Geometrically Exact Elastic Shell Crack Propagation Dynamics Based on Fiber Bundles and Differential Forms

Fracture and crack propagation in flexible shells under extreme loading represent a fundamental challenge in continuum mechanics. Traditional shell fracture theories rely heavily on local coordinate systems and asymptotic expansions, often entangled in the contradiction between three-dimensional solid fracture and two-dimensional shell theory. Taking the geometrically exact Kirchhoff-Love shell theory based on fiber bundles and differential forms previously established by the author as a starting point, this paper strictly generalizes it to a two-dimensional mid-surface manifold topology containing evolving cracks. We model through-cracks as evolving internal boundaries and one-dimensional submanifolds on the two-dimensional mid-surface manifold, introducing a rigorous kinematic mapping for crack propagation. In terms of dynamics, based on the elastic strain energy on the two-dimensional mid-surface, we derive a geometrically exact two-dimensional Eshelby configuration stress tensor and express it as a vector-valued configuration stress 1-form. Through the generalized virtual work principle applied to the variation of the crack front, a coordinate-independent J-integral (energy release rate) is naturally defined. Based on the Griffith criterion and the maximum energy release rate principle, this paper strictly derives the control equations for the crack propagation direction vector and propagation velocity on the tangent space of the manifold. This theory implicitly contains the complex curvature-fracture coupling within the structures of exterior differentiation and pullback metrics. Furthermore, we present a complete Discrete Exterior Calculus (DEC) discretization framework for the theory.