Mechanical Properties of Copper Octet-Truss Nanolattices

Full text

Mechanical Properties of Copper Octet-Truss Nanolattices
ZeZhou Hea, FengChao Wanga, YinBo Zhua, HengAn Wu*,a, Harold S. Park*,b
aCAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, CAS
Center for Excellence in Nanoscience, University of Science and Technology of China, Hefei, Anhui 230027,
China
bDepartment of Mechanical Engineering, Boston University, Boston, Massachusetts 02215, USA
Abstract
We investigate the mechanical properties of copper (Cu) octet-truss nanolattices
through a combination of classical molecular dynamics (MD) simulations and
theoretical analysis. The MD simulations show that Cu nanolattices with high relative
density are stronger than bulk Cu, while also achieving higher strength at a lower
relative density as compared to Cu meso-lattices. We demonstrate that modifying the
classical octet-truss lattice model by accounting for nodal volume and bending effects
through the free body diagram method is critical to obtaining good agreement
between the theoretical model and the MD simulations. In particular, we find that as
the relative density increases, nodal volume is the key factor governing the stiffness
scaling of the nanolattices, while bending dominates the strength scaling. Most
surprisingly, our analytic modeling shows that surface effects have little influence on
the stiffness and strength scaling of the nanolattices, even though the cross sectional
sizes of the nanowires that act as the lattice struts are on the order of 6 nm or smaller.
This is because, unlike for individual nanowires, the mechanical response of the
nanowire struts that form the nanolattice structure is also a function of bending and
nodal volume effects, all of which depend nonlinearly on the nanolattice relative
density. Overall, these results imply that nanoscale architected materials can access a
new regime of architected material performance by simultaneously achieving
ultrahigh strength and low density.
Key words: nanolattice, single crystal, surface effect, metamaterials
1. Introduction
Architected structural materials have been investigated both experimentally and
theoretically for many years. This has provided a compelling approach to obtaining
structures that are simultaneously lightweight and strong. Numerous studies on
architected structural materials have shown that the strength and stiffness of cellular
materials depends on their structural arrangement (Deshpande et al., 2001a; Fleck et
al., 2010; Gibson and Ashby, 1999). Architected materials can deform by either
bending or stretching of the individual elements, which is determined by the topology
of the lattice and its nodal connectivity, and this bending or stretching behavior
defines the deformation mechanisms of architected structural materials (Fleck et al.,
2010). Three dimensional lattice structures with connectivity of Z=12 at the nodes are

stretching-dominated and structures with are bending-dominated
6 ≤ 𝑍 < 12
(Deshpande et al., 2001a) The deformation mechanism directly impacts the effective
properties of lattice structures, such as Young’s modulus and yield strength (Fleck et
al., 2010; Gibson and Ashby, 1999). The modulus and yield strength are also linked to
the structure’s relative density , which is defined as the ratio of the structure’s
𝜌
volume within a unit cell and the volume of the unit cell (Gibson and Ashby, 1999).
The yield strength and modulus of open-cell bending-dominated structures scale as 𝜎𝑦
and , where and are the yield strength and modulus of
= 0.3𝜌1.5𝜎𝑦𝑠 𝐸=𝜌2𝐸𝑠𝜎𝑦𝑠 𝐸𝑠
the bulk material (Gibson and Ashby, 1999). For stretching-dominated structures,
such as octet-truss lattice, the yield strength and modulus scale as and
𝜎𝑦= 1/3𝜌 𝜎𝑦𝑠
(Deshpande et al., 2001b). Thus, the modulus and strength of stretching-
𝐸= 1/9𝜌𝐸𝑠
dominated structures decreases more slowly than bending-dominated ones as the
relative density decreases (Deshpande et al., 2001b; Fleck et al., 2010).
The strength and stiffness of macroscale architected structures depend as just
discussed on their relative density and follow structural scaling laws (Gibson and
Ashby, 1999). These scaling laws assume that the strength and stiffness of the
constituent materials are constant. However, when the diameter of the individual
structural elements decreases below the micron scale, many materials exhibit size
effects, i.e. the well-known “smaller is stronger” effect in single crystalline metals
(Greer and De Hosson, 2011; Jennings et al., 2010). As technology advances in
fabrication methods, especially in 3D printing and nanofabrication, architected
structural materials can be created with micro or even nanometer dimensions for the
individual structural elements (Schaedler and Carter, 2016). Recent advances in
additive manufacturing have also enabled the development of multiscale metallic
metamaterials, which have feature sizes spanning seven orders of magnitude in length
scale from tens of nanometers to centimeters, and which exhibit mechanical properties
that are not attainable through the bulk material constituent (Zheng et al., 2016). The
small sizes of these structural elements is important because these nanomaterials often
exhibit unique mechanical and physical properties not seen in the corresponding bulk
materials (Bauer et al., 2016; Jang et al., 2013; Lee et al., 2015; Meza et al., 2015;
Zheng et al., 2014; Zheng et al., 2016). For example, metallic micro-lattices can
recover under large compression strain and exhibit energy absorption capabilities
similar to rubbery polymers like elastomers (Schaedler et al., 2011). Alumina ceramic
nanolattices can recover up to 50% compressive strain while simultaneously being
lightweight and ultra-strong (Meza et al., 2014). Cu meso-lattices composed of single
crystal and nanotwinned structural elements have strength that is significantly
enhanced relative to the equivalent bulk material (Gu and Greer, 2015). Both the
ceramic nanolattices as well as Cu meso-lattices exhibit the “smaller is stronger”
behavior due to size effects (Gu and Greer, 2015; Meza et al., 2014).
While the mechanical properties of nanolattices have been investigated recently,

the size of the underlying structural elements has been on the order of a few hundred
nanometers (Lee et al., 2015; Montemayor and Greer, 2015). Therefore, it is still
unknown how surface effects, which become critical for sub-100 nm size scales,
impact the mechanical properties of the nanolattices. At these size scales, surface
effects, and not size effects, are the dominant operant physical mechanism, and it is
currently unknown how the stiffening and strengthening that has been attributed to
surface effects in individual nanowires (Park et al., 2009; Weinberger and Cai, 2012)
impacts the mechanics of nanolattices.
The objective of this work is to explore the mechanical response of octet-truss
nanolattices, illustrated in the CAD model depicted in Fig. 1, under uniaxial
compression using classical molecular dynamics (MD) simulations. The nanolattices
are constructed with single crystal FCC metal (Cu) nanowires as the individual
structural element, and because the structures have a nodal connectivity of 12, its
mechanical response is expected to be stretching-dominated (Deshpande et al., 2001b).
In doing so, size effects which emerge from grain boundaries, nano-twins, Hall-Petch
effects and which control the strengthening of meso-lattices (Gu and Greer, 2015), are
not operant. Interestingly, we find that contrary to intuition, surface effects only
weakly impact the Young’s modulus and yield strength of the nanolattices. Instead,
we find through analytic modeling that proper accounting of the nodal volume is the
key factor governing the nanolattice stiffness, and the scaling of the stiffness, while
the nanolattice strength and scaling of the strength are dominated by bending effects.
Fig. 1. (a) CAD image of the octet-truss lattice constructed by the 3D packing of unit
cell. (b) A unit cell of the nanolattice and its geometric parameters studied in this
paper.
2. Simulation Details
2.1 Molecular Dynamics
In order to investigate the mechanical properties of nanolattices constructed with

