Will the crack shape and crack tip radius affect KⅠc value and R-Curve? and why ? Thank you for your answers.
The stress intensity is increasing, therefore, crack propagation is increasing
I think it is relevant only at the stage of crack initiation - then all the geometrical parameters of element are important. When the crack is at stable development or critical stage, it is difficult to control these parameters and their meaning is rather secondary.
The shape and radius of crack tip will affect the crack initiation and consequently the evaluiation of KIC wiil be affected. The different crack tip will lead to some error, but I think the influence is limited. For example, the ASTM E1820 recommends that the fracture toughness values δQ or JQ are ascribed at the intersection of a blunting line with the power law curve where the crack extension of 0.2mm is considered.
this question and also the added answers are quite tricky in my point of view. You have to carefully consider defintions - and by definition KIc is a material property and therefore independant of geometrical parameters.
Therefore, you have to make sure to measure KIc properly to be geometry independant. However, when comparing KI with KIc the calculation of KI will depend on geometry which is considered in the geometry factor Y(a/W) for different basic geometries.
Concerning the R curve, the crack tip mainly influences the Jc value. Here you have to distinguish the technical JI_0,2 and the physical JI_SZW value. The physical value clearly is more material dependant since it represents the ability of the material to blunt the crack tip. Therefore, this value can be used for different geometries, whereas the transferability of the technical initiation value is much more questionable.
Bottom line - the crack shape must be taken into account by the appropriate stress intensity solution for your crack configuration. With respect to crack-tip radius, fracture mechanics assumes an atomically sharp crack-tip radius. This will definitely affect KIc if it is not sharp, and as one previous response indicated, erroneously increase the toughness. However, once the crack is growing on the R-curve, the crack tip will form its own radius, and provided all the other criteria for K- or J-dominance are met, e.g., limited plasticity, etc. are met, this is no longer an issue.
An excellent paper on the effect of notch radius on stress intensity factors and R-curves is: Fett, T., & Munz, D. (2006). Influence of narrow starter notches on the initial crack growth resistance curve of ceramics. Archive of Applied Mechanics, 76(11-12), 667–679. http://doi.org/10.1007/s00419-006-0055-3
The radius definitely affects fracture toughness initiation values, as shown in many papers that you may easily find in literature. However, it seems to exist a critical radius (which depends on the material being analysed) below which the notch effect is negligible (i.e., crack and notch provide the same fracture resistance). This may be important in materials where it is not easy to generate fatigue pre-cracks and other techniques must be used to generate defects (e.g., razor blade in polymers).
This is generally not taken in account as both crack-tip K- and J-field singular solutions, i.e., the linear-elastic Williams and the nonlinear-elastic HRR solutions, assume a sharp crack tip. The predicted crack profile though will depend upon which singular solution is used; for example, the linear-elastic solution predicts a parabolic crack-tip shape. Whether this is reality or not, the stress intensity K or value of J calculated for a crack of a specific size and geometry subjected to specific tractions will presume the existence of a sharp crack tip. Strictly speaking, K solutions cannot be determined for a blunt crack tip, although there are some good, yet approximate, notch-tip stress and displacement solutions available, such as by Creager and Paris.
By combining the Creager-Paris stress distribution with the Theory of Critical Distances (e.g., Point Method or Line Method), you can obtain simple expressions for the apparent fracture toughness (fracture resistance developed by the material in notched conditions). Given that Creager-Paris distribution addresses U-Shaped notches, the above mentioned expressions works for this kind of notches.
The apparent fracture toughness expressions require knowing the notch radius and the material critical distance (usually referred to as L). There is quite a lot of literature about it. I recommend you "The Theory of Critical Distances: a new perspective in Fracture Mechanics (Elsevier)", by Prof. D.Taylor (Trinity College, Dublin).
That's simply one of many approximate aspects of fracture mechanics. In linear elastic fracture mechanics, one has to ascertain that the plastic-zone size is small compared to the in-plane dimensions of crack size, a, and uncracked ligament depth, b; if that's the case, then the crack tip is presumed to be sharp enough. Ditto for nonlinear elastic fracture mechanics where one has to make sure that the unloading/non-proportional loading zone at the crack tip is small compared to the in-plane dimensions.
As Drucker and Rice once eloquently stated, "Fracture mechanics is the judicious interpretation of crack tip fields"!
