# Numerical Solution for Crack Phenomenon in Dissimilar Materials under Various Mechanical Loadings

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of the Problem

## 3. Numerical Results and Discussion

## 4. Conclusions

- For ${G}_{2}=0$ and ${G}_{1}={G}_{2}$, the elementary solution of HSIEs is reduced to two cracks in half plane and infinite plane problems, respectively.
- The equivalence of the stress acting at the crack tip due to the geometric of the cracks resulted in equal values of dimensionless SIFs.
- The value of ${G}_{2}/{G}_{1}$, types of stresses and geometry conditions of the crack affect the strength of the materials for two cracks in the upper part of dissimilar materials.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

COD | Crack Opening Displacement |

CPF | Complex Potentials Function |

HSIEs | Hypersingular Integral Equations |

MCPs | Modified Complex Potentials |

SIFs | Stress Intensity Factors |

## Glossary of Symbols

$\varphi \left(z\right)$, $\psi \left(z\right)$, $\omega \left(z\right)$ | Complex Potentials Functions |

${\varphi}_{1}\left(z\right)$, ${\psi}_{1}\left(z\right)$, ${\omega}_{1}\left(z\right)$ | Upper plane of Complex Potentials Functions |

${\varphi}_{1p}\left(z\right)$, ${\psi}_{1p}\left(z\right)$, ${\omega}_{1p}\left(z\right)$ | Principal part of Complex Potentials Functions |

${\varphi}_{1c}\left(z\right)$, ${\psi}_{1c}\left(z\right)$, ${\omega}_{1c}\left(z\right)$ | Complementary part of Complex Potentials Functions |

${\varphi}_{2}\left(z\right)$, ${\psi}_{2}\left(z\right)$, ${\omega}_{2}\left(z\right)$ | Lower plane of Complex Potentials Functions |

${\sigma}_{x}$, ${\sigma}_{y}$, ${\sigma}_{xy}$ | Stress components |

$\epsilon $ | Strain component |

G | Shear modulus |

v | Poisson’s ration |

$g\left(t\right)$ | Crack Opening Displacement function |

${\beta}_{i}$ | Elastic constant |

${K}_{{A}_{j}}$ | Stress Intensity Factors at crack tip ${A}_{j}$ |

${F}_{{A}_{j}}$ | Dimensionless Stress Intensity Factors at crack tip ${A}_{j}$ |

$\u2a0e=$ | Hypersingular integral and must be defined as a finite part integral |

≃ | Similarity equation |

${\{N\left({t}_{i0}\right)+iT\left({t}_{i0}\right)\}}_{ij}$ | The traction influence on crack i caused by COD function on crack j |

## References

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**Figure 3.**The geometry conditions for crack problems in the upper plane of dissimilar materials. (

**a**) A crack parallel to the boundary; (

**b**) Two cracks in the upper plane.

**Figure 4.**SIFs for two cracks in the upper plane of dissimilar materials under shear stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${B}_{1}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${B}_{1}$ and ${B}_{2}$.

**Figure 5.**SIFs for two cracks in the upper plane of dissimilar materials under normal stress. (

**a**) Crack tip ${A}_{1}$; (

**b**) Crack tip ${A}_{2}$; (

**c**) Crack tip ${B}_{1}$; (

**d**) Crack tip ${B}_{2}$.

**Figure 6.**SIFs for two cracks in the upper plane of dissimilar materials under tearing stress. (

**a**) Crack tip ${A}_{1}$; (

**b**) Crack tip ${A}_{2}$; (

**c**) Crack tip ${B}_{1}$; (

**d**) Crack tip ${B}_{2}$.

**Figure 7.**SIFs for two cracks in the upper plane of dissimilar materials under mixed stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${B}_{1}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${B}_{1}$ and ${B}_{2}$.

**Figure 8.**The geometry conditions for two cracks in the upper plane of dissimilar materials. (

**a**) A semi circular arc crack and an inclined crack; (

**b**) Two equal circular arc cracks on the same circle.

**Figure 9.**SIFs for two cracks in the upper plane of dissimilar materials under shear stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${B}_{1}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${B}_{1}$ and ${B}_{2}$.

