Chiral Cosserat model for architected materials constructed by homogenization

Abstract

A homogenization methodology for the construction of effective Cosserat substitution media for heterogeneous materials is proposed, combining a variational principle in linear elasticity with the extended Hill-Mandel lemma accounting for the introduced generalized kinematics. A general methodology is proposed which can be applied to a wide class of architected materials exhibiting such micropolar chiral effects. The tensors of effective micropolar moduli are formulated as integrals over a representative unit cell utilizing of the displacement localization operators, solution of classical and higher-order unit cell problems. The proposed method delivers size-independent higher-order effective moduli. The effective micropolar moduli of periodic lattice materials endowed with chirality and non-centrosymmetry are computed as an application of the developed homogenization method. The homogenized model is validated by comparing the local and global mechanical responses of fully resolved networks with those of plates and macrobeams architected with tetrachiral or anti-tetrachiral lattices.

Appendices

Appendix 1: Evaluation of the effective micropolar moduli

Since the functional defined on the right-hand side of Eq. (43) is regular in the macrostrains ${\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right)$, ${\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right)$, and macrocurvature ${\mathbf{\rm K}}\left( {\mathbf{x}} \right)$, playing the role of parameters in the bounded integrand, partial derivative and integration can be exchanged, so that the stress measures in Eq. (23) leads to

$$\begin{aligned} & {{\varvec{\Sigma}}}^{\rm sym} : = \frac{{\partial W_{M} ({\mathbf{E}}^{\rm sym} ,{\mathbf{E}}^{\rm skew} ,{\mathbf{K}})}}{{\partial {\mathbf{E}}^{\rm sym} }} = \mathop {Min}\limits_{{{\tilde{\mathbf{u}}} \in H_{per}^{1} (Y)}} \frac{\partial }{{\partial {\mathbf{E}}^{\rm sym} }}\left\{ \begin{gathered} \int\limits_{Y} {\frac{1}{2}{\mathbf{C}}\left( {\mathbf{y}} \right):\left( {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{{{\rm E}^{\rm skew} }} ({\mathbf{y}}) \cdot {\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{K}}\left( {\mathbf{x}} \right)} \right)} : \hfill \\ \left( {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{{{\rm E}^{\rm skew} }} ({\mathbf{y}}) \cdot {\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{K}}\left( {\mathbf{x}} \right)} \right)dV_{y} \hfill \\ \end{gathered} \right\} \\ & {{\varvec{\Sigma}}}^{\rm skew} : = \frac{{\partial W_{M} ({\mathbf{E}}^{\rm sym} ,{\mathbf{E}}^{\rm skew} ,{\mathbf{K}})}}{{\partial {\mathbf{E}}^{\rm skew} }} = \mathop {Min}\limits_{{{\tilde{\mathbf{u}}} \in H_{per}^{1} (Y)}} \frac{\partial }{{\partial {\mathbf{E}}^{\rm skew} }}\left\{ \begin{gathered} \int\limits_{Y} {\frac{1}{2}{\mathbf{C}}\left( {\mathbf{y}} \right):\left( {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{{{\rm E}^{\rm skew} }} ({\mathbf{y}}) \cdot {\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{K}}\left( {\mathbf{x}} \right)} \right)} : \hfill \\ \left( {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{{{\rm E}^{\rm skew} }} ({\mathbf{y}}) \cdot {\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{K}}\left( {\mathbf{x}} \right)} \right)dV_{y} \hfill \\ \end{gathered} \right\} \\ & {\mathbf{M}}: = \frac{{\partial W_{M} ({\mathbf{E}}^{\rm sym} ,{\mathbf{E}}^{\rm skew} ,{\mathbf{K}})}}{{\partial {\mathbf{K}}}} = \mathop {Min}\limits_{{{\tilde{\mathbf{u}}} \in H_{per}^{1} (Y)}} \frac{\partial }{{\partial {\mathbf{K}}}}\left\{ \begin{gathered} \int\limits_{Y} {\frac{1}{2}{\mathbf{C}}\left( {\mathbf{y}} \right):\left( {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{{{\rm E}^{\rm skew} }} ({\mathbf{y}}) \cdot {\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{K}}\left( {\mathbf{x}} \right)} \right)} : \hfill \\ \left( {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{{{\rm E}^{\rm skew} }} ({\mathbf{y}}) \cdot {\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{K}}\left( {\mathbf{x}} \right)} \right)dV_{y} \hfill \\ \end{gathered} \right\} \\ \end{aligned}$$

