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Knee finite element (FE) models are used to study tissue deformation in response to complex loads. Typically, ligaments are modeled using transversely isotropic, hyperelastic material models fitted to tension data along the predominant fiber direction (longitudinal) and, less commonly, to tension data orthogonal to the fiber direction (transverse). Currently, the shear and bulk responses of the anterior cruciate ligament (ACL) are not fitted to experimental data. In this study, a newly proposed material model was fitted to longitudinal tension, transverse tension, and shear experimental data. The matrix transverse tensile, shear, and bulk stiffnesses were then varied independently to determine the impact of each property on knee kinematics and tissue deformation in a whole-knee FE model. The range of values for each parameter was chosen based on published FE studies of the knee. For a knee at full extension under 134 N anterior tibial force (ATF), increasing matrix transverse tensile stiffness, shear stiffness, or bulk stiffness decreased anterior tibial translation (ATT), ACL longitudinal strain, and ACL shear strain. For a knee under 134 N ATF and 1600 N compression, changing the ACL matrix mechanical properties caused variations in ATT and thus changed cartilage deformation contours by changing the point of contact between the femoral and the tibial cartilage. These findings indicate that material models for the ACL must describe matrix material properties to best predict the in vivo response to applied loads.

Soft tissues of the knee contain complex physiological microstructures with nontrivial hierarchical organization that manifests as mechanical directionality and nonlinearity. For supporting ligaments in the knee, constitutive theories have been developed to address these issues—predominately assuming transverse isotropy with the preferred material direction aligned with structural collagen. However, with the abundance of constitutive theories available it can often be difficult to differentiate among them, and this selection can be complicated by new and conflicting mechanical characterization data. This work rigorously describes the process by which candidate constitutive theories can be evaluated in the context of transverse hyperelasticity using quasi-static stress–strain experimental data of the medial collateral ligament (MCL) of the knee. We show how recent data of the MCL fails to be accurately represented using previously validated and generally accepted constitutive theories. We expand the pool of candidate theories to include chain-based statistical mechanics and hybrid theories to explore the range of representative models of the MCL. This novel hybrid theory, constructed by superposing a slightly compressible, isotropic eight-chain MacKintosh network model with a phenomenological directional component, is shown to have superior performance in the representing the quasi-static stress–strain behavior of the MCL, particularly in the transverse direction. Using a simplified finite element model of ligament deformation, implicit structure among theories is examined using a straightforward framework for developing computational material models.

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.

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.

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.

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 atomistically-informed 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 multi-functional films.

The inverse Langevin function is an integral component to network models of rubber elasticity with networks assembled using non-Gaussian descriptions of chain statistics. The non-invertibility of the inverse Langevin often requires the implementation of approximations. A variety of approximant forms have been proposed, including series, rational, and trigonometric divided domain functions. In this work, we develop an error-minimizing framework for determining inverse Langevin approximants. This method can be generalized to approximants of arbitrary form, and the approximants produced through the proposed framework represent the error-minimized forms of the particular base function. We applied the error-minimizing approach to Pade approximants, reducing the average and maximum relative errors admitted by the forms of the approximants. The error-minimization technique was extended to improve standard Pade approximants by way of understanding the error admitted by the specific approximant and using error-correcting functions to minimize the residual relative error. Tailored approximants can also be constructed by appreciating the evaluation domain of the application implementing the inverse Langevin function. Using a non-Gaussian, eight-chain network model of rubber elasticity, we show how specifying locations of zero error and reducing the minimization domain can shrink the associated error of the approximant and eliminate numerical discontinuities in stress calculations at small deformations.

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.

Based on a reader comment, it has become clear that, in the originally published version of the article, Fig. 1 was published with incorrect anatomical labels.