It will be hard for me to support the statement made in the title, because I don't really know what theology is. But whatever theology is, as specified in Wikipedia, fracture mechanics is not theology. Fracture mechanics makes predictions that you and I can do experiments to verify.

Also, singularity is just an appearence. Take the familiar example of Newton's law of gravitation: Force between two massive bodies is proportional to (distance)^-2. This expression is singuar if you let distance approach zero. But we don't!

Similarly, in linear elastic fracture mechanics, the stress scales as (distance)^-1/2. But we don't let distance approach zero in using this result. The following paper summarizes this aspect of linear fracture mechanics in Section 1.2.

G. Bao and Z. Suo, " Remarks on crack-bridging concepts," Applied Mechanics Review. 45, 355-366 (1992).

Dear Ajit,

1-2 Nothing to argue about

3 LEFM gives indeed a short analytical description. Short formulas are so attractive that people (including myself) tend to compromise on their meaning. (Needless to say, only short formulas are good for gravestones )

Dear Kosta,

Zhigang is not taking your bait, but I particularly liked much of what I was reading in this thread anyway (). So, let me go a step further. Or, two. (May be three.)

1. Borrowing terminology from this physics-related blog [^] (HT to Hanning Dekant [^]), if a crack-tip in a (real) material quacks like a singularity, then...

2. But, I still do maintain that any theory that is crucially based on a singularity is theology.

3. No. No. No. LEFM is not a computational approximation---it's an analytical theoretical one.

--Ajit

- - - - -
[E&OE]

I agree with you on both. Later in the course I'll talk about notch-sensitivity. Materials like silica glass and most ceramics are extremely notch sensitive. Materials like metals and natural rubber are less notch-sensitive. Fracture mechanics can quantify notch-sensitivity.

For a visual illustration of notch-sensitivity of a soft elastomer, see Fig. 1 in the following paper:

Matt Pharr, Jeong-Yun Sun, and Zhigang Suo. Rupture of a highly stretchable acrylic dielectric elastomer. Journal of Applied Physics 111, 104114 (2012)

By coincidence, a student here has been looking at fracture of natural rubber, and has had the same experience that you mentioned.

Dear Zhigang,

1. I agree with you that “theoretical strength” based on
idealized crystal models does not exist for bulk material, which has a lot of imperfections
on various length scales. However, the samples of the same bulk material are “flawed”
similarly and, thus, can exhibit similar rupture strain/stress. For my students
I buy a set of rubber bands and stretch them up to rupture (can be painful).
The rupture occurs normally when the length of the band is about seven times its
initial length. Try yourself and you’ll see that strength for such imperfect
material exists. Your example of silica glass which is very sensitive to
imperfections is probably more an exception than a rule. I guess most engineering
materials are not sensitive to small imperfections (I am not talking about
visible cracks, notches etc).

2. People say that LEFM is useful for engineers. May be. For
my taste, the theory is highly controversial. You know, they say that all
models are wrong but some are useful. May be LEFM is a nice illustration of
this wise saying .

Dear Kosta: Thank you very much for the comments. Quick responses to your comments.

1. What's wrong with using linear elasticity in design for strength? Indeed, linear elasticity is an approximate theory, and has its merits and failings. The question in this opening lecture of fracture mechanics is specific: does linear elasticity work in design for strength. I outline the commonly used procedure, and discuss why the procedure does not work. For a brittle material like silica glass, the stress predicted from linear elasticity is accurate all the way close to atomic scale. The growth of a crack corresponds to unzipping a plane of atomic bonds. The tip of the crack concentrates stress. The stress at the tip of the crack is sensitive to flaws. Experimentally measured strength of silica glass ranges from less than 10 MPa to higher than 5 GPa.

2. Is fracture mechanics a theology? No. Linear elasticity does predict singular stress field at a sharp tip of a crack. Irwin and others, however, devised a way to use this singular crack-tip field. The outcome is linear elastic fracture mechanics. I will describe Irwin's idea later in the course. Section 1.2 of the following paper gives a brief description.

G. Bao and Z. Suo, " Remarks on crack-bridging concepts," Applied Mechanics Review. 45, 355-366 (1992).

Dear Zhigang

Welcome back to physics

Your attack on linear elasticity is incomplete:

1. Linear elasticity is a very successful computational approximation
rather than a physical theory – it prescribes stressing under rigid body
motion.

2. Linear elasticity produces singularities which do not
really exist. The whole body of linear elastic fracture mechanics is built upon
these imaginary singularities. Should we call this theology?

Strength is a separate issue. I do think that strength
exists because various samples of the same material show similar critical
stresses in tests.

I agree completely that material failure must be a part of
the constitutive description.

Kosta