Hi Mike,

cross section shrinkage is a geometrical effect,  the subsequent necking is a bifurcation of the homogeneous deformation. In our group we are usually remove the geometrical effects to capture the real material behavior. To accomplish this one may use the incompressibility assumption, which works well for many alloys and/or cross-section measurements for compressible materials like polymers.

The incompressibility assumption is the basis of the Hoff model of ductile rupture.

Dear Mike,

The problem of the reduction of effective cross-section is more complicated. In my experience, macroscopic necking (i.e. macroscopic plastic instability) occurs mainly with high strain rate creep experiments. Internal cavity growth and coalescence, which is responsible for the creep fracture, is similar process, that is microscopic plastic instability. The effect is the same: reduction of the effective cross-section. Thus the macroscopic necking is only one part of the problem, and for the low creep rates and long creep lives minor part. In my opinion, this is the reason why the necking is not extensively studied in creep.

Regards.

Luboš

I am currently working on a conversion between constant load and constant stress creep that is material independent.  As such I am looking at both cases side by side.  What I have noticed is that there is a difference between constant load and constant stress necking in prediction.  From what I observe the difference occurs at the tertiary part of the curve and possibly dictates when tertiary commences (i.e. the end of the minimum creep rate).

An experiment using s servo machine and Digital Imaging could answer the question.  A DIC could be used to measure necking and then feed into a program which changes the load as per the neck width.  By using a programmable servo your could perform the constant load constant stress tests with and without necking.

You raise and interesting question which I have also come across and would be interested in collaborating if you are looking to investigate yourself.

Dear Michael,

This is indeed a worth studying issue. I don’t know directly of similar tests to those of Link, Aktaa and Schinke, but this subject bothered me a few decades ago.

I’ve my own interpretation on it which is that the time for necking under constant creep load corresponds to an « instability time ti ». This instability time appears when the argument of the exponential in equation (1) hereunder reaches 1. In the case of constant creep stress, there would be no exponential in the equation and the instability time would be the time of transition from 2ary to 3ary creep (transition between visually linear curve and accelerating curve up to rupture). I never had the opportunity to test this approach (sorry for that !) otherwise than by fitting a lot of creep curves for different steels.

Thus theoretically I see the problem in a more general point of view of ageing systems, but which is according to me also applicable to creep as a particular case.

The approach has been published in the Belgian Journal of Engineers (in French) : Quelques réflexions autour de la courbe de fluage. Journal des ingénieurs.(50) Avril 1992, pp. 44-47 . I join a copy of this article (« Creep Curve_JDI_1992.pdf »). A translation into English has been made in my book « Quantitative General Adaptation » (2012) : chapter 3.4 « Critical time and necking ».

Here is a shortened version of this theoretical approach  :

__________________________

« In the great majority of published creep test results the strain results from applying a constant tensile load to the test piece. The stress is then obtained by definition by dividing this constant load by the initial cross-sectional area of the creep sample (called the”applied nominal stress” σN). A power relationship is then observed between the parameter ”k” in the equation (1) of the creep strain « E(t) » in function of time :

E(t) = k.exp(α.t).tβ         (1)

We have :

k = K.σNb                      (2)

With ”K, b” depending on the temperature and on the material. Actually, the true stress”σ = σ(t)”increases with the strain during the creep test (decrease of the section ”S = S (t)”), and one has following equations as long as the strain ”E = E (t)” remains homogeneous (L = L(t) : length of the sample) :

E(t) = ln(S0/S) = ln(L/L0) = ln(σ/σN)             (3)

with : the ”0” index for the initial situation. Thus, we have :

σ(t) = σN.exp(E(t))        (4)

The exponential factor in (4) can be interpreted to be due to the changes in the internal true stress. We shall therefore re-write equation (1) as follows (using equations (2) and (4)) :

E (t) = k.exp(α.t).tβ = K.σNb.exp(α.t).tβ = K.σb.tβ = K.σNb.exp(b.E(t)).tβ = k.exp(b.E(t)).tβ                            (5)

Comparing the two underlined items, we get :

α.t = b.E(t)                    (6)

At necking, any small increase of stress is balanced by a small decrease of section ”S”:

σ.δS + S.δσ = 0           (7)

We also know that :

δE =−δS/S                   (8)

Thus, combining equations (7) and (8) with the central part of equation (5) (i.e. E(t) = K.σb.tβ), we obtain (Ei = E (t) at necking):

Ei =1/b                         (9)

and, from equation (6) :

α =1/ti                         (10)

with: ti : time for necking. We see that, interpreting the time of start of instability as ”ti = 1/α” makes sense in ductile creep. If the creep tests were performed with the true stress held constant, the exponential factor of equation (1) would disappear and the acceleration of the strain up to failure would only exist after occurrence of necking at ”ti”. »

__________________________

When one fits creep curves for ductile steels using above approach, one finds that the rupture time tr comes about 10 % to 30 % after the instability time ti (see Fig. 2 of the enclosed article in French : Creep Curve_JDI_1992.pdf).