(64)

Equation (64) leads to the expressions of macro stress, anti\rm symmetric macro stress, and micropolar stress tensors after straightforward calculations:

$$\begin{aligned} & {{\varvec{\Sigma}}}^{\rm sym} = \int_{Y} {{\mathbf{C}}\left( {\mathbf{y}} \right):\left( {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{{{\rm E}^{\rm skew} }} ({\mathbf{y}}) \cdot {\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{K}}\left( {\mathbf{x}} \right)} \right)} :{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}})\,dV_{y} \\ & \quad = \left\{ {\int\limits_{Y} {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}})} dV_{y} } \right\}:{\mathbf{E}}^{\rm sym} ({\mathbf{x}}) + \left\{ {\int\limits_{Y} {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{C}}} :{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}})dV_{y} } \right\} \cdot {\mathbf{E}}^{\rm skew} ({\mathbf{x}}) + \left\{ {\int\limits_{Y} {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{\rm K} ({\mathbf{y}})} dV_{y} } \right\}:{\mathbf{\rm K}}({\mathbf{x}}) \\ & {{\varvec{\Sigma}}}^{\rm skew} = \int_{Y} {{\mathbf{C}}\left( {\mathbf{y}} \right):\left( {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{{{\rm E}^{\rm skew} }} ({\mathbf{y}}) \cdot {\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{K}}\left( {\mathbf{x}} \right)} \right)} :{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}})\,dV_{y} \\ & \quad = \left\{ {\int\limits_{Y} {{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}})} dV_{y} } \right\}:{\mathbf{E}}^{\rm sym} ({\mathbf{x}}) + \left\{ {\int\limits_{Y} {{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}}):{\mathbf{C}}} :{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}})dV_{y} } \right\} \cdot {\mathbf{E}}^{\rm skew} ({\mathbf{x}}) + \left\{ {\int\limits_{Y} {{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{\rm K} ({\mathbf{y}})} dV_{y} } \right\}:{\mathbf{\rm K}}({\mathbf{x}}) \\ & {\mathbf{M}} = \int_{Y} {{\mathbf{C}}\left( {\mathbf{y}} \right):\left( {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{E}}^{\rm sym} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{{{\rm E}^{\rm skew} }} ({\mathbf{y}}) \cdot {\mathbf{E}}^{\rm skew} \left( {\mathbf{x}} \right) + {\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{K}}\left( {\mathbf{x}} \right)} \right)} :{\mathbf{A}}^{\rm K} ({\mathbf{y}})\,dV_{y} \\ & \quad = \left\{ {\int\limits_{Y} {{\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}})} dV_{y} } \right\}:{\mathbf{E}}^{\rm sym} ({\mathbf{x}}) + \left\{ {\int\limits_{Y} {{\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{C}}} :{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}})dV_{y} } \right\} \cdot {\mathbf{E}}^{\rm skew} ({\mathbf{x}}) + \left\{ {\int\limits_{Y} {{\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{\rm K} ({\mathbf{y}})} dV_{y} } \right\}:{\mathbf{\rm K}}({\mathbf{x}}) \\ \end{aligned}$$

(65)

