# Homogenization Method to Calculate the Stiffness Matrix of Laminated Composites

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Problematic

#### 1.2. State of Art

#### 1.3. Paper Organization

## 2. Methodology

#### 2.1. UD (Unidirectional)—Homogenization:

#### 2.1.1. Algorithm Objectif Function

_{11}= V

^{f}E

^{f}

_{11}+ V

^{m}E

^{m},

_{22}= E

_{33}= E

^{m}/(1− √V

^{f}(1 − E

^{m}/(E

^{f}

_{22}))),

_{23}= G

^{m}/(1 − √V

^{f}(1 − G

^{m}/(G

^{f}

_{23}))),

_{12}= G

_{13}= G

^{m}/(1 − √V

^{f}(1 − G

^{m}/(G

^{f}

_{22}))),

_{23}= V

^{f}ν

^{f}

_{23}+ V

^{m}(2ν

^{m}− ν

_{12}·(E

_{22}/E

_{11})),

_{12}= ν

_{13}= ν

^{m}+ ν

^{f}(ν

^{f}

_{12}− ν

^{m}),

^{f}is the fiber volume fraction; E

^{f}

_{11}is the Young’s elastic modulus of the fiber in principle axis 1; E

^{f}

_{22}is the Young’s elastic modulus of the fiber in principle axis 2; G

^{f}

_{12}is the longitudinal shear modulus of the fiber; G

^{f}

_{23}is the transverse shear modulus of the fiber; ν

^{f}

_{12}is the primary Poisson’s ratio of the fiber; E

^{m}, ν

^{m}, and G

^{m}represent the Young’s elastic modulus, Poisson’s ratio, and shear modulus of the matrix, respectively.

_{i}, m

_{i}, and n

_{i}(i = 1, 2, 3) are the direction cosines and are defined as the cosines of the angle between the axes of the local and global coordinate systems before and after rotation:

- Objective function

_{i}, Eyy

_{i}… are the reel values of the mechanical properties; E

_{11}, E

_{22}… are the unknown values to be optimized. To solve and optimize the solutions from the Objectif function J, the partial derivative to each component is applied as following:

_{11}= min{Exx

_{i}}, E

_{22}= min{Eyy

_{i}}, E

_{33}= min{Ezz

_{i}},

_{11}= max{Exx

_{i}}, E

_{22}= max{Eyy

_{i}}, E

_{33}= max{Ezz

_{i}},

^{k}= λb

^{k}for each k ∈ {1, 2, ..., n}.

#### 2.1.2. Hybrid Method

- Chamis model for Hybrid Composites:

_{2}is the transverse Young’s modulus, G

_{12}is the in-plane shear modulus, V is the volume fraction, and ξ is a parameter associated with the fiber geometry (for circular fibers, it is 1.165 and 1.01).

#### 2.2. Principal Globalization

#### 2.3. Thickness

#### 2.3.1. GCD of Plies

#### 2.3.2. Stress–Strain Relation (Hook’s Law Method)

_{x}, in the x-direction, the variation of stress, σ

_{x}

## 3. Materials

#### 3.1. UD—Homogenization

- Glass/epoxy [0/0] laminate;
- Silenka E-Glass 1200 tex MY750/HY917/DY063 epoxy [0/90]s laminate;
- Silenka E-Glass 1200 tex/MY750/HY917/DY063 epoxy [+45/−45]s laminate;
- E-Glass 21xK43 Gevetex/LY556/HT907/DY063 [+90/+30/−30]s laminate.

#### 3.2. Principal Globalization

#### 3.3. Equivalence in Thickness

## 4. Results and Discussion

#### 4.1. UD—Homogenization

#### 4.2. Principal Globalization

#### 4.3. Equivalence in Thickness

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Fibers (GPa) |
---|

E-Glass E ^{f}_{11} = E^{f}_{22} = 72.4; G^{f}_{12} = 30.2; ν^{f}_{23} = 0.2 |

Silenka E-Glass 1200 tex E ^{f}_{11} = E^{f}_{22} = 74; G^{f}_{12} = 30.2; ν^{f}_{23} = 0.2 |

E-Glass 21xK43 Gevetex E ^{f}_{11} = E^{f}_{22} = 80; G^{f}_{12} = 30.2; ν^{f}_{23} = 0.2 |

Matrices |
---|

Epoxy E ^{m} = 2.76; G^{m} = 1.567; ν^{m} = 0.35 |

MY750/HY917/DY063 epoxy E ^{m} = 3.35; G^{m} = 1.24; ν^{m} = 0.35 |

LY556/HT907/DY063 E ^{m} = 3.35; G^{m} = 1.24; ν^{m} = 0.35 |

Graphite/Epoxy Laminate Properties |
---|

V^{f} = 0.7 |

E_{1} = 181 |

E_{2} = E_{3} = 10.3 |

G_{12} = G_{13} = 7.17 |

G_{23} = 5.79 |

ν_{12} = ν_{13} = 0.28 |

Property | E-Glass/Epoxy Laminate | Carbon Fiber/Epoxy Laminate |
---|---|---|

V^{f} | 0.6 | 0.6 |

E_{1} | 45.6 | 126 |

E_{2} = E_{3} | 16.2 | 11 |

G_{12} = G_{13} | 5.83 | 6.6 |

G_{23} | 5.79 | 3.93 |

ν_{12} = ν_{13} | 0.278 | 0.28 |

ν_{23} | 0.4 | 0.4 |

Type of Composite | Orientation | Experimental | Norm 0 | Norm ∞ | Norm Cart. | Norm 3 | Norm 4 | Hybrid Method |
---|---|---|---|---|---|---|---|---|