The scatter can be much bigger when the strain at rupture (here μr) is compared to the strain at instability time (here μi : « critical creep strain ») as shown on joined Fig. 17.

Finally I’d be interested to test this approach once more on the case by Link, Aktaa and Schinke. Is there any available PDF version of their article to be added to this thread? I know that there is always the possibility to request a copy of the article from the authors. But I never managed to get an answer that way till now ! Probably because the authors are too busy or overwhelmed by requests. So iIf you have a copy of the article that you would (b allowed to put on RG, it would be welcome !

Please read :

A power relationship is then observed between the parameter ”k” in the equation (1) - of the creep strain « E(t) » in function of time - and « σN » :

Dear Konstantin, Luboš, Veronica and Guibert

Thank you very much for your excellent answers.  I will get in touch with you individually to discuss specific items with you.   

Nevertheless, I thought I should explain my specific interests here:  

1. Design codes use constant load creep rupture data to define the allowable stress in the creep region.  If the life of a constant load creep rupture test is significantly shorter than that under true stress then the design codes could be too conservative.  

2. For my research work I like to calculate true stress and true strain for creep and tensile tests and I usually use the Reduction of Area to define the true local strain at failure ln(1/(1-RofA)), where RofA is in units of mm2/mm2.  (which assumes incompressibility).  Similarly, I try to make the most of the deformation data (which is measured as average homogeneous strain) to estimate the true strains in the neck.  In a tensile test it is possible to estimate the onset of necking from either work hardening rates and/or the tensile strength.  But in a creep test there is no standardised method to estimate the start of necking.  Well Swansea University have a procedure of taking necking where the tertiary creep rate vs time accelerates i.e. d2e/dt2 increases.  However, we have some tests which we interrupted very late in life and did not observe any necking, from which I concluded that necking occurred very close to the final failure (last 1hour even), which is different from that in a tensile test, where necking occurs at T.S. or Rm after which quite a lot of local strain can occur (just compare the strain at the T.S.with the true local strain at failure).  

3. That observation got me thinking about the differences between necking in a tensile test and a creep test.  We know that in a tensile test necking is strongly influenced by work hardening, such that if a strain localisation occurs the higher local stress work hardens the material and the neck spreads and may even spread all the way to the end of the specimen at which point the strain becomes homogeneous once again (like in super plasticity).  Also we know that in tensile testing, machine stiffness affects necking.    In a very stiff test machine homogeneous strain can be achieved to very large strains.  However, in a compliant machine once the T.S. or Rm is reached and a neck forms the reduction of elastic strain in the loading bars causes follow-up in the neck and the specimen breaks.  Creep tests however, are not the same as this; the loading bars are often compliant and load is applied by dead weights (not an increasing displacement as in the tensile test) and therefore the follow-up which occurs in a creep machine will be different from a tensile machine.   This effect can be seen by comparing the RofA measured in a tensile test with that measured in a short term creep test.  For example the NIMS data is very useful in this regard as both sets of data are published.  Furthermore, it is not clear if work-hardening occurs in a creep test particularly if the material is softening (tertiary creep).  So we then need to consider how softening/hardening occurs differently in plasticity and creep.  This takes us back to a previous question from Konstantin, see link, in which he considers the effect of strain localisation in elevated temperature tensile tests. 

Complicated isn't it.  

Have a nice weekend.  

https://www.researchgate.net/post/Localized_softening_in_9-12Cr_tempered_martensitic_steels_at_high_temperature

I promised to add more:

4.  My main motivation for looking at the effects of necking is to improve creep damage models for predicting failure under strain controlled conditions such as creep-fatigue and relaxation cracking.  The two most common methods for calculating creep damage are (i)  life-fraction and (ii) ductility exhaustion. 

In both methods the damage model is usually defined using the results of constant load creep tests:

(i)  In life-fraction the time at rupture is used as a function of engineering stress and during relaxation the creep damage is estimated by splitting the relaxation in small time increments and then summing the small increments of time divided by the rupture life as a function of the relevant relaxed stress during that time increment.    

The issue to be considered is that the time at rupture is defined by constant load creep tests in which engineering stress is used as the explanatory variable.  Of course we know that the true stress increases with strain and therefore the engineering stress is the lowest possible estimate of stress.  On the other hand in relaxation, creep-fatigue or relaxation cracking the total strains are always small and therefore the true stress and engineering stress are similar (ignoring thermal expansion to make things simple).  Carrying on the thought experiment if the time at rupture for an engineering stress of 100MPa is 10000h then a constant true stress test with the same 10000h life would be tested at a true stress of >100MPa.  Therefore if we use constant load creep data to calculate creep damage for creep-fatigue or relaxation cracking we would expect to overestimate the creep damage.   This sounds good; nice safe estimates of creep-fatigue damage.  But life-fraction does not generally overestimate creep damage! after all if it did ASME NH would not use bi-linear interaction diagrams!  

