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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.

The mechanical behaviors of biological soft tissues are challenging to describe abstractly, with each individual tissue potentially characterized by its own unique nonlinear, anisotropic, and viscoelastic properties. These complexities are exacerbated by patient to patient variability, both mechanically and anatomically, and by inherent constitutive heterogeneity. Despite these challenges, computational models of whole knee biomechanics can be instrumental in describing the onset and progression of injury and disease. In this work, a three-dimensional whole knee computational model was developed using patient-specific anatomy, containing tissues with constitutive relationships built from relevant experimental investigations. In an effort to address the common assumption of linear elastic descriptions of articular cartilage in whole knee models, this work investigates the implications, with respect to macroscopic kinematics and local deformation, of incorporating physiologically motivated and mechanically accurate constitutive heterogeneity in articular cartilage, highlighting the sensitivities of each corresponding level of constitutive complexity. We show how the inclusion of representative cartilage material models affects deformation distributions within the joint, as well as relative joint motion. In particular, the assumption of linear elasticity in articular cartilage results in an overprediction of joint motion and significantly affects predicted local cartilage strains, while full-field, mechanically heterogeneous cartilage descriptions have a less drastic effect at both the tissue and joint levels. Nonetheless, joints containing complete descriptions of articular cartilage heterogeneity may be an integral component in building comprehensive computational tools to advance our understanding of injury and disease mechanisms.

Although non-contact human ACL tears are a common knee injury, little is known about why they usually fail near the femoral enthesis. Recent histological studies have identified a range of characteristic femoral enthesis tidemark profiles and ligament attachment angles. We tested the effect of the tidemark profile and attachment angle on the distribution of strain across the enthesis, under a ligament stretch of 1.1. We employed a 2D analytical model followed by 3D finite element models using three constitutive forms and solved with ABAQUS/Standard. The results show that the maximum equivalent strain was located in the most distal region of the ACL femoral enthesis. It is noteworthy that this strain was markedly increased by a concave (with respect to bone) entheseal profile in that region as well as by a smaller attachment angle, both of which are features more commonly found in females. Although the magnitude of the maximum equivalent strain predicted was not consistent among the constitutive models used, it did not affect the relationship observed between entheseal shape and maximum equivalent strain. We conclude that a concave tidemark profile and acute attachment angle at the femoral ACL enthesis increase the risk for ACL failure, and that failure is most likely to begin in the most distal region of that enthesis.

Uniaxially arranged nanocomposite structures are common across biological materials. Through efficient structural ordering and hierarchies, these materials exhibit stiffnesses and strengths comparable to the better of their constituents. While much is known regarding the mechanical properties of materials with explicit stiff and compliant phases, substantially less is understood about neat (matrix-free) materials composed from high aspect ratio particles. A promising example of nanoparticle assemblies are cellulose nanocrystal (CNC) thin films, which display desirable elastic and failure properties. Despite the exceptional properties of CNC films, accurately linking their nanoscopic properties to macroscale performance remains a challenge. Therefore, this work assesses the effects of fiber geometry and in-plane topology on the elastic and failure behaviors of neat CNC thin films using an atomistically informed coarse-grained molecular dynamics model. Short fiber films show a greater dependence on their specific in-plane ordering compared to longer fiber films. Furthermore, aligned, brick, and mortar type CNC films exhibit a remarkable resiliency to random structural perturbations, particularly for films built from long fibers. Finally, simulation size is shown to affect the apparent failure properties of CNC films, with no meaningful impact on elastic property predictions. The relationships between structure and fiber length, as well as the sensitivities to structural randomness and simulation size, elucidated herein provide a comprehensive overview of the expected mechanics of high aspect ratio nanoparticle assemblies.

Articular cartilage focal defects are common soft tissue injuries potentially linked to osteoarthritis (OA) development. Although several defect characteristics likely contribute to osteoarthritis, their relationship to local tissue deformation remains unclear. Using finite element models with various femoral cartilage geometries, we explore how defects change cartilage deformation and joint kinematics assuming loading representative of the maximum joint compression during the stance phase of gait. We show how defects, in combination with location-dependent cartilage mechanics, alter deformation in affected and opposing cartilages, as well as joint kinematics. Small and average sized defects increased maximum compressive strains by approximately 50% and 100%, respectively, compared to healthy cartilage. Shifts in the spatial locations of maximum compressive strains of defect containing models were also observed, resulting in loading of cartilage regions with reduced initial stiffnesses supporting the new, elevated loading environments. Simulated osteoarthritis (modeled as a global reduction in mean cartilage stiffness) did not significantly alter joint kinematics, but exacerbated tissue deformation. Femoral defects were also found to affect healthy tibial cartilage deformations. Lateral femoral defects increased tibial cartilage maximum compressive strains by 25%, while small and average sized medial defects exhibited decreases of 6% and 15%, respectively, compared to healthy cartilage. Femoral defects also affected the spatial distributions of deformation across the articular surfaces. These deviations are especially meaningful in the context of cartilage with location-dependent mechanics, leading to increases in peak contact stresses supported by the cartilage of between 11% and 34% over healthy cartilage.

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