**Continuum models for multiscale cellular networks**

Cellular networks such as periodic micro- and nanolattices have gained renewed attention, owing to advances in additive manufacturing that allow for the careful scale-bridging architecture of metamaterials with as-designed effective, macroscopic properties. When predicting the effective behavior of such truss networks, a key challenge arises from the tremendous numbers of individual truss members (struts and junctions) that must be considered computationally. Simulations on periodic unit cells can only reveal limited information when large-scale deformation features such as instabilities or damage and fracture occur. Therefore, we develop continuum models that capture the complex mechanical behavior of (meta)materials made of small-scale truss networks.

*Figure: The concept of homogenization - as the size of individual features within a truss network becomes smaller and smaller, the effective behavior of the overall structure can increasingly be represented by an effective material model that does not involve simulating each and every truss member while predicting the same effective performance.*

Besides setting up the underlying theory, we also develop numerical tools to effectively use the new theoretical models in engineering problems. Because beam bending involves rotational degrees of freedom, the resulting theoretical models become more advanced than classical constitutive models. In addition, we aim to capture the full nonlinear, inelastic material behavior. We therefore assume a separation of scales and pass information back and forth between the macroscopic boundary value problem and the lattice's unit cell on the microscale.

*Figure: Schematic view of the multiscale simulation strategy that replaces modeling a discrete truss network by modeling a continuous block of material at the macroscale, whose effective behavior is obtained from investigating individual unit cells at the microscale.*

When applied to elastic-plastic lattices, the models can be used to predict the large-deformation behavior of periodic truss structures at a fraction of the original computational cost.

*Figure: periodic truss network described 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. For more information see our paper in CMAME.*

New theories and computational tools aim to describe not only the quasistatic mechanical behavior but also the dynamics of truss lattices. One aspect of interest is the directional guiding of acoustic waves: depending on the architecture and geometry of the truss network, mechanical waves can be guided in particular directions or attenuated completely - for applications from sound isolation to signal processing and acoustic cloaking.

*Figure: Demonstration of the directional wave guiding in a hexagonal 2D truss lattice (shown in the top left). For each direction and excitation frequency, we determine the group velocity (i.e., the velocity of energy flow; shown as color code in the bottom left image). Transient numerical calculations confirm the directional wave prediction (right images). For more information, see our paper in IJSS.*

#### References:

- L. Meza, G. Phlipot, C. M. Portela, A. Maggi, L. C. Montemayor, A. Comella, D. M. Kochmann, J. R. Greer.
Redefining the mechanical property space of architected materials,
*Acta Mater.***140**(2017), 424-432. - A. Desmoulins, D. M. Kochmann.
Local and nonlocal continuum modeling of inelastic periodic networks applied to stretching-dominated trusses,
*Comput. Methods Appl. Mech. Engng.***313**(2017), 85-105. - A. Zelhofer, D. M. Kochmann.
On acoustic wave beaming in two-dimensional structural lattices,
*Int. J. Solids Struct.***115-116**(2017), 248-269. - 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. - 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.

#### 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. Dr. Claire Lestringant's research is supported by an ETH Postdoctoral Fellowship.