**Performance through instability**

#### Multiscale Materials Systems

Like the multiscale structure of metals and ceramics (from individual atoms up to the macroscale) determines the macroscopically observable behavior, so does the performance of structural solids depend on the arrangement of structural members across all relevant scales. Recent advancements in micron-level additive manufacturing (AM) have opened a field of multiscale material design previously unattainable. This new field has the capacity to produce materials with tailored properties due to the freedom to independently control the structural architecture across many length scales as well as the base material selection. With such freedom comes immense opportunity for unprecedented effective static and dynamic mechanical properties; however, sifting through all the possible combinations is an enormous challenge. Therefore, we develop and apply computational and theoretical methods to identify particular multiscale materials systems with interesting, exceptional, or extreme properties. In the analysis of these materials we aim to capture the macroscopic properties in a computationally efficient manner. Often we may assume a separation of length scales, which allows us to use techniques of homogenization; i.e. we describe and characterize the performance at smaller length scales and pass the effective response up to the larger length scales. Of greatest interest to us is how instabilities on the small scales affect the macroscopic response.

*Figure: length scales in (micro-)structural solids from individual truss members to the periodic unit cell to the macroscopic structural material. Theory, computations and experiments aim to understand the link between microstructure and macroscale performance, in particular in case of small-scale structural instabilities.*

#### Exploiting Instability

The long history of mechanics of materials is rife with prominent examples of instabilities in solids and structures that have led to material failure or structural collapse, spanning all scales from micro-electronics to plate tectonics. Examples of mechanical instability are ubiquitous: buckling of structures or crushing of cellular solids, plastic necking, strain localization in shear or kink bands, void growth and coalescence, or material failure. Through its more than 250 year-old history, the theory of material and structural stability has resulted in analytical and numerical tools that provide engineers with safe design guidelines. Over the past decade, an exciting paradigm shift has been initiated from traditional design against instability towards novel performance through controlled instability: (in)stability is instrumentalized for beneficial material behavior such as large reversible deformation of cellular solids, or soft devices undergoing dramatic shape changes. Here, we take a slightly different route and exploit instabilities at the material level.

Our theoretical, computational, and experimental findings have confirmed an auspicious avenue for engineered materials systems, viz.
the tremendous potential of composite systems with phases undergoing instabilities.
A bistable system *A* displays unstable snapping from one stable state to another upon sufficient
perturbation. When *A* is embedded in a stiff environment *B*, the instability is constrained. Upon
perturbation, *A* now experiences stable non-positive-definite incremental elastic moduli, so-called
negative stiffness, far before the overall system *A+B* becomes unstable.
Specifically, we have shown that the constraining matrix phase in a composite can stabilize negative-stiffness inclusions, which
greatly expands the regime of linear elastic moduli traditionally studied within composite theory.
This is vital news because allowing for negative elastic moduli has been predicted before to result
in extreme increases of composite (visco)elastic moduli and damping (i.e. time-domain
harmonic phase lag) as well as further extreme physical properties, see also the work of
Prof. R. Lakes.
Following the same idea, structural instability of bistable structural members
has been functionalized in stiff high-damping vibration isolation components e.g. for scientific
instruments or for ergonomic transportation devices but has not been exploited in hierarchical or multiscale structural solids
to achieve extreme effective properties - which is exactly our approach.

*Figure: structural instabilities on the level of the periodic truss cells is embedded in a stiff constraining truss matrix which stabilizes negative-stiffness effects. Induced by thermal, electrical, or other physical stimuli, such controlled instabilities can be exploited to fine-tune the effective material response.*

Our goal is to advance current understanding and opportunities in functional material systems with superior performance based on the controlled use of mechanical instabilities. This requires a solid foundation in terms of a theoretical basis, numerical tools, and experimental avenues. We investigate the mechanical response of multiscale materials systems in order to develop the underlying stability theory and establish missing insight into effective material properties. While experiments and associated models focus on the specific example of structural materials with pre-stressed bistable elements, the envisioned new theory will be generally applicable to the mechanics of materials with more than one characteristic length scale. Our primary target mechanical properties to be investigated and ultimately to be engineered are the effective

- linear elastic moduli (in the quasistatic limit),
- frequency-dependent viscoelastic moduli for low-frequency time-harmonic excitation (i.e., incremental loss/storage moduli, or dynamic moduli and damping),
- dispersion relations for linear stress wave propagation.

Overall, we aim to answer the following open questions:

- How is microscale instability in multiscale structural materials perceived on the macroscale?
- What effective continuum theory can describe the deformation-dependent macroscale (visco)elastic properties and dispersion relations of bistable periodic structural materials, and what are the strongest possible bounds on these properties?
- Under what conditions can such materials give rise to extreme increases in static or dynamic stiffness, effective damping, or beneficial total/directional attenuation of elastic waves?
- How can the paradigm of embedding bistable elements be used to design functional material systems that dramatically change their effective mechanical behavior by the push of a button?

