Systematic design of high-strength multicomponent metamaterials

Abstract

The emergence of additive manufacturing, along with the introduction of the concept of metamaterials, allows the synthesis of high-performance materials with superior specific strength. With recent advances in printing multi-material structures, the design space of metamaterials has exponentially grown. Variation in dimensions of the printed metamaterials due to limitations of the manufacturing process can drastically offset their performance compared to their original design. So far, the impact of deviations in the manufactured metamaterials and their effect on their final performance has not been studied systematically. There are also no guidelines for selecting materials in a multi-material lattice structure to achieve higher mechanical performance. Here, the strength and toughness of printed single- and bimaterial lattice structures with a combinatorial selection of materials and their sensitivity to the printing parameters are studied. We show that the exterior elements dominate the overall mechanical performance of the metamaterial compared to the internal elements. We found two regimes of slow and fast softening in periodic lattices. We study the sensitivity of the mechanical performance of the printed metamaterial to variations in the thickness of internal and exterior elements in detail. Keywords: Metamaterial, Materials by design, Additive manufacturing, Sensitivity analysis, Multimaterial lattice structures

Full text

Systematic design of high-strength multicomponent metamaterials
Kasra Momeni
a
,
b
,
c
,
*
, S.M. Mahdi Mofidian
a
,
b
, Hamzeh Bardaweel
a
,
b
a
Department of Mechanical Engineering, Louisiana Tech University, Ruston, LA 71272, United States of America
b
Institute for Micromanufacturing, Louisiana Tech University, Ruston, LA 71272, United States of America
c
Materials Research Institute, Pennsylvania State University, University Park, PA 16802, United States of America
highlights graphical abstract
The properties of the material of
outer elements will dominate the
overall properties of the lattice
structure;
A bilinear behavior in the Force-
Displacement curve of the individ-
ual unit cells with free lateral ele-
ments is revealed
Stress-controlled instabilities are iden-
tified in octet-truss lattice structures;
A critical displacement is found at
which the elastic energy becomes in-
dependent of the printing parameters.
article info
Article history:
Received 3 June 2019
Received in revised form
12 August 2019
Accepted 13 August 2019
Available online 15 August 2019
Keywords:
Metamaterial
Materials by design
Additive manufacturing
Sensitivity analysis
Multimaterial lattice structures
Data availability
The raw data required to reproduce these
findings will be provided upon request.
abstract
The emergence of additive manufacturing, along with the introduction of the concept of metamaterials,
allows the synthesis of high-performance materials with superior specific strength. With recent ad-
vances in printing multi-material structures, the design space of metamaterials has exponentially grown.
Variation in dimensions of the printed metamaterials due to limitations of the manufacturing process
can drastically offset their performance compared to their original design. So far, the impact of deviations
in the manufactured metamaterials and their effect on their final performance has not been studied
systematically. There are also no guidelines for selecting materials in a multi-material lattice structure to
achieve higher mechanical performance. Here, the strength and toughness of printed single- and
bimaterial lattice structures with a combinatorial selection of materials and their sensitivity to the
printing parameters are studied. We show that the exterior elements dominate the overall mechanical
performance of the metamaterial compared to the internal elements. We found two regimes of slow and
fast softening in periodic lattices. We study the sensitivity of the mechanical performance of the printed
metamaterial to variations in the thickness of internal and exterior elements in detail.
Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://
creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
The advent of additive manufacturing (AM) has provided an un-
precedented opportunity for the production of functionally graded
and complex structures, including lattice structures of repeated
unit cells known as metamaterials. These materials have various
applications, including energy absorption [1], improved strength/
*Corresponding author at: Department of Mechanical Engineering, Louisiana
Tech University, Ruston, LA 71272, United States of America.
E-mail address: kmomeni@latech.edu,kzm5606@psu.edu (K. Momeni).
Contents lists available at ScienceDirect
Materials and Design
journal homepage: www.elsevier.com/locate/matdes
https://doi.org/10.1016/j.matdes.2019.108124
0264-1275/Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Materials and Design 183 (2019) 108124

