Stimuli-responsive programmable mechanics of bi-level architected nonlinear mechanical metamaterials

Abstract

Mechanical metamaterials which are often conceptualized as a periodic network of beams have been receiving significant attention over the last decade, wherein the major focus remains confined to the design of micro-structural configurations to achieve application-specific multi-functional characteristics in a passive framework. It is often not possible to actively modulate the metamaterial properties post-manufacturing, critically limiting the applications for a range of advanced intelligent structural systems. To achieve physical properties beyond conventional saturation limits attainable only through unit cell architectures, we propose to shift the design paradigm towards more innovative bi-level modulation concepts involving the coupled design space of unit cell geometries, architected beam-like members and their stimuli-responsive deformation physics. On the premise of revolutionary advancements in additive manufacturing technologies, we introduce hard magnetic soft (HMS) material architectures in the beam networks following physics-informed insights of the stress resultants. Through this framework, it is possible to achieve real-time on-demand control and modulation of fundamental mechanical properties like elastic moduli and Poisson's ratios based on a contactless far-field stimuli source. A generic semi-analytical computational framework involving the large-deformation geometric non-linearity and material non-linearity under magneto-mechanical coupling is developed for the effective elastic properties of HMS material based bi-level architected lattices under normal or shear modes of mechanical far-field stresses, wherein we demonstrate that the constitutive behavior can be programmed actively in an extreme-wide band based on applied magnetic field. Under certain combinations of the externally applied mechanical stress and magnetic field depending on the residual magnetic flux density, it is possible to achieve negative stiffness and negative Poisson's ratio with different degrees of auxecity, even for the non-auxetic unit cell configurations. The results further reveal that a single metamaterial could behave like extremely stiff metals to very soft polymers through contactless on-demand modulation, leading to a wide range of applicability in statics, stability, dynamics and control of advanced mechanical, aerospace, robotics and biomedical systems at different length scales.

Full text

Stimuli-responsive programmable mechanics of bi-level architected
nonlinear mechanical metamaterials
S. Ghuku
a
, S. Naskar
b
, T. Mukhopadhyay
b,
∗
a
Department of Mechanical Engineering, Birla Institute of Technology, Mesra, India
b
Faculty of Physical Sciences and Engineering, University of Southampton, Southampton, UK
Abstract
Mechanical metamaterials which are often conceptualized as a periodic network of beams have been
receiving signicant attention over the last decade, wherein the major focus remains conned to the
design of micro-structural congurations to achieve application-specic multi-functional characteristics
in a passive framework. It is often not possible to actively modulate the metamaterial properties post-
manufacturing, critically limiting the applications for a range of advanced intelligent structural systems.
To achieve physical properties beyond conventional saturation limits attainable only through unit cell
architectures, we propose to shift the design paradigm towards more innovative bi-level modulation
concepts involving the coupled design space of unit cell geometries, architected beam-like members and
their stimuli-responsive deformation physics. On the premise of revolutionary advancements in addi-
tive manufacturing technologies, we introduce hard magnetic soft (HMS) material architectures in the
beam networks following physics-informed insights of the stress resultants. Through this framework, it
is possible to achieve real-time on-demand control and modulation of fundamental mechanical proper-
ties like elastic moduli and Poisson's ratios based on a contactless far-eld stimuli source. A generic
semi-analytical computational framework involving the large-deformation geometric non-linearity and
material non-linearity under magneto-mechanical coupling is developed for the eective elastic prop-
erties of HMS material based bi-level architected lattices under normal or shear modes of mechanical
far-eld stresses, wherein we demonstrate that the constitutive behavior can be programmed actively
in an extreme-wide band based on applied magnetic eld. Under certain combinations of the exter-
nally applied mechanical stress and magnetic eld depending on the residual magnetic ux density, it
is possible to achieve negative stiness and negative Poisson's ratio with dierent degrees of auxecity,
even for the non-auxetic unit cell congurations. The results further reveal that a single metamaterial
could behave like extremely sti metals to very soft polymers through contactless on-demand modu-
lation, leading to a wide range of applicability in statics, stability, dynamics and control of advanced
mechanical, aerospace, robotics and biomedical systems at dierent length scales.
Keywords:
Programmable metamaterials; Hard magnetic soft beam; Stimuli-responsive mechanics;
Geometric and material nonlinearity; On-demand contactless stiness; Active mechanical
metamaterials
1. Introduction
Introduction to mechanical metamaterials and a brief literature review.
Mechanical metamaterials
are an advanced broad class of engineered materials with architected microstructures having designed
geometrical arrangements, leading to unprecedented physical and mechanical properties that are derived
∗
Corresponding author: Tanmoy Mukhopadhyay (Email address: T.Mukhopadhyay@soton.ac.uk)
Preprint submitted to Elsevier March 18, 2025

primarily based on their unique internal structures and geometry along with the intrinsic materials from
which they are made. Metamaterials are often conceptualized as a periodic network of beam-like (or
plate and shell-like) members at a relatively lower length scale to obtain eective properties at higher
length scales, and nd critical applications in a vast spectrum of structural and mechanical applications
ranging from nano and micro to macro scale systems [1, 2, 3, 4]. A typical bottom-up homogenization
framework ranging from an equivalent continuum (with eective properties) at macro-level to honey-
comb microstructures at a lower length scale is shown in Figure 1(a). Eective mechanical properties
of such periodic beam networks not only depend on the beam-level geometry and intrinsic material
characteristics but also are governed by the conguration of the network, i.e. unit cell geometry [5, 6].
Compared to the conventional naturally available materials, the lattice metamaterials have low den-
sity and they provide tunable enhanced multi-functional properties based on the application-specic
demands [7, 8, 9, 10]. Due to the advantages over the natural materials, the lattice materials have
drawn signicant attention of the material scientists and engineers for the last few decades [11, 12, 13].
Revolutionary advancements in the manufacturing technologies especially in the eld of additive man-
ufacturing elevated such interest by providing the freedom to the designers in manufacturing complex
congurations [14, 15, 16].
The major focus of the research on mechanical metamaterials has been the development of several
analytical, computational and experimental frameworks for estimation of the eective responses of
periodic beam networks under static loading [17, 18, 19], dynamic and wave propagation [20, 21],
buckling [22, 23, 24, 25], crushing [26], low-velocity impact [27] etc. Another aspect of the research area
has been the modulation of eective properties by designing the network congurations in terms of lattice
geometric parameters, like, cell angle, thickness to span ratio of the cell walls along with the aspect ratio,
etc. [28, 29, 30]. Auxetic congurations among the architected materials have drawn special attention
due to providing negative Poisson's ratio [31, 32, 33], and a range of associated mechanical advantages
including impact and indentation resistance, shape modulation, higher stiness and improved dynamic
properties. In addition to the hexagonal honeycomb and re-entrant auxetic congurations, several
other forms of lattices, like, rhombic, rectangular brick, triangular, rectangular, square, etc., have found
critical engineering applications due to their special bending or stretching dominated characteristics [34].
Manufacturing the designed complex congurations has become feasible using additive manufacturing,
followed by experimental investigations [35, 36, 37, 38] both for validating the computational frameworks
and subsequent industry-scale production.
Due to the extensive investigations on the design of network congurations for modulation of the
eective properties of lattice materials, the research area has become saturated in the past decade.
2

Figure 1: Bi-level architected lattice metamaterials with periodic network of soft beams having embedded
hard magnetic particles. (a)
A typical homogenization framework for conventional lattice metamaterials ranging
from equivalent continuum at macro-level to honeycomb microstructures at the lower length scales.
(b)
Schematic
representation of hexagonal HMS beam network with the representative unit cell to analyse multi-physical mechanics
under combined external mechanical and magnetic loads.
(c-f)
Denition of local Cartesian coordinate systems (
x
,
y
)
and representation of residual magnetic ux density
Br
0
in the unit cell of hexagonal HMS beam network to be subjected
to: (c, d) magnetic eld along direction-2 in combination with normal stress along direction-1 (
σ1
) or direction-2 (
σ2
), (e,
f) magnetic eld along direction-2 in combination with in-plane shear stress (
τ
).
(g)
Dierent other forms of periodic HMS
beam networks ((I - III) derivatives of hexagonal lattices, (IV) triangular lattice, (V) rectangular lattice) to be analysed
within the proposed multi-physical mechanics-based framework.
(h)
Large deformation multi-physical mechanics of HMS
beams representing the generalized member of periodic HMS beam networks under combined mechanical and magnetic
loading.
(i)
Deformation components of a generalized HMS beam element to derive large deformation kinematics.
3

Hence, the research area has been shifting towards more innovative designs of geometry and intrinsic
material characteristics at the elementary beam-level. One such aspect is to exploit the non-linear
characteristics of the elementary beam members undergoing large deformation. For modulation of the
eective properties of lattice metamaterials as a function of the non-linearity, several geometrically non-
linear frameworks have been developed in the last few years [39, 40, 41]. Another innovative concept
at the elementary beam-level to enhance the eective mechanical properties is providing anti-curvature
to the cell walls subjected to a particular mode of applied mechanical loading [42, 43, 44]. Signicant
enhancements in lattice stiness or exibility and elastic failure strength can be achieved due to the
introduction of anti-curvature to the cell walls [42, 43, 44]. With the revolutionary advancements in the
eld of additive manufacturing, recently lattices made of multiple intrinsic materials have been proposed
which possess unprecedented mechanical properties, attainable based on an expanded design space
[45, 46, 47, 48]. In such literature, the major focus remains conned to the design of micro-structural
congurations to achieve application-specic multi-functional characteristics in a passive framework. It
is not possible to actively modulate the metamaterial properties post-manufacturing, critically limiting
the applications for a range of advanced intelligent structural systems. To achieve physical properties
beyond conventional saturation limits attainable only through unit cell architectures, we propose to shift
the design paradigm towards more innovative bi-level modulation concepts involving the coupled design
space of unit cell geometries, architected beam-like members and their stimuli-responsive deformation
physics. We would introduce hard magnetic soft (HMS) material [49] architectures in the beam networks
following physics-informed insights of the stress resultants. The novel HMS lattice or beam network
is very light in weight but it would be able to demonstrate a wide range of stiness (including sign
reversal) depending on applied magnetic ux. The foundation of the HMS material along with the
relevant reported work in the literature on HMS beam deformations are described very briey in the
following paragraph.
Soft materials are a class of newly developed materials that have found immense technological appli-
cations in a diverse eld, especially in biomedicine [50, 51], soft robotic [52, 53], and exible electronic
devices [54, 55]. Controllable properties of soft active materials under external stimuli, like, light [56],
heat [57], electric [58], magnetic eld [59] etc., open a new avenue to design application-specic devices.
Recent advancements in 3D and 4D technologies make the innovative designs feasible and motivated the
research community [60, 61, 62]. One interesting class among such soft active materials which promises
signicant potential in critical engineering applications is the hard magnetic soft material (HMS ma-
terial) [49]. HMS material is manufactured by embedding hard magnetic particles into soft material
matrix. This newly developed active material (HMS material) shows a magnetically hard and mechani-
4

cally soft property [63]. As the beam is a very fundamental element in designing any structural device,
investigations on the response of beam made of HMS material under magnetic actuation have drawn
the attention of the research community. The complications coming from geometric non-linearity due
to large deformation and material non-linearity under magneto-mechanical coupling make the analysis
of HMS beam structures challenging [64, 65]. In the past few years, several analytical and numerical
models have been proposed by the researchers to capture the non-liner response of HMS beams under
external magnetic stimulation [66, 67, 68]. Besides the theoretical works, some experimental investiga-
tions on HMS beam responses are also reported in the literature [66, 69]. To use the devices made of
HMS beams in soft robotic and electronic applications, the deformed shapes of the HMS beam are of
interest and need to be controlled. By properly designing the residual magnetic ux density in the HMS
beam to be subjected to a particular external magnetic eld, we can design the deformed shapes [70].
As most of the structures in the biological world consist of the feature of functionally graded property,
to meet the complex demand of potential applications of HMS beam structures, recently functionally
graded HMS materials are being designed and manufactured [71, 72].
Rationale behind the proposed magneto-active metamaterials.
The above-presented literature review
reveals that despite being a topic of interest, the theoretical investigations on HMS beam structures
focus on structural characteristics under magnetic actuation only. Investigations on the multi-physical
mechanics of HMS beam structures under combined mechanical load and magnetic actuation are not
addressed in the literature suciently. Moreover, most of the reported theoretical investigations are
numerical in the framework of commercial packages which lack physical insights into the problems.
Some analytical models are also reported in the literature but they are limited to simple beam problems
in terms of loading conditions, geometry, and boundary conditions. In this paper, we consider the
complicated multi-physical mechanics of periodic HMS beam networks subjected to large deformation
under combined mechanical and magnetic loads. One major objective is to develop a physically insightful
semi-analytical framework to estimate the non-linear eective elastic moduli of the HMS beam networks
under the combined fair-eld mechanical stress and magnetic eld. By properly designing the residual
magnetic ux density in the HMS beam elements under an optimal combination of mechanical stress
and magnetic eld along with exploiting the geometric and material non-linearities, modulation of the
eective elastic moduli through the developed semi-analytical framework would be attempted in the
present work.
With the progress in manufacturing capabilities [73], active lattice metamaterials [2, 74] have started
receiving signicant attention from the scientic community. In the context of active elastic property
and stiness modulation in lattice metamaterials with distributed actuation throughout the connecting
5

beam spans, the pioneering works with detailed computational framework development can be traced in
the area of piezoelectric lattices [75, 76]. The major lacuna in piezoelectric lattices is the absence of con-
tactless modulation and involvement of wire networks for supplying voltage to each constituting beams.
Later, lattices with magnetostrictive layers (with distributed actuation throughout the connecting beam
spans) were proposed for contactless on-demand elasticity programming [77]. All these metamaterials
were developed in the regime of small deformation linear analysis framework. Some of the early research
on active control of stiness using magnetic control can be traced back to exploitation of discrete mag-
nets attached to the connecting beam members of the lattice unit cells [78]. Unlike most of the active
lattice metamaterials, Gabriel and Teng [79] presented discrete magneto-active lattices where magnetic
particles are embedded in the joints rather than the beam-like connecting elements, wherein the active
joint movement is exploited for property modulation in the proposed design. Jackson et al. [80] pro-
posed 4D eld responsive lattice metamaterials with connecting polymer tube-like elements lled with
magnetorheological uid suspensions. In general, magneto-active metamaterials have been attracting
signicant attention recently covering dierent spectrum of physical designs including elastic, impact,
vibration, wave propagation and acoustics for on-demand control [81, 82, 83, 84, 85]. Lately, HMS
material based hexagonal lattices with distributed uniform actuation along the beam-like constitut-
ing members have been investigated for active contactless property modulation considering geometric
nonlinearity [86]. In this semi-analytical framework of the earlier work, only hexagonal lattices and
their derivatives such as rhombic, rectangular brick and auxetic congurations can be analyzed. In the
present work we extend the computational framework to analyze other bending and stretching domi-
nated lattices such as triangular and rectangular congurations. Further, for enhancing the eciency
of magnetic actuation, we would introduce non-uniform residual magnetic ux to exploit the concepts
of anti-curvature [42] in metamaterials design.
Description of the bi-level architected lattices with non-uniform magnetic ux density.
A typical
hexagonal network of HMS beams is shown schematically in Figure 1(b). Within the framework of unit
cell approach, an appropriate unit cell consisting of three HMS beam members OA, OB, and OC is
chosen as shown in Figure 1(b) to analyse multi-physical lattice mechanics under combined mechanical
and magnetic load. In the gure, an enlarged view of embedded hard magnetic particles is shown
for clear understanding. From the understanding of boundary conditions for the honeycomb lattices
made of conventional elastic materials [5], denitions of local Cartesian coordinate frames (
x
,
y
) for
the inclined member OA and vertical member OC of the unit cell to be subjected to the magnetic
eld along direction-2 (
Ba
) in combination with normal mechanical stress along direction-1 (
σ1
) or
direction-2 (
σ2
) are shown in Figure 1(c). Similarly, denitions of local Cartesian coordinate frames (
x
,
6

y
) for the inclined and vertical members of the unit cell to be subjected to the magnetic eld along
direction-2 (
Ba
) in combination with in-plane shear stress (
τ
) are shown in Figure 1(e). The direction
and magnitude of residual magnetic ux density
Br
0
in the HMS beam member are controlled by the
orientation and density of the hard magnetic particles embedded in the soft material. Mathematically,
the direction and magnitude of
Br
0
are dened by a coecient
S
. If the residual magnetic ux density
Br
0
is uniform along the beam axis and directed along the
x
axis of the local Cartesian frame (
x
,
y
), the
value of
S
is unity, i.e.,
S= 1
. If the direction of uniform
Br
0
is opposite to
x
axis, then
S=−1
. For
generalized distribution of
Br
0
, the coecient
S(x)
is a function of beam length along the
x
axis of the
local Cartesian frame (
x
,
y
). For the unit cells to be subjected to the magnetic eld along direction-2 in
combination with either normal mode or shear mode of mechanical stress, generalized representations
of the residual magnetic ux density
Br
0
in the HMS beam members are shown in Figure 1(d) and (f)
corresponding to the local frames (
x
,
y
) as dened in Figure 1(c) and (e) respectively. Note that in
Figure 1(f), the direction of residual magnetic ux density
Br
0
is opposite for the inclined members OA
and OB. This opposite distribution makes the inclined members behave structurally symmetric when
subjected to in-plane shear stress
τ
in combination with external magnetic eld
Ba
. This phenomenon
will be described in more detail later through schematic diagrams in connection with the mathematical
formulation of shear modulus.
As discussed in the preceding paragraphs, we propose a novel class of metamaterials as a network of
beams made of soft material with embedded hard magnetic particles which enables real-time on-demand
control and modulation of non-linear elastic properties based on a contactless far-eld stimuli source.
The metamaterial involves a dual design space at the lower length scale (referred to as micro-scale
in the subject domain of metamaterials) as follows. (1) Architecturing of the hard magnetic particle
distribution within the HMS beam elements tailors their multi-physical large deformation mechanics
at the lower length scale (2) Architecturing of the network's periodic geometric congurations (cell
angle, vertical to inclined cell wall length ratio, thickness to inclined cell wall length ratio) further
tailors the unit cells' large deformation mechanics. Such bi-level architectures and designs at the lower
length scale (referred to as micro-scale) govern the homogenized elastic properties of the proposed HMS
metamaterials at the higher length scale (referred to as macro-scale) of the entire lattice dimension.
Hence, the developed computational framework reported in the present article is basically a multi-scale
framework starting from the magnetic particle architected HMS beams and periodic geometry of unit cell
congurations at the micro-scale yielding to tailored homogenized non-linear elastic properties of HMS
beam network at the macro-scale. Note that the computational framework for obtaining the eective
nonlinear elastic properties of the lattice would essentially involve analyzing appropriate unit cells with
7

periodic boundary conditions. The foundational concept of the multi-scale modeling of conventional
lattice metamaterials (involving unit cells that consist of homogeneous passive beams) is demonstrated
through Figure 1(a), and subsequently, the concept of the proposed bi-level architected novel class of
HMS metamaterials (involving unit cells that consist of architected magneto-active beams) is introduced
through Figure 1(b , g).
Scope of the present study.
To estimate the non-linear eective elastic moduli of the periodic HMS
beam network, a generalized multi-physical mechanics problem of HMS beam subjected to combined
mechanical and magnetic loads would be dened within the framework of the unit cell approach. The
HMS beam problem involves complex eects coming from geometric non-linearity due to large deforma-
tion and material non-linearity due to magneto-elastic coupling. A physically insightful semi-analytical
framework would be developed here through the variational principle-based energy method within the
non-linear kinematic setting of the Euler-Bernoulli beam theory using the material constitutive law
according to the Yeoh hyperelastic model. Based on the beam-level deformation results, eective elastic
moduli of the periodic HMS beam networks (i.e lattices) would be computed by accounting the unit
cell geometry and periodic boundary conditions. The semi-analytical beam model will be validated by
comparing the non-linear deformed congurations under separate mechanical load and magnetic actua-
tion with the available literature [65, 70]. After the validation study, a few critical beam-level numerical
results will be presented rst under combined mechanical and magnetic loading for HMS beams with
symmetric and asymmetric residual magnetic ux density. Through the numerical results, the eect of
asymmetry in residual magnetic ux density in dening a generalized HMS beam problem of the HMS
beam network along with the eect of centreline extensibility in analysing large deformation character-
istics of HMS beam will be investigated. A validation study of the semi-analytical framework at the
periodic beam network-level will also be carried out by comparing the non-linear eective elastic mod-
uli with the available results in the literature for honeycomb lattices under dierent modes of far-eld
mechanical loads [39, 42]. Following the validated semi-analytical framework, the eects of magnetic
eld in combination with the dierent modes of far-eld mechanical stress eld on the non-linear eec-
tive elastic moduli of periodic HMS beam network with uniform residual magnetic ux density will be
studied. Based on the kinematic and kinetic conditions of the beam elements of the hexagonal HMS
beam network, two physics-informed designs of residual magnetic ux density will further be proposed
which would signicantly inuence the non-linear eective elastic moduli. Through the numerical re-
sults, we will show that the proposed lightweight HMS beam networks or lattices possess broadband
modulation capability of the non-linear specic stiness ranging from very high stiness like hard metal
to very low stiness even lower than soft polymers depending on the residual magnetic ux density and
8

the compound eect of the externally applied mechanical load and the magnetic eld. Under certain
combinations of the mechanical and magnetic elds, it will be shown that the HMS lattices show neg-
ative stiness as well. The generality of the developed multi-physical mechanics-based semi-analytical
framework will be demonstrated by analysing non-linear elastic moduli of ve other forms of HMS beam
networks, namely, auxetic, rectangular brick, rhombic, triangular, and rectangular networks as shown
in Figure 1(g). Note that under the inuence of combined far-eld mechanical stresses and magnetic
eld, the unit cell mechanics of dierent considered lattice congurations becomes signicantly involved
that has not been investigated in the literature so far.
After presenting a brief review of literature on mechanical metamaterials and the rationale behind
proposing the present novel class of lattices in this introductory section (section 1), the mathematical
framework for the estimation of non-linear eective elastic moduli of periodic HMS beam networks under
dierent modes of far-eld mechanical stress in combination with magnetic eld will be presented in
section 2. Thereafter, section 3 will present the beam-level and periodic beam network-level results along
with the validation studies. Applicability of the proposed physically insightful framework of the periodic
HMS beam network to dierent forms of lattices will be demonstrated through numerical results. The
conclusions will be drawn in sections 4 and 5 along with the prospective future scopes of the proposed
novel class of HMS lattices.
2. Computational framework for stimuli-responsive lattices
A HMS beam multi-physical mechanics based (refer to Figure 1(h, i)) semi-analytical framework
is developed in this section to estimate the non-linear eective elastic moduli of periodic HMS beam
networks subjected to magnetic eld
Ba
along direction-2 either in combination with remote normal
stress along direction-1 (
σ1
), direction-2 (
σ2
) or remote in-plane shear stress
τ
. The combined loading
conditions of mechanical normal stress (
σ1
or
σ2
) and magnetic eld (
Ba
) for the unit cell of hexagonal
HMS beam network (refer to Figure 1(d)) are shown in Figure 2(a) and (d) respectively. Whereas,
the loading condition of mechanical in-plane shear stress (
τ
) in combination with the applied magnetic
eld (
Ba
) for the corresponding HMS unit cell (refer to Figure 1(f)) is shown in Figure 3(a). Under
the combined loading condition as shown in Figure 2(a), we obtain the longitudinal eective Young's
modulus
E1
and Poisson's ratio
ν12
of the hexagonal HMS beam network. For the combined loading
condition as shown in Figure 2(d), we obtain the transverse eective Young's modulus
E2
and Poisson's
ratio
ν21
. Whereas, under the combined loading condition of the shear mode of mechanical load and
the magnetic eld as shown in Figure 3(a), we can estimate the eective shear modulus
G12
of the
hexagonal HMS beam network. Similar loading conditions are presented for other forms of lattices in
9