#### Multiscale Modeling

Among the numerous material properties that are beneficial for a variety of applications, in this project we focus on the (visco)elastic characteristics of multiscale materials systems. This includes their quasi-static stiffness, their damping capacity (i.e., their ability to attenuate vibrations), and their frequency-dependent stiffness and damping. These macroscopic properties arise from mechanisms on the lower scales, which requires scale-bridging methodologies. For the quasistatic response, we bridge across scales by methods of theoretical and computational homogenization, which assumes a separation of scales and obtains the effective macroscale response from a representative unit cell on the lower scale(s). In addition, we calculate the material's dispersion relations which reveal the propagation and attenuation of waves in the medium. In both of these approaches we account for both material and geometric nonlinearity (the former being of importance in case of inelastic materials, the latter being crucial when large deformation or structural buckling is involved).

*Figure: homogenization methods assume a separation of scales, where the macroscale mechanics are obtained by averaging over a representative unit cell on the smaller scale (e.g. with periodic boundary conditions).*

Using first-order homogenization, we identify a periodic unit cell on the microscale whose effective (average) properties determined by applying periodic boundary conditions (periodic deformation and anti-periodic tractions). Among others, this yields the effective incremental elastic modulus tensor (i.e., the macroscopically observable stiffness of the periodic medium). For example, for structural solids such as micro- and nano-lattices, we can predict the effective modulus tensor as a function of the microscale periodic architecture, the relative density, and the base material properties. We identify the link between structure and properties and exploit it to design structural solids based on a variety of solid and hollow structural designs for optimal macroscale performance.

*Figure: illustration of band gaps in the dispersion relations of a one-dimensional chain of masses connected by linear springs (identical masses show no band gaps, while alternating different masses open up a band gap).*

The dispersion relations of a (visco)elastic medium establish the relation between the speed and the wavelength of linear stress waves propagating through the material. This reveals band gaps, i.e. ranges of frequencies at which waves cannot freely propagate through a material (either in particular directions or in all directions). These gaps of frequencies allow the material to act as a mechanical filter or as a wave guide. A simple and illustrating example, the one-dimensional chain shown above consists of point masses and elastic springs. When the masses are equal (monatomic chain), all frequencies will propagate through the chain (up to a maximum admissible frequency). If two types of masses are alternating (in a diatomic chain), then a band gap forms in which mechanical frequencies will be effectively damped out.

*Figure: example of a periodic elastic truss with possible unit cells shown on the left. Fine-tuning the angle between the trusses allows us to control the elastic properties and the dispersion relations*

(see also the Adv. Eng. Mater. paper).

(see also the Adv. Eng. Mater. paper).

High stiffness and high damping are usually exclusive in natural materials. However, interesting combinations can arise from structural solids such as the three-dimensional lattice structures shown above. We have demonstrated how a clever distribution of mass across the periodic unit cell can be used to significantly control the wave propagation with only a minimal effect on the effective stiffness (normally, stiffness and wave propagation are intimately tied to each other). Shown above are example unit cells use truss arrangement can be fine-tuned to control stiffness and wave propagation. In addition, we study the optimal arrangement of periodic lattices to result in maximum wave attenuation across wide ranges of frequency. Shown below is a small three-dimensional lattice structure which is excited by an imposed vibration at one end at two different frequencies. While in one case the wave propagates through the entire structure, the other case shows the presence of a band gap and the resulting disintegration of the wave.

*Figure: distributing extra masses within the periodic unit cell (without significantly affecting the effective stiffness) has a tremendous impact on the dispersion relations, producing pronounced band gaps*

(see also the Adv. Eng. Mater. paper).

(see also the Adv. Eng. Mater. paper).

The following movies illustrate the propagation of linear stress waves through a block of a periodic lattice, first outside all band gaps (no wave attenuation) and then at a frequency within a band gap (showing hwo the wave disappears):

Goind beyond the (visco)elastic behavior of materials, we are also interested in the inelastic response of a variety of materials, including metals, ceramics, and composites. Here more sophisticated modeling techniques are required to bridge the scales. See the summary of multiscale modeling techniques developed and/or used in our group.

#### References:

- N. Nadkarni, C. Daraio, D. M. Kochmann.
Dynamics of a periodic structure containing bistable elastic elements:
from elastic to solitary wave propagation,
*Phys. Rev. E***90**(2014), 023204. - D. M. Kochmann, G. W. Milton.
Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases,
*J. Mech. Phys. Solids***71**(2014), 46-63. - C. S. Wojnar, D. M. Kochmann.
A negative-stiffness phase in elastic composites can produce stable
extreme effective dynamic but not static stiffness,
*Philos. Mag.***94**(2014), 532-555. - S. Krödel, T. Delpero, A. Bergamini, P. Ermanni, D. M. Kochmann.
3D auxetic microlattices with independently-controllable acoustic
band gaps and quasi-static elastic moduli,
*Adv. Eng. Mater.***16**(2014), 357–363. - C. S. Wojnar, D. M. Kochmann.
Stability of extreme static and dynamic bulk moduli of an elastic two-phase composite due to a non-positive-definite phase,
*Phys. Stat. Solidi B***251**(2014), 397-405.