Constitutive behavior of asymmetric multi-material honeycombs with bi-level variably-thickened composite architecture

Abstract

Bi-level tailoring of cellular metamaterials involving a dual design space of unit cell and elementary beam level architectures has recently gained traction for the ability to achieve extreme elastic constitutive properties along with modulating multi-functional mechanical behavior in an unprecedented way. This article proposes an efficient analytical approach for the accurate evaluation of all constitutive elastic constants of asymmetric multi-material variably-thickened hexagonal lattices by considering the combined effect of bending, stretching, and shearing deformations of cell walls along with their rigid rotation. A tri-member unit cell is conceptualized, wherein all nine constitutive constants are obtained through the mechanics under one cell wall direction and subsequent repetitive coordinate transformations. We enhance the design space of lattice metamaterials substantially here by introducing multiple exploitable dimensions such as asymmetric geometry, multi-material unit cells and variably-thickened cell walls, wherein the conventional monomaterial auxetic and non-auxetic hexagonal configurations can be analyzed as special cases along with other symmetric and asymmetric lattices such as a range of rectangular and rhombic geometries. The generic analytical approach along with extensive numerical results presented in this paper opens up new avenues for efficient optimized design of the next-generation multi-functional lattices and cellular metamaterials with highly tailored effective elastic properties.

Full text

Constitutive behavior of asymmetric multi-material honeycombs with
bi-level variably-thickened composite architecture
M. Awasthi
a
, S. Naskar
b
, A. Singh
a
, T. Mukhopadhyay
b,
∗
a
Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India
b
Faculty of Engineering and Physical Sciences, University of Southampton, Southampton, UK
Abstract
Bi-level tailoring of cellular metamaterials involving a dual design space of unit cell and elementary
beam level architectures has recently gained traction for the ability to achieve extreme elastic con-
stitutive properties along with modulating multi-functional mechanical behavior in an unprecedented
way. This article proposes an ecient analytical approach for the accurate evaluation of all constitutive
elastic constants of asymmetric multi-material variably-thickened hexagonal lattices by considering the
combined eect of bending, stretching, and shearing deformations of cell walls along with their rigid
rotation. A tri-member unit cell is conceptualized, wherein all nine constitutive constants are obtained
through the mechanics under one cell wall direction and subsequent repetitive coordinate transforma-
tions. We enhance the design space of lattice metamaterials substantially here by introducing multiple
exploitable dimensions such as asymmetric geometry, multi-material unit cells and variably-thickened
cell walls, wherein the conventional monomaterial auxetic and non-auxetic hexagonal congurations
can be analyzed as special cases along with other symmetric and asymmetric lattices such as a range of
rectangular and rhombic geometries. The generic analytical approach along with extensive numerical
results presented in this paper opens up new avenues for ecient optimized design of the next-generation
multi-functional lattices and cellular metamaterials with highly tailored eective elastic properties.
Keywords:
Asymmetric honeycombs; Multi-material lattices; Programmable elastic moduli; Cellular
metamaterials; Auxetic and non-auxetic lattices; Lattice derivatives
1. Introduction
Metamaterials deal with global mechanical properties with periodic micro-structural design and
intrinsic material distribution at the micro-scale, wherein the interplay between geometry and mechanics
can bring about wonders in terms of eective physical properties [1]. Over the last decade, this eld of
architected materials has received tremendous attention from the research community for the prospect
of achieving unprecedented mechanical properties beyond the limits of conventional materials along
with tunable multi-functional abilities such as extreme eective elastic moduli and stiness [2, 3, 4,
5, 6, 7], negative Poisson's ratio [8, 9, 10, 11, 12], programmable constitutive behavior [13, 14, 15,
16], high specic energy absorption capability [17, 18, 19, 20, 21], active modulation of mechanical
∗
Corresponding author: T. Mukhopadhyay (Email address: t.mukhopadhyay@soton.ac.uk)
Preprint submitted to Elsevier June 30, 2024

properties [1, 22, 23, 24], shape morphing [25, 26, 27, 28, 29, 30], unusual acoustic parameters (for
example, negative refractive index) [31], energy harvesting [32, 33], vibration control [34, 35, 36] etc.
In practical design scenarios, most of these metamaterials need to possess multifunctionalities such as
specic energy absorption, specic strength and stiness, stability and dynamic characteristics. Many
of these properties are also interrelated; for example, the dynamic and stability characteristics are
dependent on dierent stiness components, while the specic energy absorption can be evaluated
based on the force-displacement or stress-strain constitutive behavior. For these reasons, we have
briey discussed some of the other relevant features of mechanical metamaterials as a part of the
literature review. Subsequently, we focus more on the aspect of elastic moduli and the constitutive
behavior of mechanical metamaterials. Concerning the design of metamaterials in general, due to
intense investigation of dierent unit cell level geometries over the last few years, the aspect of solely
periodic unit cell design for lattices have started to saturate. There exists a strong rationale to introduce
new dimensions in lattice metamaterial designs, which is the central focus of this paper.
Articially engineered materials with the ability of possessing passive and active programmability
in mechanical behavior to an extreme extent, along with the recent advancement in additive man-
ufacturing, have provided us with tremendous opportunities to enhance the performance of multiple
technologically demanding engineering sectors like aerospace, mechanical, civil and biomedical indus-
tries [1]. In the following discussion, we provide a very brief review of relevant research activities in the
eld of lattice metamaterials that primarily deal with eective elastic properties. Analytical expressions
for the eective elastic moduli of regular hexagonal lattices (refer to Figure 1(a)) were derived based
on a unit cell approach considering periodic boundary conditions [37]. In such honeycomb structures,
unusual eective negative Poisson's ratio (i.e. auxetic behavior), or zero Poisson's ratio can be obtained
by intuitively modifying the hexagonal unit cell architecture [38]. Unit cell based periodic lattices have
been proposed to have extreme specic stiness based on anti-curvature lattices [39, 40] and topology-
optimized cell walls [5], non-invariant elastic properties [41] and on-demand elastic property modulation
[22, 42, 43, 44]. Eective elastic properties of monomaterial asymmetric honeycombs were investigated
in [45, 46]. Formulations presented in the current work are inuenced by these papers, wherein mul-
tiple new exploitable dimensions would be introduced. Thickened joints in convenstional hexagonal
unit cell were shown to provide more strength based on optimized geometries [47]. The utilization
of buckling-induced instability has led to an enhanced energy absorption capacity and strain rate de-
pendent constitutive behavior [48]. Wei et al. [49] have proposed a lightweight cellular metastructure
incorporating coupled tailorable thermal expansion and tunable Poisson's ratio. A theoretical and nu-
2

merical investigation has been proposed by Wang et al. [50] for a hybrid hierarchical square honeycomb
to investigate eective in-plane elastic properties and their tailorable features. Ling et al. [51] have
presented an additively manufactured mechanical metamaterial with large magnitude of Poisson's ratio.
Further, apart from static properties, a range of unprecedented mechanical properties can be obtained
in metamaterials under dynamics conditions, for example, negative Young's modulus [6], negative shear
modulus [52], negative mass density [53], elastic cloaks [54] and negative bulk modulus [55]. It may be
noted here that analyzing lattice-like structural forms is relevant and useful to multiple length scales
(nano to macro) covering naturally-occurring and articial materials and structures [56, 57, 58, 59].
From the existing literature, it becomes evident that physical properties of intrinsic materials and
metamaterials are dened at two dierent length scales. Intrinsic materials properties of constituent
elements at the micro-scale generally depend upon chemical composition, atomic and molecular struc-
tures of the material. Another set of material properties related to equivalent macro-scale behavior can
be dened in metamaterials where the macroscale eective properties become a function of the intrinsic
material properties and the microstructural lattice geometry. The key factor for a metamaterial remains
that the relative length scales of the microstructural dimension of the periodic units and the overall
lattice dimensions should have a signicant dierence. However, the interplay of mechanical properties
of the intrinsic materials, geometry and the concerning mechanics can oer unparalleled opportunities
for novel material design. Generally, a unit cell-based approach is adopted to design lattice metama-
terials with a single type of intrinsic material [37] (refer to Figure 1(a, d)). Recently, multi-material
lattices have been proposed where dierent cell walls in the unit cells have dierent intrinsic material
properties (refer to Figure 1(b)) [60]. With the progress of multi-material additive manufacturing, such
lattices can be physically realized with adequate precision [61, 62, 63, 64, 65]. In the present study, to
expand the design space further, we would introduce asymmetry in the unit cell architecture of multi-
material lattices as shown in Figure 1(c). In such periodic bending-dominated architectures, we notice
that the unit cell level periodic boundary conditions can be satised by considering the cell wall (i.e.
the connecting beams) to have rotationally restrained supports at the two ends (refer to [41, 66]). In
such a scenario, the bending moment along the beam length reduces as we move away from the two
ends of a beam, and becomes zero at the mid-point. To exploit such insights in optimizing material
utilization, we would further introduce beam-level variably- thickened architectures along with unit cell
geometries (which we refer to as bi-level design), indicated in Figure 3(a). Thus, the salient features of
the current work can be summarized as the compound eect of multi-material unit cells, asymmetric
unit cell geometry, and beam-level variably-thickened architecture. This work aims to derive an e-
3

cient and insightful analytical approach for obtaining the eective elastic properties involving the entire
constitutive matrices of such architected composite lattices.
To the best knowledge of the authors, no study has been presented to date that deals with the
analytical formulations concerning entire constitutive matrices for asymmetric multi-material variably-
thickened honeycomb structures. Based on the discussions presented in the preceding paragraphs, it
becomes evident that the development of an analytical solution for accurately and eciently determin-
ing the mechanical properties including bending, stretching, and shearing deformations of cell walls
along with their rigid rotation would be crucial for achieving multifunctional properties and further
muli-objective design optimizations. In this article, we will primarily focus on hexagonal asymmetric
geometries, the special cases of which can be readily deduced as conventional non-auxetic and auxetic
hexagonal lattices, rhombic and rectangular lattices with dierent architectures (refer to Figure 1(g)).
We would follow a bottom-up approach for analyzing the eective elastic properties of lattice meta-
materials here. The local deformation behavior of the elementary beams is accounted for in the analysis
considering bending, shear and axial deformations. Note that we propose to introduce beam-level archi-
tectures at this stage, as shown later in Fig. 3(b). Subsequently, a unit cell, comprised of multi-material
variably-thickened beam elements, is analyzed for deriving the closed-form expressions of the eective
in-plane elastic moduli. We introduce the eect of asymmetry at the unit cell level. At a higher
length scale, such lattices are adopted for structural applications depending on their eective elastic
constitutive behavior (refer to Figure 1(a-f)).
The overarching contribution of this paper lies in enhancing the conventional design space of lattice
metamaterials substantially by introducing multiple exploitable dimensions such as (1) asymmetric
geometry, (2) composite multi-material unit cells, (3) variably-thickened cell walls (derived through
physics-based rationale) and (4) their coupled eect, wherein the traditional monomaterial auxetic and
non-auxetic congurations (symmetric and asymmetric) can be analyzed as special cases. Hereafter
the paper is organized as follows. In section 2, the generalized analytical formulation is presented for
determining the equivalent material properties of multi-material variably-thickened asymmetric lattices.
We have derived all elements of the elastic constitutive matrices, unlike most reported studies in this
eld. Subsequently, we show how to obtain the elastic moduli in dierent directions from the elastic
constitutive matrices. The section 3 is dedicated to extensive validation considering a range of material
distribution and unit cell architectures, followed by a detailed numerical investigation concerning the
geometrical parameters of the bi-level architecture and multi-material properties. Finally, the most
interesting ndings of the present study are summarized in the section 4 along with concluding remarks.
4

Figure 1: Bottom-up multi-scale approach for asymmetric multimaterial lattices. (a-b)
Regular mono
and multi-material hexagonal lattices with symmetric geometry.
(c)
Asymmetric hexagonal multi-material lattices with
dierent intrinsic materials in the three constituent cell walls.
(d, f)
A symmetric mono-material honeycomb unit cell is
shown in (d). The unit cell for multimaterial asymmetric lattices is shown in (f).
(e)
Prospective application of honeycomb
lattices in an aircraft following a multi-scale framework (beam-level architecture to unit cell geometries and equivalent
metamaterial properties, and subsequently structural applications).
(g)
Lattice derivatives that can be obtained from the
generic asymmetric hexagonal lattice shown in subgure (c),
(i)
Normal auxetic lattice,
(ii)
Asymmetric auxetic lattice,
(iii)
Normal rhombic lattice,
(iv)
Asymmetric rhombic lattice,
(v)
Normal Parallelogram lattice,
(vi)
Normal square
lattice,
(vii)
Normal rectangular lattice,
(viii)
Symmetric brick lattice,
(ix)
Asymmetric brick lattice.
5

2. Derivation of the full constitutive matrices
2.1. Closed-form expressions for elastic moduli
C12
,
C22
and
C32
A typical unit cell of the proposed asymmetric multimaterial hexagonal honeycombs is shown in
Fig. 2(a). In this section, the expressions of three elastic moduli have been derived by keeping one cell
wall parallel to the
y
-axis, as shown in Fig. 2(b) representing the schematic hexagonal geometry with
an edge parallel to the
y
-axis [45, 46]. The considered model has two opposite edges that have the same
length and are also parallel. The
γ12
,
γ23
, and,
γ31
represent internal angles between the hexagon edges,
as shown in Fig. 2(a).
γ12 +γ23 +γ31 = 2 π
(1)
Here
θ1
,
θ2
and
θ3
represent the angle of edges
L1
,
L2
and
L3
with the
y
-axis. For the present case, the
angle
θ1
will be zero. Subsequently the
θ2
and
θ3
can be expressed in term of
γ12
and
γ13
as,
θ2=π−γ12, θ3=π−γ13
(2)
where
γ12
is the internal angle between constitutive edges
L1
and
L2
, and
γ13
is the internal angle
between constitutive edges
L1
and
L3
.
The relationship of the relative displacement with axial strain
εx
,
εy
and with the shear strain
εxy
can be expressed as


