Dr. Pandey:

This is a very good question which I have wondered about for some time.  To my knowledge, there are no studies that I have seen that have specifically looked at the effect of stress triaxiality on fatigue.  Indeed, the effect is generally ignored.  For example, many damage accumulations models of fatigue-crack propagation, where elements are considered to fail by fatigue at the crack tip, simply use a Coffin-Manson model for the failure of the elements.  As Coffin-Manson results are based on uniaxial loading, whereas the conditions local to a crack tip would have varying degrees of triaxiality, these models assume (probably unintentionally) that stress triaxiality does not affect fatigue failure, which seems to me to be somewhat unlikely.  The effect of stress state on fatigue would definitely not be as intense as it is on cleavage fracture or microvoid coalescence, but surely one must presume that there is some form of an effect.

And so can I reiterate Dr. Pandey's question, does anyone have data or know of research which specifically addresses the effect of stress triaxiality on fatigue failure?

ROR 

Dear Vibhu and ROR

While I don't have the experience to answer this question, nevertheless, it is good fun to discuss interesting topics. 

Fatigue crack nucleation occurs at surfaces; either the outside surface of the specimen/component or internal surfaces at pores or inclusions.  Therefore, we might make the simplifying assumption that only biaxial states of stress affect fatigue crack nucleation. 

Now there are plenty of studies on specimens with V or U notches of different shapes/sizes, (often described by Kt).  Of course these specimens have biaxial (tension) states of stress at surface (and triaxial states of stress subsurface).  However, as they are only loaded axially, the maximum principal stress always dominates.  Also these specimens are very likely to be affected by plastic strains at the notch root.  Which means they might not address your question on high cycles fatigue, nevertheless, they are interesting to study.  There is also plenty of research on torsion-tension, which have biaxial (tension-compression) states of stress.  A small number of people have studied cruciform tests, again biaxial.  All in all I would suggest that there is plenty of data out there to study the effect of biaxial states of stress on fatigue crack nucleation, for a wide range of states of stress from tension-compression to tension-tension (don't forget we can include uniaxial tension and uniaxial compression in these studies). 

Please note I have deliberately ignored fatigue crack growth, which even in a smooth uniaxial specimens always occupies a very significant portion of the total endurance of the test.  I am lazy and don't want to address difficult topics, which require either fracture mechanics or even more complex analysis. 

I will finish with a question: Should we be discussing (1) the effect of triaxial states of stress at a microscopic/local scale (where cracks nucleate, at crack tips, or at a short distance ahead of a crack), or should we be discussing (2) multi-axial loading, where the loading is applied to a component/specimen in more than one direction? 

We often assume that (1) and (2) are the same, and indeed in some test specimens this assumption is reasonable, however, in complex components (1) and (2) will be somewhat different.  I am probably making this too complex.  However, as an industrial researcher my job is to take research and turn it into industrial application so I do end up thinking about these questions.   

Dr. Spindler:

Thank you for your input.  You are correct that there has been many studies focused on fatigue crack initiation that involve notches in tension and compression, biaxial cruciform testing, tension-torsion testing, etc., but few to my knowledge have isolated the specific effect of stress-state, e.g., characterized in terms of the ratio of hydrostatic to equivalent stress, on the resulting rates of crack initiation. I think that you agree with me - there must be an effect - but I have just not seen it articulated as such.

With respect to crack propagation, the only data that I have seen that addresses state-state effects on crack-growth rates comes from GE when they performed crack propagation testing on cord cracks under tension-tension loading in titanium and nickel-base alloys. I don't believe that these data were ever published but they used potential-drop techniques to monitoring the surface- and depth-wise growth of their nominally semi-elliptical surface cracks. The resulting growth rate vs. deltaK plots showed a relatively small but distinct difference between the surface-wise da/dN and much more constrained depth-wise dc/dN growth rates as a function of deltaK. This was many years ago, and I don't think that GE ever mechanistically analyzed the effect, but they definitely used the data in their life-prediction models. This, I think, is the experiment to do to determine the effect of stress-state on fatigue-crack propagation, but somewhat surprisingly I doubt if the effect will be that large.

ROR

Dear ROR

I agree fully with your points.  In my answer I was trying in my clumsy way to point out some simple observations: 

I was also trying to hint at the limitations which follow from 1. and 2.:

For crack nucleation it is only possible to study the effects of biaxial states of stress (because all crack nucleation occurs at surfaces*, either external or internal.)

{*I am open to challenge on this statement.  }

Unless, you have crack growth data (ideally for all tests) it will be problematic to separate the effect of (i) state of stress on crack nucleation and (ii) the effects of constraint on crack growth.  

