Fracture toughness is one of most important mechanical property for ceramic materials. What's the major different between normal toughness and fracture toughness? And how to conduct fracture toughness testing?
Fracture toughness is determined by making a CT specmen as per ASTM E399.Where precracking of specimen is done and then subsequently tested under plain strain till fracture.It is denoted by symbol K1c and measured in MPa root meter.
Where as impact toughness (IZOD and CHARPY) are measure of toughness of a material on impact testing machine.Impact toughness test is used to find impact toughness at room and subzero temperatures, along with finding ductile to brittle transition temperature of materials (DBTT)
Please refer ASTM A370 for impact testingand ASTM E399 for Fracture toughness.For basics refer Mechnical Metallurgy book by Dieter.
toughness is the qualitative parameter connected with plasticity of material (in Russian it is applied only to liquids), Fracture toughness - the exact parameter characterizing resistance of material to cuts depending on the direction of loading (K1c, K2c, K3c). Impact toughness - firmness of material to resist to blow.
Dr. Hu:
There is really no intrinsic difference between the terms "toughness" and "fracture toughness", other than the former is a generic term and the latter refers generally to the measurement of toughness using fracture mechanics methodologies.
Accordingly, whereas "toughness" refers to any method of assessing the stress, strain or energy to cause fracture, such as the area under the stress-strain curve, the Charpy V-notch energy which is the energy to break a V-notched specimen in three-point bending under impact, or the critical value of the stress intensity K or strain energy release rate G at fracture, the "fracture toughness" specifically refers to the latter two terms, which rely on the measurement of the conditions to cause fracture in the inevitable presence of a defect, i.e., a crack. This is the basis of fracture mechanics - fracture in the presence of defects - and the term "fracture toughness", as opposed to "toughness" specifically refers to the conditions, i.e., stress intensity or energy, for fracture in the presence of such defects!
For this reason, the fracture toughness provides both a reliable measurement of the toughness of a material, but which most importantly,
can be used predictably in design to assess the conditions for failure in a real structure.
ROR
Dear Dr. Hu,
Dr Ritchie replied very nicely.
Toughness in term of area under stress strain curve and Izod or Charpy impact energy are qualitative in nature whereas fracture toughness in term of K and J are quantitative in nature.
For fracture testing of metals we need LLD, load and crack length measurement simultaneously. And it conducted as per ASTM E-1820. For fracture testing of Ceramics, one have to follow ATM C-1421. You have to remember that pre-cracking which is a prerequisite for fracture testing, is very difficult in ceramics because samples may fracture during precracking itself. Sometimes it is recommended not to precrack the ceramic samples.
AKBIND
Very interesting discussion!
It is just necessary to consider that area under load-CMOD-curve needs to be fully recorded in oder to gain values of fracture energy G (which is totally different to strain energy release rate GIc etc.).
For that, the so called GF value means specific fracture energy indicating "fracture resistance" rather than fracture toughness.
This is an independend material parameter characterising local softening behaviour within the process zone.
Further information can be found by Elmar K. Tschegg or Hillerborg.
There are several standard types of toughness test that generate data for specific loading conditions and/or component design approaches. Two of the toughness properties that will be discussed in more detail are:
1) impact toughness,
2) fracture toughness.
3) Toughness can be determined by integrating the stress-strain curve. It is the energy of mechanical deformation per unit volume prior to fracture.
They are different terms.
Toughness is the ability of material (assumed to perfect, meaning no cracks) to absorb energy to fracture. It is measured by the area under stress strain curve to the point of fracture.
Fracture Toughness is the ability of material with pre-cracks to resist fracture by absorbing energy. It is determined by fracture tests.
Professor Robert O Ritchie ,
On a similar note, is fracture energy measurement needed alongside fracture toughness? Does the improvement of one necessarily means the other will improve?
Thank you!
Dear Anass:
Now that is a fascinating question to contemplate, whether energy measurements of toughness scale with K- or J-based characterizing parameter measurements?
As you known, there are various measures of toughness, i.e., the resistance to fracture, which can involve either unnotched samples, such as the area under the stress-strain curve, or "blunt" notched samples, such as the Charpy energy - these are both energy measurements - and those involving sharp-cracked specimens and a fracture mechanics analysis, which can be either energy-based, such as using G or J, or stress intensity based using K. To the first order, all these different measurements and parameters tend to scale pretty much directly. Essentially they give the same ranking of the toughness of various materials (although differences in the elastic modulus can change the ranking slightly).
The pertinent factor is not whether an energetic parameter, such as G, or a stress-intensity parameter such as K is used; actually the more important issue is the presence of a sharp crack, vs. a blunt notch, vs. an unnotched specimen. The issue involves the intensity of the stresses, which can be almost an order of magnitude higher ahead of a sharp crack, versus the extent (or volume) of the high stress field, which can be orders of magnitude larger ahead of a blunt notch compared to a sharp crack, and even larger in an unnotched specimen (this is the issue of the statistical sampling volume). Some materials, particularly certain polymers, can absorb lots of energy in an unnotched test, yet are very sensitive to the presence of cracks. Certain steels with coarse austenite grain sizes can have higher (sharp-crack) KIc toughness yet lower (blunt notch) Charpy toughness. This is not very common, but definitely there are certain instances where these different measures of the fracture toughness can all yield useful information. The interested reader may wish to glance through the attached file of an very old paper of mine.
ROR
Dear Dr. Ritchie:
Robert O Ritchie
When you were answering question, you said that:
"The issue involves the intensity of the stresses, which can be almost an order of magnitude higher ahead of a sharp crack, versus the extent (or volume) of the high stress field, which can be orders of magnitude larger ahead of a blunt notch compared to a sharp crack, and even larger in an unnotched specimen".
Did you mean that the stress intensity is most signifiant in unnotched specimen, and less significant in sharp crack specimen? If it is the case, why? Should the stress intensity is higher in sharp crack specimen?
Xuedong
Xuedong:
No - what I'm saying is that there are various ways to measure the toughness, primarily involving:
1. sharp crack fracture mechanics approach involving, for example, the stress intensity, e.g., the measurement of KIc,
2. a blunt notch approach, involving, for example, the measurement of the Charpy energy,
3. an unnotched specimen approach, involving, for example, the area under the stress-strain curve or a work of fracture measurement.
Although all three can involve an energy measurement, e.g., if G or J were used in the fracture mechanics approach, the question that was asked by Dr. Harmal was whether if you measured the fracture toughness using KIc or Kc, did you also need to additionally determine a fracture energy.