δix
δiy

=

εxεxy +ωxy
εxy −ωxy εy



Lix
Liy


(3)
In the above Eq. 3, the
δix
and
δiy
represent the deection for a particular cell element, due to applied
stress,
σy
in the
y
direction.
Lix
and
Liy
, represent the length of particular a unit cell wall along the
x
and
y
-axis, respectively.
The main objective of the study is to obtain the eective in-plane elastic modulus for multi-material
asymmetric hexagonal structures with variably-thickened cell walls. Hence, It is considered that the
unit cell is heterogeneous in nature and the cell walls are made up of dierent materials having elastic
modulus of
E1
,
E2
, and
E3
for walls 1, 2 and 3 respectively (refer to Fig. 2(a)). The relation of
strains
εx
,
εy
and
εxy
with the stresses
σx
,
σy
and
τxy
in term of equivalent elastic constitutive elements
is given by





εx
εy
εxy





=




C11 C12 C13
C21 C22 C23
C31 C32 C33










σx
σy
τxy





(4)
In present case, stress
σy
is only applied in
y
-direction, therefore
σx
=
τxy
=0. Then from the above Eq. 4,
6

Figure 2: Geometrical details of asymmetric hexagonal honeycombs. (a)
Unit cell of asymmetric multimaterial
hexagonal honeycombs.
(b)
Geometry of hexagonal asymmetric honeycomb unit cell with one edge parallel to y-axis.
(c,
d)
Mechanics of multi-material honeycomb cell walls subjected to stress
σy
in
y
-direction.
(e)
Stress resultants of multi-
material honeycomb cell walls subjected to stress
σy
in
y
-direction.
(f)
Displacements of the cell walls of a unit cell of
asymmetric honeycomb lattices.
7

the expressions for
Cij
can be obtained.
C12 =εx
σy
, C22 =εy
σy
, C32 =εxy
σy
.
(5)
For obtaining the values of
C12
,
C22
and
C32
, given in Eq. 5, we need to obtain the values of
εx
,
εy
and
εxy
for applied stress
σy
along
y
-direction. Subsequently, to calculate strains
εx
,
εy
and
εxy
the deections
of each constituent element of the unit cell must be obtained rst. In the following three sections, we
present three dierent approaches to determining the strain eld (
εx
,
εy
and
εxy
) in terms of cell wall
end deections (
δix
and
δiy
,
i= 1,2,3
). While we discuss all three approaches here for a comprehensive
understanding, any one of these three approaches can be adopted to establish a relationship between
the strain eld and cell wall end deections.
2.1.1. Evaluation of strain eld under the application of
σy
Here we present three dierent approaches for obtaining strain eld under the application of
σy
,
among which any one can be adopted for further calculations [45, 46]. It is veried that the results
obtained by these three approaches agree well with each other, providing a sense of condence in the
presented analytical framework.
2.1.1.1. Approach 1 for obtaining the strain eld
Overaker et al. [67] proposed a novel method to obtain the equivalent strain eld that satises the
displacements of each cell wall. The displacements
δix
and
δiy
, which are due to rigid body displacements
u0
and
v0
, rigid body rotation
ωxy
and uniform strain elds
εx
and
εy
, can be expressed as
δix =xiεx+yiεxy +yiωxy +u0,
δiy =yiεy+xiεxy −xiωxy +v0.
(6)
Here
δix
and
δiy
represent the
i
-th cell wall end deections with co-ordinates
xi
and
yi
(
i= 1,2,3
). After
obtaining deections along
x
and
y
-directions, unknown constants
εx
,
εy
,
εxy
,
ωxy
,
u0
and
v0
can be
evaluated by using the Eq. 6. Further, using these obtained strain elds, the equivalent elastic moduli
C12
,
C22
,
C32
can be obtained by using Eq. 5.
2.1.1.2. Approach 2 for obtaining the strain eld
The strain eld of the honeycomb structure can also be obtained by evaluating relative displacements
between the unit cells of the honeycomb. To demonstrate this method more, a part of the honeycomb
structure having three unit cells, shown in Fig. 2(c), is considered, which are denoted by unit 1, unit
2 and unit 3. The arrangement of these unit cells is anti-clockwise, and
O1
,
O2
and
O3
represent the
8

cell walls junction of each unit cell. Here
U21
and
V21
represent the
x
-direction and
y
-direction relative
displacement of
O2
with respect to
O1
, respectively, as shown in Fig. 2(d). Similarly,
U31
and
V31
represent the
x
-direction and
y
-direction relative displacement of
O3
with respect to
O1
, respectively,
as shown in Fig. 2. These relative displacements (
U21
,
V21
, and
U31
,
V31
) of
O2
and
O3
with respect to
O1
depend upon the strain eld caused by the application of stress
σy
in
y
-direction and given by,








U21
V21
U31
V31








=








L2x0−L2y−L2y
0−L2yL2x−L2x
L3x0L3yL3y
0L3yL3xL3x
















εx
εy
εxy
ωxy








(7)
Here,
L2x
and
L2y
are distances between
O2
and
O1
along
x
-direction and
y
-direction, respectively.
Similarly,
L3x
and
L3y
are distances between
O3
and
O1
along
x
-direction and
y
-direction, respectively.
These distance constants (
L2x
,
L3x
,
L2y
and
L3y
) are given by
L2x=L2sinθ2+L3sinθ3,
L2y=L2cosθ2−L3cosθ3,
L3x=L3sinθ3,
L3y=L1+L3cosθ3.
(8)
After obtaining,
L2x
,
L3x
,
L2y
and
L3y
, strain elds
εx
,
εy
,
εxy
,
ωxy
can be represented in terms of
L2x
,
L3x
,
L2y
and
L3y
by using Eq. 7 as,
εx=L3yU21 +L2yU31
(L2yL3x+L2xL3y),
εy=−L3xV21 +L2xV31
(L2yL3x+L2xL3y),
εxy =−L3xU21 +L3yV21 +L2xU31 +L2yV31
2(L2yL3x+L2xL3y),
ωxy =L3xU21 −L3yV21 +L2xU31 −L2yV31
2(L2yL3x+L2xL3y).
(9)
The relative displacements between unit cells can be represented in terms of the relative displacement
between junctions and the wall ends of each adjacent unit. To calculate the relative displacements, three
points `a', `b' and `c' are considered on unit cells 1, 2 and 3, respectively, as shown in Fig. 2(d). Point
`a' is considered on unit 1 at the end of wall 3 and Point `b' is considered on unit 2 at the end of wall
2. Similarly, Point `c' is on unit 3 at the end of wall 1. The relative displacements of point `a' with
respect to
O1
along
x
direction (
Ua1
) and along
y
direction (
Va1
) are equal to cell wall deformation due
9

to applied stress
σy
in
y
-direction,
Ua1=δ3x,
Va1=δ3y.
(10)
The displacements of point `b' (
Ub1
,
Vb1
) and point `c' (
Uc1
,
Vc1
) with respect to point
O1
in the respective
x
and
y
-directions can be calculated by adding the respective displacement between the units to the
displacements due to the cell wall deformation itself, as presented in Eq.11.
Ub1=δ2x+U21,
Vb1=δ2y+V21,
Uc1=δ1x+U31,
Vc1=δ1y+V31.
(11)
As shown in Fig. 2(d), the points `a', `b' and `c' are the same point which is located at the same
position (at the junction of unit 1, 2 and 3). Hence, the displacements of points `a', `b' and `c' after
the deformation will be the same and hold the conditions
UA1
=
UB1
=
UC1
and
VA1
=
VB1
=
VC1
. Based on
these conditions, the relative displacements of
O2
(
U21
,
V21
) and
O3
(
U31
,
V31
)) with respect to
O1
along
x
and
y
-directions can be expressed as,
U21 =δ3x−δ2x,
V21 =δ3y−δ2y,
U31 =δ3x−δ1x,
V31 =δ3y−δ1y.
(12)
Then, strain elds
εx
,
εy
,
εxy
,
ωxy
can be obtained in closed-form by substituting Eq.11 and 12 into Eq. 9.
2.1.1.3. Approach 3 for obtaining the strain eld
Here we present another approach of obtaining
εx
,
εy
and
εxy
in terms of
δix
and
δiy
under applied
stress
σy
. Fig. 3(c) and (d) represent unit cells of the considered asymmetric honeycomb lattice.
Depending on computational convenience any one form of these two unit cells can be considered with
appropriate periodic boundary conditions. The unit cell model of honeycomb lattice, as shown in Fig.
3(d), is considered here to formulate the expressions for the strain eld. Strains have been formulated
by the calculation of relative displacement of the constitutive cell wall elements,
OE
,
AO
, and
OD
, and
which are made of intrinsic materials elastic moduli of
E1
,
E2
, and
E3
respectively. Lengths of the cell
walls have been considered as
L1
,
L2
, and
L3
, respectively.
In the Fig. 3(d),
C′D
=
CE
. The
Y
-direction strain can be calculated from the amount of extension
10

in
LY
length along the
Y
-direction line
F E
.
UY
F E
, represents the relative
Y
-direction displacement of
point
F
with respect to point
E
.
ϵy=
UY
F E
LY
,
(13)
The value of
LY
in terms of known variables is displayed in Eq.14, and can be calculated using the
geometry presented in Fig. 3(d).
LY=L1+L2L3sin(θ2+θ3)
L2sinθ2+L3sinθ3
(14)
The relative displacement of
F E
is equal to the sum of the relative displacements of
F O
and
OE
, as
expressed in Eq.15.
UY
F E =UY
F O +UY
OE
(15)
Since line AD remains straight after deformation, this is represented as in Eq.16.
UY
F O =UY
AO
F O
AD +UY
DO
F A
AD =δ2yL3sinθ3+δ3yL2sinθ2
L2sinθ2+L3sinθ3
,
(16)
Considering
UY
OE =−δ1y
, the strain
ϵy
is provided in expression in Eq. 17.
ϵy=
−δ1y+δ2yL3sinθ3+δ3yL2sinθ2
L2sinθ2+L3sinθ3
LY
(17)
A similar procedure is used to calculate the strain,
εx
in the
X
-direction, which is dependent on the
X
-direction deformation of
BD
as depicted in Fig. 3(d), and the corresponding expression is given in
Eq.18..
ϵx=
UY
DB
LX
,
(18)
where
LX
is the length of the unit cell in the
X
-direction and is given in terms of known variables in
Eq.19.
LX=L2sinθ2+L3sinθ3,
(19)
The relative displacement of DB is given in the Eq.20, as a sum of the relative displacements of
DA
and
AB
.
UX
DB =UX
DA +UX
AB
UX
DA=δ3x−δ2x
(20)
For the calculation of relative distance
AB
, as shown in Eq.21, the displacement of each point along line
AC
from point
A
to point
C
is linearly distributed. The deformation should leave the line
AC
straight,
11

so the displacement of point
B
is provided below.
UX
AB =U
X
AC AB
AC
=UX
AC
L2cosθ2−L3cosθ3
LY
UX
AC =UX
AO +UX
OE +UX
EC =δ2x+0+UX
EC
(21)
The points
C′
and
D
in Fig. 3(d) are analogous to
C
and
E
, respectively.
UX
EC =UX
DC′
UX
DC′=UX
DA
DC′
AD = (δ3x−δ2x)L2sinθ2
L2sinθ2+L3sinθ3
UX
DB =δ3x−δ2x+δ2xL3sinθ3+δ3xL2sinθ2
L2sinθ2+L3sinθ3L2cosθ2−L3cosθ3
LXLY
(22)
Finally, the expression of strain,
epsilonx
, is given in Eq.23.
ϵx=δ3x−δ2x
LX
+δ2xL3sinθ3+δ3xL2sinθ2
L2sinθ2+L3sinθ3L2cosθ2−L3cosθ3
LXLY
(23)
The shear strain,
ϵxy
, can be calculated using the expressions in Eq.24.
ϵxy =1
2UX
F E
LY
+
UY
DB
LX
(24)
The relative displacement of
F E
is yielded as the sum of relative displacements of
F O
and
OE
, as
shown in Eq.25.
UX
F E =UX
F O +UX
OE =UX
AO
AF
AD +UX
DO
DF
AD + 0
=δ2xL3sinθ3
L2sinθ2+L3sinθ3+δ3xL2sinθ2
L2sinθ2+L3sinθ3
(25)
The term
UY
DB
shown on the right-hand side of Eq.24 can now be represented as shown in Eq.26.
UY
DB =UY
DA +UY
AB = (δ3y−δ2y) + UY
AC
AB
AC
where, AB
AC =L2cosθ2−L3cosθ3
LY
(26)
The expression of
UY
AC
is given below.
UY
AC =UY
AO +UY
OE+U
Y
EC
= (δ2y−δ1y) + UY
DC′
=δ2y−δ1y+UY
DA
DC′
AO =δ2y−δ1y+ (δ3y−δ2y)L2sinθ2
LX
=−δ1y+δ2yL3sinθ3+δ3yL2sinθ2
L2sinθ2+L3sinθ3
(27)
12