ROR
ASTM specifies requirements for how large the plastic zone can be in order to still satisfy the small-scale yielding condition - specifically that it is roughly an order of magnitude smaller than the crack size, a, and uncracked ligament width, b. That is the basis of the ASTM E399 criterion for KIc validity that a, b, B > 2.5 (KIc/Y)2, where Y is the yield strength, and B is the out-of-plane thickness which similarly sets the condition for a state of plane strain to exist. There is also a requirement that the fatigue pre-cracking Kmax value must be less than 2/3 of the KIc value over the last segment of precracking (you had better check that though), again to limit the size of the crack-tip plastic zone and hence the acuity of the precrack root.
Hence if the plastic zone is small enough to ignore, i.e., an order of magnitude small that the relevant dimensions of the sample, you are safe to use Linear Elastic Fracture Mechanics, the crack tip will be sharp enough, and fracture should ensue at the lower-bound plane-strain fracture toughness KIc value.
ROR
Dr. Hou:
Under plane-strain conditions (where the plastic-zone size ry, is small compared to the out-of plane thickness, B), there is invariably more constraint at a crack tip leading to a higher triaxial stress field there. Under small-scale yielding (SSY) conditions, where ry is also small compared to the in-plane dimensions of crack length, a, and uncracked-ligament depth, b, this will likely lead to an unstable fracture which in principle occurs at the onset of crack initiation - this is the basis of the measurement of the (lower-bound) plane-strain fracture toughness, KIc. So under these SSY and plane-strain conditions, unstable fracture will occur "nominally" at crack initiation (or at least by "pop-in" or after only limited crack extension) irrespective of the mode of fracture (although this is far more likely for brittle cleavage fracture) - that is to say that unstable fracture will be preceded by little or no subcritical crack extension.
Since plasticity in the wake of the crack is the principal factor promoting subcritical crack extension in metallic materials, this is why such ductile cracks (and cracks advancing by a nominally brittle mechanism like cleavage) can grow stably in plane strain under larger-scale yielding conditions.
Generally as short cracks may have less plasticity (and more likely less crack-tip shielding) in their wake, at the same applied (global) stress intensity the short crack may experience a higher local stress intensity actually experienced at the crack tip, as compared to a corresponding large crack subjected to the same global K - as such this will make the short crack easier to propagate. This is particularly in evidence for short fatigue cracks which often display far lower fatigue thresholds than equivalent large cracks - which is the basis of the well known "short crack effect" in fatigue failures (see the attached paper).
ROR
Article The Propagation of Short Fatigue Cracks
Dr.Robert :
1. Under plane-strain and small-scale yielding conditions,it is more likely brittle cleavage fracture,even for high fracture toughness steel. But brittle cleavage fracture means low toughness,is there some contradictory?
2.in practice, plane-strain and small-scale yielding conditions can not satisfy. thus, the KIC/JIC value can’t be used for fracture criterion.
Dr. Hou:
To answer each of your two questions:
1. Unfortunately one cannot easily make such global comparisons. The conditions of plane strain and small-scale yielding are really independent of the fracture mechanism, although we know that such constraint really promotes cleavage fracture. One can only generalize that brittle cleavage fracture is associated with relatively limited dislocation activity whereas ductile (microvoid coalescence) fracture, by necessity, requires far more plasticity - for that reason, they generally (but not alwayds) result in far higher toughnesses. There is no contradiction here. For example in a face-centered cubic (fcc) metal, such as austenitic stainless steels or aluminum alloys, there is no ductile-brittle transition to speak of, as cleavage fracture is extremely rare (due to the high symmetry of the fcc structure). Accordingly one still see a microvoid coalescence fracture at low temperatures under fully plane strain small-scale yielding conditions. Part of the problem here is the meaning of the words "ductile" and "brittle" - in the fundamental case, "brittle" means fracture by breaking a bond at the crack tip whereas "ductile" fracture means emitting a dislocation from the tip first. However, in real materials, plasticity is always involved to some degree. What this means is that the fracture of an aluminum alloy at very low temperatures may be "brittle" in so much that it fractures unstably at a low KIc value yet mechanistically the fracture mode will still be microvoid coalescence, which everyone refers to as "ductile" fracture. For real materials then, these fundamental definitions become blurred. I prefer to say that brittle fracture is fracture that occurs with limiting plasticity, on the scale of observation, whereas ductile fracture occurs with more plasticity, again on the scale of observation.