These writings together with the \rm symmetry of the average quantities therein lead to the following expression of the effective micropolar moduli:

$$\begin{aligned} & {\mathbf{C}}^{\hom } = \int\limits_{Y} {{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}})dV_{y} , \, } {\mathbf{B}}^{\hom } = \int\limits_{Y} {{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}})dV_{y} , \, } {\mathbf{D}}^{\hom } = \int\limits_{Y} {{\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{{{\rm E}^{\rm sym} }} ({\mathbf{y}})dV_{y} ,} \\ & {\mathbf{R}}^{\hom } = \int\limits_{Y} {{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}})dV_{y} ,} \, {\mathbf{G}}^{\hom } = \int\limits_{Y} {{\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{{E^{\rm skew} }} ({\mathbf{y}})dV_{y} ,} \, {\mathbf{S}}^{\hom } = \int\limits_{Y} {{\mathbf{A}}^{\rm K} ({\mathbf{y}}):{\mathbf{C}}:{\mathbf{A}}^{\rm K} ({\mathbf{y}})dV_{y} } \\ \end{aligned}$$

(66)

One can easily check from previous expressions the transpose property of the coupling tensors leading to the overall \rm symmetry of the effective stiffness rigidity tensor.

The strain energy of the effective micropolar continuum writes as the following bilinear form of the kinematic measures, as the result of the obtained homogenized constitutive law Eq. (66) and Euler’s theorem for quadratic functions in their tensor arguments (a linear theory at both microscopic and macroscopic levels is formulated):

$$\begin{aligned} & W_{M} \left( {{\mathbf{E}}^{\rm sym} ,\;{\mathbf{E}}^{\rm skew} ,\;{\mathbf{K}}} \right) = \frac{1}{2}{\mathbf{E}}^{\rm sym} :{\mathbf{C}}^{\hom } :{\mathbf{E}}^{\rm sym} + \frac{1}{2}{\mathbf{E}}^{\rm skew} \cdot {\mathbf{R}}^{\hom } \cdot {\mathbf{E}}^{\rm skew} + \frac{1}{2}{\mathbf{K}}:{\mathbf{S}}^{\hom } :{\mathbf{K}} + {\mathbf{E}}^{\rm sym} :\left( {{\mathbf{B}}^{\hom } + {\mathbf{B}}^{\hom,T} } \right) \cdot {\mathbf{E}}^{\rm skew} \\ & \quad + {\mathbf{E}}^{\rm sym} :\left( {{\mathbf{D}}^{\hom } + {\mathbf{D}}^{\hom,T} } \right):{\mathbf{K}} + {\mathbf{E}}^{\rm skew} \cdot \left( {{\mathbf{G}}^{\hom } + {\mathbf{G}}^{\hom ,T} } \right):{\mathbf{K}} = \left\langle {w_{\mu } \left( {{\varvec{\upvarepsilon}}} \right)} \right\rangle_{Y} = \left\langle {\frac{1}{2}{{\varvec{\upvarepsilon}}}\left( {\mathbf{y}} \right):{\mathbf{C}}\left( {\mathbf{y}} \right):{{\varvec{\upvarepsilon}}}\left( {\mathbf{y}} \right)} \right\rangle_{Y} \\ \end{aligned}$$

(67)

with ${\mathbf{S}}^{\hom }$, ${\mathbf{R}}^{\hom }$, and ${\mathbf{C}}^{\hom }$ therein the fourth-order tensor of couple stress moduli, the second-order tensor of anti\rm symmetric moduli, and the fourth-order tensor of micropolar moduli. ${\mathbf{B}}^{\hom }$ is a third-order tensor of coupling moduli for micropolar strain and rotation, ${\mathbf{D}}^{\hom }$ and ${\mathbf{G}}^{\hom }$ are fourth and third-order tensors of coupling moduli of micropolar strain and rotation with curvature, respectively. The presence of the sums $\left( {{\mathbf{B}}^{\hom } + {\mathbf{B}}^{\hom,T} } \right)$, $\left( {{\mathbf{D}}^{\hom } + {\mathbf{D}}^{\hom,T} } \right)$ and $\left( {{\mathbf{G}}^{\hom } + {\mathbf{G}}^{\hom ,T} } \right)$ in Eq. (67) follows from the existence of a strain energy potential which guarantees the symmetry of the effective overall stiffness matrix, by Schwarz relation. The macroscopic micropolar energy density entails the following micropolar constitutive law at the macroscopic level, relating stress tensors ${{\varvec{\Sigma}}}^{\rm sym}$, ${{\varvec{\Sigma}}}^{\rm skew}$ and ${\mathbf{M}}$ to the micropolar strain, rotation and curvature kinematic measures, respectively:

