Multiscale Fatigue Crack Growth Modeling for Welded Stiffened Panels

  • Ž. Božić
  • S. Schmauder
  • M. Mlikota
  • M. Hummel
Living reference work entry


The influence of welding residual stresses in stiffened panels on effective stress intensity factor values and fatigue crack growth rate is studied in this paper. Interpretation of relevant effects on different length scales such as dislocation appearance and microstructural crack nucleation and propagation is taken into account using molecular dynamics (MD) simulations as well as a Tanaka-Mura approach for the analysis of the problem. Mode I stress intensity factors (SIFs), K I, were calculated by the finite element method (FEM) using shell elements and the crack tip displacement extrapolation technique. The total SIF value, K tot, is derived by a part due to the applied load, K appl, and by a part due to welding residual stresses, K res. Fatigue crack propagation simulations based on power law models showed that high tensile residual stresses in the vicinity of a stiffener significantly increase the crack growth rate, which is in good agreement with experimental results.


Dislocation Microstructurally small cracks Fatigue crack growth rate Residual stress 



half crack length


initial crack length


final crack length


material constant of the Paris equation


critical resolved shear stress


slip band length


crack growth rate


Young’s modulus

\( F\left(\overrightarrow{r},t\right) \)

interatomic force


maximum applied force


minimum applied force


shear modulus


stress intensity factor (SIF)


stress intensity factor due to the applied load


stress intensity factor due to welding residual stresses


stress intensity factor threshold


total stress intensity factor


atomic mass


material constant of the Paris eq.


number of stress cycles for the fatigue crack propagation


number of stress cycles for fatigue failure


number of stress cycles required for crack nucleation in a single grain


number of stress cycles needed for the initiation of a small crack


stress ratio


effective stress intensity factor ratio

\( U\left(\overrightarrow{r}\, ,t\right) \)

interatomic embedded atom method (EAM) pair potential


specific fracture energy per unit area


applied force range


stress intensity factor range


effective stress intensity factor range


average applied stress range

\( \Delta \overline{\tau} \)

average shear stress range on the slip band


yield stress



This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. Schm 746/132-1 and as part of the Collaborative Research Centre SFB 716 at the University of Stuttgart and by the Croatian Science Foundation Grant No. 120-0362321-2198. The support is gratefully acknowledged.


