Characterizing Dissipative Elastic Metamaterials Produced by Additive Manufacturing

Polymers reveal viscoelastic response to a greater or smaller extent. This means that in addition to elastic behavior described by elastic (storage) moduli, they have viscous (loss) components. Viscous losses cause delay in the development of stress under applied strain and vice versa. Under dynamic excitation, out-of-phase stress components are dissipated through heat, thus reducing the energy of acoustic waves propagating in a viscoelastic medium. This phenomenon is referred to as viscous damping.

Viscosity originates at a molecular level due to relative motions or local rotations of bonds in polymer chains and, thus, is governed by the chemical composition, structure, and connections of polymer chains. Molecular mobility depends on temperature and deformation rate, resulting in temperature- and time-driven behavior of viscoelastic materials. All this makes viscoelasticity an inherently complex phenomenon that has a unique signature for each material. One feasible way to approximate such behavior implies modeling a viscoelastic material as a mechanical system composed of (Hookean) springs and (Newtonian) dashpots¹. Although this approach fully neglects the molecular structure of a material and all the complexity of a real relaxation process, it can provide adequate results for hard polymers with comparatively low viscous losses².

The key to obtaining an adequate mechanical model is tuning the parameters of the springs and dashpots to experimental data for the storage and loss moduli of a viscoelastic polymer^3,4,5,6,7,8. This work describes a set of methods to determine the viscoelastic moduli of additively manufactured polymers and to use them in characterizing the dynamics of elastic metamaterials. By this, we aim to bridge the gap between material properties and the structure-driven dynamics of metamaterials, enabling a more robust and reliable design of metamaterials for target working frequencies.

Elastic metamaterials are a class of engineered, often periodically structured materials that can manipulate acoustic waves in solids in an unusual yet controllable way⁹. The wave manipulation is mainly implemented by tailoring bandgaps – the frequency ranges in which wave propagation is prohibited⁴. The unique dynamics of elastic metamaterials are governed by a fine-tuned architecture represented by complex-shaped unit cells, especially for three-dimensional configurations. Such structural complexity can often be realized only using additive manufacturing which makes viscoelasticity analysis especially relevant for additively manufactured elastic metamaterials. Most current studies, however, have used oversimplified models of viscosity, such as the Maxwell^10,11 or Kelvin-Voigt model¹¹. Because these models cannot describe any real viscoelastic material², the conclusions derived by using them cannot be considered reliable. Therefore, there is a burning need for more realistic models replicating viscoelastic material properties at ultrasonic frequencies. Several studies have addressed this need^6,8,12 and reported serious limitations of commercial finite element solvers due to high¹³ computational load, especially when dealing with complex geometries and/or high frequencies¹⁴ and the restriction on considering the relaxation of a single modulus (in reality, both moduli of an isotropic medium under relaxation). Another analysis method, e.g., plane wave expansion, can reduce the computational burden¹⁵, but requires an analytical description of the scatterer geometry, limiting its applicability. The extended plane wave expansion approach^16,17 addresses this limitation but adds computational complexity. The Bloch wave expansion¹⁸ and transfer matrix methods¹⁹ can only consider periodic structures of finite dimensions, which can be described analytically. The spectral element approach^20,21 offers computational efficiency, but its applicability is limited to very low frequencies below the first bandgap. Thus, in addition to the lack of experimental data for storage and loss moduli at room temperature and high frequencies (above 100 Hz), which are common work conditions for elastic metamaterials^20,22,23,24, the analysis of their dynamics remains challenging. This work aims to fill in these gaps by summarizing the experimental (and numerical) techniques for the characterization of additively manufactured viscoelastic polymers and elastic metamaterials made of them.

This approach is illustrated by analyzing a simple one dimensional (1D) continuous analog of a periodic mass-spring model made of commonly used acrylonitrile butadiene styrene (ABS) polymer and produced by a fused-deposition modeling (FDM) 3D-printing (Section 1), for which one can experimentally determine the decomposition and glass transition temperatures (Section 2) and derive the master curves for storage and loss moduli at reference room temperature (Section 3). In addition, the quasi-static mechanical moduli can be estimated in tensile tests (Section 4) and linked to their dynamic counterparts. Next, the numerical method to model the dynamic characteristics of a metamaterial is described (Section 5), and the obtained numerical results are validated experimentally in transmission experiments (Section 6). Finally, the applicability and limitations of the proposed methods based on the findings are discussed.