stiffness to weight ratio [2e4], opposite directions of phase and
group velocities [5], polarization-independent broadband absorbers
[6], sensors [7], energy harvesting [8], and mitigation of selected
failure modes [9e11]. In addition to the synthetic metamaterials,
nature has also utilized repeated hierarchical patterns where both
strength and weight factors are taken into account, such as in nacre
shell [12] and bone [13]. The geometric design of the metamaterial
structures is based on either engineered optimization or inspired by
nature and biological architectures. The design for enhancement of
mechanical properties [14,15] of these structures is usually in a
repeated arrangement fashion rather than a stochastic one, which is
called metamaterials. Several of these metamaterials have already
been designed; for example, mechanically robust carbon nanotube
foams [16], nanoporous silica [17], and crystal lattice configurations
[1]. These engineered materials have been applied in various in-
dustries, such as aerospace, automobile, defense, or construction
[18,19]. Attempts to make materials with abnormal properties, e.g.,
negative Poisson's ratio, have also been reported [20,21].
There is a growing interest in employing these materials in engi-
neering applications. However, their fabrication process using tradi-
tional approaches prevents manufacturing designs with complex
geometries and is commonly limited to a single material. Using the
conventional methods, such as casting or forging, may adversely affect
the properties of manufactured metamaterials even for the elemen-
tary geometries, e.g., by forming residual stresses in the structure [1].
Thus, new fabrication techniques such as AM methods need to be
utilized, specifically to manufacture metamaterials of various mate-
rials and complex geometries. Moreover, the AM techniques give a
cheap and accurate product that is easy to adapt. The use of CAD
sketches presents excellent control over the editing of complicated
geometries of architectures [22]. It is also more versatile and acces-
sible rather than most of the traditional fabrication methods. The AM
techniques can also be integrated with other processes, such as
electroless deposition [23]orelectroplating[24], to make multi-
material structures.
The most commercially viable and versatile additive manufacturing
method is the Fused Deposition Modeling (FDM) process, which has
various applications ranging from printing toys [25]toarmors[26], to
in-space manufacturing [27,28]. One critical technical challenge asso-
ciated with the FDM of metamaterials is the deviation of the printed
structures from the original design, which is a strong function of the
printing parameters and their tolerances. These deviations can
significantly degrade the mechanical performance of metamaterials.
Thus, understanding the sensitivity of the metamaterial design to
variations in the printing process is crucial for designing robust ma-
terials with reliable performance. A combined experimental and
computational approach is pursued here to characterize the perfor-
mance of metamaterials and analyze the sensitivity of their mechan-
ical performance to the variations in the printing process.
In the present study, we fabricated the octet-truss structure
using the FDM process. We have investigated the strength and
energy storage capability of four different unit cell designs: (i)
Polylactic Acid (PLA), (ii) Tough Polylactic Acid (TPLA), (iii) bima-
terial unit cells of exterior elements of TPLA and interior elements
of PLA (called TO), and (iv) bimaterial unit cells of interior elements
of TPLA and exterior elements of PLA (called TI). The designed
metamaterials have implications in energy absorption, blast and
impact mitigation, and aerospace applications where there is a
need for materials with high strength to weight ratios. Their shape
recovery capability even under large deformation, allows them to
withstand sequential impacts [29]. This property also gives them an
advantage over the foams [30,31] or metals which undergo irre-
versible plastic deformation under large deformations [32e34]. We
characterized the printed samples and measured their mechanical
strength using compression tests. Two types of samples, (1) a single
unit cell and (2) a 3 3 lattice, are considered to understand the
significance of the boundary elements and interior elements on the
overall properties of the material. The single unit cell is the
building-block of the metamaterials, and investigating its behavior
is crucial for understanding the global response of the structure.
However, a large number of elements in a single unit cell are the
exterior elements with no periodicity (two-thirds or ~67% in an
octet-truss lattice structure), which may hinder generalizing the
conclusions to metamaterials with many periodic cells. To over-
come this challenge, we also studied the behavior of 3 3 lattice
structures, which are the structures with a minimum number of
cells that have a unit cell with periodic boundaries. Although lattice
structures with more than three unit cells in each direction could
provide more statistical details, we do not expect to gain any new
fundamental knowledge as each unit cell only interacts with its first
nearest neighboring cells. Furthermore, printing structures with
more unit cells will also be time-consuming and costly.
Five samples for each type of metamaterial are prepared to have
statistically representative measurements. We developed analytical
models of free and periodic unit cells, which are solved using the
finite element method. We defined two structural parameters
determining the performance of the final printed metamaterial, i.e.,
effective stiffness and stored elastic energy. Pursuing an innovative
approach, we performed a sensitivity analysis to identify the most
critical design and manufacturing parameters determining the
characteristics of the final printed metamaterial.
2. Methods
2.1. Experimental procedure
We draw the 3D CAD sketch of the specimens and created the
corresponding STL file using the SOLIDWORKS®software [25]. The
STL file was sliced into layers by Cura software [35] to generate the
g-code for printing with an Ultimaker 3 printer. The materials used
in the design are PLA (from Ultimaker) and TPLA (from KODAK)
filaments. The isotropic model is used to describe the behavior of
PLA and TPLA, which will be fully defined using two constants.
Here, we considered the elastic modulus, E, and Poisson's ratio,
g
,as
the two model material constants. The associated values are ob-
tained from the manufacturers' technical data sheets and listed in
Table 1. For printing the structures, the temperature of the build
plate was set to 60

C, the printing speed was 20 mm/s, layer height
was 0.1 mm, the line width was 0.45mm, and the infill density was
set to 100%. For PLA (TPLA), we used a nozzle with a 0.4 mm
(0.5 mm) diameter, extrusion multiplier was 100% (105%), and the
nozzle temperature for printing the first layer was 205

