Kriging for Fatigue Crack Modeling


Introduction

The accuracy of modeling fatigue crack growth is intimately related to the chosen increment of crack growth chosen for a given model. In the literature, the forward Euler method is commonly used to determine the crack growth magnitude for modeling cyclic growth or as a means to backward calculate the number of elapsed cycles for a constant growth increment. For sufficient accuracy to be obtained a very small step size much be chosen, which is not computationally attractive. Here, kriging is used as a tool to extrapolate into the future, allowing for the midpoint integration rule to be used without additional finite element simulations. The resulting prediction is more accurate than the forward Euler method, enabling larger step sizes without a loss of accuracy.

Kriging Integration Codes

The SURROGATES Toolbox is required for the use of these codes. It can be downloaded for free online. These codes were verified in MATLAB R2009a using SURROGATES Toolbox Version 2.1, Release 4.

Center Crack in an Infinite Plate (Constant da, Constant dN)

Kriging

Kriging1 is a way to fit data such that one can interpolate and extrapolate data away from the provided data points. Unlike a polynomial fit, kriging assumes that all data points are exact such that the resulting approximation passes directly through the data points. There is also a prediction of the uncertainty between data points as shown below in grey.


Kriging is used here to interpolate ahead of the current data point to allow for higher-order approximations to the governing ordinary differential equation governing fatigue crack growth2. In the case of a fixed increment of crack growth kriging can also be used to recover the elapsed cycles between data points with increased accuracy.

Example

For an example problem the center crack in an infinite plate is considered. This geometry has the well known stress intensity factor of the following form.


The Paris model3 of fatigue crack growth can be integrated giving a closed for analytical solution for the crack growth at at a given cycle as follows.


Here a comparison of the accuracy of the forward Euler approximation is compared to that of the exact midpoint and kriging assisted midpoint approximations for the case of a fixed number of elapsed cycles. The results for a variety of step sizes are presented in the following table as normalized values as the predicted value divided by the exact value from the above equation.

  Normalized Crack Size 
 Delta N Euler Midpoint     Kriging Assisted Midpoint
 1 0.998 1.001 1.000
 5 0.990 1.001 1.000
 10 0.981 1.001 1.000
 25 0.955 1.001 0.999
 50 0.916 1.001 0.999
 100 0.852 1.000 0.999

From the above table it can be noted that the kriging assisted midpoint method performs nearly as well as the exact midpoint algorithm and much better than the forward Euler method. The use of kriging in the finite element environment will allow for higher-order approximations to fatigue crack growth models without the need for additional simulations.

References

1. Kleijnen, J. (2009) "Kriging metamodeling in simulation: A review," European Journal of Operational Research, 182, 707-716.

2. Pais, M., Viana, F.A.C., Kim, N.H. (2011) "High-order integration of fatigue crack growth using surrogate model," 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, Colorado.

3. Paris, P., Gomez, M., Anderson, W. (1961) "A rational analytic theory of fatigue," The Trend in Engineering, 13, 9-14.