As per ASTM -1820, for all sample geomerty. J= Je+Jpl.

Akbind:

It is perfectly reasonable to partition the energy associated with fracture, i.e., the value of J, into elastic and plastic components, as per ASTM 1820. As was correctly stated in the Standard, the plastic component, Jpl, is calculated in terms of the plastic area under the load vs. load-line displacement curve, whereas the elastic component, Jel, is determined from the stress-intensity factor in terms of K2/E, where E is the appropriate Young's modulus (for plane stress or plane strain).  However, for many new and experimental materials (particularly biological materials), the modulus can be difficult to measure, and this can introduce errors into the calculation of Jel. Accordingly, in these instances, we often take the energy associated with total area under the load vs. load-line displacement curve, to encompass both the elastic and plastic components.  As this does not require an knowledge of the actual modulus, we believe that this approach represents a more accurate method for these cases.

ROR

Dear Dr. Ritchie,

Thanks for your details reply.

But calculation of elastic J from K will give incorrect value because any sample will never be in plane stress or plane strain condition. Due to this elastic J will be always less in case of thinner samples.

with regards

AKBIND

Akbind:

The J and K fields are planar fields - they do not speak at all to the question of the through-thickness stresses which dictate whether your sample is in plane stress or plane strain.  These are essentially completely separate issues. Although it will affect the critical values of J or K at fracture, the relationship between the crack-driving "forces" J and K is accordingly essentially independent of the plane stress vs. plane strain condition.

ROR

Rob, as alway, has provided the perfect definition for using the ASTM 1820 to calculate both Jel and Jpl. You'll also need a test method (i.e. load partial-unload method) that can provide you with detailed crack length and stress intensity factor information. There are certain limitations on the amount of unloading one can do in every loading/unloading cycle so that the compliance measurement is confident, it is worth checking! If you work with micro-scale sample, you will probably encounter the reversible plasticity effect which creates hysteresis loop upon reloading. However, this is not the scope of this discussion.  

Is J at fracture a sample or component size-dependent value ?

Dr. Hou:

Is the value of J at fracture a sample or component-size independent? For anyone who works on the fracture of real materials, this is an important question.

With linear-elastic fracture mechanics, there are size requirements on the sample dimensions to ensure a state of small-scale yielding exists at the crack tip, i.e., that the stress and displacement fields at the tip can be described by the linear elastic (Williams) 1/r1/2 singularity, i.e., in terms of the stress intensity K; specifically, the plastic-zone size (which violates this linear elastic constitutive behavior assumption) needs to be small enough to ignore - at least an order of magnitude smaller than the crack size, a, and uncracked ligament depth, b.

In the same manner, there are similar size-requirements for J-validity at the crack tip in more ductile materials, i.e., where the plastic zone size is not that small compared to the sample dimensions. Here, using nonlinear elastic fracture mechanics, the relevant crack-tip stress and displacement fields need to be described by the Hutchinson-Rice-Rosengren (HRR) singularity, which are characterized in terms of the so-called J-integral (this is the nonlinear elastic version of the strain-energy release rate G). As J is defined in terms of nonlinear elasticity, and not elastic-plastic, constitutive relationships, i.e., the material is not allowed to unload elastically (as is physically realistic) but rather is assumed to unload along the same path as it loaded, the constitutive relationships are only truly valid for monotonic, increasing and proportional loading, i.e., no unloading. As when the crack advances, the material must unload, there is a similar "zone of violation" at the crack tip where J does not describe the crack-tip fields, which is comparable to the extent of crack advance and the region where the stresses are non-proportional. This zone similarly must be small enough to ignore, specifically b > 10 J /sigma flow (where sigma flow is the mean of the yield and ultimate tensile stresses) but this J-based size-requirement is much less restrictive that the one for K; it is typically an order of magnitude smaller. Hence you can measure the fracture toughness in more ductile materials without having to use prohibitively large samples.

So if this criterion is met, yes the J at fracture should be sample size-independent, although for consistency in the toughness value, you will need to additionally satisfy that plane strain conditions exist, which will require the out-of-plane thickness B to also be larger than 10 J /sigma flow.

This is all good but the problem with this simple description of the J-approach is that because it is based on nonlinear elasticity (and not elastic-plastic deformation) and that we have ignored the so-called T-stress, which is the second term in the description of the stress fields, these size-requirements are not specimen-geometry independent, i.e., the ones that I have stated work fine for constrained geometries such as bending or compact-tension geometries, but they probably need to be much much larger for less constrained geometries like middle-cracked tension samples (center-cracked sheets). Also in ductile materials, there is invariably slow crack growth prior to unstable fracture, i.e., the crack-resistance or R-curve, and the magnitude and shape of this R-curve can be specimen-geometry dependent. So under these conditions, the theoretical foundations for the use of J to characterize the fracture toughness is much less robust, and the value of J that you measure at fracture may no longer be specimen-size or specimen geometry independent.

For more information on this, I would recommend Ted Anderson's textbook "Fracture Mechanics: Fundamentals and Applications".

ROR

Professor Ritchie:

For J and K , I have some confusions.

Even in the situation that J dominant, there is K dominant zone outside J zone. If fracture initiate in the K zone, can we say it's K dominant fracture?

Similarly, in the linear-elastic fracture mechanics, small plastic zone exsit near crack tip. Frature may initiate in this zone, thus it seems not K dominant fracture.