Eq. 28 provides the expressions for shear strain,
ϵxy
, in terms of known variables.
ϵxy =1
2δ2xL3sinθ3+δ3xL2sinθ2
LY(L2sinθ2+L3sinθ3)
+1
2δ3y−δ2y
LX
+L2cosθ2−L3cosθ3
LXLY−δ1y+δ2yL3sinθ3+δ3yL2sinθ2
L2sinθ2+L3sinθ3
(28)
Up to this point, we have formulated the expressions for
εx
,
εy
, and
εxy
in terms of
δix
and
δiy
, the
derivation of which are provided in the next section.
2.1.2. Cell wall end deections under the application of
σy
After obtaining the strain eld, as discussed in the preceding section following three dierent ap-
proaches, equivalent elastic modulus
C12
,
C22
and
C32
can be calculated from Eq. 5. But for obtaining
εx
,
εy
and
εxy
, the cell wall end deections need to be evaluated. In this section, we concentrate on
evaluating
δix
and
δiy
, which can directly be used in any one of the three approaches. Further, it may be
noted that the eects of multi-material unit cells and variably thickened cell walls would be incorporated
here.
The closed-form expressions for force, moment, and displacement are derived here under the appli-
cation of
σy
. The schematic diagram of forces and moments on each cell wall due to applied stress
σy
in
y
-direction is shown in Fig. 2(e). The expression of acting forces can be obtained from the equilibrium
condition of forces.
T1=σyh(L3sinθ3+L2sinθ2),
T2=σyhL3sinθ3,
T3=σyhL2sinθ2,
(29)
Here,
T1
,
T2
and
T3
represent the acting forces along the
y
-direction at cell wall
1
,
2
and
3
, respectively.
From the above expressions of Eq. 29, it is clear that
T1
=
T2
+
T3
. The moments that are produced due
to these forces
Ti
acting on each wall can be expressed as shown in Eq.30.
M1= 0,
M2=T2
1
2L2sinθ2=1
2σyhL2sinθ2L3sinθ3,
M3=T3
1
2L3sinθ3=1
2σyhL2sinθ2L3sinθ3
(30)
where
M2
=
M3
.
M1
,
M2
and
M3
represent the resulting moments on cell walls
1
,
2
and
3
, respectively.
The closed-form expressions are derived now for the displacements of each cell wall along the
x
and
y
directions. It is considered that the thickness of the cell walls is higher at the ends and comparatively
less at the center of the wall element (i.e. variably thickened beams), as shown in Fig. 3(a, b). Note
13

Figure 3: Beam-level variably-thickened architecture in asymmetric multimaterial lattices. (a)
Geometry of
asymmetric honeycomb hexagonal unit cell with one edge parallel to Y-axis, subjected to stress
σy
.
(b)
Geometric details
and stress resultants in a variably-thickened asymmetric unit cell.
(c)
αi
,
βi
coordinates for co-ordinate transformation.
(d)
Asymmetric unit cell for analyzing the strain eld following Approach 3.
here that we have considered the variation of thickness along the beam lengths keeping the bending
moments in mind for a both-end rotationally restrained support condition. Though we have taken the
14

stepped variation of thickness in each of the beams, continuous variation can also be considered based
on the currently proposed approach (the nal expressions for cell wall deections need to be derived
accordingly).
In a variably-thickened wall,
ηi
and
ζi
are the thickness ratio parameter and span ratio parameter,
respectively. Here,
ζi
represents the ratio of the middle part length having less thickness to the end
part's length with higher thickness, as shown in Fig 3(b). Similarly,
ηi
represents the ratio of the middle
part thickness to the end part thickness, as shown in Fig 3(b).
Based on energy method [37], the axial deection of cell wall
1
can be expressed as
δ1
axial =∂U
∂T1
, U =ZL1
0
−T1
2
2E1A1
dx
(31)
Now, the closed-form expression for axial deection of cell wall
1
having dierent thickness segments
can be expressed as,
−T1
E1h1ZL1
0
1
t1x
dx =−T1
E1h1ZL1h
ζ1
0
η1
t1h
dx +ZL1h+L1h
ζ1
L1h
ζ1
1
t1h
dx +ZL1h+2 L1h
ζ1
L1h+L1h
ζ1
1
t1h
dx,
δ1axial =−T1L1
E1h1t1h2η1+ζ1
2 + ζ1
(32)
Further, the axial deection cell wall
1
can be split into its components along the
x
and
y
direction as,
δ1x
axial = 0,
δ1y
axial =−T1L1
E1h1t1h2η1+ζ1
2 + ζ1.
(33)
It is worth noting that deection of cell wall
1
due to bending and shear will be zero because no bending
moment and shear force are acting on cell wall
1
.
Similarly, using energy method the deections of cell wall
2
and
3
can be evaluated in a closed-form
manner. Due to the presence of axial force, bending moment and shear force on cell wall
2
and
3
, the
overall deection consists of deformation due to axial, bending and shear force as,
δ2
total =δ2axial +δ2bending +δ2shear,
δ3
total =δ3axial +δ3bending +δ3shear.
(34)
15

By employing the energy method, axial deection of cell wall
2
can be expressed as
δ2axial =∂U
∂H2
, U =ZL2
0
H2
2
2E2A2
dx,
U=H2
2
2E2h2ZL2
0
1
t2x
dx =H2
2
2E2h2Z
L2h
ζ2
0
η2
t2h
dx +ZL2h+L2h
ζ2
L2h
ζ2
1
t2h
dx +ZL2h+2 L2h
ζ2
L2h+L2h
ζ2
η2
t2h
dx,
U=H2
2L2
2E2h2t2h2η2+ζ2
2 + ζ2
(35)
where,
H2=T2cos θ2
. Hence,
U=(T2cos θ2)2L2
2E2h2t2h2η2+ζ2
2 + ζ2,
δ2|axial =∂U
∂H2
=T2cos θ2L2
E2h2t2h2η2+ζ2
2 + ζ2
(36)
The axial deection of cell wall
2
, given in Eq. 36, can be split into its component along the
x
and
y
direction as
δ2x
axial =−T2L2cos θ2sin θ2
E2h2t2h2η2+ζ2
2 + ζ2,
δ2y
axial =T2L2cos2θ2
E2h2t2h2η2+ζ2
2 + ζ2,
where
, L2=L2h+L2h
ζ2
.
(37)
The energy method can also be utilized for driving closed-from expression of bending deection for
cell elements. Based on the energy method, the bending deection of cell wall
2
can be expressed as,
δ2
bending =∂U
∂V2
, U =ZL2
0
M2
2
2E2I2
dx,
where
, M2=1
2T2sin θ2x,
and
V2=T2sin θ2,
U=T2
2sin2θ2
16E2ZL2
0
x2
I2x
dx =T2
2sin2θ2
16E2Z
L2h
ζ2
0
η3
2x2
I2
dx +ZL2h+L2h
ζ2
L2h
ζ2
x2
I2
dx +ZL2h+2 L2h
ζ2
L2h+L2h
ζ2
η3
2x2
I2
dx,
U=T2
2sin2θ2
2E2h2L3
2
t3
2h8η3
2+ 9η3
2ζ2+ 3η3
2ζ2
2+ζ3
2+ 3ζ2
2+ 3ζ2
(2 + ζ2)3,
δ2
bending =∂U
∂V2
=T2sin θ2
E2h2L3
2
t3
2h8η3
2+ 9η3
2ζ2+ 3η3
2ζ2
2+ζ3
2+ 3ζ2
2+ 3ζ2
(2 + ζ2)3,
(38)
The bending deections along
x
and
y
directions can be obtained as,
δ2y
bending =T2sin2θ2
E2h2L3
2
t3
2h8η3
2+ 9η3
2ζ2+ 3η3
2ζ2
2+ζ3
2+ 3ζ2
2+ 3ζ2
(2 + ζ2)3,
δ2x
bending =T2sin θ2cos θ2
E2h2L3
2
t3
2h8η3
2+ 9η3
2ζ2+ 3η3
2ζ2
2+ζ3
2+ 3ζ2
2+ 3ζ2
(2 + ζ2)3,
(39)
16

The energy method is also utilized to obtain the deection of cell wall
2
due to shear force
V2
.
δ2
shear =∂U
∂V2
,
where
U=ZL2
0
V2
2
2G2A2
dx,
U=T2
2sin2θ2
G2h2ZL2
0
1
t2x
dx =T2
2sin2θ2
G2h2Z
L2h
ζ2
0
η2
t2h
dx +ZL2h+L2h
ζ2
L2h
ζ2
1
t2h
dx +ZL2h+2 L2h
ζ2
L2h+L2h
ζ2
η2
t2h
dx,
δ2
shear =T2L2sin θ2
G2h2t2h2η2+ζ2
2 + ζ2,
where
, V2=T2sin θ2, G2=E2
2(1 + ν2)
(40)
Now shear deection can be also split into its
y
and
x
direction components as,
δ2y
shear =T2L2sin2θ22(1 + ν2)
E2h2t2h2η2+ζ2
2 + ζ2,
δ2x
shear =T2L2sin θ2cos θ22(1 + ν2)
E2h2t2h2η2+ζ2
2 + ζ2,
(41)
In a similar way, we would calculate the deection for the third cell element. So, the nal formulation
for the deections in
x
and
y
directions for the element
3
is given below in Eq. 42.
δ3x
axial =T3L3cos θ3sin θ3
E3h3t3h2η3+ζ3
2 + ζ3,
δ3y
axial =T3L3cos2θ3
E3h3t3h2η3+ζ3
2 + ζ3,
where, L3=L3h+L3h
ζ3
.
δ3y
bending =T3sin2θ3
E3h3L3
3
t3
3h8η3
3+ 9η3
3ζ3+ 3η3
3ζ2
3+ζ3
3+ 3ζ2
3+ 3ζ3
(2 + ζ3)3,
δ3x
bending =−T3sin θ3cos θ3
E3h3L3
3
t3
3h8η3
3+ 9η3
3ζ3+ 3η3
3ζ2
3+ζ3
3+ 3ζ2
3+ 3ζ3
(2 + ζ3)3,
δ3y
shear =T3L3sin2θ32(1 + ν3)
E3h3t3h2η3+ζ3
2 + ζ3,
δ3x
shear =−T3L3sin θ3cos θ32(1 + ν3)
E3h3t3h2η3+ζ3
2 + ζ3,
where
, G3=E3
2(1 + ν3)
(42)
The total deections of unit cell walls 1, 2 and 3 along the
x
-direction due to acting axial, bending
17

and shear forces are given as,
δ1x
total = 0,
δ2x
total =−T2L2cos θ2sin θ2
E2h2t2h2η2+ζ2
2 + ζ2
+T2sin θ2cos θ2
E2h2L3
2
t3
2h8η3
2+ 9η3
2ζ2+ 3η3
2ζ2
2+ζ3
2+ 3ζ2
2+ 3ζ2
(2 + ζ2)3
+T2L2sin θ2cos θ22(1 + ν2)
E2h2t2h2η2+ζ2
2 + ζ2,
δ3x
total =T3L3cos θ3sin θ3
E3h3t3h2η3+ζ3
2 + ζ3
−T3sin θ3cos θ3
E3h3L3
3
t3
3h8η3
3+ 9η3
3ζ3+ 3η3
3ζ2
3+ζ3
3+ 3ζ2
3+ 3ζ3
(2 + ζ3)3
−T3L3sin θ3cos θ32(1 + ν3)
E3h3t3h2η3+ζ3
2 + ζ3
(43)
Similarly, the total deections of unit cell walls 1, 2 and 3 along the
y
-direction due to acting axial,
bending and shear forces are given as,
δ1y
total =−T1L1
E1h1t1h2η1+ζ1
2 + ζ1,
δ2y
total =T2L2cos2θ2
E2h2t2h2η2+ζ2
2 + ζ2
+T2sin2θ2
E2h2L3
2
t3
2h8η3
2+ 9η3
2ζ2+ 3η3
2ζ2
2+ζ3
2+ 3ζ2
2+ 3ζ2
(2 + ζ2)3
+T3L3sin2θ32(1 + ν3)
E3h3t3h2η3+ζ3
2 + ζ3,
δ3y
total =T3L3cos2θ3
E3h3t3h2η3+ζ3
2 + ζ3
+T3sin θ3cos θ3
E3h3L3
3
t3
3h8η3
3+ 9η3
3ζ3+ 3η3
3ζ2
3+ζ3
3+ 3ζ2
3+ 3ζ3
(2 + ζ3)3
+T3L3sin2θ32(1 + ν3)
E3h3t3h2η3+ζ3
2 + ζ3
(44)
2.1.2.1. Special case 1: Multimaterial asymmetric honeycombs with constant cell wall thickness
For an asymmetric multi-material honeycomb structure with a constant thickness along each consti-
tutive cell wall the value of
ηi
and
ζi
, (
i
=1, 2 and 3) are equal to unity (
ηi
=
ζi
= 1). Hence, the expression
of deections of each cell wall along
x
-direction for the asymmetric multi-material honeycomb structure
18