Maybe I wasn't too clear but what I was trying to say. My point is yes, it can the useful to have a second measure of toughness but, although all these measures of toughness should give you roughly the same ranking of the toughness of different materials, they are actually quite different and sometimes can give you different answers. However, it's nothing to do with whether an energetic or K measurement is made, it is associated with the nature (or lack thereof) of the stress concentrator.
The sharp-crack test involves very high local stresses - typically as high as 5 times the flow stress Y, but the extent of the high stress (active) zone is very small - these high stresses peak at around two CTODs from the crack tip, i.e., on the order of K2/YE, where E is the elastic modulus. This defines the sampling volume (sometimes termed the process zone) where the fracture events are likely to take place.
The blunt notch test involves lower local stresses, on the order of 3Y, but they tend to peak at the elastic-plastic interface, which is much farther from the crack tip, i.e., on the order of K2/Y2; as Y/E can be on the order of 1/100 or 1/1000. In terms of the sampling volume, which represents the region of highest stresses, this could be six orders of magnitude larger (by vol.).
The unnotched tensile test, for example, has no obvious stress concentrator and so nominally the stresses are even lower, on the order of Y, but they exist over the entire gauge length of the specimen, which represents an even larger sampling volume.
So what are the consequences of this? Let's suppose that you're examining a material with particles that can fracture, e.g., carbide particles in steels leading to cleavage fracture. There will be lots of small particles which take a high stress to fracture and much fewer large particles (the "weakest links") which take a much smaller stress to fracture. The consequences of the different toughness tests are that the sharp-crack tests, like KIc tests, can activate (i.e., fracture) the numerous small particles but may not even see a large particle as the sampling volume is so small. At the other extreme, the unnotched tensile tests represents a situation of much lower stresses but over an orders and orders larger volume of material. Now the large particles may be sampled and broken but the stresses will be too low to fracture the more numerous smaller particles. This scenario can lead to different rankings of materials in terms of their toughness depending upon the toughness measurement made, particularly in inhomogeneous materials. And so in terms of the original question, it is sometimes useful to supplement a sharp-crack assessment of the toughness with an additional blunt or unnotched test, particularly if you are evaluating inhomogeneous (e.g., cast) materials.
ROR
Tensile toughness (strength) is a measure of the maximum stress that a metal can support before starting to fracture. ... Fracture toughness is a measure of the energy required to fracture a material that contains a crack.
https://www.imetllc.com/training-article/strength-toughness/
Fracture toughness describes the ability of a material containing a crack to resist fracture. The following standards can be used to measure fracture toughness of metals: ASTM E399 Standard test method for linear-elastic plane-strain fracture toughness KIC of metallic materials.
https://www.sciencedirect.com/topics/engineering/fracture-toughness
KIc is defined as the plane strain fracture toughness. It is a measure of the resistance of a material to crack extension under predominantly linear-elastic conditions (i.e. low toughness conditions when there is little to no plastic deformation occurring at the crack tip).
https://www.twi-global.com/technical-knowledge/faqs/faq-what-is-kic
Shock toughness - the ability of a material to absorb mechanical energy during deformation and destruction under the influence of an shock load. The main difference between shock loading and tensile-compression or flexural tests is the much higher energy release rate.
Fracture toughness is a relative increase in tensile stresses at the mouth of a crack during its transition from a stable to an unstable growth stage. Fracture toughness is closely related to the strength of the material.
Dear Dr. Ritchie,
Thank you so much for the explanation!
Xuedong Robert O Ritchie
When assessing the potential for in-service fracture, standards providing guidance for assessment of performance typically advise that testing should be relevant to the specific component and cite factors such as geometry, loading rate etc to guide decisions on testing. However, experience suggests that many engineers seek to apply the results from the same type of test to a very broad range of components using analytical methods with assumptions for the judgement on how the component will perform. For Engineers involved in decisions regarding Structural Performance there would be a benefit of providing improved guidance on selection of most appropriate, or perhaps least inappropriate test methods, for obtaining component relevant fracture resistance data.
In material science, Toughness is generally the resistance to fracture.
Generally Toughness is the ability of material to absorb energy so that fracture will be delayed.
Fracture Toughness is the ability of material with indigenous cracks to resist fracture by absorbing energy. Also defined as the resistance to cracks propagation.
Very interesting discussion. I would like to ask why the integrated area under the tensile curve is never provided as a parameter in tensile tests (such as the yield point and UTS). I understand the issue that Robert O'Richie brought up, where the stresses are too low to fracture the small, hard particles. In that case, we would be measuring the toughness of the matrix only. Yet, I have never seen this parameter been used. What are the reasons for this? Could it be used as a secondary measure for toughness, together with a notched test (such as Charpy impact test)?
My research team efforts in the past 20 years concluded in the Persian curve, which provide an accurate tool for failure assessment of structures. There is no need to define toughness any more!
Toughness is the amount of stress intensity factor (SIF) that a sample can tolerate before failure. Close insight into the history of fracture mechanics, starting from Inglis going to Griffith then to westergaard and etc., shown that SIF is a man made parameter and is not a good material property! Note that by increasing defect in a sample the capacity of the sample decrease! But the SIF increase! so the SIF can not be used and is not a good parameter for failure assessment!
Detail for determination of SIF shown that it is defined in term of the energy release rate (G) which in turn is defined in terms of effective strain energy. Effective strain energy is not known so the G and the SIF are not clearly defined (Are man made parameters not related to the sample property).
Persian curve is the sole reliable solution for failure assessment of structures with defect!
Thank you Professor Robert O Ritchie for your valuable input, it has been very helpful
The next question that I have in mind is regarding hierarchical or multi scale composites (Bio inspired for instance)
Since these composites generally have toughening mechanisms in a variety of length scales, and following the information you have mentioned above, does it become a requirement to report data on fracture toughness for sharp-crack as well as blunt-crack/unnotched speciment?
If so, can each test relate to the mechanisms happening in the appropriate length scale? (Sharp crack relates to the nanoscale mechanisms and blunt relates to the microscale for example)
Is there a combination that would be representative of the composite's fracture in its entirety (since all these mechanisms work together)?
I'd like to note.
The concept of fracture toughness, crack, ets does not make sense for materials that are not prone to stress concentration - bio-inspired structural materials.