$$\begin{aligned} & {{\varvec{\Sigma}}}^{\rm sym} : = \frac{{\partial W_{M} ({\mathbf{E}}^{\rm sym} ,\;{\mathbf{E}}^{\rm skew} ,\;{\mathbf{K}})}}{{\partial {\mathbf{E}}^{\rm sym} }} \equiv {\mathbf{C}}^{\hom } :{\mathbf{E}}^{\rm sym} + {\mathbf{B}}^{\hom } \cdot {\mathbf{E}}^{\rm skew} + {\mathbf{D}}^{\hom } :{\mathbf{K}}; \\ & {{\varvec{\Sigma}}}^{\rm skew} : = \frac{{\partial W_{M} ({\mathbf{E}}^{\rm sym} ,\;{\mathbf{E}}^{\rm skew} ,\;{\mathbf{K}})}}{{\partial {\mathbf{E}}^{\rm skew} }} \equiv {\mathbf{B}}^{\hom,T} :{\mathbf{E}}^{\rm sym} + {\mathbf{R}}^{\hom } \cdot {\mathbf{E}}^{\rm skew} + {\mathbf{G}}^{\hom } :{\mathbf{K}} \\ & {\mathbf{M}}: = \frac{{\partial W_{M} ({\mathbf{E}}^{\rm sym} ,\;{\mathbf{E}}^{\rm skew} ,\;{\mathbf{K}})}}{{\partial {\mathbf{K}}}} \equiv {\mathbf{D}}^{\hom ,T} :{\mathbf{E}}^{\rm sym} + {\mathbf{G}}^{\hom ,T} \cdot {\mathbf{E}}^{\rm skew} + {\mathbf{S}}^{\hom } :{\mathbf{K}} \\ \end{aligned}$$

(68)

Appendix 2.1: Effective engineering material parameters for tetrachiral lattice

Considering the plane strain condition, and according to the 2D Cauchy stiffness matrix ${\mathbf{C}}^{\hom }$, the effective behavior of the tetrachiral lattice can be considered as a monoclinic elastic solid with five independent coefficients in 2D. The compliance matrix for the case where the $e_{3}$ plane is the plane of symmetry can be written:

$$\left( {\begin{array}{*{20}c} {E_{11}^{\rm sym} } \\ {E_{22}^{\rm sym} } \\ {E_{12}^{\rm sym} } \\ {E_{21}^{\rm sym} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\frac{{1}}{{{\text{E}}_{{1}} }}} & {{ - }\frac{{{\upnu }_{{{21}}} }}{{{\text{E}}_{{2}} }}} & {\frac{{{\upeta }_{{{61}}} }}{{{\text{G}}_{{6}} }}} & {\frac{{{\upeta }_{{{61}}} }}{{{\text{G}}_{{6}} }}} \\ {{ - }\frac{{{\upnu }_{{{12}}} }}{{{\text{E}}_{{1}} }}} & {\frac{{1}}{{{\text{E}}_{{2}} }}} & {\frac{{{\upeta }_{{{62}}} }}{{{\text{G}}_{{6}} }}} & {\frac{{{\upeta }_{{{62}}} }}{{{\text{G}}_{{6}} }}} \\ {\frac{{{\upeta }_{{{16}}} }}{{{\text{E}}_{{1}} }}} & {\frac{{{\upeta }_{{{26}}} }}{{{\text{E}}_{{2}} }}} & {\frac{{1}}{{{\text{G}}_{{6}} }}} & {\frac{{1}}{{{\text{G}}_{{6}} }}} \\ {\frac{{{\upeta }_{{{16}}} }}{{{\text{E}}_{{1}} }}} & {\frac{{{\upeta }_{{{26}}} }}{{{\text{E}}_{{2}} }}} & {\frac{{1}}{{{\text{G}}_{{6}} }}} & {\frac{{1}}{{{\text{G}}_{{6}} }}} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\Sigma_{11}^{\rm sym} } \\ {\Sigma_{22}^{\rm sym} } \\ {\Sigma_{12}^{\rm sym} } \\ {\Sigma_{21}^{\rm sym} } \\ \end{array} } \right)$$