It is actually a really complex problem, which is probably why it has not been solved at the microscopic/local scale, before.  

With regards to Vibhu Pandey's question; I can suggest that if anybody was to study this problem then they could consider:

• Collating as much published fatigue data as they can find (smooth bar, tests containing multiaxial states of stress, crack growth data, deformation data on same materials).  
• Looking at the totality of the data and deciding which sets are actually worth analysing further.   
• The notches will require FEA (elastic-plastic) to determine the state of stress at the notch root.  
• Relate the biaxial states of stress (and plasticity) at the notch root to cycles to crack nucleation. 
• Follow a fracture mechanics approach for the crack growth and take crack tip constraint into account.  
• Perform some key tests of your own to test your models.  
• Try to simplify your models and apply them to the whole dataset  to prove that the complete model set can predict fatigue endurance in all tests.  

I have considered deleting "With regards to....." onwards before pressing "Add your answer" , as by writing this down I now think my suggestions are unrealistic; they are simply too complex and there is not enough information on crack nucleation and crack growth for all of the fatigue endurance tests.  This means my suggestions are fruitless!  

Again: "It is actually a really complex problem, which is probably why it has not been solved ................before."  

Dear Mike, Dr. Ritchie and Dr. Pandey,

A very good discussion indeed, very educative and I really feel that we are fortunate to have a forum like this. I would like to share two old papers where attempts have been made to answer some of the questions if not all,

http://onlinelibrary.wiley.com/doi/10.1111/j.1460-2695.1979.tb01349.x/pdf

http://materialstechnology.asmedigitalcollection.asme.org/article.aspx?articleid=1422779

After going through these papers, I have certain doubts like

(i) I had a basic presumption that fatigue cracks undergo a multiaxial state of stress when they are considered as short cracks (i.e. short in the sense both physically and microstructurally may be shorter than the plastic zone ahead of it), but a notch creates triaxial state of stress which may help nucleating short cracks early, but what is the effect on the time period of converting it into a long crack, is it early conversion from short to long cracks or it will help the short crack to pursue its multiaxial state a little longer as their can be synergistic or anti synergistic effect of two local plastic zones, the first one is the hardened zone ahead of a blunt notch and another is the reversed plastic zone ahead of the short crack? Does these two effects coincide or they retard the crack growth?

(ii) The effective stress intensity factor inevitably contain a notch component but is n't it a residual stress component should be considered in calculations? As for strain hardening materials there is a notch root hardening and the short crack has to nucleate from a hardened state of material not from a parent material, so for accurate calculations we have to incorporate strain hardening laws also. What is the effect of this on fatigue crack threshold? is n't it a pre-strained threshold one should think of?

(iii) As from my understanding of plastic zone formation ahead of a cyclically loaded notch, there are two zones, reversed and monotonic plastic zone, if we try to pose the problem accurately I can think that crack nucleation should occur from a reversed plastic zone, what happens to crack growth rate when the crack length is greater than reversed plastic zone and smaller than monotonic plastic zone?

I am very much grateful to Dr. Pandey for addressing such an apparently simple but very complicated issue and I wish to get enlightened about these doubts.

Hi

I believe Dang-van fatigue criteria could address the tri-axial effect as it defines basically with the hydrostatic stress which has been created by the neighbouring grains among the grains at where the crack initiation (Stage 1) occurs. Hope this helps.

regards

Govind,

Dear Dr. Liu:

Definitely yes! Unlike fatigue where the role of stress triaxiality is somewhat uncertain, there is absolutely no doubt about its role in ductile fracture, specifically microvoid coalescence. Essentially if you are debonding particles or cracking them to form voids when then must grow until they impinge, or more likely form shear instabilities between them, to create a ductile fracture, the growth of the voids will occur far more readily under a triaxial stress-state as compared to a uniaxial one. That's why void growth and coalescence is a major issue after maximum load in a uniaxial tensile test as necking occurs and the presence of the neck sets up a triaxial stress-state. That's also why the value of the plane-strain ductility, measured in a circumferentially-notched tensile bar, is invariably a factor of three or more smaller than the traditional ductility measured with a smooth-bar specimen in a tensile test. The Rice and Tracey model for ductile fracture reflects this in that their estimate of the ductility of a material, based on the growth of voids until their impingement, which is an exponential function of the stress triaxiality, defined as the ratio of the hydrostatic to equivalent (Mises) stress.

So the bottom line is that yes - definitely - stress traixiality plays a very major role in ductility fracture, as it does in cleavage fracture too!

ROR