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Inglis (1913) vs. Griffith (1921)

Zhigang Suo's picture

I have updated my notes on the Griffith paper.  I added more description on the experimental determination of surface tension of solids.  Griiffith himself determined the surface tension of glass by an experimental setup.  Udin et al (1949) described a setup based on the same principle.  This setup is now known as the zero creep experiment.

In class today Chao Chen asked me to compere Inglis theory and Griffith theory, which I did at the very end of the notes.  The two theories give the same prediction: 

the strength times the squre-root of the crack length  is a material constant.

In the Inglis theory, the constant involves atomic strength and atomic size.  In the Griffith theory, the constant involves Young’s modulus and surface energy.  If we adopt any simple-minded atomic model, we can show that the two constants are essentially the same.

Both theories work well for silica glass.  Neither works for steel.  Both theories survived to this day, in somewhat different forms.  In general terms, the Inglis theory has evolved into the stress approach to fracture, and the Griffith theory has evolved into the energy approach.  The two approaches are, of course, equivalent.  We will talk more about both approaches in coming lectures.

Comments

Konstantin Volokh's picture

Dear Zhigang,


In the new notes you significantly expanded explanations concerning
surface energy in solids (and fluids) which was the subject of our discussion
on iMechanica (
notes on the Griffith
paper
).
I appreciate your effort yet I still cannot grasp the surface
energy in solids intuitively. I still think that this concept has nothing to do
with the energy necessary to advance cracks. I am sorry for my stupidity.
Probably, I need some private lessons from you Sealed

However, I would like to discuss the Griffith paper and results,
which are both original and controversial. My full length criticism of Griffith
can be found in this paper (I'll call it VT08 ). I will only strengthen the main
points.


1. The true formula derived by Griffith is not Eq.(3) or (4) of VT08 as
you and many others think. Griffith derived a more remarkable formula – Eq.(1)
of VT08. It is remarkable because the sharpness of the elliptic crack is
present there.


2. According to the original Griffith formula the sharpness of the crack
simply DOES NOT MATTER. This is truly remarkable and wrong! Every
experimentalist who tries to calibrate fracture toughness knows that the
sharpness of the notch DOES MATTER. Griffith formula directly contradicts the
experimental evidence.


3. So, what is wrong with Griffith? In my opinion, the problem is
that Griffith used global energy balance, which naturally smeared stress/strain concentration
at the tip of the crack. Contrary to Griffith the local stress/strain concentration rather than the global energy balance is responsible for the advance
of the crack.


Enough to start with Smile

Zhigang Suo's picture

Dear Kosta:  Thank you very much for the comment.  To focus the discussion, I'll restrict this response to materials like silica glass. I agree that the sharpness of the stip of a pre-existing flaw will affect the breaking stress.  For silica glass, the stress concentration factor is predicted by linear elasticity with some accuracy, so the Inglis (1913) theory will apply.  In my notes on the Inglis paper, which I titled "Trouble with linear elastic theory of strength", I included an example:

One exceptional case. Experimental strength for optic fibers. When a silica fiber develops etch pits, the stress concentration reduces the measured strength to

(S_exp) = (S_th)/C,

where C is the stress concentration factor.  For an etch pit comparable to a hemisphere, C = 2~3.  See C.R. Kurkjian and U.C. Paek, “Single-valued strength of perfect silica fibers,” Appl. Phys.Lett. 42, 251-253 (1983).

In using the Griffith (1921) theory, we tend to focus on situations that it will work.  Part of the modern fracture mechanics is about spelling out how we might use the Griffith theory.  The theory is captured by eq. (3) in your paper.  In using the Griffith theory, we require two conditions:

Small scale yielding. Any deviation from linear elasticity should be confined within a zone much smaller than macroscopic lengths, such as crack length and sample size.

Steady state.  Here is a description in my notes on the Griffith theory.  The nonlinear zone, localized around the tip of the crack, remains invariant as the crack advances. Consequently, the presence of the nonlinearity does not affect the accounting of the change in energy associated with the advance of the crack. The Griffith approach circumvents the nonlinear crack-tip behavior by invoking one quantity: the surface energy.

In the steady state, the growing crack creates its own front, and forgets whatever its initial sharpness. 

