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DTD Handbook

Handbook for Damage Tolerant Design

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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
          • 0. Failure Criteria
          • 1. Ultimate Strength
          • 2. Fracture Toughness - Abrupt Fracture
          • 3. Crack Growth Resistance – Tearing Fracture
            • 0. Crack Growth Resistance – Tearing Fracture
            • 1. The Apparent Fracture Toughness Approach
            • 2. The Resistance Curve Approach
            • 3. The J-Integral Resistance Curve Approach
        • 3. Residual Strength Capability
        • 4. Single Load Path Structure
        • 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 The J-Integral Resistance Curve Approach

The crack growth resistance curve (KR) has shown good promise for materials where limited (small-scale) yielding occurs in front of the crack tip.  Difficulties in estimating crack tip plasticity under large-scale yielding conditions, led Wilhem [1974] to an alternate failure criterion based on the J-integral [Rice, 1968].  For a basic introduction to the J-integral see Section 11.

Wilhem’s J-integral criterion is similar to the KR -curve criterion; it states that failure will occur when the following conditions are met:


where J is the value of the applied J-integral and JR is the value of the J-integral representing the material resistance to fracture.  The applied stress (sf) corresponding to Equation 4.2.5 is defined as the fracture stress.  Since the effect of large-scale yielding can be appropriately incorporated through a suitable elastic-plastic model in the estimation of J-integral, it becomes an effective parameter for predicting failure under plane stress conditions where the plastic zone size is significantly large.

The crack resistance curve for the tearing failure is now represented by ÖJR vs. Da curve instead of KR vs. Da curve.  The use of ÖJR rather than JR is justified by the fact that ÖJ is directly related to the stress-intensity factor for elastic behavior through the equation

J = K2/E¢


where E¢ is the elastic modulus (E) for plane stress conditions and E/(1-v2) for plane strain conditions.

For different levels of applied load, the J-integral can be computed using finite element techniques for the structure of interest for a series of different crack sizes; the ÖJ versus crack length curve is illustrated in Figure 4.2.12a for a constant level of applied stress.  It is noted that this curve will incorporate the influence of material properties (yield strength and strain hardening exponent) through the finite element analysis.  In a manner similar to the stress-intensity factor type resistance curve, i.e. the KR curve. The resistance curve based on ÖJR can be experimentally obtained [Griffis & Yoder, 1974; Verette & Wilhem, 1973].  A ÖJR versus crack movement (Da) curve, i.e. the J-integral resistance curve, is schematically illustrated in Figure 4.2.12b.  The failure criterion is also based on the tangency conditions between the ÖJ versus crack length curve and the ÖJR versus crack movement curve.  To obtain this condition, the ÖJR vs. Da curve can be superimposed on the plot of ÖJ vs. a curve such that at some crack length these two curves are tangent to each other as shown in Figure 4.2.12c.  The corresponding crack length then defines the critical crack size of the structure for the fracture stress, sf.

Figure 4.2.12.  Schematic Illustration of the Individual and Collective Parts of a JR Fracture Analysis