2. I don't quite understand your second question. The simple issue is that if small-scale yielding conditions apply, i.e., the plastic zone is small compared to the in-plane dimensions of crack length, a, and uncracked ligament depth, b, the K-field is valid. In plane strain, one can then measure a lower-bound KIc value, which is independent of crack size, geometry and thickness (provided it's large enough to be in plane strain). If plane strain is not satisfied, one can still measure a Kc value but it will be dependent on thickness, geometry and crack size (although the latter can be dealt with by using crack-resistance curves, also called R-curves).
Where the plasticity is more extensive such that the small-scale yielding condition is not met, then one can utilize the so-called HRR singularity and define a crack-tip field which is now a function of J, a characterizing parameter, like K, but which is equivalent to the strain-energy release rate under linear elastic conditions. Crack-tip J-field validity can be assured (according to ASTM standards) provided the uncracked ligament b > 10J/sflow, where sflow is the mean of the yield and ultimate tensile stresses. Again if this condition is met, one can still measure Jc fracture toughness values in plane stress or plane strain, although they may be a function of crack size and geometry and thickness again (the latter issue being resolved by additionally satisfying plane strain conditions again). However, nonlinear elastic fracture mechanics is far more complex to deal with, and to gain a comprehensive evaluation of toughness under such non-small scale yielding (J-field dominated) cases, one really needs to measure full R-curves to achieve some assessment of both the crack-initiation and crack-growth toughnesses.
ROR
Dear Professor Ritchie:
Suppose the fracture toughness of A steel is larger than B,both under plane-strain and small-scale yielding conditions.Is fracture toughness of A is certainly larger than B under plane stress condition or other geometric condition?
What about ductile and brittle mixture fracture mode formation? How it form ,from the view of micro fracture process?
thank you very much.
Dr. Hou:
Your first question, whether a steel with a higher fracture toughness in plane strain would have a higher toughness in plane stress, is a fascinating one. One would have to expect that this is the case. However, I know of no specific studies of this issue and since the fracture mechanism could well change between plane stress and plane strain, it does not necessarily follow that it will be the case. My take on this is that if the fracture mechanism remains unchanged, i.e., transgranular cleavage or microvoid coalescence, in the two deformation states, that your steel A. would have a higher toughness than your steel B, irrespective of whether the deformation conditions are plane stress or plane strain. However, if the fracture mechanism changes, then I would think that it is still more likely than not that steel A would remain tougher than steel B.
Regarding your second question, a mechanism change during the process of fracture, from a ductile (e.g., microvoid coalescence) to a brittle (e.g., transgranular cleavage) fracture mode, is an interesting phenomenon. One sees this during the fracture of many reasonably tough low and high strength steels where a ductile (thumbnail) crack forms at the root of the notch, which with extension transitions into a brittle cleavage fracture, with shear lips accompanying both fracture modes at the edges of the fracture surface. The presumed explanation for this is that once the ductile thumbnail crack starts to grow and accelerate, the accompanying local strain rate at the tip of the advancing crack triggers the more brittle cleavage fracture. Although this theory, which I believe was first advanced by Cottrell, seems very plausible, it has never been proven, at least to my satisfaction.
ROR
Dear Professor Ritchie:
Fracture transition from brittle to ductile, the ductile area increase from zero to hundread percent. Is this fracture process and mechanism similar as you stated?
From KIc calculation formula ,as the crack length approach zero, the fracture stress will become very large. If fracture mechanism will not applicable,when the crack is small to a certain size?
Dr. Hou:
Essentially yes. The sequence of passing through the brittle-to-ductile transition temperature in many steels (tested in notched samples) can be envisioned as follows. In the lower-shelf region, steels will fail by 100% cleavage. As the temperature is raised, the reduction in yield strength means that it becomes increasingly more difficult to initiate a stress-controlled cleavage fracture, and so the crack will form at the notch in the form of a ductile "thumbnail" and advance by microvoid coalescence until it transitions to cleavage as the crack-tip strain rate is enhanced. With further increase in temperature, the extent of this ductile "thumbnail" will increase until eventually the entire fracture occurs via this ductile mode and no cleavage is seen.