Glass/epoxy | [0/0] | 56 | 55.14 | 55.14 | 55.14 | 55.14 | 32.39 | 50 |

E-Glass 1200 tex MY750/HY917/DY063 epoxy | [0/90]s | 29.2 | 12.564 | 45.74 | 29.152 | 29.152 ± 16.58i | 24.14 | 31.19 |

E-Glass 1200 tex MY750/HY917/DY063 epoxy | [+45/−45]s | 14.4 | 12.72 | 12.72 | 12.72 | 12.72 | 7.471 | 16.511 |

E-Glass/LY556/HT907/DY063 | [+90/+30/−30]s | 27.4 | 12.52 | 29.12 | 28.8766 | 28.8766 ± 13.085i | 21.71 | 30.633 |

Type of Composite | Orientation | Experimental | Norm 0 | Norm ∞ | Norm Cart. | Norm 3 | Norm 4 | Hybrid Method |
---|---|---|---|---|---|---|---|---|

Glass/epoxy | [0/0] | 15 | 18.18 | 18.18 | 18.18 | 18.18 | 10.679 | 13 |

E-Glass 1200 tex MY750/HY917/DY063 epoxy | [0/90]s | 17.2 | 12.86 | 21.673 | 17.2665 | 17.2 ± 4.4064i | 11.155 | 21.476 |

E-Glass 1200 tex MY750/HY917/DY063 epoxy | [+45/−45]s | 17.1 | 18.67 | 18.67 | 18.67 | 18.67 | 10.966 | 21.281 |

E-Glass/LY556/HT907/DY063 | [+90/+30/−30]s | 22.3 | 12.55 | 39.154 | 24.401 | 24.401 ± 11.05i | 18.623 | 27.374 |

Type of Composite | Orientation | Experimental | Norm 0 | Norm ∞ | Norm Cart. | Norm 3 | Norm 4 | Hybrid Method |
---|---|---|---|---|---|---|---|---|

Glass/epoxy | [0/0] | 3 | 2.57 | 2.57 | 2.57 | 2.57 | 1.5 | 5.5 |

E-Glass 1200 tex MY750/HY917/DY063 epoxy | [0/90]s | 5.83 | 4.832 | 6.464 | 5.648 | 5.648 ± 0.816i | 3.428 | 3.7876 |

E-Glass 1200 tex MY750/HY917/DY063 epoxy | [+45/−45]s | 10.6 | 7.047 | 7.047 | 7.047 | 7.047 | 4.14 | 5.6186 |

E-Glass/LY556/HT907/DY063 | [+90/+30/−30]s | 5.79 | 5 | 6.43 | 5.66 | 5.66 ± 0.588i | 3.383 | 3.924 |

Engineering Constants (GPa) | Kaw | Fiber Method | PLY Method |
---|---|---|---|

Ex | 124.5 | 124.728 | 126.828 |

Ey | 67.43 | 68.27 | 69.25 |

Gxy | 7.17 | 7.55 | 7.64 |

Type of Composite | Thickness (mm) | CLT Experimental | GCD Method | Hook’s Law Method |
---|---|---|---|---|

G–G–G–G–G–G | 0.05–0.03 | 46.875 | 46.54 | 44.1 |

C–C–C–C–C–C | 0.05–0.03 | 126.857 | 127.104 | 122.458 |

G–G–C–C–G–G | 0.05–0.03 | 65.346 | 73.39 | 61.56 |

C–C–G–G–C–C | 0.05–0.03 | 108.42 | 100.25 | 101.75 |

Type of Composite | Thickness (mm) | CLT Experimental | GCD Method | Hook’s Law Method |
---|---|---|---|---|

G–G–G–G–G–G | 0.05–0.03 | 16.658 | 18.2 | 17.674 |

C–C–C–C–C–C | 0.05–0.03 | 11.07 | 10.29 | 11.358 |

G–G–C–C–G–G | 0.05–0.03 | 15.369 | 15.563 | 15.90 |

C–C–G–G–C–C | 0.05–0.03 | 12.365 | 12.92 | 12.983 |

Type of Composite | Thickness (mm) | CLT Experimental | GCD Method | Hook’s Law Method |
---|---|---|---|---|

G–G–G–G–G–G | 0.05–0.03 | 5.83 | 5.89 | 6.72 |

C–C–C–C–C–C | 0.05–0.03 | 6.6 | 6.81 | 7.61 |

G–G–C–C–G–G | 0.05–0.03 | 6 | 6.19 | 7.55 |

C–C–G–G–C–C | 0.05–0.03 | 6.423 | 6.5 | 7.63 |

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## Share and Cite

**MDPI and ACS Style**

Kaddaha, M.A.; Younes, R.; Lafon, P.
Homogenization Method to Calculate the Stiffness Matrix of Laminated Composites. *Eng* **2021**, *2*, 416-434.
https://doi.org/10.3390/eng2040026

**AMA Style**

Kaddaha MA, Younes R, Lafon P.
Homogenization Method to Calculate the Stiffness Matrix of Laminated Composites. *Eng*. 2021; 2(4):416-434.
https://doi.org/10.3390/eng2040026

**Chicago/Turabian Style**

Kaddaha, Mohamad Abbas, Rafic Younes, and Pascal Lafon.
2021. "Homogenization Method to Calculate the Stiffness Matrix of Laminated Composites" *Eng* 2, no. 4: 416-434.
https://doi.org/10.3390/eng2040026