Andrey:

Dr. Kern is absolutely correct. Any relationship between strength and fracture toughness would have to be for brittle materials as both material properties depend on the presence of cracks and on fracture. However, that's where the connection ends as the strength of a brittle material will be dependent on the bond strength and importantly the distribution of defects (and their size) that are present invariably in a macroscopic (tensile or bend) specimen (note here that brittle materials such as ceramics are weaker in larger test samples as there is a higher statistical probability of find a larger flaw. In contrast, to measure the toughness, e.g., KIc, you will have inserted a worst-case starter crack in your sample and so the distribution of inherent cracks in the material is less important; moreover, the active volume of material engaged in the fracture, i.e., within the crack-tip process zone, will be orders and orders of magnitude smaller than in your tensile or bend specimen used to measure strength.

For a ductile material, on the other hand, all bets are off! There can be no scientifically viable way, in my opinion, of ever converting fracture toughness to strength as they assess completely different properties of a material. Toughness is resistance to fracture, strength in ductile materials is resistance to plastic deformation. Indeed, they are effectively mutually incompatible properties. If you are interested, I have written a short paper on this which is attached.

ROR

According to Griffiths equation that seems to be easy practically it is impossible as strength depends on flaw size and shape, and the size and geometry of the component. And the equation is only true for really brittle materials like glass which have no R-curve behaviour.

Dear Dr. Frank Kern , thank you very much for your answer!

Andrey:

Dr. Kern is absolutely correct. Any relationship between strength and fracture toughness would have to be for brittle materials as both material properties depend on the presence of cracks and on fracture. However, that's where the connection ends as the strength of a brittle material will be dependent on the bond strength and importantly the distribution of defects (and their size) that are present invariably in a macroscopic (tensile or bend) specimen (note here that brittle materials such as ceramics are weaker in larger test samples as there is a higher statistical probability of find a larger flaw. In contrast, to measure the toughness, e.g., KIc, you will have inserted a worst-case starter crack in your sample and so the distribution of inherent cracks in the material is less important; moreover, the active volume of material engaged in the fracture, i.e., within the crack-tip process zone, will be orders and orders of magnitude smaller than in your tensile or bend specimen used to measure strength.

For a ductile material, on the other hand, all bets are off! There can be no scientifically viable way, in my opinion, of ever converting fracture toughness to strength as they assess completely different properties of a material. Toughness is resistance to fracture, strength in ductile materials is resistance to plastic deformation. Indeed, they are effectively mutually incompatible properties. If you are interested, I have written a short paper on this which is attached.

ROR

Great comment, Dr. Ritchie!

Dear Dr. Robert O Ritchie , thank you very much for your comprehensive answer!

Dr Robert O Ritchie thanks for article.

perhaps you should be more specific. Usually the two properties are unrelated or negatively related. But components can be made inhomogeneous by surface treatments, so the core is tough and the surface hard. So it is a long story...

Dr Robert O Ritchie：

For brittle material , it conflicts between strength and toughness. But for ductile material , do the toughness increase with higher strength an ductility ?

What is the purpose...? Experimentally it is more practical, for example. If you test a degraded specimens for ultimate strength and then test similar specimens for fracture toughness you could relate them over an excel sheet!!

Theoretically or empirically, it should be possible too- you will have do some math-circus with equations. In reality there has to be a relation not-necessarily linear. Also for metals at low temperatures strength improves while toughness reduces, you need to be careful there.

Yeah I go with experiment route!!

Dr. Hou:

The detailed answer to your question about the relationship between strength and toughness depends not only on whether one is dealing with a ductile or brittle material but (for the same reason) on the prevailing fracture mode in question, i.e., is it stress- or strain-controlled?

As described earlier with respect to this question, for a brittle material such as a ceramic where the tensile strength is a function of resistance to fracture, similar to the fracture toughness, the principal difference between these material parameters lies in the nature of the distribution of flaws active in the failure process: in strength tests, the initiation of fracture (which invariably is stress-controlled) depends on the distribution of flaws throughout a large sampling volume, e.g., the volume of the gauge length region of a uniaxial tensile specimen, whereas in a fracture toughness test you put in a worst-case defect, e.g., a fatigue crack, in the test sample and the sampling volume is the process zone ahead of the crack tip, which is orders of magnitude smaller.

However, for a ductile material such as a metal, where the material fails by ductile fracture, e.g., by microvoid coalescence, the prime origin of the toughness is plasticity. Since high-strength materials display limited plasticity, high-strength metallic materials are generally not tough. Another way of looking at this is if you consider a simple strain-controlled micromechanical model for ductile fracture of a critical strain across a microstructurally-significant distance (which often is some multiple of the particle spacing) ahead of the crack tip, the J-based fracture toughness scales with the strength x ductility x this characteristic distance; again as high strength is generally incompatible with high ductility, strength and toughness again can appear to be mutually incompatible.

