Arvind:

To the purist, the fracture toughness can always be described as being NOT a material property!  However, to an engineer, one must make engineering approximations that can be considered to be pertinent or relevant to real-life situations. I, for one, would rather seek approximate solutions to real problems than exact solutions to irrelevant ones!

If a large enough sample is used to measure the plane-strain fracture toughness value under small-scale yielding conditions, the resulting critical stress intensity is (as near as damn it) a material property, whether it's represents the lower shelf, upper shelf (or any other shelf for that matter!).  Of course, the purist will say that if a smaller sample was employed, the resulting measured toughness would be different - and they would be correct - but that is why ASTM and other standards institutions have described procedures to measure a reproducible value.  Why is this so troubling?  If one measures the yield stress of a nanoscale sample, e.g., a nanopillar, and compares the result to a macroscale sample, one will invariably also measure a higher strength in the smaller sample (due to mechanisms such as dislocation starvation) but few would suggest that the yield strength is not a material property.  My point here is that the fracture toughness is really no different from standard properties such as strength and ductility in that there can be size-dependent behavior; it is just that those issues can be more extreme in the measurement of toughness.

For for the usual reasons of statistical sampling, there will always be variations in measured fracture toughness, particularly if the size of the sample is not orders of magnitude larger than the variations in the microstructural length-scales.  This is compounded with a "valid" pre-cracked sample as the active volume of material subjected to the very high stresses and strains at the (nominally atomically-sharp) crack tip will be very small.  You could statistically sample many orders of magnitude more material using a blunt notch, but as most notches tend to sharpen by fatigue in real life structures, this is not necessarily a realistic solution.  

One thing that one can state though is that even for an ideal microstructure, with an ideally large enough specimen to satisfy small-scale yielding and plane strain, one can still see up to about 10% variation in the measured toughness for the valid specimen geometries, namely single-edge notched bend and tension samples, compact-tension samples and middle-cracked tension samples, based on the assumption that the Williams 1/r1/2 singular solution for the crack-tip stress and displacement fields is dominated by the first, K-dependent, term.  If the fracture processes all occurred exactly at the crack tip, this would not be a problem, but as one moves away from the crack tip, there will be a deviation from this situation that the first term dominates the actual stress/displacement field - for the constrained geometries, such as the compact-tension, this deviation is positive, for the middle-cracked tension geometry this deviation is negative.  ASTM-399 prescribes a 5% secant method to provisionally ascribe a value for the KIc, where 5% lower slope in the elastic loading line is equivalent to crack extension over 2% of the remaining uncracked ligament (this represents an operational definition for the critical stress intensity at crack initiation). However, at this distance from the crack tip, the "error" (for want of a better word) between the actual stress and that given by the first term of the singularity solution, i.e., by the K-field, will differ between the compact-tension and the middle-cracked tensions samples by about 10%.  Thus, if one uses standard geometries and satisfies all ASTM size-requirements, one still could see a 10% variation in the toughness.  Add to this the usual statistical sampling of a variable microstructure and one can see how somewhat larger variations in the measured fracture toughness values are possible.

ROR

Dear Arvind, 

The scatter of measured values in fracture toughness is controlled by the distance of cleavage initiation site from blunted fatigue crack tip. This value can reach 360 percent for normalized C-Mn steel. At higher percentage this scatter is mainly due to the variation of cleavage initiation location, especially relative to the location of maximum tensile stress.

Please look at the paper: On scattering of measured values of fracture

toughness parameters. by J.H. CHEN and G.Z.WANG. Published in International Journal of Fracture 94: 33–49. 

Dear Ravit,

Thank you very much for your reply. But I am looking for scatter in metal in upper shelf region which is not controlled by cleavage initiation sites.

Dear Arvind

This is a really good question.  Yes all the data on engineering steels shows that the scatter on the upper shelf is very large.  

Of course this tells us something very important and very challenging, it tells us that upper shelf fracture toughness is NOT A MATERIAL PROPERTY!!!   Well what is it then?  It is a useful engineering property which can be (approximately) scaled from small laboratory tests up to real full sized components.  Ok I will stop ranting and address you question:   

There are lots and lots of data around (most hidden away in industrial reports and not published in the academic literature). However, since engineering calculations usually demand lower bound values people often don't bother defining the mean property (they just describe a bound to the data), which makes it difficult to define the difference between the mean and the lower bound (1/2 of the scatter).  I will look around and see what I can find.  Sorry for not being helpful.  

Nevertheless, I will not be surprised if the answerer is close to ~100% of the mean!!!!

Dear Michael,

Thanks for your reply. I have carried out 20 fracture tests of Zr-2.5Nb pressure tube at room temperature and scatter was about +/- 30 % (95 % confidence band).

