Active mechanical cloaking for unsupervised damage resilience in programmable elastic metamaterials

Abstract

Owing to the architected void-filled low-density configurations, metamaterials are prone to defects during the complex manufacturing process, or damages under operational conditions. Recently mechanical cloaking has been proposed to shield the effect of such disorders in terms of homogenized mechanical responses. The major drawback in these studies are that the damage location should be known a priori, and the cloak is designed around that damaged zone before manufacturing. Such postulation does not allow unsupervised damage resilience during the manufacturing and service life of metamaterials by active reconfiguration of the stress field depending on the random and unpredictable evolution of damage. Here, we propose a radically different approach by introducing piezoelectric lattices where the effect of random appearance of any single or multiple disorders and damages with complex shapes, sizes and distributions can be shielded through active multi-physically controlled cloaks by voltage-dependent modulation of the stress fields within the cloaking region. Notably, this can be achieved without breaking periodicity and any additional material in the cloaking region unlike earlier studies concerning mechanical cloaks. The proposed active class of elastic metamaterials will bring a step-change in the on-demand mechanical performance of critically important structural components and unsupervised damage resilience for enhanced durability and sustainability.
This article is part of the theme issue ‘Current developments in elastic and acoustic metamaterials science (Part 1)’.

Full text

Active mechanical cloaking
for unsupervised damage
resilience in programmable
elastic metamaterials
D. Kundu1, S. Naskar2 and T. Mukhopadhyay2
1Theoretical and Applied Mechanics Program, Northwestern University, Evanston,
IL, USA
2Faculty of Physical Sciences and Engineering, University of Southampton,
Southampton, UK
SN,0000-0003-3294-8333; TM,0000-0002-0778-6515
Owing to the architected void-filled low-density
configurations, metamaterials are prone to defects
during the complex manufacturing process, or
damages under operational conditions. Recently
mechanical cloaking has been proposed to shield the
effect of such disorders in terms of homogenized
mechanical responses. The major drawback in these
studies are that the damage location should be
known a priori, and the cloak is designed around
that damaged zone before manufacturing. Such
postulation does not allow unsupervised damage
resilience during the manufacturing and service
life of metamaterials by active reconfiguration
of the stress field depending on the random
and unpredictable evolution of damage. Here, we
propose a radically different approach by introducing
piezoelectric lattices where the effect of random
appearance of any single or multiple disorders
and damages with complex shapes, sizes and
distributions can be shielded through active multi-
physically controlled cloaks by voltage-dependent
modulation of the stress fields within the cloaking
region. Notably, this can be achieved without
breaking periodicity and any additional material in
the cloaking region unlike earlier studies concerning
mechanical cloaks. The proposed active class of
elastic metamaterials will bring a step-change in
the on-demand mechanical performance of critically
important structural components and unsupervised
© 2024 The Authors. Published by the Royal Society under the terms of the
Creative Commons Attribution License http://creativecommons.org/licenses/
by/4.0/, which permits unrestricted use, provided the original author and
source are credited.
royalsocietypublishing.org/journal/rsta
Research
Cite this article: Kundu D, Naskar S,
Mukhopadhyay T. 2024 Active mechanical
cloaking for unsupervised damage resilience in
programmable elastic metamaterials. Phil.
Trans. R. Soc. A 382: 20230360.
https://doi.org/10.1098/rsta.2023.0360
Received: 18 January 2024
Accepted: 13 March 2024
One contribution of 12 to a theme issue
‘Current developments in elastic and acoustic
metamaterials science (Part 1)’.
Subject Areas:
materials science, mechanical engineering,
structural engineering, mechanics
Keywords:
mechanical cloaking, unsupervised damage
resilience, piezoelectric lattice materials, elastic
metamaterials, active cloaking, intelligent
digital twins
Author for correspondence:
T. Mukhopadhyay
e-mail: t.mukhopadhyay@soton.ac.uk
Electronic supplementary material is available
online at https://doi.org/10.6084/
m9.figshare.c.7318732.

damage resilience for enhanced durability and sustainability.
This article is part of the theme issue ‘Current developments in elastic and acoustic
metamaterials science (Part 1)’.
1. Introduction
Elastic mechanical metamaterials have been proven to be promising to develop a range of
lightweight multifunctional engineering structures. Mechanical metamaterials are artificially
engineered microstructures where the effective properties can be obtained through a condu-
cive nexus among intrinsic material properties, geometry and mechanics, leading to extreme,
tunable and multifunctional mechanical behaviour, which are not normally achievable in
naturally occurring materials. Over the last decade or so, mechanical metamaterials [1,2] have
indubitably demonstrated unprecedented abilities in terms of specific stiffness [3–5], auxetic-
ity [6], impact resistance [7,8], energy absorption [9,10], energy harvesting [11,12], fracture
toughness [13,14], variable permeability [15], wave propagation and modulation [16–18], shape
transformation [19], critical properties concerning sensors and actuators [20,21] and structural
designs with better stress distribution and deformation resistance. This has led to the accel-
erated adoption of such architected materials in technologically demanding sectors such as
aerospace, biomedical, mechanical and civil for achieving extreme or tunable mechanical
properties with multi-objective, and often conflicting demands. While elastic metamaterials
have emerged to be attractive for developing lightweight engineering structures, owing to
their architected void-filled low-density configurations, they are prone to defects during the
complex manufacturing process or damage under operational conditions. The central theme of
this paper is to propose the concept of active mechanical cloaking to shield the effect of such
damages and defects in terms of homogenized mechanical responses.
In general, lattice-based mechanical metamaterials are cellular materials with periodic
groupings of unit cells comprising unique tailor-made geometry and patterns to accomplish
exceptional bulk properties. Such materials have grown in importance recently as a result of
exceptional computational developments in optical, electromagnetic, mechanical and acous-
tic fields along with the ability to manufacture complex microstructural geometries. These
materials are manufactured at the nano-/microscales with complicated physics-informed
designs and arrangements of lattice structures, resulting in superior and often unique mechani-
cal properties at the macroscale. To be used as structural components in diverse devices and
systems, lattice-based mechanical metamaterials must have their effective elastic characteris-
tics defined. Numerous studies have been reported on various lattice geometries to find and
characterize key factors that influence elastic properties. The effective elastic characteristics are
controlled by a number of passive variables including geometry, mechanics and the constitutive
behaviour of intrinsic materials [22,23]. While a wide range of effective material properties
can be achieved in such passive material microstructures, the properties cannot be tuned in
an on-demand basis after they are manufactured. The recent developments in this direction
include bi-level topology architected optimum metamaterials [5], hierarchical metamaterials
[24], disordered metamaterials [25], anti-curvature metamaterials with programmed curva-
ture [26–29], multi-material and space-filled lattices [30,31], origami- and kirigami-inspired
metamaterials [19,32], to mention a few. Recently, the concept of pneumatic elastostatics and
deployability in mechanical metamaterials has been proposed based on inflatable lattices that
can exhibit extreme specific stiffness along with on-demand tunability [33].
Multi-physical mechanics involving external stimuli like electrical and magnetic fields, light,
temperature and shape memory effects have recently been exploited to achieve on-demand
programmability in lattice metamaterials [1]. For example, the effective elastic moduli can
be actively controlled in piezoelectric [34–36] lattices as a function of voltage, leading to
2
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

modulation of stiffer or softer behaviour of a single lattice architecture in an on-demand
framework as per operational demands, even after it is manufactured [37,38]. A magnetic field
can lead to such on-demand modulation of effective elastic properties (including sign reversal),
but in a contactless framework [39,40]. In the present work, we will introduce piezoelectric
multi-physical lattices for developing a framework for active cloaking mechanical metamateri-
als, as discussed in the following paragraphs.
Followed by the fascinating development of invisibility cloak [41], multiple studies have
been reported in the field of wave propagation concerning electromagnetics [42], fluids [43],
acoustics [44] and quantum matter [45] to develop cloaks. To cloak, an object or a phenomenon
occurring in a system is supposed to hide from the path of general perception [46]. Previous
literature also shows elastodynamic [47] and hyperelastic cloaking based on prestress [48–50].
One of the most recent trends concerning cloaking in metamaterials is the development of
mechanical cloaks dealing with primarily stress and strain fields to hide defects, damages
or other adverse features. To identify microstructures displaying such an altered (hidden)
distribution in mechanical cloaks, it has usually been pursued by applying coordinate transfor-
mation [51] to a given material–parameter distribution and then solving an inverse problem.
Mechanical cloaks are often created to alter the elastic response around objects in order to
blend them into their standardized surroundings. Some recent works have developed the
idea of cloaking the mechanical properties using metamaterials under a thermal gradient [52],
geometry transformation adding thickened cell struts [51], lattice transformation [53] and also
using optimized material addition [54] in the cloaking region. A recent study through data-
driven design approaches proposes adaptable construction of mechanical cloaks with diverse
boundary and loading conditions along with various void shapes and numbers [55].
In the above-mentioned works concerning mechanical cloaks, the cloaking is proposed
through supervised and passive frameworks, which may cost the structure its life owing to time
constraints in developing the cloaks, or alter some of the critical features of the overall structure
owing to the addition of the cloaking region after manufacturing. Furthermore, most of the
reported works lack practicality in terms of generic damage shapes. The major drawback in
such studies on mechanical metamaterials is that the damage location should be known a priori,
and the cloak is designed around that damaged zone before manufacturing. Such postulation
does not allow unsupervised damage resilience during the service life of the metamaterials by
active reconfiguration of the stress field depending on the random and unpredictable evolution
of damage. Though a recent study has proposed magnetoactive mechanical metamaterial for
tunable elastic cloaking [56], the central limitation of knowing a fixed location of damage a priori
persists. Here, we aim to propose a radically different approach by introducing piezoelectric
lattices where the effect of the random appearance of any damage can be shielded through
active multi-physically controlled cloaks by voltage-dependent modulation of the stress fields
within the cloaking region. According to the operational conditions of different structures and
mechanical systems including a range of boundary conditions, single or multiple such disorders
and damages with complex shapes and sizes can effectively be kept hidden, ensuring high
reliability in the homogenized mechanical behaviour.
For developing active mechanical cloaks, we propose piezoelectric lattice metamaterials
(refer to figures 1 and 2a) where element-wise (i.e. beam-like elements, shown as sides of the
hexagonal units) voltage modulation is conceptualized to control the stress or strain fields in
the cloaking regions. It is to be noted that in figure 2a, the tri-membered unit cell is just shown
to give a better view to the reader of the piezoelectric layer added to the substrate beam.
The tessellation of a single unit cell helps in generating the overall lattice. Also, predicting
the mechanical response from a unit cell helps understand the effective physical behaviour of
undamaged lattices in an efficient computational framework because of the periodicity concept
involved here [1,22]. However, in the current paper, we have not used the periodicity concept
of the lattice to generate our numerical results as the damaged lattice no longer remains an
exact periodic system. Instead, we have used a finite element-based scheme with optimization
3
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

