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DTD Handbook

Handbook for Damage Tolerant Design

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    • Sections
      • 1. Introduction
      • 2. Fundamentals of Damage Tolerance
        • 0. Fundamentals of Damage Tolerance
        • 1. Introduction to Damage Concepts and Behavior
        • 2. Fracture Mechanics Fundamentals
        • 3. Residual Strength Methodology
        • 4. Life Prediction Methodology
          • 0. Life Prediction Methodology
          • 1. Initial Flaw Distribution
          • 2. Usage
          • 3. Material Properties
          • 4. Crack Tip Stress Intensity Factor Analysis
          • 5. Damage Integration Models
          • 6. Failure Criteria
        • 5. Deterministic Versus Proabilistic Approaches
        • 6. Computer Codes
        • 7. Achieving Confidence in Life Prediction Methodology
        • 8. References
      • 3. Damage Size Characterizations
      • 4. Residual Strength
      • 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 2.4.5. Damage Integration Models

Rewriting Equation 2.4.1 such that the integration is conducted between the initial crack length (ao) and any intermediate crack length (aK) between ao and the critical crack length results in

(2.4.3)

where t(N) is the elapsed time (number of load cycles) corresponding to growing the crack ao to the intermediate crack length aK.  The next cycle of the applied stress (the N + 1 cycle) induces a crack length growth increment DaN+1.  The damage integration model provides the analysis capability to determine this crack length growth increment.  The growth increment DaN+1 is equated to the constant amplitude crack growth rate, which in turn is determined from a function of stress intensity factor range (DK) and stress ratio (R), i.e.,

(2.4.4)

The stress intensity factor range and stress ratio in Equation 2.4.4 are determined by using the maximum and minimum stresses in the N+1 cycle of the given stress history and evaluating the stress intensity factor coefficients associated with the given structural geometry at the crack length aK.  Subsequent to the direct calculation of the two crack tip parameters DK and R, and prior to their insertion in Equation 2.4.4, DK and R are modified to account for the effect of prior load history using retardation models.  Retardation models account for high-to-low load interaction effects, i.e., the phenomena whereby the growth of a crack is slowed by application of a high load in the spectrum.  Failure to account for high-to-low load interaction via a retardation model leads to conservative (~2 to 5 times shorter) life. 

There are numerous functional forms of Equation 2.4.4 and numerous models describing retardation.  The following list describe the general scheme of the crack growth calculation.

Step 1 - Knowing crack length aK, determines the stress intensity factor coefficient, K/s.

Step 2 - For the given stress cycle, Ds, and the coefficient K/s, determine the stress intensity factor cycle, DK, and stress ratio R.

Step 3 - Utilizing the retardation model, modify the stress-intensity cycle DK and R to account for previous load history.

Step 4 - Determine the growth rate for the stress-intensity factor cycle to establish the crack growth increment.

Section 5 provides a current state-of-the-art summary of the procedures and techniques that are used in damage integration models.