Numerical Investigation of Pentamode Mechanical Metamaterials
PANAGIOTIS N LYMPEROPOULOS, EFSTATHIOS E THEOTOKOGLOU
Department of Mechanics, School of Applied Mathematical and Physical Sciences
National Technical University of Athens
Heroes of Polytechnion Avenue, Theocharis Building, Zografou Campus
GREECE
Abstract: - Pentamodes are part of the metamaterials’ family and their main characteristics are the peculiar
properties, not found in nature, as the low ratio of shear strength to bulk strength. In addition, their lattice
nature, contribute to anisotropic behaviour of these materials. Their characteristic, to have low ratio of shear
strength to bulk strength, makes pentamodes suitable to confront a wide variety of problems in engineering
(antiseismic design, aircraft structures, etc.). Several analytical methods are proposed in order to confront
pentamodes metamaterials, but due to the complexity in nowadays engineering problems, a computational
analysis should be proposed. In this study, a computational analysis is taking place in the case of pentamodes
under quasi static conditions. The pentamodes behaviour under small and large displacements analyses for
different materials and for different loading conditions have been analysed and investigated. Analytical results
are also proposed and compared with those from numerical analyses. From our study it is observed that
pentamodes decrease their shear and compression moduli when their height increases. In addition, the ratio of
compression to shear modulus appears to coincide for the different material cases considered.
Key-Words: - metamaterials, pentamodes, finite element method, analytical solution
Received: April 24, 2021. Revised: February 12, 2022. Accepted: March 14, 2022. Published: April 14, 2022.
1 Introduction
Metamaterials is a family of materials
with properties not found in natural materials [1],
[2], [3], [4]. Mechanical metamaterials have
also peculiar properties as the negative Poisson
ratio [5] and bandgaps [6], [7], [8], [9], a band
of frequencies, that a wave with a frequency
inside the bandgap can not be propagated
through the material. Pentamodes, which are
mechanical metamaterials, are usually used for
antiseismic building design, for aircraft
structures etc. [4], [10], [11]. Their main
characteristics is the low ratio of shear
strength to stiffness modulus [1], [12], [13].
Their low-density [14], [15] makes them suitable
for aeronautical applications as the below
mentioned. Wing structures are designed to
confront the aerodynamics forces (lift and drag)
and to have low density values. Gliders
(sailplanes) aircraft designed to be lightweight
structures, and low-density materials with high
strength properties are used. The use of
pentamodes metamaterials can contribute to the
design of more lightweight structures.
Pentamodes are lattice structures as can been seen
in Fig. 1.
(a) (b)
Fig. 1 (a), Unit Cell pentamode structure [13],
(b) Pentamode structure
A lot of investigations have already been
performed for pentamodes. At first pentamodes
have been studied theoretically by Milton [1],
[2], Milton et al. [16] and Sigmud [17]. Kadic et al.
[18], have performed several experiments
calculating the bulk modulus, the shear modulus,
and they also give results about the ratio of shear
to bulk modulus, which can be up to the value of
0.001. Also, Zheng et al. [10] have performed
analyses over the bulk, shear modulus and the
density. Norris [11] and Yu
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et al. [19] have performed an analytical approach to
lattice structure, while Fabbrocino et al. [12],
Yaho et al. [20], Lymperopoulos et al. [21],
[22], have carried out several finite element
analyses over pentamode metamaterials.
An extensive review of several mechanical
metamaterials can be found in the work of Zhou
et al. [19], whereas in the study of Zadpoor [23],
[24], a review over pentamodes used for
bioengineering applications is proposed.
From previous studies, pentamodes were
analysed mostly with analytical and experimental
methods. On the contrary, computational methods
were used in a few cases, and assuming small
displacements analyses. In addition, the anisotropic
behaviour, which is mostly based on their lattice
nature, was not studied. In our study an effort has
been made in order to confront the above mentioned
problems and to investigate pentamode structures so
that they can be used for antiseismic design, aircraft
and naval applications.
As first, an analytical approach over bulk and
shear modulus of pentamodes has been
developed, based on previously proposed methods
[11], [12]. In addition, in our study a new
procedure is presented in order to calculate the
ratio (G/K), considering a linear variation of cross
section radius along the rod of pentamode (Fig. 2).
Finally, a computational analysis has been
carried out using the Finite Element Method (FEM).
The computational study is made using small and
large displacements analyses, in order to determine
the behaviour of pentamodes under different loading
conditions. In addition, a comparison of pentamodes
with isolators used in antiseismic design is made in
order to absorb the energy from the seismic shock.
The results show that pentamodes are more suitable
to confront shear displacements than isolators. Also,
the high values of compression modulus may be
used to confront heavy structures as high buildings,
dams, etc. Additionally, a comparison over shear
and compression moduli and their ratio, with
experimental results [13] has been taken place, for
pentamodes made from different materials.
Our contribution related to our previous studies
[21], [22] is the numerical study of pentamode
structures when their dimensions alter for different
material systems and loadings.
2 Theoretical Consideration
The pentamode considered in our study is shown in
Fig. 1 and Fig. 2, where the length of pentamode
rods, R, is given by [12],
3
4
a
R=
(1)
Let introduce the following notations for the
axial M and bending N compliances, of the lattice
structures [11] as,
( ) ( ) ( ) ( )
2
,
RR
oo
dx x dx
MN
E x A x E x A x
==