Arvind:

To the purist, the fracture toughness can always be described as being NOT a material property!  However, to an engineer, one must make engineering approximations that can be considered to be pertinent or relevant to real-life situations. I, for one, would rather seek approximate solutions to real problems than exact solutions to irrelevant ones!

If a large enough sample is used to measure the plane-strain fracture toughness value under small-scale yielding conditions, the resulting critical stress intensity is (as near as damn it) a material property, whether it's represents the lower shelf, upper shelf (or any other shelf for that matter!).  Of course, the purist will say that if a smaller sample was employed, the resulting measured toughness would be different - and they would be correct - but that is why ASTM and other standards institutions have described procedures to measure a reproducible value.  Why is this so troubling?  If one measures the yield stress of a nanoscale sample, e.g., a nanopillar, and compares the result to a macroscale sample, one will invariably also measure a higher strength in the smaller sample (due to mechanisms such as dislocation starvation) but few would suggest that the yield strength is not a material property.  My point here is that the fracture toughness is really no different from standard properties such as strength and ductility in that there can be size-dependent behavior; it is just that those issues can be more extreme in the measurement of toughness.

For for the usual reasons of statistical sampling, there will always be variations in measured fracture toughness, particularly if the size of the sample is not orders of magnitude larger than the variations in the microstructural length-scales.  This is compounded with a "valid" pre-cracked sample as the active volume of material subjected to the very high stresses and strains at the (nominally atomically-sharp) crack tip will be very small.  You could statistically sample many orders of magnitude more material using a blunt notch, but as most notches tend to sharpen by fatigue in real life structures, this is not necessarily a realistic solution.  

One thing that one can state though is that even for an ideal microstructure, with an ideally large enough specimen to satisfy small-scale yielding and plane strain, one can still see up to about 10% variation in the measured toughness for the valid specimen geometries, namely single-edge notched bend and tension samples, compact-tension samples and middle-cracked tension samples, based on the assumption that the Williams 1/r1/2 singular solution for the crack-tip stress and displacement fields is dominated by the first, K-dependent, term.  If the fracture processes all occurred exactly at the crack tip, this would not be a problem, but as one moves away from the crack tip, there will be a deviation from this situation that the first term dominates the actual stress/displacement field - for the constrained geometries, such as the compact-tension, this deviation is positive, for the middle-cracked tension geometry this deviation is negative.  ASTM-399 prescribes a 5% secant method to provisionally ascribe a value for the KIc, where 5% lower slope in the elastic loading line is equivalent to crack extension over 2% of the remaining uncracked ligament (this represents an operational definition for the critical stress intensity at crack initiation). However, at this distance from the crack tip, the "error" (for want of a better word) between the actual stress and that given by the first term of the singularity solution, i.e., by the K-field, will differ between the compact-tension and the middle-cracked tensions samples by about 10%.  Thus, if one uses standard geometries and satisfies all ASTM size-requirements, one still could see a 10% variation in the toughness.  Add to this the usual statistical sampling of a variable microstructure and one can see how somewhat larger variations in the measured fracture toughness values are possible.

ROR

Dear Prof. Ritchie Sir,

Thank you very much for your detailed reply.

As you said, in term of KIC scatter in fracture toughness is about 10 %.

Is same scatter found in case of JIC?

I have carried out 20 fracture tests of Zr-2.5Nb pressure tube material using 17 mm wide longitudinal curved CT specimen (crack growth direction is along the longitudinal axis of the pressure tube ) which is most common geometry for fracture testing of pressure tube. The curved CT specimens are cut directly from the pressure tube with pressure tube curvature. The curved CT means all requirement except the curvature. This curvature is going to give out of plane bending moment also. Still eta and gamma factors of CT specimen is being also used for curved CT also.

I want to know what will be effect of curvature of curved CT specimen on eta and gamma factors.

Arvind:

Your questions are difficult to answer precisely.  The inherent scatter in JIc test data would almost certainly exceed that of KIc fracture toughness measurements because nonlinear elastic fracture mechanics theory is far less robust than linear elastic fracture mechanics, although it is difficult to put a number on this.

As for your curved C(T) sample, again I don't know how much your geometry varies from the standard geometry to be able to tell whether the eta and gamma terms are changed.  If I were you, I would have someone do a full finite element analysis on your specimen geometry and determine these factors directly.

ROR

See standards: BS 7448, ASTM  E 1290 , ASTM 1823 ,ISO 12135

When measured in terms of KIC the fracture toughness fully depends on the employed standard for highly tough materials (precipitation hardened steels, beta annealed titanium alloys, etc). When measured according to ASTM E399, the fracture toughness of the material increases with increasing specimen size. This is exactly opposite to the common engineering sense. This is related to the fussy definition of "failure" in this standard (the 5% secant method). For a detailed investigation of this phenomenon, I recommend the research work "Critical Assessment of the Standard ASTM E399" from Kim Wallin.

