Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices

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Abstract

In continuum mechanics, the non-centrosymmetric micropolar theory is usually used to capture the chirality inherent in materials. However, when reduced to a two dimensional (2D) isotropic problem, the resulting model becomes non-chiral. Therefore, influence of the chiral effect cannot be properly characterized by existing theories for 2D chiral solids. To circumvent this difficulty, based on reinterpretation of isotropic tensors in the 2D case, we propose a continuum theory to model the chiral effect for 2D isotropic chiral solids. A single material parameter related to chirality is introduced to characterize the coupling between the bulk deformation and the internal rotation, which is a fundamental feature of 2D chiral solids. Coherently, the proposed continuum theory is applied for the homogenization of a triangular chiral lattice, from which the effective material constants of the lattice are analytically determined. The unique behavior in the chiral lattice is demonstrated through the analyses of a static tension problem and a plane wave propagation problem. The results, which cannot be predicted by the non-chiral model, are verified by the exact solution of the discrete model.

Highlights

► We have proposed a continuum theory to model the chiral effect in 2D isotropic chiral solids. ► A single material parameter is introduced to characterize the feature of 2D chiral solids. ► The effective material constants of the lattice are analytically determined. ► The unique coupling behavior is demonstrated, which cannot be predicted by the non-chiral model.

Introduction

An object is said to be chiral, or with handedness, if it cannot be superposed to its mirror image (Kelvin, 1904). Chirality is encountered in many branches of science, including physics, biology, chemistry and optics. A chiral material should be described by an adequate constitutive equation with handedness in order to characterize the distinct feature of such material. In continuum mechanics, chirality is considered in the context of generalized elasticity, e.g. micropolar (Cosserat) theory (Cosserat and Cosserat, 1909, Eringen, 1966). A general isotropic chiral (also known as non-centrosymmetric, acentric or hemitropic) micropolar theory introduces three additional material constants compared to the non-chiral theory to represent the effect of chirality (Nowacki, 1986, Lakes and Benedict, 1982, Lakes, 2001, Natroshvili and Stratis, 2006, Natroshvili et al., 2006, Joumaa and Ostoja-Starzewski, 2011). The additional material parameters change their signs according to the handedness of the microstructure. This theory provides an efficient tool for modeling the chiral effect presented in materials and structures, e.g., loading transfer in carbon nanotubes and chiral rods (Chandraseker and Mukherjee, 2006, Chandraseker et al., 2009, Ieşan, 2010), mechanics of bone (Lakes et al., 1983), chirality transfer in nanomaterials (Wang et al., 2011) and wave propagation in chiral solids (Lakhtakia et al., 1988, Ro, 1999, Khurana and Tomar, 2009). However, when this theory is applied to a planar isotropic case, e.g. a triangular chiral lattice, the variables describing the chiral effect disappear and the resulting theory becomes basically non-chiral (Spadoni and Ruzzene, 2012). Therefore the basic characteristic of a planar chiral solid cannot be properly modeled by the existing micropolar theory. Recently, the chiral-dependent behavior of planar solids is characterized in the context of strain gradient theory by introducing the high-order elasticity properties (Auffray et al., 2010).

On the other hand, lattice structures can be homogenized as micropolar continuum media (Bazant and Christensen, 1972, Chen et al., 1998, Kumar and McDowell, 2004), the homogenized material constants are derived directly from their microstructures. This provides a useful tool to explain the observed size effect in lattice structures. Chiral lattice structure was also proposed by Prall and Lakes (1996) to achieve a material with negative Poisson's ratio (Lakes, 1987). Among the candidates of these so-called auxetic materials (Yang et al., 2004), the triangular chiral lattice is the mostly investigated one since it is isotropic and the geometric pattern can be controlled by a single continuously varying topological parameter. Its unique mechanical behavior was examined by many researchers under both static (Alderson et al., 2010, Dirrenberger et al., 2011, Spadoni and Ruzzene, 2012) and dynamic (Spadoni et al., 2009) loading conditions with a number of targeted applications. The chiral material was recently used in designing elastic metamaterials with the negative effective bulk modulus (Liu et al., 2011b). Recently Spadoni and Ruzzene (2012) proposed a self-consistent homogenization scheme for a 2D chiral lattice in the framework of the micropolar theory in order to clarify the indeterminacy of the effective shear modulus (Liu et al., 2011a, 2011b). However, since the non-centrosymmetric isotropic micropolar model becomes non-chiral when applied to a planar problem (Spadoni and Ruzzene, 2012), the developed homogenization method in this framework cannot characterize the chiral effect inherent in the material. Therefore we are encountering a challenging problem: for planar isotropic chiral materials, e.g. triangular chiral lattices, we do not have a solid theory either in continuum formulation or in the homogenization method to characterize the chiral effects.

The objective of the paper is to propose a continuum model to capture the chiral effect in planar isotropic chiral solids, and the corresponding effective material constants will be derived for a planar triangular chiral lattice. Some typical examples are conducted to demonstrate the necessity and consistency of the theory in characterizing the chiral effect. The manuscript is organized as follows: in Section 2, a new constitutive relation for a 2D isotropic chiral solid is proposed based on a continuum formulation. In Section 3, a triangular chiral lattice structure is homogenized in the framework of the proposed theory and the effective material constants are derived. In Section 4, a tension and plane wave propagation problems are examined for a planar chiral lattice by the proposed theory. In Section 5 the main result of this work is concluded.

Section snippets

Planar isotropic micropolar model with chirality

Characterization of material chirality is closely related to the concept of pseudo (or axial) tensors, they alternate the sign with a mirror reflecting transformation or the handedness change of the underlying coordinate system, and ordinary (or polar) tensors are not affected by such actions (Borisenko and Tarapov, 1979). Both types of tensors coexist in various elastic formulations, but strain energy density must be independent of handedness.

Classical elasticity theory excludes chirality (

Homogenization for a 2D triangular chiral lattice

In this section, we will examine a 2D triangular chiral lattice, and analytically derive the five material constants proposed in the Section 2 by a homogenization method.

Discussions and applications

In this section, we will discuss in detail the obtained effective material constants for the triangular chiral lattice. A static tension problem and a plane wave analysis will also be examined in order to illustrate the proposed theory.

Conclusions

The existing micropolar theory is not able to characterize the chiral effect inherent in plane isotropic chiral solids, e.g. triangle chiral lattices. In the paper, we propose a continuum theory to capture the chiral effect in such materials. The proposed method is based on the micropolar theory and reinterpretation of in-plane isotropic tensors. The constitutive equation and the governing equation are analytically derived. Different from the existing 2D isotropic micropolar theory, a new

Acknowledgment

This work was supported in part by Air Force Office of Scientific Research under Grant no. AF 9550-10-0061 with Program Manager Dr. Byung-Lip (Les) Lee and NSF 1037569, and in part by National Natural Science Foundation of China under Grant nos. 10972036, 10832002 and 11072031, and the National Basic Research Program of China (Grant no. 2011CB610302).

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