastic moduli are derived as a function of frequency by employing the dynamic stiffness matrix of beam elements, which can exactly capture the sub-wavelength scale dynamics. It is observed that the two Poisson's ratios are not dependent on the frequency of vibration, while the amplitude of two Young's moduli and shear modulus increase significantly with the increase of frequency. The variation of frequency-dependent phase of the complex elastic moduli is studied in terms of damping factors of the intrinsic material. The tunable frequency-dependent behaviour of elastic moduli in lattice materials could be exploited in the pseudo-static design of advanced engineering structures which are often operated in a vibrating environment. The generic concepts presented in this paper introduce new exploitable dimensions in the research of engineered materials for potential applications in various vibrating devices and structures across different length-scales.
Book
Full-text available
Over the last few decades, uncertainty quantification in composite materials and structures has gained a lot of attention from the research community as a result of industrial requirements. This book presents computationally efficient uncertainty quantification schemes following meta-model-based approaches for stochasticity in material and geometric parameters of laminated composite structures. Several metamodels have been studied and comparative results have been presented for different static and dynamic responses. Results for sensitivity analyses are provided for a comprehensive coverage of the relative importance of different material and geometric parameters in the global structural responses.
Article
Full-text available
Stanene, a quasi-two-dimensional honeycomb-like structure of tin belonging to the family of 2D-Xenes (X= Si, Ge, Sn) has recently been reported to show promising electronic, optical and mechanical properties. This paper investigates the elastic moduli and crack propagation behaviour of single layer and bilayer stanene based on molecular dynamics simulations, which have been performed using Tersoff bond order potential (BOP). We have parameterized the interlayer van der Waals interaction for bilayer Lennard-Jones potential in case of the bilayer stanene. Density functional calculations are performed to fit the Lennard-Jones parameters for the properties which are not available from scientific literature. The effect of temperature and strain rate on mechanical properties of stanene is investigated for both single layer and bilayer stanene in the armchair and zigzag directions. The results reveal that both the fracture strength and strain of stanene decrease with increasing temperature, while at higher loading rate, the material is found to exhibit higher fracture strength and strain. The effect of chirality on elastic moduli of stanene is explained on the basis of a physics-based analytical approach, wherein the fundamental interaction between shear modulus and Young’s modulus is elucidated. To provide a realistic perspective, we have investigated the compound effect of uncertainty on the elastic moduli of stanene based on an efficient analytical approach. Large-scale Monte Carlo simulations are carried out considering different degree of stochasticity. The in-depth results on mechanical properties presented in this article will further aid the adoption of stanene as a potential nano-electro-optical substitute with exciting features such as 2D topological insulating properties with large bandgap, capability to support enhanced thermoelectric performance, topological superconductivity and quantum anomalous Hall effect at near-room-temperature.
Article
Full-text available
Generalized high-fidelity closed-form formulae are developed to predict the shear modulus of hexagonal graphene-like monolayer nanostructures and nano-heterostructures based on a physically insightful analytical approach. Hexagonal nano-structural forms (top view) are common for nanomaterials with monoplanar (such as graphene, hBN) and multiplanar (such as stanene, MoS2) configurations. However, a single-layer nanomaterial may not possess a particular property adequately, or multiple desired properties simultaneously. Recently a new trend has emerged to develop nano-heterostructures by assembling multiple monolayers of different nanostructures to achieve various tunable desired properties simultaneously. Shear modulus assumes an important role in characterizing the applicability of different two-dimensional nanomaterials and heterostructures in various nanoelectromechanical systems such as determining the resonance frequency of the vibration modes involving torsion, wrinkling and rippling behavior of two-dimensional materials. We have developed mechanics-based closed-form formulae for the shear modulus of monolayer nanostructures and multi-layer nano-heterostructures. New results of shear modulus are presented for different classes of nanostructures (graphene, hBN, stanene and MoS2) and nano-heterostructures (graphene-hBN, graphene-MoS2, graphene-stanene and stanene-MoS2), which are categorized on the basis of the fundamental structural configurations. The numerical values of shear modulus are compared with the results from scientific literature (as available) and separate molecular dynamics simulations, wherein a good agreement is noticed. The proposed analytical expressions will enable the scientific community to efficiently evaluate shear modulus of wide range of nanostructures and nanoheterostructures.
