An analytical model is developed to study various 'system effects' during impact of a flat-faced, cylindrical projectile into a flexible, multi-layered target with no bonding between layers. Each thin layer is assumed to have in-plane, isotropic, elastic mechanical properties. The model allows variation of the mechanical properties from layer to layer as well as the spacings between the layers in order to study their combined effects on the ballistic performance of the system. In particular, we consider such performance measures as the V 50 limit velocity, the number of layers penetrated when impacting below this limit, and the residual projectile velocity after complete penetration above this limit. The V 50 performance of the target is found to degrade progressively as the spacings between layers are increased relative to the sum of layer thicknesses without spacing. A second finding is that for a given set of layers with differing mechanical properties, both the V 50 and the residual velocity depend on the order of layer placement. A third finding is that among systems with identical layers of a given in-plane tensile strength, the V 50 velocity increases with increasing strain-to-failure of the layers. However the relative magnitude of this increase diminishes with increasing target-to-projectile areal density ratio. The model builds on the authors' previous analysis for impact into a single elastic membrane and the results have important design implications for armor design especially for hybrid material configurations.Key words: Deflection of projectile, multi-layered fibrous body armor, penetration characteristic, residual velocity after penetration, system effects in ballistic impact, V 50 performance.
We develop Monte Carlo simulation and theory to study the statistical strength characteristics of twisted fiber bundles. These consist of fibers that follow a Weibull distribution for strength with shape parameter ρ, and are arranged in an ideal helical structure with surface helix angle α s . Fiber interactions are considered in terms of frictional forces that control stress recovery along broken fibers away from the breaks. A twist-modified global load sharing (TM-GLS) rule is developed for stress redistribution from fibers that are slipping and thus only partially loaded near the breaks. Expressions for the radial pressure distribution in the yarn and corresponding lengths of frictional zones in broken fibers in the various layers are derived considering the discrete nature of the fibers in the bundle. Three different characteristic length scales of strength development for a twisted bundle are proposed, which depend on friction coefficient, f , and surface twist angle, α s . These are δ min c , δ avg c , or δ max c , arising from the consideration of the minimum, average, or maximum stress recovery length among the fibers in the bundle along its axis. We show that the normalized strengths of a twisted bundle with length equal to any one of these characteristic lengths approximately follow a Gaussian distribution. Compared to a TM-ELS (twist-modified equal load sharing) bundle, the TM-GLS bundle has improved strength because through friction a broken fiber can recover its stress within the bundle length. We also show that the relationship between the normalized bundle strength and α s depends on the characteristic length scale used: for δ min c the normalized strength drops quickly with α s ; for δ avg c it decreases as well, but at a slower rate; and for δ max c the normalized strength first attains a maximum at an optimal value of α s before ultimately decreasing with α s . Finally, we compare the simulation results for optimal twist angle with experimental data in the literature and get excellent agreement.
In this work we investigate the effects of yarn diameter and gauge length on the statistical strength of yarns spun from carbon nanotubes (CNTs). Tensile tests are conducted on a large sample set of nanostructured CNT yarns. The data show that strength varies substantially and both strength and statistical dispersion in strength decreases as yarn diameter increases. To explain these phenomena and forecast their effects on larger-scale structures, a hierarchical set of Monte Carlo simulation models is developed: the lower-scale model aims to predict the relationship between yarn nanostructure and tensile strength and the higher-scale model aims to relate the strength of CNT yarns to the strength of composites reinforced with unidirectionally aligned CNT yarns. Predictions indicate that, for both structures, the mean and statistical variation in strength will decrease as the surface twist angle, number of CNTs in cross section and gauge length of the yarn increases. The predicted reductions in variability due to yarn nanostructure will be important for determining ways to minimize the detrimental effects of increasing length scale on strength.
Equal distribution of load among fibrils in contact with a substrate is an important characteristic of fibrillar structures used by many small animals and insects for contact and adhesion. This is in contrast with continuum systems where stress concentration dominates interfacial failure. In this work, we study how adhesion strength of a fibrillar system depends on substrate roughness and variability of the fibril structure, which are modelled using probability distributions for fibril length and fibril attachment strength. Monte Carlo simulations are carried out to determine the adhesion strength statistics where fibril length follows normal or uniform distribution and attachment strength has a power-law form. Our results indicate that the strength distribution is Gaussian (normal) for both the uniform and the normal distributions for length. However, the fibrillar structure having normally distributed lengths has higher strength and lower toughness than one having uniformly distributed lengths. Our simulations also show that an increase in the compliance of the fibrils can compensate for both the substrate roughness and the attachment strength variation. We also show that, as the number of fibrils n increases, the load-carrying efficiency of each fibril goes down. For large n, this effect is found to be small. Furthermore, this effect is compensated by the fact that the standard deviation of the adhesive strength decreases as 1= ffiffiffi n p .
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