single crystal nanowires, we chose Cu square nanowires as the basic structural
element, and MD simulations were performed using LAMMPS (Plimpton, 1995) with
the embedded atom method (EAM) potential parametrized by Mishin et al (Mishin et
al., 2001). The Mishin potential has been shown to accurately capture both stacking
fault and surface energies of Cu. In addition, analysis on stress-displacement, atomic
relaxation and surface for {111}/<11 > shear indicate that the Mishin Cu
γ ‒ 2
potential captures the essential deformation behavior in keeping with ab initio
electronic structure calculations (Boyer et al., 2004).
To construct the MD models, we first used LAMMPS to generate a single crystal
cubic bulk oriented in the <100> direction as shown in Fig. 2(a). Then, nanolattice
ligaments were created by removing atoms from the bulk crystal, resulting in the
nanolattices seen in Fig. 2(b). The crystal orientation of nanowires (or struts) in the
nanolattice is different from the bulk as their axial orientations do not align with those
of the unit cell. For nanolattices constructed from <100> crystal bulk, all of the struts
(nanowires) are all along the <110> direction. In the following discussion, we refer to
nanolattices created from <100> bulk as <100> nanolattices.
Fig. 2. (a) <100> crystal bulk. (b) Unit cell structure and crystal orientations of <100>
nanolattices.
In this work, we performed MD simulations on a nanolattice unit cell with relative
densities of about 4%~40%, which is defined as the ratio of the number of atoms in
the nanolattice to the number in the bulk. The bulk cubic single crystals generated by
LAMMPS had lengths of 30 and 40 nm, containing 2.4 million and 5.6 million atoms
respectively. The resulting nanolattices contained between ~200,000 and 1.6 million
atoms, with nanowire cross sectional lengths ranging from about 2-6 nm. Periodic
boundary conditions were used in all three directions of the nanolattice unit cell.
Starting with the initial temperature at 300K, we performed simulations within the
Isothermal-Isobaric (NPT) ensemble for 600 ps with a time step of 1 fs to let the
system reach its equilibrium configuration. Then, deformation-controlled uniaxial
compression with a strain rate of was applied along the z-direction of the unit
109𝑠‒1

cell for compressive strains up to 70%, while the pressure component perpendicular to
the loading direction was controlled to maintain the uniaxial compression condition.
2.2 Relative density
A schematic model of the unit cell of the octet-truss nanolattice is shown in Fig.
1(b) together with the coordinate system. The relative density is computed by
calculating the volumes of regions occupied by material in the CAD model, and
scaling this by the unit cell volume, the octet-truss nanolattice relative density is
𝜌
given by:
(1)
𝜌= 6 2
(
𝑡
𝑙
)
2‒3
(
2 + 2
)
(
𝑡
𝑙
)
3
where t and l are the width and length of a strut defined in Fig. 1, respectively. It can
be shown that Eq. (1) reduces to the relative density expression for an octet-truss
lattice, if the node size is neglected (Deshpande et al., 2001b):
(2)
𝜌= 6 2
(
𝑡
𝑙
)
2
To simplify the volume calculation in MD simulations, bulk and surface atoms are
viewed as having the same volume. The relative density of octet-truss nanolattices can
be defined as the ratio of atoms’ number in nanolattice and bulk, and its range is from
4.3% to 39.0%. The relative densities calculated by Eq. (1) range from 4.2% for the
lowest density sample to 39.4% for the highest density sample, while relative
densities calculated by Eq. (2) range from 5.0% for lowest density sample to 56.5%
for the highest density sample. It is clear that neglecting the nodal volume leads to
significantly larger relative densities for the high density lattice structures, thus
demonstrating the relative accuracy of Eq. (1), which is used throughout this paper.
3. Compression of <100> Nanolattices
3.1 Deformation mechanisms of <100> nanolattices
Fig. 3. Compressive stress-strain responses of <100> nanolattices with different
relative densities.

We now present MD simulations of the compression of nanolattices with different
relative densities compressed to 70% strain, which as shown in Fig. 3 is the strain
level at which the nanolattice fails and loses all load carrying capacity. The z direction
compressive stress-strain response for all nanolattices we considered, with relative
densities from 5.5-39%, is shown in Fig. 3. While there are differences in response
depending on the relative density, the mechanical response of the nanolattices does
mimic the behavior previously seen in larger, bulk lattice structures. Specifically, an
initial elastic region is first observed, where the stiffness increases with increasing
relative density. This elastic region is followed by plastic yielding at compressive
strains ranging from 5.8% to 8.8%. After yielding, a plateau in the flow stress is
observed followed by failure via densification (Dong et al., 2015; Gibson and Ashby,
1999) in which the struts of the unit cells are pushed together, resulting in a sharp
increase in the stress prior to failure (Gibson and Ashby, 1999).
Fig. 4. Engineering stress-strain curve of <100> nanolattices with the relative density
of (a) 5.5% and (b) 39.0%.
To give further insights and to connect the atomic scale deformation mechanisms
to the stress-strain response, we present in Fig. 4 the engineering stress-strain curves
for the compression of <100> nanolattices with the relative density of ~5.5% and
39.0%. The aspect ratio of the struts (l/t) between relative densities of 5.5% and 39%
decreases from 13 to 4. For the lowest relative density <100> nanolattices in Fig. 4(a),
the process of compression can be divided into four stages, including elastic stage (I),
plastic stage (II), plastic buckling (III) and densification stage (IV). Following the
elastic stage (I), yielding occurs via nucleation of partial dislocations in the struts in
stage (II). As the plastic strain increases, the struts exhibit plastic buckling, which
leads to the significant decrease in flow stress in stage (III). Then, the opposing struts
in nanolattices crush together due to structural buckling, which results in the stress

increase in stage (IV) starting around 45% compressive strain.
In contrast, Fig. 4(b) shows that the nanolattices with the relative density of ~39.0%
have a different stress-strain trend comparing to lower relative density. First, the yield
strength is significantly elevated for the higher relative density, which is due to the
larger volume of material that is present to resist the compressive loading. The
process of deformation can be divided into three stages, containing elastic stage (I),
plastic stage (II) as well as densification stage (III); buckling is not observed due to
the small aspect ratios of the struts. The densification at larger relative density leads to
a rapid increase in stress above the yield stress before failure, which occurs due to the
larger amount of contact within the unit cell as compared to the lower relative density
simulation.
Fig. 5. Snapshots of <100> unit cells of nanolattices with (a) and
𝜌= 5.5% 𝜌
(b) during compression. The development of plastic truss buckling in (a) can
= 39.0%
be seen as the total strain, ε, increases.
We detail the evolution of plasticity in the <100> nanolattices in Fig. 5, where the
atoms are highlighted by the centrosymmetry parameter (Kelchner et al., 1998) and
visualized using the OVITO package (Stukowski, 2009). In the <100> nanolattices,
all struts are along the <110> direction, and nanolattices with both low and high
relative density exhibit similar deformation mechanisms. For the <110> nanowire
orientation, it was previously shown that compressive deformation leads to plasticity
via nucleation of both full and partial dislocations (Park et al., 2006). As is indicated
in Fig. 5, partial dislocations of <100> nanolattices nucleate and propagate near the
nodes during compression, which corresponds to strain bursts (Gu and Greer, 2015)

and a drop in the flow stress. With the increase of strain ( ), we observe both full
ϵ~0.4
and partial dislocation generation near the nodes, along with surface steps, which
causes strut buckling in the low relative density <100> nanolattices as shown in Fig.
5(a). At even larger compressive strains ( ), all struts in both nanolattices exhibit
ϵ~0.6
significant dislocation activity, while densification via strut contact occurs due to the
large compressive strains that are applied.
4. Analytic Modeling and Discussion
In this section, we develop analytical models for the compressive response of
nanolattices, while incorporating surface effects into the analytic models. The
mechanical properties of octet-truss lattice material have been analyzed by Deshpande
et al (Deshpande et al., 2001b). This model studied an ideal octahedral cell with cubic
symmetry that assumed pin-joined struts, and the octahedral cells can be stacked to
the octet-truss structure. It demonstrated that for small , the contribution to stiffness
𝑡/𝑙
and strength from the bending of the struts was negligible compared to from strut
stretching (Deshpande et al., 2001a). The relative density of the octet-truss lattice
𝜌
material is given by (Deshpande et al., 2001b), where t and l are the
𝜌= 6 2(𝑡/𝑙)2
width and length of a strut, respectively. As discussed previously around Eq. 2, this
equation is a first order approximation and overestimates the relative density due to a
double counting of the volume of the nodes. Under uniaxial compression, the stiffness
and strength can be expressed as (Deshpande et al., 2001b):
(3)
𝐸 =22
3
(
𝑡
𝑙
)
2𝐸𝑠
and
(4)
𝜎
𝑦= 2 2
(
𝑡
𝑙
)
2𝜎𝑦𝑠
where and are the orientation-dependent modulus and yield strength of each
𝐸𝑠𝜎𝑦𝑠
strut, where the struts in Cu <100> nanolattices are all along the <110> direction.
The ideal octet-truss lattice model works well for small t/l and relative densities,
such that nodal volume and bending effects can be neglected when the relative density
of the octet-truss lattice is smaller than about 10% (Dong et al., 2015; Gu and Greer,
2015). However, in this paper, the relative nanolattice densities range from 4-40%,
which is larger than in previous studies. Moreover, when the regions surrounding the
nodes are subject to complex stress states, the nodal volume and bending must be
accounted for (Montemayor and Greer, 2015). The nodal volume was accounted for
previously through the modification of the relative density in Eq. (1), though its effect
on the strength and stiffness will be accounted for through the analytic models
developed in this section. To account for both bending and nodal volume effects on
the struts, we utilize the free body diagram method of Finnegan et al. (Finnegan et al.,
2007), which was recently used to predict the mechanical properties of Ti-6Al-4V
octet-truss lattice structures (Dong et al., 2015). The corresponding free body diagram

