Page 48 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 28 Thursday, April 14, 2005 6:28 PM

28 Fracture Mechanics: Fundamentals and Applications

FIGURE 2.2 Elliptical hole in a flat plate.

Inglis showed that Equation (2.11) gives a good approximation for the stress concentration due to
a notch that is not elliptical except at the tip.
Equation (2.11) predicts an infinite stress at the tip of an infinitely sharp crack, where ρ = 0.
This result caused concern when it was first discovered, because no material is capable of with-
standing infinite stress. A material that contains a sharp crack should theoretically fail upon the
application of an infinitesimal load. The paradox of a sharp crack motivated Griffith [2] to develop
a fracture theory based on energy rather than local stress (Section 2.3).
An infinitely sharp crack in a continuum is a mathematical abstraction that is not relevant to
real materials, which are made of atoms. Metals, for instance, deform plastically, which causes an
initially sharp crack to blunt. In the absence of plastic deformation, the minimum radius a crack tip
can have is on the order of the atomic radius. By substituting ρ = x into Equation (2.11), we obtain
o
an estimate of the local stress concentration at the tip of an atomically sharp crack:

a
σ A σ = 2 x o (2.12)

If it is assumed that fracture occurs when σ = σ , Equation (2.12) can be set equal to Equation
c
A
(2.7), resulting in the following expression for the remote stress at failure:
E
/
σ =   4 γ  12 (2.13)
s
a 
f