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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
              • 0. Strain Energy Release Rate
              • 1. The Griffith-Irwin Energy Balance
              • 2. The Relationship between G, Compliance, and Elastic Strain Energy
            • 2. The J-Integral
            • 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 2.2.6.1.0. Strain Energy Release Rate

Paris [1960] gave one of the better descriptions of the fracture energy balance equation associated with the stability of a cracking process in a set of notes prepared for a short course given to the Boeing Company in 1960.  Paris simply described the process of determining if a crack would extend as a comparison between the Rate of Energy Input and the Rate at which Energy was absorbed or dissipated.  This comparison is similar to performing as analysis based on the Principle of virtual work.  In equation form, Paris indicated


 

 

(2.2.8)

where the left hand side of Equation 2.2.8 represents the input rate (as a function of crack area A) and the right hand side represents the dissipation rate.  If the input rate, the driving force G, is equal to the dissipation rate, the resistance R, then the crack is in an equilibrium (stable) position, i.e., it is ready to grow but doesn’t.  If the driving force exceeds the resistance, then the crack grows, an unstable position.  Since a crack will not heal itself, if the resistance is greater than the driving force, then the crack is also stable.

The basis for Equation 2.2.8 was further described [Paris, 1960] so that the components are identified as:

(2.2.9)

where < and = imply stability while > implies instability.

The driving force (input work rate, G) components are:

             = the work done by external forces on the body unit increase in crack area, dA.

             = the elastic strain energy released per unit increase in dA.

And the resistance (rate of dissipation, R) components are:

             = surface energy absorbed in creating a new surface area, dA.

             = plastic work dissipated throughout the body during an increase in surface area, dA.

While Equation 2.2.9 is most general and covers fractures that initiate in either brittle or ductile materials, it is not always possible to estimate the individual component terms.  For linear elastic materials, the terms can be estimated; and in fact, this was accomplished by Griffith [1921] forty years before Paris presented the above general work rate analysis in 1960.  Before any further discussion of the work preceeding that of Paris, however, several additional points need to be made about Equation 2.2.9.  First, the component terms of the input energy rate will be defined relative to a specific structural geometry and loading configuration: the uniaxially loaded, center cracked panel shown in Figure 2.2.10.  Then the input energy rate (G) will be related to the elastic strain energy.

 

 

Figure 2.2.10.  Finite Width, Center Cracked Panel, Loaded in Tension with Load P

The two components of the energy input rate (G) are given by

(2.2.10)

the boundary force per increment of crack extension; and by

(2.2.11)

the decrease in the total elastic strain energy of the plate.  With these additional definitions, it can be seen that G is equal to the negative of the rate of change in the potential energy of deformation (Us ), i.e.,

(2.2.12)