This module calculates the fatigue crack growth rate and the cycles-to-failure assuming a Safe-Life design approach.

The majority of the fatigue life may be taken up in the propagation of a crack. By the use of fracture mechanics principles it is possible to predict the number of cycles spent growing a crack to some specified length or to final failure. The Safe-Life (or Fail-Safe) design approach is a common design approach to predict fatigue life in structures. In this method, a component is designed such that if a crack forms, it will not grow to a critical size between specified inspection intervals. Thus, by knowing the material growth rate characteristics and with regular inspections, a cracked component may be kept in service for an extended useful life.

Constant Amplitude Crack Growth Data

To obtain a fatigue crack growth curve, it is necessary to establish reliable fatigue crack growth rate data. Typically, a cracked test specimen is subjected to a constant amplitude cyclic load with a remote stress range given by:

Typical constant amplitude crack growth data are shown in Figure 1. The crack length a is plotted versus the corresponding number of cycles N at which the crack was measured. As shown in the figure, a majority of the life of the component is spent while the crack length is relatively small.

Figure 1. Constant amplitude crack growth data

Crack growth rate is defined as crack extension per cycle, da/dN. The crack growth rate is obtained by taking the slope of the crack growth curve at the crack length, a, as shown in Figure 2. Various crack growth rate curves can be generated by varying the magnitude of the cyclic loading and/or the size of the initial crack.

Figure 2. Fatigue crack growth rate, da/dN

Crack Growth and the Stress Intensity Factor

Consider a crack that is propagating in the presence of a constant amplitude cyclic stress intensity factor DK. A cyclic plastic zone forms at the crack tip, and the growing crack leaves behind a plastic wake. If the plastic zone is sufficiently small that it is embedded within an elastic singularity zone, the stress intensity factor may still give a good indication of the stress environment at the crack tip. If two different cracks have the same stress environment (i.e., the same stress intensity factor), they behave in the same manner and show equal rates of growth. Under such conditions the rate of fatigue crack growth per cycle, da/dN, is governed by the stress intensity factor range:

A typical plot of log da/dN versus log DK is shown in Figure 3. This curve may be divided into three regions, which are discussed in the following sections.

Figure 3. Three regions of fatigue crack growth rate

Region I

At low DK values, crack propagation is extremely slow. Conceivably there is a threshold stress intensity value DK_th below which there is no fatigue crack growth or the growth is too small to measure. Although experimental verification of the existence of this threshold is difficult, DK_th is usually shown to be between 5 and 15 ksi-in^1/2 for steels and between 3 and 6 ksi-in^1/2 for aluminum alloys. The fatigue threshold decreases with an increasing stress ratio R, where R is given by:

The fatigue threshold also depends on the frequency of loading and environmental conditions. Due to the sensitivity of the fatigue threshold to the environment and load history, the best method to determine the threshold value is through testing under conditions that simulate actual service conditions. Designing a component such that the DK for the service conditions would be below the fatigue threshold is desirable. Although this would ensure a low probability of fatigue failure, this is often impractical for design conditions due to either the low level of operating stress or the small crack size required. The threshold value may be useful when a component is subjected to low stress levels and a very large number of cycles. An example would be power trains that operate at very high speeds.

Region II

In the mid-region of stress intensities, the log da/dN versus log DK curve is essentially linear with the slope roughly ranging from 10^-6 inch/cycle to 10^-3 inch/cycle. Most structures operate in this region. Most of the current applications of linear elastic fracture mechanics (LEFM) concepts used to describe fatigue crack growth behavior are associated with Region II.

Region II is generally the largest region of the fatigue crack growth rate curve and many curve fits for this region have been suggested. The Paris and Erdogan formulation, which is commonly referred to as the Paris Equation or Paris' Law, was proposed in 1963 and is the most widely accepted. The Paris Equation is:

The material constants C and n can be found in literature or by performing tests (ASTM E647 sets guidelines for these tests). Values of the exponent n can range from 2.0 to 7.0 with most values being between 3.0 and 4.0.