For bio-inspired structural materials only makes sense critical section area and fracture energy (area under the curve of deformation before fracture) of load- bearing structure element any level of hierarchy.
Dear Anass:
That is a fascinating question that you ask. I am a firm believer in a crack-based toughness measurement - be it based on a governing parameter such as the stress intensity K or an energetic (and governing) parameter such as J. Blunt notch toughness testing does sample a much larger volume of material, and this can be an advantage in a widely inhomogeneous structure, but the sharp crack test will generally be the worst-case.
Now if you wanted to sample a hierarchical multiple-scale structure as we see with many natural materials, where there may well be different toughening mechanisms originating at the different length-scales - bone is an excellent example of this - you could probe the different length-scales with different techniques, such as standard fracture mechanics test at the near macro-scale, using FIBed micro-cantilever beams at the micro-scale and maybe even indentation toughness measurements at lower scales (if your material is brittle enough), but personally I would use a sharp crack test at all the scales as your primary measurement.
Cheers
ROR
Professor Richie is right. Really, for classical homogeneous materials, it is better to start with a sharp crack.
But as the fracture energy of your materials increases, you will see that the sharpness of the notch affects the fracture energy of the specimen less and less.
In the end, you will receive a material whose strength and fracture energy will not depend on the presence of a crack/notch in the specimen. For this material, the concepts of "crack", "fracture toughness", etc. no longer make sense.
This will be the ideal structural material. Most likely, the structure of this material will be similar to the structures of biological structural materials.
Professor Robert O Ritchie I reiterate my appreciation for you valuable input
Professor Valery Borovik thank you for the example
Hi
Released energy density is defined in terms of the available energy density at the crack tip! in this case there is three sources of epistemic uncertainty and then unreliability. One: it is certain that all the available energy releases! Clearly part of it may change to heat and part of it may change to vibration! (these may be seen in fracture of rocks during earthquake).Two: stress defined as a ratio with the numerator as the stress intensity factor. then the Num and Denom of ratio may be changed without changing the stress, that is the stress intensity factor is a man made parameter and is not a property of the material. Three: according to fracture mechanics the stress at the crack tip is infinite, which is clearly wrong. As a result it is wise to forget the toughness and stick directly to the stress at the crack tip.
In material science, Toughness is generally the resistance to fracture.
Generally, Toughness is the ability of the material to absorb energy so that fracture will be delayed.
Fracture Toughness is the ability of material with indigenous cracks to resist fracture by absorbing energy. Also defined as the resistance to cracks propagation
Other than the units, there is really no difference in that they are both measures of the resistance to fracture or toughness. The difference is in how these measurements are made.
The fracture energy is a generic term though that describes any measurement of the energy to cause fracture. These include the area under the uniaxial stress-strain curve, the work of fracture which is the area under the load-displacement curve normalized by the area of fracture, the Charpy impact energy which is the energy to cause fracture due to the impact loading on a small V-notched bend sample, and the critical values at fracture of the fracture mechanics parameters Gc and Jc, as described below.
Fracture mechanics, which was largely developed starting in the 1960s, provided a more formalized and quantifiable measure of the toughness – now termed the fracture toughness – for samples containing a pre-existing sharp crack. (This latter point is a critical condition which is lost in measurements of the fracture energy using, for example, the area under the stress- strain curve – the worse-case resistance to fracture must be measured in the presence of a pre-existing sharp crack.) Two parallel approaches were developed. Under nominal conditions of linear elasticity, the fracture toughness can be measured as a critical value of (i) the so-called strain energy release rate G, which is defined as the rate of change in potential energy per unit increase in crack area in a linear-elastic solid, or (ii) the stress-intensity factor K which is a governing parameter which defines the amplitude of the local stress and strain fields at the crack tip over the dimensions where the fracture events are occurring – where K = Q sapp square root(pi a), where Q is a geometry factor of order unity, sapp is the globally applied stress and a is the crack size. These two parameters can be explicitly related, e.g., under mode I (tensile) loading, G = K2/E, where E is the elastic modulus appropriate to plane stress or plane strain.
Under nonlinear elastic conditions, the critical value at fracture of the so-called J-integral serves both purposes as (i) the rate of change in potential energy per unit increase in crack area in a nonlinear-elastic solid for fracture, and (ii) governing parameter under nonlinear-elastic conditions which defines the amplitude of the local stress and strain fields at the crack tip over the dimensions where the fracture events are occurring.
All the fracture-mechanics based fracture toughness parameters are related; as J = G under linear elastic conditions, at fracture Gc = Kc2/E, such that Kc can be back-calculated from a Jc measurement using Kc = (Jc/E)½. This is for mode I loading; under mixed-mode loading the G-J-K equivalence is a little more complicated.
The bottom line is that the term fracture energy refers to any measurement of toughness involving the energy involved in the fracture, whereas the fracture toughness refers specifically to the measurement of the critical values of Kc, Gc or Jc at fracture using the methodology of linear-elastic or nonlinear elastic fracture mechanics.
ROR
Really thanks a lot for the quick and detailed explanation. Kindly could you tell me what is the units for fracture toughness and fracture energy. Please correct me as I try to simplify it to my mind can I consider fracture toughness like stress and fracture energy as strength or this is wrong I mean that the difference between the fracture toughness and the fracture energy is like the difference between the stress and strength. According to your explanation; can I consider both the fracture toughness and the fracture energy as material properties.
For modeling fracture with XFEM and traction separation law with Abaqus I get confused about some terms what is the difference between the strain energy and damage dissipation energy, and which one of them can I consider it as fracture energy. Thanks in advance
Dr. Hanna:
The difference between the fracture toughness and the fracture energy is most definitely NOT like the difference between the stress and strength. Stress is a measure on the force per unit area; strength is the critical value of that stress to cause plastic deformation (in most materials). If you believe that my explanation led you to believe that the fracture toughness and the fracture energy can be considered as material properties, that is only partially correct. The problem is that the means of measurement can differ and so it is difficult to compare one set of data with another. That's why fracture mechanics was developed to provide a means to measure the toughness quantitatively using a standard methodology that can provide viable comparisons between different sets of experiments and which can be directly used in design.
With respect to units, if P is the force and L is length, the units of the fracture energy can be quite varied - the area under the stress-strain curve has units of P/L2, the units of the work of fracture are P/L, the units of the Charpy impact energy are P.L, and so on. In fracture mechanics terms, the units of the K-based fracture toughness (based on the stress intensity factor) are P/L3/2; the units of G conversely are P/L.
