Microtwist homogenization of three-dimensional Pyrochlore lattices on zero modes and mechanical polarization

Abstract

The mechanical Pyrochlore lattice was experimentally tested to demonstrate an intrinsically polar behavior of the material, which is soft on one side and hard on the opposite side (Bilal et al., 2017). The topological polarization in Pyrochlore lattices begs for developing a new effective medium theory because conventional Cauchy effective theories cannot predict the polarization phenomenon. In this study, we develop a 3D microtwist effective theory of Pyrochlore lattices to capture the P-asymmetric zero modes by which polarization emerges or fades on a macroscopic scale. By mapping three periodic zero modes to three macroscopic degrees of freedom, the 3D microtwist theory ends up being a kinematically enriched theory. The 3D microtwist elasticity is formulated by using two-scale asymptotic approach and its constitutive and balance equations are derived for a fairly generic isostatic lattice. Performance of the proposed theory is validated by the exact solution of the discrete model for reproducing zero modes and dispersion relations and quantitatively predicting asymmetric indentation responses. The study could shed lights on novel elastic theory of 3D polarized metamaterials outside the conventional framework of symmetry groups, which is never reported before.

Introduction

A mechanical lattice can be loosely defined as a web of beam or spring elements connecting a set of nodes or hinges. In design of the lattice material, harnessing of infinitesimal zero modes, deformation modes that cost little to no elastic energy, provides new paradigm to realize the nonstandard elastic behavior of the mechanical lattice (Lubensky et al., 2015). Although catastrophic in many scenarios, the presence of zero modes could be desired. The most spectacular application is the use of pentamode materials with five zero modes in acoustic cloaking (Milton and Cherkaev, 1995, Kadic et al., 2012, Norris and Shuvalov, 2011, Milton, 2013). Recently, polar materials with one intrinsic zero mode have been proposed in elastic cloaking (Nassar et al., 2018a, Nassar et al., 2019, Nassar et al., 2020a, Zhang et al., 2020, Xu et al., 2020). In those applications, zero modes appear and grow in the lattice material with Parity (P)-symmetry, namely the invariance of the set of solutions under the spatial inversion $\mathbf{x}\mapsto -\mathbf{x}$.

On the other hand, there are mechanical lattices or lattice materials with a broken P-symmetry, i.e., P-asymmetric, which refers to the fact that the space of solution is variant under the action of inversion $\mathbf{x}\mapsto -\mathbf{x}$. Materials with such property are polarized whose zero modes grow in amplitude in a preferential direction and decay in the opposite direction. Kagome lattices are one of the outstanding examples on topological polarization in isostatic lattices (Kane and Lubensky, 2014, Rocklin et al., 2017, Rocklin, 2017, Baardink et al., 2018, Ma et al., 2018, Mao and Lubensky, 2018, Zhang and Mao, 2018, Stenull and Lubensky, 2019, Nassar et al., 2020b). For example, a regular Kagome lattice exhibits P-symmetry bulk zero modes which maintain uniform amplitude across the whole truss. However, general geometric distortions of the lattice will make zero modes polarized where the zero modes adopt exponential profiles that decay towards the bulk and re-localize at free boundaries. Kane and Lubensky (2014) characterized the conditions under which the re-localization of zero modes towards the free boundaries of a distorted lattice happens unevenly and favors certain boundaries over their opposites. Note that the resulting P-asymmetric distribution of zero modes are topological in nature which can be quantified by a topological polarization vector, so that they are immune to continuous perturbations, small and large, as long as the signs of distortion parameters remain unchanged. This is why such Kagome lattices are qualified as “topological polarization”. The topological polarization leads to the appearance of elastic polarization effects whereby a finite sample appears hard when indented on one side and soft when indented on the opposite side. Elastic polarization effects are not restricted to boundaries and emerge in the bulk as well (Rocklin, 2017). Bilal et al. (2017) designed and tested a material made of 3D distorted Pyrochlore lattices featuring a polarized elastic behavior. A finite slab of their material appears soft when indented on one side and hard when indented on the opposite side. To capture such zero modes on the level of the material requires finer measures of strain and its gradients. It is the purpose of the present paper to propose an enriched 3D effective medium theory capable of faithfully reproducing microstructural zero modes and related polarization effects on the continuum scale. Theoretical formulations are conducted for a fairly generic 3D truss: the Pyrochlore lattice.

Mechanical lattices with no zero modes have been successfully investigated using homogenization theory based on the Cauchy continuum mechanics (Deshpande et al., 2001, Hutchinson and Fleck, 2006). However, the polarization behavior due to P-asymmetric zero modes cannot be properly captured from the perspective of conventional continuum mechanics. To address this challenge, a Cosserat micropolar continuum (Cosserat and Cosserat, 1909) was suggested to model mechanical behavior of the polarized Kagome lattices by introducing both the vector displacement of nodes and the rotational degree of freedom (DOF) to describe microrotation (Sun et al., 2012). However, the material characteristic parameter of deformation comes into effect only when the deformation with non-negligible strain gradients or non-local effects is induced. Furthermore, the polarization effects in polarized Kagome or Pyrochlore lattices are caused by the accumulation of zero modes and are usually of a stronger dominant nature. Sun and Mao (2020) and Saremi and Rocklin (2020) proposed theories for polarized effective media of the strain gradient type. As a matter of fact, the polarization described by the micropolar elasticity and strain gradient theory is not of a topological nature and its effects are weak and restricted to boundary layers. By mapping each periodic zero mode to a macroscopic degree of freedom, we recently formulated a “bottom-up” higher-order theory baptized “microtwist” theory capable of rendering polarization effects of the 2D Kagome lattice on a macroscopic scale and quantitatively predicting the polarized indentation response of finite samples (Nassar et al., 2020b). In the study, the microtwist theory is systematically extended to study the mechanical polarization and related topological behavior of 3D pyrochlore lattices. To the best of our knowledge, little to no work has been conducted on realizing polarization effects in 3D topological materials.

Microtwist elasticity is the outcome of leading order two-scale asymptotic expansions with the displacement being a fast scale variable attached to the unit cell and the position being a slow variable attached to the structure in the long wavelength limit, $\mathbf{q}\to \mathbf{0}$. By progressively perturbing the geometry of regular Pyrochlore lattices so as to transform them into distorted ones, the total displacement field is composed of the macroscopic displacement field and of three additional DOFs, namely the twisting angles, directly related to microstructural zero modes. The resulting effective 3D Microtwist Continuum is therefore an enriched continuum allowing for the presence of periodic zero modes in the form of additional DOFs and the additional odd-order tensor elasticity constants are responsible for non-standard effects accompanying them such as polarization.

The structure of the paper is as follows. In Section 2, the compatibility and equilibrium relations of general Pyrochlore lattices are introduced. The classification in terms of regular and distorted Pyrochlore lattices is then recalled based on the number of periodic zero modes they support. In Section 3, the detailed derivation of the 3D microtwist continuum for the weakly-distorted Pyrochlore lattice is presented. In Section 4, dispersion relations and static phenomena taking place in regular and weakly-distorted Pyrochlore lattices are investigated. Results are derived from the discrete model of the Pyrochlore lattice, from the microtwist model and from Cauchy’s model and then compared. The last section contains a brief conclusion.