Methods to Determine Elementary Material Block Size ρ*

Method 1: Using Fatigue Limit and Threshold Stress Intensity Range

This method requires knowledge of the fatigue limit and the threshold stress intensity range. It was postulated in this method that the fatigue crack would not grow under the applied stress intensity ranges equal to or less than the threshold stress intensity range, such as when

Δ K_{a p p l} \leq Δ K_{t h}

. Therefore, the stress range over the first elemental material block at the crack tip should be simultaneously equal to or less than the fatigue limit (that is,

Δ σ_{a p p l} \leq Δ σ_{t h}

). Since the fatigue limit (

Δ σ_{t h}

) is less than the yield limit, only the linear elastic analysis is required. According to the Creager-Paris solution the two material properties can be related as:

The elementary material block size,

ρ *

, can subsequently be expressed as:

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.

Since the mean stress effect has been already accounted for by using the SWT fatigue damage parameter, all experimental fatigue crack growth rate data points plotted as a function of the total two- parameter driving force,

Δ κ = {(K_{m a x, t o t})}^{p} {(Δ K_{t o t})}^{1 - p}

, should collapse onto one ’main’ curve. The total two-parameter driving force,

Δ κ

can be presented on the other hand as a function of the elementary material block size,

ρ *

.

Where,

$K_{r}$: Residual stress intensity factor due to plastic deformaion
$σ_{r}$: 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 $d a / d N$ vs. $Δ κ$ ‘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.

The ‘main’ curve shown in Figure 4 can be considered as theoretical idealization of the actual engineering model. In practice, it would be a set of points deviating around some mean value due to the natural scatter of the experimental fatigue crack growth data as shown in Figure 5.

Iteration process for the elementary material block size estimation based on Method 2 using linear ‘main’ curve.

Assuming some value of $ρ * = ρ *_{1}$ and performing all of the iterative steps (assume $ρ *$ , find $σ_{r}$ , find $K_{r}$ , and find $Δ κ$ ), 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 $d a / d N$ 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.

The combination of the Ramber-Osgood material stress-strain curve and the Neuber rule leads to the following equation:

This equation makes it possible to determine the maximum elastic-plastic stress over the first material block ahead of the crack tip as a function of the applied maximum stress intensity factor. A similar equation can be obtained for the elastic-plastic stress range produced by the unloading reversal:

The elastic-plastic strain range can be subsequently determined from the expanded by a factor of two cyclic stress/strain curves:

Finally, the maximum stress and the actual strain range have to be combined using the Smith-Watson- Topper (SWT) fatigue damage parameter in order to find the number of cycles required to break the first elementary material block:

The equations shown in Figure 6, Figure 9, Figure 10, Figure 11, and Figure 12 provide the solution to five unknown variables:

Δ σ

,

σ m a x

,

Δ ε

,

N

, and

ρ *

. Therefore, the elementary material block size

ρ *

can be determined for each particular point of the experimental fatigue crack growth rate curve.

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.

Subsequent solutions of the set of equations from Figure 6, Figure 9, Figure 10, Figure 11, and Figure 12 for each experimental FCG data point provide elementary material block sizes as a function of the applied stress intensity ranges (Figure 13, Figure 14) corresponding to specific measured fatigue crack growth rates. It can be observed that the elementary material block size is not constant and depends on the load level. Therefore, it contradicts the basic assumption of the model that the

ρ *

parameter was supposed to be a constant parameter characteristic for a given material. The apparent dependence of the

ρ *

parameter on the load is the result of using applied stress intensity factors in the set of equations from Figure 6, Figure 9, Figure 10, Figure 11, and Figure 12 without accounting for residual stresses induced in due course around the crack tip. Therefore, residual stresses induced ahead of the crack tip during the first iteration should be accounted for in the next iteration, resulting in a more accurate estimation of the

ρ *

parameter. This means that, after the first iteration, instead of being applied, the total stress intensity factors are used by accounting for the residual stress obtained during the preceding iteration.

This way, new values of the

ρ *

parameter are obtained after each iteration. The iteration procedure is complete when the same approximate value of the

ρ *

parameter is obtained for all experimental data points.

Surprisingly, sometimes only two iterations are sufficient in order to obtain the same value of the $ρ *$ parameter for all experimental $d a / d N - Δ K$ 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 $ρ * a v r, n$ parameter in such a way that $ρ * a v r, n \approx ρ * a v r, n + 1$ with required accuracy, where ‘n’ is the number of iterations. The average value of the $ρ * a v r, n$ 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.