Cracking elements with sequential linear analysis: An efficient event-driven numerical framework for quasi-brittle fracture

Quasi-brittle fracture simulations often lose robustness when localized cracking triggers abrupt stress redistribution and snap-back. The Cracking Elements Method (CE) captures discontinuous cracking by introducing element-wise crack-opening variables and self-propagating cracked segments, allowing multiple cracks to initiate and propagate without remeshing, nodal enrichment, or explicit crack tracking. Its conventional formulation, however, follows a continuous nonlinear traction–separation path and requires Newton iterations or arc-length-type control in unstable post-peak regimes. This paper presents a sequential-linear reformulation of CE (CE-SLA). The original cracking-element kinematics, characteristic length, crack orientation rule, mixed-mode cohesive framework, and adaptive node-insertion strategy are retained, while cohesive softening is replaced by an event-driven sequence of linear reloading problems. New cracks and cohesive traction drops are both treated as admissible events. An energy-conserving zigzag traction–separation law makes cracked elements reload linearly and lose strength through local drop events, while load-to-strength and target-opening ratios locate the next event directly from a linear state. The accepted event history is retained as the computed force–displacement response. Dogbone, L-shaped panel and notched-beam benchmarks show that the proposed CE-SLA framework reproduces reasonable crack paths, raw event responses and snap-back behavior. In our tests, CE-SLA required 31.5%–73.2% of the linear solves and 29.1%–75.1% of the CPU time of the nonlinear CE, corresponding to CPU-time reductions of 24.9%–70.9%. CE-SLA therefore preserves the main advantages of CE while converting unstable cohesive fracture into a sequence of accepted linear fracture events without using Newton iterations to trace the continuous softening law.