Abstract
Figure 1. Experimental and model stressstrain response for the Al1%Si0.3%Mg matrix material in the finite element simulations.
Monotonic and cyclic finite element simulations were performed on voids, cracked inclusions, bonded inclusions, and debonded inclusions. The local maximum and spatial average plastic shear strain were obtained as a function of equivalent overall size, loading amplitude, and load ratio. The effect of shape was studied by performing simulations for an idealized pore, star pore, idealized inclusion, and star inclusion. Based on an assessment of the local maximum plastic shear strain, voids and debonded inclusions were found to be the most likely candidates for initiating fatigue crack formation. It was also found that the equivalent overall size of the void or inclusion and the load ratio were more important in determining local plastic shear strain than shape.
Author(s): Ken Gall, Mark F. Horstemeyer, Brett W. Degner, David L. McDowell, and Jinghong Fan

Figure 2. Scanning electron microscope (SEM) images from the fracture surface of a cast A356T6 aluminum alloy showing a pore (void) and an inclusion as fatigue crack nucleation sites. (click on the image to enlarge).
Figure 3. Schematic of the different types of inhomogeneities considered in the present finite element study of fatigue crack formation. (click on the image to enlarge).

Methodology
In order to study the effect of void and inclusion shape on fatigue crack formation, SEM images were taken from a fractured A356 aluminum sample. The left image in Figure 2 shows a void with smooth edges and sharp vertices. The inclusion pictured to the right is smoother overall. A summary of the different types of inhomogeneities studied is given in Figure 3. To model these four types of inhomogeneities, the finite element method was used with meshes described in Figures 4a and 4b for different geometries. The intention is to see if the fingers of the star inclusion or the sharp vertices of the star pore produce significantly different results than the idealized inclusion or pore. The shear strain results were reported as the maximum value of the shear strain near the inclusion or void and as the average value over an area adjacent to the inclusion or void. The two forms of reporting the shear strain are depicted graphically in Figure 5. A mesh refinement was carried out to determine the mesh fineness needed for numerical convergence of the average maximum plastic shear strain. This mesh refinement is shown in Figure 6.

Material Model
The model used to study the A356 aluminum alloy was an Internal State Variable (ISV) model of the type proposed by Bammann (1993, 1996). The isotropic and kinematic hardening were the two internal state variables.

Input Data
The data used to calibrated the ISV model was experimental stressstrain data. A comparison of the experimental data and ISV model prediction is shown in Figure 1.

Results
Contour plots of the simulated maximum plastic shear strain at an applied farfield strain of 0.2% are shown in Figures 710 for the four different types of inhomogeneities listed in Figure 3. Figure 11 summarizes the results in Figures 710 and shows that voids and debonded inclusions produce the highest average maximum plastic shear strain for a given farfield total strain. Figures 1316 show the results of cyclic load experiments for load ratios of 1, 0 and 0.5. Figure 17 summarizes these results and shows a clear dependence of the average maximum plastic shear strain on load ratio. Figure 18 shows the effect of size on average maximum plastic shear strain for an ideal debonded inclusion and ideal void.
Key findings from this study include:
 Voids and debonded inclusions produced the highest plastic shear strain and are anticipated to be the key initiators of fatigue cracks
 Shape of void or inclusion is not as important as size for predicting fatigue crack initiation
 Fatigue crack initiation is significantly affected by the load ratio

Figure 4a. Overall finite element mesh and boundary conditions. 
Figure 4b. Fine mesh regions for different inclusion and void shapes. The meshed inclusions are set below the appropriate fine mesh regions. 
Figure 5. Schematic demonstrating the determination of the local average maximum plastic shear strain from contours of the maximum plastic shear strain. 
Figure 6. Plot of the (a) pointwise maximum plastic shear strain and (b) local average maximum plastic shear strain as a function of the farfield applied strain for different relative fine mesh element sizes. 
Figure 7. Contour plots of the maximum plastic shear strain for voids with different geometries. The applied farfield strain is 0.2%. 
Figure 8. Contour plots of the maximum plastic shear strain for bonded inclusions with different geometries. The applied farfield strain is 0.2%. 
Figure 9. Contour plots of the maximum plastic shear strain for cracked inclusions with different geometries. The applied farfield strain is 0.2%. 
Figure 10. Contour plots of the maximum plastic shear strain for debonded inclusions with different geometries. The applied farfield strain is 0.2%. 
Figure 11. Plot of the local average maximum plastic shear strain as a function of the farfield applied strain for the different inhomogeneities in Figures 710. 
Figure 12. Contours of the VonMises effective stresses at 0.20% strain for the (a) idealized cracked inclusion, (b) idealized debonded inclusion, and (c) idealized cracked and partially debonded inclusion. 
Figure 13. Contour plots of the maximum plastic shear strain under a load ratio of 1 and a loading amplitude of 0.2 (a) and (b) are for the void under compression and tension, while (c) and (d) are for the debonded inclusion under compression and tension. 
Figure 14. Plot of the local average maximum plastic shear strain as a function of the farfield applied strain under a load ratio of 1 for different cyclic load amplitudes. The plots are for (a) debonded idealized inclusion, and (b) idealized void. The response took nearly six cycles to show saturation, thus more cycles are shown compared to Figure 14. 
Figure 15. Plot of the local average maximum plastic shear strain as a function of the farfield applied strain under a load ratio of 0 for different cyclic load amplitudes. The plots are for (a) debonded idealized inclusion, and (b) idealized void. The response saturated in three cycles so further cycles are not shown. 
Figure 16. Plot of the local average maximum plastic shear strain as a function of the farfield applied strain under a load ratio of 0.5 for different cyclic load amplitudes. The plots are for (a) debonded idealized inclusion, and (b) idealized void. The response took nearly six cycles to show saturation, thus more cycles are shown compared to Figure 14. 
Figure 17. Plot of the local average maximum plastic shear strain as a function of the farfield applied strain amplitude for the third cycle under a load ratio of 1, 0, and 0.5 
Figure 18. Effect of inclusion size on the driving force for fatigue crack formation for the idealized debonded inclusion and the idealized void. 



Acknowledgments
This work has been sponsored by the U.S. Department of Energy, Sandia National Laboratories under contract DEAC0494AL85000. This work was performed under the leadership of Dick Osborne and Don Penrod for the USCAR Lightweight Metals Group.

References
Gall, Ken, Mark F. Horstemeyer, Brett W. Degner, David L. McDowell, and Jinghong Fan. 2001. "On the driving force for fatigue crack formation from inclusions and voids in a cast A356 aluminum alloy". International Journal of Fracture. 108 (3): 207233.
Bammann, D. J., M. L. Chiesa, M. F. Horstemeyer, and L. I. Weingarten. "Failure in ductile materials using finite element methods." Structural crashworthiness and failure (2010): 154.
Bammann, D. J., M. L. Chiesa, and G. C. Johnson. "Modeling large deformation and failure in manufacturing processes." Theoretical and Applied Mechanics (1996): 359376.