where
( )
Ex
is the Young modulus and
( )
Ax
is
the cross section of the rod.
Furthermore, the bulk modulus (K) and the ratio
of the shear to bulk modulus (G/K), are [11],
2
49
,
9 4 2
R G M
KMV K M N
==
+
(3)
and the volume V,
3
64
33
VR=
(4)
Fig. 2 Pentamode's rod, with main dimensions
The ratios G/K have been calculated using two
different methods, an analytical method [12] and a
method based on FEM [18]. According to the
analytical method, the axial N and bending M
compliances are given by [12],
( )
max min
3 2 2
max min max min
33
max min
2
3
R
MRr r
R r r r r
NRr r
=
++
=
(5)
where r is the radius of the cross section of the
pentamode’s rod, and
max min
,
22
Dd
rr==
(6)
The bulk modulus K is given by [12],
2
64 3
EdD
KR
=
(7)
Consequently, the ratio G/K is given by the
equation,
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( )
2 2 2
1
22
82
4,
9 27
D dD d R
G
K d D
−

++

=+


(8)
whereas for the method based on the FEM [18] is,
1
2
G R R
K d D
−


=




(9)
The effective shear and compression modulus
proposed by Amendola et al. [13], appear as,
,
hv
cc
hv
F H F H
GE
AA

==
(10)
where,
h
F
and
v
F
are the total lateral and vertical
forces respectivelly measured at the bottom nodes of
the pentamode structure,
H
the total pentamode’
height,
h
and
v
are the uniform horizontal and
vertical displacement respectively at top nodes of
pentamode and
A
the load area.
In this study, a new procedure in order to
calculate the ratio G/K is proposed, since from the
previous studies it does not appear that a linear
variation of the radius values from the minimum (
min
r
) to maximum (
max
r
)and vice versa has been
taken into account. Considering that the radius is
calculated using a linear interpolation function, the
axial and bending compliance are given by the
following equations,
( )
( )
( )
min max min
3 2 2
max min max min
3
3
min max min
,
33
3
R
Mr R r r
R r r r r
NRr r r
=+
++
=+
(11)
and the bulk modulus K is defined as,
( )
2
1
264 3
Ed d D
KR
+
=
(12)
Consequently, the ratio G/K becomes,
( )
( )
1
2 2 2
2
2
16 3 4
4
927
R D dD d
G
Kd D d
−