  1. 1.
    Luo C, Chattopadhyay A. Prediction of fatigue crack initial stage based on a multiscale damage criterion. Int J Fatigue. 2011;33:403–13.CrossRefGoogle Scholar
  2. 2.
    Curtin WA, Deshpande VS, Needleman A, Van der Giessen E, Wallin M. Hybrid discrete dislocation models for fatigue crack growth. Int J Fatigue. 2010;32:1511–20.CrossRefGoogle Scholar
  3. 3.
    White P. Molecular dynamic modelling of fatigue crack growth in aluminium using LEFM boundary conditions. Int J Fatigue. 2012;44:141–50.CrossRefGoogle Scholar
  4. 4.
    Horstemeyer MF, Farkas D, Kim S, Tang T, Potirniche G. Nanostructurally small cracks (NSC): a review on atomistic modeling of fatigue. Int J Fatigue. 2010;32:1473–502.CrossRefGoogle Scholar
  5. 5.
    Božić Ž, Schmauder S, Mlikota M, Hummel M. Fatigue Crack Growth Modelling in Welded Stiffened Panels under Cyclic Tension, 13th International Conference on Fracture, Beijing, China. 2013.Google Scholar
  6. 6.
    Stadler J, Mikulla R, Trebin HR. IMD: a software package for molecular dynamics studies on parallel computers. Int J Mod Phys. 1997;8:1131.CrossRefGoogle Scholar
  7. 7.
    Bonny G, Pasianot RC, Castin N, Malerba L. Ternary Fe-cu-Ni many-body potential to model reactor pressure vessel steels: first validation by simulated thermal annealing. Phil Mag. 2009;89:3531–46.CrossRefGoogle Scholar
  8. 8.
    Grottel S, Reina G, Dachsbacher C, Ertl T. Coherent culling and shading for large molecular dynamics visualization. Computer Graphics Forum Proc of EUROVIS 2010. 2010;29(3):953–62.Google Scholar
  9. 9.
    Stukowski A, Bulatov VV, Arsenlis A. Automated identification and indexing of dislocations in crystal interfaces. Modelling Simul Mater Sci Eng. 2012;20:085007.CrossRefGoogle Scholar
  10. 10.
    Stukowski A. DXA user manual Version 1.3.4; 2010.
  11. 11.
    Glodez S, Jezernik N, Kramberger J, Lassen T. Numerical modelling of fatigue crack initiation of martensitic steel. Adv Eng Softw. 2010;41(5):823–9.CrossRefMATHGoogle Scholar
  12. 12.
    Wood WA. Fatigue in aircraft structures. New York: Academic Press; 1956.Google Scholar
  13. 13.
    Fine ME, Ritchie RO. Fatigue-crack initiation and near-threshold crack growth. In: Meshii M, editor. Fatigue and microstructure. Materials Park: ASM; 1978. p. 245–78.Google Scholar
  14. 14.
    Laird C. Mechanisms and theories of fatigue. In: Meshii M, editor. Fatigue and microstructure. Materials Park: ASM; 1978. p. 149–203.Google Scholar
  15. 15.
    Klesnil M, Lukas P. Fatigue of metallic materials. New York: Elsevier; 1980. p. 57–80.Google Scholar
  16. 16.
    Mughrabi H. Rev Phys Appl. 1988;23:367–79.CrossRefGoogle Scholar
  17. 17.
    Mughrabi H. In: Chan KS, Liaw PK, Bellows RS, Zogas T, Soboyejo WO, editors. Fatigue: David L. Davidson symposium. Warrendale: TMS; 2002. p. 3–15.Google Scholar
  18. 18.
    Davidson DL, Chan KS. Crystallography of fatigue crack initiation in astrology at ambient temperature. Acta Metall. 1989;37(4):1089–97.CrossRefGoogle Scholar
  19. 19.
    Wang QY, Bathias C, Kawagoishi N, Chen Q. Effect of inclusion on subsurface crack initiation and gigacycle fatigue strength. Int J Fatigue. 2002;24(12):1269–74.CrossRefGoogle Scholar
  20. 20.
    Murakami Y, Nomoto T, Ueda T. On the mechanism of fatigue failure in the superlong life regime (N>107 cycles). Part 1: influence of hydrogen trapped by inclusions. Fatigue Fract Engng Mater Struct. 2000;23(11):893–902.CrossRefGoogle Scholar
  21. 21.
    Tanaka K, Mura T. A dislocation model for fatigue crack initiation. J Appl Mech. 1981;48:97–103.CrossRefMATHGoogle Scholar
  22. 22.
    Tanaka K, Mura T. A theory of fatigue crack initiation at inclusions. Metall Trans A. 1982;13(1):117–23.CrossRefGoogle Scholar
  23. 23.
    Brückner-Foit A, Huang X. Numerical simulation of micro-crack initiation of martensitic steel under fatigue loading. Int J Fatigue. 2006;28(9):963–71.CrossRefGoogle Scholar
  24. 24.
    Jezernik N, Kramberger J, Lassen T, Glodez S. Numerical modelling of fatigue crack initiation and growth of martensitic steels. Fatigue & Fracture of Engineering Materials & Structures. 2010;33:714–23.MATHGoogle Scholar
  25. 25.
    Broek D. The practical use of fracture mechanics. Dordrecht: Kluwer Academic Publishers; 1989.CrossRefGoogle Scholar
  26. 26.
    Paris P, Erdogan F. A critical analysis of crack propagation laws. J Basic Eng. 1963;85:528–34.CrossRefGoogle Scholar
  27. 27.
    Dexter RJ, Pilarski PJ, Mahmoud HN. Analysis of crack propagation in welded stiffened panels. Int J Fatigue. 2003;25:1169–74.CrossRefGoogle Scholar
  28. 28.
    Mahmoud HN, Dexter RJ. Propagation rate of large cracks in stiffened panels under tension loading. Mar Struct. 2005;18:265–88.CrossRefGoogle Scholar
  29. 29.