If you're really interested in understanding all this, may I humbly suggest reading my recent short book "Introduction to Fracture Mechanics" (by Ritchie & Liu, Elsevier, 2021), which can be obtained reasonably cheaply on Amazon or directly from Elsevier. (sorry for the advert!).
Thanks
ROR
Dear Dr. Robert,
Firstly thanks for your nice compliment as I am not a doctor but PhD student.
Secondly thanks for detailed explanation but really I get confused about the units as in some papers they mentioned fracture energy unit (J/m2) an in some other papers (N/mm). So which one is right but according to what you have said that it is P/L then the (N/mm) is the right one, is it. But for critical stress intensity factor KIC for concrete around 1.2 Mpa (m)^(1/2) which I don’t understand it what is the (m) and what is the units. Also which one of fracture toughness and the fracture energy can I consider as material properties, I assume fracture energy is a material property as it is required for modeling fracture by traction separation law, is it right.
It looks very interesting book and you are very welcome and I hope our university library will get it so I will be able to read it as I am looking forward for it.
Kindly could you tell me what is the difference between the strain energy and damage dissipation energy, and which one of them can I consider it as fracture energy. Thanks in advance
Regards,
John
John;
I don't mean to be unkind but we're going round in circles here. Please read my earlier messages. The fracture energy is not a material property unless it is measured as a " valid" GIc or JIc, where validity is achieved by satisfying certain size requirements for the material, specimen size and geometry. The fracture toughness KIc similarly is not a material property unless it similarly satisfies certain size requirement for the material and specimen size and geometry (specifically small-scale yielding and plane-strain conditions). Both methodologies are described in ASTM Standard 1820. None of the other fracture energy measurements that I mentioned are material properties, as they are dependent on the specimen size and how the test is performed.
The fracture toughness is not like the yield strength where you simply divide the load at yielding by the cross-sectional area of the sample. Far from it. It is a complex property to define and measure. I explained the units to you in my last answer, but in a nutshell, the units of KIc are stress times the square root of the crack size (originating first from the Griffith theory of 1921-22), simply because fracture does not simply depend upon the applied stress (like yielding) but also on the presence of pre-existing cracks of a specific size.
However, I cannot explain all the details to you in a few paragraphs on ResearchGate. If you need to proceed with this line of questioning, you will need a crash course in fracture mechanics. As I mentioned previously, my book "Introduction to Fracture Mechanics" (1st. ed. Elsevier, 2021) will provide that for you, or if you want all the "gory" details, read T. L. Anderson's book on "Fracture Mechanics - .Fundamentals and Applications" (4th ed., CRC Press, 2017).
Cheers
ROR
Second attempt: You rightfully point out that to be qualified as "fracture energy" certain test conditions must be met. Fracture energy- and its attendant plastic energy -are irreversible processes which aren't state variables. Meaning that they are path dependent which is why it isn't possible to assign an exact number and make accurate predictions about future behavior. Even if conditions define a standardized starting state and observed every part of your test sample it would't be to use all of your observations to predict future behavior? You and I both know that perhaps not today but hopefully not too far off, maybe tomorrow. I'd like to suggest something - although there's confusion on certain points (above) shouldn't there be more background information that furnishes a better understanding of the issues at hand? So far it looks as though Fracture Toughness interpretations based on engineering technology provide adequate understanding and therefore provide everything that has to be taught and learned. Apparently not.
If we talk about simple analogies, then perhaps this interpretation will be useful:
- stresses - characterize the stress state of a structural element, and strength is the ultimate stress that the material of a structural element can withstand before the fracture begins,
- stress intensity factor (SIF) - characterize the stress state at the crack tip in this element, and fracture toughness is the limiting SIF that the material can withstand until the crack begins to grow rapidly.
The energy spent on loading a structural member is characterized by the area under the diagram "load - displacement of the point of load application". This energy consists of two parts: elastic energy, which the structural element returns to the loading system during unloading and the energy irreversibly absorbed by the structural element during loading and fracture. The last energy can be considered the fracture energy of a given structural element.
Determination of the fracture energy of a material is quite simple only for brittle fracture: for brittle materials as such, under dynamic loading, at low temperatures, etc. Stress concentrators - notches - are also used to localize deformations.
In the general case, the determination of the fracture energy of a material is a rather nontrivial task. This is due to the fact that energy absorption during deformation occurs unevenly in the volume of the material of the structural element (specimen for mechanical tests).
Dear Dr. Robert,
Thanks for your reply and for sure when I say material properties I mean it has been tested according to the specification of tests like compressive strength of concrete the samples have to be in certain dimensions according to the test specifications, so both are material properties in case they satisfy the test requirements.
I see that KIc are stress times the square root of the crack size (m)^1/2 but what is the crack size I mean the initial length of pre-existing crack as the crack propagates with time so its length varies that is why I wonder which crack size shall I consider. Is the crack size means crack length or the crack area.
Ok I will try to find your book and the other mentioned book in our university library to read them. But kindly could you let me know can I model crack propagation based on Linear Elastic Fracture Mechanics (LEFM) or LEFM handle only the stationary cracks. As I see modeling crack propagation in computational fracture mechanics like XFEM but based only on cohesive zone model such as traction separation law. Thanks in advance
Regards,
John
Dear Dr. Robert,
What is the difference between fracture toughness KIc and Kss?
I tested several metals following ASTM E1820 standard and obtained the J-R curves. Interestingly, I found that the KIc values are close, but the Kss values have great difference. Should I say these metals have similar fracture resistance? What is the role of Kss play in the fracture toughness?
Regards,
Chengpeng
John:
The toughness parameters KIc and GIc are linear-elastic governing parameters which are defined at crack initiation and so the crack length in their calculation is specifically the length of the pre-existing crack from which unstable fracture ensues.
Basically most of fracture mechanics pertains to stationary cracks, meaning not necessarily dynamic but simply not moving. If you're interested in dynamic fracture mechanics, that's a whole new story, but for stable cracking, we talk about crack-resistance or R-curves where the driving force, e.g., K, G or J, is monitored as a function of stable crack extension. However, even for nonlinear-elastic fracture mechanics, the physics associated with trail of plasticity in the wake of the moving crack tip (which is the fundamental reason why the crack extension is stable) is not considered (as it would violate the assumptions of linear or nonlinear elastic constitutive behavior). There has been some attempts to consider non-stationary cracks in elastic-plastic materials, motivated by Jim Rice - this is known as the Rice, Drugan Sham analysis (see the attached description) - but this is still a relatively unexplored realm of fracture mechanics as there is no single governing parameter than can be used to characterize the moving crack-tip fields.