Figure 4 and 5. Here it should be noted that though the multi-physical mechanics of the HMS unit
cells is presented in Figures 2 - 5 for the compressive mode of normal stress and anti-clockwise mode
of shear stress respectively in combination with the generalized residual magnetic ux density having
coecient
S(x)
, the developed formulation is generalized and valid for the combination of any mode
of the normal (compressive or tensile) and shear (anti-clockwise or clockwise) mechanical stress with a
generalized magnetic eld.
Under the applied combined mechanical stress and magnetic eld, developed forces and large defor-
mation kinematics of the HMS beam elements of the unit cell are analysed rst. Based on the kinetic and
kinematic descriptions, a large deformation problem of the HMS beam representing a general beam-like
element of the periodic network is dened, wherein the boundary and loading conditions are applied
based on unit cell periodicity and applied external mechanical stress and magnetic ux, respectively.
Non-linear multi-physical mechanics of the dened generalized large deformation HMS beam problem
under combined mechanical and magnetic load is analysed subsequently through the variational energy
principle-based semi-analytical framework (with appropriate beam-level boundary condition to ensure
periodicity of the unit cells). Using the beam-level deformation results within the unit-cell framework,
the eective elastic moduli (
E1
,
ν12
,
E2
,
ν21
, and
G12
) of a periodic HMS beam network are computed.
Thus, following a multi-scale framework, the homogenized nonlinear elastic properties of the proposed
metamaterials at the higher length scale (referred to as macro-scale) are estimated in terms of the
beam-level large deformation measures coupled with unit cell geometry under combined mechanical and
magnetic loads at the lower length scale (referred to as micro-scale). In this context, it can be noted
that the proposed computational framework is scale-independent in principle; the only condition is to
maintain a substantial dierence between the unit cell dimensions and the dimension of the overall lat-
tice that leads to the computation of homogenized eective properties. In the forthcoming subsections,
followed by establishing a generic beam-level computational framework, we will rst develop a semi-
analytical formulation for the eective elastic moduli of hexagonal lattices, and subsequently dierent
other lattice geometries will be considered.
2.1. Generalized beam-level problem denition
The load-deformation characteristics of any member of the HMS beam network under any combina-
tion of the fair-eld normal or shear mode of mechanical stress and magnetic eld as presented through
Figures 2 - 5 are dened as a generalized geometrically non-linear HMS beam deformation problem.
Such a generalized large deformation HMS beam problem can be dened either as a full-beam problem
or as a half-beam problem under the specic boundary condition to ensure unit cell level periodicity (all
the beams under consideration here need to have both the edges rotationally restained). Both the type
10

of geometrically non-linear HMS beam deformation problem is presented schematically in a generalized
way in Figure 1(h, i) and described in the following two subsections.
2.1.1. Full-beam problem
When the full length of the inclined or vertical members of a periodic HMS beam network (refer to
Figures 2 - 5) is considered for the denition of the generalized beam problem, the problem is called the
full-beam problem. For example, the length
L
of the generalized HMS beam as shown in Figure 1(h),
is either equal to
l
for the inclined member or equal to
h
for the vertical member of a hexagonal HMS
beam network. For the full-length HMS beam, one end is xed with the other end being rationally
restrained but free to translation and subjected to concentrated force
F
in combination with magnetic
eld
Ba
with inclination angles
β
and
α
respectively as shown in Figure 1(h).
For the full-length HMS beam, rotation of centreline
φ
is zero at both the ends (
x= 0
and
x=L
)
(refer to Figure 1(h, i)). As the HMS beam is subjected to axial load also due to the combined eect
of the mechanical and magnetic eld, the beam centreline has non-zero axial strain
ε
at both the ends
(
x= 0
and
x=L
). The kinematic boundary conditions of the HMS full-beam problem are summarized
below.
φ= 0 at x= 0 and x=L
(1a)
ε= 0 at x= 0 and x=L
(1b)
With the proper denition of the load magnitudes (
F
and
Ba
) and their inclination angles (
β
and
α
)
as presented later in the manuscript (for example, Equations (23)-(36) for the hexagonal lattices) along
with the respective length (for example,
L=l
or
L=h
for hexagonal lattices) and boundary conditions
(Equation (1)), we can simulate deformation characteristics of each member of the HMS beam networks.
For an ordinary beam of length
Lo
with the prescribed boundary conditions undergoing small de-
formation under mechanical load only, the transverse tip-deection
δy
under transverse load
Fy
and the
axial tip-deection
δx
under axial load
Fx
are obtained analytically [5] as
δy=FyL3
o
12EsI
and
δx=FxLo
EsA
.
In these equations,
Es
denotes Young's modulus of the elastic beam material, and
I
and
A
are the rota-
tional inertia and area of the beam cross-section. Note that the above-presented analytical solutions are
not concerned with the present large deformation HMS beam problem. These analytical solutions are
only used for analogy demonstration of boundary condition modelling of the full-beam problem using
cantilevered half-beam problem as presented in the following subsection.
2.1.2. Half-beam problem
The full-beam made of ordinary elastic material undergoing small deformation under mechanical
load only as presented in the preceding subsection, can be modelled as two half-beams with cantilever
boundary conditions exploiting the physical insight that bending moment becomes zero for the full
11

beam at the midpoint here. The transverse and axial deections of the tip of the cantilevered half-beam
of length
Lo/2
are analytically [5] given by
δy=FyL3
o
24EsI
and
δx=FxLo
2EsA
. These analytical deections
obtained from the half-beam model of the ordinary beam are exactly half of the corresponding deection
results as presented in section (2.1.1). Hence, doubling the deection results coming from the half-beam
model gives the same results as the full-beam model for an ordinary beam. A similar observation also
becomes apparent for axial deformation.
Following the observations on boundary conditions, the considered large deformation HMS full-beam
problem under combined mechanical and magnetic load is modelled here as HMS half-beam problem. For
example, in the HMS half-beam problem concerning hexagonal lattices, length
L
of the generalized HMS
beam as shown in Figure 1(h) will be either
l/2
for the inclined member or
h/2
for the vertical member
of the HMS beam network. Note that consideration of the half beam will lead to more computational
eciency compared to considering a full-length beam in the nonlinear multiplysical analysis. Boundary
conditions of the generalized half-beam problem are summarized below.
φ= 0 at x= 0 and dφ
dx= 0 at x=L
(2a)
ε= 0 at x= 0 and x=L
(2b)
Note that the modelling of HMS full-beam as HMS half-beam is only possible if the residual magnetic
ux density
Br
0
is symmetric about the mid-point of the full-length beam. The statement will be proved
in section 3 through numerical results from the full-beam and half-beam models with both symmetric
and asymmetric residual magnetic ux density.
Large deformation analysis of the generalized HMS beam (refer to Figure 1(h, i)) with the above-
prescribed boundary conditions (Equations (1) and (2)) under combined mechanical and magnetic load
is not readily available in the literature. A semi-analytical beam model is developed here to analyse
such multi-physical mechanics problem as presented in the next subsection (subsection 2.2).
2.2. Large deformation analysis of generalized HMS beam problem
Large deformation characteristics of the generalized HMS beam with residual magnetic ux density
Br
0
concerning the initial conguration subjected to combined mechanical load
F
and magnetic eld
Ba
as shown in Figure 1(h, i) is analysed. Governing equation of the geometric non-linear problem
is derived in a semi-analytical framework using the variational principle-based minimization of total
potential energy method. In the derivation of the governing equation, we consider the centreline exten-
sion of the beam in addition to the bending mode of deformation within the geometrically non-linear
kinematic setting of the Euler-Bernoulli beam theory. Derivation of the governing equation through
such a generalized extensible model is presented rst in the following subsection. To investigate the
12

eect of axial rigidity of the hyperelastic HMS beam, a special form of the governing equation neglecting
centreline extension is presented in the following subsection. The nal algebraic form of the governing
equation of the HMS beam problem derived either through the extensible model or through the inexten-
sible model involves non-linearity due to the coupling of dierent deformation degrees of freedom. To
solve the coupled non-linear equation, we develop an iterative computational framework as presented
subsequently in this subsection.
2.2.1. Extensible model
2.2.1.1. Kinematics.
To account for geometrically exact non-linearity, the beam deformation is de-
scribed in terms of the rotation
φ
and strain
ε
of the beam centreline instead of the in-plane and
transverse displacement elds
u
and
v
respectively. From the geometry of deformation as presented
in Figure 1(i), the displacement elds
u
and
v
are expressed in terms of the centreline rotation
φ
and
centreline strain
ε
of the HMS beam as given below.
du
dx= (1 + ε) cos φ−1
(3a)
dv
dx= (1 + ε) sin φ
(3b)
As the left end of the beam is considered xed (refer to Figure 1(h)), the displacement elds
u
and
v
are zero at
x= 0
. With the kinematic conditions, relations of the displacement elds
u
and
v
with the
independent variables
φ
and
ε
are obtained by integrating Equation (3) as given below.
u=Zx
0n(1 + ε) cos φ−1odx
(4a)
v=Zx
0
(1 + ε) sin φdx
(4b)
2.2.1.2. Material model.
The material of the HMS beam under study is considered a soft material with
Young's modulus
Es
. The hyperelastic characteristics of the HMS beam material are modelled by the
strain energy density function
Φ
which is dened below according to the Yeoh hyperelastic model [65].
Φ=
3
X
i=1
Ci0(1 + ε)2+2
1 + ε−3i
(5)
The corresponding nominal stress, dened as
σN=dΦ
dε
, is obtained based on the Yeoh hyperelastic
model [65] using Equation (5) as given below.
σN= 2"C10 + 2C20(1 + ε)2+2
1 + ε−3+ 3C30(1 + ε)2+2
1 + ε−32#(1 + ε)−1
(1 + ε)2
(6)
13

Through Taylor expansion of Equation (6) keeping the linear term, Young's modulus of the hyperelastic
beam material is obtained as
Es= 6C10
(7)
2.2.1.3. Governing equation.
Governing equation for the large deformation characteristics of the HMS
beam under combined mechanical and magnetic load is derived through variational principle based
minimization of total potential energy, as dened mathematically by
δ(UE+UM+V)=0
(8)
In the above equation,
UE
,
UM
, and
V
are the elastic strain energy of the HMS beam, magnetic potential
energy of the HMS beam, and potential energy of the external mechanical load. The elastic strain energy
of the HMS beam
UE
consists of membrane and bending strain energies which in total is given by
UE=AZL
0
Φdx+EsI
2ZL
0dφ
dx2
dx
(9)
Magnetic potential energy
UM
of the HMS beam due to the interaction of the externally applied magnetic
eld
Ba
with the residual magnetic ux density
Br
0
(refer to Figure 1(h)) is given by [70]
UM=−A
µ0ZL
0
S|Br
0||Ba|(1 + ε) cos (φ−α) dx
(10)
In the above equation,
µ0
denotes permeability of vacuum. On the other hand, potential energy of
the externally applied mechanical load
F
is dened as
V=−Fxu|x=L−Fyv|x=L
, where
Fx
and
Fy
are the components of force
F
in the
x
and
y
directions, given by
Fx=Fcos β
and
Fy=Fsin β
respectively (refer to Figure 1(h)). Using Equation (4), the potential energy
V
is expressed in terms of
the independent variables
φ
and
ε
as given below.
V=−FxZL
0n(1 + ε) cos φ−1odx−FyZL
0
(1 + ε) sin φdx
(11)
Before going to further derivation of the governing equation through the energy principle, the physical
coordinate system (
x
,
y
) is transformed into the computational frame (
ξ
,
η
) and some other non-
dimensional geometric and material parameters are introduced as dened below.
ξ=x
L, η =y
L, Π0=AL2
I,¯σN=σN
Es
, B =|Br
0||Ba|Π0
Esµ0
, C =F L2
EsI,¯
Fx=Ccos β, ¯
Fy=Csin β
(12)
Putting the energy expressions presented in Equations (9)-(11) with respect to the normalized coor-
dinate frame (
ξ
,
η
) in terms of the normalized parameters (Equation (12)) into the energy principle
14

(Equation (8)), the governing equation is obtained in variational form as presented below.
δ"Π0Z1
0
Φdξ+1
2Z1
0dφ
dξ2
dξ−BZ1
0
S(1 + ε) cos (φ−α) dξ
−Ccos βZ1
0n(1 + ε) cos φ−1odξ−Csin βZ1
0
(1 + ε) sin φdξ#= 0
(13)
In the normalized frame (
ξ
,
η
), the unknown deformation elds
φ
and
ε
are approximated as
φ=
nb
X
i=1
c1iωi
(14a)
ε=
ns
X
i=1
c2iψi
(14b)
where,
c1i
and
c2i
are the unknown coecients to be computed, and
ωi
and
ψi
are the sets of
nb
and
ns
number of coordinate functions chosen satisfying the kinematic boundary conditions. For the full-beam
problem, the function sets are chosen by satisfying the boundary condition of Equation (1) as
ωi= sin (iπξ)
(15a)
ψi= cos {(i−1)πξ}
(15b)
Whereas, for the HMS half-beam problem, the function sets as chosen through Equation (2) are
ωi= sin 2i−1
2πξ
(16a)
ψi= cos {(i−1)πξ}
(16b)
Now substituting the approximated deformation elds as presented in Equation (14) into the gov-
erning equation (Equation (13)) and carrying out the variational operation, we derive the nal algebraic
form of the governing equation as presented below.
Kc=f
(17)
In the above equation,
K
,
c
, and
f
denote stiness matrix, set of unknown coecients
c1ic2iT
,
and load vector for the large deformation of HMS beam problem respectively. The detailed expressions
of the stiness matrix
K
and load vector
f
are given below.
K11=
nb
X
j=1
nb
X
i=1 Z1
0
ω′
iω′
jdξ
15

K12=
nb
X
j=1
ns
X
i=1 Z1
0"BS sin nb
X
k=1
c1kωk−α!+Ccos βsin nb
X
k=1
c1kωk!
−Csin βcos nb
X
k=1
c1kωk!#ψiωjdξ
K21= [0]
K22=Π0
ns
X
j=1
ns
X
i=1 Z1
0
¯σNcψiψjdξ
f1=
nb
X
j=1 Z1
0"−BS sin nb
X
k=1
c1kωk−α!−Ccos βsin nb
X
k=1
c1kωk!+Csin βcos nb
X
k=1
c1kωk!#ωjdξ
f2=
ns
X
j=1 Z1
0"BS cos nb
X
k=1
c1kωk−α!+Ccos βcos nb
X
k=1
c1kωk!+Csin βsin nb
X
k=1
c1kωk!
−Π0¯σNc(1−1
(1 + Pns
k=1 c2kψk)2)#ψjdξ
where,
¯σNc=2
Es"C10 + 2C20( 1 +
ns
X
k=1
c2kψk!2
+2
1 + Pns
k=1 c2kψk−3)+ 3C30( 1 +
ns
X
k=1
c2kψk!2
+2
1 + Pns
k=1 c2kψk−3)2#
2.2.2. Inextensible model
The governing equation (Equation (17)) presented in the previous subsection is derived considering
both the centreline rotation
φ
and centreline extension
ε
of the HMS beam. If we neglect the terms
corresponding to the centreline strain
ε
from the elements of Equation (17), we readily get the governing
equation of the HMS beam deformation problem within the framework of the inextensible model. The
elements of the stiness matrix
K
and the load vector
f
for the inextensible model are presented
below.
K=
nb
X
j=1
nb
X
i=1 Z1
0
ω′
iω′
jdξ
f=
nb
X
j=1 Z1
0"−BS sin nb
X
k=1
c1kωk−α!−Ccos βsin nb
X
k=1
c1kωk!+Csin βcos nb
X
k=1
c1kωk!#ωjdξ
Note that the Inextensible model is computationally less intensive, but it also becomes less accurate for
large deformation problems.
16

2.2.3. Iterative solution scheme
The elements of stiness matrix
K
and load vector
f
of the governing equation (Equation (17)),
either derived through the extensible model or through the inextensible model, involve unknown coef-
cients
c
. However, the degree of such non-linearity is dierent for the extensible and inextensible
models. Due to the involved non-linearity, the governing equation can not be solved directly. Hence, an
iterative computational scheme [87, 88] is developed to tackle the non-linearity involved in the governing
equation.
Under an incremental step of non-dimensional mechanical load
C
with the inclination angle
β
, the
non-dimensional magnetic load
B
is applied incrementally by a ratio
r
which is termed as magnetic load
ratio and dened by
r=B
C
(18)
Hence, the inputs of the beam model are the magnitude of the non-dimensional mechanical load
C
with
its inclination angle
β
and the magnetic load ratio
r
along with the coecient of the residual magnetic
ux density
S(ξ)
and the inclination angle of the external magnetic eld
α
.
At the incremental step of the non-dimensional mechanical load
C
and magnetic load
B=rC
, the
iterative solution process to nd the set of unknown coecients
c
starts with assumed set of the
coecients denoted as
ci−1
, where the superscript
i
denotes the iteration number. With the assumed
set of the unknown coecients
ci−1
, elements of the stiness matrix
Ki
and load vector
fi
at the
current iteration step
i
are computed. With the known
Ki
and
fi
, the set of unknown coecients
ci
are computed through the matrix inversion of the governing equation (Equation (17)) as
ci=hK−1iifi
(19)
The set of coecients
ci
computed through the above equation, is compared with its old values
ci−1
as
µ=ci−ci−1
. Until the error
µ
becomes less than its predened limit, the set of
unknown coecient
ci+1
is updated through the successive relaxation scheme presented below and
the next iteration (
i+ 1
) begins.
ci+1 =λci+ (1 −λ)ci−1
(20)
In the above equation,
λ
denotes the relaxation parameter for the successive relaxation scheme which
lies between 0 to 1. The iterative scheme to compute the large deformation characteristics of the HMS
beam under combined mechanical load and magnetic eld is presented in Algorithm 1.
Once the set of unknown coecients
c
for the current combined load step
C
and
B
is obtained
through the iterative computational scheme, the centreline rotation
φ
and the centreline strain
ε
become
17

Algorithm 1:
Beam-level computational algorithm to obtain large deformation characteristics of HMS beam under
combined mechanical load and magnetic eld.
Dene geometry:
Dene non-dimensional geometric specication of the HMS beam
Π0
.
Dene material property:
Dene the material constitutive parameters
C10
,
C20
, and
C30
in the
framework of Yeoh hyperelastic model.
Dene numerical parameters:
Dene the numerical values of the computational parameters
λ
,
µ
,
nb
, and
ns
.
Generate:
Generate the set of coordinate functions
ωi
and
ψi
through satisfaction of the boundary
conditions of the HMS beam problem under consideration.
Input load:
Input the magnitude of the non-dimensional mechanical load
C
and magnetic load
B
in terms of the magnetic load ratio
r
as
B=rC
, along with their orientation angles
β
and
α
.
Iterate:
The iterative computational scheme to obtain the set of unknown coecients
c
from the
non-linear governing equation
Kc=f
involves the following steps:

Initialize the set of unknown coecients denoted as
ci−1
.

Compute the stiness matrix
Ki
involving the set of unknown coecients
ci−1
.

Compute the load vector
fi
involving the set of unknown coecients
ci−1
under the current
step of combined mechanical and magnetic loads.

Compute the set of unknown coecients as
ci=hK−1iifi
.

Compare the computed set
ci
with its old values
ci−1
dened as
µ=ci−ci−1
.

Until the error
µ
becomes less than its predened limit, the set of coecients is updated by
ci+1 =λci+ (1 −λ)ci−1
and go for the next iteration
i+ 1
.
Note output:
Once the set of unknown coecients
c
is obtained trough the iterative computa-
tional scheme, the centreline rotation
φ
and the centreline strain
ε
become known which in turn give
the non-dimensional deection prole (
ξ
,
η
) and the tip-deections
¯
δx
and
¯
δy
.
known from Equation (14) for the extensible model. Whereas, for the inextensible model, only the
centreline rotation
φ
is obtained. With the known deformation components (
φ
and
ε
), the deection
prole (
x
,
y
) of the HMS beam is obtained which in turn provides axial deection
δx
and transverse
deection
δy
of the tip of the beam. The expressions of the axial and transverse tip-deections (
δx
and
δy
) in the normalized form as obtained from Equation (4) are given below for the extensible model.
¯
δx=δx
L=Z1
0n(1 + ε) cos φ−1odξ
(21a)
¯
δy=δy
L=Z1
0
(1 + ε) sin φdξ
(21b)
18

For the inextensible model, the normalized tip-deections (
¯
δx
and
¯
δy
) are obtained from the above
equation by neglecting the
ε
terms as
¯
δx=δx
L=Z1
0
(cos φ−1) dξ
(22a)
¯
δy=δy
L=Z1
0
sin φdξ
(22b)
Using the beam-level tip-deections, we compute unit cell level strains under a given far-eld me-
chanical stress and magnetic eld, as discussed in the following subsections considering dierent lattice
geometries.
2.3. Eective elastic moduli of hexagonal HMS beam networks
2.3.1. Beam-level forces and deformation kinematics
As described in Figure 1, the chosen unit cell in hexagonal lattices consists of three HMS beams
having residual magnetic ux density
Br
0
concerning the initial conguration. The beam-level forces de-
veloped under the two dierent combinations of normal stress and magnetic elds as shown in Figure 2(a)
and (d), and under the combination of shear stress with the magnetic eld as shown in Figure 3(a),
along with the large deformation kinematics of the HMS beam elements are described in the following
three subsections.
2.3.1.1. Mechanical normal stress along direction-1 and magnetic eld along direction-2.
Under the
combined mechanical stress
σ1
and magnetic eld
Ba
as shown in Figure 2(a), the inclined HMS beams
(OA and OB) undergo combined transverse and axial deformations with xed end O and the other end A
and B being rotationally restrained but free to translation. Whereas the vertical member OC undergoes
axial deformation only with xed end C. Due to symmetry, we concentrate on one inclined member (OA)
only along with the vertical member OC. The large deformation kinematics of the inclined member OA
and the vertical member OC are shown concerning the local Cartesian frames (
x
,
y
) in Figure 2(b) and
(c) respectively. The kinematic boundary conditions of the beam members are conceptualized from the
classical deformation analysis of conventional honeycomb lattices under mechanical stress only [5]. Note
that due to deformations of the HMS members as shown in Figure 2(b) and (c), the residual magnetic
ux density changes from
Br
0
to
Br
.
As shown in Figure 2(b), the inclined HMS beam OA is subjected to tip concentrated force
Fi
developed due to the applied stress eld
σ1
. Expression of
Fi
in terms of
σ1
is given by
Fi=σ1b(h+lsin θ)
(23)
19

Figure 2: Multi-physical mechanics of periodic hexagonal HMS beam networks under combined mechanical
normal stress and magnetic eld. (a)
Combined loading mode of the unit cell of hexagonal HMS beam network
subjected to normal stress along direction-1 (
σ1
) and magnetic eld along direction-2 (
Ba
).
(b, c)
Beam-level forces
and large deformation kinematics of the inclined and vertical members of the unit cell under the combined normal stress
σ1
and magnetic eld
Ba
. Note that under the combined loading condition (a-c), we focus on the longitudinal eective
Young's modulus
E1
and Poisson's ratio
ν12
of the HMS beam network.
(d)
Combined loading mode of the unit cell
of hexagonal HMS beam network subjected to mechanical normal stress along direction-2 (
σ2
) and magnetic eld along
direction-2 (
Ba
).
(e, f)
Beam-level forces and large deformation kinematics of the inclined and vertical members of the
unit cell under the combined normal stress
σ2
and magnetic eld
Ba
. Note that under the combined loading condition
(d-f), we focus on the transverse Young's modulus
E2
and Poisson's ratio
ν21
of the HMS beam network.
(g)
Local
coordinate systems (
x
,
y
) for the inclined and vertical members and their orientations with the global frame (1, 2).
20

