Dr. Krishnan:

If the plane-strain fracture toughness KIc is measured properly, i.e., under small-scale yielding and plane-strain conditions, meaning that the plastic-zone size remains small (by at least an order of magnitude) compared to, respectively, the in-plane uncracked ligament size and out-of-plane specimen thickness, it will be the lower-bound fracture toughness - period! This value of KIc is as close as possible to being a material property, in that its value will be independent of crack size, specimen geometry and (provided plane-strain conditions are met) the out-of-plane thickness. Conversely, the plane-stress (or more correctly stated, the non-plane strain) fracture toughness, Kc, if measured at crack initiation, will be a function of crack size, geometry and thickness, and as such it is difficult to regard Kc as a material property. However, because it there will be no plane-strain (triaxial) constraint at the crack tip, this crack-tip region will be able to deform more readily, such that the Kc toughness will always be higher than the plane-strain value (with the provision that small-scale yielding conditions are still met). The Poisson's ratio effect on the modulus has a minimal effect, and it only influences the conversion of K values to an energy, such as the strain-energy release rate, G.

However, the actual values of KIc (and Kc for that matter) for a specific material are highly unlikely to be independent of temperature, as most, but not all, metallic materials display a reduction in toughness at lower temperatures, especially in bcc and hcp metals which show a ductile-to-brittle transition (DBTT). Maybe I didn't quite understand your question, but for this reason any comparison of plane-stress and plane-strain fracture toughness values cannot be made at different temperatures. If you mean that the comparison is made at a fixed temperature, the result will still always be that the plane-strain toughness is lower, either if your test temperature is above or below the DBTT. This is simply because the crack-tip field will experience far more constraint (triaxiality) in plane-strain, compared to non-plane-strain, conditions which acts to both enhance local crack-tip stresses and inhibit crack-tip plasticity, and this represents the perfect recipe for diminishing the fracture toughness.

ROR

I think you got me wrong. After dbtt, when a material has no plasticity, i think that the distinction beween plane stress and plane strain due to axiality will diminish. In that case the plane strain formula connecting the strain energy release rate , elastic modulus and poissons ratio will depend upon poissons ratio squared which has to be subtracted from one in the denominator under a square root. So even assuming that , the poissons ratio comes down from 0.3 to 0.25 after dbtt , the plane strain fracture toughness will be more than the plane stress fracture toughness after dbtt marginally asper these formulae. But why are we not worried about the plane stress poissons ratio which is not triaxial , before or after dbtt . ?

Dr. Krishnan:

I guess that I don't understand your point.

For a start, there is no such thing as purely elastic fracture in metallic materials, such that whether fracture is occurring above or below the DBTT, it will always be proceeded by some degree of plasticity, although naturally there is less involvement of plastic deformation for fracture below the DBTT.

Secondly, Poisson's ratio and the elastic modulus will only play an indirect role when the toughness is assessed, as per usual, in terms of the critical stress intensity, Kc or KIc, for fracture instability; fracture is controlled by the magnitude of the local stress and displacement fields at the crack tip, respectively for stress-controlled brittle fracture or (stress-state modified) strain-controlled ductile fracture, coupled with the effect of the statistical size, spacing and distribution of the relevant microstructural features, such as the grain size, carbide and/or inclusions.

So I reiterate my original conclusion that above or below the DBTT, the plane-strain toughness will be lower because the crack-tip field will experience far more constraint (triaxiality) in plane strain than in plane stress, which locally enhances the crack-tip stresses and inhibits crack-tip plasticity, both of which serve to promote fracture and lower the fracture toughness.

ROR

Are there any studies on the effect of DBTT and DBT on plane and plane stress fracture toughness evaluations ? How would poissons ratio and elastic modulus vary with DBT ? The surface would see more chilling than the core, so how would the reduced surface grain size and lack of plasticity at the surface affect FT ? Now if we map K1C vs. thickness would we get the same higher plane stress fracture toughness than a plane strain fracture toughness.? I doubt ! Would there still be a noticeable difference between axialities after DBT ? I dont know ! I curious to try out ! Thanks for your thoughts.

Dr. Krishnan:

With respect to studies on the effect of plane-strain vs. plane-stress toughness for metallic materials failing by brittle (transgranular cleavage) vs. ductile (microvoid coalescence) fracture, I am certain that there are, but you would have to hunt for them; I don't believe that there is a pertinent study that specifically examines that point. Maybe it would be interesting for you to do those experiments in order to put the issue to rest.

However, having said that, I am still certain that the plane-stress (or non-plane strain) fracture toughness would always be higher, irrespective of whether the fracture mode is nominally ductile or brittle, i.e., above or below the DBTT.

ROR

Dr Krishnan,

I fully agree with ROB comment (last). First one has to understand how to determine DBTT and KIC testing procedure. So that things will be more clear, do not mix both.

