Discussion of fracture paper # 2

W. Brocks's picture

Williams derived it in 1939, Irwin addressed it in 1957 as one of two parameters characterising “the influence of the test configuration, loads and crack length upon the stresses”, and Rice used it in 1974 to calculate the effects of the specimen geometry on the plastic zone in small scale yielding: the non-singular term in the series expansion of the stress field at crack tips called T-stress. It gained importance in the early 1990s in numerous investigations on constraint effects and two-parameter approaches. With the upcoming of damage mechanics, the number of publications on T-stress went down and everything seemed to be said about their significance. The few papers on T-stresses appearing in the first decade of the 21st century mostly concerned their calculation for various configurations - even for “cracks in anisotropic bimaterials” (EFM 75, 2008), which is outside of their theoretical foundation. Surprisingly, two new (partly quite similar) papers on this subject appeared recently:

J.C. Sobotka, R.H. Dodds: Steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modelling. Engineering Fracture Mechanics, Vol. 78, 2011, pp. 343–363.J.C. Sobotka, R.H. Dodds: T-stress effects on steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modelling. Engineering Fracture Mechanics, Vol. 78, 2011, pp. 1182–1200.

Is this a renaissance of early fracture mechanics concepts or just a latecomer? Let us have a look on the details.

T-stress effects on stress fields at stationary cracks for small-scale yielding have been extensively investigated in the 1990s using a so-called boundary-layer model, i.e. a disk-shaped volume centred at the crack front, which is subjected to a K-field and a constant stress parallel to the ligament. It needs a particular Eulerian analysis to represent steady-state growth on a fixed mesh in a boundary-layer framework. The application of the respective “streamline integration” introduced by Dean and Hutchinson in 1980 to 3D panels is the basic achievement of the two papers, allowing to study thickness effects and variations of plastic zones, stress fields and crack opening displacement over the thickness, the first one for T = 0, the second for T ¹ 0.

These are thoroughly performed analyses yielding substantial information on the local fields. What they do not answer is the question on their relevance for actual fracture problems. In the extensive discussions on ductile tearing resistance, the T-stress has been proposed as a parameter characterising the “constraint”. This definitely works in small-scale yielding, but can steady state crack extension occur under small-scale yielding conditions? The authors argue with crack growth in thin panels of high-strength aluminium alloys as they are used in aerospace structures. They claim in the introduction that a T-L orientation of the cracked panel, “tends to favor a local ‘flat‘ mode I fracture process rather than a local ‘slanted‘ mixed-mode process. ... Essentially steady conditions evolve as the crack front advances further over distances of several thicknesses, characterized by a flat-to-nearly-flat tearing resistance curve.“ Both statements are indeed essential in the context of their investigations but unfortunately, they are not substantiated by experimental evidence. And a final question: how significant is the T-stress in a cracked thin panel under tension? So what about a continuation including test data and their analysis?

Luoyu Roy Xu's picture

Recent work on the interesting T-stress research topic

Dear Dr. Brocks,

I share the same interest on the T-stress  with you. In recent years, we studied 1) the T-stress history of a dynamic crack, and 2) T-stress change across the static crack kinking, which is illustrated in the figure.




Both papers could be downloaded from these web links:



There is one small correction: M. Williams published his classical paper on the stress field of a crack tip in 1950s. In 1939 (in your article), he was probably  in the elementary school.