of octet-truss nanolattices is shown in Fig. 6(a) and (b).
Fig. 6. Sketch of the deformation of a single out-of-plane strut under uniaxial
compression. (a) Equivalent unit cell for nanolattices with the Cartesian coordinate
system and loading direction also specified. (b) (c) Free body diagram for the strut
when the unit cell is under uniaxial free compression. (d) Force analysis of the strut.
4.1 Poisson effect
A key issue we consider in the analytic model is the elastic anisotropy of Cu.
Because Cu exhibits significant elastic anisotropy, it is necessary to consider axial
orientation effects on the elastic modulus of the struts (nanowires). The elastic
modulus of each crystal is calculated for uniaxial loading in a given direction [h k l]
by (Tschopp and McDowell, 2008):
(5)
1
𝐸[ℎ 𝑘 𝑙] =𝑆11 ‒
(
2𝑆11 ‒2𝑆12 ‒ 𝑆44
)
ℎ2𝑘2+𝑘2𝑙2+𝑙2ℎ2
(
ℎ2+𝑘2+𝑙2
)
2
where represents the elastic compliances for a given crystal, which were calculated
𝑆𝑖𝑗
from the elastic moduli given for the Cu EAM potential (Mishin et al., 2001).
𝐶𝑖𝑗
From Fig. 2(b), the spatial direction of all struts in nanolattices can be divided into
three directions, which are labeled as AB, AC and BC. For <100> nanolattices, struts
AB, AC and BC are all oriented along the <110> direction. To facilitate the following
analysis, we define modulus of struts AB, AD, DF, BF as , struts AC, AE, CF, EF
𝐸1

as , and struts BC, CD, BE, DE as , where E1=E2=E3=E<110>=131.45 GPa due to
𝐸2 𝐸3
all struts being oriented along the <110> direction. According to free body diagram
method, a strut of octahedral cell can be regarded as beam with length l and width t as
shown in Fig. 6(b). During uniaxial compression, both ends of the strut are able to
move because of the Poisson effect of the octahedral cell. Fig. 6(c) illustrates the
deformation behavior of a doubly clamped beam when the octahedral cell is under
uniaxial compression. By labeling the z-displacement as , the lateral (x)-
𝛿𝑧
displacement as and the lateral (y)-displacement as , the axial force and shear
𝛿𝑥 𝛿𝑦 𝐹𝐴
force in the strut are given by Timoshenko beam theory (Xia et al., 2011):
𝐹𝑆
(6)
𝐹𝐴=
𝐸𝑠𝐴(𝛿𝑧𝑠𝑖𝑛𝜃 ‒ 𝛿𝑥𝑐𝑜𝑠𝜃)
𝑙'
(7)
𝐹𝑆=
12𝐾𝑇𝐸𝑠𝐼(𝛿𝑧𝑐𝑜𝑠𝜃 +𝛿𝑥𝑠𝑖𝑛𝜃)
𝑙'3
where is the Young’s modulus of the strut along the loading direction, is strut
𝐸𝑠𝜃
inclined angle, A is the cross-sectional area, is the equivalent length of the strut,
𝑙'
where depending on the nodal volume of struts in nanolattices, I is the
𝑙 ‒ 𝑡 ≤ 𝑙'<𝑙
inertia moment of area of the beam cross section given by for square cross
𝐼=𝑡4/12
section of width t. is the shear modification of
𝐾𝑇= 1/(1 + 12𝐸𝑠𝐼/𝜅𝐺𝐴𝑙'2)
Timoshenko beam, where is the shear coefficient, G is the shear modulus. For a
𝜅
square beam, , where is the Poisson’s ratio of Cu
𝜅= 5(1 + 𝜈)/(6 +5𝜈) 𝜈
(Hutchinson, 2001). The total applied force on a single strut in the z direction is
𝐹𝑏
expressed as
(8)
𝐹𝑏=𝐹𝐴𝑠𝑖𝑛𝜃 +𝐹𝑆𝑐𝑜𝑠𝜃
Because the shear force to axial force ratio scales as , the contribution
𝐹𝑆 𝐹𝐴
(
𝑡/𝑙'
)
2
of struts’ bending can be negligible if we ignore a second-order small quantity. If
bending is neglected, we apply force equilibrium of octahedral unit cell along all three
directions, and lateral displacements and are related to the vertical displacement
𝛿𝑥 𝛿𝑦
by:
𝛿𝑧
(9a)
𝛿𝑥=
𝐸1
(
𝐸2+𝐸3
)
‒ 𝐸2𝐸3
𝐸1𝐸2+𝐸2𝐸3+𝐸3𝐸1𝛿𝑧
(9b)
𝛿𝑦=
𝐸2
(
𝐸1+𝐸3
)
‒ 𝐸1𝐸3
𝐸1𝐸2+𝐸2𝐸3+𝐸3𝐸1𝛿𝑧
For <100> nanolattices, if we neglect surface effects, , so
𝐸1=𝐸2=𝐸3=131.45 𝐺𝑃𝑎

, which is the same with octet-truss lattice (Deshpande et al., 2001b;
𝛿𝑥=𝛿𝑦= 1/3𝛿𝑧
Dong et al., 2015). However, if , as could be caused by surface effects or
𝐸1≠ 𝐸2
elastic anisotropy, the lateral displacement and are not equal and the structure
𝛿𝑥 𝛿𝑦
will deform differently compared to the macroscale octet-truss lattice.
4.2. Compressive stiffness
4.2.1 Compressive stiffness without surface effects
As is shown in Fig. 6(a), the external force applied on the octahedral unit cell
𝐹
along z direction is equal to . The compressive stress is related to the force
4𝐹𝑏 𝜎𝑧 𝐹𝑏
and displacement via:
𝛿𝑧
(10a)
𝜎𝑧=
4𝐹𝑏
𝑙2=
4𝐸𝑡2𝛿𝑧
3𝑙'𝑙2+
22𝐾𝑇
(
𝜆1+𝜆2
)
𝐸𝑡4𝛿𝑧
3𝑙'3𝑙
where
(10b)
𝐸=
3𝐸1𝐸2𝐸3
𝐸1𝐸2+𝐸2𝐸3+𝐸3𝐸1
The strain applied to the octahedral cell is related to the displacement via:
𝜖𝑧𝛿𝑧
(11)
𝜖𝑧=
2𝛿𝑧
𝑎=
2𝛿𝑧
𝑙
The effective Young’s modulus of the octahedral unit cell under uniaxial
𝐸𝑧
compression is then given by:
(12a)
𝐸
𝑧=
𝜎𝑧
𝜖𝑧=22
3𝐸
(
𝑡
𝑙
)
2𝐾𝐸
𝑉𝐾𝐸
𝐵
where
(12b)
𝐾𝐸
𝑉= 1 + 𝑙 ‒ 𝑙'
𝑙'
(12c)
𝐾𝐸
𝐵= 1 + 𝐾𝑇
(
𝜆1+𝜆2
)
(
𝑡
𝑙'
)
2
where is the equivalent beam length, which is related to the width of struts in
𝑙'
octahedral unit cell, and , .
𝜆1=𝐸1
(
1/𝐸2+ 1/𝐸3
)
/2 𝜆2=𝐸2
(
1/𝐸1+ 1/𝐸3
)
/2
Consider the strut stretched in Fig. 6(b) and the corresponding representative beam
shown in Fig. 6(c), and the equivalent length of the strut is given by
𝑙'=𝑙 ‒ 𝑡/2
(Moongkhamklang et al., 2010). Comparing Eqs. (3) and (10b), in Eq. (10b) is
𝐸
different from the modulus of the bulk material, due to the explicit incorporation of
Es
crystal orientation effects.
We note that and are the modification of nodal volume and bending effect,
𝐾𝐸
𝑉𝐾𝐸
𝐵