The symmetry of the compliance matrix requires that:

$$\frac{{{\upnu }_{{{21}}} }}{{{\text{E}}_{{2}} }}{ = }\frac{{{\upnu }_{{{12}}} }}{{{\text{E}}_{{1}} }}{;}\,\,\,\,\frac{{{\upeta }_{{{61}}} }}{{{\text{G}}_{{6}} }}{ = }\frac{{{\upeta }_{{{16}}} }}{{{\text{E}}_{{1}} }}{;}\,\,\,\,\frac{{{\upeta }_{{{62}}} }}{{{\text{G}}_{{6}} }}{ = }\frac{{{\upeta }_{{{26}}} }}{{{\text{E}}_{{2}} }}$$

So, the effective engineering material parameters are evaluated as:

$$\begin{aligned} & {\text{E}}_{{1}} =\frac{{{\text{2C}}_{{4}}^{{2}} \left( {{\text{C}}_{{1}} {\text{ + C}}_{{2}} } \right){\text{ + C}}_{{3}} \left( {{\text{C}}_{{2}}^{{2}} {\text{ - C}}_{{1}}^{{2}} } \right)}}{{{\text{C}}_{{4}}^{{2}} {\text{ - C}}_{{1}} {\text{C}}_{{3}} }}{;}\,\,\,{\text{E}}_{{1}} ={\text{E}}_{{2}} \\ & {\upnu }_{{{12}}} = \frac{{\left( {{\text{C}}_{{4}}^{{2}} {\text{ + C}}_{{2}} {\text{C}}_{{3}} } \right)\left( {{\text{C}}_{{1}} {\text{C}}_{{3}} {\text{ - C}}_{{4}}^{{2}} } \right)}}{{\left( {{\text{C}}_{{1}} {\text{ + C}}_{{2}} } \right)^{{2}} \left( {{\text{2C}}_{{4}}^{{2}} {\text{ - C}}_{{1}} {\text{C}}_{{3}} {\text{ + C}}_{{2}} {\text{C}}_{{3}} } \right)^{{2}} }}{;}\,\,\,{\upnu }_{{{12}}} ={{\upnu }}_{{{21}}} \, \\ & {\text{G}}_{{6}} =\frac{{{\text{2C}}_{{4}}^{{2}} {\text{ - C}}_{{1}} {\text{C}}_{{3}} {\text{ + C}}_{{2}} {\text{C}}_{{3}} }}{{{\text{C}}_{{2}} {\text{ - C}}_{{1}} }} \\ & {\upeta }_{{{16}}} = \frac{{{\text{C}}_{{4}} \left( {{\text{C}}_{{1}} {\text{C}}_{{3}} {\text{ - C}}_{{4}}^{{2}} } \right)}}{{\left( {{\text{C}}_{{1}} {\text{ + C}}_{{2}} } \right)\left( {{\text{2C}}_{{4}}^{{2}} {\text{ - C}}_{{1}} {\text{C}}_{{3}} {\text{ + C}}_{{2}} {\text{C}}_{{3}} } \right)^{{2}} }}{;}\,\,\,\,{\upeta }_{{{16}}} ={{ - \upeta }}_{{{26}}} \\ & {\upeta }_{{{61}}} = \frac{{{\text{C}}_{{4}} \left( {{\text{C}}_{{1}} {\text{ - C}}_{{2}} } \right)}}{{\left( {{\text{2C}}_{{4}}^{{2}} {\text{ - C}}_{{1}} {\text{C}}_{{3}} {\text{ + C}}_{{2}} {\text{C}}_{{3}} } \right)^{{2}} }}{;}\,\,\,\,{\upeta }_{{{61}}} ={{ - \upeta }}_{{{62}}} \\ \end{aligned}$$

