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elastic follow-up

W. Brocks's picture

This is the story of threefold failure, which doubtlessly is the subject of fracture mechanics, a story of failure in various regards, however. First, it comments on an article dealing with failure assessment, second it reports on the personal failure of the blogger to understand this article, and finally it bemoans the failure of a seminal idea.


Chasing for “prey”, I came upon a contribution on the assessment of “crack-like defects under combined primary and secondary loads”, namely


P.M. James: Re-derivation of plasticity interaction for combined loading under significant levels of elastic follow-up. Engineering Fracture Mechanics, Vol. 126, 2014, pp. 12–26,


and was intrigued by the expression “elastic follow-up”, of which I had never heard before. I started asking friends and colleagues who are engaged in fracture mechanics but they couldn’t help me. Collins Compact English Dictionary explains “follow-up” as “something done to reinforce an initial action” - which wasn’t really helpful, either. The author of the above contribution states that “elastic follow-up can be considered to occur in cases where the secondary load acting over a sufficiently large length scale such that localised relaxation (e.g. in the vicinity of a crack) does not diminish the influence of the remote stresses” - which left me stranded, still not knowing which “effect” is actually addressed, particularly because I do not have the slightest idea what “primary and secondary loads” are. Assuming (!) that the respective effect (which one?) “can be described by a single parameter” the author presents a quantitative measure, the “elastic follow-up factor”, Z, at least, which traces back to a preceding article of an internationally acknowledged expert of integrity assessment,


R.A. Ainsworth: Consideration of elastic follow-up in the treatment of combined primary and secondary stresses in fracture assessments. Engineering Fracture Mechanics, Vol. 96, 2012, pp. 558–569,


where I read: “when elastic follow-up is high this leads to secondary loads acting as primary”, which appeared as mystical as the explanation cited above, just inverting cause and effect. Obviously, nobody who has not internalised the concept of primary and secondary loads or stresses will ever be able to understand this “effect”.


In engineering mechanics, students are taught Cauchy’s stress principle of 1823, which was a breakthrough in the science of strength of materials enabling engineers to reduce various loading configurations to simple entities, viz. stresses, and to measure strength limits on simple test specimens. Actually, we measure deformations and relate them to stresses by constitutive laws. There is no room or need for primary and secondary stresses within in this framework, least of all for primary and secondary loads.


I scanned further literature on the problem finding numerous contributions. The whole world seemed to know what “elastic follow-up” is, except me. A contribution on “creep-fatigue tests  including elastic follow-up“ in the International Journal of Pressure Vessels and Piping of 2000 presents some uninspiring “illustration of follow-up behavior”. The essential hint resulted from the title of an article, “generalization of elastic follow-up model”, in Nuclear Engineering and Design of 1995: what, if this “effect” was not a physical phenomenon, a “behaviour”, but a model used in assessment codes? Finally, ITER Structural Design Criteria for In-Vessel Components (Appendix C) gave the enlightening explanation: “Neuber’s rule is applicable if the remote stress field away from the notch is elastic. If the remote stress-strain field itself undergoes plastic deformation, then a further correction is necessary, because the remote strain is greater than the elastically calculated strain.” It simply says that “elastic follow-up” is a correction term in an elastic analysis incorporating plasticity.


The code also gives a comprehensible definition of “primary and secondary stresses”, which appear to be model artefacts rather than having physical significance: “Consider a cylindrical bar of length L, cross-sectional area A, which is subjected to an axial load such that the extension would be uel if it behaved elastically. ... There are a number of ways of applying the specified load, the two simplest being a displacement u = εel.L and a force F = E A εel.. imposed. As long as the behaviour is linear elastic, a strain εel.. is effectively obtained for both loads. When the behaviour ceases to be linear elastic, the two loadings no longer cause the same strain. For the imposed displacement loading u, the real strain remains the same as the elastically calculated strain εel., which means that no correction is necessary and the elastically calculated stress = E u/L is a pure secondary stress. For the imposed force load, the real strain corresponds to the real stress = F/A on the stress-strain curve. This stress is a pure primary stress that can be seen to cause real strain  which is much higher than the elastically calculated εel.


This I can comprehend as it fits in my terminology and my view of the world of mechanics.


Now what is the conclusion resulting from this story?


·         If terminology creates insurmountable barriers of understanding even among people having similar scientific interests and background, namely fracture mechanics and structural integrity, we have to be concerned about the language we use in our publications.


·         If no distinction is made between models and physical phenomena, misunderstanding and misconception are programmed.


Finally, the present contribution marks the failure of the constitutive idea for the present blog. Its aim was to create a forum ofscientific exchange, realising that scientific achievements require time and chance for free, impartial and uncensored discussions among people. The European Structural Society (ESIS) and an international publisher of scientific journals appeared as an ideal combination for launching such a project. However, encouraging young scientists to frank discussions about their findings will work in a large-minded and democratic atmosphere, only, where they must not fear sanctions. Representatives of a society who themselves do not stand divergent opinions give a poor example. This is my last blog entry I shall be able to write.


Prof. Brocks,

I have enjoyed your blog posts and I'm sure others have too.  While we may not be able to respond in a substantial manner, please do not think that the reason is fear of sanctions.  It is more to do with lack of time or a deep understanding of the issues involved.

Please continue to post your findings on your iMechanica blog.



Emilio Martínez Pañeda's picture

Same here. I have learned a lot through your blog and I think that discussing published papers is a fantastic idea.

Please continue with it.

Best regards,

Emilio Martínez-Pañeda

Zhigang Suo's picture

Dear Professor Brocks:  

Thank you so much for your posts commenting on published papers.  They are entertaining and educational.  Many are provocative.    I echo Biswajit's comment:  I wish that I had followed up on the papers you commented on.  Often I said to myself, yes, I agree!  But I don't feel comfortable to comment without reading the original papers.  

It will be great if you can continue your posting.  You have got a following.  There is no need to attach your posts to a particular organization or a particular journal.  It is understandable that an organization or a journal wishes to preserve certain kind of unity.  But it is the individual posts that we read and value.    

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