With respect to your second point, you are absolutely correct. As the crack size becomes very small, eventually the failure will become immune to the presence of these cracks. In engineering structures, this is the regime where the failures will occur by plastic deformation rather than by fracture (as the fracture stress becomes too high) - these are generally limit-load failures and fracture mechanics becomes essentially inapplicable. The so-called R-6 fracture diagrams, developed in the U.K., address this very well. Nature uses this phenomenon very effectively as well. Bone contains a very brittle, and extremely low toughness, mineral called hydroxyapatite (HAP), along with low stiffness collagen. The HAP has a very high elastic modulus and this is required to give bone its stiffness. However, the HAP is very brittle and so Nature deposits it along the collagen fibrils as nanoscale precipitates. Despite their very low toughness (
Dr. Hou:
Defects are indeed a ubiquitous characteristic of all materials. However, with a fracture-mechanics evaluation of the fracture toughness, one puts in a worst-case defect - that of the fatigue pre-crack - and so the presence of the existing defects can simply be considered as a property of the material. In unnotched specimens, such as when one measures the strength of a ceramic, for example, the resulting strength is thus both a strong function of the material structure and the nature and distribution of the defects. It is for this reason that larger-sized (unnotched) specimens invariably will give lower strength values in such brittle materials because there is a greater statistical probability of finding a more severe defect.
The brittle fracture strength on an unnotched specimen is nominally the same as the cleavage fracture strength but in practice they are rarely the same, for the very reason stated above with respect to the size of the specimen and the statistical likelihood of finding a severe defect. Note that in contrast, the cleavage strength is generally a local fracture strength. I wrote a short paper about this years ago, which may be of interest to you concerning this point (T Lin, RO Ritchie, On the Effect of Sampling Volume on the Microscopic Cleavage Fracture Stress, Engin. Fract. Mech. 29, (1988) 697-703.
ROR
Dr. Hou:
You're correct - defects are sampled in the plastic zone and as they are part of the characteristics of the structure of the material, they will affect the fracture toughness. However, they won't affect the application of fracture mechanics the assess this toughness as the fracture mechanics methodology is based on the extension of a single dominant crack and that still will be provided by the fatigue pre-crack as the worst-case defect.
The bottom-line here is that defects in a material will influence the value of the fracture toughness of that material but they won't affect the validity of the use of fracture mechanics to make this evaluation.
ROR
Dear Professor Ritchie:
Different defects are sampled before fatigue pre-crack which constitute part of variation of fracture toughness,which is more severe of this effect between smooth and fatigue pre-crack sample?
Suppose a material is strong ,high strength but brittle. The surface defect is got rid of and the component is small and internal defect can be neglected. Can this material be used in static tension situation in order to utilize its high strength?
Dr. Hou:
It is difficult to generalize. The presence of defects in nominally brittle materials, presumably meaning cracks here, would most like have a more pronounced effect on the mechanical properties, e.g., strength, measured from unnotched samples, as precracked samples would contain a worst-case defect, as I wrote to you previously.
With respect to the use of high strength, brittle materials where one claims to have removed surface defects and internal defects can be neglected, I would challenge this notion. If these materials are to be used for structural applications, one needs to determine the critical crack size from the fracture toughness value to estimate the largest defect (surface and bulk) that can be tolerated in the material under its worst case in-service stress, without failure. If you can live with this defect size, i.e., it is large enough to be seen by non-destructive evaluation prior to the component entering service, and you are sure that such a defect cannot be nucleated in-service by such mechanisms as fatigue, then go ahead. However, it you don't have these added requirements, your notion of using supposedly defect-free, high strength brittle materials for structural applications can be a dangerous one.
ROR
Dr. Hou:
Clearly defects, e.g., microcracks, in close proximity to the pre-crack can definitely affect the resulting fracture toughness measurement. However, to get reliable, statistically-reproducible measurements, your test specimens need to be large (i.e., an order of magnitude larger) than the length-scales of the nano/microstructure of the material and this includes the presence, size and distribution of any pre-existing defects. If this is the case, and you perform a statistically-relevant number of tests, the presence of a defect in close proximity to the fatigue pre-crack in one particular test specimen will not overly influence your evaluation of the fracture toughness of the material in question.