Since the stress intensity factor uses a correction factor, b, that is usually a function of the crack length, the Paris Equation must often be integrated numerically. This module assumes b is a constant determined with the crack length equal to the initial crack length, a_i. Thus, the Paris Equation can be solved in closed form. It is important to note that the fatigue-life estimation is strongly dependent on a_i, and is generally not sensitive to the final crack length, a_f (when a_i << a_f). Large changes in a_f result in small changes in the calculated fatigue life. Therefore, using a constant b value does not significantly affect the accuracy of the solution. The Stress Intensity Factor module can be used to calculate values of b.

Region III

At high stress intensities, crack growth rates are extremely high and little fatigue life is involved. Region III is characterized by rapid, unstable crack growth. The crack growth rate accelerates as the maximum stress intensity factor approaches the fracture toughness of the material. In many practical engineering situations this region may be ignored because it does not affect the total crack propagation life. The point of transition from Region II and Region III behavior is dependent on the yield strength of the material, stress intensity factor, and stress ratio.

Forman's equation was developed to model Region III behavior and predicts the sharp upturn in the da/dN versus DK curve as the fracture toughness is approached. The Forman equation can be written as:

Forman's equation is more often used to model mean stress effects in Region II, which will be discussed in the next section.

Region III is of most interest when the crack propagation life is on the order of 1000 cycles or less. However, at high stress intensities, the effects of plasticity start to influence the crack growth rate because the plastic zone size become large compared to the dimensions of the crack. Thus the problem should be analyzed by an elastic-plastic fracture mechanics approach.

Stress Ratio Effects

The applied stress ratio R can have a significant effect on the crack growth rate. In general, for a constant DK, the more positive the stress ratio, the higher the crack growth rates. This stress ratio sensitivity is strongly dependent on the material. The Forman equation is consistent with test observations that as R increases, da/dN increases. Forman's equation is valid only when R > 0 (tension - tension loading).

Generally, it is believed when R < 0 (tension - compression loading), there is no significant change in growth rate compared to the R = 0 growth rate. Again this is material dependent, as some researchers have obtained data for certain materials which show higher growth rates for R < 0 loadings.

Walker Equation

Another method used to compensate for stress ratio effects is the Walker equation. The Walker equation is an enhancement to the Paris equation that provides a means to shift the crack growth rate curve as a function of R. The form of the Walker equation implemented in the ETBX Fatigue Crack Growth module is:

The shift in the crack growth rate is controlled by the term (1-R)^m-1. The material constant m is in the range (0 < m <= 1) and is obtained from curve fits of experimental data at various stress ratios. For R >= 0, (1-R) is less than 1 and thus the magnitude of shift is expected to decrease as m increases. For R < 0, Kmax is often used in place of DK so that the trend in the data shifting is consistent with respect to m.

When R = 0, the Forman and Walker equations give similar results as the Paris equation, which assumes that the da/dN curve depends only on DK.

Elber's Crack Closure Model

The Paris equation can be modified to include the dependence of the crack growth rate on the stress ratio by using Elber's Crack Closure technique. Elber noticed at low loads the compliance was close to that of an uncracked specimen. Elber believed that the low compliance was due to the contact between crack surfaces (i.e. crack closure) at low tensile loads. Elber proposed that crack closure occurs as a result of crack-tip plasticity. The plastic zone develops around the crack tip as the yield stress of the material is exceeded. As the crack grows, a wake of plastically deformed material is developed while the surrounding body remains elastic. As the component is unloaded, the plastically "stretched" material causes the crack surfaces to contact each other before zero load is reached. Elber further postulated that crack closure decreased the fatigue crack growth rate by reducing the effective stress intensity range.

