This paper presents a reliability-oriented analytical framework for the quantitative assessment of fragment-resistant multilayer protective equipment subjected to impulsive fragment loading. The study is motivated by the stochastic nature of fragment generation and impact conditions in industrial and occupational accident scenarios, where deterministic penetration criteria are insufficient to describe protective performance. Fragment interactions are modelled as stochastic spatial events, with impact locations and kinematic characteristics treated as random variables and mapped onto a predefined protected region. System failure is formulated using an energy-based limit-state criterion defined by comparison between the absorbed energy demand induced by fragment impact and a critical admissible energy threshold. The fragment–PPE interaction is described using a reduced-order dynamic formulation with concentrated parameters, capturing the dominant normal deformation response under short-duration impulsive loading. Closed-form analytical expressions are derived that relate fragment mass and velocity to impact impulse and absorbed energy. The resulting formulation establishes a direct link between impulse-driven dynamic response, progressive multilayer engagement, and failure probability under single and repeated impact events. Application of the proposed framework to a representative multilayer protective configuration demonstrates physically consistent reliability trends and confirms its computational efficiency. The framework provides a practical tool for reliability-informed assessment and preliminary design of fragment-resistant multilayer protective equipment.

This study develops an analytical description of the motion of dilute solid particles in the boundary layer of laminar horizontal flows subjected to weak transverse pulsations. The analysis is formulated for dilute spherical solid particles subjected to transverse velocity pulsations in a laminar boundary-layer flow. A coupled matrix representation of the governing equations is formulated, and closed-form solutions are obtained using Laplace transformation. The analytical expressions capture transient evolution, forced oscillations, resonance effects, and long-term behaviour for particles with different density ratios. Numerical evaluation shows that light particles migrate toward faster regions of the boundary layer and accelerate longitudinally, while heavy particles move toward slower layers and decelerate. Transverse pulsations generate oscillatory trajectories whose amplitude increases near resonance. Impulsive perturbations superimposed on the continuous motion lead to discontinuous transitions consistent with the linear matrix system. The results provide a unified physical interpretation of particle redistribution mechanisms in boundary layers and offer a compact analytical tool for dilute multiphase flow modelling.

This study develops an analytical description of the motion of dilute solid particles in the boundary layer of laminar horizontal flows subjected to weak transverse pulsations. A coupled matrix representation of the governing equations is formulated, and closed-form solutions are obtained using Laplace transformation. The analytical expressions capture transient evolution, forced oscillations, resonance effects, and long-term behaviour for particles with different density ratios. Numerical evaluation shows that light particles migrate toward faster regions of the boundary layer and accelerate longitudinally, while heavy particles move toward slower layers and decelerate. Transverse pulsations generate oscillatory trajectories whose amplitude increases near resonance. Impulsive perturbations superimposed on the continuous motion lead to discontinuous transitions consistent with the linear matrix system. The results provide a unified physical interpretation of particle redistribution mechanisms in boundary layers and offer a compact analytical tool for dilute multiphase flow modelling.