Required Material Properties
All the eN fatigue properties are required input. In addition to the eN properties, C and γ in Figure 7 also have to be input. At least two pairs of C - γ have to be input to make at least two slopes in Figure 8. In Figure 7, C and γ are a function of the elementary material block size ρ* . Once ρ* is available, C - γ pairs can be calculated using Figure 7 together with eN properties. There are three methods to estimate ρ* .
Methods to Determine Elementary Material Block Size ρ*
Method 1: Using Fatigue Limit and Threshold Stress Intensity Range


This method provides a relationship between the elementary material block size and fatigue properties of the material. Unfortunately, it requires a very accurately determined threshold stress intensity range and fatigue limit which are not always readily available. Additional care must be also taken to ensure that the fatigue limit and threshold stress intensity range are obtained under the same stress ratio R. An additional ambiguity also arises from the fact that the stress intensity range is not the only parameter driving fatigue cracks. It has been pointed out by Vasudevan et. al.4 that there are two different threshold parameters, namely the maximum threshold stress intensity factor and the threshold stress intensity range and both should be simultaneously exceeded for the crack to grow. Therefore it is not certain which threshold should be used to determine the elementary material block size.
Lastly, the method described above neither proves nor disproves whether the elementary material block size, ρ* , is only a material constant or if it also depends on the applied load and specimen/crack geometry. It must also be verified that the ρ* value obtained from Figure 2 does not depend on the stress ratio at which the fatigue crack growth threshold and the fatigue limit were determined.
Method 2: Using the Experimental Fatigue Crack Growth Data Obtained at Various Stress Ratios
Because of several uncertainties inherent to Method 1: Using Fatigue Limit and Threshold Stress Intensity Range, this alternative method based on the experimental fatigue crack growth data has been proposed.

- Kr
- Residual stress intensity factor due to plastic deformaion
- σr
- Residual stress
- m
- Weight function depending on crack geometry
Since ρ* is the only unknown parameter in the equation above, it has to be such that all experimental constant amplitude FCG data points obtained at various stress ratios R should collapse onto one da/dN vs. Δκ ‘main’ curve as shown in Figure 4.

Constant amplitude fatigue crack growth data presented in terms of the applied stress intensity range and the total driving force.
![]() Figure 5 shows the initial FCG rate data without residual stress effect. It is impossible to describe using only one 'main' curve. |
![]() Figure 6 shows the FCG rate data in terms of the total two-parameter driving force corresponding to the first approximation of ρ* . |
![]() Figure 7 shows the FCG rate data in terms of the total two-parameter driving force with snakiest scatter. |
Assuming some value of ρ*=ρ*1 and performing all of the iterative steps (assume ρ* , find σr , find Kr , and find Δκ ), it is possible to present FCG rate data in terms of the total two-parameter driving force and fit by the mean (‘main') curve using the least square method (Figure 6). However, the scatter of experimental FCG rate data shown in Figure 6 is relatively large. Therefore, the usual error minimization problem has to be solved in order to find such elementary material block size, ρ* , that experimental FCG rates presented in terms of corresponding driving force have the smallest scatter (Figure 7).
It should be mentioned that the method described above does not explicitly provide, from the fatigue fracture point of view, information about the ρ* parameter as the elementary material block size. The ρ* parameter represents only an effective crack tip radius subsequently influencing the magnitude and distribution for the residual stress field.
It is also not clear whether the ρ* parameter is unique according to this method. The method can be applied only if sufficient experimental fatigue crack growth data is available (that is, constant amplitude da/dN data obtained at three stress ratios at least).
The advantage of using the method discussed in the current section is that parameters C and γ in the crack growth equation shown in Figure 7 from the Crack Growth Mechanism page are determined from experimental fatigue crack growth data and not from approximate expressions and smooth specimen fatigue data. It is clear that, by fitting the C and γ parameters into limited amount of experimental fatigue crack growth data, the final equation simulates all other data much better than the theoretically derived approximate formula shown in Figure 7 from Crack Growth Mechanism.
The collapsed experimental fatigue crack growth data is shown together with analytical and fitted ‘main’ curves for the same ρ* . Since parameters ‘ C ’ and ‘ γ ’ for analytical curve have been estimated in two limited cases where either plasticity or elasticity effects were omitted, the curve does not fit well the experimental FCG data in the region where both plasticity and elasticity are important. Therefore, it can be concluded that the equation in Figure 7 from Crack Growth Mechanism provides only an empirical relation between the instantaneous FCG rate and total SIFs. However, it is preferable to fit parameters ‘ C ’ and ‘ γ ’ based on the experimental FCG data.
Method 3: Using the Manson-Coffin Fatigue Strain-Life Curve and Limited Fatigue Crack Growth Data
The procedure resulting in the determination of the ρ* parameter is summarized below.
Assuming that experimental constant amplitude fatigue crack growth rate data, the corresponding applied stress intensity ranges (Figure 8), and the stress ratio R are available, it is possible to determine the applied stress intensity range and the maximum applied stress intensity factor for each particular data point.

Schematic of experimental fatigue crack growth rate data
As far as the stress state over the first elementary material block is concerned, it can be noticed that there is only one non-zero stress component. Therefore, the crack tip stress/strain analysis can be reduced to the uni-axial stress state.




It is important to note that the equations in Figure 9 and Figure 10 contain the maximum total stress intensity factor and the total stress intensity range, but not the applied ones. However, since the elementary material block size and corresponding residual stresses are not known yet, the applied stress intensity factors can be used only as the input for the first iteration.
![]() Figure 13 shows the elementary material block size as a function of the applied stress intensity range after first iteration |
![]() Figure 14 shows the elementary material block size as a function of the applied stress intensity range after two iterations |

Elementary material block size as a function of applied stress intensity range after n+1 iterations when the convergence was reached
Surprisingly, sometimes only two iterations are sufficient in order to obtain the same value of the ρ* parameter for all experimental da/dN−ΔK data points (that is, the ρ* value independent of the load configuration). The iteration process must be repeated in practice as many times as it is necessary to achieve some kind of convergence (see Figure 15) measured by the variation of individual of the average ρ* avr,n parameter in such a way that ρ* avr,n ≈ ρ* avr,n+1 with required accuracy, where ‘n’ is the number of iterations. The average value of the ρ* avr,n parameter is determined as the average of all results obtained for the entire population of experimental data points.
The method described above requires solving the system of five simultaneous nonlinear equations (see Figure 6, Figure 9, Figure 10, Figure 11, and Figure 12) for each experimental fatigue crack growth data point. The number of iterations necessary for obtaining the converged value of the ρ* parameter usually ranges from five to 10 iterations. The equation system can only be solved numerically. The method does not require large amount of experimental data and sufficient estimation of the ρ* parameter can be obtained using only few data points (>3)
It has been also shown that the elementary material block size obtained by the third method does not depend on the stress ratio R. In other words, the same value of the ρ* parameter should be obtained regardless of the stress ratio R at which the experimental constant amplitude fatigue crack growth data was generated.
Summary of the Methods to Determine the Elementary Material Block Size ρ*
First method | Second method | Third method | |
Al 7075-T6 | 3.45381E-06 | 4.06184E-06 | 4.35938E-06 |
Al 2324-T3 | 2.53967E-06 | 3.94914E-06 | 3.34592E-06 |
Because the ρ* parameters are not significantly different from each other regardless of the method used, it is difficult to recommend the best method; the choice of method depends on the data available. However, if sufficient constant fatigue crack growth data is available, the use of the second method is recommended because the resulting ρ* , C and γ parameters closely approximate the trend of basic experimental da/dN vs. Δκ data.