of the displacement norm for the full lattice system to solve the problem of unsupervised
damage resilience. To demonstrate the concepts, we would primarily focus on two-dimensional
hexagonal metamaterials with different shapes of randomly placed damages having varying
numbers. However, the concept of active mechanical cloaks would be generic in nature and
it can be implemented in other two-dimensional and three-dimensional lattice metamaterials
(refer to figure 2b). Hereafter, we discuss the concept of active cloaking in elastic metamaterials
in §2, followed by the underlying mathematical formulations and computational framework
in §3. Numerical demonstrations of active cloaking are furnished in §4 considering different
shapes, numbers and randomized locations of damages along with various conditions of
far-field stresses. Finally, §5 provides concluding remarks and perspective.
2. Concept of active cloaking and damage resilience
Mechanical cloaks are material systems that are designed to alter the elastic response around
defects/voids to make them unrecognizable from their homogeneous surroundings. To cloak
a defect implies making it invisible in terms of a physically accessible field (such as strain).
Cloaking effects have shown notable progress in the design of lattice metamaterial for electro-
magnetic waves around a hidden object. Optical, thermal and electric cloaks are generally
designed using approaches based on material–parameter transformations [55]. Unfortunately,
as continuum equations are not form-invariant under generalized coordinate transformations,
these cloak design methodologies cannot be used to design mechanical cloaks. As per the
current state of knowledge, it is not possible to cloak damages or defects in metamaterials
under operational conditions as they appear in real-time after manufacturing. As a result,
existing mechanical cloak methodologies have been restricted to small pre-defined voids (or
defects) with simple shapes and limited boundary conditions.
In this paper, for on-demand cloaking, active materials will be exploited so that the
local-level mechanical constitutive laws around the damages can be modulated as a
function of external stimuli such as the electric field. The theory of cloaking will be
developed based on a computational framework of multi-physical finite element simulations
along with numerical optimization which will include the determination of element-wise
voltages in the cloaking region to negate the effect of damage in the strain field beyond
this region. By re-distributing stress and strain components in an active and on-demand
framework based on the location, nature, size and number of damages (either manufactur-
ing or operational in nature), mechanical cloaking as proposed here will facilitate hiding
essential voids and cut-outs in structural designs, or unintentional defects which are often
the sources of deviation from intended structural performance and further failure initiation
(damage resilience). It will significantly enhance the durability and lifetime of a structure
without compromising on mechanical performances like strength and stiffness. Furthermore,
since the current framework relies on external multi-physical stimuli like voltage in the
cloaking region for altering the stress and strain fields, it is not necessary to reinforce
the cloaking region through optimally introduced additional elements in the lattice, as
commonly followed in the mechanical cloaks proposed in metamaterials. From a sustaina-
bility viewpoint, mechanical cloaking without any additional weight would be crucial for
a range of engineering applications. Since the periodicity in lattice geometry is not broken
in the present framework, it will also be helpful for manufacturing and scalability. The
realization of active cloaking will bring a step-change in the field of mechanical cloaking
since the location and intensity of damage (/disorder) is not necessarily needed to be
known a priori before manufacturing. According to the service life conditions of different
structures and mechanical systems including a range of boundary conditions, single or
multiple such disorders with complex shapes and sizes can effectively be kept hidden.
To elucidate the concept of active mechanical cloaking, we consider an arbitrary domain
of a lattice metamaterial. As demonstrated schematically in figure 1, the arbitrary domain is
4
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

subjected to certain loading and boundary conditions, resulting in a stress or strain field as
shown in subfigure (c) when there is no defect or damage. If some defect evolves in the same
domain under the same loading and boundary conditions, the stress or strain field would
change, as depicted in subfigure (d). However, we propose to design the domain in such a
way that the stress or strain field will remain unaltered beyond certain boundaries around
the defects through element-wise (implying each beam-like element of the piezoelectric lattice
metamaterial) application of external stimuli in the cloaking region. The objective here is to
minimize the discrepancies between the stress or strain fields of subfigure (c, d) beyond the
cloaking region through the application of optimized voltages in each beam-like element within
the cloaking region. Note that the applied voltage would be optimum in the current framework
since we propose to apply voltages element-wise selectively and individually depending on the
respective demands to negate the effect of altered stress or strain fields owing to the introduc-
tion of defects. Further, unlike the current state of the art, we would be able to achieve active
cloaking without breaking geometric periodicity in the cloaking region, or without having
additional material therein.
In the current study, our key focus is to negate the effect of damage in edge deformation of
the damaged lattice and converge to the displacement or strain field of the undamaged lattice
beyond the cloaking region. This objective is similar to other studies discussed in the preceding
paragraphs concerning mechanical cloaks, but overcoming a range of critical limitations as
pointed out. We have achieved the stated objective in an on-demand framework by proposing
element-wise voltage optimization within the cloaking region at two levels concerning the
alteration of stress and strain fields through external stimuli. In the first stage, we obtain a
voltage matrix for on-demand activation of the cloaking region. Subsequently, we introduce an
automated voltage augmentation factor to accentuate the effect of electrical stimuli for negating
(a)
(c) (d) (e)
(bI)
(bII)
INPUT
Supply Voltage
Algorithm
Cloaking Region
Strain Increment OUTPUT
X-
Figure 1. Active mechanical cloaking in elastic metamaterials. (a) An arbitrary structural domain with Dirichlet and
Neumann boundary conditions, i.e. fixity conditions and loading. Strain measurements are sent as an output to an external
circuit board, which decides the cloaking region and the inbuilt algorithm decides the supply voltage to be sent as an
input to the structure in an unsupervised framework. (b) (I) Lattice-based hexagonal microstructure where each of the
beam-like elements is electro-active. (II) Typical representation of other prospective two-dimensional and three-dimensional
lattices where the proposed active cloaking can be implemented. (c) Undamaged lattice structure with original displacement
gradient (minimum ‘blue’ to maximum ‘red’). (d) Damaged lattice structure with the corresponding displacement gradient.
(e) Voltage-dependent modified displacement gradient of the damaged structure after cloaking. The displacement field
presented in subfigure (e) beyond the cloaking region is supposed to be close to that presented in subfigure (c).
5
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

the influence of damage in terms of the displacement or strain fields near the edges. This
seamless bi-level deformation negation algorithm allows us to save time in the overall iterative
simulation process. By limiting the activation to the beams only in the cloaking region, we take
a step towards using minimal external electrical energy.
(a) (b)
(c) (d)
(e)
(g) (h) (i) (j) (k)
(f)
Figure 2. Lattice metamaterials with piezoembedded beam-like elements and lattice-level mechanics for displacement
fields including the effect of cloaking. (a) The typical unit cell of a hexagonal lattice constructed with composite beams
with a substrate material (in violet) bonded to a piezoelectric patch (in blue), representing an unimorph configuration. (b)
Enlarged representation of a piezoembedded composite beam with the cross-sectional view shown on the right. The basic
circuit diagram is also shown along with local degrees of freedom (axial, transverse and rotation). (c) A typical representation
of a damaged lattice formed with the hexagonal unit cells is shown in subfigure (a). The bold members represent the
piezoelectric active beams (constituting the cloaking region) and the piezoelectric degrees of freedom are marked. (d)
The row-wise global node numbering is shown, numbering the odd rows only to include all nodes in the lattice. (e) An
undamaged lattice formed with the hexagonal unit cells shown in subfigure (a). The cell-wise marking is shown using the
arrows that help in the development of the finite element connectivity matrix. (f) The row-wise global element numbering
is shown separately for odd and even rows. (g–i) Uni-modal loading considering far-field axial stress along directions 1
and 2, and shear stress along direction 12. (j, k) Mixed-mode loading under complex far-field stresses. The subfigures are
represented using the general boundary conditions.
6
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

The elastic metamaterial we put forth here will be intelligent in nature to ask the energy
array to supply different voltages for activating the target lattice members within the cloaking
region. Strain gauges connected to a material or structural domain will be able to detect
damage by studying unusual strain values compared with undamaged structures as they
evolve. Based on the damaged region, the cloaking region is decided where optimally identified
electrical energy is introduced. The algorithm we develop here will determine the external
voltage required to mitigate the effect of damage by reducing the discrepancies in global strain
fields beyond the cloaking region. It is obvious that the global deflections increase within
the structural nodes when it has encountered damage owing to a reduction of stiffness. In
the current work, we have proposed to exploit the idea of the converse piezoelectric effect
for cloaking damaged honeycomb lattices to tone with the original deformation field of the
undamaged lattice.
3. Computational framework for demonstrating active and on-demand
cloaking
The computational framework for active cloaking in mechanical metamaterials involves a
seamless integration of three components: (i) piezoelectric beam elements and their voltage-
dependent deformation mechanics, (ii) lattice-level analysis to obtain the strain field under
a given loading and boundary conditions, and (iii) multi-objective discrepancy minimization
algorithm between the strain fields of damaged and undamaged lattices. A concise flowchart
in algorithmic form is presented in algorithm 1. In this study, we have considered different
damage scenarios with axis symmetry, bi-axis symmetry, and non-symmetry shapes over the
lattice. To check the validity of the proposed framework, we have expanded it further to
randomly distributed damage cases by varying the damage positions. We would begin by
considering uni-modal axial and shear loading individually, and extend our work to mixed-
mode loading with different boundary conditions.
(a) Mechanics of unimorph piezoelectric beam elements
The mathematical formulation has been presented here showing the development of a
piezoelectric stiffness matrix of a composite beam with a substrate and piezoelectric layer
having unimorph configuration (refer to figure 2b). The vectors, tensors and matrices are
written in bold case to avoid confusion. Subscripts and superscripts with  and , in gen-
eral, will represent the substrate and piezoelectric layer of the beam, respectively. T in the
superscript will indicate the transcript. Considering a beam having the local axis system as
1−2−3, the displacements of any arbitrary axis of the beam along the axial and transverse
directions at any time,  can be shown using equations (3.1) and (3.2). Because of the bending of
the beam, the axial deflection will have components of the axial deformation of the neutral axis,
i.e. 1
(1,) as well as the transverse deformation, 3
(1,) at the considered section.
(3.1)
1=1
1,−33
,11,,
(3.2)31,=3
1,.
The total potential energy of the beam with an overall volume of + can be represented
using equation (3.3). It will have the energy components both from the substrate layer as well
as the layer comprising the piezoelectric patch, and integration is done separately over the
volume. We have represented the engineering strain tensor using  and for the engineering
stress tensor, we have used,  or , depending upon the substrate or piezoelectric layer.
7
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

