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DTD Handbook

Handbook for Damage Tolerant Design

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    • About
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    • Sections
      • 1. Introduction
      • 2. Fundamentals of Damage Tolerance
      • 3. Damage Size Characterizations
      • 4. Residual Strength
        • 0. Residual Strength
        • 1. Introduction
        • 2. Failure Criteria
        • 3. Residual Strength Capability
        • 4. Single Load Path Structure
          • 0. Single Load Path Structure
          • 1. Abrupt Fracture
          • 2. Tearing Fracture
        • 5. Built-Up Structures
        • 6. References
      • 5. Analysis Of Damage Growth
      • 6. Examples of Damage Tolerant Analyses
      • 7. Damage Tolerance Testing
      • 8. Force Management and Sustainment Engineering
      • 9. Structural Repairs
      • 10. Guidelines for Damage Tolerance Design and Fracture Control Planning
      • 11. Summary of Stress Intensity Factor Information
    • Examples

Section 4.4.2. Tearing Fracture

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:

1.                  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.

2.                  Obtain the experimentally determined R-curve (KR 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 KR 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 (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.

 

 

EXAMPLE 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.

 

SOLUTION:

As the first step, the appropriate expression for the stress-intensity factor is obtained from Section 11:

           

and b is given in Section 11.

The following figure describes the variation in stress-intensity factor with crack length and stress level.

 

Stress-Intensity Factor Relationship for Various Values of Applied Stress

The 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.

For 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. 

At 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.


 

Resistance Curve for 7075-T73 Aluminum for a Thickness of 0.063 Inches

 

Matching the R-Curve and Stress-Intensity Factor Curves

The 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