++

=+ +


(13)
A wide variety of materials may be used in order to
manufacture metamaterials. In this study three
materials have been used (Table 1).
Table 1 Materials properties
Material
Young
Modulus
[GPa]
Poisson ratio
Steel [12]
206
0.29
Polymer [25]
1.4
0.4
Ti-6Al-4V
titanium alloy
[13]
120
0.342
In order to verify that our proposed equations
(12) and (13) may also be applied in antiseismic
design, a comparison has been made for the steel
material (Table 1) with
D
= 1.32mm and
d
=
1.188mm, between our equations and Fabbrocino et
al. [12] equations (Eqs. (4) and (5), [12]). It is
observed that the results almost coincide (Table 2).
Table 2 Calculation of the Bulk and Shear
Modulus
Method
Bulk Modulus
K (MPa)
Shear Modulus
G (MPa)
Fabbrocino et
al. [12]
54.3
0.411
Equations (12),
(13)
51.5
0.377
3 Computational Analysis
In our study, the finite element package ANSYS
[26] has been used for the finite element analysis. In
our study two analyses have also been performed,
with small and large displacements [26]. Quasi-
static loading conditions are assumed in order to
solve the linear elastic problem of the pentamode
structure under prescribed displacements. The
pentamode that has been considered has a total
height
z
Hn
=
, with a load area
2
xy
A n n
=
,
where
,,
x y z
n n n
denote the number of unit cells
placed along x, y and z axes respectively (
1
x y z
n n n= = =
, Fig.1).
On the contrary to previous analyses where
hexagonal 3d elements have been used [13], our
analyses have been performed with beam elements
only. Each rod (Fig. 1) has been meshed using the
beam type elements with three nodes, BEAM189
[26]. In our analyses, 2 elements on each rod have
been used. For the convergence test and increasing
the number of elements, from 96 to 384 elements,
the results for the shear and compression moduli do
not change. In Table 3, for
3
x
n=
and for the
different unit cells considered, the number of the
finite elements are given.
Table 3 Finite Elements for the different unit cells
nz=1
nz=2
nz=3
nz=4
nz=5
ny=1
96
192
288
384
480
ny=2
192
384
576
768
960
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ny=3
288
576
864
1152
1440
4 Comparison between analytical and
numerical results
At first, the ratio of G/K has been calculated for
various diameters d/D (Fig. 2). In order to verify our
analyses and the obtained results, a comparison is
made between Fabbrocino’ analytical equation [12],
the FEM based equation [18] and our proposed
equation Eq. (13), in the case that
/ 0.044D
=
(Figure 3). It is observed that our results coincide
with Fabbrocino et al. [12] results, but they have
great discrepancies with the results of Kadic et al.
[18] results. The ratio G/K is important in
pentamodes design and should be as small as
possible provided that the bulk modulus, K, has high
value whereas the shear modulus G, has very low
value [12].
A comparison between the proposed equation (13)
with Fabbrocino et al. [12] and Kadic et al. [18]
equations is presented in Fig. 3.
a
Fig.3 A comparison between the
proposed equation (13) with Fabbrocino et al. [12]
and Kadic et al. [18] equations.
In order to verify our finite element model a first
comparison with the study of Amendola et al. [13],
is carried out. The comparison is taken place for the
specimen TPM2 (Table 3, [13]), where is 30mm,
D
is 2.72mm,
d
is 1.04mm (Figs 1, 2), titanium
alloy Ti-6Al-4V is used (Table 1), and the
pentamode is considered with
2
x y z
n n n= = =
. It
is observed that our numerical results coincide
better (Table 4) with the experimental results of
Amendola et al. [13] than the numerical results of
Amendola et al. [13].
Table 4 Comparison between our numerical
results and Amendola et al. [13] results
Amendo
la et al.
[13]
Experim
ental
Results
Amen
dola et
al.
[13]
Numer
ical
Result
s
Finite
Elem
ent
Anal
yses
Percenta
ge
differen
ces of
experim
ental
work
[13] and
our
finite
element
results
Percenta
ge
differen
ces of
experim
ental
and
finite
element
results
of [13]
E
2.972
[MPa]
5.936
[MPa]
1.700
[MPa
]
0.428
0.998
G
1.062
[MPa]
2.685
[MPa]
0.926
[MPa
]
0.128
1.529
5 Results and Discussion
The shear modulus is crucial for the pentamode
structures because they have to confront seismic
waves, or waves produced by engine operation.
Consequently, the shear modulus has to be
calculated. The shear modulus of pentamodes
structures composed of pentamode layers with
variable unit cell in the y-direction
( )
1, 2,3
y
n=
and across the thickness
( )
1, 2,3, 4,5
z
n=
and with
constant unit cell in the x-direction
( )
3
x
n=
is
studied. Prescribed displacements are considered for
1mm and 7.5mm of the top nodes while keeping the
bottom nodes fully constrained. Also, small and
large displacements analyses have been applied
[26], for the case of 𝐷/𝑎 = 0.03.
At first for the titanium alloy Ti-6Al-4V material
(Table 2) and for prescribed displacements of 1mm
and 7.5mm, the Gc values are determined for
small and large displacement analyses (Fig. 4, Fig.
5, Fig. 6, Fig. 7).
Fig. 4 Comparison of the shear modulus with
prescribed displacement 1mm under small
displacements
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Fig. 5 Comparison of the shear modulus with
prescribed displacement 1mm under large
displacements
According to Fig. 4, Fig. 5, Fig. 6 and Fig. 7,
the Gc has lower value for the three
computational cases, than the numerical results
from Amendola et al. [13]. As the height of the