If the presence of a large volume fraction of such defects is ubiquitous though, e.g., in a heavily microcracked material, such a certain rocks or even bone, then the resulting effect on the toughness is more difficult to predict. A microcracked material will have a lower intrinsic toughness, yet the process of microcracking can represent a mechanism of inelasticity (akin to plasticity) and can further lead to extrinsic toughening by such mechanisms as uncracked-ligament bridging and in some materials by constrained microcrack toughening. Thus, a microcracked material can have a lower or higher toughness than the corresponding non-microcracked material, depending upon the extent and distribution of the defects. However, these latter mechanisms are mostly found in nominally brittle materials and would not likely be prominent in most metals and alloys.
ROR
Professor Ritchie:
why the presence of a defect in close proximity to the fatigue pre-crack particular test specimen will not overly influence your evaluation of the fracture toughness ,when specimens is enough large than the length-scales of the nano/microstructure of the material ?
stress intensity factor K represents the stress distribution ahead of crack. the stress distribution in bending is different from in unidirectional tension. Is the critical stress intensity factor K determined by bending same with which is by unidirectional tension for same material?
Dr. Hou:
The basis of fracture mechanics is that unique and autonomous stress and displacement fields are formed over relevant dimensions in the vicinity of the crack tip that characterize the local stresses and strains that are presumed to control fracture. This is the basis of the K-fields in linear elastic fracture mechanics (LEFM) based on the elastic constitutive laws and the J-fields in nonlinear elastic fracture mechanics based on nonlinear elastic constitutive laws. With LEFM, for example, to minimize the possibility of size- and geometry-dependent fracture toughness values, the size of the test specimen (in-plane dimensions) must be large (an order of magnitude or more larger) compared to the size of the K-field, which in turn should be an order of magnitude than the plastic-zone size, all of which should be an order of magnitude or more larger than the characteristic length-scales of the nano/microstructure and defect population. The latter requirement provides evidence that you are measuring a size-independent mechanical property; if this is not the case your resulting toughness measurements may be affected by microstructural entities or defects in close proximity to the pre-crack. This is just a statistical requirement that your measurements are not overly affected by geometry- and size-dependent effects.
These dimensional arguments are also relevant to your second question. Although the general (global) stress and displacement fields in bending and unidirectional tension are quite different, the local fields at a crack tip can be superimposed on these global fields. Provided the size-scales described above hold true, the local fields that control fracture at the crack tip will be the same in these two geometries - this is the Principle of Similitude which is fundamental to the application of fracture mechanics. However, if your plastic zone is not small enough, then the assumption of predominately linear elastic constitutive behavior becomes violated and the condition of small-scale yielding will not guarantee the existence of a unique K-field at the crack tip. By the same token, if your test specimen is not large enough compared to the extent of the K-field, then the influence of the global bending vs. tension stress fields will affect the local fields and you will no longer have unique crack-tip fields which will result in erroneous and geometry-dependent fracture toughness measurements.
ROR
Dr. Hou:
Strictly speaking, all the asymptotic crack-tip singular stress/displacement fields used in the description of the K and J fields in linear- and nonlinear-elastic fracture mechanics methodologies assume the presence of an atomically-sharp crack; if the crack is not atomically-sharp, in principle the singularity is lost (although the Creager and Paris model has addressed this point). Accordingly, ASTM has stated criteria for K- and J-field validity, meaning that the stress/displacement fields over the relevant size-scales from the crack tip (i.e., where the pertinent fracture events occurs) are (nominally) correct. In LEFM, this primary validity criteria is that of small-scale yielding, i.e., that the plastic-zone size is at least an order of smaller than the in-plane dimensions of crack size a and uncracked-ligament depth b. Thus, the ASTM E399 criterion of b > 2.5 (KIc/Y)2 can definitely be assumed to apply. Note that the corresponding out-of-plane thickness (plane strain) criterion with respect to the magnitude of B is not relevant to this argument.
The bottomline here is that if you satisfy the respective criteria for the existence of a K- or J-field at the crack tip, i.e., respectively, that b > 2.5 (KIc/Y)2 for K or b > 10 (JIc/Yflow)2 for J, then you're OK.