(3.3)=1
2Ω
TΩ+Ω
TΩ.
Considering the Young’s modulus of the substrate as , we can calculate the overall strain,
11(1,3,) of the beam and stress component, 11
(1,3,) of the substrate layer along the
direction 1 as shown in equations (3.4) and (3.5).
(3.4)11 1,3,=∂1
1,
∂1−3
∂23
1,
∂1
2,
(3.5)11
1,3,=11 1,3,.
Using the same strain component, 11(1,3,), we also calculate the stress component,
11
(1,3,) of the piezoelectric layer as shown in equation (3.6). Stress in the piezoelectric layer
will also depend on the time-dependent electric field (), i.e. /, the ratio of voltage,  with
the piezoelectric layer thickness, . Here, 31 is the effective piezoelectric stress constant.
(3.6)11
1,3,=11 1,3,+
¯31

.
The set of equation (3.7a–c) describes the relationship of permittivity constants at constant stress
and strain, 33
 and 33
 with the elastic compliance at the constant electric field, 11
 and 31
, the
piezoelectric constant. The elastic compliance at a constant electric field is the inverse of Young’s
modulus,  of the piezoelectric layer. Finally, we relate 31, the effective piezoelectric stress
constant with 11
, the elastic compliance at a constant electric field and 31
, the piezoelectric
constant.
(3.7a)33
=33
−31
2
11
,
(3.7b)=1
11
,
(3.7c)
¯31 =31

11
.
Substituting the above deductions in equation (3.3), for the total potential energy, we get
equation (3.8).
(3.8)
=1
20


∂1
(1,)
∂1
2
+
2∂23
(1,)
∂12
2
−2
1∂1
(1,)
∂1
∂23
(1,)
∂121
+1
20


∂1
(1,)
∂1
2
+
2∂23
(1,)
∂12
2
−2
1∂1
(1,)
∂1
∂23
(1,)
∂121
+
0

1∂1
(1,)
∂1−2∂23
(1,)
∂121
.
The neutral axis of the composite beam constituting the substrate material and the piezoelec-
tric patch can be given as ℎ from the bottom of the beam using equation (3.9) (refer to figure
2b).
(3.9)ℎ=
0.5+
0.5+
+

.
8
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

Here , 1 and 2 are the cross-sectional area, first and second moment of inertia separately
shown for the substrate and the piezoelectric layer with subscripts  and . The equations
for these geometric properties are shown in the set of equation (3.10a–c) and (3.11a–c) for the
substrate and piezoelectric layers of the beam.
(3.10a)=Ω
32=
0

−ℎ
−ℎ32=,
(3.10b)
1=Ω
332=
0

−ℎ
−ℎ332=
2
2−2ℎ,
(3.10c)
2=Ω
3
232=
0

−ℎ
−ℎ3
232=
3
3+ 3ℎ
2−3
2ℎ,
(3.11a)=Ω
32=
0

−ℎ
−ℎ+32=,
(3.11b)
1=Ω
332=
0

−ℎ
−ℎ+332=
22+
2−2ℎ,
(3.11c)
2=Ω
3
232=
0

−ℎ
−ℎ+3
232=
3
3+ 3
2++ℎ
2−2ℎ.
The coupling terms, 1 and 2, are shown in equations (3.12a,b). Note that these expressions
are obtained from equations (3.3) and (3.8), wherein voltage-extension and voltage-curvature
coupling and their contribution to the total potential energy can be established.
(3.12a)1=Ω

¯31

32=
0

−ℎ
−ℎ+
¯31

32=
¯31,
(3.12b)2=Ω

¯31

332=
0

−ℎ
−ℎ+
¯31

332=
¯31
22+−2ℎ.
Now, we can write the internal energy in the piezoelectric layer given by Up, as shown in
equation (3.13). The electric field vector can be represented as , whereas the vector constitut-
ing the electric displacement terms is .
(3.13)=1
2Ω

TΩ=−1
20

1∂1
1,
∂1−2∂23
1,
∂1
21+33


2.
Considering a beam element with two nodes, it will have a total of 6 degrees of freedom, 3
at each node, i.e. axial, transverse and rotation. We represent them using the vector,  with
the components as  where  stands for the node and  stands for the direction of deforma-
tion considered. Except for the third and sixth components that represent the rotations with
first-order differential terms with respect to 1, the other components represent the deforma-
tions.
(3.14)
T=11 13 13,121 23 23,1.
The deformation of the neutral axis can be represented as follows considering a shape
function matrix, . The elements of the matrix are represented as  where  represents the
node and  stands for the loading direction, either axial, transverse or rotation.
(3.15)
1

3
=10 0 20 0
011022
=.
9
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

The shape function matrix has been developed considering a beam of unit length. Hence,
scaling for the beam of length,  , we use the dummy variable, =1/. The terms of  are
shown in the set of equation (3.16a–f).
(3.16a)1= 1 −,
(3.16b)2=,
(3.16c)1= 1 −32+ 23,
(3.16d)1=  −22+3,
(3.16e)2= 32−23,
(3.16 f)2=−2+3.
Separating the neutral axis deformation terms, we have equation (3.17a,b).
(3.17a)1
= 1 0 ,
(3.17b)3
= 0 1 .
Taking variational after differentiating the above expressions, we arrive up to equation (3.18a,b).
For simplification, we create the matrices, 1 and 2.
(3.18a)1
,1=1 0 ,1 =1,
(3.18b)3
,11= 0 1 ,11 =2.
Keeping the voltage, , as constant, we take the variational of the summation of the total
potential energy in equation (3.3) and the internal energy in the piezoelectric layer. Then we get
equation (3.19).
(3.19)
=
T+11 −
1+
112 
−
T
1+
121 +
2+
222 +.
To simplify equation (3.19), we develop a few constants as shown in the set of equation (3.20a–
e). They include the integration terms separated from the material and geometrical constant
property terms.
(3.20a)11 =
0

1
T11,
(3.20b)12 =
0

1
T21,
(3.20c)12 =
0

1
T21,
(3.20d)22 =
0

2
T21,
(3.20e)=
0
1
212
T1−
0
1
222
T1.
Now, we can introduce the stiffness matrix term,  for the piezoelectric patch, as given by
equation (3.21b).
(3.21a)=
T+,
10
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

(3.21b)=+11 −
1+
112 −
1+
121 +
2+
222 .
The internal energy term can be separately written as
(3.22)=−
T.
Using Hamilton’s principle in equation (3.23), we get the final expression for the equilibrium
equation.
(3.23)
1
2
− =1
2

T+ 2  =1
2

T+ 2  = 0.
Now, the load vector can be separately represented as shown in equation (3.24a,b) and it has
terms that are linearly dependent upon the voltage value along the piezoelectric patch.
(3.24a)+ 2= 0 ⇒=−2,
(3.24b)
T=10−2−102.
Expanding the stiffness matrix, , we get the following matrix.
(3.25)
=
+
0−
1+
1
−+
0
1+
1

0 12
2+
2
36
2+
2
20−12
2+
2
36
2+
2
2
−
1+
1
6
2+
2
24
2+
2


1+
1
−6
2+
2
22
2+
2

−+
0
1+
1

+
0
1+
1

0−12
2+
2
3−6
2+
2
20 12
2+
2
3−6
2+
2
2

1+
1
6
2+
2
22
2+
2
−
1+
1
−6
2+
2
24
2+
2

.
We already know that the mechanical stiffness matrix  can be given as
(3.26)
= 1.5
+
0 0 −+
0 0
0 12
2+
2
36
2+
2
20−12
2+
2
36
2+
2
2
0 6
2+
2
24
2+
2
0−6
2+
2
22
2+
2

−+
0 0 +
0 0
0−12
2+
2
3−6
2+
2
20 12
2+
2
3−6
2+
2
2
0 6
2+
2
22
2+
2
0−6
2+
2
24
2+
2

.
We assume a small deformation field in this work. The overall displacement vector  for a
single beam can be represented as the summation of two separate displacement vectors that we
get from the mechanical and electric loading separately, i.e.  and ,
(3.27)
=+=
−1+
−1.
(b) Voltage-dependent deformation field of piezoelectric lattices
The basic development of a honeycomb lattice considering the composite piezoelectric beams is
shown in figure 2. The beam-level stiffness matrices are assembled to obtain the global stiffness
11
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

matrix of the entire piezoelectric lattice. Subsequently, the voltage-dependent displacement field
of the lattice is derived based on applied far-field stresses (resulting in lattice-level nodal load
vector) and boundary conditions through adopting appropriate matrix inversion schemes. The
global node and element numbering schemes are shown in subfigures (d, f). The global node
numbering is initiated row-wise and we move from the bottom row to the top row, in each
row we move from left to right. The order of the element numbering (counter-clockwise) is
shown in a hexagonal unit cell for the odd-numbered rows. For the even rows, only the vertical
members are marked (the inclined beams are included in the odd rows). The global element
numbering is initiated row-wise and we move from the bottom row to the top row, in each
row we move from left to right. Assuming a small deformation problem allows us to separately
calculate the displacement vectors resulting from the mechanical and piezoelectric loading, and
superimpose them to obtain the compound effect of mechanical far-field stress and electrical
stimuli. The developed lattice-level code is generic in nature, and capable of accepting the number
of cells in two perpendicular directions along with unit-cell-level intrinsic material and geometric
parameters as input to obtain the corresponding strain field as output under any given loading and
boundary conditions.
(c) Minimization solver for cloaking
A brief theory regarding fmincon solver [57,58] that follows an interior point algorithm [59,60]
has been discussed as we adopt it explicitly to achieve the minimization objectives concerning
cloaking. Here, we have limited the number of optimization variables to that in the cloaking region
(unless otherwise mentioned), leading to a reduced dependency of the displacement norm to the
minimum number of degrees of freedom. We have adopted this strategy to solve the problem with
better storage efficiency and minimum time. As it would be evident in the following section that
we are able to achieve the target modified deformation field of the damaged lattice through an
automated voltage augmentation factor.
For a particular problem, whether it is linear or nonlinear, convex or non-convex, interior
point approaches can be applied. Both huge, sparse constraints and small, dense problems can
be handled by interior point algorithms. The algorithm can recover from undefined outcomes
like complex and infinity solutions and fulfils limits at all iterations. In the current work, we
have used fmincon, a nonlinear programming solver in Matlab [61]. It uses an interior point
algorithm as a default tool to solve optimization problems (refer to equation (3.28)). To find
the restricted minimum of the nonlinear multi-variable function, fmincon is generally used as
an optimization solver. It uses sequential quadratic programming, trust areas, and a barrier
method considering slack variables (refer to equations (3.29) and (3.30)) to tackle the subpro-
blems that arise throughout the iteration. First, an ideal subproblem is put up in the sequential
quadratic programming model to look for the following feasible point in the present iteration
point. Trust regions in the domain of the problem help in converging the problem. A barrier
function in constrained optimization is a continuous function whose value on a point grows
to infinity as the point gets closer to the limit of the viable region of an optimization problem.
Such functions are employed, substituting an easier way to manage penalized terms in the goal
function for inequality restrictions. Generally, fmincon uses a logarithmic barrier function, that
impacts the objective function by considering a gradient over the values closer and away from
the extreme values. When the objective function is non-decreasing in practical directions, to
within the value of the optimality tolerance, and constraints are met, to within the value of the
constraint tolerance, it indicates that the optimization is accomplished.
12
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

