Marko K. Matikainen

Development of Elastic Forces for a Large Deformation Plate Element Based on the Absolute Nodal Coordinate Formulation

Dynamic analysis of large rotation and deformation can be carried out using the absolute nodal coordinate formulation. This formulation, which utilizes global displacements and slope coordinates as nodal variables, make it possible to avoid the difficulties that arise when a rotation is interpolated in three-dimensional applications. In the absolute nodal coordinate formulation, a continuum mechanics approach has become the dominating procedure when elastic forces are defined. It has recently been perceived, however, that the continuum mechanics based absolute nodal coordinate elements suffer from serious shortcomings, including Poissons locking and poor convergence rate. These problems can be circumvented by modifying the displacement field of a finite element in the definition of elastic forces. This allows the use of the mixed type interpolation technique, leading to accurate and efficient finite element formulations. This approach has been previously applied to two- and three-dimensional absolute nodal coordinate based finite elements. In this study, the improved approach for elastic forces is extended to the absolute nodal coordinate plate element. The introduced plate element is compared in static examples to the continuum mechanics based absolute nodal coordinate plate element, as well as to commercial finite element software. A simple dynamic analysis is performed using the introduced element in order to demonstrate the capability of the element to conserve energy.

Analysis of Stress and Strain in the Absolute Nodal Coordinate Formulation

Abstract Accurate values of stress and strain are required for the evaluation of comparative stresses in nonlinear material behavior. The absolute nodal coordinate formulation (ANCF) has been recently developed and focuses on the modeling of beams and plates under the presence of large deformation. The derivation of the equations of motion for an ANCF element is usually based on a solid finite element formulation and thus leads to finite elements that show locking behavior. While the problem of locking in the ANCF might be solved by means of standard techniques, the accuracy of stress and strain quantities within elements is still poor and needs to be improved in order to incorporate nonlinear material behavior. In the present paper, a higher order ANCF element is presented where locking is prevented by means of standard selective reduced integration techniques and the improved order and accuracy of stress and strain quantities is shown, in comparison with the original formulation. As an example of nonlinear material behavior, Prandl–Reuss plasticity is integrated in the absolute nodal coordinate formulation. Results of stress and strain components for the improved higher order element are compared to the solution of fully three-dimensional computations performed with the commercial software ABAQUS. Static and dynamic spatial examples are used to investigate the accuracy. Good agreement of the ANCF is found with the results of ABAQUS, as well as with examples of elasto-plastic multibody systems available from the literature.

Beam Elements with Trapezoidal Cross Section Deformation Modes Based on the Absolute Nodal Coordinate Formulation

In this study, higher order beam elements are developed based on the absolute nodal coordinate formulation. The absolute nodal coordinate formulation is a finite element procedure that was recently proposed for flexible multibody applications. Many different elements based on the absolute nodal coordinate formulation are introduced, but still the beam elements are not able to describe the trapezoidal cross section mode. This leads to the locking phenomena, and therefore, the beam elements based on the absolute nodal coordinate formulation with three dimensional elasticity converge to an inexact solution. In order to avoid the locking phenomena, the trapezoidal cross section deformation mode is included in the beam elements based on the absolute nodal coordinate with additional degrees of freedom. The proper description for the trapezoidal cross section deformation is important for the continuum beam elements based on three‐dimensional elasticity where the material model is often based on general continuum...

A fibre-reinforced poroviscoelastic model accurately describes the biomechanical behaviour of the rat achilles tendon.

Background Computational models of Achilles tendons can help understanding how healthy tendons are affected by repetitive loading and how the different tissue constituents contribute to the tendon’s biomechanical response. However, available models of Achilles tendon are limited in their description of the hierarchical multi-structural composition of the tissue. This study hypothesised that a poroviscoelastic fibre-reinforced model, previously successful in capturing cartilage biomechanical behaviour, can depict the biomechanical behaviour of the rat Achilles tendon found experimentally. Materials and Methods We developed a new material model of the Achilles tendon, which considers the tendon’s main constituents namely: water, proteoglycan matrix and collagen fibres. A hyperelastic formulation of the proteoglycan matrix enabled computations of large deformations of the tendon, and collagen fibres were modelled as viscoelastic. Specimen-specific finite element models were created of 9 rat Achilles tendons from an animal experiment and simulations were carried out following a repetitive tensile loading protocol. The material model parameters were calibrated against data from the rats by minimising the root mean squared error (RMS) between experimental force data and model output. Results and Conclusions All specimen models were successfully fitted to experimental data with high accuracy (RMS 0.42-1.02). Additional simulations predicted more compliant and soft tendon behaviour at reduced strain-rates compared to higher strain-rates that produce a stiff and brittle tendon response. Stress-relaxation simulations exhibited strain-dependent stress-relaxation behaviour where larger strains produced slower relaxation rates compared to smaller strain levels. Our simulations showed that the collagen fibres in the Achilles tendon are the main load-bearing component during tensile loading, where the orientation of the collagen fibres plays an important role for the tendon’s viscoelastic response. In conclusion, this model can capture the repetitive loading and unloading behaviour of intact and healthy Achilles tendons, which is a critical first step towards understanding tendon homeostasis and function as this biomechanical response changes in diseased tendons.

