Many descriptions of the function f(DK, R) in Equation 5.1.1 have been proposed.  In the early literature [Pelloux, 1970; Erdogan, 1967; Toor, 1973; Gallagher, 1974], most of the descriptions were either based on physical models of the crack growth process (referred to as “laws”) or on equations that appeared to describe the trends in the data.  Currently, the fatigue crack growth rate (FCGR) descriptions are carefully selected to provide accurate mean trend descriptions of the specific data collected to support a materials evaluation or structural design.  Before introducing these more accurate FCGR descriptions, the Paris power law [Paris, 1964], the Walker equation [Walker, 1970], and Forman equations [Forman, et al., 1964] will be reviewed.

The Paris power law equation was initially proposed to describe the crack growth rate behavior in the central region for specific values of stress ratio.  This equation is given by the general form:

(5.1.2)

where C and p are experimentally determined constants.  Equation 5.1.2 is still extensively used to develop first order approximations of life behavior when only limited amounts of data are available.  The reader is cautioned that Equation 5.1.2, as well as any other FCGR description, should not be extrapolated beyond its limits of applicability without a great deal of care and experience.  Greater life prediction errors can result from data extrapolation errors than almost all other design methodology errors combined.

The Walker equation provided one of the first simple equations that accounted for the stress ratio shift.  It is a subtle modification of Equation 5.1.2 and is given by

(5.1.3)

where C, m, and p are empirical constants.  The exponent m typically ranges from 0.4 to 0.6 for many materials.  Because Equation 5.1.3 is a power law, it has been noted to be most useful in describing the central region of the growth rate behavior.

The Forman equation was initially proposed to describe both the central and high crack growth regions of the behavior.  To account for the acceleration of the cracking rates as the stress-intensity factors levels approached critical, the Paris power law equation was divided by a factor that would reach zero when the stress-intensity factor reached a critical level.  The general form of the Forman equation is:

(5.1.4)

where C, p, and K_c are experimentally evaluated for the given material and thickness.  Equation 5.1.4 can be rearranged to yield:

(5.1.5)

which shows that the equation has the capability to describe multiple stress ratio data sets.

The empirical constants in Equations 5.1.2 - 5.1.4 are typically derived using least square fitting procedures.  Note that the simplicity of Equations 5.1.2 and 5.1.3 allow for a graphical fit to the data on log-log coordinate paper and the direct evaluation of the constants from the graph.  The usefulness of Equations 5.1.2 - 5.1.4 comes from the ease in which their constants can be evaluated from available data, as well as the direct application of the equations to simplified life integration calculations.  When considering the general expression for crack growth life (N_f)

(5.1.6)

it is seen that the function f is simple for Equations 5.1.2 - 5.1.4.

One modeling procedure that has consistently shown itself to range among the most accurate FCGR descriptions for predicting lives is the table look-up scheme .  For life prediction purposes, many aircraft companies have gone to a table look-up scheme in which they describe crack growth rate as a function of DK for specific values of fatigue crack growth rate or vice versa, i.e., da/dN is described for specific values of DK. 

Table 5.1.1 summarizes the mean trend FCGR behavior of the 2219-T851 aluminum alloy employed by the ASTM Task Group E24.04.04.  Within the main body of Table 5.1.1, da/dN are presented as a function of pre-chosen DK levels for specific levels of stress ratio (or environment, etc.).  In the rows directly above and directly below the main body of the table, the data extreme values are defined.  In the bottom rows of the table, statistical summaries that define the accuracy of the mean trend (tabular) description relative to the FCGR data and with respect to life prediction (life prediction ratios based on original a vs. N data).  The RMSPE (root mean square percentage error) is a statistic that measures the deviation of fatigue crack growth rate data from the table; and, it is somewhat akin to the coefficient of (life) variation.

