Griffith

I teach Materials Science to undergraduate engineering students. We devote 2 or 3 lectures on Fracture where we mainly introduce Griffith's formula and the stress intensity formula. I face a difficulty in reconciling the two which I would lke to present here.

The Griffiths equation gives the fracture stress s for a given crack size c:

s=sqrt[2 E g/ (p c)] (1)

The stress concentration formula gives the max stress at the crack tip as

s=s0 [1+2 sqrt (c/r)] (2)

If we equate (2) to E/6, the theoretical cohesive stress, then s0 will again be the fracture stress:

s0 [1+2 sqrt (c/r)] =E/6 (3)

Now, here is my difficulty. Since (1) involves only c whereas (3) both c and r, it is possible to select a combination (c, r) such that (3) is satisfied but not (2). In other words (3) says that there is enough stress concentration at the crack tip to break the bonds but Griffith does not permit the crack to propagate. How is this resolved?

-rajesh