Some aspects of the energy based approach to fatigue crack propagation N. Ranganathan * , F. Chalon, S. Meo Laboratoire de Me ´canique et Rhe ´ologie, Polytech Tours, DP, Universite ´ Franc ßois Rabelais de Tours, 37200 Tours, France Received 22 March 2007; received in revised form 11 January 2008; accepted 17 January 2008 Available online 26 January 2008 Abstract The energy based approach to fatigue crack growth was first proposed by Weertman. This original approach was later modified to take into account the plastic energy dissipated at the crack tip. It has also been proposed that the specific energy to create a unit surface is not a constant and depends upon the crack growth micro-mechanism. Recent studies suggest, on the contrary, that the specific energy is a constant and equivalent to the static energy release rate at rupture. Some studies have shown that the measured plastic energy is much higher than the theoretical estimates in the plastic zone. These aspects are reanalyzed to bring out the salient features of the energy based approach. The advantages of the energy based approach are brought out in the case of variable amplitude fatigue crack growth. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Energy dissipated; Plastic zone; Crack growth mechanism; Variable amplitude crack growth 1. Introduction The basics of the energetic of fracture were first laid down by Griffith [1] in his research on the fracture of glass. The equation derived by Griffith describing the stable con- dition crack growth is given by 4ac s > pa 2 r 2 =E; ð1Þ where c s is the surface energy of the material; a is the crack length; r the applied stress and E the Young modulus. This equation has to be modified for ductile materials in which plastic deformation occuring ahead of the crack tip accompanies crack growth, in contrast to fragile materials such as glass where plastic deformation is negligible. The fracture toughness of a ductile material, K c , can then be used to correlate to the dissipated energy K 2 c ¼ Eðc s þ c p Þ ð2aÞ where c p is the plastic energy dissipated. It is generally admitted that for ductile materials, c p  c s : ð2bÞ Thus in Eq. (2a), only the plastic energy term predominates. The energy based approach to fatigue crack growth was developed by Weertman [2], considering a uniform edge dislocation distribution at the crack tip. He derived the fol- lowing equation to describe fatigue crack growth: da=dN ¼ ADK 4 =ðlr 2 c U Þ; ð3Þ where da/dN is the crack growth rate per cycle; DK, the stress intensity factor amplitude; l, the shear modulus of the material; r c , the critical stress at fracture; U, the critical energy to create a unit surface and A, a constant. Previous studies concerned different techniques devel- oped in the past to determine the parameter U, such as sub grain size measurements [3], the use of micro strain gages in the plastic zone [4], microcalorimetry [5], and by direct evaluation of hysteresis energy under the loading line of a compact tension specimen [6,7]. These studies have shown that the parameter U has to be distinguished from the theo- retical energy to separate the atoms and it is associated to the energy dissipated in the plastic zone near the crack tip, for ductile materials. Pioneering work done by Fine and co- workers, have indicated that, since the crack growth rate is 0142-1123/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2008.01.010 * Corresponding author. Tel.: +33 2 47 64 36 17; fax: +33 2 47 36 13 11. E-mail address: ranganathan@univ-tours.fr (N. Ranganathan). www.elsevier.com/locate/ijfatigue Available online at www.sciencedirect.com International Journal of Fatigue 30 (2008) 1921–1929 International Journalof Fatigue