with constant thickness reduces to the following form,
δ1x= 0,
δ2x=T2sinθ2cosθ2
E2hL2
t23
+ 2(1 + ν2)T2L2sinθ2cosθ2
kE2ht2
−L2T2cosθ2sinθ2
E2ht2
,
δ3x=−T3sinθ3cosθ3
E3hL3
t33
−2(1 + ν3)T3L3sinθ3cosθ3
kE3ht3
+L3T3cosθ3sinθ3
E3ht3
(45)
Similarly, the expression of deections of each cell wall along
y
-direction for the asymmetric multi-
material honeycomb structure with constant thickness reduces to,
δ1y=−L1T1
E1hT1
,
δ2y=−T2sin2θ2
E2hL2
t23
+ 2(1 + ν2)T2L2cos2θ2
E2ht2
+L2T2cos2θ2
E2ht2
,
δ3y=T3sin2θ3
E3hL3
t33
+ 2(1 + ν3)T3L3sin2θ3
kE3ht3
+L3T3cos2θ3
E3ht3
(46)
2.1.2.2. Special case 2: Monomaterial asymmetric honeycombs with constant cell wall thickness
If the asymmetric hexagonal honeycomb structure is of mono-material (along with having constant
thickness), i.e.
E1
=
E2
=
E3
=
E
, then the above expression of displacements for
x
and
y
direction
further reduces to,
δ1x= 0,
δ2x=T2sinθ2cosθ2
Eh L2
t23
+ 2(1 + ν)T2L2sinθ2cosθ2
kEht2
−L2T2cosθ2sinθ2
Eht2
,
δ3x=−T3sinθ3cosθ3
Eh L3
t33
−2(1 + ν)T3L3sinθ3cosθ3
kEht3
+L3T3cosθ3sinθ3
Eht3
,
δ1y=−L1T1
Eht1
,
δ2y=−T2sin2θ2
Eh L2
t23
+ 2(1 + ν)T2L2cos2θ2
Eht2
+L2T2cos2θ2
Eht2
,
δ3y=T3sin2θ3
Eh L3
t33
+ 2(1 + ν)T3L3sin2θ3
kEht3
+L3T3cos2θ3
Eht3
(47)
Note that the above case is reported in literature [61], and the expressions presented here agree well
with such derivations. This provides an exact analytical validation of the current derivations.
2.1.2.3. Special case 3: Multimaterial symmetric honeycombs with constant cell wall thickness
If we further apply the condition for multi-material symmetric hexagonal structure (
θ2
=
θ3
=
θ
), then
we can obtain the expressions for deection of cell wall
2
and
3
under only bending deformation, similar
19

to the case of Mukhopadhyay et al. [60].
δ2=T2L3
2sin θ
12E2I
δ3=T3L3
3sin θ
12E3I
(48)
where,
I=ht3
12
. Though we have neglected the eect of axial and shear deformations in the above
equations for comparison with literature [60], these eects can be readily incorporated for multimate-
rial symmetric honeycombs with constant cell wall thickness.
2.1.2.4. Special case 4: Monomaterial symmetric honeycombs with constant cell wall thickness
For mono-material (
E1
=
E2
=
E3
=
E
) symmetric hexagonal structures (
L2
=
L3
=
L
and
T2
=
T3
=
T
), the deection of cell wall
2
and
3
is given by
δ2=T2L3sin θ
12EI
δ3=T3L3sin θ
12EI
(49)
Though we have neglected the eect of axial and shear deformations in the above equations for com-
parison with literature [37], these eects can be readily incorporated for monomaterial symmetric
honeycombs with constant cell wall thickness.
2.2. Evaluation of full elastic modulus matrix
In the previous section, the method to obtain three elastic moduli
C12
,
C22
and
C32
in closed-form is
proposed for muti-material variably-thickened asymmetric honeycombs having hexagon cells with cell
wall 1 parallel to the
y
-axis. But, all nine components of the constitutive matrix, mentioned in Eq.4,
must be known to model the overall elastic characteristics of such a honeycomb. In this section, other
remaining unknown elastic moduli (
C11
,
C13
and
C33
) are determined in closed-form by utilizing the
known elastic moduli
C12
,
C22
and
C32
based on co-ordinate transformation [45, 46].
In Fig. 3(c),
α1
and
β1
represent the coordinate axes for the cell wall
1
. Similarly, (
α2
,
β2
) and
(
α3
,
β3
) represent the coordinate axes for the cell wall
2
and cell wall
3
, respectively. Using these
coordinates, equivalent elastic modulus can be calculated for all the formed coordinates separately. For
such calculations, the (
x
,
y
) coordinate axes are transformed to (
α
,
β
) coordinate axes by keeping
β
axis parallel to the cell wall. Therefore,
C′
12
,
C′
22
, and
C′
32
in (
α1
,
β1
) coordinate system,
C′′
12
,
C′′
22
, and
C′′
32
in (
α2
,
β2
) coordinate system and
C′′′
12
,
C′′′
22
, and
C′′′
32
in (
α3
,
β3
) coordinate system can be found
by coordinate transformation (the upper subscripts in
Cij
represents a unit cell number). Here,
θ1
is
20

angle between the coordinate axes (
x
,
y
) and (
α1
,
β1
), as shown in Fig. 3(c). Similarly,
θ2
and
θ3
are
angles between the coordinate axes (
x
,
y
) with (
α2
,
β3
) and (
α3
,
β3
), respectively. In (
α1
,
β1
) coordinate
system
θ1
=
θ1
, whereas
θ2
=
θ1
+(
π
-
γ12
) for the (
α2
,
β2
) coordinate system, and
θ3
=
θ1
+(
π
-
γ12
)+(
π
-
γ23
)
for the (
α3
,
β3
) coordinate system. The transformation of stress and strain from the (
x
,
y
) coordinate
system to the (
α1
,
β1
) coordinate system leads to the following equation,





εα1
εβ1
εα1β1





= [T]−1




C11 C12 C13
C21 C22 C23
C31 C32 C33





[T]




σα1
σβ1
τα1β1





(50)
The coordination transformation matrix
[T]
is given as,
hTi=




cos2θ1sin2θ12sinθ1cosθ1
sin2θ1cos2θ1−2sinθ1cosθ1
−sinθ1cosθ1sinθ1cosθ1cos2θ1−sin2θ1





(51)
Here, stress and strain relationship in (
α1
,
β1
) coordinate system can also be expressed in the following
form,





εα1
εβ1
εα1β1





=




C′
11 C′
12 C′
13
C′
21 C′
22 C′
23
C′
31 C′
32 C′
33










σα1
σβ1
τα1β1





(52)
Then the Eq. 51 and 52 lead to following equation,





C′
11 C′
12 C′
13
C′
21 C′
22 C′
23
C′
31 C′
32 C′
33





= [T]−1




C11 C12 C13
C21 C22 C23
C31 C32 C33





[T].
(53)
Note in the above equation that
C′
i2
(for
i= 1,2,3
) terms are known based on the analysis presented in
the preceding section. However, the
Cij
terms (constitutive matrix as per the
x
,
y
coordinate system)
are not known and we need to obtain these terms. Based on the above equation, we have
C′
12 =C12cos4θ1−(−C13 + 2C32 )cos3θ1sin3θ1+ (C11 +C22 −2C33)cos2θ1sin2θ1
+ (C23 −C31)cos2θ1sin3θ1+C21 sin4θ1,
C′
22 =C22cos4θ1+ (C23 + 2C32 )cos3θ1sinθ1+ (C12 +C21 + 2C33 )cos2θ1sin2θ1
+ (C13 + 2C31)cosθ1sin3θ1+C11sin4θ1,
C′
32 =C32cos4θ1+ (C12 −C22 +C33 )cos3θ1sinθ1+ (−C23 +C31 −C32 +C13 )cos2θ1sin2θ1
+ (C11 −C21 −C33)cosθ1sin3θ1−C31sin4θ1.
(54)
21

In similar way, by transforming the (
x
,
y
) coordinate system to the (
α2
,
β2
) coordinate system, we have
C′′
12 =C12cos4θ2−(−C13 + 2C32 )cos3θ2sin3θ2+ (C11 +C22 −2C33)cos2θ2sin2θ2
+ (C23 −C31)cos2θ2sin3θ2+C21 sin4θ2,
C′′
22 =C22cos4θ2+ (C23 + 2C32 )cos3θ2sinθ2+ (C12 +C21 + 2C33 )cos2θ2sin2θ2
+ (C13 + 2C31)cosθ2sin3θ2+C11sin4θ2,
C′′
32 =C32cos4θ2+ (C12 −C22 +C33 )cos3θ2sinθ2+ (−C23 +C31 −C32 +C13 )cos2θ2sin2θ2
+ (C11 −C21 −C33)cosθ2sin3θ2−C31sin4θ2
(55)
Further, following a similar approach between the (
x
,
y
) coordinate system to the (
α3
,
β3
) coordinate
system, we have
C′′′
12 =C12cos4θ3−(−C13 + 2C32 )cos3θ3sin3θ3+ (C11 +C22 −2C33)cos2θ3sin2θ3
+ (C23 −C31)cos2θ3sin3θ3+C21 sin4θ3,
C′′′
22 =C22cos4θ3+ (C23 + 2C32 )cos3θ3sinθ3+ (C12 +C21 + 2C33 )cos2θ3sin2θ3
+ (C13 + 2C31)cosθ3sin3θ3+C11sin4θ3,
C′′′
32 =C32cos4θ3+ (C12 −C22 +C33 )cos3θ3sinθ3+ (−C23 +C31 −C32 +C13 )cos2θ3sin2θ3
+ (C11 −C21 −C33)cosθ3sin3θ3−C31sin4θ3
(56)
The value of local elastic moduli of element 1 (
C′
12
,
C′
22
, and
C′
32
), local elastic moduli of element 2 (
C′′
12
,
C′′
22
, and
C′′
32
) and local elastic moduli of element 3 (
C′′′
12
,
C′′′
22
, and
C′′′
32
) are already known from previous
sections. Therefore, nine components of
Cij
(refer to Eq. 4) can be determined by solving these nine
simultaneous equations given in Eqs. 54, 55 and 56.
In this context, it may be noted that the eective in-plane Young's moduli, shear modulus and
Poisson's ratios can be obtained directly from the in-plane constitutive matrix presented in Eq. 4. The
relation between elastic moduli and in-plane constitutive matrix elements is provided below in Eq. 57.
E1=1
C11
E2=1
C22
G12 =1
C33
ν12 =−C12 ×E2ν21 =−C21 ×E1
(57)
In the following section, we present numerical results to explore the bi-level expanded design space of
the proposed asymmetric lattices.
3. Results and discussion
A detailed numerical study is performed here to assess the eect of dierent design parameters cov-
ering the bi-level space of architected lattices on the eective elastic properties of regular (non-auxetic)
and auxetic honeycombs. The unit cell level architecture includes the cell angles, and lengths of dierent
22

cell walls, while the beam level architecture includes variably-thickened cell wall parameters
ζi
and
ηi
(
i= 1,2,3
) along with the multi-material ratio parameter (
q
, where
E1=E2=qE3=Es
, refer to
Figs. 1(f)) and 3(b). We have rst presented extensive validations of the proposed analytical framework
considering various cases of hexagonal honeycomb lattices available in the literature, (1) asymmetric
monomaterial honeycombs without beam-level architecture considering beam-level bending, shear and
axial deformations, (2) symmetric monomaterial honeycombs without beam-level architecture consider-
ing beam-level bending, shear and axial deformations (separate cases of non-auxetic and auxetic archi-
tectures), (3) symmetric monomaterial honeycombs without beam-level architecture considering only
beam-level bending deformations (separate cases of non-auxetic and auxetic architectures), (4) multi-
material symmetric lattices with auxetic and non-auxetic congurations considering only beam-level
bending deformation, (5) multi-material symmetric lattices with auxetic and non-auxetic congura-
tions considering beam-level bending, shear and axial deformations. Note that all these cases can be
readily evaluated based on the generic analytical formulation presented in the preceding section. Such
multi-stage validations considering the crucial factors of the proposed lattice (such as asymmetry, multi-
material architecture, unit cell level auxetic and non-auxetic geometries, and beam-level deformation
mechanics) would provide necessary condence in the developed analytical framework. Subsequently,
new results are presented concerning the bi-level architected multi-material design space.
3.1. Validation of the proposed analytical framework
The proposed analytical framework can obtain the entire elastic constitutive matrix of a generic
lattice having multi-material asymmetric unit cell with variably-thickened beam-level architecture. The
eective elastic moduli, like in-plane Young's moduli, shear modulus and Poisson's ratios can also be
readily obtained from the constitutive matrix. Plenty of special lattice classes can be obtained directly
based on the analytical framework (as shown in Fig. 1, in addition to dierent beam-level architecture
and multi-material congurations). Further, dierent levels of accuracy and computational eciency
can be ascertained by adopting the beam-level deformation mechanics accordingly involving bending,
axial and shear deformations. It can be noted here that the eect of shear and axial deformation
becomes crucial as the cell walls get thicker and axially more exible, respectively.
First, the proposed analytical framework is validated with the results of Chen and Yang [46] which
are available for asymmetric mono-material honeycombs (
t1=t2=t3= 1mm
,
η1=η2=η3= 1
, and
ζ1=ζ2=ζ3= 1
), as shown in Table 1. The results (obtained from the present formulation and available
literature) are compared for two models, such as Model A (
L1=L2=L3=L= 10mm
,
θ2= 40◦
,
θ3= 80◦
) and Model B (
L1= 24.5mm, L2= 6mm, L3= 10mm
;
θ2= 26.3◦
,
θ3= 88.7◦
) under dierent
23