C(210

C)
which changes to 215

C (220

C) for the subsequent layers.
The tests were performed using two tensile test machines: (1)
PASCO Materials Testing System for testing the single unit cell
samples, and the (2) Tinius Olsen testing machine to test the 3 3
unit cell samples. The first machine has a higher measurement
accuracy; however, its force capacity is limited (a maximum force
capacity of 7.1 kN) and only can apply a large enough force to take a
single unit cell structure to the failure point. In contrast, the latter
testing system has a higher force capacity (a maximum force ca-
pacity of 30 0 kN) but lower measurement accuracy. Thus, we only
used the second machine for compression testing of the 3 3
Table 1
Mechanical properties of PLA and TPLA.
Property PLA TPLA
Elastic Modulus, E (MPa) 2346.5 ±0.1 2447.6 ±0.1
Poisson's Ratio,
g
0.36 ±0.01 0.36 ±0.01
Density,
r
(kg/m
3
) 1240 1220
K. Momeni et al. / Materials and Design 183 (2019) 1081242

lattice structures. A schematic of the experiment apparatuses is
shown in Fig. 1. In the first step, we have connected the tensile
testing machine to the associated data acquisition software
(SPARKvue for PASCO machine and Horizon for the Tinius Olsen
machine). Then, we preloaded each sample with a compressive
load of <100Nbefore initiating the destructive compression test.
We zeroed the load cell after each test. For the PASCO machine,
several loading/unloading cycles are performed during the pre-
loading until both loading and unloading curves coincide. To avoid
dynamic loading effects, we limited the compression displacement
rate to 20 mm/min. Both testing machines measure the applied
force and the corresponding displacement. The two sample types,
i.e., single unit cell and 3 3 unit cell, and their material combi-
nations were tested (see insets in Figs. 2 and 3). We placed the
samples at the center of the loading plates to maximize the uni-
formity of the load distribution. This alignment is vital to prevent
any offset forces, such as bending forces, on the specimens. We ran
the tests till the specimens collapsed, which is equivalent to the
drop of the measured force below a threshold, or the structure of
metamaterial collapses to the point that it no longer can be
considered as a lattice structure.
2.2. Computational model
Equations governing the behavior of metamaterials are the ki-
netic, constitutive, and equilibrium elasticity equations:
ε¼1
2VuþVu
T
;ε
e
¼εε
0
;(1)
s
¼C:ε
e
;(2)
V∙
s
¼0;(3)
where uis the displacement vector, εis strain tensor, superscript
T is the transpose operation, ε
e
is the elastic strain tensor, ε
0
is
the initial strain that captures the lumped effect of thermal/trans-
formation strains,
s
is the stress, and Cis the stiffness tensors. In
Fig. 1. Schematic of the load cell. Two tensile test machines are utilized (a) PASCO for
testing single unit cells, and (b) Tinius Olsen for testing the 3 3 lattice structures.
Despite their different loading capacity, they follow the same principle that is shown
here.
Fig. 2. Experimentally measured load capacity of single unit cells. The force-displacement curves for single unit cells with free boundary condition on the lateral cells has been
considered for different materials; (a) PLA, (b) TPLA, (c) TO, and (d) TI. Using the tough material for making the exterior elements in a bimaterial unit cell is more effective in
toughening the lattice structure.
K. Momeni et al. / Materials and Design 183 (2019) 108124 3

the finite element formulation of the problem, the displacement
vector field will be solved, which subsequently gives the stresses
and strains in the lattice structure. Here, we modeled the material
as an isotropic elastic material with the mechanical properties lis-
ted in Table 1. The system of equations was solved using the finite
element technique where we used free tetrahedral meshes with
quadratic shape functions to discretize the domain. Simulations
were performed for the two classes of unit cells and the four
different material combinations, i.e., PLA, TPLA. TI, and TO. The
stress distribution in the elements, as well as the deformed
Fig. 3. Experimentally measured load capacity of a lattice structure. The force-displacement curves for 3 31 lattice structures are considered for different materials; (a) PLA, (b)
TPLA, (c) TO, and (d) TI. These results indicate that using the tough material for making the exterior elements in a bimaterial unit cell is more effective in toughening the lattice
structure.
Fig. 4. von Misses stress in individual unit cells of single and bimaterial lattice structures at an axial strain of 12.5%. (a) The stress distribution in a single non-periodic unit cell. The
structure is loaded from the top, fixed at the bottom, and there is no load on the side elements. Maximum stress occurs at the corners where elements are joined or at the middle of
the buckled elements. (b) The stress distribution in a unit cell of the single and bimaterial periodic lattice structures. The structure was loaded from the top, fixed at the bottom, and
periodic on the side elements. Maximum stress occurs at the corners where elements are joined or at the middle of the buckled elements. Two types of dihedral structures are
identified, after loading (i) with expanded volume, and (ii) contracted volume.
K. Momeni et al. / Materials and Design 183 (2019) 1081244

