# Material Symmetries in Homogenized Hexagonal-Shaped Composites as Cosserat Continua

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Framework of Cosserat Theory

## 3. Finite Element Implementation

## 4. Numerical Applications

- $A\equiv (0,0)$,
- $B\equiv ({l}_{1},0)$,
- $C\equiv ({l}_{1}+{l}_{5},{l}_{2}/2)$,
- $D\equiv ({l}_{1},{l}_{2})$,
- $E\equiv (0,{l}_{2})$,
- $F\equiv (-{l}_{4},{l}_{2}/2)$

- The upper-right area has ${\alpha}_{2}>0$ and ${\alpha}_{3}>0$ where any combination is theoretically possible in the given domain range. In this area, regular hexagons are present as well as diamond shaped ones (according to Figure 3).
- The lower-left area has ${\alpha}_{2}<0$ and ${\alpha}_{3}<0$ where not all shapes are achievable due to some self-intersections of hexagons. In this area, hourglass shaped hexagons are present (symmetric and not-symmetric ones) according to Figure 3.
- The upper-left and lower-right areas have ${\alpha}_{2}<0$ and ${\alpha}_{3}>0$ and ${\alpha}_{2}>0$ and ${\alpha}_{3}<0$, respectively. These shapes might be characterized by an asymmetric shape (according to Figure 3).

- rectangular: ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$
- hourglass: ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$
- diamond: ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$
- regular: ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$
- skew: ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$
- tip: ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Possible configurations as a function of ${\alpha}_{2}$ and ${\alpha}_{3}$ for constant value of ${l}_{r}$.

**Figure 3.**Hexagonal patterns given ${l}_{r}=\frac{100}{1/\sqrt{3}+1}$: (

**a**) rectangular ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$, (

**b**) hourglass ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$, (

**c**) diamond ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$, (

**d**) regular ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$, (

**e**) skew ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$, (

**f**) tip ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$.

**Figure 4.**Stress/strain elastic stiffness variation with ${\alpha}_{1}=0$: (

**a**) ${A}_{1111}$, (

**b**) ${A}_{1122}$, (

**c**) ${A}_{2222}$, (

**d**) ${A}_{1212}$, (

**e**) ${A}_{1221}$, (

**f**) ${A}_{2121}$.

**Figure 5.**microcouple/strain elastic stiffness variation with ${\alpha}_{1}=0$: (

**a**) ${B}_{112}$, (

**b**) ${B}_{221}$, (

**c**) ${B}_{121}$, (

**d**) ${B}_{211}$, (

**e**) ${D}_{11}$, (

**f**) ${D}_{22}$.

**Figure 6.**Hexagonal RVEs with geometric parameters ${\alpha}_{1}={0}^{\circ}$ and ${l}_{r}=\frac{100}{1/\sqrt{3}+1}$: (