Table 1: Validation for asymmetric mono-material hexagonal lattices based on comparing results obtained
from the current analytical approach, results from literature and nite element method.
Here we compare
the present results with analytical and the nite element (FE) results from literature [46] considering asymmetric mono-
material (
E1=E2=E3=Es
) hexagonal cells. The non-dimensionalized elastic parameters for non-auxetic asymmetric
honeycombs (
t1=t2=t3=t
,
η1=η2=η3= 1
, and
ζ1=ζ2=ζ3= 1
) are compared for dierent wall thickness under
the consideration of bending, axial and shear deformation of cell walls. In model A, the length of the three cell walls is
same (
L1=L2=L3=L= 10mm
), but the angles between the walls are dierent (
θ2= 40◦, θ3= 80◦
). In model B,
the lengths and angles of the cell walls are dierent (
L1= 24.5mm, L2= 6mm, L3= 10mm
;
θ2= 26.3◦, θ3= 88.7◦
).
However, in both models A and B, the thickness of the cell walls is the same,
t1=t2=t3=t
. The results are compared
for three thicknesses
t=
0.1, 0.5, 1mm under three cases: (a), b and (c). In Case (a) only the bending eect is considered
in the analysis (B). In Case (b) only bending and axial eects are considered in the analysis (Be.+Ax.). In Case (c) all
bending, axial and shear eects are considered in the analysis (Be.+Ax.+Sh.).
Model A Case (a), (Be.) Case (b), (Be.+Ax.) Case (c), (Be.+Ax.+Sh.) FEM [46]
t
Present Chen [46] Present Chen [46] Present Chen [46]
1
(C12Es)
0.1 -3.267
×10−6
-3.267
×10−6
-3.267
×10−6
-3.267
×10−6
-3.267
×10−6
-3.267
×10−6
-3.267
×10−6
0.5 -4.084
×10−4
-3.267
×10−6
-4.094
×10−4
-4.094
×10−4
-4.068
×10−4
-4.068
×10−4
-4.068
×10−4
1-3.267
×10−3
-3.267
×10−3
-3.300
×10−3
-3.300
×10−3
-3.216
×10−3
-3.216
×10−3
-3.216
×10−3
1
(C22Es)
0.1 3.111
×10−6
3.111
×10−6
3.110
×10−6
3.110
×10−6
3.109
×10−6
3.109
×10−6
3.109
×10−6
0.5 3.889
×10−4
3.889
×10−4
3.851
×10−4
3.851
×10−4
3.826
×10−4
3.826
×10−4
3.826
×10−4
13.111
×10−3
3.111
×10−3
2.991
×10−3
2.991
×10−3
2.918
×10−3
2.918
×10−3
2.918
×10−3
1
(C32Es)
0.1 4.841
×10−6
4.841
×10−6
4.841
×10−6
4.841
×10−6
4.839
×10−6
4.839
×10−6
4.839
×10−6
0.5 6.051
×10−4
6.051
×10−4
6.052
×10−4
6.052
×10−4
6.013
×10−4
6.013
×10−4
6.013
×10−4
14.841
×10−3
.841
×10−3
4.845
×10−3
4.845
×10−3
4.722
×10−3
4.722
×10−3
4.722
×10−3
Model B Case (a), (Be.) Case (b), (Be.+Ax.) Case (c), (Be.+Ax.+Sh.) FEM [46]
t
Present Chen [46] Present Chen [46] Present Chen [46]
1
(C12Es)
0.1 -1.584
×10−5
-1.58
×10−5
-1.584
×10−5
-1.584
×10−5
-1.583
×10−5
-1.583
×10−5
-1.583
×10−5
0.5 -1.980
×10−3
-1.98
×10−3
-1.993
×10−3
-1.993
×10−3
-1.959
×10−3
-1.959
×10−3
-1.959
×10−3
1-1.584
×10−2
-1.584
×10−2
-1.627
×10−2
-1.627
×10−2
-1.520
×10−2
-1.520
×10−2
-1.520
×10−2
1
(C22Es)
0.1 3.225
×10−5
3.225
×10−5
3.212
×10−5
3.212
×10−5
3.211
×10−5
3.212
×10−5
3.212
×10−5
0.5 4.031
×10−3
4.031
×10−3
3.673
×10−3
3.673
×10−3
3.637
×10−3
3.637
×10−3
3.637
×10−3
13.225
×10−2
3.225
×10−2
2.320
×10−2
2.320
×10−2
2.250
×10−2
2.250
×10−2
2.250
×10−2
1
(C32Es)
0.1 9.427
×10−6
9.427
×10−6
9.426
×10−6
9.426
×10−6
9.424
×10−6
9.424
×10−6
9.424
×10−6
0.5 1.178
×10−3
1.178
×10−3
1.178
×10−3
1.178
×10−3
1.170
×10−3
1.170
×10−3
1.170
×10−3
19.427
×10−3
9.427
×10−3
9.406
×10−3
9.406
×10−3
9.178
×10−3
9.178
×10−3
9.178
×10−3
wall thicknesses
t=
0.1, 0.5, 1mm. The non-dimensionalized elastic constitutive constants (1/(
C12 Es
),
1/(
C22 Es
) and 1/(
C32 Es
) are compared for three cases (a), (b) and (c). In Case (a) only bending
eect is considered in the analysis (Be.), and in Case (b), only bending and axial eects are considered
in the analysis (Be.+Ax.). Whereas in Case (c), all bending, axial and shear eects are considered in
the analysis (Be.+Ax.+Sh.). The corresponding nite element results [46] are also tabulated in Table 1.
The comparison presented in Table 1 shows excellent agreement for all the cases, establishing that the
24

Figure 4: Validation for regular (non-auxetic) symmetric mono-material honeycombs under axial, bending
and shear eects.
Comparison of present results with the classical approach of Gibson and Ashby [37] in which the closed-
form expression of eective Young's moduli, shear moduli and Poisson's ratios for perfectly periodic regular mono-material
(
E1=E2=E3=Es
) hexagonal cells are derived for both directions by applying stress in each direction individually. The
non-dimensionalized value of the elastic parameter for non-auxetic (regular) honeycombs (
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.1L
,
η1=η2=η3= 1
, and
ζ1=ζ2=ζ3= 1
) are compared considering dierent cell angles under the
consideration of axial, bending and shear deformations of cell walls. Variations with respect to cell angle
θ
(in degree)
are presented for
(a, b)
Young's moduli
¯
E1
and
¯
E2
(c)
shear moduli
¯
G12
or
¯
G21
(d)
Poisson's ratios
¯ν12
or
¯ν21
. The
normalization of the elastic moduli is carried out with respect to intrinsic Young's modulus (
Es
) of walls. It should be
noted that the Poisson's ratios (
¯ν12
and
¯ν21
) do not depend on the intrinsic material properties of the wall members.
eect of asymmetry can be accurately analyzed through the present analytical approach.
Further, the present analytical approach is validated with the closed-form expressions of Gibson and
Ashby [37] available for symmetric mono-material hexagonal honeycombs (
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.1L
,
η1=η2=η3= 1
, and
ζ1=ζ2=ζ3= 1
), as shown in Figs. SM1.S1, 4, SM1.S2 and
SM1.S3 considering non-auxetic and auxetic congurations along with dierent beam-level deformation
mechanics (bending, axial and shear deformations). In Fig. SM1.S1, the variation of non-dimensionalized
Young's moduli (
¯
E1/Es
,
¯
E2/Es
), shear moduli (
¯
G12/Es
and
¯
G21/Es
) and Poisson's ratios (
¯ν12
and
25

Figure 5: Validation with Timoshenko theory-based beam model for multi-material (non-auxetic) symmet-
ric honeycombs (considering beam-level bending, shear and axial deformation).
Comparison of present results
with Timoshenko theory based beam model of Mukherjee and Adhikari [68]. The non-dimensionalized value of elastic mod-
uli for non-auxetic multi-material honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.1L
,
η1=η2=η3= 1
, and
ζ1=ζ2=ζ3= 1
) are compared for dierent cell angles under the consideration of axial, shear and
bending deformations of cell walls. Variations with respect to cell angle
θ
(in degree) considering two dierent material
ratios,
q
are presented for
(a, b)
Young's moduli
¯
E1
and
¯
E2
(c)
shear moduli
¯
G12
or
¯
G21
(d)
Poisson's ratios
¯ν12
and
¯ν21
. The normalization of Young's moduli (
¯
E1
,
¯
E2
) and shear moduli (
¯
G12
,
¯
G21
) is carried out with respect to intrinsic
Young's modulus (
Es
). It should be noted that the Poisson's ratios are non-dimensional quantities.
¯ν21
) are compared for regular (non-auxetic) honeycombs by considering the bending eect only for a
wide range of cell wall angles
θ
(
10◦
to
80◦
). Here
(¯
•)
represents an eective elastic modulus of the
lattice that is non-dimensionalized with respect to the intrinsic material property
Es
of a mono-material
conguration. Note that the Poisson's ratios are already non-dimensional. Similarly, the variation of
non-dimensionalized Young's moduli, shear moduli and Poisson's ratios are compared for regular (non-
auxetic) honeycombs in Fig. 4 considering the eect axial, shear and bending deformations of the cell
walls. Further, the non-dimensionalized elastic properties for auxetic honeycombs are compared in
26

Figure 6: Validation for non-auxetic symmetric multi-material honeycombs only under beam-level bending
deformation.
Comparison of present results with the classical approach of Mukhopadhyay et al. [60] in which the closed-
form expression of Young's moduli and Poisson's ratios for multi-material (
E1=E2=qE3=Es
) hexagonal cells
are derived under bending deformation for both directions by applying stress in each direction individually. The non-
dimensionalized values of the elastic parameters for regular (non-auxetic) honeycombs (
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.1L
,
η1=η2=η3= 1
, and
ζ1=ζ2=ζ3= 1
) are compared for dierent cell angles under only bending
deformation of cell walls. The variation of equivalent elastic parameters is plotted for dierent multi-material ratios (
q=
1
to 1.6). Variations with respect to cell angle
θ
(in degree) considering dierent material ratios (
q
) are presented for
(a,
b)
Young's moduli
¯
E1
and
¯
E2
, and
(c, d)
Poisson's ratios
¯ν12
and
¯ν21
. The normalization of the Young's moduli (
¯
E1
,
¯
E2
) is carried out with respect to intrinsic Young's modulus (
Es
) of the wall with length
L
. In these gures, the arrows
depict the increasing trend of multi-material ratio (
q
) from lower value (
q= 1
, which represents mono-material) to large
value (
q= 1.6
, which represents multi-material with higher ratio). The Poisson's ratios
¯ν12,¯ν21
show independence from
the multi-material ratio (
q
) as it does not depend on the intrinsic material properties of the cell wall members. These
quantities only depend on the structural geometry of the lattice.
Figs. SM1.S2 and SM1.S3. In Fig. SM1.S2, results are compared only by condensing the bending eect
of cell walls for both cases. In Fig. SM1.S3, results are compared for the case in which the eect of
axial, shear, and bending deformation of cell walls is taken into consideration. Excellent agreement is
observed for both regular (non-auxetic) and auxetic honeycombs under the consideration of dierent
27

beam-level deformation mechanics. It is also worth noting here that the Poisson's ratios (
¯ν12
and
¯ν21
)
do not depend on the intrinsic material properties of the cell wall members.
The concept of multi-material hexagonal honeycombs (symmetric structure) and concerned analyses
of eective elastic moduli were rst presented by Mukhopadhyay et al. [60]. In that analysis, beam-level
deformation was considered based on bending with the assumption of thin cell walls. Subsequently,
the same approach was extended for thick cell walls through the incorporation of beam-level shear and
axial deformation eects by Mukherjee and Adhikari [68]. Here, the present framework is validated
with Euler-Bernoulli and Timoshenko beam theory-based approach presented by Mukhopadhyay et al.
[60] and Mukherjee and Adhikari [68] in Figs. 5, 6, and SM1.S4 to SM1.S7. The non-dimensionalized
equivalent elastic properties (
¯
E1
/
Es
,
¯
E2
/
Es
,
¯
G12
/
Es
, and
¯ν12
/
¯ν21
) are compared under dierent cell
wall angle
θ
(
10◦
to
80◦
) considering mono-material and multi-material congurations (auxetic and non-
auxetic). Excellent agreement is observed between both the approaches considering dierent beam-level
deformation mechanisms (a. only bending, b. bending and axial, c. bending, axial and shear). In this
context, it may be noted that the eective elastic moduli of the hexagonal lattices can be tuned by
introducing the proper multi-material ratio, whereas the Poisson's ratios
¯ν12,¯ν21
show independence
(primarily in the predominant beam-level transverse deformation mechanism) from the multi-material
ratio (
q
) as it does not depend on the intrinsic material properties of the cell wall members. The
Poisson's ratios of hexagonal lattices mostly depend on the unit cell level geometry.
Having adequate condence in the developed analytical framework based on extensive multi-stage
validations with existing literature, we embark on further numerical explorations in the following section
concerning the proposed bi-level architected multi-material asymmetric lattices.
3.2. Numerical investigation on the constitutive behavior
After validating the present approach, it is further utilized to obtain the complete in-plane consti-
tutive, (
Cij
) matrix of various thick and thin-walled multi-material honeycomb lattices. In this section,
we investigate the eect of dierent inuencing factors systematically, as explained below:
(1) Eect of the multimaterial parameter, considering non-auxetic and auxetic symmetric congurations
along with dierent beam-level deformation mechanics are shown in Fig. 7, and SM1.S8 to SM1.S10.
Similar sets of results for two other wall thickness values (relatively thinner and thicker, respectively)
are shown in Fig. SM2.S1 to SM2.S8.
(2) Combined eect of asymmetry and the multimaterial parameter, considering non-auxetic and aux-
etic congurations including coupled beam-level bending, axial and shear deformations are shown in
28

