# The Induced Stress Field in Cracked Composites by Heat Flow

## Abstract

**:**

## 1. Introduction

## 2. Constitutive and Governing Equations

## 3. Method of Solution

#### 3.1. Formulation in the Real Domain

#### 3.2. Formulation in the Transform Domain

- Start by assuming that ${\widehat{\mathit{\sigma}}}^{e}=\mathbf{0}$ and ${\widehat{\mathit{q}}}^{e}=\mathbf{0}$, and solve the above equations in the transform domain.
- Apply the inverse transform formula to compute the thermal and stress fields. The field variables in the actual space can be employed to compute the current eigenfields ${\mathit{\sigma}}^{e({K}_{1},{K}_{2})}$ and ${\mathit{q}}^{e({K}_{1},{K}_{2})}$.
- Solve again the equations in the transform domain. This procedure should be continued until the convergence to a desired degree of accuracy is achieved.

## 4. Verifications

## 5. Applications

#### 5.1. A Cracked Fiber-Reinforced Polymer Matrix Composite Subjected to a Remote Heat Flux

#### 5.2. A Cracked Porous Ceramic Material Subjected to a Remote Heat Flux

#### 5.3. A Cracked Periodically-Layered Ceramic Composite Subjected to a Remote Heat Flux

## 6. Conclusions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A crack of length $2a$ embedded in a composite material. (

**b**) A rectangular domain $2D\times 2H$ of the composite is divided into repeating cells. These cells are labeled by $({K}_{1},{K}_{2})$ with $-{M}_{1}\le {K}_{1}\le {M}_{1}$ and $-{M}_{2}\le {K}_{2}\le {M}_{2}$, and the size of every one of which is $2d\times 2h$ (the figure is shown for ${M}_{1}={M}_{2}=2$). (

**c**) A representative cell in which local coordinates $({X}_{1}^{{}^{\prime}},{X}_{2}^{{}^{\prime}})$ are introduced whose origin is located at the center. The cell is divided into ${N}_{\alpha}\times {N}_{\beta}$ sub-cells.

**Figure 2.**Homogenized carbon/epoxy unidirectional transversely-isotropic composite with a transverse crack, subjected to a remote normal heat flux of ${\overline{q}}_{2}$ = −1 W/m${}^{2}$. The axial direction of the homogenized composite is oriented in the X${}_{1}$-direction. Comparison between the exact solution of [7] and the present one for the temperature, normal heat flux and shear stress along the crack line.

**Figure 3.**Homogenized carbon/epoxy unidirectional transversely-isotropic composite with a transverse crack, subjected to a remote normal heat flux of ${\overline{q}}_{2}$ = −1 W/m${}^{2}$. The axial direction of the homogenized composite is oriented in the out-of-plane X${}_{3}$-direction. Comparison between the exact solution of [4] and the present one for the temperature, normal heat flux and shear stress along the crack line.

**Figure 4.**The variations along the crack line of the temperature, normal component of the heat flux and induced shear stress that develop in the carbon/epoxy fiber-reinforced composite, subjected to a remote normal heat flux of ${\overline{q}}_{2}$ = −1 W/m${}^{2}$.

**Figure 5.**Field distributions in the region $-5\le {X}_{1}/\left(2d\right)\le 5$, $-3\le {X}_{2}/\left(2h\right)\le 3$ of the cracked carbon/epoxy fiber-reinforced composite, (

**a**) Temperature T (K) distribution, (

**b**) normal component of the heat flux (W/m${}^{2}$) and (

**c**) normal stress ${\sigma}_{22}$ (MPa).

**Figure 6.**The variations along the crack line of the temperature, normal component of the heat flux and shear stress that develop in the porous alumina material, subjected to a remote normal heat flux of ${\overline{q}}_{2}$ = −1 W/m${}^{2}$.

