Geometric uncertainty can significantly affect the stability of solutions to variational problems in mechanics. We investigate fracture nucleation using a variational phase-field model posed on a domain whose boundary is represented by a stationary Gaussian random field with Matérn covariance. This formulation allows us to study how stochastic boundary morphology influences the predicted failure strength.

The problem involves an interaction between the local geometry of the random boundary and the internal phase-field length scale. We identify two distinct nucleation regimes: a stress-based regime, in which the failure load is governed by the local stress concentration, and a notch-based regime associated with steep boundary defects that progressively behave like sharp notches.

Our results show that the largest reduction in strength does not occur at either limiting geometry. Instead, it occurs for an intermediate defect geometry marking the transition from the stress-based regime to the notch-based regime. This transition is governed by the ratio between the local root radius of curvature of the boundary defect and the phase-field length scale.

The resulting dimensionless description establishes a connection between local boundary geometry and variational fracture nucleation. It also provides a basis for studying failure on statistically defined rough domains.

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