The above-presented force
Fi
is inclined by the angle
βi
concerning the local Cartesian frame (
x
,
y
)
as shown in Figure 2(b). Whereas, the inclination angle of the magnetic eld
Ba
concerning the local
frame (
x
,
y
) is denoted by
αi
. The inclination angles are expressed in terms of the inclination angle
θ
of the inclined member of the beam network as
βi=π−θ
(24a)
αi=3π
2−θ
(24b)
As shown in Figure 2(c), the vertical HMS beam OC is not subjected to any mechanical load but
subjected to magnetic eld
Ba
only with inclination angle
αv
. For the vertical HMS beam OC, the
kinetic equations similar to Equations (23) and (24) are presented below respectively.
Fv= 0
(25)
αv=π
(26)
2.3.1.2. Mechanical normal stress along direction-2 and magnetic eld along direction-2.
When the unit
cell is subjected to far-eld mechanical stress along direction-2 (i.e.
σ2
) along with the magnetic eld
Ba
as shown in Figure 2(d), the kinematic boundary conditions of the HMS members remain the same as
in the case of combined loading
σ1
and
Ba
considered in the previous subsection. The large deformation
patterns of the inclined member OA and the vertical member OC concerning the local Cartesian frames
(
x
,
y
) are shown in Figure 2(e) and (f) respectively. The tip concentrated force
Fi
developed in the
inclined member due to the mechanical stress eld
σ2
is expressed in terms of
σ2
as
Fi=σ2bl cos θ
(27)
The inclination angles of the mechanical load
Fi
and the magnetic eld
Ba
concerning the local frame
(
x
,
y
) are expressed in terms of the inclination angle
θ
as (refer to Figure 2(e))
βi=αi=3π
2−θ
(28)
The vertical HMS beam OC is subjected to mechanical concentrated force
Fv
in addition to the uniform
magnetic eld
Ba
as shown in Figure 2(f). Expression of the force
Fv
in terms of the remote stress
σ2
is given by
Fv= 2σ2bl cos θ
(29)
The inclination angles of the mechanical force
Fv
and the magnetic eld
Ba
concerning the local frame
(
x
,
y
) are given by
βv=αv=π
(30)
21

Figure 3: Multi-physical mechanics of periodic hexagonal HMS beam networks under combined mechanical
shear stress and magnetic eld. (a)
Combined loading mode of the unit cell of hexagonal HMS beam network
subjected to shear stress in 1-2 plane (
τ
) and magnetic eld along direction-2 (
Ba
).
(b)
Free body diagrams of the
inclined and vertical members of the unit cell under the combined in-plane shear stress
τ
and magnetic eld
Ba
.
(c-e)
Beam-level forces and large deformation kinematics of the inclined and vertical members of the unit cell. Note that under
this combined loading condition, we focus on the in-plane shear modulus
G12
of the HMS beam network.
2.3.1.3. Mechanical shear stress in 1-2 plane and magnetic eld along direction-2.
Under the combined
shear stress
τ
and the magnetic eld
Ba
as shown in Figure 3(a), the developed forces and end moments
at the HMS beam members are shown through free body diagrams in Figure 3(b). The forces
F1
and
22

F2
developed due to the far-eld mechanical shear stress
τ
are expressed as
F1= 2τbl cos θ
(31a)
F2=τb(h+lsin θ)
(31b)
From the moment balance condition concerning point O (refer to Figure 3(b)), the induced moment
M
in the inclined members are found to be
M=F1h/4
. Using Equation (31a), the end moment
M
is
expressed in terms of the remote stress
τ
as given below.
M=1
2τ blh cos θ
(32)
Under the mechanical forces and end moments in combination with the magnetic eld, all the HMS
beam members (OA, OB, and OC) undergo combined axial and transverse deformations with xed end
O and the other ends (A, B, and C) being rotationally restrained but free to translation. The large
deformation patterns of the inclined (OA and OB) and vertical (OC) members of the HMS unit cell
concerning the corresponding local Cartesian frames (
x
,
y
) are shown in Figure 3(c), 3(d), and 3(e)
respectively. Though the deformed geometries of the inclined members OA and OB look asymmetric,
they behave structurally (i.e. visually asymmetric, but structurally symmetric) the same under the
combined mechanical and magnetic eld due to the opposite direction of the residual magnetic ux
density
Br
0
in them. Hence, we consider the mechanics of one inclined member (OA) along with the
vertical member OC. In this context, it may be further emphasized that the direction of residual ux
densities
Br
0
is architected dierently under normal and shear far-eld stresses (refer to Figures 2(a, d)
and 3(a)) to maintain structural symmetry in the deformation behavior. Here if we keep the distribution
of residual ux densities
Br
0
same for both the far-eld normal and shear stresses, the analysis will
involve structural asymmetry in any one of cases of far-eld stress, leading to more involved unit cell
level derivation to distribute unbalanced stress resultants at joint O. In the current paper, we have
focused on demonstrating the concepts of active elasticity modulation rather than increasing unit cell
level structural complexity.
The beam-level transverse force
Fyi
for the inclined member OA as shown in Figure 3(c), is the
equivalent force of the end moment
M
derived following the typical rotationally restrained boundary
condition of the member OA as given by
Fyi=−2M/l
. Whereas, the axial force
Fxi
is obtained from
the components of
F1
and
F2
along OA as given by
Fxi=−(F1/2) cos θ−F2sin θ
. Using Equations (31)
23

and (32), the beam-level forces are expressed in terms of the applied remote shear stress
τ
as
Fxi=−τbl cos2θ+h
l+ sin θsin θ
(33a)
Fyi=−τ bh cos θ
(33b)
The inclination angle
αi
of the externally applied magnetic eld
Ba
(refer to Figure 3(c)) is given in
terms of the inclination angle
θ
as
αi=3π
2−θ
(34)
As shown in Figure 3(e), the vertical HMS beam member OC is subjected to transverse force
Fyv
which
is given by
Fyv=F1
. Hence, the expression of the transverse force
Fyv
in terms of the remote stress
τ
is obvious from Equation (31a) as
Fyv= 2τbl cos θ
(35)
In addition to the above presented mechanical force, the vertical HMS beam member OC is subjected
to the vertical magnetic eld
Ba
, inclination angle of which concerning the local Cartesian frame (
x
,
y
)
is obvious from Figure 3(e) as given below.
αv= 0
(36)
2.3.2. Eective elastic moduli
The beam model presented in the previous subsection gives non-dimensional deformation charac-
teristics (
¯
δx
and
¯
δy
) of HMS beam with non-dimensional geometric specication
Π0
for the inputs of
normalized mechanical load
C
and magnetic load
B
in terms of magnetic load ratio
r
as
B=rC
along with their orientation angles
β
and
α
respectively. To use the beam model for the estimation
of elastic moduli of hexagonal HMS beam networks following a unit cell approach (refer to Figures 2
and 3), the geometric specications and loading terms of the HMS beam network need to be dened in
non-dimensional forms. The non-dimensional geometric specications of the inclined (
Π0i
) and vertical
(
Π0v
) members of the HMS beam network are dened following Equation (12) as
Π0i=12
t
l2
(37a)
Π0v=
12 h
l2
t
l2
(37b)
Under any mode of the applied far-eld mechanical stress (
σ1
or
σ2
or
τ
), non-dimensional mechanical
force for the inclined (
Ci
) and vertical (
Cv
) members of the HMS beam network can be obtained following
24

Equation (12) from the beam-level forces (
Fi
and
Fv
) presented in subsection 2.3.1. Such expressions
of the non-dimensional mechanical loads
Ci
and
Cv
in terms of the applied stress (
σ1
or
σ2
or
τ
) are
presented in the subsequent subsections for the three dierent combinations of mechanical and magnetic
loads. Under the dened non-dimensional mechanical load
Ci
for a particular combination of mechanical
and magnetic loads, the non-dimensional magnetic load
Bi
of the inclined member is dened in terms
of the magnetic load ratio
ri
as
ri=Bi
Ci
(38)
With the known non-dimensional magnetic load
Bi
from the above equation, the non-dimensional
magnetic load
Bv
of the vertical member becomes known once we know the relationship between
Bi
and
Bv
. To derive such a relationship between
Bi
and
Bv
, let us observe their denitions from Equation (12)
as given below.
Bi=|Br
0||Ba|Π0i
Esµ0
(39a)
Bv=|Br
0||Ba|Π0v
Esµ0
(39b)
Using Equation (37), the relationship between
Bi
and
Bv
is obtained from the above equation which
gives the non-dimensional magnetic load
Bv
in terms of
Bi
as presented below.
Bv=h
l2
Bi
(40)
Now, with the dened non-dimensional geometric and load parameters, the non-dimensional tip-
deections
¯
δx
and
¯
δy
of the members of the hexagonal HMS beam network are obtained from the
generalized beam model which in turn give the non-linear eective elastic moduli following the framework
of the unit cell approach. Derivations of the eective elastic moduli for the three dierent combinations
of mechanical and magnetic loads are presented in the following three subsections. In addition, non-
dimensional forms of the eective elastic moduli are dened subsequently.
2.3.2.1. Computation of
E1
and
ν12
under combined load
σ1
and
Ba
.
Under the combined loading
of mechanical far-eld normal stress
σ1
and magnetic eld
Ba
as shown in Figure 2(a-c), the non-
dimensional mechanical loads
Ci
and
Cv
are derived using Equations (23), (25) and (12) as given by
Ci=
12 h
l+ sin θ
Est
l3σ1
(41a)
Cv= 0
(41b)
25

With the above-presented non-dimensional mechanical loads
Ci
and
Cv
under normal stress
σ1
, the
non-dimensional magnetic loads
Bi
and
Bv
are dened in terms of the magnetic load ratio
ri
using
Equations (38) and (40). With the dened mechanical and magnetic loads along with their orientation
angles (Equations (24) and (26)), the non-dimensional tip-deections of the inclined member (
¯
δxi
and
¯
δyi
) and the vertical member (
¯
δxv
) of the unit cell of hexagonal HMS beam network (refer to Figure 2(b)
and (c)) are obtained with respect to the local Cartesian frames (
x
,
y
) based on the generalized beam
model presented in subsection 2.2. Through the coordinate transformation between the local frames (
x
,
y
) and the global frame (1, 2) as shown in Figure 2(g), the resultant deection along direction-1 (
δ1
)
and direction-2 (
δ2
) are obtained as
δ1=l−¯
δxicos θ+¯
δyisin θ
(42)
δ2=−l¯
δxisin θ+¯
δyicos θ−h¯
δxv
(43)
The normal strain developed along direction-1 under the combined loading
σ1
and
Ba
is obtained by
ϵ1=δ1/l cos θ
, using Equation (42) which becomes
ϵ1=−¯
δxicos θ+¯
δyisin θ
cos θ
(44)
Similarly, the normal strain along direction-2 is obtained by
ϵ2=δ2/(h+lsin θ)
, using Equation (43)
which becomes
ϵ2=−¯
δxisin θ−¯
δyicos θ−h
l¯
δxv
h
l+ sin θ
(45)
The longitudinal eective Young's modulus of the hexagonal HMS beam network is obtained from
its fundamental denition
E1=σ1/ϵ1
using Equation (44) as
E1=σ1cos θ
−¯
δxicos θ+¯
δyisin θ
(46)
The eective Poisson's ratio
ν12
of the HMS beam network under the combined loading
σ1
and
Ba
is
obtained by the denition
ν12 =−ϵ2/ϵ1
, using Equations (44) and (45) which becomes
ν12 =¯
δxisin θ+¯
δyicos θ+h
l¯
δxvcos θ
h
l+ sin θ−¯
δxicos θ+¯
δyisin θ
(47)
The solution steps involved in the computation of the non-linear eective elastic moduli
E1
and
ν12
of
the hexagonal HMS beam network using the beam model are presented in Algorithm 2. Note that the
solution algorithm is generic and is applicable to the computations of eective elastic moduli under all
the three combined loading conditions of the magnetic eld and dierent far-eld mechanical stresses.
26

Algorithm 2:
Beam network-level computational algorithm to obtain non-linear eective elastic moduli of periodic HMS
beam networks under combined mechanical stress and magnetic eld.
Dene geometry:
Dene non-dimensional geometric parameters of the HMS beam network (such
as
t/l
,
h/l
, and
θ
for hexagonal lattices). With the dened lattice parameters, compute the geometric
specications of the constituting inclined and vertical HMS beams
Π0i
and
Π0v
.
Dene mechanical load:
Under a particular mode of applied mechanical stress (
σ1
or
σ2
or
τ
),
dene the non-dimensional mechanical force for the inclined and vertical HMS beams
Ci
and
Cv
along with their inclination angles
βi
and
βv
.
Dene magnetic load:
Dene the magnetic load ratio
ri
for the inclined HMS beam. Compute the
non-dimensional magnetic load of the inclined member in terms of
ri
and
Ci
as
Bi=riCi
. Compute
magnetic load of the vertical member as
Bv= (h/l)2Bi
along with the inclination angles
αi
and
αv
.
Compute beam deections:
Under the combined mechanical and magnetic loads, compute non-
dimensional tip-deections of the inclined and vertical HMS beams
¯
δxi
,
¯
δyi
,
¯
δxv
, and
¯
δyv
through
solution Algorithm 1.
Compute eective elastic moduli:
In terms of the tip-deections
¯
δxi
,
¯
δyi
,
¯
δxv
and
¯
δyv
, compute
the eective elastic moduli (
E1
,
ν12
,
E2
,
ν21
, and
G12
) of the periodic HMS beam network under the
corresponding mode of mechanical stress in combination with the magnetic eld.
2.3.2.2. Computation of
E2
and
ν21
under combined load
σ2
and
Ba
.
Under the applied normal far-led
stress along direction-2 (
σ2
) in combination with the magnetic eld
Ba
as shown in Figure 2(d)-(f),
the non-dimensional mechanical force for the inclined (
Ci
) and vertical (
Cv
) members of the HMS unit
cell are obtained in terms of
σ2
using Equations (27) and (29) through the normalization scheme of
Equation (12) as
Ci=12 cos θ
Est
l3σ2
(48a)
Cv=
24 h
l2
cos θ
Est
l3σ2
(48b)
The non-dimensional magnetic loads
Bi
and
Bv
are dened in terms of the magnetic load ratio
ri
and the mechanical load
Ci
using Equations (38) and (40). The inclination angles of the mechanical
and magnetic loads (
βi
,
αi
,
βv
, and
αv
) are given in Equations (28) and (30). With the dened input
parameters, the tip-deections of the HMS beam members
¯
δxi
,
¯
δyi
, and
¯
δxv
are computed through the
generalized beam model. As the coordinate systems for the current load combination of
σ2
and
Ba
is
the same with the load combination of
σ1
and
Ba
(refer to Figure 2), the expressions of the deections
δ1
and
δ2
, and the normal strains
ϵ1
and
ϵ2
are the same as presented in Equations (42)-(45). Hence,
the equations are not repeated here to maintain brevity of the paper.
27

The transverse eective Young's modulus
E2
of the hexagonal HMS beam network is dened as
E2=σ2/ϵ2
. Using Equation (45), the nal expression of the Young's modulus
E2
is obtained in terms
of the beam-level deections as
E2=
σ2h
l+ sin θ
−¯
δxisin θ−¯
δyicos θ−h
l¯
δxv
(49)
Using the strain expressions presented in Equations (44) and (45), the eective Poisson's ratio of the
hexagonal HMS beam network is obtained through its fundamental denition
ν21 =−ϵ1/ϵ2
as
ν21 =h
l+ sin θ−¯
δxicos θ+¯
δyisin θ
¯
δxisin θ+¯
δyicos θ+h
l¯
δxvcos θ
(50)
2.3.2.3. Computation of
G12
under combined load
τ
and
Ba
.
Under the combined loading condition of
the shear mode of mechanical stress (
τ
) and magnetic eld
Ba
along direction-2 as shown in Figure 3,
components of the non-dimensional mechanical force
Ci
for the inclined member of the HMS unit cell
are obtained using Equations (33) and (12) as given below.
¯
Fxi=Cicos βi=−
12 cos2θ+h
l+ sin θsin θ
Est
l3τ
(51a)
¯
Fyi=Cisin βi=−
12 h
lcos θ
Est
l3τ
(51b)
From the above set of equations, the non-dimensional mechanical force
Ci
along with its orientation
angle
βi
can be obtained. In terms of the mechanical load
Ci
and the required magnetic load ratio
ri
,
the non-dimensional magnetic loads
Bi
and
Bv
are dened using Equations (38) and (40) having the
orientation angles
αi
and
αv
as dened in Equations (34) and (36). On the other hand, non-dimensional
form of the transverse mechanical force
Fyv
having orientation angle
βv=π/2
(refer to Figure 3) is
derived from Equation (35) and (12) as
¯
Fyv=Cv=
24 h
l2
cos θ
Est
l3τ
(52)
28

Under the prescribed combined mechanical and magnetic loading, rotation
Ω
of the inclined member of
the HMS unit cell (refer to Figure 3(c)) is obtained from the generalized beam model as
Ω=−¯
δyi
(53)
Total horizontal shear deection at point C (
δ1C
) comprises of the deection of the vertical member
OC (
δyv
) and the deection component due to the rotation
Ω
(refer to Figure 3(c) and (e)) dened as
δ1C=hΩ +h¯
δyv
. Using Equation (53), the shear deection
δ1C
is obtained as
δ1C=h−¯
δyi+¯
δyv
(54)
The horizontal and vertical components of the axial deection
δxi
at point A of the inclined member
(refer to Figure 3(c)) are obtained through a coordinate transformation as given by
δ1A=−l¯
δxicos θ
(55a)
δ2A=−l¯
δxisin θ
(55b)
Due to the deections as presented in Equations (54) and (55), the total shear strain developed in the
HMS unit cell under the combined loading of
τ
and
Ba
is given by
γ12 =δ1C+δ1A
h+lsin θ+δ2A
lcos θ=
h
l−¯
δyi+¯
δyv−¯
δxicos θ
h
l+ sin θ−¯
δxisin θ
cos θ
(56)
The eective shear modulus
G12
of the hexagonal HMS beam network under the combined loading
τ
and
Ba
is dened in terms of the developed shear strain as
G12 =τ/γ12
. Using the expression of the
shear strain as presented in Equation (56), we get the nal form of
G12
as shown below.
G12 =
τh
l+ sin θcos θ
h
l−¯
δyi+¯
δyvcos θ−¯
δxicos2θ−¯
δxih
l+ sin θsin θ
(57)
From the expressions of eective elastic moduli presented in Equations (46), (47), (49), (50) and (57)
(and subsequently considering the dependencies of the tip deections), we notice nonlinear dependency of
the moduli on applied magnetic eld and far-eld stress, along with unit cell geometry, intrinsic material
properties and residual magnetic ux architecture. Such complex interplay of the inuencing parameters
in an expanded design space provides a unique scope of designing novel metamaterial functionalities
with unprecedented mechanical behavior.
2.3.2.4. Non-dimensional elastic moduli.
To observe the eect of non-linearity along with the incremen-
tal eect of the magnetic eld with the applied mechanical load on the hexagonal HMS beam network
explicitly, we present the eective elastic moduli in specic forms. Among the ve elastic moduli, the
29

Poisson's ratios
ν12
and
ν21
are already in non-dimensional forms. Hence, they are presented in their
original forms. Whereas, the other three eective elastic moduli of the HMS beam network (
E1
,
E2
,
and
G12
) are expressed in non-dimensional forms as given below.
¯
E1=E1
Esρ3,¯
E2=E2
Esρ3,¯
G12 =G12
Esρ3
(58)
Where,
ρ
is the relative density of the hexagonal HMS beam network dened as the ratio of the volume
of the total intrinsic HMS material and the volume of the equivalent plate-like object that the hexagonal
HMS beam network acquires [5]. Expression of the relative density
ρ
is given by
ρ=h
l+ 2t
l
2h
l+ sin θcos θ
(59)
2.3.2.5. Note on dierent lattice architectures.
For the hexagonal network of HMS beams, a detailed
derivation of the non-linear eective elastic moduli within the multi-physical mechanics-based semi-
analytical framework is presented in this subsection. To demonstrate the generality of the physically
insightful framework, non-linear eective elastic properties of ve other HMS beam networks, namely,
auxetic, rectangular brick, rhombic, triangular, and rectangular congurations are also analysed within
the broad framework (refer to Figure 1(g)). Among the considered ve other forms of HMS beam
networks, the eective elastic moduli of the auxetic, rectangular brick, and rhombic congurations
are readily obtained from the framework for hexagonal HMS beam network by properly selecting the
geometric parameters
h/l
and
θ
(note: for auxetic conguration
θ
is negative, for rectangular brick con-
guration
θ
is zero, for rhombic conguration
h/l
is zero). However, for the triangular and rectangular
HMS beam networks, the appropriate unit cells need to be chosen and analyzed separately. The detailed
derivations of the non-linear elastic moduli for the triangular and rectangular HMS beam networks are
presented in the following subsections. Note that under the inuence of combined far-eld mechanical
stresses and magnetic eld, the unit cell mechanics of dierent lattice congurations becomes signi-
cantly involved (due to combined bending and stretching dominance in a multi-physical environment)
that has not been investigated in the literature.
2.4. Eective elastic moduli of triangular HMS beam networks
The non-linear eective elastic moduli
E1
,
ν12
,
E2
,
ν21
, and
G12
of a triangular network of HMS
beams, as shown in Figure 1(g)IV, under dierent modes of far-eld mechanical stress (
σ1
,
σ2
, and
τ
)
in combination with the magnetic eld
Ba
are derived in this subsection. The unit cell of the triangular
HMS beam network is an equilateral triangle with side
l
having residual magnetic ux density
Br
0
. The
combined loading conditions for the triangular HMS unit cell under the longitudinal and transverse
30

normal stresses
σ1
and
σ2
in combination with the magnetic eld
Ba
along direction-2 are shown in
Figure 4(a) and (b) respectively. Whereas, the combined loading condition under the in-plane shear
stress
τ
and the magnetic eld
Ba
for the triangular HMS unit cell is shown in Figure 4(d). Note in
Figure 4(d) that the direction of residual magnetic ux density
Br
0
for the inclined members OB and AB
is opposite (unlike the unit cells considered under far-eld normal stresses). This opposite distribution
of
Br
0
makes the members OB and AB structurally symmetric under the in-plane shear stress
τ
in
combination with external magnetic eld
Ba
. This phenomenon is already described in detail for the
hexagonal HMS beam network and is not repeated here to maintain brevity.
Under only far-eld mechanical stress (
σ1
,
σ2
, and
τ
) in absence of magnetic eld
Ba
, the cell members
undergo stretch-dominated deformations [5]. Hence, the eective elastic moduli of the triangular lattice
congurations under mechanical load only are governed by the axial deformations of the members [34].
The analytical formulae for the eective elastic moduli of triangular lattices (with cell wall thickness
t
)
under mechanical load only within small deformation regime are given by [5, 34]
E1
Es
=E2
Es
=2
√3
t
l
(60a)
ν12 =ν21 =1
3
(60b)
G12
Es
=√3
4
t
l
(60c)
In this subsection, the conventional unit cell-based approach for triangular lattices [5, 34] is extended
to a magneto-active multi-physical mechanics-based semi-analytical framework following the formulation
for hexagonal HMS beam network presented in the preceding subsection, leading to the evaluation of
non-linear eective elastic moduli of the triangular HMS beam network under combined mechanical
and magnetic loads. Large deformation kinematics of the triangular HMS unit cell and the beam-level
forces developed under dierent combinations of mechanical stress and magnetic eld are described rst
in the following subsection. With the identied kinematic and kinetic conditions, the beam-level non-
linear multi-physical mechanics problems are solved through the semi-analytical HMS beam model as
presented in subsection 2.1 and subsection 2.2. Using the beam-level deformation results, computations
of the non-linear eective elastic moduli of the triangular HMS beam network under the combined
mechanical stress and magnetic eld are presented subsequently.
2.4.1. Beam-level forces and deformation kinematics
Under the combined mechanical and magnetic loads as presented in Figure 4(a), (b), and (d), the
HMS beam members undergo bending in combination with axial deformation. Kinematics and kinetics
31

