Static chiral Willis continuum mechanics for three-dimensional chiral mechanical metamaterials

Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of linear Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to linear Cauchy elasticity, of which three directly influence chiral effects. Here, we discuss the behavior of the static case of an alternative generalization of linear Cauchy elasticity, the Willis equations. We show that in the homogeneous static cubic case, only one additional parameter with respect to linear Cauchy elasticity results, which directly influences chiral effects. We show that the static Willis equations qualitatively describe the experimentally observed chiral twist effects, too. We connect the behavior to a characteristic length scale.

Figure 1

Cuboid beams with volume $L\times L\times 2L$, with side length $L=500\mu \mathrm{m}$, are subject to uniaxial loading along the negative $z$ direction. In linear Cauchy elasticity, the beam compresses and can expand or contract laterally, depending on its Poisson ratio (not depicted). However, a twist is forbidden in linear Cauchy elasticity, even if the underlying crystal symmetry would allow for a twist. (a) Finite-element calculation for a chiral metamaterial microstructure exhibiting a twist behavior [7]. (b) Same as (a), but calculated using chiral micropolar Eringen elasticity [7]. (a) and (b) are taken with permission from [7]. (c) Numerical calculations based on generalized linear Cauchy elasticity following Willis. For all three panels, the effects are calculated within the linear elastic regime and are magnified tenfold for clarity. The modulus of the displacement vector field is superimposed on a false-color scale. Parameters in (c) are $C_{11}=32.8\mathrm{MPa}$, $C_{12}=-6.1\mathrm{MPa}$, $C_{44}=19.4\mathrm{MPa}$ (in Voigt notation), and $\alpha =3\mathrm{GPa}/\mathrm{m}$.

Figure 2

(a) Same as Fig. 1 with parameter $\alpha =3\mathrm{GPa}$/m and all other parameters fixed as in Fig. 1. (b) $\alpha \to 3.3\alpha$. (c) $\alpha \to 33\alpha$. From such raw data, the twist angle at the top per axial strain can be deduced and plotted vs $\alpha$ (see Fig. 3). All deformations are magnified twofold for clarity.

Figure 3

Twist/strain vs parameter $\alpha$ (lower horizontal scale) as deduced from raw data such as the ones shown in Fig. 2. The dots are calculated; the curve is a guide to the eye. Parameters are as in Fig. 1. The upper horizontal scale shows the characteristic length $l_{\mathrm{c}}$ normalized to the sample side length $L$, $\frac{l_{\mathrm{c}}}{L}=\frac{\beta /C_{44}}{L}=\alpha \frac{L}{C_{44}}$, according to (15). Here, $C_{44}$ is an element of the elasticity tensor in Voigt notation, namely, the shear modulus $C_{2323}$.

Figure 4

(a) Twist/strain vs sample side length $L$ normalized by the characteristic length $l_{\mathrm{c}}$ on a double-logarithmic scale. The dependence on the parameter $\beta$ is implicitly contained in this normalization. The dashed straight line has a slope of $-1$, corresponding to the expected asymptotic scaling of the twist angle $\propto 1/L$ for fixed parameter $\beta$. (b) Same as in (a), but for the effective Young's modulus $E$ on a semilogarithmic scale. All other parameters are as in Fig. 1. The dots are calculated; the curves are guides to the eye.