respectively; these modifications can be neglected by setting . The nodal
𝐾𝐸
𝑉=𝐾𝐸
𝐵= 1
volume modification is purely geometric, whereas the bending effect depends
explicitly on the elastic anisotropy of the nanolattice struts. It is clear that ,
𝐾𝐸
𝐵> 1
which implies that accounting for bending will always increase . It is also clear that
𝐸𝑧
, which again will lead to an increase in compared to the octet-truss model.
𝐾𝐸
𝑉> 1 𝐸𝑧
As is expressed in Eqs. (12b) and (12c), nodal volume and bending effects are first-
and second-order quantities as compared with the compressive stress, respectively. If
the relative densities of nanolattices are less than 1%, which means l/t is higher than
25, modifications of nodal volume and bending are given by and ,
𝐾𝐸
𝑉≈1𝐾𝐸
𝐵≈1
which means ideal octet-truss lattice works well for small relative densities. Based on
these mathematical analyses, the octet-truss model is valid for low relative densities,
while nodal volume and bending corrections must be considered if the relative density
is greater than 1%.
4.2.2 Surface effects on elastic properties
Up to this point, we have not considered surface effects on the elastic properties of
the nanolattice struts, i.e. in Eqs. (6) and (7). Thus, Eq. (12) does not contain surface
effects, and all elastic stiffnesses should be modified if surface effects are operant, as
for our small cross section nanowire struts. In the past 15 years, many theories have
been developed to describe surface effects on the mechanical properties of
nanomaterials (Dingreville and Qu, 2005; Miller and Shenoy, 2000). For the
nanolattices, the octahedral unit cell is a stretching-dominated structure, which
suggests that the individual nanowires can be modeled as undergoing axial
compression. Perhaps the simplest form for capturing surface effects on the stiffness
can be written as (Miller and Shenoy, 2000): , where is the
𝐸∗=𝐸𝑠+ 4𝑆𝐸𝑡‒1 𝑆𝐸
surface modulus of nanowires, and is the Young’s modulus of the bulk material.
𝐸𝑠
For the bending effect of nanowires, Feng et al. demonstrated that the residual
surface stresses can be neglected because its influence on the effective modulus is
much weaker than that of surface elasticity (Feng et al., 2009; He and Lilley, 2008).
Surface elasticity on flexural stiffness can be given by (Feng et al., 2009; Miller and
Shenoy, 2000): . Thus, and in Eq. (13) should be
(𝐸𝐼)∗=
(
𝐸𝑠+ 8𝑆𝐸𝑡‒1
)
𝑡4/12 λ1λ2
modified as: and
λ∗
1=𝜆1(1 + 8𝑆𝐸/𝐸1𝑡)/(1 + 4𝑆𝐸/𝐸1𝑡) λ∗
2=𝜆2(1 + 8𝑆𝐸/𝐸2
. These expressions can then directly be used to account for surface
𝑡)/(1 + 4𝑆𝐸/𝐸2𝑡)
effects on the Young’s modulus of the octahedral unit cell in Eq. (12).

Fig. 7. Comparisons between the simulations and model predicted Young’s modulus
as a function of relative density . MD simulation results and theory prediction
𝐸𝑧𝜌
expressed in Eq. (12) are plotted together in this figure. The differences of theory
prediction 1~5 are listed in Tab. 1. Octet-truss lattice model is also drawn using Eq.
(3).
4.2.3 Discussion
To compare the theoretical analysis and MD simulation results, the Young’s
modulus as a function of relative density is shown in Fig. 7. Here, the Young’s
modulus of the nanolattices was obtained by taking the slope of the linear elastic
portion of the stress-strain response. Fitting the MD simulations by the functional
form (Fleck et al., 2010; Gibson and Ashby, 1999), we obtain a value for
𝐸𝑧∝ 𝐸𝑠𝜌𝑛
the exponent n=1.23, which is slightly higher than ideal octet-truss lattice (n=1)
(Deshpande et al., 2001b) and solid HDDA (n=1.1) (Zheng et al., 2014), but lower
than bending-dominated structures (n=2) (Fleck et al., 2010; Gibson and Ashby, 1999)
and hollow-tube alumina nanolattices (n=1.61) (Meza et al., 2014). From Eqs. (12b)
and (12c), we can find that the exponent n is mainly impacted by effects of nodal
volume, with bending of secondary importance, which explains why the scaling of the
Young’s modulus is closer to the stretching-limit of n=1.
To distinguish between bending, nodal volume and surface effects, Fig. 7 shows
octet-truss lattice and five modified models in comparison with the MD simulation
results. Tab. 1 lists the differences between these five models. The Young’s modulus
of single-crystalline bulk Cu along the <110> orientation is taken as 131.45 GPa
(Tschopp and McDowell, 2008). In <100> nanolattices, all <110> struts have two {1
0 0} side surfaces and two {110} side surfaces. Based on the method of Shenoy’s
atomistic simulations, MD simulations using Mishin potential estimated the surface
modulus of <110> single-crystalline Cu as on the (010) crystal
SE1 =30.61𝑁/𝑚

surface and on the surface (Shenoy, 2005). Tab. 2 and Tab.
SE2 =‒ 10.66𝑁/𝑚 (101)
3 provides relevant mechanical properties of single crystal Cu in the <100> and <110>
directions according to the Mishin potential.
Tab. 1 Differences of five theory predictions. “Yes” means this factor is included in
this theory prediction, and “No” means this factor is not contained.
Theory predictions
Bending
Nodal volume
Surface effects
1
Yes
Yes
Yes
2
Yes
Yes
No
3
Yes
No
No
4
No
Yes
No
5
No
No
Yes
Tab. 2 Relevant parameters of single crystal bulk of copper (Meyers and Chawla,
2009; Mishin et al., 2001).
Crystal orientation
Young’s Modulus
(GPa)
Shear modulus (GPa)
Poisson’s ratio
<100>
67.1
76.2
0.42
<110>
131.4
23.6
Tab. 3 Relevant parameters of surface in crystalline. Surface modulus is calculated by
MD simulations, and surface stress is taken from (Sun et al., 2008).
𝜏0
Surface (hkl)
Orientation
Surface modulus (N/m)
Surface stress (N/m)
{1 0 0}
<100>
1.5157
{1 1 0}
<100>
<110>
30.61
-10.66
1.1919
1.3793
Fig. 7 reveals that Eq. (12), which accounts for bending and nodal volume, agrees
with the MD simulations. However, the classical octet-truss model, which neglects
bending and nodal volume effects, shows increasing error compared to the MD
simulation results as the relative density increases. Performing the five theory
predictions enables us to demonstrate that accounting for nodal volume is the most
important theoretical enhancement, followed by bending, and finally surface effects.
A natural question arises: what mechanism makes nodal volume and bending
significant for nanolattices compared with previously studied macroscale octet-truss
lattice systems, i.e. the ideal octet-truss lattices of Desphande et al. (Deshpande et al.,
2001b), and Ti-6Al-4V octet-truss lattices (Dong et al., 2015)? And, why are
surface effects relatively unimportant for nanolattices even though the strut widths (2-
6 nm) lie solidly within the expected regime where surface effects are known to
strongly impact their mechanical properties (Liang et al., 2005; Park et al., 2009; Park
and Klein, 2008)?