Figure 4: Multi-physical mechanics of periodic triangular HMS beam network under combined mechanical
stress and magnetic eld. (a)
Combined loading mode of the triangular HMS unit cell under normal stress along
direction-1 (
σ1
) and magnetic eld along direction-2 (
Ba
).
(b)
Combined loading mode of the triangular HMS unit cell
under normal stress along direction-2 (
σ2
) and magnetic eld along direction-2 (
Ba
).
(c)
Deformed conguration of the
triangular HMS unit cell under combined normal stress
σ1
or
σ2
and magnetic eld
Ba
.
(d)
Combined loading mode of
the triangular HMS unit cell under shear stress in plane 1-2 (
τ
) and magnetic eld along direction-2 (
Ba
).
(e)
Deformed
conguration of the triangular HMS unit cell under combined shear stress
τ
and magnetic eld
Ba
.
(f)
Generalized forces
and large deformation kinematics of inclined and horizontal members under any of the three combined loading conditions.
32

of the beam members under the magnetic eld
Ba
in combination with the three dierent modes of the
mechanical stress
σ1
,
σ2
, and
τ
are presented in the following three subsections.
2.4.1.1. Mechanical normal stress along direction-1 and magnetic eld along direction-2.
Under the
combined loading of far-eld normal stress
σ1
and magnetic eld
Ba
as shown in Figure 4(a), all the
three members (OA, OB, and AB) of the triangular HMS unit cell undergo combined bending-stretching
deformation with one end xed, while the other ends being restrained to rotation and transverse dis-
placement but free to axial translation. The deformed conguration of the triangular HMS unit cell
under the combined loading of
σ1
and
Ba
is shown in Figure 4(c). The generalized gure also repre-
sents the deformed conguration under the combined loading of
σ2
and
Ba
. Note in the gure that the
changes in the span of the HMS beam members are shown in a generalized manner without taking into
consideration of the proper algebraic signs. Those senses of the axial deformations will be implicitly
taken care of by the generic beam model under the proper description of the sense of the beam-level
forces under a particular combined loading condition.
Due to the symmetry of the deformation under the combined loading of
σ1
and
Ba
, we concentrate
on one inclined member OB in addition to the horizontal member OA. To use the developed framework
of hexagonal HMS beam network as presented in the preceding subsection readily for the present multi-
physical mechanics of triangular HMS beam network, we consider half of the members OA and OB of
length
l/2
which have similar boundary conditions as those of the members of the hexagonal network,
i.e., one end xed with the other end being rotationally restrained but free to translation (refer to section
2.1.1). Point O is considered the xed point and origin of local Cartesian frames (
x
,
y
) for half of the
inclined and horizontal members. Large deformation kinematics along with the developed forces in half
of the inclined and horizontal HMS members under the combined loading of
σ1
and
Ba
are shown in
Figure 4(f). Note that the kinematic and kinetic descriptions of the HMS half beams in Figure 4(f) are a
generalized representation under any of the three combined loading conditions presented in Figure 4(a),
(b), and (d).
Under the remote mechanical stress
σ1
, the tip-concentrated force
Fh
developed in the horizontal
member as shown in Figure 4(f) is given by
Fh=√3
2σ1bl
(61)
Inclination angles
βh
and
αh
of the above-presented mechanical force
Fh
and the vertical magnetic eld
33

Ba
for the horizontal HMS member (refer to Figure 4(f)) are given by
βh=π
(62a)
αh=π
2
(62b)
For the inclined HMS member as shown in Figure 4(f), the developed force
Fi
and the inclination angle
αi
of the magnetic eld
Ba
are given by
Fi= 0
(63)
αi=π
6
(64)
2.4.1.2. Normal stress along direction-2 and magnetic eld along direction-2.
Under the remote nor-
mal stress
σ2
in combination with the external magnetic eld
Ba
as shown in Figure 4(b), the large
deformation kinematics of the triangular HMS unit cell and the kinetics of the HMS beam members are
already described through Figure 4(c) and (f). The concentrated force
Fh
developed in the horizontal
HMS beam due to the remote stress
σ2
is given by
Fh=1
2√3σ2bl
(65)
As observed in Figure 4(f), the inclination angles
βh
and
αh
are given by
βh= 0
(66a)
αh=π
2
(66b)
The concentrated force
Fi
developed in the inclined member is expressed in terms of the remote normal
stress
σ2
(refer to Figure 4(f)) as
Fi=1
√3σ2bl
(67)
The inclination angles
βi
and
αi
of the mechanical and magnetic loads for the inclined member as shown
in Figure 4(f) are presented below.
βi=π
(68a)
αi=π
6
(68b)
2.4.1.3. Far-eld shear stress in 1-2 plane and magnetic eld along direction-2.
When the triangular
HMS beam network is subjected to in-plane shear stress
τ
combined with the external magnetic eld
Ba
as shown in Figure 4(d), all the three members (OA, OB, and AB) of the triangular HMS unit cell
are subjected to the same boundary conditions as those under the combined normal stress (
σ1
or
σ2
)
and the magnetic eld (
Ba
) (refer to Figure 4(c)). However, under the combined load of
τ
and
Ba
,
34

the two inclined members OB and AB undergo the opposite modes of axial deformation (compression
and tension), and hence the triangular HMS unit cell becomes asymmetric as shown in Figure 4(e).
The opposite distribution of the residual magnetic ux density
Br
0
in the inclined members OB and AB
makes the structural behaviour under the mechanical and magnetic eld in phase with each other (i.e.
structurally symmetric, as discussed in the derivation of hexagonal lattices). Within the unit cell-based
approach to compute the eective shear modulus, we concentrate only on one inclined member OB in
addition to the horizontal member OA. The large deformation kinematics and force kinetics of half of
the inclined and horizontal HMS beams are presented through the generalized schematic in Figure 4(f).
Under the remote shear stress
τ
, the concentrated axial force
Fh
developed in the horizontal member
along with the inclination angle
αh
of the magnetic eld (refer to Figure 4(f)) are expressed as
Fh= 0
(69)
αh=π
2
(70)
The concentrated force
Fi
developed in the inclined HMS member as shown in Figure 4(f) is expressed
in terms of the remote shear stress
τ
as
Fi=τbl
(71)
The inclination angles
βi
and
αi
of the mechanical force
Fi
and the magnetic eld
Ba
for the inclined
HMS beam (refer to Figure 4(f)) are given below.
βi=π
(72a)
αi=π
6
(72b)
2.4.2. Eective elastic moduli
To estimate the non-linear eective elastic moduli of the triangular HMS beam network, geometri-
cally non-linear axial tip-deections
δxh
and
δxi
of the horizontal and inclined HMS beams under the
concentrated force
Fh
and
Fi
combined with the magnetic eld
Ba
as described through Figure 4(f) in
the previous subsection are computed based on the generalized HMS beam model. In the framework of
the generalized HMS beam model, the geometries of the horizontal and inclined HMS half beams shown
in Figure 4(f) are normalized as
Π0h=Π0i=3
t
l2
(73)
The non-dimensional forms of the beam-level forces in the framework of the generalized HMS beam
model are presented in the respective subsection estimating the elastic moduli of the triangular HMS
35

beam network under a particular combined loading case. Expression of the relative density and non-
dimensional forms of the eective elastic moduli are presented subsequently.
2.4.2.1. Computation of
E1
and
ν12
under combined load
σ1
and
Ba
.
Under the combined loading of
normal stress
σ1
and magnetic eld
Ba
(refer to Figure 4(a) and (f)), the non-dimensional mechanical
forces
Ch
and
Ci
for the horizontal and inclined members are derived from Equations (61) and (63)
following the normalization scheme discussed earlier as
Ch=3√3
2Est
l3σ1
(74a)
Ci= 0
(74b)
With the non-dimensional mechanical forces
Ch
and
Ci
, the non-dimensional magnetic loads
Bh
and
Bi
for the horizontal and inclined HMS beams are dened in terms of the magnetic load ratio
rh
as
Bh=Bi=rhCh
(75)
Under the non-dimensional mechanical and magnetic loads with the inclination angles presented in
Equations (62) and (64), the non-linear non-dimensional tip-deections
¯
δxh
and
¯
δxi
of the horizontal
and inclined HMS beams are computed. The normal strain in direction-1 (
ϵ1
) is obtained in terms of
the beam-level defection
¯
δxh
through a suitable coordinate transformation as given by
ϵ1=¯
δxh
(76)
The normal strain in direction-2 (
ϵ2
) is derived from the deformed geometry of the triangular HMS unit
cell as presented in Figure 4(c). By using the Pythagorean theorem on the triangle, we get
(h+δ2)2+l+δ1
22
= (l+δi)2
(77)
Noting the geometric relation of the undeformed triangular unit cell as
h2+ (l/2)2=l2
(refer to
Figure 4(a)) and neglecting the higher order terms, the above equation gives
√3δ2=−1
2δ1+ 2 δi
(78)
From the above relation, the strain
ϵ2
is obtained in terms of the beam-level displacements as
ϵ2=−1
3¯
δxh+4
3¯
δxi
(79)
With the known normal strains
ϵ1
and
ϵ2
as presented in Equations (76) and (79), the non-linear eective
elastic moduli
E1
and
ν12
are obtained as
E1=σ1
¯
δxh
(80)
36

ν12 =1
3−4
3
¯
δxi
¯
δxh
(81)
2.4.2.2. Computation of
E2
and
ν21
under combined load
σ2
and
Ba
.
Under the normal stress
σ2
com-
bined with
Ba
as shown in Figure 4(b) and (f), the non-dimensional forces
Ch
and
Ci
for the horizontal
and inclined beams are obtained from Equations (65) and (67) as
Ch=√3
2Est
l3σ2
(82a)
Ci=√3
Est
l3σ2
(82b)
In combination with the above-presented non-dimensional mechanical forces
Ch
and
Ci
, the HMS
beams are subjected to the non-dimensional magnetic loads
Bh
and
Bi
which are dened in terms of
the magnetic load ratio
ri
by
Bh=Bi=riCi
(83)
The inclination angles of the mechanical and magnetic loads are already presented in Equations (66)
and (68). Following the same procedure as in the previous combined loading case in the preceding
subsection, the non-linear non-dimensional tip-deections
¯
δxh
and
¯
δxi
are obtained which give the normal
strains
ϵ1
and
ϵ2
having the same mathematical expressions as presented in Equations (76) and (79).
Using the strain expressions, the non-linear eective Young's modulus
E2
and the Poisson's ratio
ν21
of
the triangular HMS beam network are derived as
E2=3σ2
−¯
δxh+ 4 ¯
δxi
(84)
ν21 =3¯
δxh
¯
δxh−4¯
δxi
(85)
2.4.2.3. Computation of
G12
under combined load
τ
and
Ba
.
Under the combined in-plane shear stress
τ
and magnetic eld
Ba
as presented in Figure 4(d) and (f), the non-dimensional mechanical forces
Ch
and
Ci
for the horizontal and inclined HMS beam members as derived from Equations (69) and (71)
are given by
Ch= 0
(86a)
Ci=3
Est
l3τ
(86b)
37

The non-dimensional magnetic loads
Bh
and
Bi
of the horizontal and inclined HMS beam members are
dened similarly as those for the other two combined loading cases as
Bh=Bi=riCi
(87)
Under the non-dimensional mechanical and magnetic forces with the inclination angles of Equa-
tions (70) and (72), the non-linear non-dimensional defections
¯
δxh
and
¯
δxi
are computed through the
generalized HMS beam model. To derive the in-plane shear strain
γ12
under the combined loading of
τ
and
Ba
, we concentrate on the deformed triangular HMS unit cell as presented in Figure 4(e). By using
the Pythagorean theorem on the deformed triangle, we get the following geometric relation
h2+l
2+δ1
2+δxB2
= (l+δi)2
(88)
Noting the geometric relation of the undeformed triangular unit cell as
h2+ (l/2)2=l2
(refer to
Figure 4(d)) and carrying out some mathematical manipulations by neglecting the higher order terms,
the horizontal displacement
δxB
of point B is obtained as
δxB= 2 δi−δ1
2
(89)
Due to the the horizontal displacement
δxB
, the shear strain
γ12
developed in the triangular unit cell is
given by
γ12 =δxB/h
. Using the geometric relation from Equation (89), the shear strain
γ12
is expressed
in terms of the beam-level displacements
¯
δxh
and
¯
δxi
as
γ12 =4
√3
¯
δxi−1
√3
¯
δxh
(90)
Once the shear strain
γ12
is known as presented above, the non-linear eective shear modulus
G12
of
the triangular HMS beam network is obtained through its fundamental denition
G12 =τ/γ12
as
G12 =√3τ
4¯
δxi−¯
δxh
(91)
2.4.2.4. Non-dimensional elastic moduli.
As Poisson's ratios
ν12
and
ν21
are already non-dimensional,
they are presented in their original forms. The other three eective elastic moduli
E1
,
E2
, and
G12
of
the triangular HMS beam network are presented in non-dimensional forms following the normalization
scheme as
¯
E1=E1
Esρ3,¯
E2=E2
Esρ3,¯
G12 =G12
Esρ3
(92)
Here the relative density
ρ
of the triangular HMS beam network obtained following the same denition
as the hexagonal beam network is given by
ρ= 2√3t
l
(93)
38

2.5. Eective elastic moduli of rectangular HMS beam networks
To estimate the non-linear eective elastic moduli
E1
,
ν12
,
E2
,
ν21
, and
G12
of periodic rectangular
network of HMS beams, as shown in Figure 1(g)V, the unit cell consisting of horizontal HMS beam of
length
l
and vertical HMS beam of length
h
with residual magnetic ux density
Br
0
is chosen. The three
dierent combined mechanical and magnetic loading conditions for the rectangular HMS unit cell are
shown in Figure 5(a), (b), and (d) respectively.
Under the normal modes of mechanical stress
σ1
or
σ2
in absence of magnetic eld
Ba
, the cell
members of the rectangular lattice undergo stretch-dominated deformations [5]. Whereas, under the
shear mode of mechanical stress
τ
in absence of magnetic eld
Ba
, the cell members are subjected
to bending-dominated deformations [34]. The analytical formulae for the eective elastic moduli of
rectangular lattice under mechanical load only within small deformation regime are given by [5, 34]
E1
Es
=t
l
h
l
(94a)
E2
Es
=t
l
(94b)
ν12 =ν21 = 0
(94c)
G12
Es
=t
l3
h
l1 + h
l
(94d)
In this subsection, the conventional unit cell-based approach for rectangular lattices [5, 34] is ex-
tended to a magneto-active multi-physical mechanics-based semi-analytical framework following the
formulation for hexagonal HMS beam network presented in the preceding subsection, leading to the
evaluation of non-linear eective elastic moduli of the rectangular HMS beam network under combined
mechanical and magnetic loads. Large deformation kinematics of the rectangular HMS unit cell and
the beam-level forces developed under dierent combinations of mechanical stress and magnetic eld
are described rst in the following subsection. With the identied kinematic and kinetic conditions, the
beam-level non-linear multi-physical mechanics problems are solved through the semi-analytical HMS
beam model as presented in subsection 2.1 and subsection 2.2. Using the beam-level deformation results,
computations of the non-linear eective elastic moduli of the rectangular HMS beam network under the
combined mechanical stress and magnetic eld are presented subsequently.
39

Figure 5: Multi-physical mechanics of periodic rectangular HMS beam network under combined mechan-
ical stress and magnetic eld. (a)
Combined loading mode of the rectangular HMS unit cell under normal stress
along direction-1 (
σ1
) and magnetic eld along direction-2 (
Ba
).
(b)
Combined loading mode of the rectangular HMS
unit cell under normal stress along direction-2 (
σ2
) and magnetic eld along direction-2 (
Ba
).
(c)
Generalized forces and
large deformation kinematics of the vertical and horizontal members under combined normal stress
σ1
or
σ2
and magnetic
eld
Ba
.
(d)
Combined loading mode of the rectangular HMS unit cell under shear stress in plane 1-2 (
τ
) and magnetic
eld along direction-2 (
Ba
).
(e)
Forces and large deformation kinematics of the horizontal and vertical members under
combined shear stress
τ
and magnetic eld
Ba
.
2.5.1. Beam-level forces and deformation kinematics
Under the three combined mechanical and magnetic loading conditions as presented in Figure 5(a),
(b), and (d), the HMS beam members undergo large deformation, the kinematics and kinetics of which
40

are described in the following three subsections.
2.5.1.1. Far-eld normal stress along direction-1 and magnetic eld along direction-2.
Under the com-
bined loading case of normal stress
σ1
and magnetic eld
Ba
as shown in Figure 5(a), the horizontal and
vertical HMS beam members OA and OB of the rectangular HMS unit cell undergo combined bending-
stretching deformation with xed end O and the other ends A and B being restrained to rotation and
transverse displacement but free to axial translation. The other pairs of horizontal and vertical HMS
beams BC and CA are not considered in the analysis due to the structural symmetry of the unit cell.
Following the same procedure as in the case of the triangular HMS beam network (refer to the preceding
subsection), half of the members OA and OB of length
l/2
and
h/2
respectively are considered for the
present multi-physical mechanics. The half beams are subjected to the boundary conditions of one xed
end with the other end being rotationally restrained but free to translation.
Large deformation kinematics and the force kinetics of the vertical and horizontal HMS half beams
under the combined loading of
σ1
and
Ba
are presented in Figure 5(c) concerning the local Cartesian
frames (
x
,
y
) tted at the xed point O. Note that the kinematic and kinetic descriptions of the HMS
half beams in Figure 5(c) are a generalized representation under the normal modes of mechanical stress
σ1
or
σ2
combined with the magnetic eld
Ba
as presented in Figure 5(a) and (b).
The concentrated mechanical force
Fh
developed in the horizontal HMS beam under the remote
normal stress
σ1
as shown in Figure 5(c) is given by
Fh=σ1bh
(95)
Inclination angles
βh
and
αh
of the mechanical force
Fh
and the magnetic eld
Ba
respectively for the
horizontal HMS member as shown in Figure 5(c) are given by
βh=π
(96a)
αh=π
2
(96b)
The vertical HMS beam (refer to Figure 5(c)) is only subjected to the magnetic eld
Ba
without any
mechanical force
Fv
under the present combined loading case. Hence, the kinetics of the vertical HMS
beam is represented as
Fv= 0
(97)
αv= 0
(98)
2.5.1.2. Far-eld normal stress along direction-2 and magnetic eld along direction-2.
The large defor-
mation kinematics and kinetics of the members of the rectangular HMS unit cell under the combined
41

loading of
σ2
and
Ba
are already described through the generalized schematic diagrams in Figure 5(c).
Under the present combined loading case, the horizontal HMS beams are not subjected to any mechan-
ical force
Fh
. However, the horizontal members are subjected to
Ba
with the inclination angle
αh
. The
kinetic relations for the horizontal HMS beam member are summarized as
Fh= 0
(99)
αh=π
2
(100)
The concentrated force
Fv
developed in the vertical member (refer to Figure 5(c)) is given by
Fv=σ2bl
(101)
The inclination angles
βv
and
αv
of the mechanical and magnetic loads respectively for the vertical
member as presented in Figure 5(c) are given by
βv=π
(102a)
αv= 0
(102b)
2.5.1.3. Shear stress in 1-2 plane and magnetic eld along direction-2.
Under the combined loading
of in-plane shear stress
τ
and magnetic eld
Ba
as shown in Figure 5(d), the horizontal and vertical
members OA and OB of the rectangular HMS unit cell undergo bending-dominated large deformation
with xed end O and the other ends A and B being rotationally restrained but free to translation.
Within the present multi-physical mechanics-based framework, the large deformation kinematics and
kinetics of the horizontal and vertical HMS full beam members OA and OB are analysed as presented
in Figure 5(e).
The tip-concentrated transverse force
Fh
developed in the horizontal HMS beam under the remote
shear stress
τ
is expressed as
Fh=τbh
(103)
The inclination angles
βh
and
αh
of the mechanical and magnetic loads for the horizontal HMS beam
as shown in Figure 5(e) are given by
βh=3π
2
(104a)
αh=π
2
(104b)
The concentrated force
Fv
developed in the vertical HMS beam member under the remote shear stress
42

τ
(refer to Figure 5(e)) is expressed by
Fv=τbl
(105)
The inclination angles
βv
and
αv
of the mechanical force
Fv
and the magnetic eld
Ba
for the vertical
HMS beam as shown in Figure 5(e) are summarized as
βv=π
2
(106a)
αv= 0
(106b)
2.5.2. Eective elastic moduli
To estimate the non-linear eective elastic moduli
E1
,
ν12
,
E2
, and
ν21
of the rectangular HMS
beam network under the normal modes of mechanical stress
σ1
or
σ2
in combination with the magnetic
eld
Ba
, geometrically non-linear axial tip-deections
δxh
and
δxv
of the horizontal and vertical HMS
half beams under the concentrated force
Fh
and
Fv
combined with the magnetic eld
Ba
as described
through Figure 5(c) in the previous subsection are computed through the generalized HMS beam model.
Whereas, for the estimation of the non-linear eective shear modulus
G12
under in-plane shear stress
τ
and the magnetic eld
Ba
, geometrically non-linear transverse deections
δyh
and
δyv
of the horizontal
and vertical HMS full beams as shown in Figure 5(e) are computed.
In the framework of the generalized HMS beam model, the geometries of the horizontal and vertical
HMS half beams considered for combined loading case under normal stress
σ1
or
σ2
and magnetic eld
Ba
as shown in Figure 5(c) are normalized as
Π0h=3
t
l2
(107a)
Π0v=
3h
l2
t
l2
(107b)
Whereas, the non-dimensional geometries of the HMS full beams considered for the combined loading
43

case under shear stress
τ
and magnetic eld
Ba
as shown in Figure 5(e) are given by
Π0h=12
t
l2
(108a)
Π0v=
12 h
l2
t
l2
(108b)
2.5.2.1. Computation of
E1
and
ν12
under combined load
σ1
and
Ba
.
The non-dimensional mechanical
forces
Ch
and
Cv
for the horizontal and vertical HMS beams under the combined loading of normal
stress
σ1
and magnetic eld
Ba
as shown in Figure 5(a) and (c) are obtained from Equations (95) and
(97) as
Ch=
3h
l
Est
l3σ1
(109a)
Cv= 0
(109b)
Magnitudes of the non-dimensional magnetic loads
Bh
and
Bv
for the horizontal and vertical HMS
beam members of the rectangular HMS unit cell are dened in terms of the magnetic load ratio
rh
and
the non-dimensional mechanical force
Ch
as
Bh=rhCh
(110a)
Bv=h
l2
rhCh
(110b)
Under the prescribed non-dimensional mechanical and magnetic loads with the inclination angles as
presented in Equations (96) and (98), the non-linear axial deections are computed in non-dimensional
forms
¯
δxh
and
¯
δxv
. In terms of the beam-level deections, the normal strains in direction-1 (
ϵ1
) and
direction-2 (
ϵ2
) are dened by
ϵ1=¯
δxh
(111)
ϵ2=¯
δxv
(112)
With the above-presented strains
ϵ1
and
ϵ2
, the non-linear eective Young's modulus
E1
and Poisson's
44

ratio
ν12
of the rectangular HMS beam network are obtained readily as
E1=σ1
¯
δxh
(113)
ν12 =−¯
δxv
¯
δxh
(114)
2.5.2.2. Computation of
E2
and
ν21
under combined load
σ2
and
Ba
.
When the rectangular HMS beam
network is subjected to combined loading under the normal stress
σ2
and the magnetic eld
Ba
as
shown in Figure 5(b), the concentrated forces in the horizontal and vertical HMS beams are expressed
in non-dimensional forms using Equations (99) and (101) as
Ch= 0
(115a)
Cv=
3h
l2
Est
l3σ2
(115b)
Magnitudes of the non-dimensional magnetic loads
Bh
and
Bv
are dened in terms of load ratio
rv
and non-dimensional load
Cv
in a similar way as in the case of the other previously discussed combined
loading mode as
Bh=rvCv
h
l2
(116a)
Bv=rvCv
(116b)
Under the above-presented mechanical and magnetic loads with the inclination angles presented in
Equations (100) and (102), the non-linear beam-level deections
¯
δxh
and
¯
δxv
are computed which in turn
give the normal strains
ϵ1
and
ϵ2
through Equations (111) and (112). Using the strain expressions, the
non-linear eective elastic moduli
E2
and
ν21
of the rectangular HMS beam network under the combined
loading of
σ2
and
Ba
are obtained as
E2=σ2
¯
δxv
(117)
ν21 =−¯
δxh
¯
δxv
(118)
2.5.2.3. Computation of
G12
under combined load
τ
and
Ba
.
Under the combined loading of
τ
and
Ba
as shown in Figure 5(d) and (e), the non-dimensional forces
Ch
and
Cv
for the horizontal and vertical
45

beams are derived from Equations (103) and (105) as
Ch=
12 h
l
Est
l3τ
(119a)
Cv=
12 h
l2
Est
l3τ
(119b)
The non-dimensional magnetic loads
Bh
and
Bv
for the horizontal and vertical HMS beam members
are dened as
Bh=rhCh
(120a)
Bv=h
l2
rhCh
(120b)
Under the above-presented non-dimensional mechanical and magnetic loads with the inclination
angles presented in Equations (104) and (106), non-linear transverse defections of the beam tips are
computed in non-dimensional forms as denoted by
¯
δyh
and
¯
δyv
in Figure 5(e). In terms of the transverse
tip-deections, rotations of the horizontal and vertical HMS beams are obtained as
Ωh=−¯
δyh
(121a)
Ωv=¯
δyv
(121b)
Due to the above-presented rotations
Ωh
and
Ωv
of the horizontal and vertical HMS beam members
respectively, the total shear strain
γ12
developed in the rectangular unit cell is given by
γ12 =−¯
δyh+¯
δyv
(122)
The non-linear eective shear modulus
G12
of the rectangular HMS beam network is obtained subse-
quently through the fundamental denition
G12 =τ/γ12
using Equation (122) as
G12 =τ
−¯
δyh+¯
δyv
(123)
2.5.2.4. Non-dimensional elastic moduli.
As Poisson's ratios
ν12
and
ν21
are already non-dimensional,
they are presented in their original forms. Following a similar representation framework as the other
periodic network congurations, the eective elastic moduli
E1
,
E2
, and
G12
of the rectangular HMS
beam network are normalized as
¯
E1=E1
Esρ3,¯
E2=E2
Esρ3,¯
G12 =G12
Esρ3
(124)
46