Elber introduced a new stress intensity range, the effective stress intensity range, DK_eff, to be used in the Paris equation:

As shown in Figure 4, K_max is the maximum stress intensity and K_open is the stress intensity at which the crack opens. U is the empirical relationship between DK and DK_eff derived by Elber and confirmed by many subsequent researchers. This equation can be rearranged to define opening stress intensity factor in terms of R:

For 2024-T3 aluminum, Elber determined that the effective stress intensity factor closure parameter U was related to the stress ratio R by:

Elber found this empirical relationship to hold for -0.1 < R < 0.7. Since its original introduction, the coefficients of this formulation for aluminum have changed, but the general approach for modeling U remains in use today. To use the Elber's technique in the ETBX Fatigue Crack Growth module, the user must multiply either the b value or the applied stress range, Ds, by the appropriate U value. For general applications, the crack closure technique is valid only when R > 0.

Figure 4. Comparison of Applied and Effective Stress Intensity Range

Variable Amplitude Loading

Fatigue life prediction for constant amplitude loading is reasonably straightforward, provided that the fatigue crack growth constants are known. However the majority of engineering structures are subjected to a wide spectrum of stresses over their lifetime, and the life prediction is generally much more complicated. In contrast to constant amplitude loading where crack growth Da is dependent only on the present crack size and the applied load, under variable amplitude loading crack growth is also dependent on the preceding cyclic loading history. This is referred to as load interaction and it can significantly affect fatigue crack growth rates and consequently, fatigue lives. Therefore many theories and engineering methods have been proposed to account for the effects of variable amplitude loading.

Crack Retardation

The application of a single overload (a high loading in a sequence of low amplitude cycles) was observed to cause a significant decrease in the crack growth rate for a large number of cycles subsequent to the overload. This phenomenon is referred to as crack retardation. If the overload is large enough, crack arrest can occur and the growth of the fatigue crack stops completely. Crack retardation remains in effect for a period of loading after the overload. The number of cycles in this period has been shown to correspond to the plastic zone size developed due to the overload. The larger the overload plastic zone, the longer the crack retardation stays in effect. The growth rate resumes its earlier value once the crack has grown through the overload plastic zone.

The crack growth rate does not reach a minimum immediately after the overload is applied. Rather, the minimum is reached after the crack has grown a distance approximately one-eighth to one-fourth the distance into the overload plastic zone. This behavior is known as delayed retardation. A single compressive overload (termed underload) generally causes the acceleration of the crack growth rate. In addition, when an underload follows an overload, the amount of crack growth retardation is significantly reduced. It should be noted that periodic overloads are not always beneficial. In some low cycle fatigue tests, periodic overloads have been found to cause crack growth acceleration.

Several theories have been developed to explain crack retardation, including:

• Crack-tip blunting
  The crack-tip blunting theory states that the crack tip blunts during the overload and the stress concentration associated with the crack becomes less severe, resulting in a slower crack growth rate. However, this theory is not consistent with the delayed retardation behavior.
  
  
• Compressive residual stresses at the crack tip
  The compressive residual stresses at the crack tip theory states that after the overload, compressive residual stresses are developed at the crack tip due to the large plastic zone. The material in this zone undergoes a permanent stretching (deformation). Upon the removal of the overload, compressive stresses are developed as the elastic body surrounding the crack tip "squeezes" the overload plastic zone to its original size. This theory also does not predict delayed retardation, and predicts that maximum retardation occurs immediately after the overload.
  
  
• Crack closure effects
  The crack closure theory assumes that crack retardation and acceleration are caused by crack closure effects discussed above. Changes in load affect both DK_eff and K_open. The crack closure theory successfully predicts the observed delayed retardation behavior after a single overload. While the overload plastic zone is in front of the crack tip, the compressive stresses do not affect the crack opening stress. Although, once the crack has propagated into the overload plastic zone the compressive stresses act on the crack surfaces and retardation occurs.

Retardation following an overload is a complicated phenomenon. There are a number of empirical models for retardation, which contain one or more curve-fitting parameters that must be obtained experimentally. The ETBX Crack Growth Module provides two commonly used crack-tip plasticity models to determine the crack growth rate under variable amplitude loading: the Wheeler model and the Willenborg model. Other statistical models and crack closure models exist (see references). Crack tip plasticity models are based on the assumption the crack growth rates under variable amplitude loading can be related to the interaction of the crack tip plastic zones. These models predict the retardation of the crack growth rate by modifying the constant amplitude growth rates discussed above.