structure of the unit cells, was calculated.
Two performance measures, (1) average stored elastic energy
per unit volume, and (2) effective stiffness, are defined as follows:
E¼0:5ε:C:ε(4)
S
eff
¼F
z
.U
z
;(5)
where F
z
is the applied compressive load, and U
z
is the displace-
ment in the direction of force.
Sensitivity of the metamaterial characteristics to the variations
of the manufactured unit cells, i.e., deviation in the diameter of
interior (t
in
) and exterior (t
out
) elements, has been calculated at
different displacements using two scalar-valued performance
measures, Eand S
eff
. The two control variables (t
in
and t
out
) are not
directly related to these sensitivity objective functions (Eand S
eff
),
but they are rather connected through the displacement vector
field u. Mathematically speaking, we have E¼Eðuðt
x
Þ;t
x
Þand
S
eff
¼S
eff
(u(t
x
),t
x
) where x¼in or out. Assuming the unique solution
to Eq. (3) to be u¼L
1
(t
x
), the sensitivity of these objective func-
tions can be calculated using the chain rule as,
d
dt
x
Eðuðt
x
Þ;t
x
Þ¼vE
vt
x
þvE
vu
∙
vu
vL
∙
vL
vt
x
;(6)
d
dt
x
S
eff
ðuðt
x
Þ;t
x
Þ¼vS
eff
vt
x
þvS
eff
vu
∙
vu
vL
∙
vL
vt
x
:(7)
The first terms in these equations can be computed using
symbolic differentiation if an analytical correlation between the
objective-functions and the control variables t
x
is established.
Nevertheless, finding the second term is more difficult. Assuming
that the solution has Ndegrees of freedom,
vE
vu
and
vS
eff
vu
are N-by-1
matrices, and vu/vL¼(vL/vu)
1
is an N-by-Nmatrix where vL/vuis
the Jacobian, and
vL
vt
x
is an N-by-2matrix. However, the calculation of
the Jacobian matrix is computationally expensive. An auxiliary
linear problem has been introduced to avoid such expensive cal-
culations Jacobian matrix, where we used the Forward sensitivity
method [36e38] and introduced the N-by-2matrix of solution
sensitivities
vu
vt
x
¼vL
vu
1
∙
vL
vt
x
;(8)
which can be evaluated by solving the following two linear systems
of equations using the same Jacobian, vL/vu, evaluated at the
nominal element thicknesses,
vL
vu
∙
vu
vt
in
¼vL
vt
in
;(9)
vL
vu
∙
vu
vt
out
¼vL
vt
out
:(10)
Substituting Eqs. (9)e(10) in Eqs. (6)e(7), we will be able to
calculate sensitivity of the average elastic energy and effective
stiffness to variation in the diameter of the printed elements.
3. Results and discussion
Experimental measurements are performed for five different
samples for each case study, which are designated by S1 to S5 in
Figs. 2 and 3. For the developed numerical model, the simplifying
assumptions made in developing the corresponding mathematical
models, including assuming an isotropic material model, uniform
Fig. 5. Experimentally maximum load-carrying capacity per unit cell of the printed lattice structures. Measurements are performed for both (a) single unit cell structures, and (b)
33 unit cell structures.
Fig. 6. Force-displacement curves for building blocks of metamaterials. The force-displacement curves for single and bimaterial unit cells with free (a) and periodic (b) circum-
ferential boundaries. The force increases monotonously with increasing displacement for the unit cells with free circumferential boundaries, while the unit cells with periodic
circumferential boundaries show a drop in the carried load after the load reaches a maximumvalue. This indicates a structural instability in the periodic lattice materials beyond the
critical displacement of U
z
¼2 mm.
K. Momeni et al. / Materials and Design 183 (2019) 108124 5