Here the relative density
ρ
of the rectangular HMS beam network is derived as
ρ=t
l1 + h
l
h
l
(125)
Having established the semi-analytical large-deformation computational frameworks for dierent
magneto-active periodic beam networks, we present numerical results in the following section to demon-
strate active broadband elasticity programming as a function of the externally applied magnetic eld
and bi-level (unit cell geometry and beam-level spatially-varying residual magnetic ux direction) meta-
material architectures.
3. Results and discussion
The generalized HMS beam model is the backbone of the present semi-analytical framework to
estimate the non-linear eective elastic moduli of hexagonal HMS beam networks under combined me-
chanical and magnetic loads. Hence, before going to investigate the eective elastic moduli of HMS beam
networks, the HMS beam model is validated rst, as presented in the rst subsection here. Thereafter,
critical numerical beam-level results are furnished with symmetric and asymmetric residual magnetic
ux density under dierent combinations of mechanical and magnetic loads. Note that modulation
capability of the shapes of such architected beams will constitute the foundation for bi-level design
of lattices, as discussed later in this section. Applicability of the full-beam and half-beam model for
symmetric and asymmetric residual magnetic ux density of HMS beam is also investigated along with
the inuence of centreline extensibility on the load-deformation characteristics of HMS beam.
Following the beam-level results, the geometrically non-linear semi-analytical framework estimating
the eective elastic moduli of the HMS beam networks is validated, as presented in the third subsection.
Validations of the present framework at the beam-level as well as at the beam network-level would
provide adequate condence in the proposed computational models. Subsequently, the eect of magnetic
eld in combination with the dierent modes of mechanical load on the non-linear eective elastic
moduli of hexagonal HMS beam network with uniform residual magnetic ux density is investigated, as
presented in the fourth subsection. Based on the kinematic and kinetic conditions of the beam elements
of the hexagonal HMS beam network, two intuitive designs of residual magnetic ux density
S(ξ)
(beam-level architecture) are proposed in the fth subsection which would signicantly inuence the
eective elastic moduli of the HMS beam network under combined mechanical and magnetic loads. In
the following subsection, we demonstrate the applicability of the concept of active broad-band elasticity
modulation for dierent other forms of lattice geometries, as presented in Figure 1(g).
47

For all the computations at beam-level and beam network-level, the material constitutive parameters
in the framework of the Yeoh hyperelastic model are considered as
C10 = 0.2712
,
C20 = 0.0305
, and
C30 =−0.004
[89]. The numerical value of the computational parameter
λ
and the limit of
µ
are
considered as 0.9 and 0.05% respectively. The number of functions for the centreline rotation
φ
and
centreline strain
ε
are selected as
nb =ns = 5
, based on a convergence study.
3.1. Beam-level validation
Though large deformation analysis of HMS beam structures has become a topic of interest for the
last few years, the studies focus on structural characteristics separately under mechanical load only
and magnetic actuation only. Hence, comparable results for our multi-physical mechanics-based beam
model for coupled mechanical and magnetic loading conditions are not readily available in the literature.
Thus, the current geometrically non-linear HMS beam model is rst validated under mechanical load
only by comparing it with the results presented by Chen and Wang [65]. Whereas, for the loading case
of magnetic actuation only, we validate our model with the paper by Chen et al. [70]. The validation
studies for both the mechanical and magnetic loading cases are performed for the non-dimensional
geometric specication of the HMS beam
Π0= 10000
.
The validation study of the generalized HMS beam model under mechanical load only is carried
out for the cantilever boundary conditions subjected to tip-concentrated non-dimensional load
C
with
inclination angle
β
as considered in the paper [65]. The non-dimensional deformed congurations (
ξ
,
η
)
of the cantilever beam under dierent values of
C
for inclination angle
β
of
π/4
,
π/2
,
3π/4
, and
9π/10
as obtained from the present model are shown through solid lines in Figure S1(a)-(d) respectively.
Whereas, the corresponding deformation results reported in the literature [65] are also plotted through
dotted points in Figure S1. As obvious from Figure S1, an excellent agreement between the present semi-
analytical HMS beam model and the model presented in literature [65] is found for all the considered load
magnitudes
C
and the orientation angles
β
. Hence, the comparison studies in Figure S1 clearly show the
capability of the present HMS beam model in predicting highly non-linear deformation characteristics
of the soft beam under mechanical load only.
The validation study of the present non-linear beam model under magnetic load only is carried out
for four dierent deformed shapes obtained under dierent designs of residual magnetic ux density
S(ξ)
of the HMS beam subjected to multiple boundary conditions as considered in literature [70]. The
rst considered case among them is the m-shape deformed conguration which is obtained for the design
of
S(ξ)
as given below with the free-free boundary conditions (
θ′(0) = 0
and
θ′(1) = 0
) and inclination
48

angle
α=π/2
[70].
S=




1,0≤ξ≤0.25 or 0.5≤ξ≤0.75
−1,0.25 < ξ < 0.5 or 0.75 < ξ ≤1.0
With the above-presented residual magnetic ux density
S(ξ)
, the prescribed boundary conditions and
inclination angle, the m-shape deformed congurations of HMS beam under non-dimensional magnetic
actuations
B= 30
,
B= 100
, and
B= 300
are obtained from the present non-linear model as shown
through solid lines in Figure S2(a)-(c) respectively. The corresponding deformed shapes as reported in
literature [70] are also shown in the plots through dotted points.
The second shape we concentrate on is the s-shape conguration which is obtained under the same
boundary conditions and inclination angle
α
as in the case of m-shape congurations but with the
following design of
S(ξ)
[70]
S=








1,0≤ξ < 1
3or 2
3≤ξ≤1
−1,1
3≤ξ < 2
3
The comparison plots between the present model and the results reported in literature [70] for the s-
shape congurations under the non-dimensional magnetic actuation
B
of 30, 100, and 300 are presented
in Figure S2(d)-(f) respectively.
The third type of deformed shape considered for the validation study of the HMS beam model under
magnetic actuation only is the n-shape conguration. The n-shape conguration is achieved for the
same boundary conditions and inclination angle
α
as those of the m-shape and s-shape congurations
but with the coecient of residual magnetic ux density [70]
S=




1,0≤ξ < 0.5
−1,0.5≤ξ≤1
Comparisons of n-shape deformed congurations from the present semi-analytical model with the results
reported in literature [70] are shown in Figure S2(g)-(i) for the magnetic actuation
B= 30
,
B= 60
,
and
B= 100
respectively.
The fourth type of the deformed shape of the HMS beam under magnetic actuation we consider is
the
Ω
-shape conguration. The conguration is achieved for the same design of
S(ξ)
as that for the
n-shape congurations but under the boundary conditions of
θ(0) = 0
and
θ(1) = 0
with the inclination
angle of the magnetic eld
α=π
[70]. The
Ω
-shape deformed congurations of HMS beam under
magnetic actuation
B
of 60, 100, and 200 are compared with the present non-linear model and the
reported results in literature [70] as presented in Figure S2(j)-(l) respectively.
49

The excellent matching of the deformation results obtained from the present semi-analytical model
and literature [70], as shown in Figure S2, validates our non-linear model in predicting complex cong-
urations of HMS beam with designed spatially-varying residual magnetic ux densities under dierent
magnetic actuation.
3.2. Beam-level numerical results under coupled mechanical and magnetic loads
Once the developed geometrically non-linear HMS beam model is validated for separate loading
conditions of mechanical load only and magnetic load only, as presented in the previous subsection,
benchmark numerical results under coupled mechanical and magnetic loading conditions are presented
here. Note that the such coupled eect of magneto-mechanical loading has not been investigated in the
literature through the development of a comprehensive computational framework for HMS beams.
An HMS beam representing the generalized element (full or half length) of the HMS beam network
having length
L
with non-dimensional geometric specication
Π0= 10000
is considered here. Non-
linear deformation characteristics of the HMS beam are simulated through the full-beam and half-beam
models within the extensible and inextensible versions of the present semi-analytical framework. The
typical boundary conditions (as considered here) of the HMS beam as a full-beam problem and as a
half-beam problem have been already described in detail in subsection 2.1.
The considered HMS full-beam is xed at one end with the other end being rotationally restrained
but free to translation and subjected to non-dimensional mechanical force
C= 10
applied incrementally
in 50 steps. At each incremental step of
C
, ve non-dimensional magnetic loads
B=rC
are applied by
ve magnetic load ratio
r
of 0.8, 1.6, 2.4, 3.2, and 4 for two dierent cases of uniform residual magnetic
ux density with
S= 1
and
S=−1
. For the considered HMS full-beam problem, four dierent
inclination angles of the mechanical and magnetic loads are considered as
α=β=π/2
,
α=β=π/3
,
α=β=π/4
, and
α=β=π/6
. The non-dimensional deformed congurations (
ξ
,
η
) of the HMS
beam with residual magnetic ux density
S= 1
and
S=−1
under the mechanical load
C= 10
in
combination with dierent magnetic load ratios
r
are presented in Figure 6(a)-(d) for the considered four
sets of inclination angles respectively. The solid lines in the plots represent the results obtained from the
extensible model. Whereas, the results obtained from the inextensible version of the non-linear model
are plotted through dotted points in the gure. To observe the eect of magnetic load in combination
with the mechanical loading on the non-linear deformation characteristics of the HMS beam with
S= 1
and
S=−1
, variations of the non-dimensional tip-deection
¯
δy
with the non-dimensional mechanical
load
C
for the considered dierent magnetic load ratio
r
are shown in Figure S3(a)-(d) corresponding
to four sets of inclination angles.
50

Figure 6: Deformed shapes of HMS full-beam congurations with symmetric uniform residual magnetic
ux density about the mid-point under combined mechanical and magnetic load.
Non-dimensional deformed
congurations (
ξ
,
η
) of HMS full beams with the coecient of residual magnetic ux density
S= 1
and
S=−1
under
non-dimensional mechanical force
C= 10
in combination with dierent magnitudes of non-dimensional magnetic load
B=rC
in terms of the magnetic load ratio
r
with the inclination angles of the mechanical and magnetic loads of
(a)
α=β=π/2
,
(b)
α=β=π/3
,
(c)
α=β=π/4
, and
(d)
α=β=π/6
.
51

Figures 6 and S3 clearly show that for the residual magnetic ux density with the coecient
S= 1
,
deection under combined mechanical load
C
and magnetic eld
B
for all the considered inclination
angles
β
and
α
increases with magnetic load ratio
r
compared to the deection under mechanical load
only (
r= 0
). Whereas, for the residual magnetic ux density having coecient
S=−1
, the deection
decreases with
r
for the same combination of mechanical and magnetic loads. Hence, it is clear from
the results that we can modulate stiness characteristics of HMS beam as per our requirements by
applying a magnetic eld in combination with mechanical load through proper design of the residual
magnetic ux density
S(ξ)
of the HMS beam. Such eects are exploited in the current design of lattice
metamaterials for broadband elasticity programming.
Now the HMS full-beam of length
L
is modelled as two HMS half-beams with length
L/2
subjected
to cantilever boundary conditions. To apply the same dimensional force
F
as that of the full-beam, the
maximum value of the non-dimensional force
C
for the half-beam is taken as 2.5. At each incremental
step of mechanical force
C
, the same ve magnetic load ratios
r
as those for the full-beam problem
are considered as 0.8, 1.6, 2.4, 3.2, and 4. The deformed congurations of the HMS half-beam in the
non-dimensional plane (
ξ
,
η
) under the maximum step of the mechanical load
C= 2.5
in combination
with the considered dierent magnetic loads are shown in Figure S4(a)-(d). Whereas, the non-linear
variations of the non-dimensional tip-deection
¯
δy
with the non-dimensional mechanical load
C
for the
considered dierent magnetic load ratio
r
are presented in Figure S5(a)-(d).
It is evident from Figures 6-S5 that the eects of the magnetic eld in combination with the mechan-
ical load on the deformation characteristics of the HMS half-beam are the same as the HMS full-beam.
The overall deections of the HMS half-beam are exactly half of the deections for the HMS full-beam
under the same condition of combined mechanical and magnetic loads. Hence, it is proved that an HMS
full-beam with one xed end and the other end being rotationally restrained but free to translation can
be modelled as an HMS half-beam with cantilever boundary conditions when the HMS beam has sym-
metric residual magnetic ux density about the mid-point. However, for asymmetric residual magnetic
ux density, the applicability of such a modelling concept is investigated in the following paragraphs.
Two dierent asymmetric distributions of residual magnetic ux density about the mid-point are
considered for HMS full-beam by the following
S(ξ)
.
S=




1,0≤ξ < 0.5
−1,0.5≤ξ≤1
52

S=




−1,0≤ξ < 0.5
1,0.5≤ξ≤1
With the above-presented designs of
S(ξ)
for the same geometric and loading parameters as those for the
HMS full-beam with symmetric uniform residual magnetic ux density, load-deformation characteristics
of HMS full-beam are computed. Deformed congurations of the HMS full-beam having asymmetric
magnetic ux density are presented in Figure 7. The gure depicts some non-conventional typical
complex shapes of HMS beam achieved for the considered designs of
S(ξ)
. Though the curvatures of
the deformed congurations are dierent for the two considered distributions of residual magnetic ux
density, the endpoints undergo the same deections. Variations of such common tip-deection
¯
δy
with
the mechanical load
C
for the considered dierent magnetic load ratios
r
are shown in Figure S6. The
gure clearly shows that for the considered two designs of
S(ξ)
, the deections got reduced compared
to the loading condition of mechanical load only (
r= 0
).
The HMS full-beam with the considered two asymmetric distributions of residual magnetic ux
density is tried to be modelled now as two HMS half-length beams either with
S= 1
or with
S=−1
.
Load-deformation characteristics of such HMS half-length beams are already presented in Figures S4 and
S5. Comparisons of the deection results for the HMS full-beam with asymmetric residual magnetic ux
density as presented in Figures 7 and S6 with those for the HMS half-beam as presented in Figures S4
and S5 depicts that the deections through the half-beam model are not half of the deections obtained
through the full-beam model. However, for symmetric residual magnetic ux density, we got exactly the
half deections from the half-beam model compared to the full-beam model under the same condition
of combined mechanical and magnetic loading as described through comparisons between Figures 6-S5.
Hence, it is concluded from the comparison studies that modelling of HMS full-beam with one xed
end and the other end being rotationally restrained but free to translation as two half-length cantilever
beams is only possible when the residual magnetic ux density is symmetric about the mid-point of
the HMS full-beam. As we focus on both symmetric and asymmetric designs of
S(ξ)
for modulation of
eective elastic moduli of HMS beam networks, the two beam models are applied carefully for analyzing
nonlinear hexagonal lattices in the following subsections.
Comparisons of the deection results between the extensible and inextensible versions of the present
semi-analytical HMS beam model as presented in Figures 6-S6 clearly show that the eect of centreline
extension is not signicant for the considered HMS beam under combined mechanical and magnetic
loads. For achieving higher level of accuracy, we will consider the generalized extensible model in the
53

Figure 7: Deformed shapes of HMS full-beam congurations with asymmetric residual magnetic ux
density about the mid-point under combined mechanical and magnetic load.
Non-dimensional deformed
congurations (
ξ
,
η
) of HMS full-beam with asymmetric residual magnetic ux density under non-dimensional mechanical
force
C= 10
in combination with dierent magnitudes of non-dimensional magnetic load
B=rC
in terms of the magnetic
load ratio
r
with the inclination angles of the mechanical and magnetic loads as
(a)
α=β=π/2
,
(b)
α=β=π/3
,
(c)
α=β=π/4
, and
(d)
α=β=π/6
.
54

further computations of eective elastic moduli of the HMS beam networks.
3.3. Periodic beam network-level validation
As the hexagonal lattice consisting of HMS beam members subjected to combined mechanical and
magnetic loads is not investigated in the literature, directly comparable results for the presently devel-
oped semi-analytical framework are not readily available for reference and validation. Hence, the current
semi-analytical framework estimating non-linear eective elastic moduli of hexagonal HMS beam net-
work under combined mechanical and magnetic loads is validated for the special case of zero magnetic
eld (
ri= 0
) subjected to dierent modes of mechanical stress only (
σ1
or
σ2
or
τ
). Validations for
the non-linear eective elastic moduli
E1
and
ν12
under normal mechanical stress
σ1
and for the elastic
moduli
E2
and
ν21
under normal mechanical stress
σ2
are carried out by comparing with the results
presented by Ghuku and Mukhopadhyay [42]. Whereas, for the non-linear eective shear modulus
G12
under the shear mode of mechanical stress
τ
, the semi-analytical framework is validated by comparing
with the paper by Fu et al. [39].
The validation study for non-linear elastic moduli (
E1
,
ν12
,
E2
, and
ν21
) of the hexagonal HMS beam
network under the normal modes of mechanical stress only (
σ1
and
σ2
) [42] is carried out for the lattice
conguration with the geometric specications
h/l = 2
,
t/l = 0.01
, and
θ=π/6
. Young's modulus
of the intrinsic material is taken as
Es= 200
GPa in the reference literature [42]. Whereas, for the
present semi-analytical model, the material constitutive parameters are considered as
C10 = 0.2712
,
C20 = 0.0305
, and
C30 =−0.004
within the framework of the Yeoh hyperelastic model [89]. In the
reference literature [42], the non-linear results are presented as the variations of the non-dimensional
elastic moduli
¯
E1
,
ν12
,
¯
E2
, and
ν21
with the dimensional input normal stress
σ1
and
σ2
. As the elastic
moduli are presented in non-dimensional forms, they are independent of the intrinsic material property
Es
. However, the dimensional form of the input normal stress
σ1
and
σ2
makes the results dependent on
the intrinsic material property
Es
. Hence, to make the input normal stress independent of the material
property
Es
, the stresses
σ1
and
σ2
are also expressed in non-dimensional forms following Equation (58)
as
¯σ1=σ1/Esρ3
and
¯σ2=σ2/Esρ3
. Variations of the non-dimensional eective Young's modulus
¯
E1
and the Poisson's ratio
ν12
of the considered hexagonal lattice conguration with the non-dimensional
compressive and tensile modes of normal stress
¯σ1
are compared considering the present model, the
results reported in the paper [42], and the linear small-deformation analytical model [5] as presented
in Figure S7(a) and (b). The similar comparison plots for the the non-dimensional eective Young's
modulus
¯
E2
and the Poisson's ratio
ν21
under the non-dimensional compressive and tensile modes of
normal stress
¯σ2
are presented in Figure S7(c) and (d).
55

The comparison plots in Figure S7 depict that the non-dimensional eective elastic moduli
¯
E1
,
ν12
,
¯
E2
, and
ν21
of the hexagonal HMS beam network under normal modes of mechanical stress
¯σ1
and
¯σ2
as
estimated by the present model match exactly with the non-linear model in literature [42] at lower input
stress level. However, the dierences between them increase with the input stress level. The geometric
exactness in non-linear kinematics and the hyperelastic material model of the present framework is
the possible cause of this dierence with the model reported in [42]. However, the dierences in the
elastic moduli at the higher stress levels are not very signicant. Moreover, the increasing or decreasing
trends of the elastic moduli with the input stress magnitudes agree well between the present model
and the non-linear model reported in literature [42]. As also observed from Figure S7 that within the
small deformation regime, the non-linear elastic moduli match exactly with the conventional analytical
solutions [5]. Dierences between the elastic moduli estimated by the present framework and the linear
solutions [5] increase with input stress magnitude due to the non-linearity in the system which is not
considered in the conventional linear analytical solutions [5].
The validation study of the present non-linear framework for the eective shear modulus
G12
of
hexagonal HMS beam network under shear mode of mechanical stress
τ
is carried out for the auxetic
conguration with
θ=−π/6
in terms of shear strain
γ12
versus non-dimensional shear stress
τ/Es
curve
and shear strain
γ12
versus non-dimensional shear modulus
G12/Es
curve following similar representation
scheme of the reference literature [39]. The shear strain
γ12
versus shear stress
τ/Es
curves for the auxetic
lattice conguration with
h/l = 2
and
t/l = 0.1
as obtained from the present model, the model reported
by Fu et al. [39], and the analytical model [5] are compared in Figure S8(a). Whereas, the similar
comparison of stress-strain curves under the shear mode of mechanical stress for the auxetic lattice
conguration with
h/l = 2
and
t/l = 0.12
is shown in Figure S8(b). On the other hand, variations of
the non-dimensional eective shear modulus
G12/Es
with the shear strain
γ12
are compared considering
the present model, the model reported by Fu et al. [39], and the analytical model [5] in Figure S8(c)
and (d) for two lattice congurations with
h/l = 1.5
,
t/l = 0.1
and
h/l = 2
,
t/l = 0.1
respectively.
The comparison plots in Figure S8(a) and (b) show that the stress-strain curves (
γ12
versus
τ/Es
) of
the HMS beam network under the shear mode of mechanical stress
τ
as estimated by the present semi-
analytical framework match exactly with the analytical solutions [5] within the small deformation regime.
The non-linear stress-strain curves estimated by the present framework also match with the non-linear
model [39] at the lower shear strain levels within the non-linear zone. However, with the increase in the
shear strain
γ12
, the dierences between the non-linear stress-strain curves increase. Similar observations
are found from the comparison plots of variations of the non-dimensional eective shear modulus
G12/Es
of the HMS beam network with the shear strain
γ12
in Figure S8(c) and (d). The dierences between the
56

present framework and the non-linear model reported in [39] arise due to the fundamental dierences
in their respective formulations. The present framework is developed in the geometrically exact non-
linear kinematic setting considering combined bending and axial deformations with the hyperelastic
constitutive material model. Whereas, the model reported in the reference literature [39] is developed
within the geometric non-linear kinematic setting excluding the axial deformation considering linear
elastic constitutive material characteristics. Though the non-linear shear stiness of the HMS beam
network as predicted by the present framework has some dierence at the higher strain levels, the
trends are the same with the non-linear model reported in literature [39]. Within the framework of the
existing fundamental dierences in the formulations (where the present model is more accurate), the
validation study of the present model with the non-linear model from literature [39] for the eective
shear stiness of the HMS beam networks can be considered quite satisfactory.
In this subsection, we have primarily concentrated on the hexagonal lattices with non-auxetic and
auxetic geometries for lattice-level validation, depending on the availability of reference literature. While
rectangular brick, re-entrant auxetic and rhombic geometries are direct derivatives of hexagonal lattices
(thus no need for additional validation), the triangular and rectangular lattice congurations are further
validated later in their respective subsections.
3.4. Hexagonal periodic HMS beam networks under uniform residual magnetic ux density
Eect of the magnetic eld
Ba
along direction-2 in combination with a particular mode of mechanical
stress (
σ1
or
σ2
or
τ
) on the non-linear eective elastic moduli of the hexagonal HMS beam network
having uniform residual magnetic ux density
S= 1
and
S=−1
is investigated in this subsection.
As mentioned earlier, under the combined loading of normal stress
σ1
and magnetic eld
Ba
, we will
focus on the longitudinal non-dimensional Young's modulus
¯
E1
and Poisson's ratio
ν12
. Under the
combined loading of
σ2
and
Ba
, we will focus on the transverse non-dimensional Young's modulus
¯
E2
and Poisson's ratio
ν21
. Whereas, under the combined loading of shear stress
τ
and magnetic eld
Ba
, we will investigate the eective non-dimensional shear modulus
¯
G12
. For a particular mechanical
loading mode in combination with the magnetic eld, the hexagonal HMS beam network is subjected
to mechanical stress incrementally in 50 steps. At each step of mechanical loading, the incremental
magnetic load is applied to the hexagonal HMS beam network in terms of the magnetic load ratio
ri
through 100 steps.
Variations of the non-dimensional eective Young's modulus
¯
E1
of the hexagonal HMS beam network
having the uniform residual magnetic ux density
S= 1
as a function of the magnetic load ratio
ri
at dierent stress levels under the compressive mechanical stress
σ1
in combination with the magnetic
eld
Ba
are shown in Figure 8(a). Under the same combined loading conditions for the hexagonal HMS
57