ROR
Dr. Huang:
The way that I view KIc and Kss (or JIc and Jss) is that the former define the crack-initiation toughness where the latter is one measure of the crack-growth toughness. Both are important, the same way for strength that the yield strength and the ultimate tensile strength are both important, as the latter is some measure of the effect of strain hardening. However, the measurement of the initiation toughness is quite rigorously defined by the ASTM Standard E1820 and so is relatively independent of such factors as specimen size and geometry, provided the K- or J-dominance criteria are met together with plane-strain conditions.
Crack-growth toughness is somewhat more problematic as once the crack starts to move, you have unloading at the crack tip due to its advance and the crack will leave a trail of plastic zone in its wake, both of which must be ignored in both linear- (LEFM) and nonlinear-elastic (NLEFM) fracture mechanics methodologies (specifically the size of the plastic zone in LEFM or the unloading/nonproportional loading zone in NLEFM must be small enough to ignore). Essentially this means that J-controlled stable crack growth is an oxymoron! For these reasons, unlike KIc or JIc, the crack-growth toughness cannot be as rigorously defined; the result of this is that its measurement can be more affected by specimen size and especially specimen geometry effects.
You can measure the crack-growth toughness by using the Kss or Jss values at the ASTM limit of crack extension for your particular specimen (as I do), or alternatively in terms of the initial slope of the R-curve (this is the basis of Paul Paris' tearing modulus) but both measurements can be somewhat compromised by specimen geometry and size effects (particularly geometry effects), and so one must expect some degree of variability in the resulting values.
ROR
Dear Dr. Robert,
Thanks a lot for your kind detailed explanation about the LEFM. But if I remember during our master studies we always heard that crack analysis is nonlinear analysis, so I am wondering why it is called Linear Elastic Fracture Mechanics (LEFM) is that means in this theory the crack is handled as linear elastic analysis without considering nonlinearity.
Is the nonlinear-elastic fracture mechanics is another name for elastic-plastic fracture mechanics, if not kindly let me know what is the difference between them.
Kindly could you let me know what is the crack size in KIcI mean is it means the length of pre-existing crack or the crack surface area (crack length * depth of the sample).
Also I found in a paper in order to obtain the fracture energy they have used this equation: G= KIc ^2/E is that right. As when I use this equation to calculate the fracture energy value in other papers it gives me a complete different value from what they have considered in their paper. Thanks in advance
Regards,
John
John:
You really need to look at a book or section of a book on fracture mechanics to answer to your questions - at least read the article that I attached to my last message - as I cannot keep fielding these questions.
Fracture mechanics was first developed and is most used in the form of linear-elastic fracture mechanics. This discipline was further developed circa 1970s and later to include the effect of plasticity which was modeled as nonlinear elasticity - hence the name nonlinear elastic fracture mechanics (this was done to approximate elastic-plastic conditions). In the strict analytical sense, elastic-plastic fracture mechanics hardly exists - except in the form that I talked about in my last answer as the Rice-Drugan-Sham analysis of nonstationary cracks.
The crack size in the expression for the stress-intensity factor K is generally the crack length (or half length of an internal crack) but it depends somewhat on what the crack configuration is and the mode of loading. You need to use the appropriate K solution for whatever geometry and crack configuration that you're considering.
The equivalence of G and K is indeed G = K2/E', where E' = E/(1-v2) in plane strain and = E (Young's modulus) in plane stress (v is Poisson's ratio), but this only applies under linear-elastic conditions for mode I (tensile) loading. The full expression is G = KI2/E' + KII2/E' + KIII2/2mu, where mu is the shear modulus, KI is the mode I stress intensity, KII is the stress intensity in mode II (in shear), and KIII is the stress intensity in mode III (in anti-plane shear).
There are definitive restrictions on how you measure these quantities under specific conditions. If you compute the fracture energy based on other measurements procedures, naturally you will get different answers. That's exactly what I was trying to tell you in my first answer to your initial question.
May I respectively ask that you please pick up an introductory book on fracture mechanics or read a chapter in any engineering text book that covers the topic before you ask further questions.
Thanks
ROR
Dear Dr Robert,
Thanks for your comprehensive explanation on the KIc and Kss, and I am very grateful and benefit greatly from your answer. I now know the difference between KIc (or JIc) and Kss (or Jss), but still have some confusions on the relationship between them.
For example, recently I tested two different types of steels. I found that the stress-strain curves of these two steels are nearly identical, so are the JIc values. However, the Jss value of one steel is two times that of the other. Furthermore, the room-temperature Charpy impact energy of the two steels are also identical. (The CT specimens for fracture toughness tests of the two steels have the same geometry and size, and the Jss value is measured at the ASTM limit of crack extension. I think the specimen size and geometry effects are limited in my case.)
So my question is that, what is the relationship between the following four toughness, (i) toughness defined by the area under stress-strain curve, (ii) crack-initiation toughness KIc (or JIc), (iii) crack-growth toughness Kss (or Jss) and (iv) Charpy impact toughness. Especially from my case, it seems that the toughness (i)(ii)(iv) have similar trend, but the toughness Kss (or Jss) are completely different.
Finally, what is the practical significance (or benefit) of high Kss (or Jss) for a material as structural application? When we choose a material as structural application, what are the role of KIc and Kss play, respectively, for preventing structure failure during service? Which one is more important?
Regards,
Chengpeng Huang
Dr. Huang:
With respect to your multiple questions, let me make the following points:
1. The area under the stress/strain curve involves a measurement of toughness in an unnotched sample. This clearly is not worst-case and cannot give much information about when a large real-life structure might fail due to the presence of a pre-existing defect. How would you propose to design such a large structure to be able to be damage-tolerant to any incipient defects based on the area under a stress-strain curve on a unnotched sample of a size of the order of your finger? You can't!
2. The crack-growth toughness is far more sensitive to microstructure than the crack-initiation toughness. This is an established fact, as described in the attached 1985 Metall. Trans. paper.
