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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 2.2.6.1.2. The Relationship between G, Compliance, and Elastic Strain Energy

If one defines the relationship between the force (P) applied to the structure shown in Figure 2.2.10 and the deformation it induces in the direction of load as

DL = C × P

(2.2.25)

where

C = C(A)

(2.2.25a)

is the compliance, the inverse structural stiffness, which varies as a function of crack length (area).  With the definitions given by Equation 2.2.25, the elastic strain energy (V) can be written as

(2.2.26)

The change in V simultaneous to dA and dP is

(2.2.27)

which leads to

(2.2.28)

Similar operations on changes in dL (=d(DL)) lead to

(2.2.29)

So that the input energy rate (G) based on Equation 2.2.9 becomes

(2.2.30)

Showing that the input energy rate is independent of the variation of force during any incremental crack extension.  Thus, Equation 2.2.30 reduces to

(2.2.31)

Equation 2.2.31 provides the basis for experimentally evaluating the crack driving force using compliance measurements and clearly shows that the rate of energy input is identically equal to the change in elastic strain energy considering the loading force constant.  When one conducts a similar analysis with the displacement (D) and crack area (A) as independent variables, one finds that

(2.2.32)

which means that the input energy rate is the negative of the areal derivative of elastic strain energy considering the displacement constant during crack extension.  This is the so-called fixed displacement condition.  The term strain energy release rate was assigned to G, the input energy rate, when it was realized that for cracked elastic bodies Equation 2.2.30 and 2.2.31 were generally applicable.

Figure 2.2.12 describes the change in elastic strain energy that occurs when a crack grows under fixed load and fixed displacement conditions.  It can be noted that the difference between the change in elastic strain energy for the two cases is the infinitesimal area 1/2dP*DL, shown cross-hatched in Figure 2.2.12a.  For the case of the fixed load condition (Figure 2.2.12a), the elastic strain energy is seen to increase as the crack grows; the gain in elastic strain energy is greater than the indicated loss (by a factor of 2).  For the case of the fixed displacement condition (Figure 2.2.12b), the elastic strain energy is seen to decrease as the crack grows; only a loss is indicated.

 

Figure 2.2.12.  Load-Displacement Diagrams for the Structure Illustrated in Figure 2.2.10.  The Diagram Shows the Changes that Occur in the Elastic Strain Energy as a Crack Grows Under the Two Defined Conditions

Some important observations presented in the subsection are:

(a)          the general form of Equation 2.2.24 can be utilized to relate G and K;

(b)         G is equal to the negative rate of change in the potential energy of deformation (Equation 2.2.12); and

(c)          G is related to the areal rate of change in compliance (Equation 2.2.31). 

Note that by combining Equations 2.2.24 and 2.2.12 or 2.2.31 the analyst and/or experimentalist have energy-based methods for obtaining estimates of the stress-intensity factor.  These combinations are discussed in Section 11.2.1.4 (see, for example, Equation 11.2.25).