# Stochastic Thermal Properties of Laminates Filled with Long Fibers

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{0}of the specimen in order to simplify further calculations. To describe the behavior of the structure at each point, we can use some typical relations described by the following heat equation and Fourier’s law [15]:

**q**and f denote a heat flux intensity and a heat source, respectively;

**λ**is the matrix of thermal conductivity coefficients and ∇T denotes the gradient of the temperature field.

_{t}, q

_{n}

^{iso}and q

_{n}

^{heat}denote the prescribed values of temperature and the heat fluxes, and

**n**is the normal unit vector of the boundary line at a chosen point. By considering the heat flux in the y direction, following 1

^{2}and taking into account the uniform temperature along the boundary Γ

_{0}, we can write the following:

_{ey}denotes the wanted effective thermal conductivity in the y direction and T

_{0}is the temperature (along boundary Γ

_{0}) that should be determined during the FEM calculations.

## 3. Methods

#### 3.1. Finite Element Model

_{ifc}, y

_{ifc}) was randomly chosen.

^{6}W/(mK) for the matrix, the fiber and the diffuser materials, respectively. The diffuser material was artificial. The acting heat flux q

_{n}

^{heat}was equal to 100 W/m

^{2}. For each of the examined numbers of layers and saturation densities, 10,000 virtual specimens were created and tested. Each separate RVE was discretized with 820 triangular elements, which means that in the case of a specimen built with 8 layers, 65,600 triangular finite elements were used to analyze the specimen’s behavior. The obtained results were the basis for a further investigation, during which the best fitted probability distributions were chosen.

#### 3.2. Probability Distribution Fitting

#### 3.2.1. Log-Logistic Distribution

#### 3.2.2. Log-Normal Distribution

#### 3.2.3. Gamma Distribution

#### 3.2.4. Fitting Process

**v**= {α,β,σ}

^{T}and it was treated as an individual of a population during the evolutionary process. According to the used fitting method, the following function was chosen as a fitness function:

_{id}and y

_{ie}are values of the probability density function and the frequency of occurrence of an effective thermal conductivity in the i-th class, respectively.

## 4. Results

## 5. Discussion

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 9.**Modes for different numbers of layers and a different saturation density of matrix with fibers.

**Figure 10.**Standard deviations for different numbers of layers and a different saturation density of matrix with fibers.

**Table 1.**Modes and standard deviations for the saturation density of 0.25 and the given number of layers.

Number of Layers | Modes | Standard Deviations |
---|---|---|

2 | 1.609 | 1.08 × 10^{−2} |

3 | 1.602 | 8.00 × 10^{−3} |

4 | 1.599 | 6.44 × 10^{−3} |

5 | 1.597 | 5.46 × 10^{−3} |

6 | 1.596 | 4.86 × 10^{−3} |

7 | 1.596 | 4.34 × 10^{−3} |

8 | 1.595 | 3.97 × 10^{−3} |

**Table 2.**Modes and standard deviations for the saturation density of 0.30 and the given number of layers.

Number of Layers | Modes | Standard Deviations |
---|---|---|

2 | 1.778 | 1.39 × 10^{−2} |

3 | 1.771 | 1.01 × 10^{−2} |

4 | 1.767 | 8.27 × 10^{−3} |

5 | 1.765 | 7.09 × 10^{−3} |

6 | 1.764 | 6.37 × 10^{−3} |

7 | 1.763 | 5.56 × 10^{−3} |

8 | 1.762 | 5.16 × 10^{−3} |

**Table 3.**Modes and standard deviations for the saturation density of 0.35 and the given number of layers.

Number of Layers | Modes | Standard Deviations |
---|---|---|

2 | 1.972 | 1.72 × 10^{−2} |

3 | 1.964 | 1.25 × 10^{−2} |

4 | 1.960 | 1.02 × 10^{−2} |

5 | 1.957 | 8.67 × 10^{−3} |

6 | 1.955 | 7.69 × 10^{−3} |

7 | 1.954 | 7.03 × 10^{−3} |

8 | 1.953 | 6.43 × 10^{−3} |

**Table 4.**Modes and standard deviations for the saturation density of 0.40 and the given number of layers.

Number of Layers | Modes | Standard Deviations |
---|---|---|

2 | 2.197 | 2.03 × 10^{−2} |

3 | 2.187 | 1.50 × 10^{−2} |

4 | 2.183 | 1.23 × 10^{−2} |

5 | 2.179 | 1.06 × 10^{−2} |

6 | 2.178 | 9.39 × 10^{−3} |

7 | 2.176 | 8.44 × 10^{−3} |

8 | 2.175 | 7.73 × 10^{−3} |

**Table 5.**Modes and standard deviations for the saturation density of 0.45 and the given number of layers.

Number of Layers | Modes | Standard Deviations |
---|---|---|

2 | 2.461 | 2.42 × 10^{−2} |

3 | 2.450 | 1.77 × 10^{−2} |

4 | 2.444 | 1.47 × 10^{−2} |

5 | 2.441 | 1.26 × 10^{−2} |

6 | 2.438 | 1.09 × 10^{−2} |

7 | 2.437 | 1.00 × 10^{−2} |

8 | 2.436 | 9.34 × 10^{−3} |

**Table 6.**Modes and standard deviations for the saturation density of 0.50 and the given number of layers.

Number of Layers | Modes | Standard Deviations |
---|---|---|

2 | 2.778 | 2.76 × 10^{−2} |

3 | 2.765 | 2.04 × 10^{−2} |

4 | 2.759 | 1.67 × 10^{−2} |

5 | 2.755 | 1.46 × 10^{−2} |

6 | 2.752 | 1.30 × 10^{−2} |

7 | 2.751 | 1.19 × 10^{−2} |

8 | 2.749 | 1.10 × 10^{−2} |

**Table 7.**Modes and standard deviations for the saturation density of 0.55 and the given number of layers.

Number of Layers | Modes | Standard Deviations |
---|---|---|

2 | 3.167 | 3.02 × 10^{−2} |

3 | 3.154 | 2.28 × 10^{−2} |

4 | 3.147 | 1.93 × 10^{−2} |

5 | 3.143 | 1.66 × 10^{−2} |

6 | 3.140 | 1.47 × 10^{−2} |

7 | 3.138 | 1.35 × 10^{−2} |

8 | 3.136 | 1.24 × 10^{−2} |

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**MDPI and ACS Style**

Turant, J.
Stochastic Thermal Properties of Laminates Filled with Long Fibers. *Materials* **2021**, *14*, 2511.
https://doi.org/10.3390/ma14102511

**AMA Style**

Turant J.
Stochastic Thermal Properties of Laminates Filled with Long Fibers. *Materials*. 2021; 14(10):2511.
https://doi.org/10.3390/ma14102511

**Chicago/Turabian Style**

Turant, Jan.
2021. "Stochastic Thermal Properties of Laminates Filled with Long Fibers" *Materials* 14, no. 10: 2511.
https://doi.org/10.3390/ma14102511