3. The Charpy toughness is different on multiple counts. Obviously it's measured at much higher strain rates but also important is the fact that the initial stress concentrator is not a crack, but a blunt notch. This means that it's not worst-case for any design (particularly as notches in real structures in practice invariably get sharpened by fatigue). Also the local stresses at the notch tip are significantly lower than ahead of a sharp crack, but interestingly extend over a much larger volume - meaning that the mechanisms of fracture, in terms of what entities in the microstructure are activated, may be different than those in sharp-crack tests. This occasionally can induce anomalies in the relationship between KIc and the Charpy energy as shown in the attached 1976 Metall. Trans. paper where increasing the prior austenite grain size in a quenched and tempered alloy steel can result in an increase in KIc yet a decrease in Charpy energy.
The bottom line here is that correlations between these various measures of fracture toughness can be problematic as they are not always focused on the same microstructural events causing the fracture. This is where fracture mechanics has the advantage. Provided fracture mechanics based toughness parameters, i.e., KIc or JIc, are measured properly, specifically conforming to ASTM Standard 1820 (or equivalent) in terms of satisfying K- or J-dominance at the crack tip under plane-strain conditions, these fracture toughness values can be usefully compared with other researchers' measurements and most importantly can be directly used in design.
4. Finally with respect to the practical significance of a high Jss. Let me draw the analogy to the uniaxial tensile properties. We invariably design structural materials to their yield strength, but if they fractured directly afterwards, especially catastrophically like a ceramic, they would be useless structurally. For that reason the UTS is important too, as the region between the yield stress and UTS provides the uniform ductility, which is absolutely critical to the designer. Of course, you're not going to utilize the material above the yield strength (unless something goes horribly wrong) but the fact that you have that region of strain up to the UTS before the material will start to fail unstably provides your margin of safety - your "parachute" (so to speak) - so that any impending failure is not be sudden and catastrophic. This is the main reason that ceramics are now being used in commercial gas turbine engines - not as monolithic ceramics which would suddenly fracture catastrophically but as ceramic-fiber reinforced ceramics - the ceramic matrix still breaks catastrophically at the "yield" strength but as the fiber/matrix interface is relatively weak, the fibers hold the material together (by bridging) up the the UTS. The ceramic components are designed for use below the "yield" strength but you again have created some degree of "ductility" up to the UTS to avert catastrophic fracture with no warning.
The measurement of the crack-initiation and crack-growth toughnesses provides exactly the same function. If your material broke catastrophically at KIc, then this is potentially a dangerous situation (unless the value of KIc is very large) as fracture would be sudden and catastrophic. For this reason, the rising R-curve between KIc and Kss is so vital for most structural materials in safety-critical applications. A high KIc is important in design but a high Kss after a rising R-curve, like ductility in a tensile test, serves the essential purpose of providing your margin of safety, your "parachute", which is absolutely critical for major structures.
ROR
Dear Dr. Robert,
Thanks a lot for your kind detailed explanation and yes I have started to read your recommended book T. L. Anderson's book on "Fracture Mechanics - Fundamentals and Applications" it is very interesting book and I will read your interesting paper which you have sent me. But really I get confused is the nonlinear-elastic fracture mechanics is another name for elastic-plastic fracture mechanics or if I understood you correctly the nonlinear-elastic fracture mechanics is a generic theory which can cover the elastic-plastic conditions is it right, as I have found elastic-plastic fracture mechanics in T. L. Anderson's book but I didn’t find nonlinear-elastic fracture mechanics.
Regards,
John
Under conditions where there is too much plasticity at the crack tip to use linear-elastic fracture mechanics, the definition of the single governing parameter which characterizes the crack-tip fields, namely J, is defined in terms of the deformation theory of plasticity, NOT the flow theory. Whatever anyone else choses to call it, this is nonlinear-elasticity.
Dear Dr. Ritchie
Are the terms plasticity and nonlinear elasticity synonymous? In my opinion, the absorption of energy during fracture is due to plasticity, and a linearly or nonlinearly elastic body does not absorb energy during the fracture. Am I right?
The relationship between elastic and inelastic deformation (brittle and non-brittle fracture) of a material leads to difficulties in understanding the relationship between toughness and fracture energy (area under the deformation diagram).
Dr. Borovik:
You are absolutely right. That is the paradox of nonlinear-elastic fracture mechanics. The terms "plasticity" and "nonlinear elasticity" are only synonymous under monotonic, increasing, proportional loading; provided this is the case, it works - this is the so called deformation theory of plasticity, where the plastic strain is proportional to the deviatoric stress. However, once you unload, which obviously occurs elastically in physical reality, you violate the assumption of reversible deformation. To deal with this, you need to use the so-called flow theory of plasticity which is based on reality, i.e., the incremental plastic strains being related to the deviatoric stress.
The path-independent definition of J is formulated on the deformation theory - this is way it is known as nonlinear-elastic fracture mechanics. The zone of unloading and non-proportional loading that occurs when a crack advances thus violates that concept - hence the extent of that zone has to be small enough to ignore - akin to the crack-tip plastic zone being small enough to ignore in linear elastic fracture mechanics. These conditions respectively are the basis of the size requirements for the use of J and K as singular parameters describing the amplitude of the local stresses and strains in the vicinity of the crack tip.
ROR
Dear Dr. Ritchie
thanks for the detailed explanations.
It follows from them that in the development of non-brittle structural materials that are not prone to stress concentration (bio-inspired structural materials), linear and nonlinear fracture mechanics turns out to be of little use.
Dr. Borovik:
I couldn't disagree with you more. If you believe that engineering is finding exact solutions to irrelevant problems, then I might agree, but the best engineering is finding approximate solutions to relevant problems, and that is where fracture mechanics has excelled. Nonlinear-elastic fracture mechanics is a complex subject but is utilized every day in the design and maintenance of nuclear reactors, aerospace and the like. The problem is most people do not bother to understand its origins and the critical assumptions enabling its development and use, and for that reason it is "one of the most abused form of mechanics". I am sorry to be frank but to say that the discipline of linear and nonlinear fracture mechanics is "of little use" is a totally misguided conclusion which flies in the face of reality.
ROR
Dr. Ritchie:
I have not argued that fracture mechanics are useless.
From my point of view, fracture mechanics applicable only to materials prone to stress concentration.
The most important feature of the next generation structural materials is delocalization of deformations, i.e. minimal tendency to stress concentration. It is for these materials that I find the classical fracture mechanics of little use, especially at the upper levels of their hierarchy.
