Paris  gave one of the better descriptions of the
fracture energy balance equation associated with the stability of a cracking
process in a set of notes prepared for a short course given to the Boeing
Company in 1960. Paris simply described
the process of determining if a crack would extend as a comparison between the
Rate of Energy Input and the Rate at which Energy was absorbed or dissipated. This comparison is similar to performing as
analysis based on the Principle of virtual work. In equation form, Paris indicated
where the left hand side of Equation 2.2.8 represents the input
rate (as a function of crack area A)
and the right hand side represents the dissipation rate. If the input rate, the driving force G, is equal to the dissipation rate, the
resistance R, then the crack is in an
equilibrium (stable) position, i.e., it is ready to grow but doesn’t. If the driving force exceeds the resistance,
then the crack grows, an unstable position.
Since a crack will not heal itself, if the resistance is greater than
the driving force, then the crack is also stable.
The basis for Equation 2.2.8 was further described [Paris,
1960] so that the components are identified as:
where < and = imply stability while > implies
The driving force (input work rate, G) components are:
= the work done
by external forces on the body unit increase in crack area, dA.
= the elastic strain
energy released per unit increase in dA.
And the resistance (rate of dissipation, R) components are:
= surface energy absorbed
in creating a new surface area, dA.
= plastic work dissipated
throughout the body during an increase in surface area, dA.
While Equation 2.2.9 is most general and covers fractures that
initiate in either brittle or ductile materials, it is not always possible to
estimate the individual component terms.
For linear elastic materials, the terms can be estimated; and in fact,
this was accomplished by Griffith  forty years before Paris presented the
above general work rate analysis in 1960.
Before any further discussion of the work preceeding that of Paris,
however, several additional points need to be made about Equation 2.2.9. First, the component terms of the input
energy rate will be defined relative to a specific structural geometry and
loading configuration: the uniaxially loaded, center cracked panel shown in Figure 2.2.10.
Then the input energy rate (G)
will be related to the elastic strain energy.
Figure 2.2.10. Finite Width, Center Cracked Panel, Loaded
in Tension with Load P
The two components of the energy
input rate (G) are given by
the boundary force per increment of crack extension; and by
the decrease in the total elastic strain energy of the
plate. With these additional
definitions, it can be seen that G is
equal to the negative of the rate of change in the potential energy of