The ballistic limit velocity, v_bl, is a central parameter in dynamic fracture mechanics, as it defines the threshold impact velocity that separates perforation from non-perforation for a given projectile-target pair (Backman and Goldsmith, 1978). Beyond its role in assessing protective systems for personal, structural, or vehicular applications, v_bl can also serve as a proxy for transitions between fracture mechanisms in ductile plate perforation, including ductile hole enlargement, shear plugging, and mixed-mode responses under dynamic loading. Capturing how v_bl varies with projectile orientation is therefore essential not only for practical design but also for advancing the understanding of failure-mode transitions in high-rate fracture.

Classical work by Recht and Ipson (1963) established a kinetic-energy framework to describe residual velocity and perforation dynamics, while Lambert and Jonas (1976) extended this approach through an empirical velocity law that remains widely used in ballistic analysis. These velocity-based models are complemented by constitutive and fracture formulations, most notably the Johnson–Cook plasticity law (Johnson and Cook, 1983) and its companion failure model (Johnson and Cook, 1985), which explicitly incorporate strain hardening, strain-rate sensitivity, thermal softening, and stress triaxiality effects. Together, these approaches allow finite element simulations to capture both global ballistic resistance and the local progression of fracture under large plastic strains and high strain rates (Chen, 1990; Dey et al., 2004; Dolinski and Rittel, 2015; Deng et al., 2022; Janda et al., 2023).

Projectile orientation is classically defined by three angles: roll, pitch, and yaw (McCoy, 1999; Carlucci and Jacobson, 2008). Due to axisymmetry, pitch and yaw are often combined into the total yaw angle (α). While the effect of obliquity (β) on v₍bl₎ has been widely studied (Recht and Ipson, 1963; Backman and Goldsmith, 1978; Anderson Jr., 2017), the role of total yaw has received far less systematic attention. Numerical and experimental studies have reported that yaw increases the apparent ballistic limit velocity (Goldsmith, Tam and Tomer, 1995; Li and Goldsmith, 1996a–c; Bless, Satapathy and Normandia, 1999; Satapathy, Bedford and Bless, 1998; Vayig and Rosenberg, 2021, 2023), but most of these works considered yaw ≤ 20° and focused primarily on penetration depth, projectile deformation, or trajectory stabilization. Moreover, standards such as NIJ 0101.06 (NIJ, 2008) and MIL-STD-662F (DoD, 1997) restrict valid ballistic tests to yaw ≤ 5°, which has limited experimental exploration of yaw-induced fracture phenomena. From a fracture-mechanics perspective, however, yaw directly alters local stress triaxiality, shear localization, and crack initiation, all of which are central to ductile failure processes.

This lack of systematic evaluation of yaw-induced fracture-mechanism transitions constitutes a significant gap. Although incremental increases in v_bl with yaw have been reported (Goldsmith, Tam and Tomer, 1995; Goldsmith, 1999), the magnitude of these effects, their non-monotonic behavior at large angles, and their connection to underlying failure modes remain poorly documented. In particular, whether yaw can trigger a transition from ductile enlargement to shear-dominated plugging in metallic plates has not been clarified. The present work addresses this gap through a computational study of blunt cylindrical 4340 steel projectiles impacting 4340 steel plates at yaw angles from 0° to 90°. Using three-dimensional finite element simulations with the Johnson–Cook plasticity and failure models, v_bl was obtained via Lambert–Jonas regressions for multiple mesh densities.

The dynamic perforation of metallic plates by high-velocity projectiles can be interpreted as a fracture process controlled by a balance between global energy absorption and local failure mechanisms. From the perspective of fracture mechanics, the ballistic limit velocity v_bl represents a critical threshold: the minimum impact velocity at which the target undergoes complete perforation, leaving a nonzero residual velocity in the projectile (Backman and Goldsmith, 1978). This parameter has long been used as a performance index for protective systems, but in computational and mechanistic studies it also acts as a boundary condition for fracture-mode transitions, separating cases of ductile hole enlargement from shear plugging or mixed mechanisms.

Numerical simulations were carried out with the ANSYS/Explicit Dynamics package (ANSYS, 2022a), which solves transient dynamic problems using explicit time integration. This method is particularly suitable for high-velocity impacts, where severe nonlinearities arise due to large deformations, contact, and material failure. Given that non-zero yaw angles break axial symmetry, the problem was modeled in three dimensions, since two-dimensional planar or axisymmetric formulations cannot capture the asymmetric stress and failure fields associated with yaw-induced impacts.

The target was defined as a square SAE 4340 steel plate, 200 mm × 200 mm × 10 mm. Its constitutive and failure behavior followed the Johnson-Cook plasticity law (Johnson and Cook, 1983) and the Johnson–Cook failure model (Johnson and Cook, 1985), with parameters given in Table 1.

These values include elastic modulus (E), density (ρ), Poisson’s ratio (ν), and specific heat (Cₚ), in addition to the constants required for the constitutive and fracture models. The projectile was modeled as a rigid blunt-faced cylinder with diameter 7.62 mm, length 22.86 mm (aspect ratio 3.0), and mass 8.16 g, also made of SAE 4340 steel. Treating the projectile as rigid is a common approximation when the penetrator material is significantly harder than the target and deformation is negligible compared to the plate response (Li and Goldsmith, 1996b). This assumption was later verified by inspecting deformation levels in control runs. Figure 2 illustrates the mesh scheme of the target, while Fig. 3 shows the projectile mesh, both generated in ANSYS.