Wheeler Model

In the Wheeler model, the constant amplitude da/dN relationship is multiplied by a retardation parameter, Cp:

The retardation parameter is a function of the ratio of the current plastic zone size, r_yi, to the plastic zone size created by the overload, r_o:

where:

r_yi =

cyclic plastic zone size due to the ith loading cycle

a_p =

sum of the crack length at which the overload occurred (a_o) and the overload plastic zone size (r_o)

a_i =

crack length at ith loading cycle

p =

empirically determined shaping parameter

First order estimates, ry, of the corresponding plastic zone sizes should be used.

The Wheeler model predicts that retardation decreases proportionally to the penetration of the crack into the overload plastic zone with the maximum retardation occurring immediately after the overload. As shown in Figure 5, crack retardation is assumed to occur as long as the current plastic zone size is within the plastic zone created by the overload. As soon as the boundary of the current plastic zone touches the boundary of the overload zone, retardation is assumed to cease and Cp = 1.0. A major disadvantage of this model is the empirical constant, p, which is required to curve fit the retardation parameter to test data.

Figure 5. The Wheeler model for crack retardation

Willenborg Model

The Willenborg Model is based on the assumption that crack growth retardation is caused by compressive residual stresses acting at the crack tip. The Willenborg model uses an effective stress intensity factor, which is the applied stress intensity reduced by the compressive residual stress intensity K_red:

The model states that the compressive residual stress K_red is caused by the elastic body surrounding the overload plastic zone and is equal to the difference between the maximum stress intensity K_max occurring at cycle i and the stress intensity K_max* required to produce a yield zone whose boundary r_p* just touches the overload yield zone boundary r_o:

and:

This relationship is shown graphically in Figure 6. Assuming that the plastic zone r_p at cycle i is:

the compressive residual stress intensity factor K_red can be expressed in terms of K_max:

In the Generalized Willenborg Model, K_red can be modified by a shaping factor f to improve correlation to crack growth data for a particular material:

Crack growth rates are calculated by substituting the effective (local) stress intensity factors K_max,eff and K_min,eff and effective stress ratio R_eff into a crack growth equation that relates the influence of stress intensity range and stress ratio such as Walker's equation or Foreman's equation.

Note that the effective (local) stress intensity range equals the remote stress intensity range:

Thus the Willenborg Model does not affect crack growth rates calculated from the basic Paris Equation, which assumes that the da/dN curve depends only on DK.

Figure 6. The Willenborg model load interaction zone

Recall from the introduction to the Willenborg Model the following equations:

and

The Willenborg Model predicts zero effective stress intensity, or shut-off, when Kred is equal in magnitude to the remote stress intensity Kmax. Immediately following an overload, Da is usually close to zero such that:

Thus Kmax,eff is approximately zero when the maximum stress intensity due to the overload (Kmax,OL) is twice Kmax. Test results have shown that the actual crack growth shut-off ratio is somewhat greater than two. The generalized Willenborg Model corrects the prediction of the overload to maximum load ratio required to halt crack growth by introducing a correlation factor f:

When shut-off occurs, Da is equal to zero and Kmax,eff = Kmax,th (threshold stress intensity for R=0) so that:

Expressing f in terms of the shut-off overload ratio (Kmax,OL / Kmax) gives:

The shut-off overload ratio is typically obtained from test for a specific material and thickness. While ideally it would not be affected by load spectrum or stress level, that is not always the case. The exact value may need to be adjusted to correlate life predictions with test results. Shut-off overload ratios for some common engineering materials are:

• Aluminum = 3.0
• Steel = 2.0
• Titanium = 2.7

The Fatigue Crack Growth module input form is shown in Figure 7. The fatigue crack growth problem is defined by first selecting the proper type of solution from the Solution Type pull-down menu. ETBX will automatically update the inputs fields to match the required parameters for the selected solution.

