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Probabilistic Fracture Mechanics for Analysis of Longitudinal Cracks in Pipes Under Internal Pressure

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Abstract

In this paper, we present a probabilistic fracture mechanics methodology to analyze elastic and elastic–plastic fracture of semi-elliptical longitudinal cracks in pipes under internal pressure. Numerical results are acquired using three-dimensional finite element simulations. Analytical expressions are proposed with unknown coefficients obtained by nonlinear fitting to the numerical results. For the elastic case, results of the shape function using the newly proposed expression are found to be in a good agreement with those found in the literature. In the elastic–plastic case, estimates of the J-integral are presented for various ratios including crack depth to pipe thickness (a/t), reference stress to material yield stress (σref/σy) and mean pipe radius to its thickness (Rm/t). It is found that the range of applicability of the proposed expressions is extended even beyond those found in the literature. Finally, failure probability is accessed by a statistical analysis for uncertainties in loads and material properties, and structural reliability. The probability density function is estimated by the Monte Carlo Method. It is shown from the present results that the crack size is an important factor influencing the distribution function of (J/Je), failure and reliability rates.

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Abbreviations

a :

Crack depth

c :

Crack half length

E :

Young’s modulus

F :

Shape function of the stress intensity factor

J :

Total J-integral, including elastic and plastic components

Je :

Elastic component of the J-integral

Jp :

Plastic component of the J-integral

K I :

Stress intensity factor

n :

Strain hardening index in the Ramberg–Osgood (R-O) stress–strain model

R i, R m, R o :

Inner radius, mean radius and outer radius of the cylinder

P :

Internal pressure

t :

Pipe thickness

α :

Coefficient for the Ramberg–Osgood (R–O) law

ε y :

Yield strain

ε ref :

Reference strain

\(\phi\) :

Angle reflecting the location along the semi-elliptical crack front

\(\theta\) :

Half of total crack angle in the part circumferentially cracked cylinder

\(\gamma\) :

Non-dimensional factor relating the limit load and the optimized reference load

υ :

Poisson’s ratio

σ ap :

Applied stress

σ ref :

Reference stress

σ u :

Ultimate stress

σ y :

Yield stress

FEM:

Finite element method

ERSM:

Enhanced reference stress method

EPFM:

Elastic–plastic fracture mechanic

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Authors and Affiliations

  1. LMPM, University Djillali Liabes, 22000, Sidi Bel Abbès, Algeria

    Belaïd Mechab & Boualem Serier

  2. Laboratory of Aero-Hydrodynamics, USTO MB, 31000, Oran, Algeria

    Nadji Chioukh

  3. Laboratory of Statistics and Stochastic Processes, UDL, 22000, Sidi Bel Abbès, Algeria

    Boubaker Mechab

Authors
  1. Belaïd Mechab
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  2. Nadji Chioukh
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  3. Boubaker Mechab
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  4. Boualem Serier
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Correspondence to Nadji Chioukh.

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Mechab, B., Chioukh, N., Mechab, B. et al. Probabilistic Fracture Mechanics for Analysis of Longitudinal Cracks in Pipes Under Internal Pressure. J Fail. Anal. and Preven. 18, 1643–1651 (2018). https://doi.org/10.1007/s11668-018-0564-8

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  • DOI: https://doi.org/10.1007/s11668-018-0564-8

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