• DTDHandbook
• Contact
• Contributors
• 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.