Page 59 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Linear Elastic Fracture Mechanics 39

The conditions for stable crack growth can be expressed as follows:

G = R (2.32a)

and

dG ≤ dR
da da (2.32b)

Unstable crack growth occurs when

dG > dR
da da (2.33)

When the resistance curve is flat, as in Figure 2.10(a), one can define a critical value of
energy release rate G , unambiguously. A material with a rising R curve, however, cannot be
c
uniquely characterized with a single toughness value. According to Equation (2.33) a flawed
structure fails when the driving force curve is tangent to the R curve, but this point of tangency
depends on the shape of the driving force curve, which depends on the configuration of the
structure. The driving force curve for the through crack configuration is linear, but G in the DCB
2
specimen (Example 2.2) varies with a ; these two configurations would have different G values
c
for a given R curve.
Materials with rising R curves can be characterized by the value of G at the initiation of the
crack growth. Although the initiation toughness is usually not sensitive to structural geometry, there
are other problems with this measurement. It is virtually impossible to determine the precise moment
of crack initiation in most materials. An engineering definition of initiation, analogous to the 0.2%
offset yield strength in tensile tests, is usually required. Another limitation of initiation toughness
is that it characterizes only the onset of crack growth; it provides no information on the shape of
the R curve.

2.5.1 REASONS FOR THE R CURVE SHAPE

Some materials exhibit a rising R curve, while the R curve for other materials is flat. The shape of
the R curve depends on the material behavior and, to a lesser extent, on the configuration of the
cracked structure.
The R curve for an ideally brittle material is flat because the surface energy is an invariant
material property. When nonlinear material behavior accompanies fracture, however, the R curve
can take on a variety of shapes. For example, ductile fracture in metals usually results in a rising
R curve; a plastic zone at the tip of the crack increases in size as the crack grows. The driving
force must increase in such materials to maintain the crack growth. If the cracked body is infinite
(i.e., if the plastic zone is small compared to the relevant dimensions of the body) the plastic zone
size and R eventually reach steady-state values, and the R curve becomes flat with further growth
(see Section 3.5.2).
Some materials can display a falling R curve. When a metal fails by cleavage, for example,
the material resistance is provided by the surface energy and local plastic dissipation, as illustrated
in Figure 2.6(b). The R curve would be relatively flat if the crack growth were stable. However,
cleavage propagation is normally unstable; the material near the tip of the growing crack is subject
to very high strain rates, which suppress plastic deformation. Thus, the resistance of a rapidly
growing cleavage crack is less than the initial resistance at the onset of fracture.