The Langevin function is a non-invertible function, whose inverse commonly appears in statistical mechanics problems, particularly network models of rubber elasticity. This non-invertibility often results in the use of approximations. Owing to the prevalence of the inverse Langevin function, numerous forms of approximate have been proposed. Rational approximates are often employed because of their ability to admit asymptotic behavior in finite domains, similar to the exact inverse Langevin function. Despite the desired asymptotics of rational approximates, there is unavoidable error associated with the approximate within its domain. In this work, an error-minimizing approach for determining specific forms of rational approximates is generalized to approximates of arbitrary numerator and denominator orders. By expanding to general orders of rational approximates, the best approximate can be selected for an application based on either maximum relative error or function form considerations.
The Bouligand structure features a helicoidal (twisted plywood) layup of fibers that are uniaxially arranged in-plane and is a hallmark of biomaterials that exhibit outstanding impact resistance. Despite its performance advantage, the underlying mechanisms for its outstanding impact resistance remain poorly understood, posing challenges for optimizing the design and development of bio-inspired materials with Bouligand microstructures. Interestingly, many bio-sourced nanomaterials, such as cellulose nanocrystals (CNCs), readily self-assemble into helicoidal thin films with inter-layer (pitch) angles tunable via solvent processing. Taking CNC films as a model Bouligand system, we present atomisticallyinformed coarse-grained molecular dynamics simulations to measure the ballistic performance of thin films with helicoidally assembled nanocrystals by subjecting them to loading similar to laser-induced projectile impact tests. The effect of pitch angle on the impact performance of CNC films was quantified in the context of their specific ballistic limit velocity and energy absorption. Bouligand structures with low pitch angles (18-42 ) were found to display the highest ballistic resistance, significantly outperforming other pitch angle and quasi-isotropic baseline structures. Improved energy dissipation through greater interfacial sliding, larger in-plane crack openings, and through-thickness twisting cracks resulted in improved impact performance of optimal pitch angle Bouligand CNC films. Intriguingly, decreasing interfacial interactions enhanced the impact performance by readily admitting dissipative inter-fibril and inter-layer sliding events without severe fibril fragmentation. This work helps reveal structural and chemical factors that govern the optimal mechanical design of Bouligand microstructures made from high aspect ratio nanocrystals, paving the way for sustainable, impact resistant, and multifunctional films.
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