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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

2VuþVu

T

;ε

e

¼εε

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 31 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

¼vL

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)

33 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

11/3 3

TPLA

11/3 3

TO

11/3 3

TI

11/3 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

33 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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