Page 49 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 29 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 29

Equation (2.13) must be viewed as a rough estimate of failure stress, because the continuum
assumption upon which the Inglis analysis is based is not valid at the atomic level. However, Gehlen
and Kanninen [3] obtained similar results from a numerical simulation of a crack in a two-
dimensional lattice, where discrete “atoms” were connected by nonlinear springs:

/
γ 
σ f α =   E a  12 (2.14)
s
where α is a constant, on the order of unity, which depends slightly on the assumed atomic force-
displacement law (Equation (2.2)).

2.3 THE GRIFFITH ENERGY BALANCE

According to the first law of thermodynamics, when a system goes from a nonequilibrium state to
equilibrium, there is a net decrease in energy. In 1920, Griffith applied this idea to the formation
of a crack [2]:
It may be supposed, for the present purpose, that the crack is formed by the sudden annihilation of
the tractions acting on its surface. At the instant following this operation, the strains, and therefore
the potential energy under consideration, have their original values; but in general, the new state is
not one of equilibrium. If it is not a state of equilibrium, then, by the theorem of minimum potential
energy, the potential energy is reduced by the attainment of equilibrium; if it is a state of equilibrium,
the energy does not change.

A crack can form (or an existing crack can grow) only if such a process causes the total energy
to decrease or remain constant. Thus the critical conditions for fracture can be defined as the point
where crack growth occurs under equilibrium conditions, with no net change in total energy.
Consider a plate subjected to a constant stress σ which contains a crack 2a long (Figure 2.3).
Assume that the plate width >> 2a and that plane stress conditions prevail. (Note that the plates
in Figure 2.2 and Figure 2.3 are identical when a >> b). In order for this crack to increase in size,
sufficient potential energy must be available in the plate to overcome the surface energy of the
material. The Griffith energy balance for an incremental increase in the crack area dA, under
equilibrium conditions, can be expressed in the following way:
dE = d ∏ + dW s =

dA dA dA 0 (2.15a)
or
d ∏ dW
− = s (2.15b)
dA dA

where
E = total energy
Π = potential energy supplied by the internal strain energy and external forces
W = work required to create new surfaces
s
For the cracked plate illustrated in Figure 2.3, Griffith used the stress analysis of Inglis [1] to
show that
aB
πσ 22
∏= ∏ − E (2.16)
o