diameter across the printed elements, and uniform loading of the
structure. However, these assumptions are relaxed in the experi-
mental characterizations reported in Figs. 2 and 3 and are the
source of the variation in the measurements reported for different
samples S1 to S5. We may refer to the nonuniform diameter of the
elements along the axis of the elements as a result of layer by layer
printing of the structure, nonideal mixing of PLA and TPLA in the
nodes for the bimaterial unit cells, and nonuniform loading of the
structure that may result in an equivalent torque during testing.
The experimental measurements on single unit cells (Fig. 2a,b)
indicate that PLA and TPLA have comparable elastic moduli. How-
ever, the TPLA is more sensitive to printing conditions where it led
to a larger variation in the elastic modulus of the printed unit cells
(Fig. 2b). There are also cases that the printed TPLA unit cells had
even lower toughness than the PLA itself, which was due to the
uncertainty in the manufactured unit cells. Although the maximum
strain that TPLA sustains is generally larger than the PLA, it has a
lower capacity for carrying loads. We further tested bimaterial unit
cells where TPLA was used to make the exterior and interior ele-
ments, i.e., TO and TI, respectively. Our results (Fig. 2c,d) indicate
that the maximum strain that the bimaterial structures can sustain
is determined by the lower toughness constituent, which was PLA
in our case. However, the sensitivity to the printing parameters was
reduced in the bimaterial lattice structure in comparison with the
TPLA unit cells, and less variation in the elastic moduli was
observed. The TO composite unit cell had also demonstrated a
multistage failure mode, indicating a more effective toughening
when TPLA was used for making the exterior elements.
In contrast to the single-material individual unit cells, single-
material lattice structures of several unit cells of PLA show a
larger variation in their elastic moduli (Fig. 3a,b). This variation
indicates that volumetric defects are more susceptible to form
in PLA, while TPLA is more sensitive to the variations in the
manufacturing parameters. The TPLA sustains a smaller force
compared to the PLA. However, TPLA has more active dissipative
mechanisms, which results in higher toughness. The TI lattice
structures have a higher load-carrying capacity than the TO. On the
other hand, the toughness of the TO lattice structures is higher than
the TI. Thus, the properties of the exterior elements dominate the
overall response of the composite lattices.
The maximum load per unit cell, F
z
max
, that each structure can
carry was measured for all the material combinations (Fig. 5). The
results show that for the single unit cells, the maximum load-carrying
capacity is highest for PLA, and it is reduced by ~20% for the other
materials. Also, TPLA has the largest variation in the F
z
max
, indicating
its sensitivity to the printing parameters. The same conclusion can be
made about the maximum load-carrying capacity of the 3 3 lattice
structure. The variation in the measured F
z
max
values of 3 3 lattices is
comparable in all materials except the TI lattice structure where the
variation is half the other materials. The overall variation in the
measured F
z
max
is larger for the 3 3 unit cells compared to the single
unit cells, due to the higher probability for the presence of structural
defects due to the larger volume. Comparing the F
z
max
for a single unit
cell and a 3 3 lattice structure, the load-carrying capacity of single
unit cell composite structures lays between the load-carrying ca-
pacity of monomaterial single unit cells. However, the maximum
load-carrying capacity of the lattice structures of composite materials
will be determined by the weakest elements. Furthermore, the
experimental measurements indicate a ~20% increase in the
maximum load-carrying capacity per unit volume of the lattice
structure compared to the single unit cells, indicating the profound
effect of boundary elements on the overall load-carrying capacity.
The stress analysis (von Mises stress) for the unit cells of single-
and bimaterial components is performed for two cases of unit cells
with (i) no load on lateral boundary elements, and (ii) with periodic
boundary conditions using finite element simulations, Fig. 4.In
these simulations, we have neglected the effect of variation in the
diameter of each element that may occur during the actual printing
process and assumed uniform loading of the unit cells. Thus, all the
side elements have the same stress distribution due to symmetry.
The first case represents unit cells that construct the boundary cells
of the metamaterial, while the latter represents the unit cells in the
core of the structure. Here, the bottom surface is fixed, and the top
surface has been displaced under quasi-static loading. In the case
Table 2
Performance of the designed metamaterials. The experimentally measured relative density
r
/
r
s
and transmitted stress
s
tr
of the designed metamaterials are compared with
other lattice structures.
Al Foam Duocel [39] EPS Foam [40] Al Honeycomb [41] Twin Hemispheres Skydex [39] PLA
11/3 3
TPLA
11/3 3
TO
11/3 3
TI
11/3 3
r
(kg/m
3
) 185 66 35 213 141 139 140.1 140.2
r
/
r
s
0.068 0.062 0.013 0.079 0.115 0.113 0.114 0.114
s
tr
(MPa) 2.67 1.40 1.20 0.21 2.65 2.1 2.23 2.35
2.94 2.45 2.38 2.36
Fig. 7. Experimentally measured average elastic energy stored in the printed lattice structures. Measurements are performed for both (a) single unit cell structures, and (b) 3 3
unit cell structures.
K. Momeni et al. / Materials and Design 183 (2019) 1081246

