ANCF Finite Element/Multibody System Formulation of the Ligament/Bone Insertion Site Constraints

Abstract

The focus of this investigation is to study the mechanics of the knee joint using new ligament/bone insertion site constraint models that require the integration of multibody system and large displacement finite element algorithms. Two different sets of clamped end conditions at the ligament/bone insertion site are examined using nonlinear large displacement absolute nodal coordinate formulation (ANCF) finite elements. The first set of end conditions, called the partially clamped joint, eliminates only the translations and rotations at a point, allowing for the cross section stretch and shear at the ligament/bone connection. The second joint, called the fully clamped joint, eliminates all the translation, rotation, and deformation degrees of freedom of the cross section at the ligament/bone insertion site. In the case of the fully clamped joint, the gradient vectors do not change their length and orientation, allowing for the use of the constant strain assumptions. The partially clamped joint, on the other hand, allows for the change in length and relative orientation of the gradient vectors at the bone/ligament insertion site, leading to the cross section deformation induced by knee movements. Nanson’s formula is applied as a measure of the deformation of the cross section in the case of the partially clamped joint. In this study, the major bones in the knee joint consisting of the femur, tibia, and fibula are modeled as rigid bodies while the ligaments structures are modeled using the large displacement ANCF finite elements. Two different ANCF finite element models are developed in this investigation: the first model employs the fully parameterized three-dimensional beam element while the second model employs the three-dimensional cable element. The three-dimensional fully parameterized beam element allows for a straight forward implementation of a neo-Hookean constitutive model that can be used to accurately predict the large displacement as experienced in knee flexation and rotation. At the ligament bone insertion site, the ANCF fully parameterized beam element is used to define a fully or partially constrained joint while the ANCF cable element can only be used to define one joint type. The fully and partially clamped joint constraints are satisfied at the position, velocity, and acceleration levels using a dynamic formulation that is based on an optimum sparse matrix structure. The approach described in this investigation can be used to develop more realistic models of the knee and is applicable to future research studies on ligaments, muscles, and soft tissues. In particular, the partially clamped joint representation of the ligament/bone insertion site constraints can be used to develop improved structural mechanics models of the knee.