Fig. 8 and SM1.S11. Similar sets of results for two other wall thickness values (relatively thinner and
thicker, respectively) are shown in Fig. SM2.S9 to SM2.S12.
(3) Eect of the beam-level architecture in terms of the thickened length ratio
ζi
, considering non-auxetic
and auxetic symmetric congurations along with dierent beam-level deformation mechanics are shown
in Fig. 9, and SM1.S12 to SM1.S14. Similar sets of results for two other wall thickness values (relatively
thinner and thicker, respectively) are shown in Fig. SM2.S13 to SM2.S20.
(4) Combined eect of asymmetry and beam-level architecture in terms of the thickened length ratio
ζi
, considering non-auxetic and auxetic congurations including coupled beam-level bending, axial and
shear deformations are shown in Fig. 10 and SM1.S15. Similar sets of results for two other wall thickness
values (relatively thinner and thicker, respectively) are shown in Fig. SM2.S21 to SM2.S24.
(5) Eect of the beam-level architecture in terms of the thickness ratio
ηi
, considering non-auxetic and
auxetic symmetric congurations along with dierent beam-level deformation mechanics are shown in
Fig. 11, and SM1.S16 to SM1.S18. Similar sets of results for two other wall thickness values (relatively
thinner and thicker, respectively) are shown in Fig. SM2.S25 to SM2.S32.
(6) Combined eect of asymmetry and beam-level architecture in terms of the thickness ratio
ηi
, con-
sidering non-auxetic and auxetic congurations including coupled beam-level bending, axial and shear
deformations are shown in Fig. 12 and SM1.S19. Similar sets of results for two other wall thickness
values (relatively thinner and thicker, respectively) are shown in Fig. SM2.S33 to SM2.S36.
(7) Eect of cell wall thickness, considering non-auxetic and auxetic symmetric congurations along
with dierent beam-level deformation mechanics are shown in Fig. 13, and SM1.S20 to SM1.S22.
(8) Combined eect of asymmetry and cell wall thickness, considering non-auxetic and auxetic cong-
urations including coupled beam-level bending, axial and shear deformations are shown in Fig. 14 and
SM1.S23.
(9) Eect of cell wall length ratio, considering non-auxetic and auxetic symmetric congurations along
with dierent beam-level deformation mechanics are shown in Fig. 15, and SM1.S24 to SM1.S26. Sim-
ilar sets of results for two other wall thickness values (relatively thinner and thicker, respectively) are
shown in Fig. SM2.S37 to SM2.S44.
(10) Combined eect of asymmetry and cell wall length ratio, considering non-auxetic and auxetic con-
gurations including coupled beam-level bending, axial and shear deformations are shown in Fig. 16
and SM1.S27. Similar sets of results for two other wall thickness values (relatively thinner and thicker,
respectively) are shown in Fig. SM2.S45 to SM2.S48.
(11)
Supplementary material:
The results mentioned in points (1) to (10) above for the main manuscript
29

and supplementary material 1 (SM1) are obtained for
t/L = 0.2
, unless otherwise mentioned. For a com-
prehensive understanding, we provide additional numerical results in supplementary material 2 (SM2)
for the above cases considering two other cell wall thicknesses, as
t/L = 0.1
(thinner) and
0.3
(thicker).
In Fig. SM1.S8 and 7, the variation of all in-plane constitutive constants for non-auxetic multi-
material honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1
mm,
t1=t2=t3= 0.2L
,
η1=η2=η3= 1
, and
ζ1=ζ2=ζ3= 1
) is plotted considering dierent cell angles (
θ= 10◦
to
80◦
).
Similar variations of in-plane constitutive constants for other thicknesses
t1=t2=t3= 0.1L, 0.3L
are
shown in Fig. SM2.S1, SM2.S2, SM2.S3 and SM2.S4 of the supplementary material. The behavior of
each in-plane constitutive constant is plotted for dierent multi-material ratios (
q=
1 to 2) and cell angle
θ
. The normalization of the constitutive constants (
Cij
) is carried out with respect to intrinsic Young's
modulus (
Es
) of the vertical wall with length
L
. In these gures, the arrows depict the increasing values
of multi-material ratio (
q
) from lower value (
q= 1
, which represents mono-material) to large value
(
q= 2
, which represents multi-material with higher ratio). In Figs. SM1.S8, SM2.S1 and SM2.S2, the
variation of all in-plane constitutive constants are presented by considering only bending deformations
of cell walls in the analysis. Similarly, corresponding variations of in-plane constitutive constants are
presented in the Figs. 7, SM2.S3 and SM2.S4 under the consideration of axial, bending and shear
deformations of cell walls. In general, it is observed that all in-plane constitutive constants of honeycomb
structures are aected signicantly by multi-material ratio (
q
) for dierent cell angles
θ
. The results
demonstrate that the elastic properties of a honeycomb can be eectively modulated by introducing
multi-material unit cells (
q=E1/E3=E2/E3
) without altering the unit cell shapes. The eect of
multi-material ratio is more inuential for
C12 / C21
,
C13 / C31
and
C23 / C32
, whereas it's eect on the
other elastic constitutive constants like
C11
,
C22
and
C33
is comparatively less. The values of
C12
and
C21
decrease by 1.5 times as the value of
q
is increased from 1 to 2, whereas the value of
C13 / C31
decreases
and
C23 / C32
increases by
3
to
6
times as
q
is increased from 1 to 2. The corresponding eect of multi-
material ratio (
q
) on
C11
,
C22
and
C33
are comparatively less and their values increase by
50%
to
100%
for dierent cell angle
θ
as
q
is increased from 1 to 2. The results show that the eect of multi-material
ratio depends upon the cell angle
θ
, the trends of which change depending on the constitutive constant
under consideration. From the comparison of Figs. SM1.S8, SM2.S3 and SM2.S4 with Figs. 7, SM2.S1
and SM2.S2, it is also observed that each elastic constitutive constants have signicant contribution of
axial and shear eect of cell walls. The magnitude of each in-plane constitutive constant increases by
10-20
%
for thinner-walled (
t/L = 0.1
), 20-30
%
for moderately thick-walled ((
t/L = 0.2
) and 30-40
%
thicker-walled honeycombs ((
t/L = 0.3
), when the eects of axial and shear deformations of cell walls
30

are considered along with the bending deformation. However, the trend of variation of all constitutive
constants with respect to cell wall angle (
θ
) remains unchanged under axial, shear and bending eects
consideration as compared to the only bending case.
The variation of all in-plane constitutive constants for auxetic multi-material honeycombs is plotted
for dierent cell angles in Fig. SM1.S9 and SM1.S10 considering a range of multi-material ratios (
q=
1
to 2) and
t/L = 0.2
. Further, similar results for auxetic honeycombs are presented for two other cell wall
thicknesses (
t/L = 0.1,0.3
) in supplementary material, Figs. SM2.S5 to SM2.S8. In the Figs. SM1.S9,
SM2.S5 and SM2.S6, variations of all in-plane constitutive constants are presented by considering only
the bending deformation eect of cell walls. Similarly, corresponding variations of constitutive constants
under axial, bending and shear deformation eects of cell walls are shown in Figs. SM1.S10, SM2.S7
and SM2.S8. It is noted that all the in-plane constitutive constants get aected signicantly for auxetic
congurations with the variation of cell angle
θ
and the multimaterial parameter
q
. For
C11
,
C12
and
C21
,
the maximum eect of multi-material ratio is observed at cell wall angle
45◦
, whereas the eect is most
notable in
C13
,
C31
,
C23
and
C32
for the cell wall angles more than
60◦
. Maximum change for
C22
is noted
for the cell wall angles lower than
40◦
. The trends of variations are similar under the consideration of
only bending deformation, and the combined eect of bending, axial and shear deformations. However,
the magnitude of each constitutive constant increases by 10-20
%
for thinner-walled (
t/L = 0.1
), 20-30
%
for moderately thick-walled ((
t/L = 0.2
) and 30-40
%
for thicker-walled honeycombs (
t/L = 0.3
) with
the consideration of axial and shear eect of cell walls along with the bending deformation.
In addition to the regular symmetric case, the asymmetric multi-material lattices with nonauxetic
and auxetic congurations are considered in Figs. 8 and SM1.S11 (also refer to Figs. SM2.S9 to SM2.S12
of the supplementary material) to plot the in-plane constitutive behavior of
Cij
elements with respect
to angle ratio, (
θ3/θ2
= 0.5 to 2) and multi-material ratio, (
q
= 1 to 2) under the axial, bending and
shear mode of deformations. When the ratio of
θ3/θ2
= 1, it indicates the symmetric multi-material
case, as discussed in the preceding paragraphs. In-plane constitutive elements
C11
,
C22
, and
C33
show
increasing trends with the enhancement of
q
from 1 to 2 (33
%
, 26
%
, and 11
%
increament, respectively).
However,
C12
/
C21
and
C13/C31
show a decreasing trend with respect to changing values of
q
from 1 to
2. Further
C23
and
C32
show increasing trends with up to 200
%
variation with respect to variation in
q
from 1 to 2. The in-plane constitutive elements
Cij
for auxetic asymmetric multi-material lattices show
an increasing trend with the variation of
q
from 1 to 2.
We now explore the eect of variably-thickened cell walls in multimaterial lattices for non-auxetic
and auxetic congurations. In Fig. SM1.S12, the non-dimensionalized values of elastic constitutive
31

Figure 7: Complete in-plane constitutive (elastic modulus) matrix for non-auxetic symmetric multi-
material honeycombs under axial, bending and shear eects.
The non-dimensionalized value of elastic con-
stitutive parameters for non-auxetic honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
,
η1=η2=η3= 1
, and
ζ1=ζ2=ζ3= 1
) are plotted for dierent cell angles under the consideration of axial, bending and
shear deformations of cell walls. The variation of constitutive constants is plotted for dierent multi-material ratios (
q=
1
to 2). The normalization of the constitutive constants (
Cij
) is carried out with respect to the intrinsic Young's modulus
(
Es
) of the vertical wall. Variations with respect to cell angle
θ
(in degree) are presented considering dierent values of
q
for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In these gures, the arrows depict the
increasing trend of multi-material ratio (
q
) from lower (
q= 1
, which represents mono-material) to higher values (
q= 2
,
which represents multi-material with higher ratio).
parameters (
Cij
) are plotted for non-auxetic honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=
L= 1mm
,
t1=t2=t3= 0.2L
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ=
1 to 10,
q= 2
)
considering dierent cell angles based on only bending deformations of cell walls. Further, the axial,
and shear deformations are also considered along with the bending deformation of cell walls, and the
32

Figure 8: Complete in-plane constitutive (elastic modulus) matrix for nonauxetic asymmetric multi-
material honeycombs under axial, bending and shear eects.
The non-dimensionalized value of elastic constitutive
parameters for nonauxetic honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
,
η1=η2=η3= 1
, and
ζ1=ζ2=ζ3= 1
) are plotted for angle ratio
θ3/θ2
under the consideration of axial, bending
and shear deformations of cell walls. The variation of constitutive constants is plotted for dierent multi-material ratios
(
q=
1 to 2). The normalization of the constitutive constants (
Cij
) is carried out with respect to the intrinsic Young's
modulus (
Es
) of the vertical wall. Variations with respect to angle ratio
θ3/θ2
are presented considering dierent values
of
q
for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In these gures, the arrows depict the
increasing trend of multi-material ratio (
q
) from lower (
q= 1
, which represents mono-material) to higher values (
q= 2
,
which represents multi-material with higher ratio).
corresponding elastic constitutive parameters are plotted in Fig. 9. In the supplementary material,
additional results are presented for non-auxetic variably-thickened multimaterial lattices considering
thinner and thicker cell walls with
t1=t2=t3= 0.1L, 0.3L
(refer to Fig. SM2.S13 to SM2.S16). In
these gures, the variation of constitutive constants is plotted for dierent
ζ
ratios (
ζ=
1 to 10), where
ζ
, the thickened-length ratio, represents the ratio of the length of thickened cell wall to the middle part
33

length. The cell wall thickness ratio (
η
) is taken as 0.5 (
η= 0.5
), which means that the thickness of
the thicker part is double as compared to the middle part thickness of the walls. The normalization
of the constitutive constants (
Cij
) is carried out with respect to the intrinsic Young's modulus (
Es
) of
the vertical wall as before. It is observed that all the constitutive elastic constants can be modulated
eectively by controlling length of the thickened cell wall. For
C11
,
C13
, and
C31
the maximum eect
of
ζ
is observed at higher cell wall angles, whereas for
C12
and
C21
the eect is maximum at cell wall
angle
45◦
. On the other hand, eect of
ζ
on
C22
,
C23
,
C32
and
C33
is more prominent at lower cell wall
angles. It can be noted that the contribution of considering the shear and axial deformations along with
the bending deformation is signicant on all the constitutive constants. The magnitude of constitutive
constants increases by 10-20
%
(for
t/L = 0.1
), 15 to 30
%
(for
t/L = 0.2
), 30-45
%
(for
t/L = 0.3
) under
the consideration of axial, shear and bending eect compared to the only bending case.
Similarly to explore the eect of variably-thickened cell walls on auxetic multi-material honeycombs,
the in-plane constitutive behavior under dierent cell angles is plotted in Fig. SM1.S13 considering
only bending (
t/L = 0.2
). To asses the eect of shear and axial eect along with bending, a similar
variation is presented in Fig. SM1.S14 for
t/L = 0.2
. Further, we have explored the variably-thickened
auxetic multi-material honeycombs considering thinner and thicker cell walls with
t/L = 0.1,0.3
(Figs.
SM2.S17 - SM2.S20 in the supplementary material). It is observed variably-thickened cell wall length
signicantly aects all the in-plane constitutive matrix elements depending on the cell wall angle. In
the auxetic case, the eect of
ζ
on
C11
,
C12
,
C21
and
C22
is more prominent at cell wall angle
45◦
. The
maximum eect of
ζ
for all other constitutive elastic constants is observed at a higher cell wall angle. It
is noted that the consideration of axial and shear deformation signicantly aects the magnitude of each
constitutive element. It is also worth noting here that the eect of variably-thickened cell walls (
ζ
) for
both regular and auxetic honeycombs becomes more prominent when its value increases from 1 to 5 and
this eect gradually becomes less as its value further increases from 5 to 10. Such observations can play
an important role in achieving optimized properties by controlling the variably-thickened beam-level
architecture in multi-material honeycomb lattices.
After investigating symmetric lattices, we have presented results for variably-thickened cell walls
considering asymmetric multi-material nonauxetic and auxetic lattices (Figs. 10, SM1.S15, and SM2.S21
- SM2.S24). It can be noted, when
θ3/θ2
is equivalent to 1, then the variation of
Cij
elements becomes
similar to regular symmetric multi-material honeycombs, as discussed in the preceding paragraphs. The
Cij
elements are plotted for asymmetric designs concerning
ζ
and angle ratio
θ3/θ2
, wherein an intricate
interplay among the prospective design parameters can be noticed that would lead to important insights
34