In the analysis of ideal octet-truss lattice, truss elements are slender, and the nodal
connectivity is assumed to be pin-jointed, so nodal volume and bending, which scales
as , are negligible (Deshpande et al., 2001a). For Ti-6Al-4V octet-truss lattices,
(𝑡/𝑙)2
their relative densities are between 2.4% and 15.9%, so the modification of nodal
volume is taken into account for the relative density calculations. Furthermore, the
nodal connectivity is pin-jointed for Ti-6Al-4V lattices, which cannot support a
bending moment; this decreases the effect of bending during deformation (Dong et al.,
2015). However, the Cu nanolattices studied here are single crystalline and have
smaller aspect ratio struts (between 4 and 13) but larger relative density (between 4%
and 40%), so nodal connectivity can be treated as rigid-jointed, which enhances the
effect of bending. Moreover, the aspect ratio of the struts within the nanolattices is
small compared with previously studied octet-truss lattices, so the effects of nodal
volume and bending are enhanced in the nanolattices.
Fig. 8 Young’s modulus of nanolattices as a function of strut width (dashed lines) and
length of unit cell (solid lines) from theory prediction 1 and MD simulations (solid
points).
To explain why surface effects do not strongly impact the nanolattice elastic
modulus, we calculate the difference between the modulus of the nanolattice with and
without surface effects as , where is the
Δ=𝐸∗
𝑧‒ 𝐸𝑧≈1.05𝜌1.23 𝑆𝐸𝑡‒1𝐸∗
𝑧
nanolattice stiffness with surface effects and is the nanolattice stiffness neglecting
𝐸𝑧
surface effects. This difference demonstrates that nanolattices have lower free surface
to volume ratio in unit cell (the order of ) compared with an individual
𝜌1.23/𝑡
nanowire (the order of 1/t) (Miller and Shenoy, 2000). For the strut widths in our
work, which range from 2.1 to 6.0 nm, the difference is 0.1 GPa for the smallest
relative density of and 0.61 GPa for the largest relative density of .
𝜌= 0.043 𝜌= 0.4
Given that the nanolattice stiffness is calculated from the MD simulations to be 0.797
GPa at the smallest relative density and 11.8 GPa at the largest, the error from
neglecting surface effects is largest (~14%) for the smallest relative density, and

decreasing to ~5% for the largest relative density nanolattice we considered. Given
that the classical octet-truss model predicts a nanolattice stiffness of 8.64 GPa at the
highest relative density of , for an error of 26.8%.
𝜌= 0.4
The impact of surface effects can also be interpreted using Fig. 8, where the dashed
lines represent the Young’s modulus as function of strut width with different relative
densities, and the solid lines show the Young’s modulus varying with length of unit
cell. For each relative density (dashed lines), surface effects increase with decreasing
strut width, which is consistent with size-dependent stiffening previously predicted
for <110> FCC metal nanowires (Liang et al., 2005). Furthermore, for each unit cell
length, as the relative density (or strut width) increases, surface effects tend to
decrease. Finally, as the strut length, a increases, surface effects have decreasing
impact. All of these trends are consistent to above analysis. It is clear that, particularly
at the highest relative densities, surface effects play a very minor role in explaining
the difference between the MD simulation results and the classical octet-truss solution.
4.3. Compressive strength
4.3.1 Elastic buckling
A stretching-dominated lattice structure may fail under compressive loading by
elastic buckling or by plastic yielding depending on the aspect ratio (l/t) of the struts.
At low relative densities, struts are flexible enough to collapse by elastic buckling
before plastic yielding. The elastic buckling stress including surface effects is given
by (Wang and Feng, 2009):
(13)
σE=
𝑛2𝜋2
(
𝐸𝑠𝐼
)
∗
𝐴𝑙'2 +
2𝜏0𝑡
𝐴=𝑛2𝜋2
12 𝐸𝑠
[
1 +
8𝑆𝐸
𝐸𝑠𝑡+24
𝑛2𝜋2
(
𝑙
𝑡
)
2
(
𝜏0
𝐸𝑠𝑡
)
]
(
𝑡
𝑙
)
2
(
1 + 𝑙 ‒ 𝑙'
𝑙'
)
2
where A is the strut cross-section area given by , and is equal to .
𝐴=𝑡2𝐸𝑠𝑚𝑎𝑥
[
𝐸1,𝐸2
]
The factor n is determined by the end conditions on the buckling struts. We simplify
the problem by assuming that the struts are fixed at both ends. Thus, the rotational
stiffness of the nodes is zero and n=2 (Deshpande et al., 2001b). In the case of failure
via elastic buckling:
(14)
σ𝐸
z=22
3𝜋2𝐸𝑠
[
1 +
8𝑆𝐸
𝐸𝑠𝑡+6
𝜋2
(
𝑙
𝑡
)
2
(
𝜏0
𝐸𝑠𝑡
)
]
(
𝑡
𝑙
)
4
(
1 + 𝑙 ‒ 𝑙'
𝑙'
)
2
For the lowest relative density (4.3%) simulated in this paper, the strut length and
width of strut are 2.1 nm and 28.3 nm, respectively. According to Eq. (14), the elastic
buckling stress of the nanolattices is 55.5MPa, which is slightly higher than the yield
strength (45.6MPa) of nanolattices, so elastic buckling is not observed in this work.
However, if surface effects are neglected, the buckling stress is lower than the yield
strength, and nanolattices exhibit elastic buckling. Therefore, surface effects enhance
the rigidity of the struts, which improves the stability of nanolattices with low relative
density.
4.3.2 Yield strength
At high relative densities, l tends towards t, and a lattice structure will fail by

yielding of the low aspect ratio struts. As shown in Fig. 6(d), if we remove the
clamped constraint at one end of the beam, the equilibrium deformation of struts
under uniaxial compression can be replaced by constraint reaction forces. Because
struts in nanolattices are rigid-jointed, the bending moment at both clamped boundary
cannot be neglected. From Timoshenko beam theory, we can conclude the
equilibrium moment is expressed as:
(15)
𝑀=1
2𝐹𝑆𝑙'
The maximum normal stress in struts is given by:
(16a)
𝜎𝑚=
𝐹𝐴
𝐴+𝑀𝑡
2𝐼 =𝜎𝐴
[
1 + 6𝜆
(
𝑡
𝑙'
)
]
and
(16b)
𝜎𝐴=
𝐹𝐴
𝐴=2
3𝐸
𝛿𝑧
𝑙'
where is equal to . The maximum normal stress in beam is at the
𝜆𝑚𝑎𝑥
[
𝜆1,𝜆2
]
clamped ends, which means the collapse of struts always occurs at clamped ends as is
indicated in Fig. 6(d). This also corresponds to the MD simulations where plastic
deformation originates at or near the nodes in the nanolattices, as shown in Fig. 9(b),
which shows the von Mises stress distribution calculated from six components of
virial stress of each atom (Zhou, 2003).
The maximum shear stress in struts is given by (Timoshenko and Goodier, 1970):
, where is a constant depending on the cross sectional
𝜏𝑚= 3(1 + 𝜒)𝐹𝑠/2𝑡2𝜒
geometry and Poisson’s ratio. Combining Eqs. (7) and (9), the maximum shear stress
can be expressed as: . This equation demonstrates that is
𝜏𝑚= 3(1 + 𝜒)𝜆
(
𝑡/𝑙'
)
2𝜎𝐴τm
a second order small quantity compared with the axial stress. In addition, the MD
simulations show that dislocations nucleate from the surfaces of the struts at the nodes,
while the maximum shear stress occurs at the neutral axis of the struts, or away from
the surfaces (Timoshenko and Goodier, 1970), as showed in Fig. 9(c) and (d). For
both of these reasons, shear stresses have little influence on dislocation nucleation in
struts and are neglected in the following discussion.
Combining Eqs. (10) and (16), the stress applied to the octahedral unit cell can
𝜎𝑧
be expressed in terms of the axial stress in the strut:
𝜎𝐴
(17)
𝜎𝑧=22𝑡2
𝑙2𝜎𝐴+
22𝐾𝑇
(
𝜆1+𝜆2
)
𝑡4
𝑙'2𝑙2𝜎𝐴
The yielding condition of strut can be effectively predicted by the von Mises criterion:
, where is the von Mises stress at yielding. As the loading increases, the
σv=𝜎𝑚σv
von Mises stress gradually reaches its yield strength . Therefore, the lattice
𝜎𝑦𝑠

strength is expressed as:
(18a)
𝜎𝑦= 2 2
(
𝑡
𝑙
)
2𝜎𝑦𝑠𝐾𝑌
𝑉𝐾𝑌
𝐵
and
(18b)
𝐾𝑌
𝐵=
1 + 𝐾𝑇
(
𝜆1+𝜆2
)
(
𝑡
𝑙'
)
2
1 + 6𝜆
(
𝑡
𝑙'
)
and are the modification of nodal volume and bending effect, respectively. In
𝐾𝑌
𝑉𝐾𝑌
𝐵
Eq. (17), the modification of equivalent length is offset by the axial stress expressed
in Eq. (16b). Therefore, different from Eq. (12b), in Eq. (18a) is always given by:
𝐾𝑌
𝑉
.
𝐾𝑌
𝑉= 1
The strut bending effects on the yield strength are shown in Eq. (18b), which is a
first order quantity compared with axial stress. An important difference in the bending
effects on yield strength as compared to stiffness in Eq. (12c) is that the denominator
of Eq. (18b) is always greater than 1, which is caused by moments at both end of the
struts shown in Fig. 6(d) and expressed by Eq. (16). Therefore, , which
𝐾𝑌
𝐵< 1
implies that bending always decreases . Similarly, if the relative densities are less
𝜎𝑦
than 1%, the bending effect modification is given by , and the yield strength of
𝐾𝑌
𝐵≈1
nanolattices reduces to those of the ideal octet-truss lattice (Deshpande et al., 2001b).

