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Curvature-Corrected Crack-Tip Fields on Curved Manifolds: A Geometric Framework for Fracture
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DOI：10.1016/j.jmps.2026.106667.png)
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Abstract
En 中文
Classical fracture mechanics assumes crack-tip stress fields derived on flat Euclidean domains. However, many natural and engineered systems—from biological membranes to thin-film devices and shell structures— fail on curved surfaces, where planar near-tip fields can become incomplete or insufficient. This work presents the first crack-tip asymptotic solution derived directly on a curved manifold. Starting from differential-geometric formulations of in-plane elasticity, the Airy stress function is constructed using the Laplace–Beltrami operator and expanded asymptotically via perturbation analysis supported by Fredholm solvability. We demonstrate that curvature preserves the universal r−1/2 stress singularity while perturbing the angular eigenfunctions and modifying the near-tip stress distribution. Closed-form curvature-corrected singular fields are obtained and used to construct a curvature-dependent maximum tangential stress criterion for crack trajectory prediction. Experiments on elastomeric sheets conformed to 3D-printed curved substrates validate the theory: predicted crack paths quantitatively match measured trajectories, whereas classical flat-surface solutions accumulate significant path errors during propagation. These results establish surface curvature as an intrinsic geometric control parameter in fracture and provide a unified mechanics framework for crack evolution on curved surfaces.
Keywords:
crack-tip fields
curved manifolds
fracture mechanics
differential geometry
stress singularity
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