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Microcantilever bending experiments in NiAl — Evaluation, size effects, and crack tip plasticity

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Abstract

For a better understanding of the local fracture behavior of semi-brittle materials, we carried out bending experiments on notched microcantilevers of varying sizes in the micrometer range using NiAl single crystals. Smaller and larger beams were milled with a focused ion beam in the so-called “soft” <110> and “hard” <100> orientation and were tested in situ in a scanning electron microscope and ex situ with a nanoindenter, respectively. The measurements were evaluated using both linear-elastic fracture mechanics and elastic–plastic fracture mechanics. The results show that (i) the fracture toughness is in the same range as the macroscopically determined one which is around \(3.5\;{\rm{MPa}}\sqrt {\rm{m}}\) for the soft orientation and around \(8.5\;{\rm{MPa}}\sqrt {\rm{m}}\) for the hard orientation, that (ii) there is a strong influence of the anisotropic behavior of NiAl on the fracture toughness values, and that (iii) the J-integral technique is the most accurate quantification method.

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ACKNOWLEDGMENTS

The authors gratefully acknowledge the funding of the German Research Council (DFG), which within the framework of its ‘Excellence Initiative’ supports the cluster of Excellence ‘Engineering of Advanced Materials’ at the University of Erlangen-Nürnberg. Johannes Möller is thanked for his support concerning the anisotropy calculations.

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Correspondence to Johannes Ast.

APPENDIX: EVALUATION OF CRACK GROWTH FOR NOTCHED MICROCANTILEVERS

APPENDIX: EVALUATION OF CRACK GROWTH FOR NOTCHED MICROCANTILEVERS

Crack growth was calculated by the determination of the unloading stiffness recorded during the loading–unloading experiments. In general, a growing crack corresponds to an increased a/W ratio and a decreasing bending stiffness. For the understanding of the correlation between the stiffness of a notched cantilever and the a/W ratio, FEM calculations were performed. The stiffness decrease is plotted as a function of the a/W ratio in Fig. A1 for the applied geometry according to these simulations. For comparison, the analytical solution of a clamped bending beam with respective dimensions is also shown. The correlation between the stiffness k and the geometrical dimensions is according to Ref. 28:

$${W^\prime } = W - a = \root 3 \of {{{4k{L^3}} \over {BE}}} .$$
(9)
FIG. A1
figure A1

(a) Stiffness decrease as a function of individual a/W ratios for the tested geometry (from FEM simulations, green circles) and for the analytical solution (black squares) of a fixed bending beam according to Eq. (9). Schematics are shown in (b) for the notched geometry from the FE simulations where the support (in green) is around four times thicker than the thickness B of the cantilever and for the clamped beam where the width W is by definition reduced by the respective notch length a for a better comparison.

The stiffness is shown in Fig. A1 for the two geometries and normalized to each maximum stiffness value. The decrease in stiffness with increasing a/W ratio is clearly visible. It is apparent that the stiffness decrease is more pronounced for the unnotched geometry which is explained by the existing support for the notched cantilever. Only at very large a/W ratios when both structures become very slim, the stiffness values are comparable. It is concluded that the stiffness reduction due to a change in the a/W ratio which represents crack growth depends strongly on the boundary conditions and the sample geometry.

When an initial crack to width ratio a0/W = 0.35 is chosen, which is approximately the experimental case, a stiffness decrease in percent is calculated by dividing the stiffness value at any a/W ≥ 0.35 by the stiffness value of a0/W = 0.35. By calculating the stiffness values from all the unloading sequences in the experiments, a direct correlation between these values and the modeled a/W ratios is possible and crack growth can be determined. It becomes obvious that the analytical solution for the clamped beam underestimates the crack growth because for various stiffness decreases the notched configuration provides larger a/W ratios and hence larger crack lengths than the analytical solution. This emphasizes again the need for detailed knowledge of the geometry and the boundary conditions of the model to determine the correct crack growth from the unloading stiffness.

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Ast, J., Przybilla, T., Maier, V. et al. Microcantilever bending experiments in NiAl — Evaluation, size effects, and crack tip plasticity. Journal of Materials Research 29, 2129–2140 (2014). https://doi.org/10.1557/jmr.2014.240

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  • DOI: https://doi.org/10.1557/jmr.2014.240

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