Fig. 9 Generation and propagation of partial dislocations at nodes of nanolattices
during compression. (b) Schematic diagram of stress distribution. The red areas
indicate yielding areas and are marked by box. (c), (d) Dislocation distribution of
zoom areas signed by solid and dashed box, respectively. The green lines in (c) and (d)
represent 1/6<112> dislocation, and the purple arrows are Burgers vector of
dislocation (Stukowski et al., 2012). The white surface represents the surface of
nanolattices.
4.3.3 Surface effects on yield strength of struts
So far, surface effects on the yield strength of struts has not been considered in
𝜎𝑦𝑠
discussing the yielding of nanolattices. Numerous studies have demonstrated that
yield strength of FCC metal nanowires depends on the orientation and size of the
nanowires (Cao and Ma, 2008; Diao et al., 2004; Park et al., 2006). The yield stress
varies with nanowire width t, and a behavior of the form is often
σys =𝜎0
𝑦𝑠 +𝐾𝑡‒1
observed, where and K are independent of t (Yang et al., 2009; Zhang et al., 2008).
σ0
𝑦𝑠
To determine the parameters in this equation, MD simulations using Mishin potential

were performed to estimate the yield strength of <110> nanowires with different
width (Park et al., 2006). Fitting the data from MD simulations, we obtain =9.48
𝜎0
𝑦𝑠
GPa and K=9.48 N/m. This expressions can then directly be applied to account for
surface effects on the yield strength of the octahedral unit cell in Eq. (18) (Hodge et
al., 2007).
4.3.4 Discussion
The yield point was chosen as the stress at the first peak after the linear elastic
loading region. The yield strengths of the nanolattices are plotted as a function of
relative density in Fig. 10 with unit cell size of 40 nm for strut size of 2.2-6 nm, and
unit cell size of 30 nm for strut size of 4.3-5.6 nm. As is shown in Fig. 10, size-
dependent strengthening is revealed by comparing the nanolattice yield strength
against the bulk yield strength. In the paper of Gu et al. (Gu and Greer, 2015), bulk
yield strength of Cu was calculated to be 133 MPa, and the maximum yield strength
of meso-lattices is about 332 MPa with relative density of 0.8 (Gu and Greer, 2015).
For <100> nanolattices, the maximum yield strength is about 586 MPa with relative
density of 0.39. As shown in Fig. 12, comparison of the yield strengths of the
nanolattices and the electroplated Cu thin film reveals that nanolattices with
𝜌> 0.1
are stronger than the bulk, with the high-density <100> nanolattices ( having
𝜌~0.26)
the same strength as the densest meso-lattices ( that were studied
𝜌~0.8)
experimentally (Gu and Greer, 2015).
Fig. 10. Comparisons between the simulations and model predicated yield strength 𝜎𝑦
as a function of relative density . MD simulation results and theory prediction
𝜌
expressed in Eq. (18) are plotted together in this figure. The differences of theory
prediction 1~5 are listed in Tab. 1. Octet-truss lattice model is also drawn using Eq.

(4).
In comparing the MD results for yield strength with the analytic theory predictions
summarized in Tab. 1, we find that bending is the key contributor to enable the octet
truss model to match the MD simulation results. Specifically, the bending term shifts
the octet truss yield stress curve down as compared to the octet truss result in Fig. 10
because bending is a first order effect that scales as according to Eq. (16). The
𝑡/𝑙’
yield strength of struts not considering surface effects is 7.11 GPa, which is the
compressive strength of <110> nanowires with width of 4 nm (the median width of
struts).
While bending alone leads to a slight underprediction of the yield strength,
additionally incorporating nodal volume effects (theory prediction 2 in Fig. 10) shifts
the yield stress curve up close to the MD results. A detailed analysis demonstrates that,
however, theory prediction 2, which accounts for both bending and nodal volume,
deviates from the MD yield stress curve at low relative densities, or equivalently
struts with small widths. This is due to the size-dependent compressive strength of
<110> nanowires, which increases with increasing nanowire width; as discussed with
regards to the nanolattice modulus, surface effects had the largest effect on the low
relative density nanolattices, which also manifests itself here with regards to the yield
strength, as shown in Fig. 11. By incorporating surface effects in conjunction with
bending and nodal volume via theory prediction 1, we are able to find good agreement
with the MD simulations in Fig. 10 and 11. Thus, bending, surface effects and nodal
volume contribute in decreasing order to the strength of nanolattices.
A further refinement in the ranges that the various mechanisms (bending, nodal
volume, surface effects) dominate can be obtained by considering different
nanolattice relative densities, as seen in Fig. 10. For low relative densities (under
5.5%), surface effects have a significant effect on the yield strength, due to the fact
that the effects of bending and nodal volume are small as shown in Eq. (18), which
can be simplified as: in the case of very small t/l (or very low relative density).
𝐾𝑌
𝐵≈1
This can also be observed in comparing theory predictions 1 and 2 in Fig. 10, where
for low relative densities theory prediction 1, which includes surface effects, better
matches the MD simulation results.
As the relative density increases, bending effects gradually become dominant, so
the green points in the middle relative density regime (between 10% and 20%) closely
match theory prediction 3, which considers only bending, as shown in Fig. 10. For
high relative densities (over 25%), bending and nodal volume both contribute because
of enhanced constraints at the nodes and the increase of nodal volume, while surface
effects become negligible. Thus, all blue points are closer to theory prediction 2,
which considers both nodal volume and bending. While nodal volume was most
important to match the nanolattice stiffness scaling, in both cases surface effects,
despite the small strut widths considered, have the least effect on the observed scaling
behavior with increasing relative density.
From the analysis of nanolattice stiffness, we found that due to the small aspect

ratios of the struts and stronger kinematic constraints at the nodes, the effects of nodal
volume and bending have a significant influence compared with ideal octet-truss
lattices. Similarly, these effects also impact the yield strength of nanolattices.
However, what should be emphasized is that the rigid joint of the node generates
bending moment during compression of nanolattices. Bending moment is first order
quantity of axial force, while shear force is second order quantity, which means
bending moment is a critical factor decreasing the yield strength of nanolattices. As is
shown in Fig. 9, due to bending moment given by Eq. (16), yielding areas are
distributed at or near the nodes in the nanolattices, and partial dislocations nucleate
and propagate from these areas where compressive stress is maximum in theory
analysis. This also explains why most of deformation is localized to the nodes during
previous experimental studies of the compression of alumina nanolattices and meso-
lattices (Gu and Greer, 2015; Meza et al., 2014). For previously studied octet-truss
lattices, nodal connectivity is assumed to be pin-jointed, where their nodes cannot
form bending moment during compression, so bending effects are negligible
(Deshpande et al., 2001a; Dong et al., 2015). Therefore, our analytic studies
demonstrate that reducing the nodal constraint in an octet truss lattice is a feasible
approach to enhancing the scaling of strength with density.
Moreover, surface effects have been established to play a key role on the yield
strength and mechanisms of individual nanostructures like nanowires (Park et al.,
2009; Park et al., 2006; Weinberger and Cai, 2012). Compared to individual
nanowires, however, we find that surface effects have a smaller influence on
nanolattices. This is because surface effects compete with bending and nodal volume
effects, all of which depend nonlinearly with the nanolattice relative density as
discussed above, and shown in Fig. 10. Thus, while surface effects do improve the
accuracy of the theoretical models, particularly at lower relative densities, they are not
the dominant effect in governing the mechanics of nanolattices.
Fig. 11 Yield strength of nanolattices as a function of strut width (dashed lines) and

