Capturing Finite Target Dynamics: Phase-Delayed Analytic Modeling of Multi-Layer Penetration Events

Technical Summary: Capturing Finite Target Dynamics via Phase-Delayed Analytic Modeling

Problem Statement
The Walker-Anderson half-space penetration model has long served as an efficient tool for calculating penetration depths of rigid and eroding rods in thick targets. However, existing extensions of this model to finite targets struggle to accurately capture nose-tail velocity profiles, particularly in thinner targets. In multi-layer, thin-walled target stacks, this deficiency leads to mischaracterized rod erosion compared to detailed hydrodynamic simulations or experimental predictions. Specifically, previous finite domain models (such as those by Ravid and Walker) often trigger target weakening and projectile recovery phases prematurely, failing to account for the time required for wave propagation effects to traverse the target thickness.

Methodology
The authors propose an updated modification to the Walker-Anderson model that incorporates insights from detailed ALE3D hydro-simulations to correctly account for wave propagation within finite targets. The methodology involves three primary technical developments:

1. Updated Initial Conditions: The paper revises the starting conditions for hemispherical-headed projectiles. Instead of relying solely on the traditional 1-D shock jump condition ($P_{shock}$), the authors introduce a "stagnation shock" condition ($P_{stag}$) for initial impacts. This condition assumes the target more quickly adopts a stagnating flow form, introducing a factor of 1/2 relative to the original shock equation. This better initializes the interface velocity for pristine impacts, while the traditional $P_{shock}$ condition is retained for subsequent impacts in multi-target scenarios.
2. Wave Propagation Analysis: Through post-processing of ALE3D simulations, the authors analyzed the evolution of velocity and plasticity characteristics within a finite target. They determined that while shock speeds are initially over-driven, the propagation of plastic effects is best approximated by the target's bulk sound speed ($c_0$). This analysis revealed that finite target effects (such as back-face bulging and weakening) do not influence the projectile interface instantaneously but are delayed by the wave transit time.
3. Phase-Delayed Coupling: The core contribution is the implementation of a phase delay in the finite target model. The authors utilize characteristic forms of the 1D wave equation to link the front and back faces of the target. The model calculates forward ($t_+$) and backward ($t_-$) characteristic times to delay the application of back-face properties (such as back-face velocity $u_b$ and the proximity parameter $\tilde{\alpha}$) to the interface drag terms. The model only switches from infinite target assumptions to finite target assumptions once the wave has had sufficient time to traverse the target thickness, ensuring the projectile interface is not affected by back-face softening until physically justified.

Key Contributions

• Phase-Delay Mechanism: The introduction of a phase delay based on bulk wave propagation allows the analytic model to correctly time the onset of target weakening. This prevents the premature re-acceleration of the projectile nose observed in previous finite domain models.
• Zero Additional Parameters: The updated model improves accuracy for thin targets while retaining good behavior for thick targets without requiring any new tunable model parameters. The modifications rely on existing material properties (bulk sound speed) and the geometry of the impact.
• Improved Initial Conditions: The distinction between stagnation shock initialization for pristine impacts and standard shock initialization for subsequent impacts provides a more accurate early-time velocity profile.
• Experimental Validation: The model was validated against experimental data for thick finite targets (2.90 cm steel), demonstrating that omitting an explicit projectile recovery phase (relying instead on the natural momentum balance) yields better agreement with exit velocities than models that force a recovery trigger.

Results
The updated model was benchmarked against detailed ALE3D hydro-simulations across various scenarios:

• Thin Targets: For targets as thin as 0.5 cm and 1 cm, the new model significantly outperforms the original Walker back-face model. It correctly captures the delay in nose velocity recovery and the sustained target strength, whereas the original model predicted premature weakening.
• Multi-Target Impacts: In simulations involving stacks of six 4 cm plates, the updated model demonstrated superior agreement with hydro-simulations regarding both the initial response and the sustained erosion of the projectile. It successfully captured the timing of target weakening and the resulting projectile velocity profiles without sensitivity to specific failure strain criteria.
• Velocity Range: The model maintained high agreement across a range of impact velocities (1200 m/s to 3000 m/s) without re-tuning parameters.
• Experimental Correlation: Comparisons with experimental exit velocity data for 2.90 cm steel targets confirmed the model's efficacy, particularly in capturing the transition through the V50 regime and post-recovery velocities.

Significance
The paper claims that these corrections offer a "remarkable accuracy" in comparison to high-fidelity hydrodynamic simulations, which require significantly more computational resources and operator expertise. By leveraging the accuracy of detailed simulations to refine semi-analytic methods, the authors provide a tool that retains the speed and simplicity of analytic models while correcting their primary failure mode in thin-target dynamics. The work suggests that the original Walker half-space model assumptions can be re-interpreted: the model applies immediately to the projectile interface (justifying early-time immediacy) but requires a delayed application of finite back-face conditions to account for the realistic development of the velocity field within the target depth. This approach allows engineers to better understand coupled impactor-target interactions in high-rate penetration events without the computational penalty of full hydrodynamic codes.