For ductile materials that fail by brittle fracture, e.g., for body-centered cubic metallic materials (e.g., ferritic or martensitic steels) below their ductile-to-brittle transition temperature), the situation is similar to that in ceramics. Fracture is stress-controlled, e.g., by transgranular cleavage, and high-strength materials simply generate much higher local stresses at a crack tip, which induce the fracture. (Depending on the degree of strain hardening, the local crack-tip stresses can be 3 to 5 times the flow stress, e.g., the yield or tensile strength). Consequently, once more strength and toughness will generally be mutually incompatible.

ROR

From the R-curve is possible. I am working in a paper about this, correlating the strength from ball-on-ring and r curve from double cantilever.

Dr. Boccaccini:

Yes, you are correct, it is possible to deduce strength from the R-curve - I wrote a paper on this with Jay Kruzic and Rowland Cannon some 10 years ago (attached) - but as I mentioned before in my responses to this question, relationships between strength and toughness only work for brittle materials such as ceramics, where both properties are both associated with cracking behavior. There is no meaningful way to convert the fracture toughness to strength for a ductile material.

ROR

Dear Robert O Ritchie

Firstly thanks for that paper, it will be an interesting read for me for sure..

Just looking at the fundamental aspect of fracture, Ultimate Strength and Fracture Toughness are two distinct derived quantities, which represent failures-although in two different modes (needless to define)..

So if you perform an experiment even for ductile material for a number of specimens and failing them in two different modes of failures (one for ultimate strength and second for fracture toughness), we can deduce not necessarily a linear but certainly a relation based on type of material, correct me if I am wrong.

Thank you.

Vikram

Vikram:

You can always develop some form of correlation and you're right there is clearly a connection between the ultimate tensile strength and the fracture toughness - as plasticity plays a major role in the definition of each parameter, but to attempt to calculate one property from the other for a ductile material (which is the question being addressed) would be to my mind a flawed exercise (if you'll excuse the pun!). The UTS defines the point of necking instability - this clearly would be relevant to the initiation and/or propagation of a ductile crack (by, say, microvoid coalescence), which defines the fracture toughness, but clearly not the whole story - other factors are involved as well.

ROR

Dear Dr Ritchie,

Thank you very much for your answer. I have read your paper and it has been a source of inspiration for my research on solid mechanics of porous Y-PSZ (please find attached a paper with the first part of this work). In the second paper I will address the correlation between r-curve and strength in porous PSZ, considering the influence of transformation toughening.

Kind regards,

Dino

Dino:

Thanks for your publication. I appreciate it. Good luck with your project.

Cheers ROR

In my opinion, the answer of this question is concludable in completely brittle materials without R-curve behavior! Because any type of structural defects (distribution and size), which one of the most important of them is crack, can be influenced on strength and fracture toughness.

I believe that this article contains a equation for the conversion for bonding strength. Hope it helps

Data Inter-wafer bonding strength characterization by laser-induc...

Dear Qinwu:

There are literally no realistic, physics-based, equations to accurately relate strength to the fracture toughness (in modes I, or II or III) that you can rely on to give you a sound estimate of the toughness, simply because these two properties rely on different phenomena.

In ductile materials such as metals, strength is resistance to deformation and toughness is resistance to fracture; they are of course related but there is no universal or simple relationship that is very meaningful, except for micromechanical models that would predict that that J-based toughness is proportional to the strength x ductility x a microstructural characteristic distance (if you're interested, see Ritchie & Thompson, Metall. Trans. A 16A (1985) 233). But these models depend on the prevalent fracture mechanism; the one mentioned above is based on a strain-controlled ductile fracture; there is a completely different relationship for stress-controlled cleavage fracture. Moreover, they involve a fitting parameter in the form of the microstructural characteristic distance, which invariably needs to be set empirically.

For brittle materials, such as ceramics, it should be an easier proposition as the strength and toughness both depend on the material's resistance to fracture. However, the toughness is controlled by the local stress to fracture the material ahead of a known defect - the precrack in a fracture mechanics sample - whereas the strength depends on the corresponding stress to fracture an unnotched uniaxial tensile sample in the presence of an unknown distribution of pre-existing defects in the sample. Clearly, one could derive a relationship here but you would need to know the size distribution of these inherent defects and solving the geometry effects in this instance would be quite complex and make any prediction highly approximate - in essence, "the Devil is in the details" here!

I think that your best plan of action would be to perform indentation fracture toughness tests on your samples. As you're working with ceramics, the cracks that emanate from the corners of the indents can then be used to estimate an approximate (mode I) fracture toughness. There are lots of papers written on this method, e.g., Brian Lawn's text book on "Fracture in Brittle Solids" would be one place, but if this book is difficult to track down, I have written a few papers on this method myself that you can find on ResearchGate, e.g., Kruzic & Ritchie, J. Am. Ceram. Soc. 86 (2003) 1433; Kruzic, et al., J. Mech. Bev. Biomed. Matls. 2 (2009) 384.

I hope that this helps.

ROR

Dear Quinwu,

I really do not understand for which strength (impact strength, breaking strength or fracture strength or yield strength or ultimate tensile strength) you are talking. Determination of fracture toughness from charpy is most common, impact strength vs. fracture toughness. Impact strength and fracture toughness are two completely different properties. Fracture toughness should be considered if the part is subjected to constant loading. It depends on the various materials,

For rock materials, σt= 8.88KIC 0.62, tensile strength can be directly calculated.