Dr. Hou:

You are absolutely correct. For the use of linear-elastic fracture mechanics, we assume that the Williams 1/r1/2 - based singular solution for the crack-tip stress and displacement fields, the amplitude of which is controlled by K, is dominant over a distance ahead of the tip where the active fracture events are occurring. Similarly for nonlinear elastic fracture mechanics, the J-based crack-tip fields are assumed to be relevant over a region where the controlling fracture events are active. If you examine what the critical fracture events are and the size-scales of these crack-tip fields, for the best part they are, but this is why that I regard Dan Drucker's and Jim Rice's quotation of this endeavor, that "fracture mechanics is the judicious interpretation of crack-tip fields", as the most honest and relevant description of the discipline.

ROR

Hi Robert O Ritchie and Hou Wang

I am not sure if you are following also the big debate about strain gradient plasticity, for which all this classical theory of plasticity doesn't seem sound.... Curiously, the "corrected" plastic tips seem to return to the Williams singularity, so the matter is extremely confusing. Also, notice that all this requires additional parameters (length scales for strain gradient), and it all sound very curve-fitting in the end...

Article Mode I crack tip fields: Strain gradient plasticity theory v...

regards

mike

An interesting question. Rob is quite right, both in his discussion of the ASTM Standard and that in that, for a material that behaves in a way that is consistent with the assumptions underpinning the HRR solution, under monotonic loads the stress and strain fields at the tip of a crack can be expressed as a function of J. However, for cyclic loading (even for a metal with valid LEFM conditions) crack growth is not a function of delta(J). Indeed, plotting da/dN v Delta(J) leads to anomalous plots. This anomaly vanishes if you plot da/dN versus Delta(sqrt(J)).

Thus it it is best to limit the use of J to static problems. If you ever have a fatigue problem where there is a significant plastic component then it is going to have a short life and the way forward is to redesign your structure.

Thank you, Mike and Rhys, for your comments.

Taking the easier one first, I couldn't agree more about the use of delta(J) to correlate to fatigue crack growth rates. Although Dowling, who first proposed this, saw some reasonable correlation with da/dN, it seems to me to be fundamentally incorrect to relate a parameter such as J, which is defined under nonlinear elastic (not elastic-plastic) conditions, i.e., under monotonic, increasing and proportional loading, to cyclic crack advance where you unload each cycle. However, I heard recently that the French Atomic Energy people were contemplating the use of delta(J) for the prediction of safe fatigue lifetimes in their nuclear reactors, and so as a correlation, the approach is apparently still "alive and well".

On the second point on the use of strain gradient plasticity to define crack-tip fields, this is a fascinating topic which has been percolating through the field for several years now. I appreciate the need to predict higher local crack-tip stresses to satisfy local fracture criteria, e.g., in the presence of hydrogen, and as much as I admire the work of Norman Fleck, it is difficult for me to believe that local stresses can reach as high as ~50 times the flow stress within 5 microns of the crack tip (as I saw in one of their presentations recently). As these size-scales (on the order of several microns) are quite large, surely this can be experimentally measured to put this issue "to rest"!

ROR

Hi Rob

The experimental data conclusively disproves the hypotgesis that da/dN can be related to delta(J). The evidence is so strong that it just cant be debated. Since I am primarily an aerospace person I have not followed developments with regard to large scale plasticity. That said, when I was following this in the 90's I found the debate to be very interesting, and I thought that Albert Kobayasi's experimental measurements had also put the topic to bed. I seem to recall that he had a paper on this at ICF in Sydney. That said, my memory could be playing games with me.

Rhys

I did use Delta J to correlated the fatigue crack growth rates many years ago.

In thick walled plant operating at High Temperatures the crack tip relaxation will be dominated by creep mechanisms. Historically the stress exponent ,n, developed from relatively short term round bar uniaxial testing has been used to describe this relaxation. There are several assumptions in making this calculation not least of which link to stress state creep effects. Is there need to revisit the traditional models of cyclic stress relaxation, sometimes called creep - fatigue?

This is a very important question.

Unfortunately, in a very large number of articles, a measure of the fracture toughness of a material mistakenly considers the area under the deformation diagram, which includes elastic energy.

A simple and correct analysis of the deformation diagram can be found, for example, in the article: OZAWA K, KATOH Y, NOZAWA T, et al. Evaluation of damage tolerance of advanced SiC/SiC composites after neutron irradiation [J]. IOP Conference Series: Materials Science and Engineering, 2011, 18: 162005. doi:10.1088/1757-899X/18/16/162005

A very interesting, bright and even funny example of an incorrect analysis of the tensile diagrams of two fibers and more parallel carbon fibers (a fiber bundle) can be found in the work: S. Kumari, S. Nithya, N. Padmavathi, N. Eswara Prasad, J. Subrahmanyam, "Tensile properties and fracture behaviour of carbon fibre filament materials", J. Mater. Sci. 45(2010) 192-200.

According to Arne Hillerborg from Lund University, the area under a fully-recorded and steady-stable load-CMOD-diagram is called specific fracture energy GF, which is a non-linear elastic-plastic fracture criterion and very popular in fracture mechanics of concrete.

See file attached.

Also, this is used for cohesive zone modelling (CZM). Wrongly, often GIc is used for this from DCB tests instead of GF.

For further questions you can contact me.

Prof..Ritchie has answered beautifully. Nothing more to add.