Figure 9: Complete in-plane constitutive (elastic modulus) matrix for non-auxetic multi-material symmet-
ric variably-thickened cell wall honeycombs under bending, axial and shear eects.
The non-dimensionalized
values of elastic constitutive parameters for non-auxetic honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ=
1 to 10,
q= 2
) are plotted for dierent cell angles
under the consideration of bending, axial and shear deformations of cell walls. The variation of constitutive constants is
plotted for dierent
ζ
ratios (
ζ=
1 to 10). Here,
ζ
represents a ratio of the length of the thickened portion near the joint
to the less-thick middle part. The cell wall thickness ratio (
η
) is considered as 0.5 (
η= 0.5
), which represents that the
thickness near the end part is double as compared to the middle part thickness. The normalization of the constitutive
constants (
Cij
) is carried out with respect to the intrinsic Young's modulus (
Es
) of the vertical wall. Variations with
respect to cell angle
θ
(in degree) considering dierent values of
ζ
are presented for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In these gures, the arrows depict the increasing trend of
ζ
from lower (
ζ= 1
,
which represents that the length of the thickened part is equal to middle part) to higher values (
ζ= 10
, which represents
that the length of the thickened part is 1/10 of the middle part).
for developing multifunctional lightweight architected materials.
The eect of another parameter in variably-thickened cell walls,
η
indicating the cell wall thickness
ratio (refer to Fig. 3(b)), is studied further on multi-material (
q= 2
) non-auxetic and auxetic hon-
eycombs (
ζ1=ζ2=ζ3=ζ= 1
). In Fig. 11 and SM1.S16 to SM1.S18 (also refer to Figs. SM2.S25
35

Figure 10: Complete in-plane constitutive (elastic modulus) matrix for nonauxetic asymmetric multi-
material variably-thickened cell wall honeycombs under axial, bending and shear eects.
The non-
dimensionalized values of elastic constitutive parameters for non-auxetic honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ=
1 to 10,
q= 2
) are plotted for dierent cell angle ratios
θ3/θ2
under the consideration of bending, axial and shear deformations
of cell walls. The variation of constitutive constants is plotted for dierent
ζ
ratios (
ζ=
1 to 10). Here,
ζ
represents
a ratio of the length of the thickened portion near the joint to the less-thick middle part. The cell wall thickness ratio
(
η
) is considered as 0.5 (
η= 0.5
), which represents that the thickness near the end part is double as compared to the
middle part thickness. The normalization of the constitutive constants (
Cij
) is carried out with respect to the intrinsic
Young's modulus (
Es
) of the vertical wall. Variations with respect to angle ratio
θ3/θ2
considering dierent values of
ζ
are presented for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In these gures, the arrows
depict the increasing trend of
ζ
from lower (
ζ= 1
, which represents that the length of the thickened part is equal to
middle part) to higher values (
ζ= 10
, which represents that the length of the thickened part is 1/10 of the middle part).
to SM2.S32 of the supplementary material), the elastic constitutive constants (
Cij
) are plotted for
multi-material regular (non-auxetic) and auxetic honeycomb structures by considering dierent
η
val-
36

ues (
η1=η2=η3=η=
0.5 to 1). In these gures, the arrows depict the increasing trend of cell wall
thickness ratio (
η
) from
η= 0.5
(which represents the thickness of cell wall is double near the joints
compared to the middle part) to
η= 1
(which represents the thickness of cell wall is same near the
joints compared to the middle part). In Fig. SM1.S16, the non-dimensionalized value of elastic consti-
tutive constants (
Cij
) for non-auxetic honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
and
ζ1=ζ2=ζ3=ζ= 1
,
q= 2
) is plotted at dierent cell angles (
θ
) under the
consideration of bending deformation of cell walls. Similarly, the non-dimensionalized value of elastic
constitutive constants (
Cij
) for non-auxetic honeycomb is plotted in Fig. 11 at dierent cell angles (
θ
)
under the consideration of combined axial, shear and bending deformations of cell walls. For a more
comprehensive analysis, the similar variations of in-plane constitutive constants for thinner-walled and
thicker-walled non-auxetic multi-material honeycombs are presented in the supplementary material (re-
fer to Figs. SM2.S25 to SM2.S28). It is observed that all the elastic constitutive constants (
Cij
) are
signicantly aected by
η
. The magnitude of elastic constitutive constants (
Cij
) increases by approxi-
mately 3 to 4 times as the value of
η
increases from 0.5 to 1. It is noted that the eect of
η
on
C11
,
C13
and
C31
is more prominent when the cell wall angle is more than
60◦
, whereas this eect for
C12
and
C21
is maximum at cell wall angle
45◦
. The eect of
η
on
C22
,
C23
,
C32
and
C33
is comparatively high at
lower cell wall angles. From the gures, it can also be noted that the consideration of axial and shear
deformations of cell walls plays a signicant role in the characterization of constitutive constants (
Cij
).
The elastic constitutive constants (
Cij
) under axial, shear and bending eects become 20-30
%
more
than the only bending case. In Fig. SM1.S17 and SM1.S18, the non-dimensionalized value of elastic
constitutive constants (
Cij
) for auxetic honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
and
ζ1=ζ2=ζ3=ζ= 1
,
q= 2
) is plotted at dierent cell angle (
θ
) un-
der consideration of only bending deformations and the combined eect of axial, shear and bending
deformations. For a more comprehensive analysis, the variation of elastic constitutive constants is pre-
sented further considering thinner and thicker walled honeycombs in the supplementary material (refer
to Figs. SM2.S29 to SM2.S32). It is noted that the eect of
η
on
C11
,
C12
and
C21
is more prominent
when the cell wall angle is
45◦
, whereas this eect for
C22
is maximum at cell wall angle in the range of
10◦
to
40◦
. The eect of
η
on
C13
,
C31
,
C23
,
C32
and
C33
is signicantly high at lower cell wall angles.
Subsequently, the eect of cell wall thickness ratio
η
is studied for asymmetric multi-material nonaux-
etic and auxetic honeycombs in Fig. 12, and SM1.S19 (also refer to Figs. SM2.S33 to SM2.S36). The
in-plane constitutive constants are plotted with respect to
η
and cell angle ratio
θ3/θ2
. It can be noted
that when the ratio of
θ3/θ2
is equivalent to 1, the results obtained are similar to the symmetric case
37

Figure 11: Eect of variably-thickened cell wall thickness ratio (
η
) on complete in-plane constitutive
(elastic modulus) matrix for non-auxetic multi-material symmetric honeycombs under bending, axial and
shear deformations.
The non-dimensionalized values of elastic constitutive parameters for non-auxetic honeycombs
(
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
,
η1=η2=η3=η=
0.5 to 1, and
ζ1=ζ2=ζ3=ζ= 1
,
q= 2
) are plotted for dierent cell angles under the consideration of bending, axial and shear
deformation of cell walls. The variation of constitutive constants is plotted for dierent
η
ratios (
η=
0.5 to 1) considering
ζ= 1
. The normalization of the constitutive constants (
Cij
) is carried out with respect to the intrinsic Young's modulus
(
Es
) of the vertical wall. Variations with respect to cell angle
θ
(in degree) considering dierent values of
η
are presented
for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In these gures, the arrows depict the
increasing trend of cell wall thickness ratio (
η
) from lower (
0.5
, which represents that the thickness of cell walls is twice
near the joints compared to the middle part) to higher values (
η= 1
, which represents that the thickness of the cell walls
is uniform throughout the length).
of nonauxetic and auxetic lattices. With the introduction of asymmetry in the geometry, an intricate
interplay among the prospective design parameters can be noticed that would lead to important insights
for developing multifunctional lightweight architected materials in an expanded design space.
The eect of wall thickness corresponding to the middle part of a variably-thickened cell wall (indi-
38

Figure 12: Eect of variably-thickened cell wall thickness ratio (
η
) on complete in-plane constitutive
(elastic modulus) matrix for nonauxetic asymmetric multi-material honeycombs under axial, bending and
shear eect.
The non-dimensionalized values of elastic constitutive parameters for nonauxetic honeycombs (
E1=E2=
qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
,
η1=η2=η3=η=
0.5 to 1, and
ζ1=ζ2=ζ3=ζ= 1
,
q= 2
) are plotted with the angle ratio
θ3/θ2
under the consideration of bending, axial and shear deformations of cell
walls. The variation of constitutive constants is plotted for dierent
η
ratios (
η=
0.5 to 1) with
ζ= 1
. The normalization
of the constitutive constants (
Cij
) is carried out with respect to the intrinsic Young's modulus (
Es
) of the vertical wall.
Variations with respect to angle ratio
θ3/θ2
considering dierent values of
η
are presented for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In these gures, the arrows depict the increasing trend of cell wall
thickness ratio (
η
) from lower (
0.5
, which represents that the thickness of cell walls is twice near the joints compared to
the middle part) to higher values (
η= 1
, which represents that the thickness of the cell walls is uniform throughout the
length).
cated as
tih
in Fig 3(b); note that we have dropped the subscript in most discussions in this section as
thickness of all the three members are taken as same) is studied further on the in-plane constitutive be-
havior of multi-material (
ζ1=ζ2=ζ3=ζ= 5
,
η1=η2=η3=η= 0.5
,
q= 2
) non-auxetic and auxetic
honeycombs. In Fig. 13, and SM1.S20 to SM1.S22, the elastic constitutive constants (
Cij
) are investi-
gated considering dierent thickness of cell walls (
t= 0.06L
to
t= 0.2L
, ranging from thin to moderately
39

thick). Such an analysis would further emphasize the importance of considering shear and axial beam-
level deformations for thicker cell wall honeycombs. In Fig. SM1.S20, the non-dimensionalized value
of elastic constitutive constants (
Cij
) for thin to thick-walled non-auxetic multi-material honeycombs
(
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ= 5
,
q= 2
) are plotted for dierent cell angles under the consideration of only bending deformations of cell
walls. Similarly, the non-dimensionalized value of elastic constitutive constants (
Cij
) for thin to thick-
walled non-auxetic honeycomb is plotted in Fig. 13 at dierent cell angles by consideration of axial,
shear and bending deformations of cell walls. It is observed that all the in-plane elastic constitutive
constants (
Cij
) are signicantly aected by the thickness of cell walls (
t
). In general, the magnitude of
elastic constitutive constants (
Cij
) decreases signicantly as the value of the thickness of cell walls (
t
)
increases from 0.06 to 0.2. It is noted that the eect of cell wall thickness on
C11
,
C13
and
C31
becomes
more prominent as the cell wall angle increases, whereas this eect on
C12
and
C21
is maximum at cell
wall angle
45◦
. Similarly, the eect of cell wall thickness
t
on
C22
,
C23
,
C32
and
C33
decrease as the
cell wall angle approaches higher values. From Fig. SM1.S20 and 13, it can be noted that the in-plane
constitutive constants (
Cij
) for thin-walled honeycomb is not aected much by considering the of axial
and shear deformations of cell walls. The inuence of axial and shear deformations becomes higher in
the case of thicker cell walls. In Fig. SM1.S21, the non-dimensionalized value of elastic constitutive
constants (
Cij
) for thin to thick-walled auxetic multi-material honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ= 5
,
q= 2
) is plotted
at dierent cell angles (
θ
) under the consideration of only bending deformations of cell walls. Further,
the additional eect of shear and axial deformations in the case of auxetic honeycombs is included in
Fig. SM1.S22. In multi-material auxetic honeycombs, it is noted that the eect of wall thickness on
C11
,
C12
and
C21
is more prominent when cell wall angle is around
45◦
, whereas this eect of wall thickness
on
C22
is maximum for cell wall angle between
10◦
to
40◦
. Similarly, the eect of wall thickness on
C13
,
C31
,
C23
,
C32
and
C33
becomes prominent as the cell wall angle increases.
The eect of wall thickness is further explored for nonauxetic and auxetic congurations of the
asymmetric multi-material lattices, as shown in Figs. 14 and SM1.S23. Here, if the angle ratio
θ3/θ2
is
considered as 1, then the results are obtained same as symmetric multi-material nonauxetic and auxetic
congurations. In general, increasing cell wall thickness leads to a lesser value of the constitutive
constants, while there exists an interesting interplay among the other design parameters for developing
multifunctional optimized lattice congurations.
To understand the interplay among unit cell geometry in terms of side wall lengths, cell angles and
40