pentamode increases, the results appear to be more
close to experimental results. It is also observed
that for the prescribed displacement of 1 mm, the
results from small and large displacements
analyses almost coincide.
Fig. 6 Comparison of shear modulus with
prescribed displacement 7.5mm under small
displacements
Fig. 7 Comparison of shear modulus with
prescribed displacement 7.5mm under large
displacements
It is observed that our results almost coincide with
the experimental results of Amendola et al. [13] for
𝐷/𝑎 = 0.03. Small and large displacement analyses
for the prescribed displacements of 1 mm and 7.5
mm give almost similar results. Also, the analyses for
the titanium alloy material present almost the same
results. According to Fig. 6, Fig. 7, the results from
small and large displacements analyses
are quite similar. But for the pentamodes structures
with height nz=1, the results from large
displacements analysis are slightly increased when
compared to small displacements analysis.
In order to examine the behaviour of the
pentamodes structures under a different material, the
shear modulus Gc of the polymer pentamodes (Table
1) under prescribed displacements 1 mm and 7.5
mm, are also plotted in Fig. 8, Fig. 9, Fig. 10,
Fig. 11. The polymer material is widely used
in bioengineering applications [23].
Fig. 8 Comparison of the shear modulus with
prescribed displacement 1mm under small
displacements
Fig. 9 Comparison of the shear modulus with
prescribed displacement 1mm under large
displacements
According to Figs. 8, 9 the results for the
polymer material and the prescribed displacement of
1 mm coincide for small and large displacements
analyses. The results in the case where
1
y
n=
,
appear to have lower value than in the cases
2,3
y
n=
. In addition it is observed that the results
for the shear modulus of the polymer material are
lower than the results for the shear modulus of the
titanium alloy Ti-6Al-4V material.
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Fig. 10 Comparison of the shear modulus with
prescribed displacement 7.5mm under small
displacements
Fig. 11 Comparison of shear modulus with
prescribed displacement 7.5mm under large
displacements
It is also observed that the results from the large
displacements analysis under the prescribed
displacement of 7.5 mm, appear to have
discrepancies comparing with the results of the
small displacements analysis (Figs 10,11), and the
results with the prescribed displacement of 1 mm. In
addition, pentamodes with height
1
z
n=
, for the
prescribed displacement of 7.5 mm, present
different behaviour under small and large
displacement analyses.
It should be mentioned that, when the width ny of
the pentamode structure increases, the results for the
shear modulus appear to increase (Figs. 4-11),
whereas the results are decreasing when the height
of pentamodes structure nz increases. In addition,
the results from the FEM analyses provide higher
values for the shear modulus, than the analytical
approach, possibly due to assumptions made for the
analytical approach.
In addition, according to Figs 8, 10 the results for
the prescribed displacements of 1 mm and 7.5 mm,
almost coincide for small displacements analyses.
On the contrary the results for large displacements
analyses (Figs. 9, 11) for 1 mm and 7.5 mm, appear
to have discrepancies for
1
z
n=
.
It has to be mentioned that the results for
pentamodes with polymer material (Figs. 8-11),
have lower value than the results from pentamodes
with titanium alloy (Figs. 4-7). This comes in
agreement with the strengths of the aforementioned
materials.
In order to be able to apply pentamodes in
buildings, airplane engines, bioengineering
applications and in general in structures, it is also
very important to study the compression moduli of
them. Thus, a study of the compression modulus, Ec,
of pentamodes structures using titanium alloy Ti-
6Al-4V (Table 1) and applying a prescribed
displacement of 1mm, is performed (Figs.12,13).
Fig. 12 Comparison of the compression modulus
with prescribed displacement 1mm under small
displacements
Fig. 13 Comparison of the compression modulus
with prescribed displacement 1mm under large
displacements
According to Fig. 12, Fig. 13,
pentamodes compression modulus, Ec, is very
close to the experimental results given by
Amendola et al. [13].
Furthermore, pentamodes with width
2,3
y
n=
appear to have a slightly increase at the values of the
compression modulus for small displacements.
A similar comparison over pentamodes
compression modulus, Ec is made for the polymer
material (Table 1), (Figs. 14, 15).
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Fig. 14 Comparison of the compression modulus
with prescribed displacement 1mm under small
displacements
Fig. 15 Comparison of the compression modulus
with prescribed displacement 1mm under large
displacements
According to Fig. 14 and Fig. 15, pentamodes with
width
2,3
y
n=
appear to have a slightly increase in
compression modulus; related to the width
1
y
n=
.In addition the values for small displacements are a
little higher than the values for large displacements
analyses.
Finally, a comparison of the ratio Ec/Gc is made
for the titanium alloy Ti-6Al-4V material and for
the polymer material (Table 1), in the following
Fig. 16, Fig. 17, Fig. 18 and Fig. 19.
Fig. 16 Comparison of the compression to shear
modulus with prescribed displacement 1mm under
