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DTD Handbook

Handbook for Damage Tolerant Design

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    • About
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    • Sections
      • 1. Introduction
      • 2. Fundamentals of Damage Tolerance
      • 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
        • 0. Summary of Stress Intensity Factor Information
        • 1. Background of Stress Intensity Factors
        • 2. Methodology For Determining Stress Intensity Factors
          • 0. Methodology For Determining Stress Intensity Factors
          • 1. Principle of Superposition
          • 3. Finite Element Methods
            • 0. Finite Element Methods
            • 1. Direct Methods
            • 2. Indirect Methods
            • 3. Cracked Element Methods
        • 3. Selected Stress Intensity Factor Cases
        • 5. Computer Codes
        • 6. References
    • Examples

Section Direct Methods

The direct methods use the results of the general elastic solutions to the crack-tip stress and displacement fields.  For the Mode 1, the crack tip stresses can always be described by the equations



for plane stress


plane strain


where r and q are polar coordinates originating at the crack tip, and where x is the direction of the crack, y is perpendicular to the crack in the plane of the plate, and z is perpendicular to the plate surface.

If the stresses around the crack tip are calculated by means of finite-element analysis, the stress-intensity factor can be determined as


where i and j are used to represent various permutations of x and y.

By taking the stress calculated for an element not too far from the crack tip, the stress intensity follows from a substitution of this stress and the r and q of the element into Equation 11.2.30.  This can be done for any element in the crack tip vicinity.

Ideally, the same value of K should result from each substitution; however, the stress field equations are only valid in an area very close to the crack tip.  Also at some distance from the crack tip, nonsingular terms should be taken into account.  Consequently, the calculated K differs from the actual K.  The result can be improved [Chan, et al., 1970] by refining the finite-element mesh or by plotting the calculated K as a function of the distance of the element to the crack tip.  The resulting line should be extrapolated to the crack tip, since the crack tip equations are exact for r = 0.  Usually, the element at the crack tip should be discarded.  Since it is too close to the singularity, the calculated stresses are largely in error.  As a result, Equation 11.2.30 yields a K value that is more in error than those for more remote element, despite the neglect of the nonsingular terms.

Instead of the stresses, one can also use the displacements for the determination of K.  In general, the displacements of the crack edge (crack-opening displacements) are employed.  The Mode 1 and Mode 2 plane strain displacement equations are given by


and by


respectively.  The functions u and v represent the displacements in the x and y direction, respectively.  The crack tip polar coordinates r and q are chosen to coincide with the nodal points in the finite element mesh where displacements are desired.  Since the above elastic field equations are only valid in an area near the tip of the crack, the application should be restricted to that area.