Figure 13: Eect of wall thickness on complete in-plane constitutive (elastic modulus) matrix for non-
auxetic multi-material variably-thickened cell wall symmetric honeycombs under bending, axial and shear
deformations.
The non-dimensionalized values of elastic constitutive parameters for thin to thick-walled non-auxetic
honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ= 5
,
q= 2
)
are plotted considering dierent cell angles under bending, axial and shear deformations of cell walls. The variation of
constitutive constants is plotted for dierent wall thicknesses to length ratio
t/L
, thin (
t
=0.06
L
) to thick (
t
=0.2
L
) with
η= 0.5
and
ζ= 5
. The normalization of the constitutive constants (
Cij
) is carried out with respect to the intrinsic
Young's modulus (
Es
) of the vertical wall. Variations with respect to cell angle
θ
(in degree) considering dierent values
of
t/L
are presented for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In these gures, the
arrows depict the increasing trend of thickness (
t/L
) from lower to higher values.
multimaterial parameters along with beam-level architectures, the eect of vertical wall length to slant
walls length ratio (
H/L
) on the eective elastic properties of honeycomb is studied for variably-thickened
multi-material non-auxetic and auxetic honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ= 5
,
q= 2
). In Fig. 15, and
SM1.S24 to SM1.S26, the elastic constitutive constants (
Cij
) are presented for dierent
H/L
values
41

Figure 14: Eect of cell wall thickness on complete in-plane constitutive (elastic modulus) matrix for
non-auxetic asymmetric multi-material variably-thickened cell wall honeycombs under axial, bending and
shear eects.
The non-dimensionalized values of elastic constitutive parameters for thin to thick-walled nonauxetic
honeycombs (
E1=E2=qE3=Es
,
L1=L2=L3=L= 1mm
,
t1=t2=t3= 0.2L
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ= 5
,
q= 2
) are plotted with cell angle ratios
θ3/θ2
under the consideration of axial, bending and shear
deformations of cell walls. The variation of constitutive constants is plotted for dierent wall thicknesses to length ratio
t/L
, thin (
t
=0.06
L
) to thick (
t
=0.2
L
), with
η= 0.5
and
ζ= 5
. The normalization of the constitutive constants (
Cij
) is
carried out with respect to the intrinsic Young's modulus (
Es
) of the vertical wall. Variations with respect to angle ratio
θ3/θ2
considering dierent values of
t/L
are presented for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In these gures, the arrows depict the increasing trend of thickness (
t/L
) from lower to higher values.
considering
t1=t2=t3= 0.2L
, ranging from 1 to 1.5 (non-auxetic congurations). Figs. SM2.S37
to SM2.S40 in the supplementary material elaborate the variation of in-plane constitutive constants
for relatively thinner-walled and thicker-walled non-auxetic congurations. In Fig. SM1.S24, the non-
dimensionalized elastic constitutive constants (
Cij
) for non-auxetic variably-thickened multi-material
honeycombs are plotted for dierent
H/L
ratios under dierent cell angle by considering only bending
42

deformations of cell walls. Similarly, the non-dimensionalized value of elastic constitutive constants
(
Cij
) for non-auxetic honeycombs with dierent
H/L
ratios is plotted in Fig. 15 at dierent cell angles
by considering the axial, shear and bending deformations of cell walls. It is observed that all the elastic
constitutive constants (
Cij
) are signicantly aected by
H/L
ratio. The eect of
H/L
ratio on
C11
,
C13
and
C31
becomes more prominent at higher cell wall angles, whereas its eect on
C22
,
C23
,
C32
and
C33
is maximum when cell wall angle is smaller. Similarly, eect of
H/L
ratio on
C12
and
C21
is
maximum at the cell wall angle of
45◦
. It can further be noted that the contribution of the shear and
axial deformation eect is signicant on all the constitutive constants. The magnitude of all in-plane
constitutive constants is 10 to 15
%
higher under the consideration of axial, shear and bending eects
compared to thr only bending case.
Fig. SM1.S25 shows the variation
Cij
of auxetic variably-thickened multi-material honeycombs (
E1=
E2=qE3=Es
,
L1=H= 1
to 1.5mm,
L2=L3=L= 1
mm,
t1=t2=t3= 0.2L
,
η1=η2=η3=
η= 0.5
, and
ζ1=ζ2=ζ3=ζ= 5
,
q= 2
) for dierent
H/L
ratios and cell angles under the
consideration of only bending deformations of cell walls. Such variation for auxetic variably-thickened
multi-material honeycombs is also plotted in Fig. SM1.S26 under the combined eect of axial, shear
and bending deformations. For a more comprehensive analysis, the variation of in-plane constitutive
constants for thinner-walled and thicker-walled variably-thickened auxetic multi-material honeycombs
considering dierent
H/L
values is further elaborated in the supplementary material (refer to Figs.
SM2.S41 to SM2.S44). From the numerical results, it is noted that the eect of
H/L
ratio on
C11
,
C13
,
C31
and
C33
becomes more prominent as the cell wall angle increases, whereas the eect of
H/L
ratio
on
C12
,
C21
and
C22
is maximum for cell wall angle at
45◦
. Similarly, the eect of
H/L
ratio on
C23
and
C32
is maximum when the cell wall angle is in the range of
40◦
to
60◦
. The eect of
H/L
ratio on
C12
and
C21
is found to be maximum at the cell wall angle of
45◦
. From the comparative results of dierent
cell wall thicknesses, it becomes evident that the eect of axial and shear deformation becomes more
signicant for thicker-walled honeycombs.
The eect of the vertical wall-to-slant wall-length ratio (
H/L
) is studied further for asymmetric multi-
material nonauxetic and auxetic cases (refer to Fig. 16, SM1.S27, and SM2.S45 to SM2.S48). It can be
noted here that when the value of angle ratio
θ3/θ2
equals 1, the results are similar to symmetric multi-
material nonauxetic and auxetic congurations. In general, we show that the proposed bi-level design
space, including unit cell level geometric parameters such as a cell wall angles constituting asymmetry
in the hexagonal units, cell wall (multi-)material properties, thickness and dimensions along with beam
level architecture, can lead to optimum design of multi-functional metamaterials. In this context, it may
43

Figure 15: Eect of
H/L
ratio (ratio of the length of vertical and slant walls) on the complete in-plane
constitutive (elastic modulus) matrix for non-auxetic multi-material variably-thickened cell wall symmetric
honeycombs under bending, axial and shear deformations.
The non-dimensionalized values of elastic constitutive
parameters for non-auxetic honeycombs (
E1=E2=qE3=Es
,
L1=H= 1
to 1.5mm,
L2=L3=L= 1
mm,
t1=t2=t3= 0.2L
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ= 5
,
q= 2
) are plotted for dierent cell angles under
the consideration of bending, axial and shear deformations of cell walls. The variation of constitutive constants is plotted
for dierent H/L values with
η= 0.5
and
ζ= 5
. The normalization of the constitutive constants (
Cij
) is carried out
with respect to the intrinsic Young's modulus (
Es
) of the vertical wall. Variations with respect to cell angle
θ
(in degree)
considering dierent values of
H/L
are presented for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In these gures the arrows depict an increasing trend of
H/L
from
H/L = 1
to
H/L = 1.5
.
be noted that the design space can be signicantly expanded further by considering dierent values of
ti
,
ζi
and
ηi
for the three cell walls (
i= 1,2,3
). In addition, a range of other symmetric and asymmetric
architectures can be introduced at the beam level. Such results can be directly obtained (or with slight
modications) following the proposed analytical framework.
44

Figure 16: Eect of
H/L
ratio (ratio of the length of vertical and slant walls) on the complete in-plane
constitutive (elastic modulus) matrix for non-auxetic multi-material variably-thickened cell wall asym-
metric honeycombs under bending, axial and shear deformations.
The non-dimensionalized values of elastic
constitutive parameters for nonauxetic honeycombs (
E1=E2=qE3=Es
,
L1=H= 1
to 1.5mm,
L2=L3=L= 1
mm,
t1=t2=t3= 0.2L
,
η1=η2=η3=η= 0.5
, and
ζ1=ζ2=ζ3=ζ= 5
,
q= 2
) are plotted with angle ratio
θ3/θ2
under
the consideration of axial, bending and shear deformations of cell walls. The variation of constitutive constants is plotted
for dierent
H/L
values with
η= 0.5
and
ζ= 5
. The normalization of the constitutive constants (
Cij
) is carried out with
respect to the intrinsic Young's modulus (
Es
) of the vertical wall. Variations with respect to angle ratio
θ3/θ2
considering
dierent values of
H/L
are presented for
(a)
C11
(b)
C12
(c)
C13
(d)
C21
(e)
C22
(f)
C23
(g)
C31
(h)
C32
(i)
C33
. In
these gures, the arrows depict an increasing trend of
H/L
from
H/L = 1
to
H/L = 1.5
.
4. Conclusions and perspective
A computationally ecient, yet insightful analytical approach is developed here to determine the
eective elastic constitutive properties of a novel class of bi-level architected lattice metamaterials.
There are multiple notable facets (individually and their coupled inuence) in the proposed lattice
metamaterial that would improve the design space signicantly: (1) the entire elastic constitutive matrix
has been formulated considering the combined eect of bending, shear and axial deformation of the cell
45

walls (applicable for a wide range of lattices with thin to thick cell walls), (2) eect of multimaterial in
the unit cells is introduced, (3) asymmetric designs in the unit cells are coupled with the multimaterial
architecture, (4) beam-level variably-thickened architectures are proposed based on physical insights of
stress resultants. In the proposed framework, rst, the equivalent elastic moduli
C12
,
C22
, to
C23
are
determined for directions parallel to one cell wall of the variably-thickened multi-material honeycomb
based on the beam level deformation mechanics. Further, the coordinate transformation is exploited to
obtain all nine in-plane constitutive constants for asymmetric hexagonal honeycomb lattices.
A multi-stage validation approach, involving dierent beam-level deformation mechanics, multi-
material parameters and asymmetry, is adopted to ascertain adequate condence in the proposed an-
alytical framework. Subsequently, a detailed numerical study is performed considering auxetic and
non-auxetic congurations to explore the expanded design space of variably-thickened cell wall parame-
ters
η
and
ζ
, multi-material parameter
q
, cell wall thickness and side lengths, and geometric parameters
concerning asymmetry of the unit cell. The importance of considering axial and shear deformations
along with bending is demonstrated for thicker cell walls in the lattices. The numerical results reveal
that the variability-thickened cell wall parameters and the multimaterial parameter can alter all the
in-plane constitutive constants of the non-auxetic and auxetic lattices without changing the basic unit
cell geometry. Along with that, the coupled design space of asymmetry with auxetic and non-auxetic
congurations would play a vital role in the design of next-generation highly optimized honeycomb
structures for static and dynamic applications in various engineering elds.
Future research in this eld will cover aspects beyond just weight reduction and stiness-to-weight
ratio, including the enhancement and control in a more expanded design space for specic strength and
possibilities of localized failure in asymmetric geometries, stability, vibration and damping character-
istics. Moreover, eective control over the auxetic behaviour and the prospect of achieving any type
of functional properties gradation along the span can allow for desired eects such as improved impact
resistance and energy absorption, enhanced structural stability under dynamic loads and shape con-
formability. In this context, the proposed concept of variably-thickened honeycomb design can further
be investigated to enhance failure resistance through localized stiening and strengthening along with
improving the energy absorption capacity.
Besides the non-auxetic and auxetic congurations, the concept of current bi-level design can be
exploited directly for analyzing a range of lattice forms such as rectangular and rhombic lattices with
symmetric and asymmetric geometries, and further extended to other 2D and 3D lattices by considering
appropriate unit cells. Following the proposed analytical framework, the design space can be signicantly
46

expanded further by considering dierent cell wall thicknesses and parameters of variably-thickened cell
walls (including consideration of continuous cell wall thickness variation) for the three cell walls. In
addition, a range of other symmetric and asymmetric architectures can be introduced at the beam
level as an extension of the current analytical framework. The disseminated generic concepts of this
paper (including the computationally ecient analytical framework) would lead to exploitable physics-
based insights for innovating topologically optimized multi-functional heterogeneous metamaterials with
tailorable and direction-dependent elastic properties.
Supplementary materials
Supplementary Material 1 (SM1):
In this Supplementary Material 1 (SM1), we present sup-
plementary numerical results considering
t/L = 0.2
, encompassing (1) non-auxetic lattices with only
beam-level bending deformation, and (2) auxetic lattices considering only beam-level bending deforma-
tion and the combined eect of beam-level bending, axial and shear deformations. Further, additional
validation results are presented to reinforce the condence in the current analytical framework.
Supplementary Material 2 (SM2):
The results presented in the main paper and Supplementary
Material 1 (SM1) are obtained for
t/L = 0.2
, unless otherwise mentioned. For a comprehensive un-
derstanding, we provide additional numerical results here for non-auxetic and auxetic honeycombs
considering two other cell wall thicknesses, as
t/L = 0.1,0.3
(relatively thinner and thicker cell walls)
with the variation of dierent bi-level geometric parameters such as
θ
,
ζ
,
η
,
q
, and
H/L
etc. (symmetric
and asymmetric congurations).
Acknowledgments
MA acknowledges the nancial support from the Ministry of Education, India through a doc-
toral scholarship. TM would like to acknowledge the Initiation grant received from the University
of Southampton during the period of this research work.
Data availability
All data sets used to generate the results are available in the main paper. Further details could be
obtained from the corresponding authors upon reasonable request.
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