Materials with medium or high fracture toughness exhibit a type
of subcritical crack extension behavior prior to reaching the maximum load
carrying capacity of the structure.
When a limited amount of yielding occurs in front of the crack tip, the
initial extension of an existing crack in these materials will be slow and
stable threshold values of the stress-intensity factor (KONSET).
To understand this behavior,
consider an unreinforced, center-cracked panel. The stress-intensity factor (K)
at the crack tip increases linearly with the value of the normal tensile stress
component acting on the structure for a stationary crack. As the K
level increases, some point (point A) will be reached at which the crack length
will begin to extend as shown in Figure 4.4.2. The crack will extend gradually as the load
continues to increase, until reaching the critical size at which the crack
extension becomes unstable (point B in Figure 4.4.2). The point of crack initiation and
instability are determined by the appropriate failure criteria.
Figure 4.4.2. Diagrams Showing Onset of Unstable Crack
Growth for Conditions of Limited or Extensive Crack Extension
When the subcritical growth of the crack, as shown in Figure 4.4.2a between the points A and B, is not
significant, the fracture toughness criterion KCR values can be used in the analysis. In this case, fracture is assumed to occur
immediately after the start of crack extension as under abrupt failure
conditions. However, for materials
exhibiting substantial crack growth between points A and B as shown in Figure 4.4.2b, the crack resistance curve approach can
be used in the residual strength analysis.
The crack resistance (R) curve
approach might be based on either KR
vs. Da or ÖJR
vs. a. The KR vs. Da curve is normally used when the fracture strength
is associated with stress levels below net section yield conditions; in other
words, when limited crack tip plasticity occurs prior to fracture. The ÖJR vs. a curve is used for those conditions
where the fracture strength is expected to result in gross yielding.
In the calculation of residual strength when the cracked
structure exhibits a tearing instability, one normally follows these steps:
Obtain Keff values for the
structure for various crack lengths and applied stresses using a suitable
plastic zone model (e.g. Dugdale Model).
Evaluation of the K values involves methods described in Section
11. Plot K versus a curves for
various applied stresses as shown in Figure 4.4.3a.
Obtain the experimentally determined R-curve (KR
versus Da) for the sheet
material (Figure 4.4.3b).
Determine the point of instability from the K curves of the structure and the KR curves of the material as
shown schematically in Figure 4.4.3c.
Obtain different values for the fracture strength and the
corresponding crack lengths from step 3 and plot these points to establish the
failure strength (sf) crack
length (ac) curve. This provides the necessary residual
strength diagram of the structure.
Figure 4.4.3. Steps Associated with Calculating Residual
Strength of Cracked Structures with Tearing Fractures
The residual strength diagram for intermediate or high fracture
toughness materials can be constructed by using either the KR curve or the ÖJR method. To understand the use of the R-curve failure criteria in evaluating
the residual strength, consider the following example in which failure
criterion based on the KR
curve is applied.
4.4.2 Residual Strength of
Tearing Radial Hole Crack
Construct the residual
strength diagram for a large and relatively thin (0.063 in.) plate of 7075-T73
aluminum alloy having a through crack emanating (radially) from a hole with a
diameter (D) equal to one inch, such
as illustrated here. Assume the
material inhibits a limited amount of crack tip yielding. Also calculate the crack length associated
with a fracture strength associated with a crack length of 2.0 inch.
As the first step, the
appropriate expression for the stress-intensity factor is obtained from Section
and b is given in Section 11.
following figure describes the variation in stress-intensity factor with crack
length and stress level.
Factor Relationship for Various Values of Applied Stress
next step is to consider the appropriate failure criterion. The given geometry is a thin sheet and the
material exhibits limited crack tip yielding behavior. Therefore, the R-curve method based on KR
values can be applied to evaluate the fracture strength.
the given 7075-T73 aluminum alloy material (0.063 inch thick), an
experimentally obtained R-curve is shown here.
By superposing the R-curve
onto the plot obtained in step one, as explained in Section 4.2.1, the points
where the R-curve is tangent to the K-curves are obtained.
these points the failure criterion, i.e. K
- KR and ÐK/Ða = ÐKR/Ða, is satisfied. The corresponding stress sc is the critical (fracture) stress at
which the initiation of rapid fracture will occur. From a diagram like this, we can obtain the critical initial
sizes of the crack and the respective fracture stresses.
Curve for 7075-T73 Aluminum for a Thickness of 0.063 Inches
the R-Curve and Stress-Intensity Factor
final step is to plot the sf
vs. ac curve. The required residual strength diagram is
shown next for the 7075-T73 aluminum plate with a crack emanating radially from
a hole. It can be seen from this figure
that the critical crack size for a 20 ksi operating stress level is equal to
4.0 inches. As can also be seen from
the figure, for an observed crack of 2.0 inches, the residual strength
available is 27 ksi.
Residual Strength Diagram
Obtained for Structure in Example 4.4.4