Journal of Failure Analysis and Prevention

, Volume 19, Issue 1, pp 85–97 | Cite as

Assessment of Mode I and Mode II Stress Intensity Factors Obtained by Displacement Extrapolation and Interaction Integral Methods

  • Reza Ghafoori AhangarEmail author
  • Yves Verreman
Technical Article---Peer-Reviewed


A numerical study of the parameters influencing the accuracy of computed stress intensity factor (SIF) values is presented in this article. Mode I SIFs of a through-thickness center crack are calculated by employing the displacement extrapolation method (DEM) of near-tip field extrapolation and the energy-based interaction integral method (IIM). The results are compared with an analytical solution, and some recommendations are provided to increase the accuracy of the calculated mode I SIFs. Optimized influencing parameters can minimize the errors obtained with DEM, whereas the errors obtained with IIM are negligible for a variety of crack lengths. Based on the recommendations made for the mode I SIF calculation, assessment of the mode I and mode II SIFs is performed at the weld root crack tip of a load-carrying cruciform fillet welded joint loaded under a three-point bending condition. The accuracy of the mode I SIF calculated with DEM is independent of the crack length, whereas that of mode II is highly dependent. The IIM provides accurate SIFs for both modes for various crack lengths.


Stress intensity factor (SIF) Center crack Displacement extrapolation method (DEM) Interaction integral method (IIM) Load-carrying cruciform fillet welded joint 

List of symbols


Crack length (mm)


Area inside the contour (m2)


Half specimen width (mm)


Percent difference (%)


Elastic modulus (GPa)

\(E^{\prime }\)

Effective elastic modulus (GPa)


Bending load (kN)


Shear modulus (GPa)


Half specimen height (mm)


Interaction integral (MPa m)


J-integral (MPa m)


Equivalent stress intensity factor (MPa m1/2)

\(K_{\text{I}} ,K_{\text{II}}\)

Mode I and mode II stress intensity factors (MPa m1/2)


Element size (mm)


Element size in the body (mm)


Element size around the crack tip (mm)

\(L_{\text{B}} /L_{\text{T}}\)

Mesh size ratio


Angular discretization around the crack tip


Weight function in the domain integral


Plate thickness (mm)


Strain energy density (J/m3)


Geometry factor

(u, v)

Displacements parallel and perpendicular to the crack growth direction

(r, θ)

Polar coordinates with the origin at the crack tip


Crack length dimensionless parameter


Elastic parameter


Poisson’s ratio


Tensile stress (MPa)



This work was supported by the Consortium de Recherche en Fabrication et Réparation des Roues d’Eau (CReFaRRE), the Natural Sciences and Engineering Research Council of Canada (NSERC), General Electric Renewable Energy, Hydro-Québec, and the Mathematics of Information Technology and Complex Systems (MITACS).


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Copyright information

© ASM International 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringÉcole Polytechnique de MontréalMontréalCanada

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