small displacements
Fig. 17 Comparison of the compression to shear
modulus with prescribed displacement 1mm under
large displacements
According to Fig. 16, Fig. 17, the results of our
study, almost coincide with the experimental
results from the work of Amendola et al. [13],
Fig.7, for
1,2.
z
n=
But for
3,4,5
z
n=
discrepancies are observed.
In the sequel the results for the polymer
material (Table 1), are given in Fig. 18, Fig. 19.
Fig. 18 Comparison of compression to shear
modulus with prescribed displacement 1mm under
small displacements
Fig. 19 Comparison of compression to shear
modulus with prescribed displacement 1mm under
large displacements
It is observed that the results for the three cases of
pentamodes with
1, 2,3
y
n=
almost coincide. It has
also to be mentioned that according to Fig. 16,
Fig. 17, Fig. 18, Fig. 19, the ratio Ec/Gc coincide for
the cases of the titanium alloy Ti-6Al-4V and
the polymer material pentamodes (Table 1)
whereas the shear and compression moduli
differ for the two materials. This comes in
agreement with the study of Fabbrocino et al.
[12],
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6 Conclusions
Pentamodes metamaterials structures can be
designed in order to confront different problems in
different applications. Therefore, a wide field of
applications appears from biomechanical
applications, to building structures and also to naval
and aeronautical applications. The different
properties, that pentamodes may have, make the
pentamodes analysis challenging. In biomechanics
applications, pentamode should have high strength
for application regarding bones, and be more elastic
for applications of surgical wound treatment, like
chest and belly. For antiseismic design, the main
goal is to confront the shear displacement of seismic
phenomena in heavy structures. For aeronautical
and naval applications, lightweight structures with
high strength are the intention of designers. A first
study over pentamodes behaviour in compression
and shear loading conditions in order to calculate
elastic moduli is essential for a first evaluation of
pentamodes.
In this paper, a study over the shear and the
compression moduli, which are very characteristic
pentamodes properties, has been carried out.
According to our study, the results proposed by
Fabbroccino et al. [12] for the shear modulus are
close to the results obtained in our study under large
displacement analyses. The shear modulus
calculated by the theoretical formulas presented in
our study, are close to the results that are obtained
for pentamodes with unit cell along the thickness
nz(= 4, 5), and a single unit cell ny(= 1) in the y-
direction.
It should also be mentioned that the results for the
shear moduli from small displacements analyses are
almost coincide with those from large displacements
analyses, but when the prescribed displacements at
pentamodes top nodes increase over 15 mm, in the
case of low height pentamodes structures, the results
present differences and do not coincide. The
variation in results, in low height pentamode
structures and for high prescribed displacements at
the top nodes, may occur from the nonlinear
behaviour and probably from material failure. In the
case of pentamodes with structures height over nz =
3, the shear strength values are close.
Furthermore, the results for the compression
moduli show the reasonable result that pentamodes
with a wide surface load have a large compression
modulus. In addition, high structure pentamodes
have a lower compression modulus which seems to
converge to a value. Τhe ratio Ec/Gc coincide for the
cases of the titanium alloy Ti-6Al-4V and polymer
manufactured pentamodes whereas the shear and
bulk strengths differ for the two cases of materials.
Consequently, in our study in contrast to previous
studies the width and the height of pentamodes
structures are investigated in terms of the influence
of the shear and compression moduli for the
titanium alloy Ti-6Al-4V and for the polymeric
material under small and large displacements
analyses. The aforementioned results constitute our
contribution related to previous studies. In order to
verify our results and to extend them to structures
from polymeric material, they have also been
compared in the case of the titanium alloy, with the
experimental results obtained by Amendola et al.
[13].
Further analysis considering non linear material
behaviour and probably higher prescribed
displacements should be carried out, in order to
better understand pentamodes.
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Efstathios E. Theotokoglou
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Efstathios E Theotokoglou is responsible for overall
supervision, writing- original draft and the writing-
review & editing.
Panagiotis N Lymperopoulos is responsible for the
Formal analysis, Investigation, Validation, writing-
original draft and the writing- review & editing.
Sources of funding for research
presented in a scientific article or
scientific article itself
This research was funded by the Research
Committee of the National Technical University of
Athens.
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License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.7
Panagiotis N. Lymperopoulos,
Efstathios E. Theotokoglou
E-ISSN: 2224-3429
55
Volume 17, 2022
Conflicts of Interest
The authors have no conflicts of interest to
declare .