length of unit cell (solid lines) from theory prediction 1 and MD simulations (solid
points).
4.4 Lightweight and strong nanolattices
To compare the yield strength of the Cu nanolattices with other, macroscale
nanolattices and other metallic engineering materials, we plot them together on the
material property chart in Fig. 12, which shows the density dependent yield strength
for existing materials (Ashby, 2011) and other space filling lattice structures,
including alumina nanolattices (Meza et al., 2014), Ti-6Al-4V octet-truss lattice(Dong
et al., 2015) and Cu meso-lattices (Gu and Greer, 2015). The strengths of both foams
and lattices scale with those of the materials from which they are made (Ashby, 2011;
Fleck et al., 2010).
As shown in Fig. 12, Cu nanolattices provide higher strength at a lower density
compared with Cu meso-lattices, and are about an order of magnitude stronger for the
same relative density, which marks a new entry in the high-strength lightweight
material parameter space. Compared with meso-lattices, Cu nanolattices have smaller
length scales and lower densities but provide higher strength, which exemplifies the
well-known “smaller is stronger” effect (Dou and Derby, 2009; Uchic et al., 2004).
While we have performed comparison with existing experimental studies, there are
clearly differences between the MD simulations performed in this work and those
experimental studies. First, the strain rates in the MD simulations are significantly
higher than in experiment, which may lead to higher yield stresses in the simulations
as compared to those expected at experimental strain rates (Zhu et al., 2008).
Furthermore, the struts in our nanolattices are defect free and pristine, whereas in the
experimental study of meso-lattices by Gu et al. (Gu and Greer, 2015), the
microstructure analysis revealed that substantial regions within the meso-lattices
consisted of grain and twin boundaries, which leads to different strengthening
mechanisms as compared to the defect-free and pristine struts that are considered in
the present MD simulations. Therefore, there are both simulation artifacts (higher
strain rates), as well as real structural and size effects (smaller, defect-free struts in
nanolattices compared to larger struts with significant microstructure in meso-lattices)
that contribute to the higher strength seen in the present work as compared to
previously studied metallic meso-lattices(Greer and De Hosson, 2011; Gu and Greer,
2015).

Fig. 12. Material property chart comparing material compressive strength against
density. The simulation data and linear fitting of strength for <100> nanolattices has
been included for comparison with other octet-truss lattice materials.
Furthermore, fitting the MD simulations by the functional form (Fleck
𝜎𝑦∝ 𝜎𝑦𝑠𝜌𝑚
et al., 2010; Gibson and Ashby, 1999), we obtain the exponent m=1.17, which is
slightly higher than ideal octet-truss lattice (m=1) (Deshpande et al., 2001b) and solid
HDDA (m=1.1) (Zheng et al., 2014), but lower bending-dominated structures (m=3/2)
(Fleck et al., 2010; Gibson and Ashby, 1999) and hollow-tube alumina nanolattices
(m=1.76) (Meza et al., 2014). Our scaling is slightly different than the analytic
prediction for ideal stretching-dominated structures. This deviation can be explained
by Eq. (18), which can be expanded as:
(19)
𝜎𝑦≈22𝜎𝑦𝑠
(
𝑡
𝑙
)
2
[
1‒ 𝛽
(
𝑡
𝑙
)
+
(
𝛼 ‒ 𝛽
2+𝛽2
)(
𝑡
𝑙
)
2+
(
𝛼 ‒ 𝛽
4‒ 𝛼𝛽 +𝛽2‒ 𝛽3
)(
𝑡
𝑙
)
3
]
where , and . Eq. (19) demonstrates that the principle part of
𝛼=𝐾𝑇(𝜆1+𝜆2)𝛽=6𝜆
Eq. (18) is the same with Eq. (4), which means our nanolattices are still stretching-
dominated, so the exponent m should be lower than 3/2. Besides, the contribution of
bending effects reduces the exponent m due to the modification of negative (t/l). If we
ignore the surface effects and fit theory prediction 2 (which includes bending and
nodal volume) with a power law, we find m=0.986, while m=1.17 if theory prediction
4 (only considering nodal volume) is fitted by power law. However, if surface effects

are considered and theory prediction 1 is fitted by power law, m=1.16. This result is
close to MD simulations, which means bending effects decrease the scaling exponent.
Nevertheless, octet-truss nanolattices are mainly stretching-dominated and are
simultaneously lightweight and strong, while outperforming bulk copper and copper
meso-lattices by reaching a previously unexplored property range in the Ashby plot in
Fig. 12.
5. Conclusion
In summary, we performed MD simulations in conjunction with the development
of analytic models to study the compressive mechanical properties of octet-truss Cu
nanolattices. Significant property enhancements as compared to bulk Cu and Cu
meso-lattices were observed, where the strength of nanolattices with high relative
density were larger than that of bulk Cu, while nanolattices achieved superior yield
strengths with lower relative densities as compared to meso-lattices.
We modified the classical octet truss model to account for nodal volume, bending,
and surface effects on the mechanical properties of the nanowire struts. The analytic
models were found to be in good agreement with the MD simulation results. These
enabled new insights into the mechanisms governing the strength and stiffness scaling
of the nanolattices, where nodal volume is the key factor governing the stiffness
scaling, while bending dominates the strength scaling.
Perhaps most surprisingly, surface effects, which have previously been shown to
cause significant changes in both the elastic and inelastic behavior and properties of
individual nanowires, have a smaller impact on the stiffness and strength scaling of
nanolattices. This is because, unlike for individual nanowires, the mechanical
response of the nanowire struts that form the nanolattice structure is also a function of
bending and nodal volume effects, all of which depend nonlinearly on the nanolattice
relative density.
Overall, this work demonstrates the promise of nanoscale cellular materials for
accessing previously untapped regimes of mechanical performance by simultaneously
achieving high strength and low density.
Acknowledgments
This work was jointly supported by the National Natural Science Foundation of China
(11525211, 11472263, 11572307) and the Strategic Priority Research Program of the
Chinese Academy of Sciences (XDB22040502). H.S.P acknowledges the support of
the Mechanical Engineering Department at Boston University.
Corresponding Authors:
*E-mail: wuha@ustc.edu.cn