    Sumi Y, Božić Ž, Iyama H, Kawamura Y. Multiple fatigue cracks propagating in a stiffened panel. Journal of the Society of Naval Architects of Japan. 1996;179:407–12.CrossRefGoogle Scholar
  30. 30.
    Elber W. The significance of fatigue crack closure, Damage tolerance in aircraft structures. ASTM STP 486. American Society for Testing & Materials; 1971. p. 230–242.Google Scholar
  31. 31.
    Donahue RJ, Clark HM, Atanmo P, Kumble R, McEvily AJ. Crack opening displacement and the rate of fatigue crack growth. Int J Fract Mech. 1972;8:209–19.CrossRefGoogle Scholar
  32. 32.
    Swanson Analysis System (2009). Inc. ANSYS User’s Manual Revision 11.0.Google Scholar
  33. 33.
    Han T, Luo Y, Wang C. Effects of temperature and strain rate on the mechanical properties of hexagonal boron nitride nanosheets. J Phys D Appl Phys. 2014;47:025303.CrossRefGoogle Scholar
  34. 34.
    Tapasa K, Bacon DJ, Osetsky YN. Simulation of dislocation glide in dilute Fe-cu alloys. Materials Science & Engineering A. 2005;400-401:109–13.CrossRefGoogle Scholar
  35. 35.
    Kohler C, Kizler P, Schmauder S. Atomistic simulation of precipitation hardening in α-iron: influence of precipitate shape and chemical composition. Model Simul Mater Sci Eng. 2005;13:35–45.CrossRefGoogle Scholar
  36. 36.
    Molnar D, et al.. Unpublished research. 2014.Google Scholar
  37. 37.
    Naveen Kumar N, Durgaprasad PV, Dutta BK, Dey GK. Modeling of radiation hardening in ferritic/martensitic steel using multi-scale approach. Comput Mater Sci. 2012;53:258–67.CrossRefGoogle Scholar
  38. 38.
    Latapie A, Farkas D. Molecular dynamics simulations of stress-induced phase transformations and grain nucleation at crack tips in Fe. Modelling Simul. Mater. Sci. Eng. 2003;11:745–53.CrossRefGoogle Scholar
  39. 39.
    Nakai Y. Evaluation of fatigue damage and fatigue crack initiation process by means of atomic-force microscopy. Mater Sci Res Int. 2001;7(2):1–9.Google Scholar
  40. 40.
    Zabett A, Plumtree A. Microstructural effects on the small fatigue crack behaviour of an aluminum alloy plate. Fatigue & Fracture of Engineering Materials & Structures. 1995;18(7–8):801–9.Google Scholar
  41. 41.
    Taylor D, Knott JF. Fatigue crack propagation behaviour of short cracks; the effect of microstructure. Fatigue & Fracture of Engineering Materials & Structures. 1981;4(2):147–55.CrossRefGoogle Scholar
  42. 42.
    Miller KJ. The behaviour of short fatigue cracks and their initiation part II-A general summary. Fatigue & Fracture of Engineering Materials & Structures. 1987;10(2):93–113.CrossRefGoogle Scholar
  43. 43.
    Bao R, Zhang X, Yahaya NA. Evaluating stress intensity factors due to weld residual stresses by the weight function and finite element methods. Eng Fract Mech. 2010;77:2550–66.CrossRefGoogle Scholar
  44. 44.
    Croatian Register of Shipping. Rules for the Classification of Ships, Part 25 – Metallic Materials. 2012.Google Scholar
  45. 45.
    Faulkner D. A review of effective plating for use in the analysis of stiffened plating in bending and compression. J Ship Res. 1975;19:1–17.Google Scholar
  46. 46.
    Barsoum RS. On the use of Isoparametric finite elements in linear fracture mechanics. Int J Numer Methods Eng. 1976;10:25–37.CrossRefMATHGoogle Scholar
  47. 47.
    Henshell RD, Shaw KG. Crack tip finite elements are unnecessary. Int J Numer Methods Eng. 1975;9:495–507.CrossRefMATHGoogle Scholar
  48. 48.
    Božić Ž, Mlikota M, Schmauder S. Application of the ΔK, ΔJ and ΔCTOD parameters in fatigue crack growth modelling, Technical. Gazette. 2011;18(3):459–66.Google Scholar
  49. 49.
    Božić Ž, Schmauder S, Mlikota M. Fatigue growth models for multiple long cracks in plates under cyclic tension based on ΔKI, ΔJ-integral and ΔCTOD parameter. Key Eng Mater. 2012;488-489:525–8.Google Scholar
  50. 50.
    Liu Y, Mahadevan S. Threshold stress intensity factor and crack growth rate prediction under mixed-mode loading. Eng Fract Mech. 2007;74:332–45.CrossRefGoogle Scholar
  51. 51.
    Glinka G. Effect of residual stresses on fatigue crack growth in steel weldments under constant and variable amplitude load, Fracture mechanics, ASTM STP 677, American Society for Testing and Materials; 1979. p. 198–214.Google Scholar
  52. 52.
    Servetti G, Zhang X. Predicting fatigue crack growth rate in a welded butt joint: the role of effective R ratio in accounting for residual stress effect. Engng Fract Mech. 2009;76:1589–602.CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Ž. Božić
    • 1
  • S. Schmauder
    • 2
  • M. Mlikota
    • 2
  • M. Hummel
    • 2
  1. 1.Faculty of Mechanical Engineering and Naval ArchitectureUniversity of ZagrebZagrebCroatia
  2. 2.IMWFUniversity of StuttgartStuttgartGermany

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