Algorithm 1: Active cloaking in mechanical met a materials. Algorithmic flowchart concisely describing
the development of active cloaking based on voltage-sensitive piezoelectric lattices.
stress constant, p
¯31.
Calculate beam level geometrical properties: Cross-sectional area, first and second moment of inertia for substrate and
piezoelectric layers {As, Ap, I1
s, I1
p, I2
s, I2
p}, and coupling terms {C1, C2} using the set of Equation 11, Equation 12 and Equation 13.
degrees of freedom at the element and global levels. Develop generic expressions for geometrical coordinates of the nodes.
load vectors {Lm}. First, create them at the local element level using Equation 25 then use the connectivity conditions to create
them at global level.
LatKm
−1LatLm evaluate the original displacement field of the undamaged lattice. Here, the matrix terms with “Lat” as a front
Introduce damage and cloaking region: Specify the shape, size, and location of the damage (which we assume, may appear
due to manufacturing irregularity, or at any point during the service life). Remove the damaged beams from the lattice. Subse-
quently, specify the cloaking region, where to activate the piezoelectric beams.
at the local element level then use the connectivity conditions to create them at the global level.
Minimization algorithm: Start off creating [0]np×1 voltage vector, Vi, with np as the number of activated piezoelectric
beams. Use fmincon with displacement norm, Dnorm as the scalar optimization objective function for the cloaking region,
and V as the multi-variable optimization parameter. After the optimization iterations, we get a final Vck for the cloaking at
the primary level. At every step, Dnorm is calculated using the selected elements from the displacement vector, Latvm for the
undamaged lattice and Latv = LatKm
−1LatLm + LatKp
−1LatLp for the damaged lattice. It is to be noted that Latv updates after
each iteration of the fmincon solving process as the voltage vector, V changes, modifying the global piezoelectric load vector,
LatLp. To converge the maximum edge deflection of the damaged lattice to that of the edge deflection of the original undamaged
lattice, we use an automated voltage augmentation factor for multiplying it as a scalar to the Vck that we achieved after primary
storage space.
Define lattice properties: Length ratio, cell wall angle { , θ} and lattice size.
Define intrinsic material properties: Young’s modulus of substrate and piezoelectric layer {Es, Ep}. Effective piezoelectric
Define FEM parameters: Develop strategies for generic cell, element, and node numbering at the local and global level. Define
Define mechanical stiffness and force vectors: Calculate the mechanical stiffness matrices {Km}. Calculate the mechanical
Calculate displacement field for undamaged lattice: First, eliminate the elements relating to the fixed degrees of freedom
considering the boundary conditions for the global stiffness matrix and load vector at the lattice level. Using, Latvm =
superscript indicate the contracted stiffness, load, and displacement vectors after we apply the boundary conditions.
Modify the mechanical stiffness matrix: Update the global stiffness matrix by modifying the local stiffness matrix to [0]6×6
for a group of damaged beams, constituting the damage with different shapes at any random locations.
Define piezoelectric stiffness and force vectors: Depending on the cloaking region, specify the piezoelectric degrees of
freedom. Calculate the piezoelectric stiffness matrices {Kp}. Calculate the piezoelectric load vectors {Lp}. First, create them
level cloaking. The second stage of operation improves the efficiency in terms of the time of optimization and requirement of
h
l
(3.28)
min  
¯ such that
 
¯≤0
 
¯= 0
.
¯≤
 .
¯≤
≤
¯≤
.
In the above set of equation (3.28), () and () represent the set of nonlinear inequality
and equality constraints through arrays. In a similar manner, ,  and ,  represent the
set of linear inequality and equality constraints, respectively. Here,  and  are the lower
and upper bounds for the minimization problem.
(3.29)min

¯  
¯ subject to 
1
¯= 0 and 
2
¯≤0,
(3.30)min

¯, ,
¯= min

¯−Σ 
 subject to ≥0, 
1
¯= 0 and 
2
¯+= 0.
The number of slack variables, 
 is equal to that of the number of inequality constraints, 
2.
Now, the slack variables have to be positive to keep the iterations running in the feasible
domain. For each of > 0, we have equation (3.30).
(3.31)
L
¯,= 
¯+ Σ
1
1
¯+ Σ
2
2
¯,
(3.32)∇,
¯
¯
2L
¯,=∇2 
¯+ Σ
1∇2
1
¯+ Σ
2∇2
2
¯.
13
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

Fmincon solver uses finite differences to estimate gradients if derivatives are not provided by
the user. While a finite-difference estimate may be quicker for difficult derivatives, if we supply
gradients, the solver need not conduct this finite-difference estimation, allowing it to run faster
and be more accurate. Additionally, solvers employ an approximation of the Hessian, which
might differ greatly from the actual Hessian. A Hessian can speed up the process of finding a
solution. The Hessian of the Lagrangian in fmincon algorithm is represented by equations (3.31)
and (3.32). Here, the terms 
1 and 
2 represent the Lagrangian multipliers. The fmincon solver
generally has different algorithms with diverse Hessian options.
(i) Definition of displacement norm and cloaking region
For the current optimization problem performed in this paper, we have tried to minimize ,
which is given by equation (3.33). Here  stands for the degree of freedoms (all including
axial, transverse and rotational) that is included in the cloaking region constituting the activated
piezoelectric beams. Now, it is to be noted that we have used a scalar objective function, but
the optimization problem has multiple design variables, i.e. the different voltage values of the
activated beams stored in . The design variables are related to the global displacement vector
of the lattice, nonlinearly through the finite element formulation.
(3.33) =∑
= 1

Lat−Lat   2
The constraint is provided here in terms of the range of applied voltage. In the above equation
for , Lat and Lat represent the displacements of undamaged lattices without any applied
voltage and the displacements of damaged lattices with externally applied voltage, respectively.
This leads to a cloaking effect with minimal difference in the displacement fields of undamaged
and damaged lattices beyond the cloaking region.
It can be noted that the entire lattice is made of piezoelectric components here but the
members are electrically activated selectively based on the respective value of supplied voltage.
Depending on the location and nature of the damage, the cloaking region can be defined in
an on-demand framework by supplying necessary electrical stimuli to the members within that
region only, leading to an unprecedented active and on-demand cloaking. We have chosen the
cloaking region to be around the damage (unless otherwise mentioned) instead of activating
(i.e. applying voltage) all the undamaged beams. This results in a reduction of the dependence
of the norm,  on more degrees of freedom, instead, we only use the degrees of freedom
inside the cloaking region. It also reduces the optimization time and energy requirement
and does not affect our results. The shape of the cloaking region is determined based on
an algorithmic understanding coupled with intuition. For example, as it would be evident in
the results section, while considering unidirectional loading along direction 1 and for shear
loading along direction 12, we have shifted the cloaking region along the loading direction.
This has been adopted to increase the cloaking effect by providing more space in the highly
affected regions to negate the effect of altered strain fields owing to damage. Our final aim is to
converge the deformation field beyond the cloaking region (and near the loading edge) of the
damaged lattice to that of the edge deformation in an undamaged lattice. We achieve the final
cloaking using an automated voltage augmentation factor to modify the voltage vector resulting
from the fmincon solver, leading to reduction in the time of optimization and requirement of
storage space.
In this context, it is worth mentioning that the number of design variables (i.e. number
of electrically active beam members having non-zero voltage) in the minimization algorithm
increases significantly as the area of the cloaking region increases, leading to an escalation in
the computational expenses. However, a bigger cloaking region may provide better cloaking
effect beyond this region. Thus, there exists a trade-off between these two aspects, and we have
14
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

tried to minimize the cloaking area through an automated and iterative algorithm in the current
study. In case of multiple damages, if they are too scattered as an extreme case, the cloaking
region can be considered throughout the lattice (except for the edge rows), as depicted later in
figure 8.
(ii) Stopping criteria of the minimization solver
It is important to precisely define when the iterations in the optimization algorithm stop, which
depends on different threshold values and tolerances. The adopted minimization solver allows
the user to set different tolerance values including the number of iterations, optimality measure,
constraints, norm value, etc. For the fmincon solver, the tolerance values are generally relative,
i.e. scaling is done with respect to the problem size. We should make sure that the optimization
curve has converged. Depending on the nonlinear optimization problem, the objective function
may not minimize as we expect. Also, results are not always precise when very small tolerances
are set. Instead, a solution might not be able to tell when it has converged, leading to pointless
repetitions. Fmincon iterations will generally stop when any of the function values may surpass
the evaluation limit or when the objective function plot converges and stops reducing.
4. Results and discussion
The design parameters for the lattice construction are briefly discussed first, followed by
boundary and loading conditions. In this context, it may be noted that the proposed method-
ology for active cloaking is generic in nature and it can be implemented in any other lattice
forms. Before presenting the results concerning active cloaking, we have adopted a multi-
stage validation approach involving the voltage-dependent response of the unimorph beam
elements, lattice-level effective mechanical properties without damage and effective electrome-
chanical properties of metamaterials. The results concerning such validation with the literature
[22,38,62,63] are presented in electronic supplementary material, figure S1. Subsequently, we
would demonstrate in the following subsection the concept of active cloaking in lattice-based
mechanical materials considering a range of defect shapes and sizes, along with their numbers,
random locations and multiple scenarios of normal, shear and compound mixed-mode far-field
externally applied stresses.
(a) Lattice- and beam-level geometrical and material properties
In the present work, we have considered honeycomb lattices having composite beams of width,
 as 200 × 10−3 mm with 100 × 10−3-mm thick () substrate of Young’s modulus,  as 70
GPa, and a 50 × 10−3-mm thick () piezoelectric layer of Young’s modulus,  as 127 GPa,
bonded above the substrate. The piezoelectric layer has an effective piezoelectric constant, 31
 of
−2.74 × 10−10 m/V. The size of the honeycomb lattice is taken as 25 × 25 with a cell wall angle of
50° for demonstrating the proposed concepts numerically. The length of the inclined members is
considered as 20 mm and that of the vertical members as 30 mm.
(b) Loading and boundary conditions
As discussed in the earlier sections, the main objective of the current work is to hide (or
minimize) the additional deformation perceived by a hexagonal lattice when subjected to
different types of loading. To start off, we try uni-modal axial loading (or in other words,
far-field applied external stresses to the honeycomb lattice) along direction 1, direction 2 and
15
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