Figure 8: Eective Young's modulus of hexagonal HMS beam networks having uniform residual magnetic
ux density under combined mechanical normal stress along direction-1 and magnetic eld along direction-
2.
Variations of the non-dimensional eective Young's modulus
¯
E1
of the hexagonal HMS beam network having uniform
residual magnetic ux density
(a, c)
S= 1
and
(b, d)
S=−1
as function of the magnetic load ratio
ri
at dierent
mechanical stress levels
σ1
under the
(a, b)
compressive and
(c, d)
tensile modes of the mechanical stress
σ1
in combination
with the magnetic eld
Ba
along direction-2.
beam network having the negative uniform residual magnetic ux density
S=−1
, variations of the
Young's modulus
¯
E1
with the magnetic load ratio
ri
are shown in Figure 8(b). Whereas, under the
tensile mode of the mechanical normal stress
σ1
in combination with the magnetic eld
Ba
, the similar
58

plots of the non-dimensional Young's modulus of the hexagonal HMS beam network with
S= 1
and
S=−1
are shown in Figure 8(c) and (d) respectively. Variations of the eective Poisson's ratio
ν12
of the hexagonal HMS beam network having the uniform residual magnetic ux density
S= 1
and
S=−1
as a function of the magnetic load ratio
ri
for the same combined loading conditions as of
Figure 8(a)-(d) are presented in Figure 9(a)-(d).
Eects of the magnetic eld along with the residual magnetization pattern in combination with
dierent modes of far-eld mechanical loading on the non-linear variations of the elastic moduli as
function of the input stress magnitude are investigated here. As observed in Figure 8(a), (c), (d), and
Figure 9(a), (c), (d), singularity points for the eective Young's modulus
¯
E1
and Poisson's ratio
ν12
arise
at some magnetic load ratios
ri
for the hexagonal HMS beam network with
S= 1
under both tension
and compression and for the hexagonal HMS beam network with
S=−1
under tensile mode only. The
beam-level deections under the magnetic load
Ba
corresponding to singular magnetic load ratios
ri
balance the deections under the corresponding far-eld mechanical stress levels
σ1
. Hence, at those
magnetic load ratios
ri
, the eective Young's modulus
¯
E1
and Poisson's ratio
ν12
of the hexagonal HMS
beam network become undened due to no eective lattice-level strain. However, such singularity points
for the eective Young's modulus
¯
E1
and Poisson's ratio
ν12
do not arise for the hexagonal HMS beam
network with
S=−1
under the compressive mode of the mechanical stress
σ1
in combination with the
magnetic eld
Ba
as observed in Figure 8(b) and Figure 9(b). As also observed from Figure 8 that under
certain combinations of the mechanical and magnetic loading, negative stiness of the hexagonal HMS
beam network can be achieved. To observe the eect of the magnetic load in terms of the magnetic load
ratio
ri
on the eective stiness of the hexagonal HMS beam network, variations of the non-dimensional
Young's modulus
¯
E1
with the input stress
σ1
for equally spaced magnetic load ratios
ri
are further
presented in Figure S9(a)-(d). For the same magnetic load ratios
ri
, variations of the Poisson's ratio
ν12
with the input stress
σ1
are presented in Figure S10(a)-(d). The variations of the elastic moduli
with the input stress magnitude is coming from the geometric non-linearity due to large deformation
and material non-linearity under magneto-mechanical coupling.
As observed from Figure S9(a), the eective non-dimensional Young's modulus
¯
E1
of the hexagonal
HMS beam network with
S= 1
decreases with the input stress magnitude under the compressive
mechanical stress
σ1
in combination with the magnetic load having the magnetic load ratio
0≤ri≤0.4
.
Under the same loading condition for the magnetic load ratio
0.6≤ri≤0.7
, negative stiness of the
HMS beam network is observed. The negative stiness initially increases with the stress magnitude
σ1
and then starts decreasing at the higher stress levels. However, both the positive and negative non-
dimensional Young's modulus increases with the magnetic load ratio
ri
. Maximum 225.5% enhancement
59

Figure 9: Eective Poisson's ratio of hexagonal HMS beam networks having uniform residual magnetic ux
density under combined mechanical normal stress along direction-1 and magnetic eld along direction-2.
Variations of the eective Poisson's ratio
ν12
of the hexagonal HMS beam network having uniform residual magnetic ux
density
(a, c)
S= 1
and
(b, d)
S=−1
as function of the magnetic load ratio
ri
at dierent mechanical stress levels
σ1
under the
(a, b)
compressive and
(c, d)
tensile modes of the mechanical stress
σ1
in combination with the magnetic
eld
Ba
along direction-2.
in the positive Young's modulus
¯
E1
is observed from Figure S9(a) compared to the only mechanical
loading condition (
ri= 0
). Whereas, the maximum enhancement in the negative Young's modulus
¯
E1
is achieved as 74.2% for
ri= 0.7
compared to
ri= 0.6
. Under the compressive stress
σ1
in combination
60

with the magnetic load having
0≤ri≤3
for the hexagonal HMS beam network with
S=−1
as
observed from Figure S9(b),
¯
E1
decreases with the input stress magnitude
σ1
for lower
ri
. However,
for higher
ri
,
¯
E1
initially decreases and then increases with
σ1
. The overall non-dimensional Young's
modulus
¯
E1
decreases with the the magnetic load ratio
ri
. A maximum 84% reduction in
¯
E1
is observed
in Figure S9(b) for
ri= 3
compared to
ri= 0
.
As evident from Figure S9(c), for the hexagonal HMS beam network with
S= 1
under the tensile
mode of mechanical normal stress
σ1
in combination with the magnetic load having
0≤ri≤1.5
, the
non-dimensional Young's modulus
¯
E1
increases with the stress amplitude. The overall
¯
E1
decreases
with the magnetic load ratio
ri
at the lower stress zone, however, at the higher input stress level
σ1
, it
has some mixed trend with
ri
. Maximum enhancement and reduction in the non-dimensional Young's
modulus
¯
E1
compared to the only mechanical loading condition (
ri= 0
) are obtained as 44.1% and
72.1% respectively. Under the combined tensile stress
σ1
and magnetic eld with
0≤ri≤0.4
for
the HMS beam network with the negative residual magnetic ux density
S=−1
, the positive non-
dimensional Young's modulus
¯
E1
increases with the stress amplitude as observed from Figure S9(d).
For the magnetic load ratio
1≤ri≤2
, the non-dimensional Young's modulus
¯
E1
is negative which
decreases with
σ1
. However, both the positive and negative Young's modulus
¯
E1
increases with
ri
. As
obtained from Figure S9(d), the maximum enhancements in the positive and negative
¯
E1
are found to
be 189.1% and 67.6% respectively.
As observed from Figure S10(a), for the hexagonal HMS beam network with
S= 1
under the
combined compressive stress
σ1
and magnetic load, the eective Poisson's ratio
ν12
decreases with
σ1
for
0≤ri≤0.4
and increases with
σ1
for
0.6≤ri≤0.7
. However, for both the ranges of
ri
, the overall
Poisson's ratio
ν12
has an increasing trend with the magnetic load ratio
ri
. The maximum enhancements
in
ν12
for the two ranges of
ri
are found to be 29.8% and 232.8% respectively. Under the same combined
loading conditions for the HMS beam network with
S=−1
as presented in Figure S10(b), the eective
Poisson's ratio
ν12
has decreasing trends with both
σ1
and
ri
. A maximum 29.8% reduction in
ν12
is
observed compared to the only mechanical loading condition
ri= 0
. As evident from Figure S10(c),
the eective Poisson's ratio
ν12
of the HMS beam network with
S= 1
increases with both input tensile
stress magnitude
σ1
and the magnetic load ratio
ri
. The maximum enhancement in
ν12
compared to
the loading condition of
ri= 0
is found to be 449.2%. Under the combined loading of tensile
σ1
and
ri
within the range
0≤ri≤0.4
,
ν12
of the HMS beam network with
S=−1
increases with
σ1
as observed
from Figure S10(d). For the range
1≤ri≤2
,
ν12
decreases with
σ1
. For both the ranges of
ri
, the
overall eective Poisson's ratio
ν12
has decreasing trends with
ri
. The maximum reductions in
ν12
for
the considered two ranges of
ri
are obtained from Figure S10(d) as 20.6% and 21.9% respectively.
61

Figure 10: Eective Young's modulus of hexagonal HMS beam networks having uniform residual magnetic
ux density under combined mechanical normal stress along direction-2 and magnetic eld along direction-
2.
Variations of the non-dimensional eective Young's modulus
¯
E2
of the hexagonal HMS beam network having the
uniform residual magnetic ux density
(a, c)
S= 1
and
(b, d)
S=−1
as function of the magnetic load ratio
ri
at
dierent mechanical stress levels
σ2
under the
(a, b)
compressive and
(c, d)
tensile modes of the mechanical stress
σ2
in
combination with the magnetic eld
Ba
along direction-2.
Under the compressive and tensile normal stress along direction-2 (
σ2
) in combination with the mag-
netic eld along direction-2 (
Ba
), eects of the magnetic load ratio
ri
and input stress magnitude
σ2
on the non-dimensional elastic moduli
¯
E2
and
ν21
of the hexagonal HMS beam network with uniform
62

Figure 11: Eective Poisson's ratio of hexagonal HMS beam networks having uniform residual magnetic
ux density under combined mechanical normal stress along direction-2 and magnetic eld along direction-
2.
Variations of the eective Poisson's ratio
ν21
of the hexagonal HMS beam network having the uniform residual magnetic
ux density
(a, c)
S= 1
and
(b, d)
S=−1
as function of the magnetic load ratio
ri
at dierent mechanical stress levels
σ2
under the
(a, b)
compressive and
(c, d)
tensile modes of the mechanical stress
σ2
in combination with the magnetic
eld
Ba
along direction-2.
residual magnetic ux density
S= 1
and
S=−1
are shown in Figures 10-S12 following the repre-
sentation scheme for the combined loading
σ1
and
Ba
(refer to Figures 8-S10). Figure 10(b), (c), and
Figure 11(b), (c) depict that for the hexagonal HMS beam network with
S=−1
under compression
and the hexagonal HMS beam network with
S= 1
under tension, singularity points on the eective
¯
E2
63

and
ν21
arise at some magnetic load ratios
ri
. However, for the other two congurations as presented in
Figure 10(a), (d), and Figure 11(a), (d), such phenomena are not observed.
Figure 12: Eective shear modulus of hexagonal HMS beam networks having uniform residual magnetic
ux density under combined mechanical shear stress in plane 1-2 and magnetic eld along direction-2.
Variations of the non-dimensional eective shear modulus
¯
G12
of the hexagonal HMS beam network having the uniform
residual magnetic ux density
(a, c)
S= 1
and
(b, d)
S=−1
as function of the magnetic load ratio
ri
at dierent
mechanical stress levels
τ
under the
(a, b)
anti-clockwise and
(c, d)
clockwise modes of the mechanical stress
τ
in
combination with the magnetic eld
Ba
along direction-2.
As observed from Figure S11(a), the non-dimensional eective Young's modulus
¯
E2
of the hexagonal
HMS beam network with
S= 1
decrease with compressive stress magnitude
σ2
for lower values of
ri
.
64

However, for higher values of
ri
,
¯
E2
initially decreases and then increases with
σ2
. The overall stiness
decreases with
ri
and maximum 83.9% reduction in
¯
E2
is observed. Under the same compressive mode
of mechanical loading, the positive and negative
¯
E2
of the hexagonal HMS beam network with
S=−1
for the ranges of the magnetic load ratio
0≤ri≤0.7
and
1.5≤ri≤2.5
respectively decreases with
stress magnitude
σ2
and increases with
ri
as observed in Figure S11(b). The maximum enhancements
in the positive and negative
¯
E2
due to the magnetic eld are achieved as 233.7% and 66.6% respectively.
As observed from Figure S11(c) and (d), under the tensile mode of the normal stress
σ2
, the eective
Young's modulus
¯
E2
increase with
σ2
for both the hexagonal HMS beam networks with
S= 1
and
S=−1
. However, for the HMS beam network with
S= 1
, the positive and negative non-dimensional
¯
E2
increases with
ri
in the considered ranges
0≤ri≤0.7
and
1.5≤ri≤2.5
respectively. Maximum 232.6%
and 66.8% enhancements in the positive and negative
¯
E2
are achieved as obtained from Figure S11(c).
Whereas, for the HMS beam network with the negative residual magnetic ux density
S=−1
, opposite
eect of
ri
is observed in Figure S11(d) with the 83.1% maximum reduction with respect to the only
mechanical loading condition,
ri= 0
.
As evident from Figure S12(a), the eective Poisson's ratio
ν21
of the hexagonal HMS beam network
with
S= 1
decreases with both the compressive stress
σ2
and magnetic load ratio
ri
. A maximum
129.4% reduction in
ν21
is observed for
ri= 5
compared to
ri= 0
. For the HMS beam network with
S=−1
under tensile mode of normal stress as presented in Figure S12(d), completely opposite eects
of
σ2
and
ri
are observed with the maximum 55% enhancement. As obvious from Figure S12(b), for
the HMS beam network with
S=−1
under compressive stress
σ2
in combination with the magnetic
load
0≤ri≤0.7
, the eective Poisson's ratio
ν21
decreases with stress magnitude. For the magnetic
load range
1.5≤ri≤2.5
, an opposite eect of the non-linearity is observed. However, for both
the considered magnetic load ranges,
ν21
increases with
ri
having the maximum 35.1% and 21.9%
enhancements respectively. Completely opposite eects of
σ2
and
ri
are observed in Figure S12(c) for
the HMS beam network with
S= 1
under tensile stress
σ2
. The corresponding reductions in the eective
ν21
due to the application of magnetic eld are found to be 15.1% and 39% respectively.
Under the anti-clockwise and clockwise modes of the shear stress
τ
in combination with the mag-
netic eld
Ba
along direction-2, combined eects of the magnetic load ratio
ri
and the input stress
magnitude
τ
on the non-dimensional shear modulus
¯
G12
of the hexagonal HMS beam network with
uniform residual magnetic ux density
S= 1
and
S=−1
are shown in Figures 12 and S13 following
similar representation scheme for the combined loading condition of normal stress and magnetic eld.
As obvious from Figure 12(b) and (c), for the HMS beam network with
S=−1
under anti-clockwise
shear stress and the HMS beam network with
S= 1
under clockwise shear stress, singularity points
65

arise at some
ri
values. For these combined loading cases, negative shear modulus is observed under
certain combinations of
τ
and
ri
. Whereas, for the other two combined loading conditions as presented
in Figure 12(a) and (d), such singularity points of the shear modulus do not arise.
As obvious from Figure S13(a) and (d), for the hexagonal HMS beam network with
S= 1
under
anti-clockwise shear stress and the hexagonal HMS beam network with
S=−1
under clockwise shear
stress, the eective non-dimensional shear modulus
¯
G12
increases with stress magnitude
τ
for the lower
values of
ri
. Whereas, for the higher magnetic loading
ri
, mixed increasing-decreasing eects of the
stress magnitude are observed. However, for both the congurations,
ri
has the same decreasing eects
with the corresponding 41.8% and 68.4% maximum reductions in
¯
G12
. For the HMS beam network
with the negative magnetization
S=−1
under the anti-clockwise mode of shear stress
τ
as presented
in Figure S13(b), some irregular eects of the stress magnitude
τ
and the magnetic load ratio
ri
are
observed on the non-dimensional positive
¯
G12
for
0≤ri≤3
and the mixed negative-positive
¯
G12
for
5≤ri≤6
. The maximum enhancement and reduction in the positive
¯
G12
are found to be 339.6% and
56.8% respectively. Whereas, the maximum enhancement in the negative
¯
G12
is observed as 47.3%. For
the HMS beam network with
S= 1
under the clockwise shear stress
τ
as presented in Figure S13(c), the
positive non-dimensional shear modulus
¯
G12
for
0≤ri≤1.5
increases with the input stress amplitude.
However, for the magnetic load range
5≤ri≤6
, the negative
¯
G12
initially increases with
τ
but at the
higher stress level becomes almost independent of
τ
. Both the positive and negative
¯
G12
of the HMS
beam network increase with
ri
resulting in maximum 463.4% and 43.2% enhancements respectively. It
is interesting no note from the trends presented for the elastic moduli, the value of applied magnetic
eld can be actively modulated (and optimized) based on the applied external mechanical stresses to
achieve a target level of certain elastic modulus and stiness.
3.5. Periodic HMS beam network with optimally-architected residual magnetic ux density
As described in the mathematical formulation in subsection 2.3.1, the beam elements of the hexagonal
HMS beam network are subjected to nite moments at the ends with zero moment at the mid-point due
to the typical rotationally boundary conditions. Based on the kinetic conditions, two sets of intuitive
designs of the residual magnetic ux density (
S(ξ)
) are proposed having maximum hard particle density
at the endpoints with zero at the mid-point of the HMS beam elements. In the rst set of design, we
consider either
S= 1
or
S=−1
at both the ends
ξ= 0,1
with
S= 0
at the mid-point
ξ= 0.5
. The
variation of
S(ξ)
along the normalized coordinate
ξ
is dened by the following equation with the degree
66

of non-linearity
n
.
S(ξ) = 




±(1 −2ξ)n,0≤ξ < 0.5
±(−1+2ξ)n,0.5≤ξ≤1
For the second set of design,
S(ξ)
is varying either from
S=−1
to
S= 1
or from
S= 1
to
S=−1
Figure 13: Physics-informed intuitive designs of spatially-varying residual magnetic ux density in the
HMS beam elements of the hexagonal HMS beam network.
Distribution of the coecient of residual magnetic
ux density
S(ξ)
along the normalized coordinate
ξ
with the degree of non-linearity
n= 0,0.1,0.25,0.5,1
, and
3
for:
(a,
b)
the rst set of design of
S(ξ)
having (a) positive and (b) negative distribution, and
(c, d)
the second set of design of
S(ξ)
varying (c) from
S=−1
to
S= 1
and (d) from
S= 1
to
S=−1
.
67

between the ends
ξ= 0,1
with
S= 0
at the mid-point
ξ= 0.5
. The variation of
S(ξ)
along the
normalized coordinate
ξ
for the second set of design of
S(ξ)
is expressed mathematically below with the
degree of non-linearity
n
.
S(ξ) = 




∓(1 −2ξ)n,0≤ξ < 0.5
±(−1+2ξ)n,0.5≤ξ≤1
The positive and negative distributions of the rst designed set of
S(ξ)
along the normalized coordinate
ξ
with the degree of non-linearity
n= 0,0.1,0.25,0.5,1
, and
3
are shown in Figure 13(a) and (b)
respectively. Similarly, for the two cases of the second designed set of
S(ξ)
, the distribution of
S(ξ)
along the normalized coordinate
ξ
are presented in Figure 13(c) and (d) respectively. The eect of the
degree of non-linearity
n
for the two sets of designed
S(ξ)
on the non-linear variation of the elastic
moduli of the hexagonal HMS beam network as functions of the input stress are investigated here as
presented in the following paragraphs.
Variations of the non-dimensional eective Young's modulus
¯
E1
of the hexagonal HMS beam network
with the input stress
σ1
for the considered six degrees of non-linearity
n
(
0,0.1,0.25,0.5,1
, and
3
) of
the positive and negative distribution of the rst set of designed
S(ξ)
(refer to Figure 13(a) and (b))
under the combined compressive stress along direction-1 (
σ1
) and the external magnetic eld
Ba
along
direction-2 are shown in Figure 14(a). Whereas, the variations of
¯
E1
under the tensile mode of the
normal stress
σ1
in combination with the magnetic eld
Ba
are presented in Figure 14(b). The similar
plots showing the eects of the degree of non-linearity
n
on the eective Poisson's ratio
ν12
of the
hexagonal HMS beam network with the rst set of designed
S(ξ)
are shown in Figure 14(c) and (d)
respectively. The results are compared in Figure 14 for the magnetic load ratio
ri= 0.4
. Under the
combined loading of normal stress along direction-2 (
σ2
) and the magnetic eld
Ba
along direction-2,
eects of the the degree of non-linearity
n
on the non-linear variations of the eective Young's modulus
¯
E2
and Poisson's ratio
ν21
of the hexagonal HMS beam network with the rst set of designed
S(ξ)
are
shown in Figure S14 for the magnetic load ratio
ri= 0.5
. Whereas, similar variations of the non-linear
shear modulus
¯
G12
of the HMS beam network with the degree of non-linearity
n
for the rst set of
designed
S(ξ)
under the anti-clockwise and clockwise modes of shear stress (
τ
) in combination with the
external magnetic eld
Ba
are shown in Figure 15 for the magnetic load ratio
ri= 1.5
.
As observed from Figure 14(a) and (c), the non-dimensional Young's modulus
¯
E1
and the Poisson's
ratio
ν12
non-linearly decreases with compressive stress
σ1
for both the positive and negative distribution
of the rst set of design of
S(ξ)
. Such non-linearity in the system stiness is coming from the inherent
geometric non-linearity due to large deformation and material non-linearity due to magneto-elastic
68