**Figure 10.**SIFs for two cracks in the upper plane of dissimilar materials under normal stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${B}_{1}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${B}_{1}$ and ${B}_{2}$.

**Figure 11.**SIFs for two cracks in the upper plane of dissimilar materials under tearing stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${B}_{1}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${B}_{1}$ and ${B}_{2}$.

**Figure 12.**SIFs for two cracks in the upper plane of dissimilar materials under mixed stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${B}_{1}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${A}_{2}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${B}_{1}$ and ${B}_{2}$.

**Figure 13.**Comparison of dimensionless SIFs between present study and Chen and Hasebe [26] at all crack tips.

**Figure 14.**SIFs for two cracks in the upper plane of dissimilar materials under shear stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${B}_{1}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{2}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${B}_{1}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{2}$ and ${B}_{2}$.

**Figure 15.**SIFs for two cracks in the upper plane of dissimilar materials under normal stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${B}_{1}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{2}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${B}_{1}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{2}$ and ${B}_{2}$.

**Figure 16.**SIFs for two cracks in the upper plane of dissimilar materials under tearing stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${B}_{1}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{2}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${B}_{1}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{2}$ and ${B}_{2}$.

**Figure 17.**SIFs for cracks in the upper plane of dissimilar materials under mixed stress. (

**a**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{1}$ and ${B}_{1}$; (

**b**) Dimensionless SIF ${F}_{1}$ at crack tips ${A}_{2}$ and ${B}_{2}$; (

**c**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{1}$ and ${B}_{1}$; (

**d**) Dimensionless SIF ${F}_{2}$ at crack tips ${A}_{2}$ and ${B}_{2}$.

**Table 1.**Stress intensity factors (SIFs) for a crack parallel to the boundary of dissimilar materials (Figure 3a).

SIF | $\mathit{h}/2\mathit{R}$ | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{0}.\mathbf{1}$ | $\mathbf{0}.\mathbf{2}$ | $\mathbf{0}.\mathbf{3}$ | $\mathbf{0}.\mathbf{4}$ | $\mathbf{0}.\mathbf{5}$ | $\mathbf{0}.\mathbf{6}$ | $\mathbf{0}.\mathbf{7}$ | |

${F}_{1{A}_{1}}$* | 0.8083 | 0.8357 | 0.8568 | 0.8752 | 0.8920 | 0.9071 | 0.9203 |

${F}_{2{A}_{1}}$* | 0.1119 | 0.0811 | 0.0634 | 0.0500 | 0.0391 | 0.0304 | 0.0236 |

${F}_{1{A}_{2}}$* | 0.8083 | 0.8357 | 0.8568 | 0.8752 | 0.8920 | 0.9071 | 0.9203 |

${F}_{1{A}_{2}}$** | 0.8080 | 0.8360 | 0.8570 | 0.8750 | 0.8920 | 0.9070 | 0.9200 |

${F}_{2{A}_{2}}$* | −0.1119 | −0.0811 | −0.0634 | −0.0500 | −0.0391 | −0.0304 | −0.0236 |

${F}_{2{A}_{2}}$** | −0.1120 | −0.0810 | −0.0630 | −0.0500 | −0.0390 | −0.0300 | −0.0240 |

**Table 2.**SIFs for two cracks in the upper plane of dissimilar materials under various mechanical loadings (Figure 3b).