shear loading along direction 12. Subsequently, more complex mixed-mode loading scenar-
ios are considered. Schematic representations of all the uni-modal or mixed-mode loading
scenarios are furnished in figure 2g–k.
For axial loading on the lattice along direction 1, the degree of freedom along direction 1 of
the nodes spanning the left edge of the lattice is fixed and the loading is applied to the nodes
of the opposite edge along the same direction. For axial loading on the lattice along direction 2,
the degree of freedoms along direction 2 of the nodes spanning the bottom edge of the lattice
is fixed and the loading is applied to the nodes of the top edge along the same direction. For
shear loading on the lattice along direction 12, both the degrees of freedom along directions
1 and 2 of the nodes spanning the bottom edge of the lattice are fixed, and the loading is
applied to the nodes of the top edge along direction 1. The loading and boundary conditions
for mixed-mode loading scenarios are obtained by superimposing the cases of normal and shear
modes, as depicted in figure 2j,k. To consider the stability of the lattice system (such that it does
not displace as a rigid body), we fix both the translational degrees of freedom of a corner node
in all the loading scenarios. For all the loading cases that we have considered, either uni-modal
or mixed-mode, we have applied a stress magnitude of 25 Pa.
(c) Displacement field representation
The colour gradient ranging from dark blue to red indicates deformation ranging from zero
to the maximum value. It is clear from figures 3–8 that when the damage is introduced
without activating the piezopatches, the deformation increases which is indicated by the red
region. Later, different activating voltage values on the piezopatches allow the reduction of
the deformation close to the original undamaged lattice. For a better understanding, the edge
deformation of the loaded edges is shown separately. We have used four coloured legends for
all these figures, yellow for the undamaged case, orange for the damaged case without cloaking
and blue for the cloaking case when the exact fmincon voltage values are used. Finally, we have
shown in purple colour the edge deformation when an automated voltage augmentation factor
is multiplied by the fmincon voltage matrix. This factor allows us to converge the final edge
deformation to the edge deformation of the undamaged case, to the extent that the maximum
amount of the additional edge deformation has been reduced. Even though the optimization
problem is nonlinear, fmincon manages to give a voltage matrix that when operated linearly
with the automated voltage augmentation factor leads to the additional edge deformation
reduction. We have seen earlier in §3 that the piezoelectric load vector in equation (3.24a,b)
is linearly dependent upon the voltage of the beam. Now considering the overall lattice with
the cloaking region with different beam voltages, the automated voltage augmentation factor
allows voltage enhancement of each of the beams such that the global deformation reduces.
It is to be noted that, for the case of uni-modal and mix-modal loading (i.e. normal far-field
loading in direction 1, direction 2, or far-field shear loading), the deformed nodal positions
indicate the final position of the disturbed nodes. However, the colour gradient in case of
uni-modal loading is shown depending explicitly on the loading direction, i.e. for loading
along direction 1, the colour gradient ranges only for the direction 1 deformation of the nodes.
Similarly, for loading along direction 2, the colour gradient ranges only for the direction 2
deformation of the nodes. For shear loading along direction 12, the colour gradient ranges only
for the direction 1 deformation of the nodes. For the mixed-mode loading cases, as there is
not a single direction of externally applied stress similar to the uni-modal case, we present the
colour gradient considering the values of normal displacements both in directions 1 and 2 for
each of the nodes. A similar colour bar with shades of purple has been shown to properly
depict the voltage values after we activate the piezopatches. It can be noted that the presented
voltage values are for the final case, when we use the automated voltage augmentation factor to
converge the edge deformation to the maximum possible extent.
16
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

(d) Optimization iterations for different damage shapes
The subfigures (d, h, l) in figures 3–8 show the optimization iterations indicating the norm
reduction as the simulation progresses. For most of the cases, the solver settings allow us to get
the final optimized voltage values on the beams iterating below 600 times. It has been found
that bi-symmetrical damage shapes like a hexagon (figure 3) and star (electronic supplementary
material, figure S3) give better optimization to both the axial loading types. Damage shapes that
are symmetrical only along one axis, for example triangle (electronic supplementary material,
figure S4) and arrow shapes (electronic supplementary material, figure S5) give relatively better
deformation reduction when the loading direction is parallel to the direction of the damage
symmetry. Non-symmetrical shapes like the rhombic (electronic supplementary material, figure
S6) in this case do not show any specific trends as compared with the other ones. Nevertheless,
using fmincon, we are able to achieve a satisfactory reduction in the maximum deformation
for the central damage location for the cases we have considered. In general, considering the
subfigures (a, e, i), the mechanical properties and the configuration of the lattice are such
that the overall moduli of the undamaged lattice are higher along direction 2 than that along
directions 1 and 12. We can infer about this considering the maximum edge deformation values
that are the least for the case of direction 2 loading. It can be seen that the optimization gives
0.00
0.00
1.20 × 10–3
Displacement (m)
0.00
1.19 × 10–4
Displacement (m)
3.61 × 10–3 (i)
Displacement (m)
(a)(b)(c) (d)
2.1034
Voltage (V)
–1.5133
(e)(f)(g)(h)
0.0142
Voltage (V)
–0.0212
(j)(k)(l)
2.2700
Voltage (V)
–0.9263
Function value Function value
2
1.5
1
0.5
0
30
×10–4
Norm: 5.6483e-06
1.05
20
10
0810
Deform (m) ×104
Deform (m) ×10–3
200
Iteration
400
1.5
1
×10–5
Norm: 3.56697e–07
1.2
1
×10–4
0.8
Function value
0.5
2.5
2
1.5
1
0.5
×10–4
Deform (m) Unit Cells
0.80 10 20
Unit Cells
500
Iteration
100
Norm: 8.79427e–06
30
Unit Cells
20
10
00 2
2.85
4
0100 200 300
Iteration
Figure 3. Active mechanical cloaking of a lattice with single hexagonal damage at the centre under uni-modal loading. (a, e,
i) Deformation field of a lattice metamaterial without damage subjected to axial loading along directions 1 and 2, and shear
loading along direction 12. (b, f, j) Deformation field of the same lattice metamaterial with damage subjected to axial loading
along directions 1 and 2, and shear loading along direction 12. The black thickened beams around the damage represent the
cloaking region. (c, g, k) Deformation field of the same lattice metamaterial with damage, subjected to axial loading along
directions 1 and 2, and shear loading along direction 12 (under active piezoelectric cloaking). The thickened beams around
the damage with shades of purple represent the electrically activated cloaking region with different voltage values supplied
to the beams (note the purple colour bars). (d, h, l) Iteration and convergence of the minimization algorithm along with
deformation of the loaded edges for different cases (yellow: undamaged case, orange: damaged case without cloaking, blue:
cloaking with exact fmincon voltage output, purple: cloaking with the voltage output of the bi-level minimization algorithm
involving fmincon and automated voltage augmentation factor).
17
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

lower voltage values for direction 2 loading and it is because of the higher lattice moduli
along that direction. On the other hand, the voltage values are higher in case of damage when
introduced for the shear loading case owing to the lower overall shear moduli of the hexagonal
lattice that we have considered.
(e) Discussion on progressive complexity in damage scenarios
In this section, we demonstrate active cloaking considering progressively complicated damage
scenarios and boundary conditions. In figures 3–8 and electronic supplementary material,
figures S2–S14, the subfigures (a, e, i) show the displacement fields of undamaged lattices,
subfigures (b, f, j) show the displacement fields of damaged lattices without any external
electrical stimuli and subfigures (c, g, k) show the displacement fields of damaged lattices
with activated piezoelectric elements (i.e. when external voltage is applied for cloaking). The
objective here is to minimize the difference between the deformation fields beyond the cloaking
region, and near the loading edges for the undamaged metamaterial (without any applied
voltage) and damaged metamaterial (with applied voltage). Thus, if the displacement fields
0.00
1.62× 10–3
Displacement (m)
0.00
1.16 × 10–4
Displacement (m)
0.00
3.66 × 10–3
Displacement (m)
(b)(c)(d)
2.7180
–2.0498
(e)
(a)
(f)(g)(h)
0.0148
–0.0221
(i)(j)(k)(l)
–0.5430
1.5964
Voltage (V)
Function value
Voltage (V)
Function value
Voltage (V)
Function value
1
1.5 × 10–4
× 10–4
× 10–3
Norm: 5.50956e–06
Unit Cells
30
20
0.5
1.25
0
10
8 9 10
Deform (m)
1.5
0.5
0
1
0
×10–5
×10–4
200 400 600
Iteration
Norm: 3.66947e–07
× 10–4
0.8
Deform (m)
0.8
1
1.2
0 10 20
Unit Cells
50 100 150
Iteration
2.5
2
1.5
Norm: 8.17262e–06
Unit Cells
30
20
10
1
0.5
00
Deform (m)
2
2.35
4
0100 200
Iteration
300
Figure 4. Active mechanical cloaking of a lattice with a single hexagonal damage at any random position under uni-modal
loading. (a, e, i) Deformation field of a lattice metamaterial without damage subjected to axial loading along directions 1 and
2, and shear loading along direction 12. (b, f, j) Deformation field of the same lattice metamaterial with damage subjected
to axial loading along directions 1 and 2, and shear loading along direction 12. The black thickened beams around the
damage represent the cloaking region. (c, g, k) Deformation field of the same lattice metamaterial with damage, subjected
to axial loading along directions 1 and 2, and shear loading along direction 12 (under active piezoelectric cloaking). The
thickened beams around the damage with shades of purple represent the electrically activated cloaking region with different
voltage values supplied to the beams (note the purple colour bars). (d, h, l) Iteration and convergence of the minimization
algorithm along with deformation of the loaded edges for different cases (yellow: undamaged case, orange: damaged case
without cloaking, blue: cloaking with exact fmincon voltage output, purple: cloaking with the voltage output of the bi-level
minimization algorithm involving fmincon and automated voltage augmentation factor).
18
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