Konstantin Volokh's picture

Dear Zhigang,

You are talking about
the crack steady propagation. This is not about Griffith. He
considered only the critical condition of the initiation of propagation of pre-existing
cracks. Whatever short and elegant his analysis is, it is wrong. The global
energy balance used by Griffith cannot decide the fate of the crack. The crack
fate is decided locally by the strain/stress concentration at its tip.

Since the results of the
experimental calibration of fracture toughness scatter (because the experiments
are based on a wrong theory) the engineers decided to standardize the fracture
toughness tests and strict prescriptions are given in the literature. What a
waste of money and time! I suggest the engineers save money on such
experiments and I would gladly accept 5% of the saved amount for this wise
advice Cool

Zhigang Suo's picture

Dear Kosta:  Section 2 of the Griffith paper is titled "A Theoretical Criterion of Rupture".  Here are a few sentences from this section:

If the body is such that a crack forms a part of its surface in the unstrained state, it is not to be expected that the spreading of the crack, under a load sufficient to cause rupture, will result in any large change in the shape of its extremities.  If, further, the crack is of such a size that its width is greater than the radius of molecular action at all points except very near its ends, it may be inferred that the increase of surface energy, due to the spreading of the crack, will be given with sufficient accuracy by the product of the increment of surface into the surface tension of the material.

Here, as well as at many points of the paper, Griffith made it very clear that he was talking about the spreading of a crack, and the tip of the crack is of molecular dimension.

Konstantin Volokh's picture

Dear
Zhigang,

1. You say
that Griffith means crack with the tip on the length scale of one molecule.
That completely contradicts his formula (Eq. (1) of VT08). Again, the great
thing about his formula is that it does not depend on the tip of the crack.
This result would be great, indeed, if it were correct.

2. There are highly ideal crystals in which cracks are
separations of adjacent atoms along atomic planes. However, I think that cracks
in most bulk materials are created by simultaneous breakage of millions of
atomic bonds at the tip – it is not a peaceful atom-by-atom in-chain separation
– it is a huge local explosion of the bonds. In 
this post  I discuss the
estimates of the thickness (not the opening!) of the crack – the area where the
explosion takes place.

Zhigang Suo's picture

Dear Kosta:  I need to think about your Point 1.  I agree with your Point 2.  In many materials, atoms and molecules off the crack plane will undergo dissipative processes, such as plastic flow, unzip of weak bonds, and microcracking.  I will describe this Irwin-Orowan generalization of the Griffith theory in the next lecture, as well as in many of future lectures.

Engineers are required to produce results in a short period of time without having to justify the validity of the approach taken.  In the absence of better approaches we are forced to fall back on approaches, such as fracture mechanics, that are relatively easy to characterize and have a scientific basis.  Determination of whether that scientific basis is truly sound is typically left to academics.  My opinion is that some fracture mechanics is better than none in the absence of better models.

To quote Rajagopal:http://dx.doi.org/10.1016/j.ijengsci.2013.06.002

" A problem that is at the heart of damage and failure of solids is the
problem of fracture and crack propagation. The problem of fracture has
been the object of study from very ancient times; it was studied by Galilei (1658)
and before him it was discussed by Aristotle. Modern mathematical
studies of cracks and their propagation, within the context of continua,
assume that a crack ends at a “point”, and in the case of classical
linearized elasticity leads to mathematical singularities at the tip of a
crack. Of course, one can question the appropriateness of assuming that
the body in question is elastic or investigating whether one could
avoid such singularities by using different constitutive relations to
model the response of the body,35
but this is not the real issue. A more fundamental question to grapple
with is how reasonable is the assumption that the tip of crack is a
point?"

-- Biswajit

Zhigang Suo's picture

Dear Biswajit:  I share your view.  Fracture mechanics is a way to deal with practical problems.  It certainly does not solve all our problems, but it is the best choice under certain conditions.  In teaching fracture mechanics, one strives to teach what fracture mechanics is, and where it is applied more effectively than alternative approaches.

In the text that you quoted, Rajagopal wrote, "A more fundamental question to grapple with is how reasonable is the assumption that the tip of crack is a point?"

I have not read the paper by Rajagopal, but have grappled with the assumption that the tip of crack is a point.  This is an assumption in a model.  Together with the assumption of linear elasticity, the assumption of a point-sharp crack simplifies the mathematical analysis, but also leads to the prediction of singular stress.

