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DTD Handbook

Handbook for Damage Tolerant Design

  • DTDHandbook
    • About
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    • PDF Versions
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    • Sections
      • 1. Introduction
      • 2. Fundamentals of Damage Tolerance
      • 3. Damage Size Characterizations
      • 4. Residual Strength
      • 5. Analysis Of Damage Growth
        • 0. Analysis Of Damage Growth
        • 2. Variable-Amplitude Loading
          • 0. Variable-Amplitude Loading
          • 1. Retardation
          • 2. Integration Routines
          • 3. Cycle-by-Cycle Analysis
        • 3. Small Crack Behavior
        • 4. Stress Sequence Development
        • 5. Crack Growth Prediction
        • 6. References
      • 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 5.2.3. Cycle-by-Cycle Analysis

In general, the crack-growth-damage-integration procedure consists of the following steps, schematically outlined in Figure 5.2.12.

 

Figure 5.2.12.  Steps Required for Crack Growth Integration

Step 1.    The initial crack size follows from the damage tolerance assumptions as a1.  The stress range in the first cycle is Ds1.  Then determine DK1 = bDs1Öpa1by using the appropriate b for the given structural geometry and crack geometry.  Computer programs generally have a library of stress-intensity factors or schemes for tabular data input for determining the appropriate b.

Step 2.    Determine (da/dN)1, at DK1 from the da/dN -DK baseline information, taking into account the appropriate R value.  The da/dN - DK baseline information may use one of the crack growth equations discussed in Section 5.1.2.  The computer program may contain options for any of these equations, or it may use data in tabular form and interpolate between data points.  The crack extension Da1 in cycle 1 is

                       

               The new crack size will be a2 = a1 + Da1

Step 3.    The extent of the yield zone in Cycle 1 is determined as

                         

                        where  

                         for plane stress

                        or  for plane strain.

Step 4.    The crack size is now a2.  The stress range in the next cycle is Ds2.  Calculate DK with

                        .

Step 5.    Calculate the extent of the yield zone
         Y22 = a2 + rp2 .

Step 6.    If Y22 < Y2:

                  When using the Wheeler model, calculate Cp according to Equation 5.2.2.

                  When using the Generalized Willenborg model, calculate or and Reff according to Equations 5.2.3, and 5.2.5.

               Go to Step 9, skipping steps 7 and 8.

Step 7.    If Y22 > Y2, determine (da/dN)2 from DK2.  Determine the new crack size a3

                       

Step 8.    Replace Y2 by Y22 , which is now called Y2.  Replace aoL =a1 with aoL =a2.  Go to Step 10, skipping Step 9.

Step 9.    When using the Wheeler model, determine the amount of crack growth on the basis of DK2 from the da/dN - DK data.  Find the new crack size from

                       

When using the Generalized Willenborg model, determine the amount of crack growth using the DK and Reff value determine in Step 6 from the da/dN - DK data.  Determine the new crack size as

                       

Step 10.  Repeat Steps 4 through 9 for every following cycle, while for the ith cycle replacing a2 by ai and a3 by ai+1.

 

This routine of cycle-by-cycle integration is not always necessary.  The integration is faster if the crack size is increased stepwise in the following way.

·        At a certain crack size, the available information is ai, aoL, Y2.

·        Calculate Dai for the ith cycle in the same way as in Steps 4 through 9.

·        Calculate Daj+1, . . . , Daj, . . . , Dan for the following cycles but let the current crack size remain ai constant.  This eliminates recalculation of b every cycle.

·        Calculate Y2k for every cycle.  If Y2k > Y2, then replace Y2 by Y2k and call it Y2.  Then replace aoL by ai and call it aoL.

·        Sum the crack-growth increments to give:
           

·        Continue increasing j until Da exceeds a previously determined size or until j = n and the cycles are exhausted.  Then increment the crack size by 

                  a= ai + Da,

and repeat the procedure.

 

A reasonable size for the crack-growth increment is Da = 1/20 ai; this choice of increment typically keeps the change in K small.  It can also be based on the extent of the yield zone, e.g., Da = 1/10 (Y2 - ai).  The advantage of the incremental crack-growth procedure is especially obvious if series of constant-amplitude cycles occur.  Since the crack size (ai) is fixed, the stress intensity does not change.  Hence, each cycle produces the same amount of growth.  This means that all n constant-amplitude cycles can be treated as one cycle to give

           

The integration scheme is a matter of individual judgment, but may be dictated by available computer facilities.