Luis G. Maqueda

Use of general nonlinear material models in beam problems: Application to belts and rubber chains

Accurate modeling of many engineering systems requires the integration of multibody system and large deformation finite element algorithms that are based on general constitutive models, account for the coupling between the large rotation and deformation, and allow capturing coupled deformation modes that cannot be captured using beam formulations implemented in existing computational algorithms and computer codes. In this investigation, new three-dimensional nonlinear dynamic rubber chains and belt drives models are developed using the finite element absolute nodal coordinate formulation (ANCF) that allows for a straight forward implementation of general linear and nonlinear material models for structural elements such as beams, plates, and shells. Furthermore, this formulation, which is based on a more general kinematic description, can be used to predict the cross section deformation and its coupling with the extension and bending of the belt drives and rubber chains. The ANCF cross section deformation results are validated by comparison with the results obtained using solid finite elements in the case of a simple tension test problem. The effect of the use of different linear and nonlinear constitutive laws in modeling belt drive mechanisms is also examined in this investigation. The finite element formulation presented in this paper is implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing detailed models of mechanical systems subject to general loading conditions, nonlinear algebraic constraint equations, and arbitrary large displacements that characterize belt drives and tracked vehicle dynamics. The successful integration of large deformation finite element and multibody system algorithms is shown to be necessary in order to be able to study the dynamics of complex tracked vehicles with rubber chains. A computer simulation of a three-dimensional multibody tracked vehicle model that consists of twenty rigid bodies and two flexible rubber chains is used in order to demonstrate the use of the formulations presented in this investigation.

Nonlinear Constitutive Models and the Finite Element Absolute Nodal Coordinate Formulation

In this investigation, the use of three different nonlinear constitutive models based on the hyper-elasticity theory with the absolute nodal coordinate formulation is considered. These three nonlinear constitutive models are based on the Neo-Hookean constitutive law for compressible materials, the Neo-Hookean constitutive law for incompressible materials, and the Mooney-Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior and are considered incompressible or nearly incompressible. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. The results show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.Copyright

A Multibody/Finite Element Non-Linear Formulation of a Two-Ligament Knee Joint

The focus of this investigation is to study the mechanics of the human knee using a new method that integrates multi-body system and large deformation finite element algorithms. The major bones in the knee joint consisting of the femur, tibia, fibula are modeled as rigid bodies. The ligaments structures are modeled using the large deformation finite element Absolute Nodal Coordinate Formulation (ANCF) with an implementation of a Neo-Hookean constitutive model that allows for large deformations as experienced in knee flexation and rotation. The Neo-Hookean strain energy function used in this study takes into consideration the near incompressibility of the ligaments. The ANCF is used in the formulation of the algebraic equations that define the ligament/bone rigid connection. A unique feature of the ANCF is that it allows for the deformation of the ligament cross-section. At the ligament/bone connection, the ANCF is used to define a fully constrained joint. This aspect of the model reflects the actual structural mechanics of the knee. In addition, this model will reflect the fact that the geometry, placement and attachment of the two collateral ligaments (the LCL and MCL), are significantly different from what has been used in most knee models developed in previous investigations. The approach described in this paper will provide a more realistic model of the knee and thus more applicable to future research studies. The obtained preliminary results of other applications show that the ANCF can be an effective tool for modeling very flexible structures like ligaments subjected to large deformations. In the future, the ANCF models could assist in examining the mechanics of the knee to study knee injuries and possible prevention means, as well as an improved understanding of the role of each individual ligament in the diagnosis and assessment of disease states, aging and potential therapies.Copyright

Three-dimensional Large Deformation Finite Element Analysis of Belt Drives

In this paper, methods for the large deformation finite element analysis of belt drives are presented. The new nonlinear dynamic formulations for belt drives are based on the three-dimensional large deformation absolute nodal coordinate formulation. Two different belt drive models that have different numbers of degrees of freedom and different modes of deformation are presented. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. In the first model, a thin plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects is used. In the second model, a cable element obtained from a more general threedimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications is used. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling one to systematically examine the effect of bending on the nonlinear dynamics of belt drives. The finite element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multi-body algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The plate formulation also allows using a surface distribution of the contact forces; such a distribution can not be obtained using beam elements since this element is represented by its centerline. Contact forces on the surface are compared to analytical results of similar but twodimensional model. The friction and normal force distributions are in agreement with analytical models. Some differences in results between the plate, cable and analytical formulations are obtained and discussed.