Stress intensity correction factors b can be input in two ways. If the Constant Beta button is selected, a Beta input field will appear in the Crack Growth Model input group as shown in Figure 7 below. To define multiple values of b as a function of crack size, select the Table button and open the Beta Table by clicking on the button labeled Define Beta vs Crack Size. An example of the Beta table is shown in Figure 8. Data entry rules are as follows:

• Crack sizes must be entered in ascending order
• Crack sizes must be numeric values that are greater than or equal to zero
• For each crack size, a corresponding beta factor must be entered
• At least one beta factor must be defined, in the first row of the table
• An empty row terminates input (no data is read after an empty row)
• A maximum of 500 beta factors can be entered in the table

ETBX uses linear interpolation to obtain Beta factors for a crack size within the bounds of the supplied beta table. Therefore the accuracy of the solution will depend on the number of data points in the table. For crack sizes outside the range of the table, the nearest value in the table will be used. Data will not be extrapolated. It is important that the beta table covers the expected range of crack sizes in the solution.

Figure 7: The Fatigue Crack Growth module input form

Figure 8: Stress intensity correction factor Beta table

Solutions

There are currently 13 solution types offered by the Fatigue Crack Growth module. Some solutions are known to work better than others in correlating to actual fatigue test data, but there are academic and practical benefits associated with producing results for various types of crack growth models. Whenever possible, verification tests should be used to test the soultions and determine the appropriate parameters. These solutions and the underlying models are provided at the user’s discretion and responsibility.

The 13 available solutions are:

1. Paris Equation - Find Crack Length af
2. Paris Equation - Find Fatigue Life Nf
3. Forman Equation - Find Fatigue Life Nf
4. Walker Equation - Find Crack Length af
5. Walker Equation - Find Fatigue Life Nf
6. Paris/Wheeler - Find Crack Length af
7. Paris/Wheeler - Find Fatigue Life Nf
8. Forman / Wheeler - Find Fatigue Life Nf
9. Forman / Willenborg - Find Fatigue Life Nf
10. Walker / Wheeler - Find Fatigue Life af
11. Walker / Wheeler - Find Fatigue Life Nf
12. Walker / Willenborg - Find Fatigue Life af
13. Walker / Willenborg - Find Fatigue Life Nf

Input Parameters

Ni =

Initial number of cycles

Nf =

Final number of cycles

ai =

Initial crack length

af =

Ffinal crack length

Ns =

Number of steps in the solution

Smax =

Maximum applied (remote) stress

DS =

Stress range Ds

R =

Stress ratio

Beta =

Dimensionless stress intensity correction factor b, which is dependent on specimen and crack geometry

C =

Crack growth rate coefficient

m =

Crack growth rate exponent (used in Walker equation)

n =

Paris crack growth rate exponent

Kc =

Fracture toughness

rp =

Plastic zone size

ro =

Overload plastic zone size

p =

Wheeler exponent

phi =

Empirically determined shaping parameter f

Units

Due to the empirical nature of crack growth rate equations, the Fatigue Crack Growth module does not support ETBX unit conversions. The user must be consistent in the use of units to get correct answers. The output data will be consistent with the input units. For example, if the stress were entered in ksi, the initial crack length in inches, and C in units of ksi and inches, then the final crack size will be calculated in inches.

• All problem inputs, including material properties and load spectrum
• The final crack length, af, or the fatigue life, Nf, depending on the selected solution
• The stress intensity range DK and maximum stress intensity Kmax
• The crack growth rate da/dN

If crack retardation is selected, the following additional data is output:

• Plastic zone size rp* required to reach the overload yield zone boundary (rp* = ao+ro-ai)
• The retardation parameter Cp (Wheeler model)
• The reduction in local stress intensity Kred (Willenborg model)
• The effective stress ratio Reff and the effective stress intensity Kmax,eff (Willenborg model)

Figure 9. Module tabulated results.

This section provides links to sample problems to show users how to use some of the features described in the documentation.

• Example 1 - Paris solution with constant beta
• Example 2 - Paris solution with constant beta
• Example 3 - Paris solution with beta table
• Example 4 - Walker solution at multiple R values