If one defines the relationship between the force (P) applied to the structure shown in
Figure 2.2.10 and the deformation it induces in the direction of load as
is the compliance, the inverse structural stiffness, which
varies as a function of crack length (area).
With the definitions given by Equation 2.2.25, the elastic strain energy
(V) can be written as
The change in V
simultaneous to dA and dP is
which leads to
Similar operations on changes in dL (=d(DL))
So that the input energy rate (G) based on Equation 2.2.9
Showing that the input energy rate is independent of the
variation of force during any incremental crack extension. Thus, Equation 2.2.30 reduces to
Equation 2.2.31 provides the basis for experimentally
evaluating the crack driving force using compliance measurements and clearly
shows that the rate of energy input is identically equal to the change in
elastic strain energy considering the loading force constant. When one conducts a similar analysis with
the displacement (D) and crack area (A) as independent variables, one finds
which means that the input energy rate is the negative of the
areal derivative of elastic strain energy considering the displacement constant
during crack extension. This is the
so-called fixed displacement condition.
The term strain energy release rate was assigned to G, the input energy rate, when it was realized that for cracked
elastic bodies Equation 2.2.30 and 2.2.31 were generally applicable.
Figure 2.2.12 describes the change
in elastic strain energy that occurs when a crack grows under fixed load and
fixed displacement conditions. It can
be noted that the difference between the change in elastic strain energy for
the two cases is the infinitesimal area 1/2dP*DL, shown
cross-hatched in Figure 2.2.12a. For the case of the fixed load condition (Figure 2.2.12a), the elastic strain energy is seen to
increase as the crack grows; the gain in elastic strain energy is greater than
the indicated loss (by a factor of 2).
For the case of the fixed displacement condition (Figure
2.2.12b), the elastic strain energy is seen to decrease as the crack grows;
only a loss is indicated.
2.2.12. Load-Displacement Diagrams
for the Structure Illustrated in Figure 2.2.10. The Diagram Shows the Changes that Occur in the Elastic Strain
Energy as a Crack Grows Under the Two Defined Conditions
Some important observations presented in the subsection are:
the general form of Equation 2.2.24 can be utilized to relate G and K;
G is equal to the
negative rate of change in the potential energy of deformation (Equation
G is related to the
areal rate of change in compliance (Equation 2.2.31).
Note that by combining Equations 2.2.24 and 2.2.12 or 2.2.31
the analyst and/or experimentalist have energy-based methods for obtaining
estimates of the stress-intensity factor.
These combinations are discussed in Section 18.104.22.168 (see, for example,