(i), the maximum stress occurs at the corners, where different el-
ements are joined, as well as the center of surrounding exterior
elements. Both sublattices in the unit cell, tetrahedral and octahe-
dral sites, expand outward during the loading. The von Mises stress
in the exterior elements is generally higher than its value in the
internal elements, except for the TI unit cell. In contrast, in the case
(ii), for the unit cells with periodic circumferential elements, the
octahedral element is distorted, and the tetrahedral sites can be
divided into two classes: (a) sites which their volume has expanded
and (b) sites which their volume has been contracted.
We found a good correspondence between the simulations
and experiments, Figs. 5 and 6. The maximum F
z
occurs at
U
z
¼1e1.2 mm and at U
z
¼1.2 e1.5 mm for the single unit cell and
33 lattice structures, respectively. Our simulations predict 0.7
kN <F
z
max
<0.9 kN for the single unit cells, and 1.4 kN <F
z
max
<1.7
kN for the 3 3 lattice structures. The slightly higher values of F
z
max
predicted by simulations compared to the experimental measure-
ments for the lattice structure can be interpreted using the
following two reasons: (i) the printed lattice has a finite number of
cells, and (ii) the presence of defects in the printed structure. One of
the key performance measures for metamaterials is the transmitted
stress,
s
tr
, versus relative density of the metamaterial,
r
/
r
s
, where
r
is the density of the metamaterial and
r
s
is the density of the solid
base material. We have calculated these two performance measures
and compared them with some of the lattice structures in Table 2.
Here, we assumed
r
s
to be the average density of PLA and TPLA, i.e.,
1.23 g/cm
3
. We need to know the volume of the metamaterial to
calcualte
r
. There are two different ways for calculating the volume
of metamaterial: (i) calculating the volume of actual printed ma-
terial (excluding the empty spaces in the cell), and (ii) calculating
the volume of the whole structure (including the empty space in
the lattice). The latter approach will be used for calculating the
value of
r
of the metamaterials designed here. The volume of
exterior and interior elements are 477.84 mm
3
and 433.76 mm
3
,
respectively. The volume of a unit cell is 8 cm
3
.
The free exterior elements will buckle at a lower load, which will
cause the change in the slope of the F
z
curve. This is in effect of the
drop in the strength of the structure. The results presented in Fig. 6a
indicates that the response of single- and bimaterial unit cells, can
be effectively modeled with a bi-linear material. For periodic cells
in the core of the lattice material, a linear response has been
detected for both single and bimaterial sample unit cells over a
wide loading range, up to U
z
¼2 mm, which is followed by a drastic
drop in strength at higher loads. Comparing Fig. 6a and b reveal that
unit cells with free lateral elements, i.e., boundary unit cells, have
lower load-carrying capacity but are less sensitive to buckling of the
elements compared to the core unit cells. This is evident from the
sudden drop in the F
z
in Fig. 6b.
The average elastic energy that is stored in a unit cell up to a
displacement of U
z
¼2.5 mm was measured by calculating the area
under the force-displacement curves for different samples. In
Figs. 7 and 8, the average elastic energy per volume of material for a
unit cell, which was measured experimentally and calculated from
the numerical model, are presented, respectively. In these figures,
the volume considered is the volume of base material used in the
base structure, and is V¼0.91161 cm
3
.
The results presented in Fig. 7 indicates that for the single unit cell
structures, the TPLA stores the maximum elastic energy while the
composite cells have an energystorage capacity that is comparable to
that of PLA. Surprisingly, for a periodic lattice structure, the stored
elastic energy per unitcell in the composite lattices is comparable to
the energy stored in the TPLAlattice, and the TO has a slightly higher
toughness than the TI. The periodic lattices can also store a larger
elastic energy per unit cell compared to the single unit cells, i.e., at
least 60% higher. This is consistent with computational calculations
presented in Fig. 8. However, the experimentally measured specific
stored elastic energy for the PLA lattice structure is far smaller than
the predictions, due to the fact that the PLA structure fracture far
before reaching to a 2.5 mm displacement.
The average elastic energy stored in the system calculated from
the finite element model is plotted in Fig. 8, where we revealed the
higher load bearing capacity of the unit cell with periodic boundary
conditions under the same displacement. While the elastic energy
of the single unit cell structures with free boundary conditions
increases monotonously with increasing the applied displacement,
the elastic energy of periodic unit cells drops after reaching a large
Fig. 8. Average elastic energy in the lattice materials. (a) Average energy for single and bimaterial unit cells with free boundary conditions, showing a monotonous increase with increasing
the axial stress. (b) Average for single and bimaterial unit cells with periodic boundaries, revealing their higher load bearing capacity as well as their failure under large elastic energies.
Fig. 9. Effective stiffness of different metamaterials. (a) For an isolated unit cell with free boundary conditions, and (b) for a unit cell of periodic lattice structure. The stiffness
increases initially, followed by a reduction.
K. Momeni et al. / Materials and Design 183 (2019) 108124 7

