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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
          • 0. Fracture Mechanics Fundamentals
          • 1. Stress Intensity Factor – What It Is
          • 2. Application to Fracture
          • 3. Fracture Toughness - A Material Property
          • 4. Crack Tip Plastic Zone Size
          • 5. Application to Sub-critical Crack Growth
          • 6. Alternate Fracture Mechanics Analysis Methods
            • 0. Alternate Fracture Mechanics Analysis Methods
            • 1. Strain Energy Release Rate
            • 2. The J-Integral
              • 0. The J-Integral
              • 1. J-Integral Calculations
              • 2. Engineering Estimates of J
            • 3. Crack Opening Displacement
        • 3. Residual Strength Methodology
        • 4. Life Prediction Methodology
        • 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 2.2.6.2.0. The J-Integral

In 1968, Rice [1968b] published a paper describing a path independent integral (J) which was noted to be equal to the negative of the change in potential energy of deformation occurring during the infinitesimal growth of a crack in a nonlinear elastic material, i.e. he showed that

(2.2.33)

Rice’s path independent integral J was defined by [Rice, 1968a; 1968b]

(2.2.34)

where G is any contour surrounding the crack tip, traversing in a counter clockwise direction (see Figure 2.2.13), W is the strain energy density,  is the traction on G, and  is the displacement on an element along arc s.

 

Figure 2.2.13.  J-Integral Parameters Illustrated

Before elaborating on a detailed description of the parameters involved in the calculation of the J-Integral, it is useful to note that Equation 2.2.33 is the nonlinear elastic equivalent of Equation 2.2.12.  Thus, for linear elastic materials, J reduces to the value of the strain energy release rate, G, i.e.

J = G

(2.2.35)

and the J-integral is related to the stress-intensity factor through the expression

(2.2.36)

where E¢ is given by Equation 2.2.18.

Equations 2.2.35 and 2.2.36 are noted to be valid only when the material is behaving in a linear elastic fashion.  When values of the J-Integral are determined via Equation 2.2.34 using finite element methods applied to linear elastic cracked structures, Equation 2.2.36 provides the engineer with a simple energy-based method for obtaining stress-intensity factors as a function of crack length.

In the first subsection below, the calculations associated with developing the J-Integral for an elastic-plastic material are detailed.  In the second subsection, some engineering approximation methods for calculating the J-Integral are outlined.