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 (K_ONSET). 

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 K_CR 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 K_R vs. Da or ÖJ_R vs. a.  The K_R 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 ÖJ_R 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:

1.                  Obtain K_eff  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.

2.                  Obtain the experimentally determined R-curve (K_R versus Da) for the sheet material (Figure 4.4.3b).

3.                  Determine the point of instability from the K curves of the structure and the K_R curves of the material as shown schematically in Figure 4.4.3c.

4.                  Obtain different values for the fracture strength and the corresponding crack lengths from step 3 and plot these points to establish the failure strength (s_f) crack length (a_c) 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 K_R curve or the ÖJ_R 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 K_R curve is applied.