Let us come to DBTT, which depends on Temperature, loading rate and geometry of the sample. Now let us do the DBTT test and find the Transition Temperature, let say -50deg C with 68J. The sample type is three point bend type(Charpy-sample). The sample exposed to DBTT or do the test in transition temperature with varying thickness of the sample. It has to obey the ROB comment mentioned above. You will find the answer

Nani Babu

I think you have to read the entire conversations again. We are not discussing Charpy impact DBTT tests or K1d ( Dynamic fracture toughness). I was curious to know whether after DBTT, which one, Plane stress or plane Strain fracture toughness would be higher . There is no dynamic test involved here. We are not going to first of all determine the transition through a dynamic test. There is no comparison here as our interest is quasistatic fracture toughness. We go on measuring K1C at various temperatures and do a mapping. At low and embrittled temperatures we make a comparison with respect to thickness and plane stress and plane strain. That 's all. Now, when we plot k1c vs. thickness how will the picture be ? It has to be experimented. Thanks. There is no mix up.

Plain stress fracture toughness is always higher.

I wont put the cart before the ox. I would find out through experimentation and not judge. Thanks , who knows what is in store ?

It is great if you can find different from fracture mechanics principles.

with out doing how can we call them principles. Thanks , I think that at low temperatures there could be some surprises.

As per the fracture mechanics principles the plain stress fracture toughness must be high irrespective of anything. If you can prove that is different, I think all fracture mechanics scientist will see new directions in the fracture mechanics.

Iam only posing a question. There are very few papers on fracture mechanics of low and cryogenic temperatures. Looking forward to some papers on this.

I think we can always do Kjc analysis near DBTT or even below that, Always there are some samples those are not having any DBTT, DBTT does not always mean that you load your crack in a very slow loading rate as the crack feels it, it will instantaneously propagate. There is a term called cleavage initiation probability if we see crack initiation from probabilistic viewpoint of fracture mechanics. And this is not always 1 below DBTT. DBTT always correspond to a particular loading rate and it is dynamic in nature. But if we change the loading rate the Kjc changes from material to material. You can read a good report in

https://www.vtt.fi/inf/pdf/publications/1998/P367.pdf

I hope this will clear many concepts regarding this crack initiation under DBTT.

Padmanabhan Krishnan Nani Babu Matcha

Plain strain fracture toughness parameter (K1c) is an important material parameter and very useful for designers. It incorporates combination of stress level as well as crack length for safe design. For K1c testing, we can vary displacement/pulling rate and temperature. For a given operating condition, K1c can give you information like range of stress/ load level and crack sizes, where crack growth is stable (fail-safe) and crack growth is unstable.

However, DBTT and impact testing gives you idea about mechanical behavior of material only under sub-zero temp. and impact kind of loading, an operating condition applicable to limited components .

In short, though DBTT/ impact testing is important parameter under subzero temp/impact loading, K1c has much more wider area of application. Even at sub-zero temp, it gives you idea about the range of/ combination of applied stress and flaw size which can allowed for safe operation. This important aspect is missing in DBTT/ impact testing.

First of all, the statement that below DBT there is no plasticity is Wrong.

The crack initiation even in lower shelf, for conventional steels, is slip induced, so plasticity is not zero.

Secondly, the fracture specimen is a 3D entity, which shows a gradient of high constraint (plane strain) to low constraint (plane stress) from the centre of the specimen ahead crack tip to the surface.

The stresses at the crack tip can be described for SSY as,

Sij = K/(2pi*r)^2 *f(theta), and K = f(Sapplied, crack length)

lets assume that the material is homogenous plastic and we apply a stress = yield strength (YS),

Sij = {[YS(a)^0.5]/(2r)^2}*f(theta)

r = [(YS*a^0.5)/(4Sij)]*f(theta)

So far we did not consider any effect of plane stress or strain on the above equations,

for plane stress the stresses are going to be smaller in comparison to plane strain due to lower constraint in PS., therefore, the radius of plastic zone is always large for plane strain condition.

This conclusion is not based on Poison's ratio, or DBT or any other parameter. This is true for any condition.

Now just relate it intutively (or if you wish you may do some more analytics) that when plastic zone is large the fracture toughness is ought to be more, and plastic zone is larger for PS condition, at any temperature, below or above DBT.

The lower temperature experimentation can surprise only when YS shows an unconventional behaviour and affects the amount of plasticity.

Unless we touch the lower shelf, there is going to be a significant difference in plasticity in plane strain and plane stress condition, and when there is very limited plasticity, then it is a higher constraint regime, why would I apply a plane stress condition then? As @Ritchie said there cannot be a pure elastic fracture. Even ceramics show plasticity before initiation of fracture.

For some polymers, although, the plane stress condition is realized even at room temperature and in that case we cannot apply plane strain fracture toughness calculation.

Whatever be the material or condition, when more plastic deformation is allowed the load-displacement area will increase, the material will resist owing to easy defromation and so more energy will be consumed in breaking the material.

I think there are lot of anomalies in your perception. I am not able to understand the coherence behind it.