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Fatigue crack growth :: Total Materia Article
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Te aircraft industry has leaded te effort to understand and predict fatigue crack growth. They have developed te safe-life or fail-safe design approach. In this method, a component is designed in a way that if a crack forms, it will not grow to a critical size between specified inspection intervals. Thus, by knowing te material growth rate characteristics and with regular inspections, a cracked component may be kept in service for an extended useful life.
Te major share of te fatigue life of te component may be taken up in te propagation of crack. By applying fracture mechanics principles it is possible to predict te number of cycles spent in growing a crack to some specified length or to final failure.
Te aircraft industry has leaded te effort to understand and predict fatigue crack growth. They have developed te safe-life or fail-safe design approach. In this method, a component is designed in a way that if a crack forms, it will not grow to a critical size between specified inspection intervals. Thus, by knowing te material growth rate characteristics and with regular inspections, a cracked component may be kept in service for an extended useful life. This concept is shown schematically in Fig. 1.
Figure 1. Extended service life of a cracked component
Fatigue Crack Growth Curves
Typical constant amplitude crack propagation data are shown in Fig. 2. Te crack length, a, is plotted versus te corresponding number of cycles, N, at which te crack was measured.
Figure 2. Constant amplitude crack growth data
As shown, most of te life of te component is spent while te crack length is relatively small. In addition, te crack growth rate increases with increased applied stress.
Te crack growth rate, da/dN, is obtained by taking te derivative of te above crack length, a, versus cycles, N, curve. Two generally accepted numerical approaches for obtaining this derivative are te spline fitting method and te incremental polynomial method. These methods are explained in detail in many numerical methods textbooks. Values of log da/dN can then be plotted versus log DK, for a given crack length, using te equation
where Ds is te remote stress applied to te component as shown in Fig. 3.
Figure 3. Remote stress range
A plot of log da/dN versus log DK, a sigmoidal curve, is shown in Fig. 4. This curve may be divided into three regions. At low stress intensities, Region I, cracking behavior is associated with threshold, DKth, effects. In te mid-region, Region II, te curve is essentially linear. Many structures operate in this region. Finally, in te Region III, at high DK values, crack growth rates are extremely high and little fatigue life is involved.
Figure 4. Three regions of crack growth rate curve
Most of te current applications of LEFM concepts to describe crack growth behavior are associated with Region II. In this region te slope of te log da/dN versus log DK curve is approximately linear and lies roughly between 10-6 and 10-3 in/cycle. Many curve fits to this region have been suggested. Te Paris equation, which was proposed in te early 1960s, is te most widely accepted. In this equation
where C and m are material constants and DK is te stress intensity range Kmax - Kmin.
Values of te exponent, m, are usually between 3 and 4. These range from 2,3 to 6,7 with a sample average of m = 3,5. In addition, tests may be performed. ASTM E647 sets guidelines for these tests.
Te crack growth life, in terms of cycles to failure, may be calculated using Eq. (2). Te relation may be generally described by
Thus, cycles to failure, Nf, may be calculated as
where ai is te initial crack length and af is te final (critical) crack length. Using te Paris formulation,
Because DK is a function of te crack length and a correction factor that is dependent on crack length [see Eq. (1)], te integration above must often be solved numerically. As a first approximation, te correction factor can be calculated at te initial crack length and Eq. (4) can be evaluated in closed form.
As an example of closed form integration, fatigue life calculations for a small edge-crack in a large plate are performed below. In this case te correction factor, f(g) does not vary with crack length. Te stress intensity factor range is
Substituting into te Paris equation yields
Separating variables and integrating (for m
Before this equation may be solved, te final crack size, af, must be evaluated. This may be done using as follows:
For more complicated formulations of DK, where te correction factor varies with te crack length, a, iterative procedures may be required to solve for af in Eq. (8).
It is important to note that te fatigue-life estimation is strongly dependent on ai, and generally not sensitive to af (when ai«af). Large changes in af result in small changes of Nf as shown schematically in Fig. 5.
Figure 5. Effect of final crack size on life
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