Article
Full-text available
Two-dimensional and quasi-two-dimensional materials are important nanostructures because of their exciting electronic, optical, thermal, chemical and mechanical properties. However, a single-layer nanomaterial may not possess a particular property adequately, or multiple desired properties simultaneously. Recently a new trend has emerged to develop nano-heterostructures by assembling multiple monolayers of different nanostructures to achieve various tunable desired properties simultaneously. For example, transition metal dichalcogenides such as MoS2 show promising electronic and piezoelectric properties, but their low mechanical strength is a constraint for practical applications. This barrier can be mitigated by considering graphene-MoS2 heterostructure, as graphene possesses strong mechanical properties. We have developed efficient closed-form expressions for the equivalent elastic properties of such multi-layer hexagonal nano-hetrostructures. Based on these physics-based analytical formulae, mechanical properties are investigated for different heterostructures such as graphene-MoS2, graphene-hBN, graphene-stanene and stanene-MoS2. The proposed formulae will enable efficient characterization of mechanical properties in developing a wide range of application-specific nano-heterostructures.
Article
Full-text available
An analytical framework is developed for investigating the effect of viscoelasticity on irregular hexagonal lattices. At room temperature many polymers are found to be near their glass temperature. Elastic moduli of honeycombs made of such materials are not constant, but changes in the time or frequency domain. Thus consideration of viscoelastic properties are essential for such honeycombs. Irregularity in lattice structures being inevitable from practical point of view, analysis of the compound effect considering both irregularity and viscoelasticty is crucial for such structural forms. On the basis of a mechanics based bottom-up approach, computationally efficient closed-form formulae are derived in frequency domain. The spatially correlated structural and material attributes are obtained based on Karhunen-Loève expansion, which is integrated with the developed analytical approach to quantify the viscoelastic effect for irregular lattices. Consideration of such spatially correlated behaviour can simulate the practical stochastic system more closely. The two effective complex Young’s moduli and shear modulus are found to be dependent on the viscoelastic parameters, while the two in-plane effective Poisson’s ratios are found to be independent of viscoelastic parameters and frequency. Results are presented in both deterministic and stochastic regime, wherein it is observed that the amplitude of Young’s moduli and shear modulus are significantly amplified in the frequency domain. The response bounds are quantified considering two different forms of irregularity, randomly inhomogeneous irregularity and randomly homogeneous irregularity. The computationally efficient analytical approach presented in this study can be quite attractive for practical purposes to analyse and design lattices with predominantly viscoelastic behaviour along with consideration of structural and material irregularity.
Article
Full-text available
An analytical framework is developed for predicting the effective in-plane elastic moduli (longitudinal and transverse Young's modulus, Poisson's ratios and shear modulus) of irregular hexagonal lattices with generalized form of spatially random structural geometry. On the basis of a mechanics based bottom-up multi-step approach, computationally efficient closed-form formulae are derived in this article. As a special case when there is no irregularity, the derived analytical expressions reduce to the respective well known formulae of regular honeycombs available in literature. Previous analytical investigations include the derivation of effective in-plane elastic moduli for hexagonal lattices with spatially random variation of cell angles, which is a special case of the generalized form of irregularity in material and structural attributes considered in this paper. The present study also includes development of a highly generalized finite element code for obtaining equivalent elastic properties of random lattices, which is employed to validate the proposed analytical formulae. The statistical results of elastic moduli obtained using the developed analytical expressions and using direct finite element simulations are noticed to be in good agreement affirming the accuracy and validity of the proposed analytical framework. All the in-plane elastic moduli are found to be significantly influenced by spatially random irregularity resulting in a decrease of the mean values for the two Young's moduli and two Poisson's ratios, while an increase of the mean value for the shear modulus.
Article
Full-text available
A generalized analytical approach is presented to derive closed-form formulae for the elastic moduli of hexagonal multiplanar nano-structures. Hexagonal nano-structural forms are common for various materials. Four different classes of materials (single layer) from a structural point of view are proposed to demonstrate the validity and prospective application of the developed formulae. For example, graphene, an allotrope of carbon, consists of only carbon atoms to form a honeycomb like hexagonal lattice in a single plane, while hexagonal boron nitride (hBN) consists of boron and nitrogen atoms to form the hexagonal lattice in a single plane. Unlike graphene and hBN, there are plenty of other materials with hexagonal nano-structures that have the atoms placed in multiple planes such as stanene (consists of only Sn atoms) and molybdenum disulfide (consists of two different atoms: Mo and S). The physics based high-fidelity analytical model developed in this article are capable of obtaining the elastic properties in a computationally efficient manner for wide range of such materials with hexagonal nano-structures that are broadly classified in four classes from structural viewpoint. Results are provided for materials belonging to all the four classes, wherein a good agreement between the elastic moduli obtained using the proposed formulae and available scientific literature is observed.
Article
We explore the dynamics of spontaneous breakdown of mirror symmetry in a pair of identical optomechanical cavities symmetrically coupled to a waveguide. Large optical intensities enable optomechanically-induced nonlinear detuning of the optical resonators, resulting in a pitchfork bifurcation. We investigate the stability of this regime and explore the possibility of inducing multistability. By injecting proper trigger pulses, the proposed structure can toggle between two asymmetric stable states, thus serving as a low-noise nanophotonic all-optical switch or memory element.