ROR
Dr. Hou:
Can crack length influence fracture mode? Well yes and no! The mere length of the crack shouldn't make a difference but of course, if the stress intensity is increased by a longer crack length, thus could trigger alternative modes of fracture. For example, fatigue cracks can grow at stress intensities below the fracture toughness KIc; for conditions of constant stress, for example, as the cracks grow longer the stress intensity at their crack tip will increase until KIc is reached and overload fracture, e.g., cleavage or microvoid coalescence, will ensue. This is one rather obvious example where the crack length will change the fracture mode from fatigue to an overload fracture.
Another example, that I believe that I mentioned to you earlier, is when cracks can initiate under quasi-static loading by a microvoid coalescence fracture, for example in steels where a ductile thumbnail forms at the tip of a notch or precrack; however, as the ductile crack extends subcritically it can transition to an unstable cleavage overload fracture. This is thought to be associated with the increasing strain rate developed at the tip of the extending ductile crack, which then promotes conditions for the onset of cleavage fracture, but I have never been completely convinced of this particular theory.
ROR
Dr. Hou:
Further to what I described in my previous response with respect the the effect of crack length on the mode of fracture, I don't believe that your assertion is strictly correct - that a ductile material will fracture by cleavage simply because the initial crack size is over a certain value. A lower temperature or a higher strain rate could certainly induce cleavage fracture in a nominally ductile material, such as a steel or any material that displays a ductile-to-brittle transition, but not merely an increase in crack length.
However, as I noted before, if your (nominally) ductile material is at a temperature or strain rate where it is already below this ductile-to-brittle transition, and the crack size is larger than the critical crack size for overload fracture, typically where the stress intensity exceeds the fracture toughness KIc, then the material will fail by a brittle mode such as cleavage.
ROR
Dear Professor Ritchie:
What’s the meaning of KA(apparent fracture toughness) in your article” Evaluation of Toughness in AISI 4340 Alloy Steel Austenitized at Low and High Temperatures”?
If the sample don’t meet the KIC sample's requirement, will the measured K value be larger than actual KIC?and why?
Dear Prof. Ritchie,
I have a question on J-integral.
It is said that J-integral is path independent. What is meant by it? Is it meant by path independency of J corresponding to a point on load displacement curve? Does it also include the path independency of the crack growth? Also is path independency of J-integral experimentally verified?
Thanks you
with regards
AKBIND
Dear Akbind:
The simplest proof of the path independence of the J-integral is best described in T.L. Anderson's textbook Fracture Mechanics: Fundamentals and Applications. Essentially if you take any closed path around a crack , you can define a value for the J-integral, simply defined as a line integral involving the strain energy density and a term representing the traction vector of the stresses normal to this path, which always comes to zero for any closed path. That was the original definition of J given by Rice in his 1967 paper. Hutchinson and Rice & Rosengren, in back-to-back papers in J. Mech. Phys. Solids the following year, were able to relate this to the existence of a unique stress/displacement field created at the crack tip for a nonlinear-elastic solid (the so-called HRR singular field) where J was identified as the amplitude (or characterizing parameter), akin to the role of K in the Williams' linear elastic singular field. Perhaps even more importantly, J was also identified as the rate of change in potential energy per unit increase in crack area for a nonlinear-elastic solid, i.e., equal to the strain energy release rate G under linear elastic conditions. This provided the link between J and K, between nonlinear and linear elastic fracture mechanics, which served as the initial basis for small-sample fracture mechanics. The uniqueness of J and its use as a potential parameter to describe the fracture toughness of a material which showed some degree of plasticity was later verified experimentally by Begley and Landes in 1972, who tested highly constrained deep-cracked bend SE(B) and compact-tension C(T) samples and compared their JIc measurements for those of the relatively unconstrained center-cracked sheet sample (now called a middle-cracked tension MC(T) sample). They got the same answer and J-based nonlinear-elastic fracture mechanics was born but they made a couple of mistakes and calculated the wrong toughness for the MC(T) sample!! All this was subsequently resolved some five years later by the numerical studies of Parks and McMeeking who showed that you can get the same JIc toughness for measurements on the constrained and unconstrained samples but the size requirement for J-field validity in the MC(T) sample had to be much much larger than for the SE(T) and C(T). This is one reason why nonlinear elastic fracture mechanics can be so much more complex as, unlike linear elastic fracture mechanics, the size-requirements for valid J measurements also differ with specimen geometry. In fact the extent of J-dominance at the crack tip in a middle-cracked tension sample is very restricted.