In using the model, we never let the distance approach zero, and never use the singular stress.  In fact, we always use the solution when the crack is not point-sharp, and material is not linearly elastic.  The main points are discussed in many places.  A short description can be found in Section 1.2 of the following review:

G. Bao and Z. Suo, " Remarks on crack-bridging concepts,Applied Mechanics Review45, 355-366 (1992). 

Many other models remove these assumptions, to allow inelastic deformation and to make the tip of the crack more realistic.  These models are more complex, but also shed light.  But they have not replaced the model of a point-sharp crack in a linearly elastic material.

Perhaps the situation parallels that the theory of relativity has not replaced Newton's laws.  We use Newton's laws in their domain of validity, where the theory of relativity adds nothing.

Zhigang Suo's picture

The Inglis (1913) paper evolves into the stress approach.  In modern time, to calculate stress accurately, we add many ingredients to the calculation:  elasticity, plasticity, viscoelasticity, interatomic potentials...  In some sense, this is a more basic approach than the Griffith approach, but the stress approach rarely yields results of practical use.  Not today.  Maybe someday.

By contrast, the Griffith (1921) paper  focusses on a more limited question:  the condition for the growth of a crack.  This approach has led to the definition of the energy release rate as the driving force for the growth of crack.  For example, it has led to a practical solution to the problem of fatigue crack growth.  This phenomenon remains a challenge to the detailed stress approach.

Incidentally, in the Introduction of the paper, Griffith mentioned that object of his work was to understand fatigue of steel.  

Konstantin Volokh's picture

I agree completely

Zhigang Suo's picture

I have just updated again my notes on the Griffith paper.  I added an item that addresses a question raised by Kejie Zhao.  To relate the Inglis (1913) and Griffith (1921), one needs a relation among several quantities:  theoretical strength, elastic modulus, atomic spacing, and surface tension. I found an old paper by Michael Polanyi that provided such a relation in 1921.

Roberto Ballarini's picture

Many years ago I was fortunate enough to get to know Bill Brown at NASA-Lewis (now NASA-Glenn). He was one of the leaders of the fracture mechanics group at Lewis that contributed to ASTM E-399. The foundation of this standard is Linear Elastic Fracture Mechanics (LEFM), a theory whose foundation is Griffith and its generalization is Orowan. This testing standard for metals spells out conditions under which fracture toughness-based analyses, predictions, etc. can be performed with a decent level of confidence (including the Small Scale Yielding stated by Zhigang). According to Konstantin, all of these efforts were a waste of time and money. This I consider insulting to the memory of excellent engineers that had a mission; to get ships to the Moon and back. Their research was instrumental in achieving that goal; they did not WASTE THEIR TIME. In fact one day a keen undergraduate student and I sat down and were mesmerized by Bill as he told us about the early days of fracture mechanics, and in particular how LEFM analysis/predictions/etc. enabled safer structures. 

Konstantin, you may also want to look at data for the strength of quasi-brittle materials and structures. The size effect observed in such systems (transition from plastic collapse to brittle failure) is captured beautifully by simple cohesive models (in the concrete business there are numerous names to size effect laws, including Bazant's size effect law; but ultimately the transition is captured qualitatively by the results of the simple Dugdale model, which too relies on the concept of (removing) singularities). And these ideas are helping civil engineers develop better structures, including the prediction of the pull-out capacity of anchors embedded in concrete. I do not think this is a waste of time or "wrong." LEFM turns out in this case to predict with extremely high accuracy the dependence on depth of the pull-out force. 

The modeling of materials and structures is ever-evolving (mostly in the positive direction). I believe it is not fair to belittle previous-generation models as we search for better models. Newton was not "wrong" because Einstein extended our understanding to situations for which relativistic effects are significant. Similarly Griffith was not wrong because there are many problems for which LEFM breaks down. 

Konstantin Volokh's picture

Dear Roberto,

Thanks for your defense of LEFM. I am
sure LEFM will withstand my remarksLaughing. I ensure you that I appreciate Griffith contributions.
Nonetheless, without clear understanding of what was/is correct and incorrect
it is difficult to move forward.

I was discussing with Zhigang a very specific issue
– Griffith formula (even Griffith vs. Inglis). I precisely explained why the
formula is incorrect in my opinion. Your post is emotional yet not specific
enough…Frown

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