# Numerical Construction of the Homogenized Strength Criterion for Fiber-Reinforced Composite

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## Abstract

**:**

## 1. Introduction

## 2. Homogenization Method as Applied to Composite Reinforced by Systems of Fibers

## 3. The Strength of the Composite

## 4. Construction of HSC in the Form of a Computer Program

Block of micro-structural analysis: solve PCP (4) for the unit value strains ${\epsilon}_{mn}^{}={\delta}_{mn}$, compute the local stresses ${\sigma}_{pq}^{mn}(y)={a}_{pqkl}(y){Z}_{k,l}^{mn}(y)$ in all finite elements, and save ${\sigma}_{pq}^{mn}(y)$ into files. |

$\downarrow $ |

Block HSC: compute the local stresses ${\sigma}_{pq}^{loc}(y)$ in all finite elements for the given macroscopic strains ${\epsilon}_{mn}^{}$ following Formula (1) and check the condition (5). |

## 5. An Example. Construction of the Failure Surfaces and the Safety Zones

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Layers of fibers in “slow” variables $x$ and PC $P$ of composite in the “fast” variables $y$.

**Figure 4.**Failure surface for the tension-shift modes: (

**a**)—in plane $O{\epsilon}_{11}^{}{\epsilon}_{13}^{}$, (

**b**)—enlarged, and (

**c**)—in plane $O{\epsilon}_{13}^{}{\epsilon}_{33}^{}$, (

**d**)—enlarged.

**Figure 5.**Failure surface: (

**a**)—for the tension modes; (

**b**)—for tension-shift mode; (

**c**)—for composite as a whole (the central part of Figure (

**b**), enlarged).

**Figure 6.**The failure surfaces when the epoxy strength limit ${\sigma}_{M}^{*}$ = $85\cdot {10}^{6}$ Pa.

Young’s Modulus GPa | Poisson’s Ratio | Strength Limit Pa | |
---|---|---|---|

Fibers | ${E}_{F}^{}=170$ GPa | ${\nu}_{F}^{}=0.3$ | ${\sigma}_{F}^{*}$ = $1.5\cdot {10}^{9}$ |

Matrix | ${E}_{M}^{}=2$ GPa | ${\nu}_{M}^{}=0.36$ | ${\sigma}_{M}^{*}$ = $60\cdot {10}^{6}$ |

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**MDPI and ACS Style**

Kolpakov, A.G.; Rakin, S.I.
Numerical Construction of the Homogenized Strength Criterion for Fiber-Reinforced Composite. *J. Compos. Sci.* **2023**, *7*, 145.
https://doi.org/10.3390/jcs7040145

**AMA Style**

Kolpakov AG, Rakin SI.
Numerical Construction of the Homogenized Strength Criterion for Fiber-Reinforced Composite. *Journal of Composites Science*. 2023; 7(4):145.
https://doi.org/10.3390/jcs7040145

**Chicago/Turabian Style**

Kolpakov, Alexander G., and Sergei I. Rakin.
2023. "Numerical Construction of the Homogenized Strength Criterion for Fiber-Reinforced Composite" *Journal of Composites Science* 7, no. 4: 145.
https://doi.org/10.3390/jcs7040145