Calculation of the Wöhler (S-N) Curve Using a Two-Scale Model


Calculation of the Wöhler (S-N) Curve Using a Two-Scale Model

M. Mlikota, S. Schmauder, Ž. Božić

This chapter deals with the initiation of a short crack and subsequent growth of the long crack in a carbon steel under cyclic loading, concluded with the estimation of the complete lifetime represented by the Wöhler (S-N) curve. A micro-model containing the microstructure of the material is generated using the Finite Element Method and the according non-uniform stress distribution is calculated afterwards. The number of cycles needed for crack initiation is estimated on the basis of the stress distribution in the microstructural model and by applying the physically-based Tanaka-Mura model. The long crack growth is handled using the Paris law. The analysis yields good agreement with experimental results from literature.

Multiscale Modelling, Fatigue, Crack Initiation, Lifetime Estimation, Wöhler (S-N) Curve

Published online , 21 pages

Citation: M. Mlikota, S. Schmauder, Ž. Božić, Calculation of the Wöhler (S-N) Curve Using a Two-Scale Model, Materials Research Foundations, Vol. 114, pp 16-36, 2022


Part of the book on Multiscale Fatigue Modelling of Metals

[1] Mlikota M, Schmauder S, 2017, ‘Numerical determination of component Wöhler curve’, DVM Bericht / Anwendungsspezifische Werkstoffgesetze für die Bauteilsimulation 1684, 111-124.
[2] Sangid M. D, 2013, ‘The physics of fatigue crack initiation’, International Journal of Fatigue 57(0), 58-72.
[3] Polak J, Man, J, 2014, ‘Fatigue crack initiation – The role of point defects’, International Journal of Fatigue 65(0), 18-27.
[4] Ewing J. A, Humfrey J. C. W, 1903, ‘The fracture of metals under repeated alternations of stress’, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 200(321-330), 241-250.
[5] Santus C, Taylor D, 2009, ‘Physically short crack propagation in metals during high cycle fatigue’, International Journal of Fatigue 31(8-9), 1356-1365.
[6] Kujawski D, 2001, ‘Correlation of long- and physically short-cracks growth in aluminum alloys’, Engineering Fracture Mechanics 68(12), 1357-1369.
[7] Lorenzino P, Navarro A, 2015, ‘Growth of very long “short cracks” initiated at holes’, International Journal of Fatigue 71(Supplement C), 64 – 74.
[8] Klesnil M, Lukas P, 1980, ‘Fatigue of metallic materials’, Elsevier Scientific.
[9] Glodez S, Jezernik N, Kramberger J, Lassen T, 2010, ‘Numerical modelling of fatigue crack initiation of martensitic steel’, Advances in Engineering Software, 41(5), 823 -829.
[10] Mughrabi H, 2015, ‘Microstructural mechanisms of cyclic deformation, fatigue crack initiation and early crack growth’, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 373(2038).
[11] Suresh S, Ritchie R O, 1984, ‘Propagation of short fatigue cracks’, International Metals Reviews 29(1), 445-475.
[12] McEvily A.J, 1989, ‘On the growth of small/short fatigue cracks’, JSME International Journal 32(2), 181-191.
[13] Christman T, Suresh S, 1986, ‘Crack initiation under far-field cyclic compression and the study of short fatigue cracks’, Engineering Fracture Mechanics 23(6), 953-964.
[14] Newman J, Phillips E, Swain, M, 1999, ‘Fatigue-life prediction methodology using small-crack theory’, International Journal of Fatigue 21(2), 109-119.
[15] Tanaka K, Mura T, 1981, ‘A dislocation model for fatigue crack initiation’, Journal of Applied Mechanics 48(1), 97-103.
[16] Tanaka K, Mura T, 1982, ‘A theory of fatigue crack initiation at inclusions’, Metallurgical Transactions A 13(1), 117-123.
[17] Bozic Z, Schmauder S, Mlikota M, Hummel M, 2014, ‘Multiscale fatigue crack growth modelling for welded stiffened panels’, Fatigue & Fracture of Engineering Materials & Structures 37(9), 1043-1054.
[18] Paris P, and Erdogan F, 1963, ‘A critical analysis of crack propagation laws’, Journal of Basic Engineering 85(4), 528-533.
[19] Bozic Z, Mlikota M, & Schmauder S, 2011, ‘Application of the K, J and CTOD parameters in fatigue crack growth modelling’, Technical Gazette 18(3), 459-466.
[20] Broek D, 1988, The pratical use of fracture mechanics, Kluwer Academic Publishers, Dordrecht, The Netherlands.
[21] Branco R, Antunes F, Ferreira J. M, Silva M, 2009, ‘Determination of Paris law constants with a reverse engineering technique’, Engineering Failure Analysis 16,631-638.
[22] Branco R, Antunes F, Costa D, Yang F. P, & Kuang Z. B, 2012, ‘Determination of the Paris law constants in round bars from beach marks on fracture surfaces’, Engineering Fracture Mechanics 96, 96-106.
[23] Ancona F, Palumbo D, Finis R. D., Demelio G, Galietti U, 2016, ‘Automatic procedure for evaluating the Paris Law of martensitic and austenitic stainless steels by means of thermal methods’, Engineering Fracture Mechanics 163, 206-219.
[24] Szata M, Lesiuk G, 2009, ‘Algorithms for the estimation of fatigue crack growth using energy method’, Archives of Civil and Mechanical Engineering 9(1), 119-134.
[25] Mlikota M, Staib S, Schmauder S, and Bozic Z, 2017, ‘Numerical determination of Paris law constants for carbon steel using a two-scale model’, Journal of Physics: Conference Series 843(1), 012042.
[26] Mlikota M, Schmauder S, Bozic Z, and Hummel M, 2017, ‘Modelling of overload effects on fatigue crack initiation in case of carbon steel’, Fatigue & Fracture of Engineering Materials & Structures 40(8), 1182-1190.
[27] Fatemi A, Zeng Z, & Plaseied A, 2004, ‘Fatigue behavior and life predictions of notched specimens made of QT and forged microalloyed steels’, International Journal of Fatigue 26(6), 663-672.
[28] SIMULIA ABAQUS Documentation.
[29] Mirzazadeh M. M, and Plumtree A, 2012, ‘High cycle fatigue behavior of shot-peened steels’, Metallurgical and Materials Transactions A 43(8), 2777-2784.
[30] Deimel P, & Sattler E, 1998, ‘Non-metallic inclusions and their relation to the J-integral, Ji, phys, at physical crack initiation for different steels and weld metals’, Journal of Materials Science 33(7), 1723-1736.
[31] Siegfried S, and Immanuel S, 2016, Multiscale materials modeling, approaches to full multiscaling, De Gruyter, Berlin, Boston.
[32] Tsach U, 1981, ‘Locking of thin plate/shell elements’, International Journal for Numerical Methods in Engineering, 17(4), 633-644.
[33] Jezernik N, Kramberger J, Lassen T and Glodez S, 2010, ‘Numerical modelling of fatigue crack initiation and growth of martensitic steels’, Fatigue & Fracture of Engineering Materials & Structures 33(11), 714-723.
[34] Yang L, and Fatemi A, 1996, ‘Impact resistance and fracture toughness of vanadium-based microalloyed forging steel in the as-forged and Q&T conditions’, Journal of Engineering Materials and Technology 118, 71-79.
[35] Hui W, Zhang Y, Zhao X, Xiao N, Hu F, 2016, ‘High cycle fatigue behavior of V-microalloyed medium carbon steels: A comparison between bainitic and ferritic-pearlitic microstructures’, International Journal of Fatigue 91(1), 232 – 241.