Take it further! Turn it around and it means that if we did things properly and used true stress creep tests to estimate creep damage and then we would underestimate creep damage under creep-fatigue.  Does this not tell us something about life-fraction?  Would it not suggest that life-fraction might be generally non-conservative and therefore not a good method?  Two wrongs don't make a right. 

I would like to understand the effect on necking on rupture life as I think it might help prove that life-fraction is a poor method to calculate creep damage! 

Why else do I want to understand the effect of necking on creep? 

Well the alternative to life-fraction is ductility exhaustion.  But it is a hard sell.  I think ductility exhaustion is better and have evidence to prove it but life-fraction is just so appealing and I need to really stamp hard on life-fraction to stop people from using it!  I also need to really explain ductility exhaustion to a doubting audience.  One common problem people have with ductility exhaustion is: What value of ductility should be used?  In a creep test there is (a) elongation at failure or (b) reduction of area.  But RofA is not strain it is mm2/mm2 so (c) convert it to true local strain ln(1/(1-RofA)).  Fine but look at the typical values A=30%, Z=80%,  ln(1/(1-Z))=161% so what is the ductility 30, 80 or 161%; it is a big range!  People start to question the approach.  Then there are other options such as (d) uniform strain.  Furthermore, what about the difference between plastic strain and creep strain?  What do you do with the initial plastic strain on loading, do you include it or exclude it?  What about necking is that plastic strain or creep strain?  Now you see where I am coming from.  How does necking affect elongation at failure ?  Also how consistent and reproducible is RofA or Z? 

I need to understand these things to make a good strong case for ductility exhaustion. 

Dear Michael,

I can follow you very well when you question the life fraction rule (in creep and low-cycle fatigue). Although it has been improved since the beginning (see the discussion you participated in in relationship with another RG question by Naveen Bhatt : https://www.researchgate.net/post/What_is_physical_interpretation_of_damage ),

there still seems to be problems matching the reality of creep-fatigue life exhaustion in power plants. I also understand you very well on the issue of exhaustion ductility. Both problems bothered me in the past as an engineer having to propose criteria for residual life assessment in the Belgian fossil fired power plants. Reading your comments, I’ve the feeling that it is still a bit of a « day-to-day navigation » (« de la navigation à vue » as we say in French). Although tools have much improved meanwhile (R5, …). But as you mention the problem is complex (and there is scatter : data must be shown in logarithmic diagrams…..).

So, in addition to my former posts, I give you some more results of curve fittings I did in the year 1989. Maybe this would interest you and other followers of the question although I couldn’t further deepen the issue (mainly because of new professional duties, but also nobody was interested in this matter that time).

This is a presentation made in French at the : Tenth International Conference on Power Stations (25.09-29.09.1989), Liège (Belgium), Part III - General Reports and Discussions - Session 1.4 pp. 4-6 and 12-15.

It has the shape of an answer to 4 previously received questions (rule of the game for this conference). I join the scan of the presentation (because of the figures) : Life extension_10th Power Station_1989.pdf

and also a free translation into English of the answer to one of the questions (question 2) more related to the present thread :

Life extension_10th Power Station_1989_extract.pdf

So do just read the translation into English together with the figures of the original version in French (the presentation was made in French but the slides for the figures were made in English)

Please take care of the following remarks :

1)      As mentioned by Dr Luboš Kloc in a previous post, for higher rupture times the instability is due to cavity growth and coalescence not to necking. But the influence on the true stress is similar : reduction of the effective cross-section. And one can speak of instability time and instability strain for both phenomena.

2)      For clarity, in fig.6, the intermediate times on the abscissa correspond to 10, 100, 1000, … hours : thus the sequence is : 3, 10, 32, 100, 316, 1000, 3162, 10000 …. hours. For instance, for the 2 points corresponding to a rupture time around 100 hours, the instability times as shown on fig. 6 are 58 hours and 72 hours.

Finally here is a small summary of the enclosed extract :

An amount of 41 creep curves of 2.25Cr1Mo low-alloy steel have been checked. The instability and rupture times have been determined as well as the corresponding strains at instability and at rupture. Comparisons could be made allowing to define criteria. For instance, if the instability is found to settle at 275.000 hours, there could still remain minimum 25,000 hours before rupture. In average, the remaining life could even be more going up to 83,000 hours. Concerning the strain, it is a.o. found that after 100,000 hours the differences between the strain at instability and the strain at rupture becomes soon smaller.

https://www.researchgate.net/post/What_is_physical_interpretation_of_damage

Thank you Guibert

I have lots of useful directions for taking my research.