**Figure 7.**Field distributions in the region $-5\le {X}_{1}/\left(2d\right)\le 5$, $-3\le {X}_{2}/\left(2h\right)\le 3$ of the cracked porous alumina material. (

**a**) Temperature T (K) distribution, (

**b**) normal component of the heat flux ${q}_{2}$ (W/m${}^{2}$), (

**c**) shear stress ${\sigma}_{12}$ (MPa) and (

**d**) normal stress ${\sigma}_{22}$ (MPa).

**Figure 8.**The variations along the crack line of the temperature, normal component of the heat flux and shear stress that develop in the periodically-layered alumina/zirconia, subjected to a remote normal heat flux of ${\overline{q}}_{2}$ = −1 W/m${}^{2}$.

**Figure 9.**Field distributions in the region $-1.5\le {X}_{1}/\left(2d\right)\le 1.5$, $-1.5\le {X}_{2}/\left(2h\right)\le 1.5$ of the cracked layered ceramic composite. (

**a**) Shear stress ${\sigma}_{12}$ (MPa) and (

**b**) normal stress ${\sigma}_{22}$ (MPa).

**Table 1.**Material properties of carbon, epoxy, alumina and zirconia constituents (Columns 1, 2, 4 and 5, respectively). In Column 3, the effective properties of carbon/epoxy composite (${v}_{f}=0.5$) are presented. Here, E${}_{A}$, E${}_{T}$, ${\nu}_{A}$, ${\nu}_{T}$, G${}_{A}$, ${\alpha}_{A}$, ${\alpha}_{T}$, ${\kappa}_{A}$ and ${\kappa}_{T}$ denote the axial and transverse Young’s moduli, axial and transverse Poisson’s ratios, axial and transverse coefficients of thermal expansion, axial and transverse conductivities, respectively.

Property | Carbon T300 | Epoxy | Homogenized Carbon/Epoxy ${\mathit{v}}_{\mathit{f}}$ = 0.5 | Alumina Al${}_{2}$O${}_{3}$ | Zirconia ZrO${}_{2}$ |
---|---|---|---|---|---|

E${}_{A}$(GPa) | 220 | 3.45 | 111.9 | 393 | 207 |

E${}_{T}$(GPa) | 22 | 3.45 | 8.49 | 393 | 207 |

${\nu}_{A}$ | 0.3 | 0.35 | 0.32 | 0.27 | 0.32 |

${\nu}_{T}$ | 0.35 | 0.35 | 0.39 | 0.27 | 0.32 |

G${}_{A}$(GPa) | 22 | 1.28 | 3.16 | 154.7 | 78.4 |

${\alpha}_{A}\left({10}^{-6}\phantom{\rule{5.0pt}{0ex}}{K}^{-1}\right)$ | −1.3 | 54 | −0.42 | 8.4 | 11 |

${\alpha}_{T}\left({10}^{-6}\phantom{\rule{5.0pt}{0ex}}{K}^{-1}\right)$ | 7 | 54 | 37.1 | 8.4 | 11 |

${\kappa}_{A}$(W/(mK)) | 20.5 | 0.18 | 10.35 | 35 | 2.7 |

${\kappa}_{T}$(W/(mK)) | 1.46 | 0.18 | 0.39 | 35 | 2.7 |

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**MDPI and ACS Style**

Aboudi, J. The Induced Stress Field in Cracked Composites by Heat Flow. *J. Compos. Sci.* **2017**, *1*, 4.
https://doi.org/10.3390/jcs1010004

**AMA Style**

Aboudi J. The Induced Stress Field in Cracked Composites by Heat Flow. *Journal of Composites Science*. 2017; 1(1):4.
https://doi.org/10.3390/jcs1010004

**Chicago/Turabian Style**

Aboudi, Jacob. 2017. "The Induced Stress Field in Cracked Composites by Heat Flow" *Journal of Composites Science* 1, no. 1: 4.
https://doi.org/10.3390/jcs1010004