Appendix 2.2: Effective engineering material parameters for Anti-tetrachiral lattice

Considering the plane strain condition, and according to the 2D Cauchy stiffness matrix ${\mathbf{C}}^{\hom }$ the effective behavior of the anti-tetrachiral lattice can be considered as a orthotropic elastic solid with three independent coefficients in 2D. The compliance matrix for the case where the $e_{3}$ plane is the plane of symmetry can be written:

$$\left( {\begin{array}{*{20}c} {E_{11}^{\rm sym} } \\ {E_{22}^{\rm sym} } \\ {E_{12}^{\rm sym} } \\ {E_{21}^{\rm sym} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\frac{{1}}{{{\text{E}}_{{1}} }}} & {{ - }\frac{{{\upnu }_{{{21}}} }}{{{\text{E}}_{{2}} }}} & 0 & 0 \\ {{ - }\frac{{{\upnu }_{{{12}}} }}{{{\text{E}}_{{1}} }}} & {\frac{{1}}{{{\text{E}}_{{2}} }}} & 0 & 0 \\ 0 & 0 & {\frac{{1}}{{{\text{G}}_{{6}} }}} & {\frac{{1}}{{{\text{G}}_{{6}} }}} \\ 0 & 0 & {\frac{{1}}{{{\text{G}}_{{6}} }}} & {\frac{{1}}{{{\text{G}}_{{6}} }}} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\Sigma_{11}^{\rm sym} } \\ {\Sigma_{22}^{\rm sym} } \\ {\Sigma_{12}^{\rm sym} } \\ {\Sigma_{21}^{\rm sym} } \\ \end{array} } \right)$$

The symmetry of the compliance matrix requires that:

$$\frac{{{\upnu }_{{{21}}} }}{{{\text{E}}_{{2}} }}{ = }\frac{{{\upnu }_{{{12}}} }}{{{\text{E}}_{{1}} }}{;}$$

So, the effective engineering material parameters are evaluated as:

$$\begin{aligned} & {\text{E}}_{{1}} = \frac{{\left( {{\text{C}}_{1}^{{2}} {\text{ - C}}_{2}^{{2}} } \right)}}{{{\text{C}}_{{1}} }}{;}\,\,\,{\text{E}}_{{1}} ={\text{E}}_{{2}} \\ & {\upnu }_{{{12}}} = \frac{{{\text{C}}_{{1}} {\text{C}}_{2} }}{{\left( {{\text{C}}_{1}^{2} {\text{ - C}}_{2}^{2} } \right)^{{2}} }}{;}\,\,\,{\upnu }_{{{12}}} ={{ \upnu }}_{{{21}}} \, \\ & {\text{G}}_{{6}}= {\text{C}}_{3} \\ \end{aligned}$$

Appendix 3: Boundary constraint equations and corresponding classical rigidity components