Fracture toughness in terms of JIC is much more consistently defined and therefore you can expect substantially less scatter. Here you should be careful about the applied standard, too (BS, ASTM, ETM, etc). Each will yield another result. Therefore, you can compare only the values measured acc. to the same standard.

Volkan

Dear Volkan,

Thanks for your reply

If I am using same standard, how much scatter I will get?

Depends on the material and the processing route. For aerospace grade steel and titanium alloys (almost always vacuum melted, generally two or three times) I can very roughly say about 20% variation in Kic for the specimen of the same type, size and loading orientation would be normal. For Jic I expect less than this. 

If you have some hard low alloy steel that in air processed, the variation will be much higher. In such a case the fracture becomes a "weakest link" problem. That is, it depends on the probability of finding a suitable failure initiation point in the vicinity of the starter crack tip. Specimen thickness will have a dominant role as it directly changes the sampled volume near the crack front. Variation of toughness with temperature will also show a larger scatter. Even for the equal crack geometry/configuration you can see a big scatter band with the upper limit as big as 2-3 times of the lower one.

Dear Volkan,

Can you provide me some references?

I case of Zr-2.5Nb pressure tube (quadruple vacuum melted), I got +/-30 % scatter in JIC.

No idea about Zirconium

Dear Volkan,

Can you provide references from Titanium alloys?

Check open literature

Rob has provided the beautiful answer. It is a question of whether the crack samples the microstructure representative of the whole material - if not then the value - any property will not correctly represent the bulk material property - represents only the local material property. Additional constraints also arise due to surface conditions, and these effects become magnified if the surface to volume sampled is high as in plane stress or nano-size specimens.

Based on 20 fracture tests of Zr-2.5Nb pressure tube carried out using curved CT specimens at room temperature, I have found scatter of +/-30% in JQ values. But scatter in Jmax (J value corresponding to maximum load) is +/-16% only. It supports finding of Pehrson et al. that Jic calculated from Jmax gives lower scatter.

B.P. Pehrson and J.D. Landes, A direct estimate of J Ic from the load versus load-line displacement record, Fatigue & Fracture of Engineering Materials & Structures 30(2):73-86, 2007.

It means that about +/-14% scatter is due to error in crack length measurement. 

Dear Ritchie,

I want to know your view on this.

with regards

AKBIND

Akvind:

I think that I gave you my best answer on this on Nov. 26, 2016.  The systematic error in fracture toughness values will certainly be 10% or more (and definitely be larger for nonlinear elastic fracture mechanics measurements).  However, experimental error in measurement and more importantly variation in microstructure (remember that a nominally atomically sharp crack will "sample" only a tiny volume of material) could increase this scatter significantly, certainly to 20% or more variation in measured toughness values.

However, I certainly don't subscribe to the Pehrson et al. notion that scatter in toughness measurements will be reduced by measurements of Jmax as opposed to measurements at crack initiation. That may well be true, but measurements of the toughness at maximum load are rather dicey in terms of what they represent - that was one of the mistakes that Begley and Landes made in their initial two papers of JIc measurement.  It is generally OK for brittle fracture where fracture instability occurs at that point, but the condition of maximum load in a more ductile material showing subcritical crack extension will invariably not pertain to crack initiation and is often associated with the onset of failure of the shear ligaments. Such measurements do not represent a true, reproducible toughness.

ROR

Dear Prof. Ritchie,

In the paper mentioned above, Pehrson and Landes calculated initiation toughness, JQ from a relation between JQ and Jmax given by: JQ=√(W/50)*Jmax, they observed that for 1CT JQ=Jmax. On the basis of this relation, they showed that JQ showed less scatter than that calculated using ASTM standard. 

with regards

AKBIND

Dear Prof. Ritchie,

I have carried out fracture test of Zr-2.5Nb pressure tube material in axial (crack growth along axial direction of the tube) and transverse direction of the pressure tube using CT specimen. The CT specimens are cut after cold flattening of the pressure tube. The transverse CT specimens got buckled during fracture test whereas there was no buckling in axial CT specimens. The strength in transverse direction is about 20 % higher than that in axial direction. I want to if there is any relation between buckling load and strength for CT specimen for a given geometry.

With regards

AKBIND   

Akbind:

Buckling is an elastic phenomenon; it is not a function of strength, or any plastic properties, but rather of the elastic modulus, asymmetry of loading, and specific dimensions of your test sample.  Is there any change in elastic stiffness between the transverse and axial directions?

ROR