However, if you're interested, Anderson does the best job of any textbook of trying to explain it all in a clear and simple way. If you haven't already done so, I would strongly advise you to read this chapter.
ROR
Dear Prof. Ritchie,
Thank you very much for such a detailed reply. I know the path independency of J in mathematical term. Is this path independency of J applicable during fracture testing?
with regards
AKBIND
Dear Akbind:
I don't think that the path-independence of J is specifically considered during experimental fracture testing anymore other than to set the size-requirements for J-field dominance, and I suspect that ASTM doesn't even do that by this approach. However, as the path-independence breaks down when you're too close to the crack tip and encroach the unloading and non-proportional loading zones - this, of course, does speak to the size-requirements for J-dominance but I suspect that the current requirement of the uncracked-ligament b exceeding 10J/sigma flow, is as much set - at least the factor of 10 - by comparison to experiments to see when the JIc value starts to become size-dependent.
ROR
The stress intensity is increasing, therefore, crack propagation is increasing
Dear Professor Ritchie:
What’s the meaning of KA(apparent fracture toughness) in your article” Evaluation of Toughness in AISI 4340 Alloy Steel Austenitized at Low and High Temperatures”?
If the sample don’t meet the KIC sample's requirement, will the measured K value be larger than actual KIC?and why?
Dr. Hou:
The so-called "apparent" fracture toughness, KA, is largely a "fictitious" material parameter (another example is the ultimate tensile strength) which describes the critical stress intensity at fracture, i.e., nominally the fracture toughness, but which is measured ahead of the stress concentrator which is not a (nominally) atomically sharp crack, i.e., a fatigue crack, but is generally a rounded notch. It is thus fictitious as it is calculated assuming a sharp crack stress field which does not actually exist (akin to the UTS being calculated as the maximum load divided by the initial cross-sectional area, which not longer exists to due the reduction in area). Myself and others, such as Tetelman and co-workers, used this parameter to assess the effect of the root radius of notches on the required "stress intensity" for fracture, which is inflated in direct proportion to the square root of the radius of the notch. It provided a useful assessment of how blunt notches (as opposed to sharp cracks) can seriously and erroneously inflate the toughness (a mistake that still occurs today, particularly for the toughness assessment of brittle materials, such as ceramics, that are difficult to fatigue crack). I specifically used the approach over 40 years ago to explain why the (sharp-crack) fracture toughness KIc of quenched and tempered steels austenitized at increasingly high temperatures is increased, whereas as the rounded-notch Charpy Energy toughness is correspondingly decreased (which is a fascinating and relatively unusual effect). My old publication is reproduced below.
The approach of using the apparent toughness at rounded notches, KA, as a function of the notch root radius has been used much more recently to examine the toughness behavior of bulk-metallic glasses, but for a different reason. Here the rounded notches tend to promote excessive shear-band formation, which is the essence of plasticity in these amorphous alloys. Accordingly, the increase in apparent toughness with root radius is far larger in these materials.
ROR
Dear Professor Ritchie:
In actual situation, the crack geometry, tip radius and component dimension are different from KIC test condition , Is the KIC value still useful in fracture judgement?
What’s the mechanism of root radius of notches affect the fracture toughness? There seem have two different effects. it can reduce stress intensity as radius increase, on the other hand it increase sample area with radius increase.
Dr. Hou:
The power of fracture mechanics is that it is conservative - it aims to give you an indication of the worst-case toughness - the lower-bound - and so a nominally atomically-sharp crack is used in the measurement of the fracture toughness. This simulates many defects that are found in service components. If these fracture mechanics based toughness values are applied to predict when a blunt notch will fail, clearly it will be conservative - which of course is good in engineering design and failure prediction. But remember, in practice, notches, due to their stress concentration effect, are sites where fatigue cracks can initiate, and so many of these notches sharpen by fatigue in practice. Consequently, toughness parameters such as KIc are exceedingly valuable for the judgement of when fracture can occur. The form a quantitative basis for predicting worst-case conditions.
