Stability of Extrinsic Cohesive-zone Model with Penalty-based Contact in Explicit Dynamic Fragmentation Simulations☆

Dynamic fragmentation simulations are essential for predicting material response at high strain rates, yet explicit dynamic simulations that combine an extrinsic cohesive-zone model (CZM) with penalty-based contact often exhibit severe instabilities. In a two-dimensional benchmark, we observe exponential energy growth and resulting artificial fragmentation under standard contact penalty settings and time step choices, which motivates a systematic analysis of instability sources. Three mechanisms are isolated and quantified: (i) diverging initial cohesive stiffness, which constrains the stable time step; (ii) discontinuous stiffness jumps at the cohesive-contact interface; and (iii) discontinuity introduced by cohesive softening. Analytical error estimates, phase-space diagnostics, and energy growth metrics reveal that repeated cohesive-contact switching can accumulate small per-step energy errors into long-term energy drift. Within the explored parameter space, maintaining stability requires time steps well below the usual limit. To mitigate these energy artifacts, we assess an adaptive penalty strategy that ties the contact stiffness to the evolving cohesive stiffness. This modification eliminates the discontinuity and restores energy conservation, but it allows larger interpenetration, making it suitable as a diagnostic rather than a definitive remedy. Overall, our study identifies the root causes of unphysical energy drift and demonstrates that penalty-based contact is not a viable approach for long-term, energy-consistent fragmentation simulations with physically meaningful fragment statistics.