The mean trend data presented in the Damage Tolerant Design (Data) Handbook [1994] can be directly utilized with table look-up algorithms in crack growth life prediction computer codes.  These data might also be utilized with least square fitting procedures to generate wider ranging predictive schemes that account for the effects of stress ratio, frequency, environment, temperature, and other controlling conditions.

The Damage Tolerant Design (Data) Handbook provides crack-growth data for a variety of materials.  The data are presented in the form of graphs and tables, as shown in Figure 5.1.3.  Multiple parameter equation fitting should not be attempted if only limited sets of data are available.  In case limited data sets have to be used, a comparison should be made with similar alloys for which complete data are available, and curves may be fitted through the limited data sets on the basis of this comparison.

Table 5.1.1.  Example Fatigue Crack Growth Rate Table (2219-T851 Aluminum)

 

ASTM Task Group E24.04.04 on FCGR descriptions conducted two analytical round robin investigations of the utility of various FCGR descriptions that describe crack growth behavior [Miller, et al., 1981; Mueller, et al., 1981].  These round robin investigations have clearly demonstrated that FCGR descriptions which are classified as “good” from a life analysis standpoint must adequately represent the mean trend of the FCGR data.  Figure 5.1.7 outlines a general procedure whereby the FCGR behavior is first described by least square regression analysis (Figure 5.1.7a) and then the regression equation, in conjunction with the stress-intensity factor analysis for the test geometry, is used in integral form to obtain an estimate of the fatigue crack growth life N_f (Figure 5.1.7b).  In Figure 5.1.7a, the mean trend behavior is described along with bounds on the regression equation.  Those descriptions which fail to model the mean trend of the FCGR data, either because they are preconceived to have a specific form (sinh, power law, Forman, etc.) or due to a lack of care in performing the regression analysis, lead to life prediction errors that are biased or exhibit significant scatter.

Figure 5.1.7.  Description of FCGR Data Fitting and the Comparison of Predicted to Actual Behaviors

To support the first round robin, FCGR data from compact and center crack test geometries fabricated from 0.25 inch thick 2219-T851 aluminum alloy were supplied to the participants.  The tests were conducted between threshold and fracture toughness levels for five separate stress ratios (-1, 0.1, 0.3, 0.5, and 0.8).  A number of individuals from government, industry, and academia participated in the round robin (see Table 5.1.2) and chose to evaluate the ten (10) descriptions defined in Table 5.1.3.  Each participant was given FCGR data and asked to describe the mean trend of the behavior using equations or other procedures.  The participants then integrated their mean trend analysis to establish predicted life values.  They were each given the initial and final crack sizes as well as the loading conditions for these life predictions of center crack specimens and compact specimens.

Table 5.1.2.  Active Participants and their Organizations for

Round Robin Investigation [Miller, et al., 1981]

^*Chairman, ASTM Task Group E24.04.04 on FCGR Descriptions (1975 - 80)

⁺Chairman, ASTM Task Group E24.04.04 on FCGR Descriptions (1980 - 83)

 

One of the procedures utilized to evaluate the ten descriptions was to summarize the sixteen (16) life prediction ratios (life predicted divided by life measured, N^P_f /N_f, see Figure 5.1.7b) associated with each description.  The means and standard deviations for the life prediction ratios associated with each participant/FCGR description is presented in Table 5.1.4.

Table 5.1.3.  FCGR Descriptions for Round Robin Investigation

+ The hyperbolic sine model is listed twice because two separate organizations
 chose to evaluate this description.

 

The life prediction ratio (LPR) numbers in Table 5.1.3 can be interpreted by comparing the mean LPR to 1.0 and the standard deviation to 0.0.  A mean LPR less than 1.0 implies a conservative prediction.  A further interpretation of the results of the round-robin are also presented in Table 5.1.3 with the percentage of life prediction ratios that fall within the ranges of 0.80 and 1.20 and of 0.90 and 1.10.  Note that five descriptions were able to achieve LPR numbers between 0.80 and 1.20 for at least 80 percent of the number of predictions made.

Table 5.1.4.  Comparison of FCGR Descriptions