*E-mail: parkhs@bu.edu
The authors declare no competing financial interest.
References
Ashby, M.F., 2011. Materials selection in mechanical design, fourth edition.
Bauer, J., Schroer, A., Schwaiger, R., Kraft, O., 2016. Approaching theoretical strength in glassy
carbon nanolattices. Nature materials.
Boyer, R.D., Li, J., Ogata, S., Yip, S., 2004. Analysis of shear deformations in Al and Cu: empirical
potentials versus density functional theory. Modelling and Simulation in Materials Science and
Engineering 12, 1017
Cao, A., Ma, E., 2008. Sample shape and temperature strongly influence the yield strength of metallic
nanopillars. Acta Materialia 56, 4816-4828.
Deshpande, V.S., Ashby, M.F., Fleck, N.A., 2001a. Foam topology: bending versus stretching
dominated architectures. Acta Materialia 49, 1035-1040
Deshpande, V.S., Fleck, N.A., Ashby, M.F., 2001b. Effective properties of the octet-truss lattice
material. Journal of the Mechanics and Physics of Solids 49, 1747-1769
Diao, J., Gall, K., Dunn, M.L., 2004. Yield strength asymmetry in metal nanowires. Nano Letters 4,
1863-1867.
Dingreville, R., Qu, J., 2005. Surface free energy and its effect on the elastic behavior of nano-sized
particles, wires and films. Journal of the Mechanics and Physics of Solids 53, 1827-1854.
Dong, L., Deshpande, V., Wadley, H., 2015. Mechanical response of Ti–6Al–4V octet-truss lattice
structures. International Journal of Solids and Structures 60-61, 107-124.
Dou, R., Derby, B., 2009. A universal scaling law for the strength of metal micropillars and nanowires.
Scripta Materialia 61, 524-527.
Feng, X.-Q., Xia, R., Li, X., Li, B., 2009. Surface effects on the elastic modulus of nanoporous
materials. Applied Physics Letters 94, 011916.
Finnegan, K., Kooistra, G., Wadley, H.N.G., Deshpande, V.S., 2007. The compressive response of
carbon fiber composite pyramidal truss sandwich cores. International Journal of Materials Research 98,
1264-1272.
Fleck, N.A., Deshpande, V.S., Ashby, M.F., 2010. Micro-architectured materials: past, present and
future. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466,
2495-2516.
Gibson, L.J., Ashby, M.F.X., 1999. Cellular solids: structure and properties. Cambridge university
press.
Greer, J.R., De Hosson, J.T.M., 2011. Plasticity in small-sized metallic systems: Intrinsic versus
extrinsic size effect. Progress in Materials Science 56, 654-724.
Gu, X.W., Greer, J.R., 2015. Ultra-strong architected Cu meso-lattices. Extreme Mechanics Letters 2,
7-14
He, J., Lilley, C.M., 2008. Surface effect on the elastic behavior of static bending nanowires. Nano
Letters 8, 1798-1802.
Hodge, A.M., Biener, J., Hayes, J.R., Bythrow, P.M., Volkert, C.A., Hamza, A.V., 2007. Scaling
equation for yield strength of nanoporous open-cell foams. Acta Materialia 55, 1343-1349.
Hutchinson, J.R., 2001. Shear coefficients for Timoshenko beam theory. Journal of Applied Mechanics

68, 87-92.
Jang, D., Meza, L.R., Greer, F., Greer, J.R., 2013. Fabrication and deformation of three-dimensional
hollow ceramic nanostructures. Nat Mater 12, 893-898.
Jennings, A.T., Burek, M.J., Greer, J.R., 2010. Microstructure versus size: mechanical properties of
electroplated single crystalline Cu nanopillars. Physical review letters 104, 135503.
Kelchner, C.L., Plimpton, S.J., Hamilton, J.C., 1998. Dislocation nucleation and defect structure during
surface indentation. Physical Review B 58, 11085.
Lee, S.W., Jafary-Zadeh, M., Chen, D.Z., Zhang, Y.W., Greer, J.R., 2015. Size Effect Suppresses
Brittle Failure in Hollow Cu60Zr40 Metallic Glass Nanolattices Deformed at Cryogenic Temperatures.
Nano Lett 15, 5673-5681.
Liang, H., Upmanyu, M., Huang, H., 2005. Size-dependent elasticity of nanowires: nonlinear effects.
Physical Review B 71, 241403.
Meyers, M.A., Chawla, K.K., 2009. Mechanical behavior of materials. Cambridge university press
Cambridge.
Meza, L.R., Das, S., Greer, J.R., 2014. Strong, lightweight, and recoverable three-dimensional ceramic
nanolattices. Science 345, 1322-1326
Meza, L.R., Zelhofer, A.J., Clarke, N., Mateos, A.J., Kochmann, D.M., Greer, J.R., 2015. Resilient 3D
hierarchical architected metamaterials. Proceedings of the National Academy of Sciences 112, 11502-
11507.
Miller, R.E., Shenoy, V.B., 2000. Size-dependent elastic properties of nanosized structural elements.
Nanotechnology 11, 139
Mishin, Y., Mehl, M.J., Papaconstantopoulos, D.A., Voter, A.F., Kress, J.D., 2001. Structural stability
and lattice defects in copper: Ab initio, tight-binding, and embedded-atom calculations. Physical
Review B 63, 224106.
Montemayor, L.C., Greer, J.R., 2015. Mechanical Response of Hollow Metallic Nanolattices:
Combining Structural and Material Size Effects. Journal of Applied Mechanics 82, 071012.
Moongkhamklang, P., Deshpande, V.S., Wadley, H.N.G., 2010. The compressive and shear response
of titanium matrix composite lattice structures. Acta Materialia 58, 2822-2835.
Park, H.S., Cai, W., Espinosa, H.D., Huang, H., 2009. Mechanics of crystalline nanowires. MRS
bulletin 34, 178-183.
Park, H.S., Gall, K., Zimmerman, J.A., 2006. Deformation of FCC nanowires by twinning and slip.
Journal of the Mechanics and Physics of Solids 54, 1862-1881.
Park, H.S., Klein, P.A., 2008. Surface stress effects on the resonant properties of metal nanowires: the
importance of finite deformation kinematics and the impact of the residual surface stress. Journal of the
Mechanics and Physics of Solids 56, 3144-3166
Plimpton, S., 1995. Fast parallel algorithms for short-range molecular dynamics. Journal of
computational physics 117, 1-19.
Schaedler, T.A., Carter, W.B., 2016. Architected Cellular Materials. Annual Review of Materials
Research.
Schaedler, T.A., Jacobsen, A.J., Torrents, A., Sorensen, A.E., Lian, J., Greer, J.R., Valdevit, L., Carter,
W.B., 2011. Ultralight metallic microlattices. Science 334, 962-965.
Shenoy, V.B., 2005. Atomistic calculations of elastic properties of metallic fcc crystal surfaces.
Physical Review B 71.
Stukowski, A., 2009. Visualization and analysis of atomistic simulation data with OVITO–the Open

Visualization Tool. Modelling and Simulation in Materials Science and Engineering 18, 015012.
Stukowski, A., Bulatov, V.V., Arsenlis, A., 2012. Automated identification and indexing of
dislocations in crystal interfaces. Modelling and Simulation in Materials Science and Engineering 20,
085007.
Sun, Z.H., Wang, X.X., Wu, H.A., 2008. Surface relaxation effect on the distributions of energy and
bulk stresses in the vicinity of Cu surface: An embedded-atom method study. Journal of Applied
Physics 104, 033501.
Timoshenko, S.P., Goodier, J.N., 1970. Theory of elasticity,3rd edition. MacGraw Hill International
Book.
Tschopp, M.A., McDowell, D.L., 2008. Influence of single crystal orientation on homogeneous
dislocation nucleation under uniaxial loading. Journal of the Mechanics and Physics of Solids 56,
1806-1830.
Uchic, M.D., Dimiduk, D.M., Florando, J.N., Nix, W.D., 2004. Sample dimensions influence strength
and crystal plasticity. Science 305, 986-989.
Wang, G.-F., Feng, X.-Q., 2009. Surface effects on buckling of nanowires under uniaxial compression.
Applied physics letters 94, 141913.
Weinberger, C.R., Cai, W., 2012. Plasticity of metal nanowires. Journal of Materials Chemistry 22,
3277.
Xia, R., Li, X., Qin, Q., Liu, J., Feng, X.Q., 2011. Surface effects on the mechanical properties of
nanoporous materials. Nanotechnology 22, 265714.
Yang, Z., Lu, Z., Zhao, Y.-P., 2009. Atomistic simulation on size-dependent yield strength and defects
evolution of metal nanowires. Computational Materials Science 46, 142-150.
Zhang, W., Wang, T., Chen, X., 2008. Effect of surface stress on the asymmetric yield strength of
nanowires. Journal of Applied Physics 103, 123527
Zheng, X., Lee, H., Weisgraber, T.H., Shusteff, M., DeOtte, J., Duoss, E.B., Kuntz, J.D., Biener, M.M.,
Ge, Q., Jackson, J.A., 2014. Ultralight, ultrastiff mechanical metamaterials. Science 344, 1373-1377.
Zheng, X., Smith, W., Jackson, J., Moran, B., Cui, H., Chen, D., Ye, J., Fang, N., Rodriguez, N.,
Weisgraber, T., 2016. Multiscale metallic metamaterials. Nature Materials
Zhou, M., 2003. A new look at the atomic level virial stress: on continuum-molecular system
equivalence. The Royal Society, pp. 2347-2392.
Zhu, T., Li, J., Samanta, A., Leach, A., Gall, K., 2008. Temperature and strain-rate dependence of
surface dislocation nucleation. Physical Review Letters 100, 025502.