**Mechanics of hierarchical nanolattices**

Over the past decade, fabrication techniques have made significant advancements to the nanoscale, which enables us
to manufacture material systems with feature sizes comparable to the characteristic microstructure.
The nanotrusses fabricated by our collaborators in
Prof. Julia Greer's group at Caltech show
intriguingly small features: each truss member is on the order of 1 micron in length, and the hollow-tube
truss cross-sections come with wall thicknesses on the order of less than 50 nm. At these scales, strong
size effects
become apparent, i.e., these nanoscale structures show effects that are very different from those observed in
similar structures on larger scales. In collaboration with Prof. Greer, we investigate these
size effects through the combination of coarse-grained atomistic and continuum modeling (*Kochmann group*) and
nanotruss fabrication plus in-situ experimental characterization (*Greer group*).

*Figure: micrographs of example nanotrusses fabricated by Prof. Julia Greer's group at Caltech.*

The mechanical behavior of this class of architected solids is of enormous interest, owing to the extreme level of tunability that
admits the creation of novel materials with optimal elastic and inelastic properties. We have investigated various interesting
aspects of nanotruss deformation mechanisms, including elastic and inelastic properties as well as wave propagation.
Studies revealed two types of size effects that compete in nanolattices.
Intrinsic size effects arise from the material's microstructure and, e.g.,
show that a base material's strength increases with decreasing average grain size, thus demonstrating a
*smaller is stronger* behavior. This phenomenon is reversed at small grain sizes on the nanoscale near 20 nm,
below which the behavior transforms into *smaller is weaker*. In contrast,
extrinsic size effects are observed in
small-scale structures and arise from the abundance of free surfaces and the small volumes
in nanoscale structural members. Here, strength increases with decreasing feature size of single-crystalline
smaples (*smaller is stronger*) due to dislocation starvation and the attraction of existing defects to
free surfaces. In polycrystalline samples, size effects and their underlying mechanisms are much more complex due
to the interplay between competing effects arising from the characteristic microstructural sizes (e.g., grain size)
and the characteristic structural feature sizes (e.g., truss diameter and wall thickness). This competition
results in interesting size effects to be proliferated to the macroscale. We investigate these competing deformation
mechanisms and the resulting macroscopic response through a mixture of techniques described in the following.

In order to understand the fundamental deformation and failure mechanisms in nanoscale metallic solids, we use the quasicontinuum (QC) method to simulate the mechanical behavior of individual nanoscale trusses and truss junctions. The QC method bridges across scales from atomistics to the continuum and enables us to extend local atomistic accuracy to the micronscale by providing finite-element efficiency (see here for details). This is crucial when investigating inelastic nanolattices, in which continuum mechanics and the finite element method break down but the characteristic structural sizes are too large for atomistic simulations. A sample QC representation of a single-crystalline nanotrusses is shown below (full atomistic regions is confined to where it is indeed needed, e.g., near defects or free surfaces; the remaining simulation domain is efficiently coarsened). Such simulations have revealed the effective force-displacement relation (which allows for one-to-one comparison with experimental data), and the nucleation and evolution of microstructure such as the formation of dislocations and slip bands.

*Figure: a simulated nanotruss junction made up of four nanotruss members that are radially coarsened. The left image shows the initial geometry of the junction, while the images on the right illustrate various cross-sections of beams, displaying their QC meshes.*

When considering the elastic nanotruss behavior up to plastic yielding, we showed that methods of continuum mechanics are indeed appropriate to predict the effective structural response. In collaboration with the group of Prof. Julia Greer, we have investigated the mechanical response of complex nanolattices. In particular, we studied the behavior of hierarchical lattice networks, which are commonly found in biomaterials such as bone and wood. Hierarchical structures are hypothesized to be robust and damage-tolerant due to their scale-spanning architecture, but they are exceedingly difficult to model since they can have length scales spanning many orders of magnitude and they may require the representation of thousands to millions of truss members. We have used the finite element method and devised reduced-order models of solid, hollow, and composite hierarchical nanolattices by kinematically constraining the deformation of fully-resolved lattice beams and nodes (see figure below). By condensing the degrees of freedom from the kinematic constraints, we created a simplified model that accurately matches the linear elastic behavior of a variety of hierarchical nanolattices fabricated and tested by the Greer group.

