Stability of Extrinsic Cohesive-Zone Model with Penalty-Based Contact in Explicit Dynamic Fragmentation Simulations

TL;DR

This work investigates why coupling an extrinsic cohesive-zone model (CZM) with penalty-based contact in explicit dynamic fragmentation leads to unphysical energy drift and spurious fragmentation. By isolating three mechanisms—diverging initial cohesive stiffness, abrupt cohesive–contact transitions, and cohesive softening—the authors quantify their contributions using analytical results, a 1D spring–mass surrogate, and phase-space energy diagnostics. They demonstrate that energy drift arises primarily from cohesive–contact switching during repeated transitions, and that an adaptive penalty can restore energy balance but at the cost of significant interpenetration, limiting physical fidelity. The paper argues that penalty-based contact is not viable for robust, long-term fragmentation simulations and advocates nonsmooth time-stepping as a more stable, energy-consistent alternative, with potential to improve fragment statistics and scalability. This has practical significance for reliable debris-fragmentation modeling where energy conservation and accurate fragment statistics are essential for risk assessment.

Abstract

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.

Figures (14)

• Figure 1: (a) Initiation, propagation, and coalescence of microcracks in the fracture process zone $l_c$, ahead of the main crack tip. (b) Idealized version of the crack where all inelastic processes are represented via a cohesive zone model. $\Gamma^+$ represents the upper surface of the crack and $\Gamma^-$ the lower surface.
• Figure 2: Linear irreversible TSL for extrinsic cohesive elements, following the formulation of Camacho and Ortiz camacho1996computational, with a penalty-based contact enforcement. (a) Normal component. (b) Tangential component.
• Figure 3: (a) Physical system: expanding hollow sphere. (b) Planar analog: square plate of side $L$, with velocity field $\textbf{v}(r)=\dot\varepsilon\,\textbf{r}$.
• Figure 4: Fragmentation snapshot taken at $t=3.75\times10^{-5}\text{ s}$. The magnified view reveals partially damaged cohesive elements within an individual fragment.
• Figure 5: (a) Time history of energy components, kinetic $\mathcal{K}$, fracture $\mathcal{G}$, and elastic energies, normalized by the total injected energy $\mathcal{E}_\mathrm{inj} = \mathcal{W}_\mathrm{ext}(t_f) + \mathcal{E}_0$. Rather than settling to steady values after fragmentation, all components exhibit continuous growth, revealing unphysical energy drift. (b) Evolution of the fragment count over time. After an initial plateau, spurious high‐frequency oscillations induce non‐physical additional fragments, underscoring the impact of numerical instabilities on fragment statistics.