beyond the cloaking region and near the loading edges, presented in subfigures (a, e, i) and
(c, g, k), respectively, agree closely, the cloaking algorithm would be treated to have worked
satisfactorily. In addition, the insets in subfigures (d, h, l), primarily the curves representing
the loading edge deformations of undamaged lattice (yellow) and damage lattice with bi-level
optimized voltage input (purple), also ascertains the accuracy of the cloaking algorithm. In
lattice metamaterials, having the loading edge deformations unaltered in damaged lattices
would mean that the critical properties like effective stiffness or effective Young’s moduli
remain unchanged as a result of the mechanical cloaking. In doing so, we need to find out
a set of voltage values applied in the piezoelectric members (obtained through the bi-level
automated minimization algorithm discussed in the preceding sections) so that such mechanical
cloaking can be achieved. It can be noted in this context that the entire lattice is made of
piezoelectric components here, but the members are electrically activated selectively based on
the respective value of supplied voltage. Depending on the location and nature of damage,
the cloaking region can be defined here in an on-demand framework by supplying necessary
electrical stimuli to the members within that region only, leading to an unprecedented active
and on-demand cloaking.
First, we have investigated single damage at the centre covering different shapes like
hexagonal-, rhombic-, triangular-, star- and arrow-shaped damages as shown in figure 3,
1.46 × 10–3
Displacement (m)
0.00
1.75 × 10–4
Displacement (m)
0.00
4.60 × 10–3
Displacement (m)
0.00
(a)(b)(c) (d)
0.2670
Voltage (V)
6
4
2
4
3
2
1
0
30
×10–4
×10–5
×10–4
×10–3
×10–4
×10–3
Norm: 6.74905e–05
1.35
Function value
Function value
Function value
Unit Cells
20
10
01.51
Deform (m)
–0.3591
(e)(f)(g)(h)
0.0254
(i)(j)(k)
Voltage (V)
–0.0616
10.1908
Voltage (V)
–7.1565
(l)
Iteration
Norm: 5.40883e–07
Deform (m)
1.6
1.4
1.2
0.8
1
0 10 20
Unit Cells
0 100 200 300
0.85
6
4
2
0
Iteration
Norm: 1.16175e–05
30
Unit Cells
20
10
00
1.65
5
Deform (m)
200
Iteration
400
300100 200
Figure 5. Active mechanical cloaking of a lattice with a single arrow-shaped damage at any random position under
uni-modal loading. (a, e, i) Deformation field of a lattice metamaterial without damage subjected to axial loading along
directions 1 and 2, and shear loading along direction 12. (b, f, j) Deformation field of the same lattice metamaterial with
damage subjected to axial loading along directions 1 and 2, and shear loading along direction 12. The black thickened
beams around the damage represent the cloaking region. (c, g, k) Deformation field of the same lattice metamaterial with
damage, subjected to axial loading along directions 1 and 2, and shear loading along direction 12 (under active piezoelectric
cloaking). The thickened beams around the damage with shades of purple represent the electrically activated cloaking region
with different voltage values supplied to the beams (note the purple colour bars). (d, h, l) Iteration and convergence of the
minimization algorithm along with deformation of the loaded edges for different cases (yellow: undamaged case, orange:
damaged case without cloaking, blue: cloaking with exact fmincon voltage output, purple: cloaking with the voltage output
of the bi-level minimization algorithm involving fmincon and automated voltage augmentation factor).
19
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

electronic supplementary material, figures S2–S6 (under unidirectional normal and shear
loading). In electronic supplementary material, figure S2, we have increased the damage size
as compared with that in figure 3 to check whether our cloaking algorithm works. The second
set of figures is for uni-modal axial and shear loading but with non-central damage position as
shown in figures 5 and 6 and electronic supplementary material, figures S7–S9. In the previous
set of studies with centrally located damage, the edge deformation is likely to be symmetrical
before we activate the piezopatches. When we change the position of the damage to any
random position, the edge deformation is not symmetrical and the position of the maximum
deformation also changes. Hence, the optimization does not give symmetric results as well,
similar to that of the central damage with a rhombic shape. However, there is a satisfactory
reduction in the deformation value when the piezopatches are activated with the voltage values
as given by fmincon. Considering the cases for the shear loading for the non-central loading
shapes, the edge deformation patterns are similar to those of the central loading shapes. It
can be observed that the fmincon solver iterates the least before convergence for the uni-modal
loading in direction 2 both for central and non-central single loading types. The value of
0.00
0.00
(a)
Displacement (m)Displacement (m)
0.00
3.11 × 10–4
5.81 × 10–3
2.21 × 10–3
(i)
Displacement (m)
(b)(c) (d)
0.1274 ×10–3
×10–3
–0.2970
(e)(f)(g)(h)
0.0527
Function value
Voltage (V)Voltage (V)
Function value
Norm: 0.000145907
30 1.05
Unit Cells
0
10
20
1 1.5 2
Deform (m)
1000
1
2
3
4
Iteration
200
5
4
3
2
1
×10–4
×10–3
Norm: 1.14198e–06
×10–4
0.9
0
–0.0427
100
Iteration
200
(j)(k)0.4461 (l)
–0.2632
Voltage (V)
Deform (m)
0
3
2
1
10 20
Unit Cells
0.01 Norm: 9.35279e–05
30
0.008
0.006
0.004
Function value
0.002
Unit Cells
20
10
00 5
1.1
Deform (m)
1000
Iteration
200
Figure 6. Active mechanical cloaking of a lattice with a random damage shape at any random position under uni-modal
loading. (a, e, i) Deformation field of a lattice metamaterial without damage subjected to axial loading along directions
1 and 2, and shear loading along direction 12. (b, f, j) Deformation field of the same lattice metamaterial with damage
subjected to axial loading along directions 1 and 2, and shear loading along direction 12. The black thickened beams around
the damage represent the cloaking region. In this case, the damage configuration is developed using the overlap of multiple
hexagonal shapes. (c, g, k) Deformation field of the same lattice metamaterial with damage, subjected to axial loading along
directions 1 and 2, and shear loading along direction 12 (under active piezoelectric cloaking). The thickened beams around
the damage with shades of purple represent the electrically activated cloaking region with different voltage values supplied
to the beams (note the purple colour bars). The cloaking region is formed with the union of cloaking regions corresponding to
each of the damaged shapes. (d, h, l) Iteration and convergence of the minimization algorithm along with deformation of the
loaded edges for different cases (yellow: undamaged case, orange: damaged case without cloaking, blue: cloaking with exact
fmincon voltage output, purple: cloaking with the voltage output of the bi-level minimization algorithm involving fmincon
and automated voltage augmentation factor).
20
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

automated voltage augmentation factor is less than one for uni-modal loading in direction 2,
and it is the highest for the shear loading in direction 12.
In the next stage, we consider more complicated damage scenarios concerning random
damage over the lattice domain as shown in figure 6, electronic supplementary material,
figures S10 and S11. Here, we have created random damage patterns considering the overlap
of the existing generalized damage shapes. Separate arrays storing the damaged beams and
corresponding activated piezoelectric beams are created for each generalized damage shape.
Now, let us consider some  number of that similar shape distributed randomly over the lattice.
Hence, we have  arrays, each for the damaged and cloaking cells. In mathematical terms, we
(a) (b) (c) (d) (i)
(e) (f) (g) (h)
(j) (k) (l) (m) (r)
(n) (o) (p) (q)
5.17 × 10–3
Displacement (m)
0.00
3.31 × 10–3
Displacement (m)
0.00
4.24 × 10–3
Displacement (m)
0.00
2.61 × 10–3
Displacement (m)
0.00
30
Unit Cells
0.04
20 Norm: 0.000764644
10
0 5
Deform (m)
Function value
0.03
0.02
0.01
×10–3
×10–3
Deform (m)
2
1000
Iteration
200
–0.5220 0.2352
010 20 Voltage (V)
Unit Cells
30
Unit Cells
0.025 Norm: 0.00019427
20
10
0 5
Deform (m) ×10–3
Function value
0.02
0.015
0.01
0.005
Deform (m)
×10–3
0 100 200
Iteration
–0.2589 Voltage (V) 0.3484
010 20
2
1
Unit Cells
Figure 7. Active mechanical cloaking of a lattice with a random damage shape at any number of random positions under
mixed-mode loading. (a, e) Deformation field of a lattice metamaterial without damage subjected to axial loading along
direction 1 along with shear loading along direction 21. (j, n) Deformation field of a lattice metamaterial without damage
subjected to axial loading along direction 2 along with shear loading along direction 12. (b, k, c, l) Deformation gradient
along direction 1 for the uncloaked and cloaked lattices with damage. (f, o, g, p) Deformation gradient along direction 2 for
the uncloaked and cloaked lattices with damage. Note that the column of subfigures (a, e, j, n) represents the deformation
field of undamaged lattices, while the column of subfigures (b, f, k, o) and (c, g, l, p) represent damaged uncloaked and
damaged cloaked lattices, respectively. The black thickened beams around the damage represent the cloaking region. In
this case, the damage configuration is developed using the overlap of multiple hexagonal shapes. The thickened beams
around the damage with shades of purple represent the electrically activated cloaking region with different voltage values
supplied to the beams (note the purple colour bars). The cloaking region is formed with the union of cloaking regions
corresponding to each of the damaged shapes. (d, m, h, q) Edge deformation for the right and top sides of the lattice.
(i, r) Iteration and convergence of the minimization algorithm along with deformation of the loaded edges for different
cases (yellow: undamaged case, orange: damaged case without cloaking, blue: cloaking with exact fmincon voltage output,
purple: cloaking with the voltage output of the bi-level minimization algorithm involving fmincon and automated voltage
augmentation factor).
21
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

take their union separately which removes the repeating cell numbers. The presented results
ascertain that the cloaking algorithm works satisfactorily, albeit the number of optimization
iterations to converge depends on the complexity of random damage scenarios.
So far, we have discussed the cloaking under uni-modal far-field applied normal or shear
stresses, as depicted in figure 2g–i. In the next stage, we consider mixed-mode far-field loading
scenarios (refer to figure 2j,k). In the first case, we apply an axial load along direction 1, along
with a shear loading along direction 12. Here, we resist the transnational degree of freedoms
(directions 1 and 2) of the left edge. In the second case, we apply an axial load along direction
2, along with a shear loading along direction 12. Here, we resist the transnational degrees of
freedom (directions 1 and 2) of the bottom edge. Another important point to be taken care of
is the magnitude of the stress we apply. Earlier, we mentioned that 25 Pa of stress was used
for both the uni-modal, axial and shear loading cases. Here, for the first mixed-mode loading,
25 Pa has been used for the direction 1 load and 25 Pa for the direction 21 load. However,
for the second mixed-mode loading, 250 Pa has been used for the direction 2 load and 25 Pa
0.00
(a)
Displacement (m)
0.00
5.32 × 10–4
4.09 × 10–3
Displacement (m)
0.00
7.18 × 10–3
Displacement (m)
(b)(c) (d)
–0.1508
(e)(f)(g)(h)
0.0199
Voltage (V)
(i)(j)(k)(l)
0.2323
–0.1697
Voltage (V)
–0.0574
Function value
0.0800
Voltage (V)
0.03 Norm: 0.000204069
0.02
Function value
Unit Cells
30 1.0
20
10
0
0.01
0
02 4
Deform (m)
20 40
Iteration
60
2.5
2
1.5
1
× 10–3
× 10–3
× 10–3
Norm: 3.45497e–06
Deform (m)
×10–4
6
5
4
3
2
1
1.0
0 10 20
0.5 Unit Cells
0 20 40 60
Iteration
0.03 Norm: 0.00038425
0.02
Function value
Unit Cells
30
20
10
00
0.01 Deform (m)
1.05
5
0 20 6040
Iteration
Figure 8. Active mechanical cloaking of a lattice with multiple randomly shaped damages at any random position scattered
around a large part of the lattice domain under uni-modal loading. (a, e, i) Deformation field of a lattice metamaterial
without damage subjected to axial loading along directions 1 and 2, and shear loading along direction 12. (b, f, j)
Deformation field of the same lattice metamaterial with damage subjected to axial loading along directions 1 and 2, and
shear loading along direction 12. The black thickened beams around the damage represent the cloaking region. In this
case, the damage configuration is developed using the overlap of multiple hexagonal shapes. (c, g, k) Deformation field
of the same lattice metamaterial with damage, subjected to axial loading along directions 1 and 2, and shear loading
along direction 12 (under active piezoelectric cloaking). The thickened beams around the damage with shades of purple
represent the electrically activated cloaking region with different voltage values supplied to the beams (note the purple
colour bars). The cloaking region is formed with the union of cloaking regions corresponding to each of the damaged shapes.
(d, h, l) Iteration and convergence of the minimization algorithm along with deformation of the loaded edges for different
cases (yellow: undamaged case, orange: damaged case without cloaking, blue: cloaking with exact fmincon voltage output,
purple: cloaking with the voltage output of the bi-level minimization algorithm involving fmincon and automated voltage
augmentation factor).
22
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

