# Factors of Stress Concentration around Spherical Cavity Embedded in Cylinder Subjected to Internal Pressure

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Eshelby’s Homogenization Principle

#### 2.2. Jump Condition across Interface $\Omega $

#### 2.3. Inhomogeneity in the Form of Spherical Cavity

## 3. Spherical Cavity Embedded in a Cylinder Wall

## 4. Factors of Stress Concentration around a Cavity within the Tube Wall

## 5. Numerical Validation of the Presented Methodology

_{t}

_{1}and K

_{t}

_{2}, respectively, in the following form:

_{t3}stress concentration factor can be derived from Formula (48) to obtain

_{m}= 0.25, 0.3, and 0.35. Two geometric ratios for $\kappa $ have been taken into account: $\kappa $ = 15.5 for a thin-walled cylinder and $\kappa $ = 3.1 for a thick tube. For each geometric ratio $\kappa $, three various dimensionless porosity depths were examined Δ/t = 5$\rho $/t, Δ/t = 0.5, and Δ/t = 1−5$\rho $/t.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\mathit{\u03f5}}_{ij}^{*}\left(X\right)$ | Eigenstrain |

${\mathit{\Omega}}_{m}$,${\mathit{\Omega}}_{i}$ | Domain occupied by the matrix and the inclusion, respectively |

${\mathit{C}}_{ijkl}^{m}$,${\mathit{C}}_{ijkl}^{i}$ | Elastic stiffness of the matrix and the inclusion |

${\mathit{\sigma}}_{ij}^{0}$,${\mathit{\u03f5}}_{ij}^{0}$ | Applied stress and the corresponding deformation |

${\mathit{\sigma}}_{ij}^{d}\left(X\right)$,${\mathit{\u03f5}}_{ij}^{d}\left(X\right)$ | Total stress and the corresponding deformation |

${\mathit{\sigma}}_{ij}\left(X\right)$,${\mathit{\u03f5}}_{ij}\left(X\right)$ | Perturbation stress and the corresponding deformation |

${\mathit{S}}_{klmn}$ | Fourth rank Elshby’s tensor |

$\Delta {\mathit{\sigma}}_{ij}$ | Stress jump across the interface |

$n$ | Unit vector normal to the interface |

${\overline{\mathit{\sigma}}}_{ij}^{0}$,${\overline{\mathit{\u03f5}}}_{ij}^{0}$ | Average applied stress and corresponding deformation |

${\mathit{\sigma}}_{r}$ | Reference stress |

$P$ | Applied internal pressure |

${r}_{i}$,${r}_{e}$ | Internal and external radius of the tube |

${r}_{C}$ | Cavity position with respect to the cylinder axis |

$\rho $ | Porosity radius |

${\nu}_{m}$,${E}_{m}$,${\mu}_{m}$ | Poisson ratio, Young and shear modulus of the cylinder |

${\nu}_{i}$,${E}_{i}$,${\mu}_{i}$ | Poisson ratio, Young and shear modulus of the inclusion |

${K}_{t}$ | Stress concentration factor |

$\kappa $ | Geometric ratio of the cylinder |

$\Delta $ | Cavity depth |

$d$ | Distance between two inclusions |

$t$ | Cylinder wall thickness |

## Appendix A

#### Appendix A.1. Expressions of the Coefficients ${C}_{0}$ to ${C}_{5}$

#### Appendix A.2. Components of Normal Constraint ${\mathit{\sigma}}_{11}$ across a Spherical Porosity

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**Figure 1.**Distance between two spherical inclusions of different radii embedded in an isotropic-elastic material.

**Figure 4.**Spherical cavity embedded in the weld of an isotropic-elastic long cylinder: (

**a**) elementary solid volume surrounding the porosity and (

**b**) components of the average stress acting on the faces of the elementary volume of the solid body.

**Figure 5.**A typical mesh in the pipe containing a spherical cavity: (

**a**) depth of the flaw on the thickness of the cylinder and (

**b**) magnification of the area occupied by the spherical porosity.

**Figure 7.**Variations of stress components around the spherical cavity: (

**a**) equator ${x}_{1}=0$, (

**b**) equator ${x}_{2}=0$, and (

**c**) equator ${x}_{3}=0$. Porosity radius $\rho =0.5\mathrm{mm}$, dimensionless geometric ratio of the cylinder $\kappa =15.5$, and dimensionless depth $\mathsf{\Delta}/t=0.75$.