Rigidity components	Periodic boundary conditions ($E_{ij} = 1;\,\,\,\,i,j = x,\,y,\,z$)
$C_{11} = \frac{{2U^{cell} }}{{V^{cell} }}$	$u_{x}^{F2} - u_{x}^{F1} - l_{x}\epsilon_{xx} = 0,\,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0$
$C_{22} = \frac{{2U^{cell} }}{{V^{cell} }}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{yy} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0$
$C_{33} = \frac{{2U^{cell} }}{{V^{cell} }}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} - l_{z}\epsilon_{zz} = 0$
$C_{44} = \frac{{U^{cell} }}{{2V^{cell} }}$	$u_{x}^{F2} - u_{x}^{F1} - l_{x}\epsilon_{xz} = 0,\,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} - l_{z}\epsilon_{xz} = 0$
$C_{55} = \frac{{U^{cell} }}{{2V^{cell} }}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{yz} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} - l_{z}\epsilon\epsilon_{yz} = 0$
$C_{66} = \frac{{U^{cell} }}{{2V^{cell} }}$	$u_{x}^{F2} - u_{x}^{F1} - l_{x}\epsilon_{xy} = 0,\,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{xy} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0$
$C_{12} = \frac{{U^{cell} }}{{V^{cell} }} - \frac{{C_{11} + C_{22} }}{2}$	$u_{x}^{F2} - u_{x}^{F1} - l_{x}\epsilon_{xx} = 0,\,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{yy} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0$
$C_{13} = \frac{{U^{cell} }}{{V^{cell} }} - \frac{{C_{11} + C_{33} }}{2}$	$u_{x}^{F2} - u_{x}^{F1} - l_{x}\epsilon_{xx} = 0,\,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} - l_{z}\epsilon_{zz} = 0$
$C_{14} = \frac{{U^{cell} }}{{2V^{cell} }} - \frac{{C_{11} }}{4} - C_{44}$	$u_{x}^{F2} - u_{x}^{F1} - l_{x}\epsilon_{xx} = 0,\,\,\,\,u_{z}^{F2} - u_{z}^{F1} - l_{x}\epsilon_{xz} = 0,\, u_{y}^{F4} - u_{y}^{F3} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0,\,\,\,\,u_{x}^{F6} - u_{x}^{F5} - l_{z}\epsilon_{xz} = 0$
$C_{15} = \frac{{U^{cell} }}{{2V^{cell} }} - \frac{{C_{11} }}{4} - C_{55}$	$u_{x}^{F2} - u_{x}^{F1} - l_{x}\epsilon_{xx} = 0,\,\,u_{y}^{F4} - u_{y}^{F3} = 0\,,\, u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{yz} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0,\,\,\,\,u_{x}^{F6} - u_{x}^{F5} - l_{y}\epsilon_{yz} = 0$
$C_{16} = \frac{{U^{cell} }}{{2V^{cell} }} - \frac{{C_{11} }}{4} - C_{66}$	$u_{x}^{F2} - u_{x}^{F1} - l_{x}\epsilon_{xx} = 0,\,\,\,\,u_{y}^{F2} - u_{y}^{F1} - l_{x}\epsilon_{xy} = 0,\, u_{y}^{F4} - u_{y}^{F3} = 0,\,\,\,\,u_{x}^{F4} - u_{x}^{F3} - l_{y}\epsilon_{xy} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0$
$C_{23} = \frac{{U^{cell} }}{{V^{cell} }} - \frac{{C_{22} + C_{33} }}{2}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,u_{y}^{F2} - u_{y}^{F1} - l_{y}\epsilon_{yy} = 0,\,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} - l_{z}\epsilon_{zz} = 0$
$C_{24} = \frac{{U^{cell} }}{{2V^{cell} }} - \frac{{C_{22} }}{4} - C_{44}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,u_{z}^{F2} - u_{z}^{F1} - l_{x}\epsilon_{xz} = 0,\,u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{yy} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0,\,\,\,\,u_{x}^{F6} - u_{x}^{F5} - l_{z}\epsilon_{xz} = 0$