Stress | ${\mathit{G}}_{2}/{\mathit{G}}_{1}$ | SIF | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{F}}_{\mathbf{1}{\mathit{A}}_{\mathbf{1}}}$ | ${\mathit{F}}_{\mathbf{2}{\mathit{A}}_{\mathbf{1}}}$ | ${\mathit{F}}_{\mathbf{1}{\mathit{A}}_{\mathbf{2}}}$ | ${\mathit{F}}_{\mathbf{2}{\mathit{A}}_{\mathbf{2}}}$ | ${\mathit{F}}_{\mathbf{1}{\mathit{B}}_{\mathbf{1}}}$ | ${\mathit{F}}_{\mathbf{2}{\mathit{B}}_{\mathbf{1}}}$ | ${\mathit{F}}_{\mathbf{1}{\mathit{B}}_{\mathbf{2}}}$ | ${\mathit{F}}_{\mathbf{2}{\mathit{B}}_{\mathbf{2}}}$ | ||

Shear | 0.2 * | 0.4634 | −0.3358 | 0.0612 | −0.1974 | 1.2272 | −0.0336 | 1.1004 | 0.0510 |

0.2 ** | 0.4670 | −0.3333 | 0.0420 | −0.2010 | 1.2140 | −0.0180 | 1.1010 | 0.0650 | |

1.0 * | 0.4250 | −0.3226 | 0.0420 | −0.2341 | 0.8909 | −0.0364 | 0.9844 | 0.0536 | |

1.0 ** | 0.4250 | −0.3230 | 0.0420 | −0.2360 | 0.8910 | −0.0360 | 0.9850 | 0.0530 | |

5.0 * | 0.4219 | −0.3000 | 0.0571 | −0.2352 | 0.6666 | −0.0486 | 0.9128 | 0.0472 | |

5.0 ** | 0.4020 | −0.2830 | 0.0370 | −0.2400 | 0.6600 | −0.0440 | 0.9030 | 0.0460 | |

Normal | 0.2 * | 0.7333 | 0.5407 | 0.5561 | 0.6535 | −0.1501 | −0.1375 | 0.1483 | −0.0758 |

1.0 * | 0.5177 | 0.4920 | 0.4833 | 0.5594 | −0.1097 | −0.0805 | 0.1152 | −0.0270 | |

5.0 * | 0.3974 | 0.4609 | 0.4327 | 0.5093 | −0.0649 | −0.0472 | 0.1122 | −0.0079 | |

Tearing | 0.2 * | −1.3748 | 0.2490 | −1.0732 | 0.0420 | 0.4691 | −1.1090 | −0.0698 | −1.1981 |

1.0 * | −1.1290 | 0.1528 | −0.9643 | 0.0614 | 0.3696 | −0.7944 | −0.0589 | −1.1132 | |

5.0 * | −0.9501 | 0.0443 | −0.8446 | 0.0524 | 0.2347 | −0.5754 | −0.0770 | −1.0218 | |

Mixed | 0.2 * | 1.1967 | 0.2050 | 0.6173 | 0.4562 | 1.0771 | −0.1711 | 1.2488 | −0.0247 |

1.0 * | 0.9427 | 0.1694 | 0.5253 | 0.3254 | 0.7812 | −0.1169 | 1.0996 | 0.0266 | |

5.0 * | 0.8194 | 0.1609 | 0.4898 | 0.2741 | 0.6017 | −0.0958 | 1.0250 | 0.0393 |

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**MDPI and ACS Style**

Hamzah, K.B.; Nik Long, N.M.A.; Senu, N.; Eshkuvatov, Z.K.
Numerical Solution for Crack Phenomenon in Dissimilar Materials under Various Mechanical Loadings. *Symmetry* **2021**, *13*, 235.
https://doi.org/10.3390/sym13020235

**AMA Style**

Hamzah KB, Nik Long NMA, Senu N, Eshkuvatov ZK.
Numerical Solution for Crack Phenomenon in Dissimilar Materials under Various Mechanical Loadings. *Symmetry*. 2021; 13(2):235.
https://doi.org/10.3390/sym13020235

**Chicago/Turabian Style**

Hamzah, Khairum Bin, Nik Mohd Asri Nik Long, Norazak Senu, and Zainidin K. Eshkuvatov.
2021. "Numerical Solution for Crack Phenomenon in Dissimilar Materials under Various Mechanical Loadings" *Symmetry* 13, no. 2: 235.
https://doi.org/10.3390/sym13020235