For different mode of fracture toughness,

1. For ductile plastic fracture mode, (KIC/σYS,. > 1.5)

2. Elastic-plastic fracture mode (1.5 > KIC/ σYS > 0.5).

3. Brittle elastic instability fracture mode (KIC/ σYS < 0.5).

For tensile toughness,

Unit of tensile toughness can be easily calculated by using area underneath the stress–strain (σ–ε) curve, which gives tensile toughness value, as given below: UT = Area underneath the stress–strain (σ–ε) curve = σ × ε UT

As I understand strength and fracture toughness are two important mechanical properties. Yield strength is the measure of the stress that a metal can withstand before deforming and tensile 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. Fracture toughness, KIC, values for metal, ceramic, polymer and composites are different. Fracture toughness can be determined from R-Curve(fracture toughness vs. crack length). There are so many mechanisms behind the fracture toughness and those are importantly intrinsic, extrinsic and crack repair.

For two-dimensional problems (plane stress, plane strain, antiplane shear) involving cracks that move in a straight path, the mode I stress intensity factor KI is related to the energy release rate, also in mode I, G1by

G1= KI2/ E’ E is the Young's modulus and E’=E for plane stress andE’= E/(1-ν2) for plane strain.

Basic differences between fracture toughness and strength

"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. While Fracture strength, also known as breaking strength, is the stress at which a specimen fails via fracture. This is usually determined for a given specimen by atensile test, which charts the stress–strain curve (see image). The final recorded point is the fracture strength.

Calculation

One way to measure toughness is by calculating the area under the stress strain curve from a tensile test. This value is simply called “material toughness” and it has units of energy per volume. Material toughness equates to a slow absorption of energy by the material.

As fracture toughness increases, the energy required to cause a crack to grow to fracture increases. Low fracture toughness corresponds to low ductility. Suppose for example, glass has very low toughness and is very brittle.

For a certain load, as the fracture toughness increases, a component can tolerate a longer crack before fracturing. For any particular alloy, toughness decreases as strength increases (many references can be cited in support- more information read Deformation and Fracture Mechanics of Engineering Materials by R.W. Hertzberg).

Fatigue stress is also another cause of cracks. The fracture toughness is required to determine how long the component can remain in service before a crack grows so long that the intact cross-section of the component cannot support the load, and the component fractures. This applies to aerospace components and pressure vessels such as boilers.

Hope it work out for you.

Ashish

Dear every one.

The fracture toughness and strength can be classified as two types: static type and dynamic type. I agree with Dr. Ritchie, that there are no realistic, physics-based equations to accurately relate strength to the fracture toughness, even though there are some relationships between them in some cases. I think this is the same in the two types in principle. When the fracture toughness Kc is determined by static fracture tests where specimens have fatigue pre-cracking, their values are about 50-200 MPam1/2, in the steels with tensile strengths of 500-1000 MPa. However, in dynamic fracture tests under cyclic loading, the stress intensity factors Ka, Ka(0) are very small and decline to about 2.5 MPam1/2 in the fatigue crack initiation (Ka(0)). The fatigue limits are about half values of tensile strengths in smooth specimens, but decrease greatly in notch specimens relating to the stress concentration factors. The tensile strengths are low sensitive to the stress concentration factors, with relating to the ductility. The values of Ka, Ka(0), and Kc are necessary to be used separately in the dynamic loading and static loading conditions without getting them mixed up. The Kc means an instability criterion to the final stage of fractures, and it is different from the fatigue crack initiation and stable propagation.

You can make correlation for this

Fracture toughness can be classified into three types:

1. In general, the static fracture toughness, an indicator for unstable breakage criteria in plastic constraint state. For example in ASTM E 399, E 1820 – 01.

2. Dynamic fracture toughness. In cases of high speed loading, impact loading etc.

3. Fatigue fracture toughness. In the case of that the cyclic loading is applied throughout the crack initiation, static propagation, and unstable breakage process.

The values of these three fracture toughnesses are greatly different from each other, and the strengths with many kinds are different from each other as well.

Dear Naseem Ahmad , do you think they have correlations? I don't think so in principal aspect, because they are different properties for materials and structures and the fracture mechanisms are different in principal as shown in the above discussions. Fracture mechanics would be developed for these reasons.

There is no direct correlation between the two but you will adopt some vendor factors for the correlation Gyoko Oh

Naseem Ahmad , will you show any example from your data please?

Three explanations below may be useful for the question:

Resilience indicates how much energy a material can absorb within the elastic limit.

Strength indicates how much force the material can support.

Toughness indicates how much energy a material can absorb till the point of fracture or before rupturing.

These are greatly different between ductile materials and brittle materials as well.

I agree with Robert and thank him for his comprehensive answer.

In my opinion, even if there was a relationship between these parameters, it would not be the same for brittle and ductile materials

I agree with Dr. Ritchie

I agree with the answers of @Robert Ritchie and @Gyoko oh and the relation is not possible even with factors. These are two different characteristics. As it is impossible to arrive with (Best fit) generic factors for different situations and purposes used.