**Figure 8.**Repartition of the components of stress around the spherical cavity and the stress values at $\theta =0$, $\pi /2$, and $-\pi /2$, (

**a**) component of hoop stress ${\mathit{\sigma}}_{11}$, (

**b**) component of axial stress ${\mathit{\sigma}}_{22}$, and (

**c**) component of radial stress ${\mathit{\sigma}}_{33}$.

**Table 1.**Comparison of analytical and finite element analysis (FEA) results of stress concentration factors (SCFs) near the cavity of radius of 0.5 mm and κ = 15.5.

SCF | $\Delta /\mathit{t}$ | Proposed Approach (Equations (49)–(51)) | FEA Results | ||||
---|---|---|---|---|---|---|---|

${\mathit{\nu}}_{\mathit{m}}$ | ${\mathit{\nu}}_{\mathit{m}}$ | ||||||

0.35 | 0.3 | 0.25 | 0.35 | 0.3 | 0.25 | ||

${K}_{t1}$ | $5\rho /t$ | 2.190 | 2.124 | 2.063 | 2.181 | 2.116 | 2.056 |

0.50 | 2.201 | 2.134 | 2.073 | 2.189 | 2.124 | 2.064 | |

$1-5\rho /t$ | 2.212 | 2.144 | 2.082 | 2.217 | 2.149 | 2.087 | |

${K}_{t2}$ | $5\rho /t$ | 2.533 | 2.345 | 2.174 | 2.520 | 2.333 | 2.162 |

0.50 | 2.568 | 2.374 | 2.196 | 2.550 | 2.358 | 2.182 | |

$1-5\rho /t$ | 2.604 | 2.404 | 2.220 | 2.601 | 2.398 | 2.213 | |

${K}_{t3}$ | $5\rho /t$ | 43.913 | 39.855 | 36.149 | 43.155 | 39.165 | 35.516 |

0.50 | 22.881 | 20.835 | 18.967 | 22.030 | 20.094 | 18.310 | |

$1-5\rho /t$ | 15.872 | 14.496 | 13.241 | 15.661 | 14.307 | 13.066 |

**Table 2.**Comparison of analytical and FEA results of SCFs near the porosity of radius of 0.5 mm and κ = 3.1.

SCF | $\Delta /\mathit{t}$ | Presented Solution (Equations (49)–(51)) | FEA Results | ||||
---|---|---|---|---|---|---|---|

${\mathit{\nu}}_{\mathit{m}}$ | ${\mathit{\nu}}_{\mathit{m}}$ | ||||||

0.35 | 0.3 | 0.25 | 0.35 | 0.3 | 0.25 | ||

${K}_{t1}$ | $5\rho /t$ | 2.190 | 2.124 | 2.063 | 2.189 | 2.125 | 2.066 |

0.50 | 2.297 | 2.220 | 2.151 | 2.304 | 2.228 | 2.158 | |

$1-5\rho /t$ | 2.416 | 2.329 | 2.249 | 2.444 | 2.355 | 2.227 | |

${K}_{t2}$ | $5\rho /t$ | 2.533 | 2.345 | 2.174 | 2.492 | 2.305 | 2.134 |

0.50 | 2.912 | 2.663 | 2.427 | 2.909 | 2.651 | 2.414 | |

$1-5\rho /t$ | 3.579 | 3.201 | 2.857 | 3.615 | 3.232 | 2.834 | |

${K}_{t3}$ | $5\rho /t$ | 43.913 | 39.855 | 36.149 | 43.519 | 39.463 | 35.921 |

0.50 | 6.076 | 5.638 | 5.238 | 5.980 | 5.549 | 5.154 | |

$1-5\rho /t$ | 4.115 | 3.865 | 3.636 | 4.095 | 3.845 | 3.617 |

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**MDPI and ACS Style**

Abdelghani, M.; Tewfik, G.; Witek, M.; Djahida, D.
Factors of Stress Concentration around Spherical Cavity Embedded in Cylinder Subjected to Internal Pressure. *Materials* **2021**, *14*, 3057.
https://doi.org/10.3390/ma14113057

**AMA Style**

Abdelghani M, Tewfik G, Witek M, Djahida D.
Factors of Stress Concentration around Spherical Cavity Embedded in Cylinder Subjected to Internal Pressure. *Materials*. 2021; 14(11):3057.
https://doi.org/10.3390/ma14113057

**Chicago/Turabian Style**

Abdelghani, Mechri, Ghomari Tewfik, Maciej Witek, and Djouadi Djahida.
2021. "Factors of Stress Concentration around Spherical Cavity Embedded in Cylinder Subjected to Internal Pressure" *Materials* 14, no. 11: 3057.
https://doi.org/10.3390/ma14113057