for the direction 12 load, to investigate the accuracy of the cloaking algorithm under higher
far-field applied stress. The results presented in figure 7, electronic supplementary material,
figures S12 and S13 (displacement fields along with the edge deformations for the right and top
edges) establish the accuracy of the proposed active cloaking algorithm under random multiple
damage scenarios and mixed-mode far-field applied stresses.
Though we have tried to restrict the area of cloaking region thus far keeping the computa-
tional expenses in mind, it may sometimes be necessary to consider almost the entire lattice as
cloaking region in certain extreme cases of dense random damage distribution spread through-
out the entire domain. Notably, the necessary area of cloaking region can be determined by
the intelligent metamaterial system through an automated iterative algorithm. Thus, even in
such extreme cases, active mechanical cloaking can be achieved in an unsupervised framework.
Such extreme damage scenarios are shown in figure 8 (under uni-modal loading) and electronic
supplementary material, figure S14 (under mixed-mode loading), where the cloaking region is
considered to be the entire lattice except the outer edges. The comparative displacement fields
near the edges for undamaged and cloaked damaged lattices show a satisfactory degree of
accuracy. In this context, it may be noted that we have not considered failure of individual
beam-like members under the compound effect of mechanical and electrical loading in the
simulation. Such aspects should be included further while designing active mechanical cloaking
for future practical implementations. Consideration of a wider cloaking region would reduce
the possibility of element-level failure owing to the higher scope of stress re-distribution for
negating the effect of damage.
5. Conclusions and perspective
This paper proposes a novel concept of active mechanical cloaking for unsupervised damage
shielding in elastic metamaterials. A radically different approach by introducing piezoelectric
lattices is developed with element-wise multi-physically controlled cloaks by voltage-depend-
ent modulation of the stress and strain fields. Based on the nature and location of damage(s),
the cloaking region can be determined by the intelligent metamaterial system along with the
necessary external voltage demand in each of the beam-like members within the cloaking
region through an automated algorithm. We envisage in a physical system that the strain
measurements can be sent as an output to an external circuit board, which would decide
the cloaking region and the inbuilt algorithm would decide the supply voltage to be sent as
an input to the lattice structure in an unsupervised framework. Subject to a wide range of
uni-modal or mixed-mode loading and boundary conditions, single or multiple disorders and
damages with complex shapes, sizes, distribution and numbers can effectively be kept hidden,
ensuring extended lifetime and high reliability in the homogenized mechanical behaviour of
mechanical metamaterials.
The proposed computational framework for active cloaking in mechanical metamateri-
als involves a seamless integration of three components: (i) piezoelectric beam elements
and their voltage-dependent deformation mechanics, (ii) lattice-level mechanics to obtain
the strain field under a given loading and boundary conditions, and (iii) multi-objective
discrepancy minimization algorithm between the strain fields of damaged and undam-
aged lattices. We have adopted a multi-stage validation approach involving the voltage-
dependent response of the unimorph beam elements, lattice-level effective mechanical
properties without damage and effective electromechanical properties of metamaterials.
Subsequently, a detailed numerical investigation is presented to demonstrate the proposed
active mechanical cloaking algorithm considering the progressive level of complexity in
damage scenarios.
By re-distributing stress and strain components in an active and on-demand framework
based on the location, nature, size and number of damages, mechanical cloaking as proposed
here will facilitate in hiding essential voids and cut-outs in structural designs, or unintentional
23
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

defects (either manufacturing or operational in nature) which are often the sources of deviation
from intended structural performance and further failure initiation. With improved damage
resilience, it will significantly enhance the durability and lifetime of a structure without
compromising on mechanical performances like strength and stiffness. Furthermore, since
the current framework relies on external multi-physical stimuli like voltage in the cloaking
region for altering the stress and strain fields, it is not necessary to reinforce the cloaking
region through optimally introduced additional elements in the lattice, as commonly followed
in the mechanical cloaks proposed in literature. From a sustainability viewpoint, mechanical
cloaking without any additional weight would be crucial for a range of lightweight engineering
applications. Since the periodicity in lattice geometry is not broken in the present framework
and no additional material is added in the cloaking region, it will be crucial for efficient
manufacturing and achieving scalability in production.
Owing to the architected void-filled low-density configurations of metamaterials, they are
prone to defects during the complex manufacturing process, or damages under operational
conditions. The major drawback in the studies concerning mechanical cloaks so far to neg-
ate the effect of damages is that the damage location should be known a priori, and the
cloak is designed around that damaged zone before manufacturing. Such postulation does
not allow unsupervised damage shielding during the manufacturing or service life of the
metamaterials by active reconfiguration of the stress fields depending on random and unpre-
dictable evolution of damage. The realization of active cloaking will bring a step-change in
the field of mechanical cloaking since the location and intensity of damage (/disorder) do not
necessarily need to be known a priori before manufacturing. The current elastostatic cloaking
algorithm can be extended to the dynamic regime to protect lattices and metamaterials from
incident elastic waves by controlling the stiffness locally in a programmable framework. The
proposed active cloaking models will further lead to advanced digital twins, wherein damages
in intelligent mechanical, aerospace and biomedical structures can be identified in real-time and
subsequently the effect of such damages can be eliminated for an uninterrupted mechanical
performance.
Data accessibility. All data sets used to generate the results are available in the main paper. Further details could
be obtained from the corresponding authors upon reasonable request.
Supplementary material is available online [64].
Declaration of AI use. We have not used AI-assisted technologies in creating this article.
Authors’ contributions. D.K.: data curation, formal analysis, validation, visualization, writing—original draft;
S.N.: investigation, supervision, writing—review and editing; T.M.: conceptualization, formal analysis,
funding acquisition, investigation, methodology, resources, supervision, validation, writing—review and
editing.
All authors gave final approval for publication and agreed to be held accountable for the work
performed therein.
Conflict of interest declaration. We declare we have no competing interests.
Funding. DK acknowledges the financial support from the Ministry of Education, India through a Masters
Scholarship. TM would like to acknowledge the Initiation grant received from University of Southampton.
References
1. Sinha P, Mukhopadhyay T. 2023 Programmable multi-physical mechanics of mechanical
metamaterials. Mater. Sci. Eng. R Rep. 155, 100745. (doi:10.1016/j.mser.2023.100745)
2. Jiao P, Mueller J, Raney JR, Zheng X, Alavi AH. 2023 Mechanical metamaterials and beyond.
Nat. Commun. 14, 6004. (doi:10.1038/s41467-023-41679-8)
3. Berger JB, Wadley HNG, McMeeking RM. 2017 Mechanical metamaterials at the theoretical
limit of isotropic elastic stiffness. Nature 543, 533–537. (doi:10.1038/nature21075)
24
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

4. Mukhopadhyay T, Kundu D. 2022 Mixed‐mode multidirectional poisson’s ratio
modulation in auxetic 3D lattice metamaterials. Adv. Eng. Mater. 24, 2101183. (doi:10.1002/
adem.202101183)
5. Kundu D, Ghuku S, Naskar S, Mukhopadhyay T. 2023 Extreme specific stiffness through
interactive cellular networks in bi‐level micro‐topology architected metamaterials. Adv.
Eng. Mater. 25, 2201407. (doi:10.1002/adem.202201407)
6. Ren X, Das R, Tran P, Ngo TD, Xie YM. 2018 Auxetic metamaterials and structures: a review.
Smart Mater. Struct. 27, 023001. (doi:10.1088/1361-665X/aaa61c)
7. Tancogne-Dejean T, Karathanasopoulos N, Mohr D. 2019 Stiffness and strength of
hexachiral honeycomb-like metamaterials. J. Appl. Mech. 86, 11. (doi:10.1115/1.4044494)
8. Zheng X
et
al
. 2014 Ultralight, ultrastiff mechanical metamaterials. Science 344, 1373–1377.
(doi:10.1126/science.1252291)
9. Scarpa F, Ciffo LG, Yates JR. 2004 Dynamic properties of high structural integrity auxetic
open cell foam. Smart Mater. Struct. 13, 49–56. (doi:10.1088/0964-1726/13/1/006)
10. Lakes RS, Elms K. 1993 Indentability of conventional and negative poisson’s ratio foams. J.
Compos. Mater. 27, 1193–1202. (doi:10.1177/002199839302701203)
11. Erturk A, Inman DJ. 2011
Piezoelectric energy harvesting
. Hoboken, NJ: John Wiley & Sons.
(doi:10.1002/9781119991151)
12. Dwivedi A, Banerjee A, Bhattacharya B. 2020 Simultaneous energy harvesting and vibration
attenuation in piezo-embedded negative stiffness metamaterial. J. Intell. Mater. Syst. Struct.
31, 1076–1090. (doi:10.1177/1045389X20910261)
13. Lakes R. 1987 Foam structures with a negative poisson’s ratio. Science 235, 1038–1040. (doi:
10.1126/science.235.4792.1038)
14. Choi JB, Lakes RS. 1992 Non-linear properties of polymer cellular materials with a negative
poisson’s ratio. J. Mater. Sci. 27, 4678–4684. (doi:10.1007/BF01166005)
15. Zeng C, Wang W. 2022 Modeling method for variable and isotropic permeability design of
porous material based on TPMS lattices. Tribol. Int. 176, 107913. (doi:10.1016/j.triboint.2022.
107913)
16. Phani AS, Woodhouse J, Fleck NA. 2006 Wave propagation in two-dimensional periodic
lattices. J. Acoust. Soc. Am. 119, 1995–2005. (doi:10.1121/1.2179748)
17. Gonella S, Ruzzene M. 2008 Analysis of in-plane wave propagation in hexagonal and re-
entrant lattices. J. Sound Vib. 312, 125–139. (doi:10.1016/j.jsv.2007.10.033)
18. de Moura BB, Machado MR, Dey S, Mukhopadhyay T. 2024 Manipulating flexural waves to
enhance the broadband vibration mitigation through inducing programmed disorder on
smart rainbow metamaterials. Appl. Math. Model. 125, 650–671. (doi:10.1016/j.apm.2023.10.
011)
19. Mukhopadhyay T, Ma J, Feng H, Hou D, Gattas JM, Chen Y, You Z. 2020 Programmable
stiffness and shape modulation in origami materials: emergence of a distant actuation
feature. Appl. Mater. Today 19, 100537. (doi:10.1016/j.apmt.2019.100537)
20. Pan Q, Chen S, Chen F, Zhu X. 2020 Programmable soft bending actuators with auxetic
metamaterials. Sci. China Technol. Sci. 63, 2518–2526. (doi:10.1007/s11431-020-1741-2)
21. Crawley EF, De Luis J. 1987 Use of piezoelectric actuators as elements of intelligent
structures. AIAA J. 25, 1373–1385. (doi:10.2514/3.9792)
22. Gibson L, Ashby MF. 1999
Cellular solids structure and properties
. Cambridge, UK:
Cambridge University Press.
23. Sinha P, Walker MG, Mukhopadhyay T. 2023 Non-invariant elastic moduli of bi-level
architected lattice materials through programmed domain discontinuity. Mech. Mater. 184,
104691. (doi:10.1016/j.mechmat.2023.104691)
24. Mazzotti M, Foehr A, Bilal OR, Bergamini A, Bosia F, Daraio C, Pugno NM, Miniaci M. 2023
Bio-inspired non self-similar hierarchical elastic metamaterials. Int. J. Mech. Sci. 241, 107915.
(doi:10.1016/j.ijmecsci.2022.107915)
25. Zaiser M, Zapperi S. 2023 Disordered mechanical metamaterials. Nat. Rev. Phys. 5, 679–688.
(doi:10.1038/s42254-023-00639-3)
26. Prajwal P, Ghuku S, Mukhopadhyay T. 2022 Large-deformation mechanics of anti-curvature
lattice materials for mode-dependent enhancement of non-linear shear modulus. Mech.
Mater. 171, 104337. (doi:10.1016/j.mechmat.2022.104337)
25
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