Figure 14: Modulation of the eective elastic moduli of hexagonal HMS beam networks with the rst set
of designed
S(ξ)
under the normal stress along direction-1 in combination with the magnetic eld along
direction-2.
Variations of the
(a, b)
non-dimensional eective Young's modulus
¯
E1
and
(c, d)
eective Poisson's ratio
ν12
of the hexagonal HMS beam network as function of the input stress
σ1
for the considered six degrees of non-linearity
n
(
0,0.1,0.25,0.5,1
, and
3
) of the positive and negative distributions of the rst set of designed
S(ξ)
under the
(a, c)
compressive and
(b, d)
tensile mode of normal stress
σ1
along direction-1 in combination with the magnetic eld
Ba
along direction-2. The results are compared for the magnetic load ratio of the inclined member
ri= 0.4
.
69

Figure 15: Modulation of the eective shear modulus of hexagonal HMS beam networks with the rst
set of designed
S(ξ)
under the shear stress in plane 1-2 in combination with the magnetic eld along
direction-2.
Variations of the non-dimensional eective shear modulus
¯
G12
of the hexagonal HMS beam network as
function of the input stress
τ
for the considered six degrees of non-linearity
n
(
0,0.1,0.25,0.5,1
, and
3
) of the positive and
negative distributions of the rst set of designed
S(ξ)
under the
(a)
anti-clockwise and
(b)
clockwise mode of shear stress
τ
in plane 1-2 in combination with the magnetic eld
Ba
along direction-2. The results are compared for the magnetic
load ratio of the inclined member
ri= 1.5
.
coupling under the combined mechanical and magnetic loading. For the positive distribution of the
rst set of designed
S(ξ)
, the overall non-linear Young's modulus
¯
E1
and Poisson's ratio
ν12
decrease
with the degree of non-linearity
n
as observed in Figure 14(a) and (c). Whereas, for the negative
distribution of
S(ξ)
, the degree of non-linearity
n
shows the opposite increasing eect on the non-
linear Young's modulus
¯
E1
and Poisson's ratio
ν12
. Maximum 56% and 11% enhancements in the
non-dimensional Young's modulus
¯
E1
and Poisson's ratio
ν12
are achieved respectively for
n= 3
of
the negative distribution of
S(ξ)
compared to the uniform distribution (
S=−1
) for
n= 0
(refer to
Figure 14(a) and (c)). Whereas, maximum 66.4% and 21% reductions in
¯
E1
and
ν12
are obtained for
n= 3
of the positive
S(ξ)
with respect to the uniform distribution (
S= 1
) for
n= 0
.
Under the tensile mode of the normal stress
σ1
in combination with the external magnetic eld
Ba
as presented in Figure 14(b) and (d), a completely opposite eect of the inherent system non-linearity is
observed compared to the compressive mode of
σ1
as shown in Figure 14(a) and (c). The non-dimensional
Young's modulus
¯
E1
and the Poisson's ratio
ν12
increase with increase in the tensile
σ1
for both the
70

positive and negative distribution of the rst set of design of
S(ξ)
. As shown in Figure 14(b), the overall
non-linear Young's modulus
¯
E1
increases with the degree of non-linearity
n
for the positive distribution
of
S(ξ)
, whereas, it decreases with
n
for the negative distribution of
S(ξ)
. Whereas, as observed from
Figure 14(d), the degree of non-linearity
n
has the opposite eect on the non-linear Poisson's ratio
ν12
compared to the Young's modulus
¯
E1
. The maximum 31.1% and 22.7% enhancements in
¯
E1
and
ν12
are
achieved respectively for the non-linear
S(ξ)
with
n= 3
compared to the uniform
S
with
n= 0
under
the tensile mode of normal stress
σ1
in combination with the external magnetic eld
Ba
as observed
from Figure 14(b) and (d). Whereas, the maximum reductions in the elastic moduli
¯
E1
and
ν12
are
observed as 62.6% and 23.3% respectively from Figure 14(b) and (d) for the non-linear
S(ξ)
with
n= 3
compared to the uniform
S
with
n= 0
.
Under the normal stress along direction-2 (
σ2
) in combination with the magnetic eld
Ba
along
direction-2, eects of non-linearity on the non-dimensional elastic moduli
¯
E2
and
ν21
in terms of their
variations with input stress magnitude
σ2
are observed from Figure S14 similar to the combined loading
of
σ1
and
Ba
as presented in Figure 14. However, the eects of the degree of non-linearity
n
of the
rst set of designed
S(ξ)
are found opposite for the combined loading of
σ2
and
Ba
compared to the
combined loading of
σ1
and
Ba
. As evident from Figure S14(a) and (c), the maximum enhancements in
the non-dimensional Young's modulus
¯
E2
and Poisson's ratio
ν21
under the compressive mode of
σ2
are
achieved as 42.4% and 27.5% respectively for the positive
S(ξ)
with
n= 3
compared to the uniform
S
for
n= 0
. Whereas, 47.2% and 18% reductions in
¯
E2
and
ν21
are obtained for the negative distribution
of
S(ξ)
with
n= 3
compared to
n= 0
. Under the tensile mode of
σ2
in combination with
Ba
, the
maximum enhancement and reduction in
¯
E2
for
n= 3
with respect to the uniform
S
(
n= 0
) are found
to be 41% and 46.6% respectively from Figure S14(b). Whereas, as evident from Figure S14(d), the
enhancement and reduction in
ν21
for the non-linear
S(ξ)
with
n= 3
compared to
n= 0
under the
tensile mode of
σ2
in combination with
Ba
are obtained as 10.4% and 7.3% respectively.
As evident from Figure 15(a) and (b), under both the anti-clockwise and clockwise modes of shear
stress
τ
in combination with the magnetic eld
Ba
, the non-dimensional shear modulus
¯
G12
increases
with the input stress
τ
for the positive distribution of the rst set of design of
S(ξ)
. Whereas, for the
HMS beam network with the negative distribution of the rst set of designed
S(ξ)
, the non-dimensional
shear modulus
¯
G12
initially decreases and then increases with
τ
for the lower values of
n
. However, for
the highest value of the degree of non-linearity
n= 3
,
¯
G12
has an increasing trend with the input stress
τ
amplitude. As observed from Figure 15(a), under the anti-clockwise mode of
τ
in combination with
the magnetic eld
Ba
, the non-dimensional shear modulus
¯
G12
increases with the degree of non-linearity
n
for the positive distribution of the rst designed set
S(ξ)
. However, for the negative distribution of
71

S(ξ)
,
¯
G12
decreases with
n
at the lower stress level, whereas, it increases with
n
at the higher stress zone.
Maximum 30.9% enhancement in the non-dimensional shear modulus
¯
G12
is achieved for
n= 3
of the
positive distribution of the rst set of designed
S(ξ)
compared to the uniform
S
with
n= 0
. Whereas,
the maximum reduction and enhancement in
¯
G12
for the negative
S(ξ)
are observed as 35.6% and 50.9%
respectively. On the other hand, under the clockwise mode of
τ
in combination with the magnetic eld
Ba
as observed in Figure 15(b), the non-dimensional shear modulus
¯
G12
decreases with the degree of
non-linearity
n
for the positive distribution of
S(ξ)
. Whereas, for the hexagonal HMS beam network
with negative designed
S(ξ)
,
¯
G12
increases with
n
. The maximum enhancement and reduction in
¯
G12
are observed from Figure 15(b) as 104.3% and 80.4% respectively compared to the uniform
S
.
For the second set of design of the residual magnetic ux density (refer to Figure 13(c) and (d)),
the two opposite distributions of
S(ξ)
varying from
S=−1
to
S= 1
and from
S= 1
to
S=−1
cause
the same eects on the non-linear elastic moduli of the hexagonal HMS beam network under each mode
of the mechanical stress in combination with the magnetic eld. Despite of the opposite curvatures at
the deformed state, the same tip-deections of HMS beam for the two opposite distributions of
S(ξ)
varying from
S=−1
to
S= 1
and from
S= 1
to
S=−1
is the cause behind such phenomenon.
Such a phenomenon is already described in connection with Figures 7 and S6 for a HMS beam with the
opposite signs of
S(ξ)
in the two halves. Hence, for the two opposite distributions (varying from
S=−1
to
S= 1
and from
S= 1
to
S=−1
) of the second set of designed
S(ξ)
as shown in Figure 13(c) and (d),
we get single set of results. Eects of the degree on non-linearity
n
for the second set of designed
S(ξ)
on the non-linear elastic moduli of the hexagonal HMS beam network under the loading combinations
of
σ1
,
σ2
, and
τ
with the magnetic eld
Ba
are shown in Figures S15-S17 respectively for the magnetic
load ratio
ri= 2.5
,
ri= 4
, and
ri= 4
.
Under the compressive mode of normal stress along direction-1 (
σ1
) in combination with the external
magnetic eld
Ba
for the second set of designed
S(ξ)
, the non-dimensional Young's modulus
¯
E1
initially
decreases with the input stress magnitude
σ1
as observed from Figure S15(a). At the higher magnitude
of the applied stress
σ1
,
¯
E1
increases with
σ1
for the lower values of
n
and goes on decreasing for the
higher values of
n
. Under the same combination of mechanical and magnetic loading, the Poisson's
ratio
ν12
decreases with the applied stress
σ1
as evident from Figure S15(c). Negative Poisson's ratio
is obtained for
n= 0
, and
0.1
even for the non-auxetic conguration of the hexagonal HMS beam
network under consideration. Under the tensile mode of the normal stress
σ1
in combination with
Ba
as
observed from Figure S15(b) and (d), both Young's modulus
¯
E1
and Poisson's ratio
ν12
increase with an
increase in the magnitude of the applied stress
σ1
. The overall non-linear Young's modulus
¯
E1
decreases
with the degree of non-linearity
n
of the second set of designed
S(ξ)
under both the compressive and
72

tensile modes of
σ1
as observed from Figure S15(a) and (b). The maximum reductions in
¯
E1
for
n= 3
compared to
n= 0
are observed to be 86.9% and 63.9% under the compression and tension respectively.
As observed in Figure S15(c), the Poisson's ratio
ν12
has an increasing trend with
n
at the lower range
of the compressive stress
σ1
. However, at the higher range of
σ1
, some mixed trend is observed. The
maximum enhancement of 143.5% in
ν12
for
n= 3
compared to
n= 0
is achieved. Whereas, under the
tensile mode of
σ1
, Poisson's ratio
ν12
decreases with
n
as shown in Figure S15(d), and the maximum
reduction in
ν12
is found to be 73.9%.
As shown in Figure S16(a) and (c), the non-dimensional Young's modulus
¯
E2
and Poisson's ratio
ν21
of the hexagonal HMS beam network with the second set of designed
S(ξ)
decrease with the applied
stress input under the combined loading condition of compressive normal stress along direction-2 (
σ2
)
and magnetic eld along direction-2 (
Ba
). The Overall non-linear elastic moduli
¯
E2
and
ν21
increase
with the degree of non-linearity
n
. The maximum enhancements in the elastic moduli
¯
E2
and
ν21
for
the non-linear
S(ξ)
with
n= 3
with respect to the linear
S(ξ)
with
n= 0
are found to be 23% and
68.5% respectively. Eects of the inherent system non-linearity and the degree of non-linearity
n
of the
second set of designed
S(ξ)
on the elastic moduli
¯
E2
and
ν21
are found exactly the opposite under the
tensile mode of normal stress
σ2
as observed from Figure S16(b) and (d) compared to the compressive
mode (refer to Figure S16(a) and (c)). The maximum reductions of 63.3% and 35.8% are obtained in
the elastic moduli
¯
E2
and
ν21
for the non-linear
S(ξ)
with
n= 3
compared to the linear
S(ξ)
with
n= 0
.
Under both the anti-clockwise and clockwise modes of shear stress
τ
in combination with the external
magnetic eld
Ba
, the non-dimensional eective shear modulus
¯
G12
of the hexagonal HMS beam network
with the second set of designed
S(ξ)
initially decreases and then increases with the input stress
τ
for the
lower values of
n
as observed from Figure S17(a) and (b). Whereas, for the highest value of the degree
of non-linearity
n= 3
,
¯
G12
has an increasing trend with the magnitude of the input stress
τ
. The plots
in Figure S17(a) and (b) also depict that the non-linear shear modulus
¯
G12
increases with the degree of
non-linearity
n
of the second set of deigned
S(ξ)
. The maximum enhancements in the non-dimensional
shear modulus
¯
G12
are achieved to be 68.9% and 57.5% for the non-linear
S(ξ)
with
n= 3
compared
to the linear
S(ξ)
with
n= 0
under the anti-clockwise and clockwise mode of shear stress respectively.
The numerical results presented in the preceding subsection (subsection 3.4) demonstrate on-demand
magneto-active modulations (enhancements and reductions) of the eective nonlinear elasticity of hexag-
onal HMS beam networks through uniform residual magnetic ux density design in the cell walls under
far-eld magnetic eld in combination with externally applied mechanical stresses. Physics-informed
(nite moments at the ends with zero moment at the mid-point due to the typical rotationally re-
strained beam boundary conditions for periodic lattices) architecturing of the residual magnetic ux
73

density pattern in the cell walls as proposed in the present subsection results further augmentations in
the deformation components due to far-eld magnetic eld compared to uniform residual magnetic ux
density which are in-phase or out-of-phase with the deformations caused by mechanical stresses only.
The in-phase and out-of-phase deformations coming from magnetic eld and mechanical stresses respec-
tively results augmented anti-curvature or pro-curvature eects [42, 43] to the cell wall deformations
compared to the uniform residual magnetic ux density of the cell walls. Such active anti-curvature
or pro-curvature eects cause further enhancements or reductions of the HMS beam network stiness
compared to the uniform residual magnetic ux density design as demonstrated through the numerical
results in the present subsection (subsection 3.5). In turn this will lead to improved energy eciency
in achieving a target on-demand stiness, resulting in sustainable programmable metamaterials with
minimum utilization of the intrinsic materials.
3.6. Applicability to other forms of periodic HMS beam networks
Within the developed multi-physical mechanics-based semi-analytical framework, modulations of
the elastic moduli of hexagonal HMS beam networks with uniform and two intuitively designed residual
magnetic ux densities are extensively investigated in the preceding two subsections. To demonstrate the
generality of the proposed concept of modulating elastic properties through an external magnetic eld
within the developed physically insightful computational framework, non-linear eective elastic moduli
of ve other forms of HMS beam networks, namely, auxetic, rectangular brick, rhombic, triangular,
and rectangular networks as shown in Figure 1(g) are analysed in this subsection considering uniform
residual magnetic ux density in combination with dierent modes of far-eld mechanical stresses. Note
that the concept of beam-level architecturing the residual magnetic ux density can also be implemented
to dierent other unit cell architectures for more accentuated elasticity modulation as demonstrated in
the case of hexagonal lattices (refer to section 3.5). However, we limit the current demonstration to
uniform residual magnetic ux density for other lattices in order to maintain the brevity of this paper.
3.6.1. Auxetic HMS beam networks
For the auxetic HMS beam network, as shown in Figure 1(g)I, the geometric parameters are con-
sidered as
h/l = 2
and
θ=−π/6
. The unit cell conguration of the auxetic HMS beam network with
residual magnetic ux density
S= 1
subjected to normal (
σ1
or
σ2
) and shear (
τ
) stresses in combina-
tion with the external magnetic eld
Ba
is shown in Figure 16(a). Variations of the non-dimensional
elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
with dierent modes of input stress magnitude under dierent
magnetic load levels are presented in Figure 16(b)-(f) respectively. It is evident from the gure that
within a small deformation regime in absence of the external magnetic eld, all the results obtained
74

from the present framework agree well with the analytical solutions from literature [5]. This provides
a degree of condence and validation to the present computational framework before exploiting it for
further investigation.
As observed from Figure 16(b), the eective non-dimensional Young's modulus
¯
E1
of the auxetic
HMS beam network decreases with compressive stress
σ1
and magnetic load ratio
ri
. Whereas, under
the tensile mode of the normal stress
σ1
, Young's modulus
¯
E1
increases with the stress magnitude and
the magnetic load ratio
ri
for
0≤ri≤0.4
. Under the same loading condition for the magnetic load ratio
1≤ri≤2
, negative stiness is observed which decreases with stress magnitude but increases with
ri
.
Maximum 201.9% enhancement and 46.4% reduction in the positive Young's modulus
¯
E1
are achieved
concerning the only mechanical loading condition (
ri= 0
). Whereas, the maximum enhancement in
the negative Young's modulus
¯
E1
is obtained as 68.8% for
ri= 2
compared to
ri= 1
. Figure 16(c)
depicts that the eective Poisson's ratio
ν12
increases with magnetic load ratio
ri
with dierent degrees
of auxecity under the compressive and tensile modes of normal stress
σ1
. A maximum 19% enhancement
in
ν12
for the considered ranges of
ri
can be obtained from Figure 16(c).
For the combined loading under normal stress
σ2
and magnetic eld
Ba
along direction-2 as presented
in Figure 16(d) and (e), eects of non-linearity in terms of variations of the elastic moduli
¯
E2
and
ν21
with stress magnitude are found opposite compared to the loading combination under
σ1
and
Ba
.
However, decreasing and increasing eects of the magnetic loading under the compressive and tensile
loading modes are the same for
¯
E2
as that of
¯
E1
, with maximum 400.4% and 66.49% enhancement and
reduction respectively. However, for
ν21
, the eect of magnetic load ratio is found opposite to that of
ν12
with a maximum 40% reduction. Notably the degree of auxeticity for
ν12
and
ν21
can be actively
controlled in a wide band as a function of the magnetic eld.
As obvious from Figure 16(f), under the anti-clockwise mode of shear loading, the non-dimensional
shear modulus
¯
G12
increases with stress magnitude
τ
and decreases with magnetic load
ri
. Under the
clockwise mode of shear loading,
¯
G12
increases with stress magnitude
τ
for a lower range of
ri
. However,
for a higher range of
ri
under the clockwise loading, negative
¯
G12
are observed having mixed increasing-
decreasing trends with the stress magnitude. However, for both the ranges of
ri
under the clockwise
loading mode,
ri
has increasing eects on
¯
G12
. The maximum enhancement and reduction in the positive
non-dimensional
¯
G12
concerning the only mechanical loading condition
ri= 0
are observed as 248.3%
and 62.7% respectively. Whereas, in the negative shear modulus
¯
G12
, a maximum 46% enhancement is
achieved for
ri= 10
compared to
ri= 8
.
75

Figure 16: Modulation of the eective elastic moduli of auxetic HMS beam networks having uniform
residual magnetic ux density under dierent modes of mechanical stress in combination with magnetic
eld. (a)
The unit cell of auxetic HMS beam network with
h/l = 2
and
θ=−π/6
having residual magnetic ux
density
S= 1
subjected to (1) normal stress
σ1
or
σ2
, and (2) shear stress
τ
in combination with magnetic eld
Ba
along
direction-2.
(b-f)
Variations of the non-dimensional eective elastic moduli of the auxetic HMS beam network as function
of the dierent modes of the mechanical stress at equally spaced magnetic load levels
ri
. The dotted points represent the
analytical solutions [5] without magnetic eld under small deformation regime.
76

Figure 17: Modulation of the eective elastic moduli of rectangular brick HMS beam networks having
uniform residual magnetic ux density under dierent modes of mechanical stress in combination with
magnetic eld. (a)
The unit cell of rectangular brick HMS beam network with
h/l = 1
having residual magnetic ux
density
S= 1
subjected to (1) normal stress
σ1
or
σ2
, and (2) shear stress
τ
in combination with magnetic eld
Ba
along direction-2.
(b-f)
Variations of the non-dimensional eective elastic moduli of the rectangular brick HMS beam
network as function of dierent modes of the mechanical stress at equally spaced magnetic load levels
ri
. The dotted
points represent the analytical solutions [5] without magnetic eld under small deformation regime.
77

3.6.2. Rectangular brick HMS beam networks
The rectangular brick HMS beam network as shown in Figure 1(g)II is derived readily from the
hexagonal HMS beam network by taking
θ= 0
. The unit cell conguration of the rectangular brick
HMS beam network with
h/l = 1
having uniform residual magnetic ux density (
S= 1
) is shown
schematically in Figure 17(a). Variations of the non-dimensional eective elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
,
and
¯
G12
of the rectangular brick HMS beam network as functions of the dierent modes of normal and
shear stresses combined with external magnetic eld are presented in Figure 17(b)-(f). Comparisons of
each set of results with the corresponding analytical solutions from literature [5], as presented through
the large dotted points in the plots, validate our framework for the special case in absence of the
magnetic eld within a small deformation regime. This provides a degree of condence to the present
computational framework before exploiting it for further investigation.
As in cases of the other HMS beam networks, modulations of the non-linear elastic moduli of the
rectangular brick HMS beam network in terms of the external magnetic eld are evident from Fig-
ure 17(b)-(f). Eects of geometric and material non-linearity on the elastic moduli in terms of their
variations with stress magnitude
σ1
,
σ2
or
τ
and magnetic load ratio
ri
can be readily noticed Fig-
ure 17(b)-(f). Interestingly, from Figure 17(b)-(f) it becomes obvious that depending on the combina-
tion of the magnetic load with a particular mode of the mechanical stress, negative Young's modulus,
negative Poisson's ratio and negative shear modulus can be achieved. Maximum enhancements in
¯
E1
,
¯
E2
, and
¯
G12
are noted to be 64.4%, 150%, and 162.1% respectively. Whereas, maximum 32%, 54.5%,
91.7%, and 48.5% reductions in
¯
E1
,
¯
E2
,
ν21
, and
¯
G12
are obtained respectively under the considered
ranges of the magnetic load ratio
ri
.
Note in Figure 17(c) that under the combined loading of normal stress
σ1
and magnetic eld
Ba
, the
magnitudes of the negative or positive Poisson's ratio
ν12
of the rectangular brick HMS beam network
are very large compared to the unity. As obvious from Figure 17(a-1), under the combined loading of
normal stress
σ1
and magnetic eld
Ba
, the normal strain in direction-2 (
ϵ2
) is governed by the bending-
dominated deformation of the horizontal cell walls. Whereas, the normal strain along direction-1 (
ϵ1
)
is governed by the stretching-dominated deformation of the horizontal cell walls. Due to the dierence
in the order of magnitudes of the bending and stretching dominated axial strains along direction-1 (
ϵ1
)
and direction-2 (
ϵ2
), such large magnitudes of Poisson's ratio
ν12
is achieved for the rectangular brick
HMS beam network under the present loading combination. As
ν12
is zero under the only mechanical
load in absence of the magnetic eld, the enhancement and reduction in it are noted in terms of their
absolute values instead of percentage and they are 240.4 and 109.3 receptively.
78

3.6.3. Rhombic HMS beam networks
The rhombic HMS beam network as shown in Figure 1(g)III is obtained from generic hexagonal
HMS beam lattices by putting
h/l = 0
and
θ=π/4
. The unit cell conguration of the rhombic
HMS beam network with uniform residual magnetic ux density (
S= 1
) is shown in Figure 18(a).
Variations of the non-dimensional eective elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
of the rhombic HMS
beam network with combined stress and external magnetic eld along with the comparisons with the
respective analytical results from literature [5] are shown in Figure 18(b)-(f). The good agreement with
literature provides a degree of condence and validation to the present computational framework before
exploiting it for further investigation.
The self-explanatory plots in Figure 18(b)-(f) establish the idea of modulating the non-linear elastic
moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
of the rhombic HMS beam network by external magnetic eld in
combination with the dierent modes of the mechanical stress. The gure also depicts that under
certain combinations of mechanical and magnetic loads, negative stiness of the rhombic network can
be achieved. Maximum 233%, 36.8%, 232.7%, and 77.6% enhancements in the non-dimensional elastic
moduli
¯
E1
,
ν12
,
¯
E2
, and
¯
G12
of the rhombic HMS beam network are obtained respectively under the
considered ranges of the magnetic loads. Whereas, the maximum reductions in the non-dimensional
elastic moduli
¯
E1
,
¯
E2
,
ν21
, and
¯
G12
are achieved to be 58%, 60.2%, 37%, and 36.6% respectively.
3.6.4. Triangular HMS beam networks
The non-linear elastic moduli of the triangular HMS beam network (refer to Figure 1(g)IV) is not
readily derivable from the multi-physical mechanics-based semi-analytical framework for the hexagonal
HMS beam lattices. However, by selecting the proper unit cell as shown in Figure 19(a), the eective
elastic moduli of the triangular HMS beam network are derived following a similar computational
framework. A detailed derivation of non-linear elastic moduli
E1
,
ν12
,
E2
,
ν21
, and
G12
of the triangular
HMS beam network under the combined mechanical stress and magnetic eld is presented in section
2.4.
Variations of the non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
of the triangular HMS
beam network with dierent modes of mechanical stress in combination with the magnetic eld are
shown in Figure 19(b)-(f). The corresponding analytical results from literature [5, 34] in absence of the
magnetic eld within a small deformation regime are also plotted in the gure through the large dotted
points. The comparison studies successfully validate our proposed semi-analytical framework for the
special case of small deformation in absence of the magnetic eld.
Figure 19(b)-(f) depicts that the non-linear non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
79