**a**) rectangular ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$, (

**b**) hourglass ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$, (

**c**) diamond ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$, (

**d**) regular ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$, (

**e**) skew ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$, (

**f**) tip ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$.

**Figure 7.**Current mesh used $16\times 32$ Q4 elements with displayed applied load and boundary conditions.

**Figure 8.**Horizontal displacement ${u}_{1}$ for ${l}_{r}=\frac{100}{1/\sqrt{3}+1}$: (

**a**) rectangular ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$, (

**b**) hourglass ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$, (

**c**) diamond ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$, (

**d**) regular ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$, (

**e**) skew ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$, (

**f**) tip ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$.

**Figure 9.**Vertical displacement ${u}_{2}$ for ${l}_{r}=\frac{100}{1/\sqrt{3}+1}$: (

**a**) rectangular ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$, (

**b**) hourglass ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$, (

**c**) diamond ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$, (

**d**) regular ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$, (

**e**) skew ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$, (

**f**) tip ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$.

**Figure 10.**Horizontal stress ${\sigma}_{11}$ for ${l}_{r}=\frac{100}{1/\sqrt{3}+1}$: (

**a**) rectangular ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$, (

**b**) hourglass ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$, (

**c**) diamond ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$, (

**d**) regular ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$, (

**e**) skew ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$, (

**f)**tip ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$.

**Figure 11.**Vertical stress ${\sigma}_{22}$ for ${l}_{r}=\frac{100}{1/\sqrt{3}+1}$: (

**a**) rectangular ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$, (

**b**) hourglass ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$, (

**c**) diamond ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$, (

**d**) regular ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$, (

**e**) skew ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$, (

**f**) tip ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$.

**Figure 12.**Relative rotation $\theta -\omega $ for ${l}_{r}=\frac{100}{1/\sqrt{3}+1}$: (

**a**) rectangular ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$, (

**b**) hourglass ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$, (

**c**) diamond ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$, (

**d**) regular ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$, (

**e**) skew ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$, (

**f**) tip ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$.

**Figure 13.**Shear strain ${\epsilon}_{12}$ for ${l}_{r}=\frac{100}{1/\sqrt{3}+1}$: (

**a**) rectangular ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$, (

**b**) hourglass ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$, (

**c**) diamond ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$, (

**d**) regular ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$, (

**e**) skew ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$, (

**f**) tip ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$.

**Figure 14.**Shear strain ${\epsilon}_{21}$ for ${l}_{r}=\frac{100}{1/\sqrt{3}+1}$: (

**a**) rectangular ${\alpha}_{2}={\alpha}_{3}={0}^{\circ}$, (

**b**) hourglass ${\alpha}_{2}={\alpha}_{3}=-{20}^{\circ}$, (

**c**) diamond ${\alpha}_{2}={\alpha}_{3}={70}^{\circ}$, (

**d**) regular ${\alpha}_{2}={\alpha}_{3}={30}^{\circ}$, (

**e**) skew ${\alpha}_{2}=-{\alpha}_{3}={70}^{\circ}$, (

**f**) tip ${\alpha}_{2}={70}^{\circ}$, ${\alpha}_{3}={0}^{\circ}$.

**Table 1.**This is a table caption. Tables should be placed in the main text near to the first time they are cited.

Rect | Hour | Diam | Reg | Skew | Tip | |
---|---|---|---|---|---|---|

${A}_{1111}$ | 287.3 | 209.2 | 2823 | 496.88 | 835.37 | 1123 |

${A}_{1122}$ | 0 | −0.5 | 1.4894 | 0.79247 | 0 | 0.6714 |

${A}_{2222}$ | 711.8 | 1059.7 | 333.57 | 496.88 | 1127.3 | 419.94 |

${A}_{1212}$ | 710 | 1056.8 | 331.73 | 495.29 | 1121 | 418.1 |

${A}_{1221}$ | 0 | −0.5 | 1.4894 | 0.79247 | 0 | 0.6714 |

${A}_{2121}$ | 285.5 | 208.2 | 2836.9 | 495.29 | 839.47 | 1124.2 |

${B}_{112}$ | 0 | 0 | 0 | 0 | 0 | 30.03 |

${B}_{222}$ | 0 | 0 | 0 | 0 | −216.71 | −88.869 |

${B}_{121}$ | 0 | 0 | 0 | 0 | 0 | 30.03 |

${B}_{211}$ | 0 | 0 | 0 | 0 | 0 | −103.94 |

${D}_{11}$ | 14.4 | 11.9 | 1219.3 | 33.338 | 360.81 | 340.19 |

${D}_{22}$ | 44.2 | 67.7 | 89.941 | 33.338 | 303.95 | 79.488 |

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**MDPI and ACS Style**

Fantuzzi, N.; Trovalusci, P.; Luciano, R.
Material Symmetries in Homogenized Hexagonal-Shaped Composites as Cosserat Continua. *Symmetry* **2020**, *12*, 441.
https://doi.org/10.3390/sym12030441

**AMA Style**

Fantuzzi N, Trovalusci P, Luciano R.
Material Symmetries in Homogenized Hexagonal-Shaped Composites as Cosserat Continua. *Symmetry*. 2020; 12(3):441.
https://doi.org/10.3390/sym12030441

**Chicago/Turabian Style**

Fantuzzi, Nicholas, Patrizia Trovalusci, and Raimondo Luciano.
2020. "Material Symmetries in Homogenized Hexagonal-Shaped Composites as Cosserat Continua" *Symmetry* 12, no. 3: 441.
https://doi.org/10.3390/sym12030441