*Figure: model flowchart showing a representative lattice geometry section (top left), simplified bar-truss model (top middle), refined model containing geometrically unique super-nodes (SN) and super-beams (SB) (bottom left), and refined nanolattice model colored by unique beam or node geometries (right).*

We currently explore the fundamental stiffness scaling of various nanolattice topologies with the objective of relating it to existing cellular solids theory. Importantly, experimentally-fabricated lattices oftentimes do not admit analytical treatment as slender elastic beams, which renders most available classical theories invalid. We have developed computational models that achieve excellent agreement with experiments performed by the Greer group within and beyond the realm of classical beam models. These models are also being used to predict the stiffness of nanolattices currently unable to be tested experimentally, entering regimes that have not been studied before and where interesting mechanisms appear.

*Figure: Tetrakai lattices - A) Micrograph of a nanolattice manufactured by Prof. Julia R. Greer's Group;*

B) Timoshenko beam simulation using in-house finite element code; C) Full-resolution simulation of a single unit cell.

B) Timoshenko beam simulation using in-house finite element code; C) Full-resolution simulation of a single unit cell.

We have also begun to expand upon the knowledge gained from linear elastic models by studying the nonlinear response of complex architected structures. Specifically, we focus on defect sensitivity and failure of nanolattices through finite element modeling in order to better understand the mechanics and enhance the design of nano-architected materials. Imperfections play an important role in realistic, fabricated samples and must be accounted for in the design process in order to produce optimal nanolattices exhibiting the desired mechanical performance.

*Figure: Timoshenko beam simulation of notched octet-truss lattice in tension with contours of the von Mises stress before failure (right) and damage after failure (left).*

In addition to the above finite element simulations, we develop nonlocal continuum theories (combined with QC coarse-graining techniques) which enable us to efficiently model periodic truss networks with enormous numbers of truss members by using a continuum approximation to replace the exact discrete lattice description. This technique, which is based on the extended Cauchy-Born rule applied to a representative unit cell of the lattice, has been applied to auxetic and other lattice types.

*Figure: describing a periodic truss network by a continuum model: results for the deformed shape of an auxetic X-braced square lattice and the associated force-displacement curve; comparison of the exact discrete truss model (right images) and the FEM-discretized continuum model (left images); force-displacement curves are shown for different FEM meshes in comparison with the exact discrete solution.*

#### References:

- L. R. Meza, A. J. Zelhofer, N. A. Clarke, A. J. Mateos, D. M. Kochmann, J. R. Greer.
Summation rules for a fully-nonlocal energy-based quasicontinuum method,
*Proc. Nat. Acad. Sci.***112**(2015), 11502-11507. - J. S. Amelang, G. N. Venturini, D. M. Kochmann.
Summation rules for a fully-nonlocal energy-based quasicontinuum method,
*J. Mech. Phys. Solids***82**(2015), 378-413. - J. S. Amelang, D. M. Kochmann.
Surface effects in nanoscale structures investigated by a fully-nonlocal energy-based
quasicontinuum method,
*Mech. Mater.***90**(2015), 166-184. - A. Desmoulins, A. J. Zelhofer, D. M. Kochmann.
Auxeticity in truss networks and the role of bending vs. stretching deformation,
*Smart Mater. Struct.***25**(2016), 054003. - See also the publications of Prof. Julia R. Greer's group.

#### Research Sponsors:

We gratefully acknowledge the support from the**National Science Foundation**(NSF) through award CMMI-123436 as well as from the

**Office of Naval Research**(ONR) through award N00014-16-1-2431.