Figure 18: Modulation of the eective elastic moduli of rhombic HMS beam networks having uniform
residual magnetic ux density under dierent modes of mechanical stress in combination with magnetic
eld. (a)
The unit cell of rhombic HMS beam network with
θ=π/4
having residual magnetic ux density
S= 1
subjected to (1) normal stress
σ1
or
σ2
, and (2) shear stress
τ
in combination with magnetic eld
Ba
along direction-
2.
(b-f)
Variations of the non-dimensional eective elastic moduli of the rhombic HMS beam network as function of
the dierent modes of the mechanical stress at equally spaced magnetic load levels
ri
. The dotted points represent the
analytical solutions [5] without magnetic eld under small deformation regime.
80

Figure 19: Modulation of the eective elastic moduli of triangular HMS beam networks having uniform
residual magnetic ux density under dierent modes of mechanical stress in combination with magnetic
eld. (a)
The unit cell of triangular HMS beam network having residual magnetic ux density
S= 1
subjected to (1)
normal stress
σ1
or
σ2
, and (2) shear stress
τ
in combination with magnetic eld
Ba
along direction-2.
(b-f)
Variations
of non-dimensional eective elastic moduli of the triangular HMS beam network as function of the dierent modes of the
mechanical stress at equally spaced magnetic load levels
rh
or
ri
. The dotted points represent the analytical solutions
[5, 34] without magnetic eld under small deformation regime.
81

of the triangular HMS beam network can be modulated as per requirement through the magnetic load
in terms of ratio
rh
or
ri
. Under certain combinations of mechanical stress with the magnetic eld,
even a negative Poisson's ratio is achievable with dierent degrees. The maximum enhancements in
the non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
of the triangular HMS beam network are
attainable as 14.1%, 27.5%, 44.5%, 865.5%, and 154% respectively. Whereas, maximum 11.6%, 27.6%,
32%, 1523.5%, and 65.8% reductions in the non-dimensional elastic moduli are obtained respectively.
Note the exceptional enhancement (865.5%) and reduction (1523.5%) in the Poisson's ratio
ν21
as
observed from Figure 19(e). As obvious from Figure 19(a-1), under the combined loading of normal stress
σ2
and magnetic eld
Ba
, the inuence of bending due to the magnetic eld is more on the horizontal
member OA compared to the inclined member OB. Such a deformation pattern of the triangular HMS
unit cell creates a dierence in the order of magnitudes of the normal strains along direction-1 (
ϵ1
)
and direction-2 (
ϵ2
) which in turn results in an exceptionally large enhancement and reduction in the
Poisson's ratio
ν21
as noted in the numerical results.
3.6.5. Rectangular HMS beam networks
As in the case of the triangular HMS beam network, derivation of the non-linear elastic moduli of
the rectangular HMS beam network (refer to Figure 1(g)V) by considering appropriate unit cell (refer
to Figure 20(a)) within the current semi-analytical framework is presented in section 2.5. Variations of
the non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
of the rectangular HMS beam network with
dierent modes of mechanical stress in combination with the magnetic eld along with the comparisons
with the respective analytical results (concerning only small deformation mechanical stresses) from
literature [5, 34] are presented in Figure 20(b)-(f). As in the case of the other congurations, the
comparison studies between the present semi-analytical framework and the analytical models [5, 34] are
found quite satisfactory in absence of the magnetic eld within the small deformation regime.
The concept of modulating non-linear elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
through applying an
external magnetic eld is demonstrated in Figure 20(b)-(f) for the rectangular HMS beam network. The
gure also depicts that by controlling the external magnetic eld in combination with the mechanical
load, mode-dependent negative Poisson's ratio and negative shear modulus can be achieved. Maximum
111.1%, 66.7%, and 102.1% enhancements in the non-dimensional elastic moduli
¯
E1
,
¯
E2
, and
¯
G12
are
obtained respectively. Whereas, the maximum reductions in the elastic moduli are found to be 38.8%,
28.6%, and 50% respectively. As the Poisson's ratios
ν12
and
ν21
are zero under only mechanical load,
their enhancements and reductions under magnetic eld are expressed by absolute values instead of
percentage values, and they are 0.1 and 13.1 in enhancement and 0.2 and 30.7 in reduction respectively.
82

Figure 20: Modulation of the eective elastic moduli of rectangular HMS beam networks having uniform
residual magnetic ux density under dierent modes of mechanical stress in combination with magnetic
eld. (a)
The unit cell of rectangular HMS beam network with
h/l = 0.5
having residual magnetic ux density
S= 1
subjected to normal stress
σ1
or
σ2
and shear stress
τ
in combination with magnetic eld
Ba
along direction-2.
(b-
f)
Variations of non-dimensional eective elastic moduli of the rectangular HMS beam network as function of dierent
modes of the mechanical stress at equally spaced magnetic load levels
rh
or
rv
. The dotted points represent the analytical
solutions [5, 34] without magnetic eld under small deformation regime.
83

The large magnitudes of the positive and negative Poisson's ratio
ν21
of the rectangular HMS beam
network are caused by the dierence in the order of magnitudes in the normal strains
ϵ1
and
ϵ2
under
the combined loading of normal stress
σ2
and magnetic eld
Ba
due to dierent respective modes of
predominant beam deformations.
In general, the numerical results demonstrate the on-demand active modulation of eective elastic
moduli in a wide band (i.e. broadband stiness and exibility programming) as a function of the unit
cell geometry, beam-level architecture of residual magnetic ux density and nonlinear intrinsic material
properties along with the applied far-eld mechanical stresses and magnetic eld. The eectiveness
of applied magnetic eld can be further optimized (including target attainment) corresponding to a
particular mode and level of applied far-eld stress depending on the unit cell geometry (such as dierent
bending and stretching dominated unit cells and dimensions of the beam-like members) and beam-level
residual magnetic ux density.
4. Summary and perspective
In the paper, we have proposed a novel class of lattice metamaterials as periodic networks of beams
made of soft material with embedded hard magnetic particles (HMS beam networks) subjected to large
deformation under combined remote mechanical stress and magnetic eld. The architected networks
of HMS beams are very light in weight and provide excellent modulation capability of the non-linear
eective elastic properties depending on the hard magnetic particle distribution in the HMS beam
elements, unit cell geometry and the combination of applied mechanical stress with the external magnetic
eld. To actively modulate the metamaterial properties post-manufacturing enabling applications for
a range of advanced intelligent structural systems, we propose here to adopt an innovative bi-level
modulation concept involving the coupled design space of unit cell geometries, architected HMS beam-
like members and their stimuli-responsive deformation physics. We have exploited the geometric non-
linearity due to large deformation and material non-linearity under magneto-mechanical coupling to
modulate the eective elastic properties of the novel class of architected HMS beam networks ranging
from very high stiness like sti metal to very low stiness, even lower compared to the soft polymers.
By externally applying dierent values of the magnetic eld intensity, dierent elastic properties and
stiness can be achieved, and that too from a distance (i.e. on-demand contactless elasticity control).
Essentially, this will help in minimizing the material utilization to an extreme extent by controlling the
stiness of a structure based on active operational demands. For example, the stiness corresponding
to target modes and direction of a structure can be actively increased during an operational condition
when higher magnitudes of loads are experienced to keep the deformations under control or the natural
84

frequencies need to be increased to avoid resonance under dynamic loading. The stiness can also be
actively reduced to allow large deformation and shape control for (soft-)robotic motions or increased
energy absorption and avert sudden failure.
To estimate the non-linear eective elastic moduli under the normal or shear mode of mechanical
stress in combination with the external magnetic eld, a physically insightful semi-analytical framework
is developed for periodic HMS beam networks. Within the unit cell-based framework, the non-linear
multi-physical mechanics of rotationally restrained HMS beams subjected to combined mechanical and
magnetic loads representing generalized elements of the architected beam network is dened. Gov-
erning equation of the non-linear HMS beams is derived using the variational principle-based energy
method within the non-linear kinematic setting of the Euler-Bernoulli beam theory and the material
constitutive law of the Yeoh hyperelastic model. To deal with the non-linearity involved in the govern-
ing equation of the multi-physical mechanics problem, a successive two-stage iterative computational
scheme is developed as an integral part of the semi-analytical framework.
Considering the aim of this paper, we have limited the scope to 2D lattices with dierent bending and
stretching-dominated periodic congurations (as shown in gure 1(b, g)) to demonstrate the concept
of post-manufacturing contactless active mechanical property modulation. Extension of the 2D lattice
framework into 3D lattices can be readily performed by considering the same HMS beam model and
appropriate 3D unit cells with appropriate boundary conditions (for example, refer to [90]).
Within the developed semi-analytical framework, we rst investigate the eect of external magnetic
eld in combination with dierent modes of remote mechanical stress on the non-linear eective elastic
moduli of the architected hexagonal HMS beam network having uniform residual magnetic ux den-
sity. Based on the observations along with the kinematics and kinetics of the HMS beam elements, we
have proposed two physics-informed beam-level designs of residual magnetic ux density for the hexag-
onal HMS beam network, leading to enhanced eciency of the magnetic eld. Further to demonstrate
the generality of the proposed multi-physical mechanics-based framework, dierent other HMS beam
based lattice geometries, namely, auxetic, rectangular brick, rhombic, triangular, and rectangular con-
gurations are investigated considering uniform residual magnetic ux density. Before presenting the
numerical results, the developed semi-analytical framework has been thoroughly validated to ascertain
adequate condence, considering (1) HMS beam-level deformation under mechanical and magnetic actu-
ation (note that the lattice-level homogenized mechanical behavior depends on beam-level deformation
physics), (2) eective elastic moduli of dierent lattice geometries considering the conventional linear
regime, and (3) eective nonlinear elastic moduli of hexagonal lattices under large deformation. Such
multi-level validations at the beam and lattice level considering the linear and non-linear deformation
85

regimes along with multi-physical loading conditions provide adequate condence in the proposed com-
putational framework. A full-scale nite element modelling can be carried out to compare the current
results. But considering the complexity of modelling such HMS beam-based lattices in the nite element
framework, it is beyond the scope of this manuscript. Further, a detailed nite element model of the
lattice is also not strictly necessary considering the extensive multi-level validation approach adopted
for the proposed computational framework.
For the hexagonal HMS beam network with the uniform residual magnetic ux density, the maximum
enhancements in the non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
under the compressive
normal modes and anti-clockwise shear mode of the mechanical stress in combination with the magnetic
eld are achieved as 225.5%, 232.8%, 233.7%, 35.1%, and 339.6% respectively compared to the only
mechanical loading condition without any magnetic eld. Under the same combined loading conditions,
the maximum reductions in the ve elastic moduli are observed to be 84%, 29.8%, 83.9%, 129.4%, and
56.8% respectively. Whereas, under the tensile modes of normal stress and the clockwise mode of shear
stress in combination with the magnetic eld, 189.1%, 449.2%, 232.6%, 55%, and 463.4% enhancements
and 72.1%, 21.9%, 83.1%, 39%, and 68.4% reductions in the ve elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
are achieved respectively.
The eectiveness of on-demand elasticity modulation can further be enhanced through beam-level
spatially-varying architectures of the residual magnetic ux density. For the hexagonal HMS beam
network with the rst set of designed residual magnetic ux density, 56%, 11%, 42.4%, 27.5%, and 50.9%
enhancements in the non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
are achieved respectively
compared to the uniform magnetization under the compressive modes of normal stress and anti-clockwise
mode of shear stress in combination with the external magnetic eld. Whereas, the maximum reductions
in the non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
under the compressive normal modes
and the anti-clockwise shear mode of the mechanical stress in combination with the magnetic eld are
found to be 66.4%, 21%, 47.2%, 18%, and 35.6% respectively. Under the tensile modes of the normal
stress and the clockwise mode of shear stress in combination with the external magnetic eld, 31.1%,
22.7%, 41%, 10.4%, and 104.3% enhancements and 62.6%, 23.3%, 46.6%, 7.3%, and 80.4% reductions
in the ve elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
of the hexagonal HMS beam network with the rst
designed set of residual magnetic ux density are obtained respectively.
For the hexagonal HMS beam network with the second set of design (beam-level spatial variation) of
the residual magnetic ux density under the compressive modes of normal stress and the anti-clockwise
mode of shear stress in combination with the magnetic eld, maximum 86.9% reduction in
¯
E1
and max-
imum 143.5%, 23%, 68.5%, and 68.9% enhancements in
ν12
,
¯
E2
,
ν21
, and
¯
G12
are achieved respectively
86

with respect to uniform designs. Whereas, under the tensile normal modes and the clockwise shear
mode of the mechanical stress in combination with the magnetic eld, maximum 63.9%, 73.9%, 63.3%,
and 35.8% reductions in
¯
E1
,
ν12
,
¯
E2
, and
ν21
and maximum 57.5% enhancement in
¯
G12
are achieved
respectively. It is worthy to mention that we have explored here two dierent classes of architectures
for spatially varying residual ux density, while there exist a vast scope of further optimization follow-
ing single and multi-objective optimization algorithms along with unit cell geometry for enhancing the
eectiveness of broad-band elasticity modulation.
For the auxetic HMS beam network with the uniform residual magnetic ux density, the maximum
enhancements in the non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
, and
¯
G12
are achieved to be 201.9%,
19%, 400.4%, and 248.3% respectively compared to the only mechanical loading condition. Whereas,
maximum 46.4%, 66.49%, 40%, and 62.7% reductions are obtained in the non-dimensional elastic moduli
¯
E1
,
¯
E2
,
ν21
, and
¯
G12
respectively. For the rectangular brick HMS beam network with the uniform
residual magnetic ux density, maximum 64.4%, 150%, and 162.1% enhancements are achieved in
¯
E1
,
¯
E2
, and
¯
G12
respectively compared to the only mechanical loading condition. Whereas, the maximum
reductions in
¯
E1
,
¯
E2
,
ν21
, and
¯
G12
are obtained to be 32%, 54.5%, 91.7%, and 48.5% respectively. As
ν12
is zero for rectangular brick lattices under the only mechanical load in absence of the magnetic eld,
the enhancement and reduction in it are noted in terms of their absolute values instead of percentage
and they are 240.4 and 109.3 receptively.
For the rhombic HMS beam network with the uniform residual magnetic ux density, maximum
233%, 36.8%, 232.7%, and 77.6% enhancements in the non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
,
and
¯
G12
are obtained respectively compared to the only mechanical loading condition. Whereas, the
maximum reductions in the non-dimensional elastic moduli
¯
E1
,
¯
E2
,
ν21
, and
¯
G12
are achieved to be
58%, 60.2%, 37%, and 36.6% respectively. For the triangular HMS beam network with the uniform
residual magnetic ux density, the maximum enhancements in non-dimensional elastic moduli
¯
E1
,
ν12
,
¯
E2
,
ν21
, and
¯
G12
are achieved to be 14.1%, 27.5%, 44.5%, 865.5%, and 154% respectively compared to
the only mechanical loading condition in absence of magnetic eld. Whereas, maximum 11.6%, 27.6%,
32%, 1523.5%, and 65.8% reductions in the non-dimensional elastic moduli are obtained respectively.
For rectangular HMS beam network with the uniform residual magnetic ux density, maximum 111.1%,
66.7%, and 102.1% enhancements in the non-dimensional elastic moduli
¯
E1
,
¯
E2
, and
¯
G12
are obtained
respectively compared to the only mechanical condition. Whereas, the maximum reductions in the
elastic moduli are found to be 38.8%, 28.6%, and 50% respectively. As the Poisson's ratios
ν12
and
ν21
are zero for rectangular lattices under only mechanical load, their enhancements and reductions under
magnetic eld are expressed by absolute values instead of percentage values, and they are 0.1 and 13.1
87

in enhancement and 0.2 and 30.7 in reduction respectively.
The numerical investigations on the eective elastic moduli of the HMS beam networks depict an
excellent modulation capability of the elastic properties in an extremely wide band for the proposed
novel class of lightweight lattice metamaterials through designing the beam-level distribution of residual
magnetic ux density, unit cell geometry and nonlinear coupled material physics, along with controlling
the external magnetic eld in combination with the mechanical mode of loading. The numerical results
exhibit non-invariant elastic properties [91] of the periodic HMS beam networks under the anti-clockwise
and clockwise modes of shear stress in addition to the tensile and compressive modes of normal stress.
Moreover, under certain combinations of the externally applied mechanical stress and magnetic eld
depending on the residual magnetic ux density, it is possible to achieve negative stiness and negative
Poisson's ratio with dierent degrees of auxecity, even for the non-auxetic unit cell congurations. The
reported numerical results would provide a foundation for more innovative designs of the residual mag-
netic ux density of the HMS beam elements along with the interactive inuence of unit cell geometry
to increase the spectrum of modulated eective elastic properties.
In this paper, we have considered dierent modes of far-eld in-plane mechanical stresses (normal
stress along the horizontal and vertical direction (direction-1 and 2) and shear stress in plane 1-2) in
combination with remote magnetic eld along direction-2. It can be noted that there are three aspects
of magnetic stimuli in the context of the proposed active metamaterials (1) distribution of residual
magnetic ux density along the length of the constituting beams that form a unit cell, leading to
beam-level magnetic particle distribution architecture, (2) direction of the externally applied magnetic
eld, and (3) intensity of externally applied magnetic eld. In the analysis of the multi-physical large
deformation mechanics of HMS beam representing the generalized member of periodic HMS beam
networks under the combined mechanical and magnetic loading as presented in subsection 2.1 and
subsection 2.2, generalized direction (inclination angle
α
) of the externally applied magnetic eld
Ba
is considered in combination with the generalized mechanical force. Hence, the multi-scale framework
estimating the non-linear elastic properties of the proposed HMS metamaterials under the far-eld
mechanical and magnetic elds is generalized for considering any arbitrary direction of the external
magnetic eld in combination with the dierent modes of the in-plane mechanical stresses. Though
we have concentrated on the remote magnetic eld along direction-2 considering dierent intensities in
combination with normal and shear modes of the in-plane mechanical stresses, the framework can easily
be extended to consider other directions of magnetic elds. In fact, this will give a scope of achieving
tunable normal-shear lattice level coupling behavior for a given bi-level designed lattice architecture just
by changing the direction of external magnetic eld [92]. The eect of intensity of externally applied
88

magnetic eld is investigated throughout the presented results for multi-physical property modulation
of lattices, while the beam-level architecture based on the distribution of residual magnetic ux density
is explored in subsection subsection 3.5.
We would conclude this section by highlighting, summarizing and justifying some of the keywords and
concepts of the presented research, as reected in the discussions throughout this paper. (1)
Metamate-
rials
: The work deals with the development of a new class of mechanical metamaterials conceptualized
as a periodic network of hard magnetic soft beams that can change their properties in real-time based
on external stimuli. (2)
Magneto-active
: The proposed novel class of metamaterials under consideration
is magneto-active because their mechanical properties can be actively altered by applying an external
magnetic eld. The title includes this term to signify the magneto-mechanical interaction that under-
pins the unique homogenized behavior and active eective elastic moduli of these metamaterials. (3)
Nonlinear
: The metamaterials' homogenized constitutive response under the combined mechanical and
magnetic elds is non-linear due to geometric nonlinearity coming from the large deformation of the
beam-like soft cell walls and material nonlinearity of the considered materials. (4)
Bi-level architected
:
The paper introduces the concept of bi-level modulation of the eective elastic properties of the novel
class of metamaterials under the far-eld combined mechanical stress and magnetic stimuli, where the
design incorporates both the unit cell periodic geometries, and the deformation physics of the beam-like
members based on the hard magnetic particle distribution patterns within the soft cell walls. This
term in the title refers to this dual-level design approach, integrating geometric and multi-physical as-
pects (both at unit cell level and beam level) to control the eective lattice-level material behavior.
(5)
Multi-physically programmable
: The paper discusses the ability to actively modulate the physical
properties of metamaterials, such as elastic moduli and Poisson's ratios, through contactless far-eld
stimuli (magnetic eld). This shows that the metamaterials can be programmed post-manufacturing
to exhibit dierent mechanical behaviors depending on external stimuli as per application-specic op-
erational demands. The term multi-physical highlights the fact that active on-demand elastic moduli
tailoring is achieved here through dierent physics involving mechanical and magnetic deformations.
(6)
Stimuli-responsive
: The work emphasizes the stimuli-responsive nature of the metamaterials, where
the mechanical properties change in response to external magnetic elds and mechanical stresses. This
term reects the adaptability of the metamaterials to dierent external stimuli, which is a key focus
of the paper. (7)
Multi-scale mechanics
: The research focuses on the development of a multi-physical
mechanics-based framework for the estimations and modulations of the homogenized mechanical prop-
erties of the proposed metamaterials considering geometric and material non-linearities due to large
deformation and magneto-mechanical coupling. The developed computational framework involves the
89

deformation mechanics of hard magnetic soft beams and subsequent integration of that in the unit cell
mechanics to obtain the homogenized mechanical behavior of the lattices. In essence, it may be noted
that the computational mechanics framework developed here entails components and understanding at
dierent length scales (i.e. multi-scale) to obtain the eective elastic properties: hard magnetic parti-
cles and their distribution at the beam level (i.e. beam-level architecture), unit cell geomety, eective
material properties (i.e. the eective elastic moduli) at continuum level and subsequently design of
structures (such as an aircraft) based on such continuum level eective elastic properties.
5. Conclusions
The current work addresses a critical limitation in conventional mechanical metamaterials in terms of
contactless broad-band programming of elastic moduli based on on-demand operational requirements.
This is achieved through shifting the design paradigm towards more innovative bi-level modulation
concepts involving the coupled design space of unit cell geometries, architected beam-like members and
their stimuli-responsive deformation physics. We have introduced graded hard magnetic soft (HMS)
material architectures in the periodic beam networks following physics-informed insights of the stress
resultants depending on uni cell geometry. The compound eect of spatially-graded residual magnetic
ux density and unit cell geometries lead to improved stimuli eciency in achieving a target on-demand
stiness, resulting in programmable and sustainable metamaterials with minimal utilization of the
intrinsic materials. Moreover, under certain combinations of the externally applied mechanical stress
and magnetic eld depending on the residual magnetic ux density, it is possible to achieve negative
stiness and negative Poisson's ratio with dierent degrees of auxecity, even for the non-auxetic unit
cell congurations. A generic semi-analytical computational framework involving the large-deformation
geometric non-linearity and material non-linearity under magneto-mechanical coupling is developed
here for the eective elastic moduli of HMS material based bi-level architected lattices under normal
or shear modes of mechanical far-eld stresses. Eective elastic moduli being a critically fundamental
property of materials, the capability of having extreme-broadband active control would essentially lead
to on-demand programming of a range of static, stability and dynamic structural behavior, including
direction-dependent deformation, vibration and control, wave propagation, impact and penetration
resistance, energy absorption, shape morphing, robotic motion and actuation at multiple length scales.
Data availability
All data sets used to generate the results are available in the paper. Further details could be obtained
from the corresponding author upon reasonable request.
90

Acknowledgements
TM would like to acknowledge the initiation grant received from University of Southampton during
the period of this research work.
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