$C_{25} = \frac{{U^{cell} }}{{2V^{cell} }} - \frac{{C_{22} }}{4} - C_{55}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{yy} = 0\,,\, u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{yz} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0,\,\,\,\,u_{x}^{F6} - u_{x}^{F5} - l_{y}\epsilon_{yz} = 0$
$C_{26} = \frac{{U^{cell} }}{{2V^{cell} }} - \frac{{C_{22} }}{4} - C_{66}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,u_{y}^{F2} - u_{y}^{F1} - l_{x}\epsilon_{xy} = 0,\, u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{yy} = 0,\,\,\,\,u_{x}^{F4} - u_{x}^{F3} - l_{y}\epsilon_{xy} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} = 0$
$C_{34} = \frac{{U^{cell} }}{{2V^{cell} }} - \frac{{C_{33} }}{4} - C_{44}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,u_{z}^{F2} - u_{z}^{F1} - l_{x}\epsilon_{xz} = 0,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} = 0, u_{z}^{F6} - u_{z}^{F5} - l_{z}\epsilon_{zz} = 0,\,\,\,\,u_{x}^{F6} - u_{x}^{F5} - l_{z}\epsilon_{xz} = 0$
$C_{35} = \frac{{U^{cell} }}{{2V^{cell} }} - \frac{{C_{33} }}{4} - C_{55}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,u_{y}^{F4} - u_{y}^{F3} = 0\,,\,u_{y}^{F4} - u_{y}^{F3} - l_{y}\epsilon_{yz} = 0, u_{z}^{F6} - u_{z}^{F5} - l_{z}\epsilon_{zz} = 0,\,\,\,\,u_{x}^{F6} - u_{x}^{F5} - l_{y}\epsilon_{yz} = 0$
$C_{36} = \frac{{U^{cell} }}{{2V^{cell} }} - \frac{{C_{33} }}{4} - C_{66}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,u_{y}^{F2} - u_{y}^{F1} - l_{x}\epsilon_{xy} = 0,\,\,\,\,u_{y}^{F4} - u_{y}^{F3} = 0,u_{x}^{F4} - u_{x}^{F3} - l_{y}\epsilon_{xy} = 0,\,\,\,\,u_{z}^{F6} - u_{z}^{F5} - l_{z}\epsilon_{zz} = 0$
$C_{45} = \frac{{U^{cell} }}{{4V^{cell} }} - \frac{{C_{44} + C_{55} }}{2}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,u_{z}^{F2} - u_{z}^{F1} - l_{x}\epsilon_{xz} = 0,\,\,\,\, u_{y}^{F4} - u_{y}^{F3} = 0,\,\,\,u_{z}^{F4} - u_{z}^{F3} - l_{y}\epsilon_{yz} = 0,\,\,\, u_{z}^{F6} - u_{z}^{F5} = 0,\,\,\,\,u_{x}^{F6} - u_{x}^{F5} - l_{z}\epsilon_{xz} = 0,\,\,\,\,u_{y}^{F6} - u_{y}^{F5} - l_{z}\epsilon_{yz} = 0$
$C_{46} = \frac{{U^{cell} }}{{4V^{cell} }} - \frac{{C_{44} + C_{66} }}{2}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,u_{y}^{F2} - u_{y}^{F1} - l_{x}\epsilon_{xy} = 0,\,\,\,u_{z}^{F2} - u_{z}^{F1} - l_{x}\epsilon_{xz} = 0, u_{y}^{F4} - u_{y}^{F3} = 0,\,\,\,u_{x}^{F4} - u_{x}^{F3} - l_{y}\epsilon_{xy} = 0,\,\,\, u_{z}^{F6} - u_{z}^{F5} = 0,\,\,\,\,u_{x}^{F6} - u_{x}^{F5} - l_{z}\epsilon_{xz} = 0$
$C_{56} = \frac{{U^{cell} }}{{4V^{cell} }} - \frac{{C_{55} + C_{66} }}{2}$	$u_{x}^{F2} - u_{x}^{F1} = 0,\,\,\,\,u_{y}^{F2} - u_{y}^{F1} - l_{x}\epsilon_{xy} = 0,\,\,\,\, u_{y}^{F4} - u_{y}^{F3} = 0,\,\,\,u_{x}^{F4} - u_{x}^{F3} - l_{y}\epsilon_{xy} = 0,\,\,\,u_{z}^{F4} - u_{z}^{F3} - l_{y}\epsilon_{yz} = 0, u_{z}^{F6} - u_{z}^{F5} = 0,\,\,\,\,u_{y}^{F6} - u_{y}^{F5} - l_{z}\epsilon\epsilon_{yz} = 0$

In above table, the following notations are introduced: the superscript of the displacement indicates the faces over the boundary of the unit cells shown in Fig. 4, and the subscript of the displacement field indicates the displacement component. As an example of the notations indicated in Appendix 1 and Fig. 4, $u_{x}^{F2}$ shows the displacement of the nodal points on the face $F_{2}$ in ‘x’ direction. Also, scalar quantities $U^{cell}$ and $V^{cell}$ are the strain energy and volume of the unit cell, respectively.