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Fracture in sheets draped on curved surfaces

Abstract

Conforming materials to rigid substrates with Gaussian curvature—positive for spheres and negative for saddles—has proven a versatile tool to guide the self-assembly of defects such as scars, pleats1,2,3,4,5, folds, blisters6,7, and liquid crystal ripples8. Here, we show how curvature can likewise be used to control material failure and guide the paths of cracks. In our experiments, and unlike in previous studies on cracked plates and shells9,10,11, we constrained flat elastic sheets to adopt fixed curvature profiles. This constraint provides a geometric tool for controlling fracture behaviour: curvature can stimulate or suppress the growth of cracks and steer or arrest their propagation. A simple analytical model captures crack behaviour at the onset of propagation, while a two-dimensional phase-field model with an added curvature term successfully captures the crack’s path. Because the curvature-induced stresses are independent of material parameters for isotropic, brittle media, our results apply across scales12,13.

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Figure 1: Gaussian curvature—positive for caps and negative for saddles—governs the behaviour of cracks.
Figure 2: Curvature stimulates or suppresses fracture initiation.
Figure 3: Kinking and curving of crack paths in sheets conformed to a bump.
Figure 4: Curvature arrests a centre crack.
Figure 5: Tuning crack paths with the curvature landscape.

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Acknowledgements

The authors thank E. Efrati, H. Kedia, D. Kleckner, M. Driscoll, S. Nagel, T. Witten and R. Scott for interesting discussions and J. Mazor for assistance with some supplementary experiments. Some simulations were carried out on the Midway Cluster provided by the University of Chicago Research Computing Center. We acknowledge the Materials Research and Engineering Centers (MRSEC) Shared Facilities at The University of Chicago for the use of their instruments. This work was supported by the National Science Foundation MRSEC Program at The University of Chicago (Grant DMR-1420709) and a Packard Fellowship. V.K. and V.V. acknowledge funding from FOM and NWO.

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Contributions

W.T.M.I. and V.V. initiated this study. N.P.M. and W.T.M.I. designed experiments. N.P.M. performed and analysed the experiments and simulations. N.P.M. and V.K. constructed the analytical model. All authors interpreted the data. N.P.M., V.V. and W.T.M.I. wrote the manuscript.

Corresponding authors

Correspondence to Noah P. Mitchell or William T. M. Irvine.

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The authors declare no competing financial interests.

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Mitchell, N., Koning, V., Vitelli, V. et al. Fracture in sheets draped on curved surfaces. Nature Mater 16, 89–93 (2017). https://doi.org/10.1038/nmat4733

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