27. Ghuku S, Mukhopadhyay T. 2022 Anti-curvature honeycomb lattices for mode-dependent
enhancement of nonlinear elastic properties under large deformation. Int. J. Non Linear
Mech. 140, 103887. (doi:10.1016/j.ijnonlinmec.2021.103887)
28. Ghuku S, Mukhopadhyay T. 2023 On enhancing mode-dependent failure strength under
large deformation: the concept of anti-curvature in honeycomb lattices. Compos. Struct. 305,
116318. (doi:10.1016/j.compstruct.2022.116318)
29. Tiwari P, Naskar S, Mukhopadhyay T. 2023 Programmed out-of-plane curvature to enhance
multimodal stiffness of bending-dominated composite lattices. AIAA J. 61, 1820–1838. (doi:
10.2514/1.J062573)
30. Mukhopadhyay T, Naskar S, Adhikari S. 2020 Anisotropy tailoring in geometrically
isotropic multi-material lattices. Extreme Mech. Lett. 40, 100934. (doi:10.1016/j.eml.2020.
100934)
31. Mukhopadhyay T, Naskar S, Kundu D, Adhikari S. 2023 Effective elastic moduli of space-
filled multi-material composite lattices. Compos. Commun. 42, 101656. (doi:10.1016/j.coco.
2023.101656)
32. Sinha A, Mukhopadhyay T. 2022 Kirigami-inspired Metamaterials for programming
constitutive laws: mixed-mode multidirectional auxeticity and contact-induced stiffness
modulation. Iscience 25, 105656. (doi:10.1016/j.isci.2022.105656)
33. Sinha P, Mukhopadhyay T. 2024 Pneumatic elastostatics of multi-functional inflatable
lattices: realization of extreme specific stiffness with active modulation and deployability. R.
Soc. Open Sci. 11, 231272. (doi:10.1098/rsos.231272)
34. Chopra I, Sirohi J. 2013
Smart structures theory
. vol. 35. Cambridge, UK: Cambridge
University Press. (doi:10.1017/CBO9781139025164)
35. Khandelwal RP, Chakrabarti A, Bhargava P. 2013 An efficient hybrid plate model for
accurate analysis of smart composite laminates. J. Intell. Mater. Syst. Struct. 24, 1927–1950.
(doi:10.1177/1045389X13486713)
36. Khandelwal RP, Chakrabarti A, Bhargava P. 2014 Static and dynamic control of smart
composite laminates. AIAA J. 52, 1896–1914. (doi:10.2514/1.J052666)
37. Singh A, Mukhopadhyay T, Adhikari S, Bhattacharya B. 2021 Voltage-dependent
modulation of elastic moduli in lattice metamaterials: emergence of a programmable state-
transition capability. Int. J. Solids Struct. 208–209, 31–48. (doi:10.1016/j.ijsolstr.2020.10.009)
38. Singh A, Mukhopadhyay T, Adhikari S, Bhattacharya B. 2022 Active multi-physical
modulation of poisson’s ratios in composite piezoelectric lattices: on-demand sign reversal.
Compos. Struct. 280, 114857. (doi:10.1016/j.compstruct.2021.114857)
39. Sinha P, Mukhopadhyay T. 2023 On-demand contactless programming of nonlinear elastic
moduli in hard magnetic soft beam based broadband active lattice materials. Smart Mater.
Struct. 32, 055021. (doi:10.1088/1361-665X/acc43b)
40. Singh A, Mukhopadhyay T, Adhikari S, Bhattacharya B. 2022 Extreme on-demand
contactless modulation of elastic properties in magnetostrictive lattices. Smart Mater. Struct.
31, 125005. (doi:10.1088/1361-665X/ac9cac)
41. Fleury R, Monticone F, Alù A. 2015 Invisibility and cloaking: origins, present, and future
perspectives. Phys. Rev. Appl. 4, 037001. (doi:10.1103/PhysRevApplied.4.037001)
42. Schurig D, Mock JJ, Justice BJ, Cummer SA, Pendry JB, Starr AF, Smith DR. 2006
Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977–980. (doi:10.
1126/science.1133628)
43. Zhang S, Xia C, Fang N. 2011 Broadband acoustic cloak for ultrasound waves. Phys. Rev.
Lett. 106, 024301. (doi:10.1103/PhysRevLett.106.024301)
44. Farhat M, Enoch S, Guenneau S, Movchan AB. 2008 Broadband cylindrical acoustic cloak for
linear surface waves in a fluid. Phys. Rev. Lett. 101, 134501. (doi:10.1103/PhysRevLett.101.
134501)
45. Greenleaf A, Kurylev Y, Lassas M, Uhlmann G. 2008 Approximate quantum cloaking and
almost-trapped states. Phys. Rev. Lett. 101, 220404. (doi:10.1103/PhysRevLett.101.220404)
46. Martinez F, Maldovan M. 2022 Metamaterials: optical, acoustic, elastic, heat, mass, electric,
magnetic, and hydrodynamic cloaking. Mat. Today Phys. 27, 100819. (doi:10.1016/j.mtphys.
2022.100819)
47. Milton GW, Briane M, Willis JR. 2006 On cloaking for elasticity and physical equations with
a transformation invariant form. New J. Phys. 8, 248–248. (doi:10.1088/1367-2630/8/10/248)
26
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360

48. Parnell W, Norris A, Shearer T. Elastodynamic cloaking: transformation elasticity with pre-
stressed hyperelastic solids. In
Proceedings of meetings on acoustics
, vol. 17. Melville, NY:
AIP Publishing. (doi:10.1121/1.4795345)
49. Norris AN, Parnell WJ. 2012 Hyperelastic cloaking theory: transformation elasticity with
pre-stressed solids. Proc. R. Soc. A Math. Phys. Eng. Sci. 468, 2881–2903. (doi:10.1098/rspa.
2012.0123)
50. Norris AN, Amirkulova FA, Parnell WJ. 2014 Active elastodynamic cloaking. Math. Mech.
Solids 19, 603–625. (doi:10.1177/1081286513479962)
51. Hai L, Zhao Q, Meng Y. 2018 Unfeelable mechanical cloak based on proportional parameter
transform in bimode structures. Adv. Funct. Mater. 28, 1801473. (doi:10.1002/adfm.
201801473)
52. Álvarez Hostos JC, Fachinotti VD, Peralta I. 2019 Metamaterial for elastostatic cloaking
under thermal gradients. Sci. Rep. 9, 3614. (doi:10.1038/s41598-019-40517-6)
53. Bückmann T, Kadic M, Schittny R, Wegener M. 2015 Mechanical cloak design by direct
lattice transformation. Proc. Natl Acad. Sci. USA 112, 4930–4934. (doi:10.1073/pnas.
1501240112)
54. Sanders E, Aguiló M, Paulino G. 2021 Optimized lattice-based metamaterials for elastostatic
cloaking. Proc. R. Soc. A 477, 20210418. (doi:10.1098/rspa.2021.0418)
55. Wang L, Boddapati J, Liu K, Zhu P, Daraio C, Chen W. 2022 Mechanical cloak via data-
driven aperiodic metamaterial design. Proc. Natl Acad. Sci. USA 119, e2122185119. (doi:10.
1073/pnas.2122185119)
56. Zhang Q, Hu G, Rudykh S. 2024 Magnetoactive asymmetric mechanical metamaterial for
tunable elastic cloaking. Int. J. Solids Struct. 289, 112648. (doi:10.1016/j.ijsolstr.2024.112648)
57. Byrd RH, Hribar ME, Nocedal J. 1999 An interior point algorithm for large-scale nonlinear
programming. SIAM J. Optim. 9, 877–900. (doi:10.1137/S1052623497325107)
58. Waltz RA, Morales JL, Nocedal J, Orban D. 2006 An interior algorithm for nonlinear
optimization that combines line search and trust region steps. Math. Program. 107, 391–408.
(doi:10.1007/s10107-004-0560-5)
59. Coleman TF, Li Y. 1994 On the convergence of interior-reflective Newton methods for
nonlinear minimization subject to bounds. Math. Program. 67, 189–224. (doi:10.1007/
BF01582221)
60. Han SP. 1977 A globally convergent method for nonlinear programming. J. Optim. Theory
Appl. 22, 297–309. (doi:10.1007/BF00932858)
61. MathWorks. 2022
Matlab version: 9.13.0 (r2022b)
. Natick, Massachusetts: The Mathworks
Inc. See https://www.mathworks.com.
62. Mukhopadhyay T, Adhikari S. 2017 Effective in-plane elastic moduli of quasi-random
spatially irregular hexagonal lattices. Int. J. Eng. Sci. 119, 142–179. (doi:10.1016/j.ijengsci.
2017.06.004)
63. COMSOL Documentation. 2023 Comsol multiphysics reference manual. Comsol inc. See
https://www.comsol.com/.
64. Kundu D, Naskar S, Mukhopadhyay T. 2024 Data from: Active mechanical cloaking for
unsupervised damage resilience in programmable elastic metamaterials. Figshare. (doi:10.
6084/m9.figshare.c.7318732)
27
royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 382: 20230360