For your second question, you are correct on both counts. The effect of a notch will both lower the value of the local stresses and strain generated in the vicinity of the stress concentrator and extend the region over which is enhanced stress/strain is felt, i.e., increase the extent of the sampling value (sometimes called the process zone). The stresses in the vicinity of a sharp crack can, depending upon factors like the degree of strain hardening, can approach some 5 times the yield or flow strength (sigma/y) - higher if you believe in strain-gradient plasticity, but this occurs very close to the crack within two crack-tip opening displacements (~K2/sigma/y E), where E is Young's modulus. For a blunt notch, the maximum stresses are typically on the order of 3 times the yield or flow strength, but now this peaks close to the elastic-plastic interface which is much further away from the stress concentrator (~K2/sigma/y2). As E can be up to three orders of magnitude larger than sigma/y, this means that the blunt notch will samples a volume of material which is orders and orders of magnitude larger than a sharp crack, but with a much lower stress. And so sharp crack and blunt notch toughness measurements have their respective merits, but the sharp crack values are most used become of their conservative nature.
ROR
Dr. Hou:
There are indeed both global and local criteria for crack initiation and crack extension where the KIc or JIc requirements are not met, by which I assume that you mean the requirements are not met of K-dominance (small-scale yielding) or J-dominance, and/or of plane strain.
If the condition of plane strain is not met (but the K- or J-dominance criteria are), it is perfectly OK to use the critical values of K or J to describe the onset of fracture in plane stress (or generally more correctly) non-plane strain conditions. Such global criteria will invariably require R-curve analysis to define instability and/or subcritical crack extension, but as long as the K- or J-fields are valid, then the use of global fracture mechanics is fine, as both these singular fields - the linear-elastic Williams singularity and the nonlinear-elastic HRR singularity - are planar fields. You may well have to take care of other variables though, such as specimen size, as the critical K or J values for the initiation and growth of cracks will seldom be independent of sample thickness and geometry under non-plane strain conditions.
If, however, the K- or J-dominance criteria are not met, then it becomes difficult to use global fracture mechanics as neither of these singular fields will provide an accurate description of the crack-tip stress and displacement fields over the dimensions where the relevant fracture events occur. Here local criteria can be used. There are several: (i) the notion of a critical fracture stress being exceeded over a characteristic distance (related to grain size) ahead of the crack tip, as used in the RKR criterion for brittle (cleavage) fracture; (ii) the corresponding notion of a stress-state modified critical strain being exceeded over a different characteristic distance (related to the particle or inclusion spacing) ahead of the crack tip, as used in criteria for ductile (microvoid coalescence) fracture, and (iii) a criterion for ductile crack extension, e.g., along the R-curve, involving the attainment of a critical crack-tip displacement at some characteristic dimension (again related to the particle spacing) but behind the crack tip. These are all described in the attached article of mine from the mid 1980s (Ritchie & Thompson, On macroscopic and microscopic analyses for crack initiation and crack growth toughness in ductile alloys, Metall. Trans. A, (1985) 16A, 233-248.
ROR
Article On Macroscopic and Microscopic Analyses for Crack Initiation...
Dr. Hou:
This is an important question. As you know, if you satisfy small-scale yielding (i.e., a valid K field) and plane-strain conditions, the resulting fracture toughness, KIc, will be independent of (i) crack size, (ii) sample/component geometry and (iii) sample/component thickness (provided it remains in plane strain). It will be essentially a geometry- and size-independent material property.
However, this is no longer the case for non-plane strain conditions, where the resulting stress intensity at crack instability with be a function of crack size, sample geometry and sample thickness.
How then can you handle this? Well, the effect of crack size can be dealt with by measuring the crack-resistance curve (R-curve) and determining the point of tangency at crack instability (allowing for either load- or displacement control conditions); the critical K at crack instability will no longer be independent of crack size but the R-curve itself is considered to be a material property (for the specific sample geometry and thickness studied).
What about sample/component thickness (i.e., the through-thickness dimension)? Well if plane-strain conditions are not met, you really need to test a sample with the same thickness as your actual structure. This is the most reliable course of action.
What about sample/component geometry? Again there are no reliable theoretical ways to handle this other than to test a sample with the same geometry as your structure. Of course, in many cases this may not the possible and here one probably show take a conservative approach and use a three- or four-point bend (or compact-tension) test-piece sample to measure your R-curve as this will give you the most constraint and hence a lower-bound (worst-case) R-curve.
If possible though, it is always wise to test a sample with a size and geometry as close as possible as your actual structure, but if this is impractical, it is best to take an engineering conservative approach when it comes to predicting fracture.
ROR
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