Dr. Borovik:
OK - I would agree with that. Fracture mechanics does focus on the notion of a single dominant crack and the general concept of self-similar crack growth, i.e., to localized damage such as fatigue, etc. If you're dealing with a problem of distributed damage, such as certain forms of distributed creep damage, then alternative methodologies may be more appropriate. Damage mechanics is one such methodology - I don't personally like this approach because it tends to be rather empirical and lacking in much physical interpretation, but it certainly is a viable procedure for distributed damage failures.
ROR
Dr. Ritchie:
Now, if we return to the question of the ratio of toughness, fracture toughness and fracture energy, it turns out: the better the structural material (the higher the synergy of strength and fracture energy), the less the fracture mechanics approaches are applicable to this material. In particular, it is practically impossible to fulfill the dimensional ratios for test specimens due to the size of the plastic zone (energy absorption) at the tip of the notch or crack.
Which method of measure the fracture energy do you consider the most correct for materials with a reduced tendency to stress concentration (with delocalization of deformation)?
Dr. Borovik:
If by "materials with a reduced tendency to stress concentration" you mean ductile materials, then nonlinear-elastic fracture mechanics in the form of a J-based resistance curve (J-R curve) measurements, where you determine a crack-initiation toughness, e.g., JIc, and a crack-growth toughness, e.g., Jss or the tearing modulus (which is the normalized slope of the R-curve), are undoubtedly the best way to go. These represent an energy for fracture but also can be used to back-calculate KIc and Kss, so that you can estimate a stress and crack size for failure (which is more useful and design or for failure analysis). If you're concerned about excessive local plasticity and geometry effects lessening the crack-tip constraint, there is also a two-parameter approach involving both J-fields and their modification due to the T-stress (or the elastic-plastic parameter Q) which can take lessen crack-tip constraint into account. As these methods have been used for decades for the design and in-service fracture control of nuclear pressure vessel and piping steels, which represent ductile and very tough steels (KIc ~ 250 MPa/m) that are used in the ultimate safety-critical structural applications, I find it hard to understand how you can conclude that "classical fracture mechanics [is] of little use".
However, if you mean by "materials with a reduced tendency to stress concentration" those materials that don't fail from a single dominant crack but rather from distributed damage, such as excessive microcracking, fiber failure in certain reinforced composites, or from creep cavitation damage in steel pressure vessels, then fracture mechanics used to predict their failure at the global scale is far less useful. You can use it to model local failures, such as the breaking of a fiber or for coalescence of cavities along a boundary, but as a general criterion to describe eventual failure of a component undergoing distributed damage, some form of damage-mechanics represents a more appropriate methodology.
ROR
Dr. Ritchie:
We are trying to investigate structural materials similar to fiber composites only without a matrix for maximum weight efficiency (see photo of the fracture "surface"). With close to optimal shear resistance of the interfaces between fibers, specimens of these materials are not sensitive to notching. And here the problem of comparison with traditional homogeneous structural materials arises, for which fracture toughness is a measure of brittleness.
Dr. Borovik:
For these types of materials, you're right that the adoption of a fracture mechanics approach would be difficult. I think that the best way here would simply to measure the fracture energies of your materials and to compare them with identical tests on traditional homogeneous structural materials. For example, you could make the comparison with Charpy impact tests, or simply quasi-static bend or tensile tests where you measure the work of fracture (area under the load-displacement curve normalized by the area of the fracture surface). There is absolutely nothing wrong with doing it this way. Personally, I would make those measurements for both notched and unnotched specimens; in that way you can better demonstrate that your materials are less sensitive to the presence of stress concentrations.
ROR
Dr. Ritchie:
We are approximately doing that. Determination of fracture energy of the test speciement in the area under the deformation diagram does not cause problems.
I doubt that it is correct to determine the fracture energy of our materials by applying the normalization of the work of specimen fracture to its cross-sectional area. The purpose of the development of these structural materials is to achieve delocalization of deformations and fracture in volume.
The toughness, which can be calculated from the mechanical properties (yield stress, tensile strength, and elongation), was found to correlate with the Charpy impact energy, and it is known that CVN is in good agreement with the fracture toughness KIC. These relationships may link the Toughness to KIC. A ref.:
Preprint A simplified toughness estimation method based on standard t...
I'd like to rise an issue related to this topic, since experts are involved in this talk, conducting a fracture toughness experiment for metallic materials, a dependency occur between computed or measured fracture toughness and the normal dimension of specimen (thickness), till a certain value of this geometric properties, so a special care is needed when we test thin structures and thick structures, knowing that there is a difference between the two situations where a triaxiality occur in region surrounding crack front in thick walled plates which is absent in thin walled plates and plane strain/stress condition dominates, the point is what would be a good practice to treat these situation in order to make a good fracture testing?
Brick:
I am uncertain about exactly what your question is but if I believe that you are asking about the sample size requirements for "valid" fracture toughness measurements. These are explicitly stated in the relevant standards, e.g., ASTM E1820. Let me consider here the criteria required for the measure of the crack-initiation toughness.
In a nutshell, for any stress-intensity K-based toughness measurements, you need to ensure that K characterizes the linear-elastic crack-tip stress and strain fields, i.e., you must satisfy the small-scale yielding requirement that the plastic-zone size remains small compared to the in-plane dimensions of crack size a and the uncracked ligament b. [b here is (W - a), where W is the specimen width] If you want to measure the plane-strain fracture toughness KIc, you must additionally satisfy the plane-strain requirement that the plastic-zone size remains small compared to the out-of-plane dimension of specimen thickness B. For LEFM KIc measurements, ASTM combines both these criteria into one: that your specimen dimensions must be such that a, b & B > 2.5 (KIc/Y)2, where Y is the yield strength. Under these conditions, you have the constraint to provide the appropriate triaxiality of stresses at the crack tip such that your KIc should reflect a lower-bound toughness.
For nonlinear- elastic fracture mechanics J-based testing, you similarly need to first guarantee that J actually characterized the crack-tip stress and strain fields over the dimensions of the fracture events. This condition of J-dominance is provided when the uncracked ligament b > 10J/Yeff, where Yeff is the effective yield stress (or flow stress) given by the mean of the yield and ultimate tensile strengths. For plane-strain measurements, B > b. Note that there are a few additional criteria for valid measurements, such as a more severe J-validity size criterion for the required uncracked ligament size for brittle cleavage fracture.
As I have noted previously, satisfying these size requirements is absolutely vital for reliable fracture toughness testing. There are numerous examples in the literature where they have been essentially ignored such that the resulting toughness values are at worst, "dead wrong", or at best, "size- and geometry-dependent".
