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There has been significant recent interest in the mechanics community to design structures that can either violate reciprocity, or exhibit elastic asymmetry or odd elasticity. While these properties are highly desirable to enable mechanical metamaterials to exhibit novel wave propagation phenomena, it remains an open question as to how to design passive structures that exhibit both significant non-reciprocity and elastic asymmetry. In this paper, we first define several design spaces for chiral metamaterials leveraging specific design parameters, including the ligament contact angles, the ligament shape, and circle radius. Having defined the design spaces, we then leverage machine learning approaches, and specifically Bayesian optimization, to determine optimally performing designs within each design space satisfying maximal non-reciprocity or stiffness asymmetry. Finally, we perform multi-objective optimization by determining the Pareto optimum and find chiral metamaterials that simultaneously exhibit high non-reciprocity and stiffness asymmetry. Our analysis of the underlying mechanisms reveals that chiral metamaterials that can display multiple different contact states under loading in different directions are able to simultaneously exhibit both high non-reciprocity and stiffness asymmetry. Overall, this work demonstrates the effectiveness of employing ML to bring insights to a novel domain with limited prior information, and more generally will pave the way for metamaterials with unique properties and functionality in directing and guiding mechanical wave energy.

Fullerene-based sandwiches have emerged as new candidates for potential applications of two-dimensional nanomaterials in electronics or energy storage. Recently, experimentalists have observed the evolution of boundaries for fullerene clusters sandwiched by two graphene layers, while vacuum space with typical dimension of 30 {\AA} was found within the fullerene layer. Because the pattern of the fullerene cluster impacts the physical properties of the sandwiches, it is important to understand the mechanisms for their structural transformations. In the present work, we find that the graphene/fullerene/graphene sandwich structure transforms among three configurations, depending on the fullerene to graphene area ratio. Molecular dynamics simulations show that there are two critical values for the area ratio. The fullerene pattern transforms from circular to rectangular at the first critical area ratio of 1/{\pi}. The critical value of 1/{\pi} is successfully derived by comparing the geometrical perimeter of the circular and rectangular shapes for the fullerene cluster without any physical parameters. At the second critical area ratio, the graphene layers surrounding the fullerene cluster become separated due to the competition between the bending energy and the cohesive energy. Based on the analytic model, the vacuum space is predicted to be 34 {\AA}, which agrees quite well with the experimental result. These findings provide fundamental insights into the mechanisms driving structural transformation of fullerene-based sandwiches, which will enable the selection of appropriate structures and materials to guide the design of future electronic and energy storage applications.

Flexoelectricity is a form of electromechanical coupling that has recently emerged because, unlike piezoelectricity, it is theoretically possible in any dielectric material. Two-dimensional (2D) materials have also garnered significant interest because of their unusual electromechanical properties and high flexibility, but the intrinsic flexoelectric properties of these materials remain unresolved. In this work, using atomistic modeling accounting for charge-dipole interactions, we report the intrinsic flexoelectric constants for a range of two-dimensional materials, including graphene allotropes, nitrides, graphene analogs of group-IV elements, and the transition metal dichalcogenides (TMDCs). We accomplish this through a proposed mechanical bending scheme that eliminates the piezoelectric contribution to the total polarization, which enables us to directly measure the flexoelectric constants. While flat 2D materials like graphene have low flexoelectric constants due to weak $\pi-\sigma$ interactions, buckling is found to increase the flexoelectric constants in monolayer group-IV elements. Finally, due to significantly enhanced charge transfer coupled with structural asymmetry due to bending, the TMDCs are found to have the largest flexoelectric constants, including MoS$_{2}$ having a flexoelectric constant ten times larger than graphene.

Due to their combination of mechanical stiffness and flexibility, two-dimensional (2D) materials have received significant interest as potential electromechanical materials. Flexoelectricity is an electromechanical coupling between strain gradient and polarization. Unlike piezoelectricity, which exists only in non-centrosymmetric materials, flexoelectricity theoretically exists in all dielectric materials. However, most work on the electromechanical energy conversion potential of 2D materials has focused on their piezoelectric, and not flexoelectric behavior and properties. In the present work, we demonstrate that the intrinsic structural asymmetry present in monolayer Janus transition metal dichalcogenides (TMDCs) enables significant flexoelectric properties. We report these flexoelectric properties using a recently developed charge-dipole model that couples with classical molecular dynamics simulations. By employing a prescribed bending deformation, we directly calculate the flexoelectric constants while eliminating the piezoelectric contribution to the polarization. We find that the flexoelectric response of a Janus TMDC is positively correlated to its initial degree of asymmetry, which contributes to stronger $\sigma-\sigma$ interactions as the initial degree of asymmetry rises. In addition, the high transfer of charge across atoms in Janus TMDCs leads to larger electric fields due to $\pi-\sigma$ coupling. These enhanced $\sigma-\sigma$ and $\pi-\sigma$ interactions are found to cause the flexoelectric coefficients of the Janus TMDCs to be several times higher than traditional TMDCs such as MoS$_{2}$, whose flexoelectric constant is already ten times larger than graphene.

There has been a massive increase in research interest towards applying data driven methods to problems in mechanics. While traditional machine learning (ML) methods have enabled many breakthroughs, they rely on the assumption that the training (observed) data and testing (unseen) data are independent and identically distributed (i.i.d). Thus, traditional ML approaches often break down when applied to real world mechanics problems with unknown test environments and data distribution shifts. In contrast, out-of-distribution (OOD) generalization assumes that the test data may shift (i.e., violate the i.i.d. assumption). To date, multiple methods have been proposed to improve the OOD generalization of ML methods. However, because of the lack of benchmark datasets for OOD regression problems, the efficiency of these OOD methods on regression problems, which dominate the mechanics field, remains unknown. To address this, we investigate the performance of OOD generalization methods for regression problems in mechanics. Specifically, we identify three OOD problems: covariate shift, mechanism shift, and sampling bias. For each problem, we create two benchmark examples that extend the Mechanical MNIST dataset collection, and we investigate the performance of popular OOD generalization methods on these mechanics-specific regression problems. Our numerical experiments show that in most cases, while the OOD generalization algorithms perform better compared to traditional ML methods on these OOD problems, there is a compelling need to develop more robust OOD generalization methods that are effective across multiple OOD scenarios. Overall, we expect that this study, as well as the associated open access benchmark datasets, will enable further development of OOD generalization methods for mechanics specific regression problems.