Computational Modeling, Inverse Design, and Characterization of Electro-Mechanical Willis Metamaterials

Artificially engineered materials, known as metamaterials, have been developed to generate extraordinary effective properties useful for modulating dynamic material responses. A specific class of metamaterials, referred to as Willis materials, is characterized by an effective description termed Willis coupling, which is absent in their microstructures and enables coupling between momentum and strain, as well as stress and velocity. Recently, the understanding of Willis materials has evolved, leading to the development of composites in which mechanical and non-mechanical fields are highly coupled. Particularly, when applying the Willis homogenization approach to piezoelectric composites by Salomon and Shmuel [1], the momentum is also macroscopically coupled to the electrical field, resulting in a new form of electro-momentum coupling. The emergence of this Willis-type cross-coupling offers an additional approach to controlling elastic wave responses through non-mechanical stimuli and supports an alternative way to excite wave propagation via electrical means. Building on the closed-form expressions derived by Shmuel et al. [1], this dissertation develops a numerical framework for ensemble averaging that facilitates the calculation of effective dynamic properties of two-dimensional periodic piezoelectric composites and supports materials design applications. The subsequent task is to design a heterogeneous structure, assisted by topology optimization, to maximize electro-momentum (EM) coupling resulting from the effective electro-mechanical constitutive parameters. Further analysis is performed on the optimized unit cells to demonstrate asymmetric wave propagation due to EM coupling induced by mechanical and electrical loads. Based on the form of effective properties of such piezoelectric metamaterials, the scattering experiment approach is developed to homogenize a heterogeneous medium, where the local (imaginary) parts of Willis coupling as well as EM coupling cause asymmetric responses of the reflection waves. When accounting for the asymmetry of reflection amplitudes, a topology optimization approach is employed to design a one-way zero reflection system. The analysis is conducted on layered piezoelectric media with introduction of material loss as an additional factor to control the asymmetric reflection amplitudes and to effectively embrace the non-Hermitian physics. In addition to the framework of dynamic homogenization methods, the micromechanical technique applied in elastodynamics [2] and electromagnetism [3] is extended to periodic heterogeneous piezoelectric media. This method is derived based on the spatial transformations of field variables and Eshelby tensors into Fourier series, and the conversion of integrals into a system of linear equations. Since the Bloch form is specialized to mechanical and electrical fields, the resultant macroscopic dispersion relations are exactly reproduced in the microscopic counterparts. The propagation of free waves is accordingly examined by comparing their microscopic responses with effective ones to verify the homogenization work in the case of long wavelengths. The concept of Willis materials is notably extended to flexoelectric composites [4], whose non-mechanical fields are additionally induced by strain gradients. Their homogenized constituent shows that the momentum is also coupled to the strain gradient term through an effective description termed gradient elasto-momentum (GEM) coupling. These heterogeneities are homogenized using ensemble averaging, which is incorporated into the dynamic homogenization approach of Willis to deliver a metamaterial system that possesses this coupling term. The existence of strain gradients, which can break the inversion symmetry of a unit cell, is thus significant for generating the imaginary parts of all cross-coupling terms without requiring asymmetry in microstructures. The study also finds an interesting feature that EM coupling not only arises from intrinsic piezoelectric parameters but is also generated by intrinsic flexoelectric couplings. Successive to the introduction of such flexoelectric Willis materials, physical restrictions on their properties are derived from assumptions about reciprocity, passivity, and causality. In particular, reciprocity imposes symmetric constraints on high-order elasticity, flexoelectric tensor, and GEM coupling, in addition to classical terms in piezoelectricity. For passivity, restrictions are placed on the real part of the power flow, ensuring that it is non-negative definite. Inequality relations between the Hermitian and skew-Hermitian parts of the effective material parameters are then defined, followed by the derivation of bounds for the GEM response. Causality, associated with the concept of passivity, imposes restrictions on the flexoelectric Willis constitutive relations, which are modeled as history functions to represent cause-effect relationships. This leads to the definition of complex-valued material parameters in which the real and imaginary parts are interdependent. Since the physical principles underlying the new Willis medium are theoretically understood, it is of interest to explore its applications in wave manipulation. Specifically, the scattering behavior in layered flexoelectric Willis structures is analyzed to show the role of GEM coupling in inducing asymmetry in reflection phases. Moreover, by adjusting external circuit conditions, the intrinsic non-centrosymmetric feature of dielectric materials causes non-symmetries in the effective properties between a given flexoelectric composition and its inversion, which can be harnessed to further modulate dynamic material responses and asymmetric wave phenomena.