ROR
Mr Robert:
Thank you for this explicit response, you've literally summarized the standards related to that specific case.
The choice of a relevant standard is important, a special care is always needed when performing fracture testing of metallic specimens of thin sheets, metals are often ductile and exhibit an elastic-plastic behavior under tensile loading, so during fracture testing it is important to make sure small-scale yielding condition dominates so LEFM relationships can be used to compute K-based fracture toughness (a tensile testing is important).
We initiated recently a project in which we'd like to study the sensitivity of fracture to natural and artificial hardening in a 2xxx series aluminum alloy, it is available in the form of thin (2mm) sheets, the choice of an appropriate standard has to be made, an experimental and numerical approach is considered for now, well implemented elastic-plastic models in commercial finite elements codes allow computation of J-integral based fracture toughness parameters, fracture experiments can be assisted by the relatively recent 2D Digitale Images Correlation Technique DIC which can return full-field displacement and strain field and J-integral computations can be performed using DIC strain field, although these results can be compared to numerically determined displacement and strain fields.
We would like to ask you some guidance for this project especially with the choice of specimen and fracture standard type.
Dear everyone,
In general, Fracture toughness can not be obtained through experiments of a thin sheet of ductile metal under the static loading. Fatigue fracture toughness under the cyclic loading may be useful. Meanwhile, the toughness may be a failure criterion for a thin sheet. Ref.:
https://www.researchgate.net/post/What_are_the_differences_between_Toughness_Fracture_toughness_and_Fatigue_fracture_toughness
Some time ago I did this work with Ludwig Schoening on synthetic diamonds that may be relevant to this question: (SEE ATTACHED) An X-ray diffraction investigation of FeCo, graphite and compressed graphite inclusions in heat-treated single crystals of synthetic diamonds
To cite this article: F R L Schoening and D C Levendis 1994 J. Phys. D: Appl. Phys. 27 2223
The presence of sqrt(r) in the SIF is a matter of definition by pioneers! It can be removed as is done in the attached paper. Regards
s. Hamed Ebrahimi
Dear sir
In reality all materials are, more or less, cracked. Consequently, the load bearing capacity is a unique characteristics for all systems. The capacity is independent of the method used! There is no singularity at the crack tip (infinite stress). If one assume that singularity exist, then a cracked structure cannot exist, because under a load the material break under a limited load much less than infinity.
I have treated all systems with the same formulation.
In the case of classical fracture mechanics, I found that the classical crack compliance (CF) should be replaced by the fled function (FR) as follows!
dy=CF*y(2) Classical FM CF=0 to infinity
dy=FR*y(2). Persian FM FR= 0 to 1
In governing equation they appear as follows
(y(2)-CF*y(2))(2)+…=0 Classical FM
(y(2)-FR*y(2))(2)+…=0 Persian FM
It is clear that the governing equation for cracked system in terms of CF is wrong. That is because the terms in braces soon become negative and the equation loose it’s validity! Not that the number in braces, (2) denotes second derivative.
Dear Dr Yingbin Hu . See the link: https://www.sciencedirect.com/topics/chemistry/fracture-toughness
See the following link: https://dl.asminternational.org/handbooks/book/47/chapter/537912/Fracture-Toughness-and-Fracture-Mechanics
See also the following link: http://iopscience.iop.org/article/10.1088/1757-899X/431/6/062007/pdf
Also see the following RG link: Article Fracture and fracture toughness of nanopolycrystalline metal...
In structural design, earthquake engineering, and etc. more than 99 % are not elastic. All design codes that supposed to provide safety against failure are based on inelastic behavior. The fracture mechanics is also 100 % inelastic. The reason for presence of more than one formulation , for example for crack compliance of an axial bar, is a clear reason for errors in LEFM.
Dear Yingbin Hu,
Thank you for your interesting question. I am pleased to borrow the following articles and you can consult the relevant references.
1- Schneibel, J. H., Horton, J. A., & Munroe, P. R. (2001). Fracture toughness, fracture morphology, and crack-tip plastic zone of a Zr-based bulk amorphous alloy. Metallurgical and Materials Transactions A, 32(11), 2819-2825.
2- To, T., Célarié, F., Roux-Langlois, C., Bazin, A., Gueguen, Y., Orain, H., ... & Rouxel, T. (2018). Fracture toughness, fracture energy and slow crack growth of glass as investigated by the Single-Edge Precracked Beam (SEPB) and Chevron-Notched Beam (CNB) methods. Acta Materialia, 146, 1-11.
3-Zindal, A., Jain, J., Prasad, R., Singh, S. S., & Cizek, P. (2017). Correlation of grain boundary precipitate characteristics with fracture and fracture toughness in an Mg-8Al-0.5 Zn alloy. Materials Science and Engineering: A, 706, 192-200.
4- Zhang, J., Tan, H., Pei, J., Qu, T., & Liu, W. (2019). Evaluating crack resistance of asphalt mixture based on essential fracture energy and fracture toughness. International Journal of Geomechanics, 19(4), 06019005.
Enjoy reading,
IDIR.
Dear Hu
Normal Toughness refers to the energy absorbed by the material before undergoing fracture generally given by area under stress-strain curve in tensile test while fracture toughness is specific term used in Fracture Mechanics which refers to the resistance offered by the material for the crack propagation.
thanks & regards,
g. sudhakar
phd(materials engg),hcu.
Toughness: Toughness of the materials is the ability of a perfect or a flawless system to resist both fracture and deformation or breaking under impact loading by absorption of impact energy. It deals with an ideal system
Fracture toughness: It refers to the ability of material having manufacturing defects (may be micro- cracks or micro- or nano- size holes) to resist failure/fracture by absorption of impact energy. Thus it applies to actual systems.
Area under the stress-strain curves for a particular material gives the toughness of that material
Dr. B R Gupta, Retired Professor, RTC, I I T Kharagpur, India, [email protected]
I am fully in agreement with Dr. Gupta.Thanks Dr. Gupta for such nice and clear differentiation between two characterstics of materials.
Thanks and Regards
Dr. S.S.Kasana
There are many flaws such as microvoids and inclusions in a material even though it is a smooth specimen. The value of toughness can be obtained from a smooth specimen, where that of fracture toughness from a notched or precracked specimen. The relationship between two is shown in the link above at room temperature.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Materials Science