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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 2.2.6.1.1. The Griffith-Irwin Energy Balance

The earliest analysis along the above lines was conducted by Griffith [1921] in 1920.  Griffith used the crack geometry and loading configuration shown in Figure 2.2.11 and assumed that the stress would be constant during any incremental growth of the crack.  Griffith also neglected the plastic work term in Equation 2.2.9 since he was trying to test his fracture hypothesis with a brittle material, glass.  Griffith’s analysis showed that the input work rate (G) was equal to the negative of the derivative of potential energy of deformation (Us ) as shown by Equation 2.2.12, and the resistance (R) was equal to the rate of increase in potential energy due to surface energy (UT ) during crack extension:

(2.2.13)

 

Figure 2.2.11.  Griffith Crack and Loading Configuration, Uniformly Loaded, Infinite Plate with a Center Crack of Length 2a

The potential energy of deformation (Us ) was found to be

(2.2.14)

while the potential energy due to surface tension (UT ) was given by

(2.2.15)

with surface tension T, and for plate thickness B.

The crack area A is given by

(2.2.16)

So the energy balance equation becomes

(2.2.17)

where E¢ is dependent on the stress state in the following way

E¢ =

E / (1-n2), for plane strain

(2.2.18)

E, for plane stress

 

Solving Equation 2.2.17 for the critical stress (scr) associated with the point at which the crack (a) would grow, one finds

(2.2.19)

Later, Irwin [1948] and Orowan [1949] incorporated the effects of crack tip plasticity into the analysis by taking the plastic dissipation term in Equation 2.2.9 as a constant, i.e. they assumed that

(2.2.20)

so that the resistance in Equation 2.2.17 was defined as the combination of surface energy absorbed and plastic work dissipated.  Thus, the Griffith-Irwin-Orowan energy balance equation became

(2.2.21)

and the critical stress was

(2.2.22)

Both Irwin and Orowan noted that the plastic dissipation rate for metals was at least a factor of 1000 greater than the surface energy absorption rate so that Equation 2.2.22 could be approximated by

(2.2.23)

Irwin also noted that the driving force or input energy rate G was directly related to the square of the magnitude of the crack tip stress field for the Griffith center crack geometry (Figure 2.2.11), i.e., that

(2.2.24)

Later, Irwin [1960] reported this result to be general for any cracked elastic body based upon a virtual work analysis of the stresses and displacements associated with crack tip behavior during an infinitesimal crack extension.