enough elastic energy indicating buckling of the elements. Fig. 8b
indicates that the instabilities induced in the structure are stress-
controlled rather than strain-controlled because for the same
strain the free-boundary single unit cell (Fig. 8a) does not show any
drop in the average elastic energy curve. Furthermore, the elements
of bimaterial periodic lattice structure TI are more susceptible to
become unstable under the compressive loading.
The effective stiffness, Eq. (5), at different displacements is also
studied. It has a higher value for periodic lattice structures than for
the free unit cells. The former has a maximum of 1 kN/mm while
the latter has a maximum of 1.3 kN/mm. The softening behavior of
the unit cells with free boundary conditions, which represents the
boundary unit cells in a lattice structure, is entirely different from
the unit cells with periodic boundaries, which represent the inte-
rior unit cells in the structure. The free boundary unit cell has a
nonlinear softening response with the fastest softening rate at the
beginning. The periodic lattice shows two different rates of
reduction in effective stiffness. The first region has an almost linear
reduction rate, which is followed by a region of fast softening at
larger displacements. The softening in the cells with free bound-
aries occurs at a higher rate than the softening of the periodic cells
in the linear region. For example at the end of the linear softening
region, U
z
¼2mm(Fig. 9b), the axial compressive force has reduced
from a maximum of 1.3kN/mm to 1 kN/mm in the unit cell of pe-
riodic, which indicates ~24% reduction in the stiffness. In contrast,
the stiffness of the unit cell with free boundaries reduces from
1 kN/mm to 0.5 kN/mm at U
z
¼2 mm. That is equivalent to a 50%
reduction in the stiffness and is twice the corresponding reduction
in the unit cells with periodic boundaries. Among the considered
unit cell configurations, the unit cell of TI with periodic boundaries
Fig. 10. Sensitivity of the stored elastic energy to variations in the diameters of the interior and exterior elements. Sensitivity of the stored elastic energy in a unit cellto variation in
the diameter of interior elements when the surrounding boundary elements are not loaded (a) and for periodic boundary conditions (b). Same sensitivity analysis has been
performed with respect to variation in the diameter of exterior elements for free and periodic circumferential boundaries (c, d).
Fig. 11. Sensitivity of the effective stiffness to variations in the diameters of the interior and exterior elements. Sensitivity of the effective stiffness to variation in the diameter of
interior elements when the surrounding boundary elements are not loaded (a) and for periodic boundary conditions (b). Same sensitivity analysis has been performed with respect
to variation in the diameter of exterior elements for free and periodic circumferential boundaries (c, d).
K. Momeni et al. / Materials and Design 183 (2019) 1081248