Mesh design plays a central role in impact simulations, as local stress and strain gradients around the perforation zone require fine resolution. To address this, the plate was divided into two regions: a central square region (40 mm side) with refined elements, ensuring accurate tracking of material deformation in the perforation zone; and an outer region meshed more coarsely, to reduce the total number of elements and simulation time. Three characteristic element sizes were investigated in the refined region: 0.625 mm, 0.500 mm, and 0.400 mm, while the outer region was fixed at 10 mm. Hexahedral elements were employed to minimize volumetric locking and ensure accurate integration (ANSYS, 2022b).

Mesh quality was assessed using the standard element quality metric Q = kV()^–3/2, where V is element volume, l_i are edge lengths, and k is a constant, which for the hexahedron is . Values above 0.2 were considered acceptable (ANSYS, 2022c). The resulting element counts and quality metrics are reported in Table 2. The projectile mesh was generated by the axisymmetric sweep algorithm, ensuring consistent refinement along its geometry. This procedure, shown in Fig. 3, produced meshes with increasing element counts proportional to refinement levels.

The projectile was initially positioned 5 mm from the plate surface, with a prescribed yaw angle α ∈ {0°, 5°, 10°, 15°, 20°, 25°, 30°, 45°, 60°, 75°, 85°, 90°}. This gap ensured proper separation of bodies during meshing and allowed the projectile to approach at its defined orientation (Fig. 4). Impact velocities ranged from 580 m/s to 2600 m/s, encompassing sub-ballistic and fully perforating regimes. The plate was clamped along all four lateral edges, imposing zero displacement, consistent with previous ballistic simulations (Dey et al., 2004; Børvik et al., 2009). Each simulation was run for 200 µs of physical time, sufficient to capture projectile perforation and stabilize residual velocity.

Contact interactions were modeled using the penalty algorithm with the trajectory method for detection. Sliding was handled via the discrete surface method, and self-contact was activated with a tolerance of 0.2. Frictionless contact was assumed in the baseline simulations, while sensitivity to finite friction was assessed in supplementary runs.

Failure-induced element erosion followed the Johnson-Cook cumulative damage criterion (Section 2.3). Additionally, a geometric deformation limit was applied with a factor of 2.0, preventing element distortion from destabilizing the solution. Eroded elements were retained with mass and inertia to conserve system energy, following best practices in explicit dynamics.

For each case, the projectile’s center-of-mass velocity vector was extracted at the end of the simulation. The residual velocity (v_r) was then computed as its magnitude. These data were used to calibrate the Lambert–Jonas model (Eq. 1) and determine the ballistic limit velocity (v_bl) for each yaw angle and mesh size.

All numerical simulations successfully converged across the investigated range of impact velocities and yaw angles. The results are presented below, organized into qualitative analysis of penetration kinematics, residual velocity curves and ballistic-limit extraction, mesh dependence and normalization, and evolution of Lambert-Jonas parameters as indicators of fracture-mechanism transition.

This work presented a computational investigation into the role of total yaw angle α on the ballistic limit velocity of blunt cylindrical projectiles impacting 4340 steel plates. Using three-dimensional finite element simulations with Johnson-Cook plasticity and failure models, it was possible to capture the combined effects of projectile orientation, material response, and mesh discretization on perforation mechanisms. The results demonstrate that yaw is not a minor perturbation but a governing variable in the fracture mechanics of ductile plate perforation.

The simulations revealed that increasing yaw generally elevates the ballistic limit compared to normal impact, but this effect is non-monotonic. A pronounced maximum was observed between 60° and 75°, where the ballistic limit approximately doubled relative to α = 0°. At higher angles, however, the ballistic limit decreased, showing that yaw does not indefinitely enhance resistance but instead triggers distinct perforation regimes. This non-monotonic trend was robust across mesh densities, and normalization by the normal-impact case collapsed the results into a mesh-independent master curve. Such normalization not only overcomes numerical sensitivity but also provides a practical predictive framework, which was successfully represented by a cubic polynomial correlation in yaw angle.

Beyond quantitative trends, the study provided mechanistic insights into the transition of fracture modes. For yaw angles up to 45°, the projectile tended to realign during penetration, producing ductile hole enlargement governed by tensile-dominated states. Between 45° and 60°, however, abrupt variations in the Lambert-Jonas parameters indicated a shift in governing mechanisms. At these orientations, projectiles penetrated laterally with limited reorientation, consistent with shear localization and plugging. The Johnson-Cook failure criterion, with its explicit dependence on stress triaxiality, explains this behavior: low yaw favors void growth and ductile tearing, while high yaw enhances shear-driven fracture. The agreement with available experimental data for small yaw angles further strengthens the reliability of these interpretations, while the extension to large yaw angles reveals phenomena not previously documented.

In conclusion, yaw emerges as a critical parameter for both the quantitative description and the mechanistic understanding of ballistic plate perforation. By combining mesh-independent normalization, regression with the Lambert-Jonas model, and fracture-based interpretation of simulation results, this work establishes a bridge between empirical ballistic-limit analysis and the theoretical framework of fracture mechanics. Future efforts should expand this approach to multilayer targets, different material systems, and dedicated experiments at large yaw, consolidating yaw as a central design variable in the mechanics of protective structures.