A constrained variational method for crack networks with criterion-based evolution

Abstract A constrained Variational Crack Element Method (VCEM) for interacting crack networks in two-dimensional isotropic elasticity is formulated. The method is variational at the level of crack equilibrium for fixed geometry, while crack advance is treated separately using near-tip propagation criteria. Cracks are represented by distributed Burgers-density fields, preserving the exact crack-tip singular structure within a unified energy-minimization framework. The governing equations follow from a constrained variational problem in which equality constraints enforce Burgers-vector closure and displacement single-valuedness at junctions, while inequality constraints enforce crack-face non-penetration. The resulting system has a Karush–Kuhn–Tucker (KKT) saddle-point structure. Crack geometry is represented by graph-based polylines composed of polarity-carrying half-crack elements, allowing the treatment of tips, kinks, branching, and multi-degree junctions. Elastostatic interactions are expressed through tensor kernels derived from Kelvin’s fundamental solutions, and both strong (collocation) and weak (Galerkin/least-squares) traction enforcement are implemented. The framework provides a sharp-interface, computationally scalable approach for equilibrium and criterion-based evolution of complex crack networks in finite or infinite domains.