reaches the instability point at a lower displacement, U
z
¼2 mm.
Among the considered unit cells, the one built from TPLA has the
largest stiffness, and the one made up of PLA has the lowest stiff-
ness except for the second softening region of the periodic cell
where the periodic TI unit cell becomes the softest structure. The TI
and TO unit cells have relatively the same stiffness that is between
TPLA and PLA except in the second region of softening of the pe-
riodic cells.
We also investigated the sensitivity of the energy stored in the
lattice material using the forward sensitivity method, see Eqs.
(6)e(10), for different unit cells to variation in the thickness of the
interior and exterior elements of the cell, i.e., t
in
and t
out
, respec-
tively. The objective function for this analysis is the elastic energy of
the cell, Eq. (4). The results are shown in Fig. 10. The sensitivity of
the elastic energy to variation of the interior elements' diameter of
the unit cells with free surrounding boundaries, Fig. 10a, monoto-
nously increases with increasing displacement. At small displace-
ments, the sensitivity of all structures is comparable. At larger
displacement, the sensitivity of PLA and TO will still be comparable
but slightly smaller than the TPLA and TI, which are relatively the
same except at extensive compressive strains where TI unit cells
become the most sensitive (30% higher). The sensitivity of elastic
energy to the variation of exterior elements of the unit cells with
free surrounding boundaries is shown in Fig. 10c, which shows a
monotonous increase in sensitivity by increasing displacement. At
any particular displacement, the sensitivity of unit cells in the
ascending order is PLA, TPLA, and bimaterial unit cells (TI and TO). It
is worth noting that the sensitivity of the stored elasticenergy with
respect to the diameter of exterior elements for bimaterial unit cells
is independent of the order of the material. The sensitivity of the
stored elastic energy to variation in the diameter of interior and
exterior elements of periodic lattice structures are shown in Fig. 10b
and d. The sensitivity with respect to changes in the diameter of the
interior elements, Fig. 10b, increases monotonously by increasing
displacement U
z
. The absolute value of sensitivity with respect to
the changes in the diameter of exterior elements increases by
increasing displacement till it reaches a maximum value, where it
starts to decrease to zero followed by an increase in the opposite
direction. Thus, there is critical displacement at which the elastic
energy stored in the unit cell will not be sensitive to the changes in
the diameter of the exterior elements of the lattice structure. The
S
E
out
for single material lattice structures follow the same trend,
while TPLA is slightly more sensitive than PLA. The bimaterial lat-
tice structures have the same S
E
out
. They are less sensitive to varia-
tions in the diameter of exterior elements, for small to moderate
displacements, compared to single material lattice structures.
However, at large displacements, the bimaterial lattice structures
become more sensitive than the single material lattice structures.
We further investigated the sensitivity of effective stiffness for
different unit cells to variation in the diameter of the interior and
exterior elements of the cell, i.e., S
stf
in
and S
stf
out
, respectively. The
objective function for this analysis is the effective stiffness, Eq. (5).
The results are shown in Fig. 11. The sensitivity of the interior and
exterior elements of the unit cells with free surrounding bound-
aries (Fig.11a,c) are comparable, except for TI at high loading strains
where the sensitivity increases by 50%. The sensitivity of the
effective stiffness to variations in the diameter of interior and
exterior elements for the periodic lattice material, Fig. 11b and d,
follow the same trend for both single-material and bimaterial
structures. It decreases linearly at low to medium strains and fol-
lows a nonlinear pattern at high strains. The sensitivity of effective
stiffness to variations in the diameter of the interior elements is the
same for the bimaterial periodic lattice materials TI and TO for low
to medium strains. However, at larger displacements, the sensi-
tivity of the TO periodic lattice is higher than TI, but they will
coincide again at large displacements (U
z
¼2.5 mm). This
sensitivity for the exterior elements of bimaterial periodic lattices
TI and TO has the same trends, Fig. 11d, although the unit cells of
TPLA are slightly more sensitive than the unit cells of PLA.
4. Conclusion
Variations of octet-truss metamaterial design have been
considered here, and the effect of combinatorial changes in mate-
rials on the load-carrying capacity, stored elastic energy, and
effective stiffness of these structures is studied. We developed a
numerical model and performed the sensitivity analysis using the
Forward sensitivity method, where we defined two scalar-valued
performance measures, i.e., the stored elastic energy Eand effec-
tive stiffness S
eff
. We studied the sensitivity of these parameters
concerning variations of manufacturing parameters. Different sur-
prising and new phenomena were captured (listed below), which
can guide the optimum design of high-performance metamaterials.
We have shown that for the octet-truss metamaterial, the ma-
terial of exterior elements will dominate the overall properties of
the lattice structure. We have shown that the maximum load-
carrying ability per unit volume of the lattice structure is higher
than the single unit cell. This finding indicates the crucial role of
boundary elements on the overall load-carrying capacity of the
metamaterials. The higher F
z
max
calculated from the simulations
compared to the experimental measurements for the lattice
structure was interpreted based on the finite number of cells in the
printed lattice, and the presence of defects. We also showed that
the periodic lattices store a more considerable elastic energy per
volume than the single unit cells. While the octahedral elements in
the periodic lattice structures are distorted, two classes of tetra-
hedral sites exist, i.e., expanded and contracted. We also revealed a
bilinear behavior for the individual unit cells of both single- and
bimaterial lattice structures with free lateral elements.
We discovered the higher load-bearing capacity of the unit cells
with periodic boundary conditions under the same displacement.
Furthermore, stabilities induced in the structure are stress-
controlled rather than strain-controlled as for the same strain the
free-boundary single unit cells with the lower applied force do not
show any drop in the stored elastic energy. In the linear regime, the
rate of softening for unit cells with free boundaries is higher than
periodic cells. The core unit cells of a periodic lattice structure show
a linear force-displacement relation over a wide loading range,
which dropped as loading increased. However, the individual unit
cells showed a bilinear response with no sign of drastic energy
drops at higher displacements.
We analyzed the sensitivity of the stored elastic energy and the
effective stiffness of the printed parts within the finite element
framework. Our results indicated that the sensitivity of the stored
elastic energy concerning variations in the diameter of the exterior
elements of bimaterial unit cells is independent of the order of the
materials. Furthermore, we revealed a critical displacement, where
the stored elastic energy in the unit cell becomes insensitive to
variation of the exterior elements' thickness.
In conclusion, we developed a novel sensitivity model for octet-
truss metamaterials and revealed several key phenomena which
guide the design of these materials that are summarized below:
1) properties of the material of outer elements will dominate the
overall properties of the lattice structure;
2) the maximum load-carrying capacity and elastic energy storage
per unit volume for the inner unit cells is higher than the
boundary unit cells;
3) the force-displacement curve of the individual unit cells of both
single- and bi-material lattice structures with free lateral ele-
ments follow a bilinear behavior;
4) instabilities in the structure are stress-controlled;
K. Momeni et al. / Materials and Design 183 (2019) 108124 9

5) the sensitivity of stored elastic energy to manufacturing varia-
tions in bimaterial lattices is independent of the order of the
selected materials;
6) there is a critical displacement at which the elastic energy be-
comes independent of the printing parameters.
Similar guidelines could be derived for other metamaterials
using the framework developed here. The structures considered
here could also be fabricated with different dimensions, which may
affect the load-carrying capacity and stored energy capacity as the
density of the metamaterial will change by changing its unit cell
dimensions. We will study the effect of unit cell dimensions on the
effective properties of the metamaterial in our future publications.
Acknowledgement
This project is supported by Louisiana Tech University, the DoE-
ARPA-E OPEN, and NASA-EPSCoR. This project is also partly supported
by Louisiana EPSCoR-OIA-1541079 (N SF(2 018) -CIM MSee d-18 an d
NSF(2018)-CIMMSeed-19)and LEQSF(2015-18)-LaSPACE. Calculations
are performed using Louisiana Optical Network Initiative (LONI).
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