Inclusion of Transverse Shear Deformation in a Beam Element Based on the Absolute Nodal Coordinate Formulation

In this study, a procedure to account for transverse shear deformation in the absolute nodal coordinate formulation is presented. In the absolute nodal coordinate formulation, shear deformation is usually defined by employing the slope vectors in the element transverse direction. This leads to the description of deformation modes that are, in practical problems, associated with high frequencies. These high frequencies, in turn, complicate the time integration procedure burdening numerical performance. In this study, the description of transverse shear deformation is accounted for in a two-dimensional beam element based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the orientation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. Numerical results are presented in order to demonstrate the accuracy of the introduced element in static and dynamic cases. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable beam elements.

The Simplest 3- and 4-Noded Fully-Parameterized ANCF Plate Elements

In this research, the simplest kinematical models of triangular and rectangular plate finite elements using the absolute nodal coordinate formulation (ANCF) are presented. The ANCF is the finite-element large-displacement-and-rotation approach, which uses the inertial-frame nodal position vectors and their derivatives (slopes) only, without employing any rotation parameters or their equivalent. As a consequence, the kinematics of the elements becomes linear, simplifying the inertia part of the equations of motion, which is also linear. In contrary, due to the need for employing the Green-Lagrange strain tensor, the elastic forces normally appear in a more complicated highly-nonlinear manner than in other large-rotation formulations. In this research, to reduce the computational burden, two new plate elements are proposed that are the simplest possible triangular and rectangular elements in the fully-parameterized ANCF: they employ transverse slopes only, without using longitudinal slopes.Copyright

A Procedure for the Inclusion of Transverse Shear Deformation in a Beam Element Based on the Absolute Nodal Coordinate Formulation

In this study, a procedure to account for transverse shear deformation in the absolute nodal coordinate formulation is presented. In the absolute nodal coordinate formulation, shear deformation is usually defined by employing the slope vectors in the element transverse direction. This leads to the description of deformation modes that are, in practical problems, associated with high frequencies. These high frequencies, in turn, complicate the time integration procedure burdening numerical performance. In this study, the description of transverse shear deformation is accounted for in a two-dimensional beam element based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the rotation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. Numerical results are presented in order to demonstrate the accuracy of the introduced element in static and dynamic cases. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable beam elements.Copyright

Stresses of an AMB-Supported Rotor Arising From the Sudden Contact With Backup Bearings

High speed technology has a fundamental advantage when compared to conventional motors. Active magnetic bearings (AMBs) are often needed in the high speed motors, as they can provide almost frictionless support and allowing control for the dynamics of the system. AMBs require backup bearings to avoid damage resulting from a failure in the component itself, in the power system, or in the control system. During a rotor-bearing contact event, substantial impact forces may occur between the backup bearing and the rotor.In this study the rotor model is implemented using a finite element approach including backup bearings which are modelled as simplified cageless ball bearings. Based on results of this study, the increase in backup bearing misalignment can have proportional impact on the rotor bending and shear stresses in touchdown situation.Copyright

Improved Description of Elastic Forces for the Absolute Nodal Coordinate Based Plate Element

In this study, the improved description of elastic forces for the absolute nodal coordinate based plate element is introduced. The absolute nodal coordinate formulation, which utilizes global displacements and slope coordinates as nodal variables, can be used in large rotation and deformation dynamic analysis of beam and plate structures. The formulation avoids difficulties that arise when a rotation is interpolated in three-dimensional applications. In the absolute nodal coordinate formulation, a continuum mechanics approach has become the dominating procedure when elastic forces are defined. It has recently been perceived, however, that the continuum mechanics based absolute nodal coordinate elements suffer from serious shortcomings, including Poisson’s locking and poor convergence rate. These problems can be circumvented by modifying the displacement field of a finite element in the definition of elastic forces. This allows the use of the mixed type interpolation technique, leading to accurate and efficient finite element formulations. This approach has been previously applied to two- and three-dimensional absolute nodal coordinate based finite elements. In this study, the improved approach for elastic forces is extended to the absolute nodal coordinate plate element. The introduced plate element is compared in static examples to the continuum mechanics based absolute nodal coordinate plate element, as well as to commercial finite element software.Copyright

Analysis of high-order quadrilateral plate elements based on the absolute nodal coordinate formulation for three-dimensional elasticity

The absolute nodal coordinate formulation is a computational approach to analyze the dynamic performance of flexible bodies experiencing large deformations in multibody system dynamics applications. In the absolute nodal coordinate formulation, full three-dimensional elasticity can be used in the definition of the elastic forces. This approach makes it straightforward to implement advanced material models known from general continuum mechanics in the absolute nodal coordinate formulation. As, however, pointed out in the literature, the use of full three-dimensional elasticity can lead to severe locking problems, already present in simple, static tests. To overcome these drawbacks and to get a better understanding of these behaviors in the case of absolute nodal coordinate formulation elements, this study introduces and carefully analyses several high-order three-dimensional plate elements based on the absolute nodal coordinate formulation, primarily in meaningful static scenarios. The proposed elements are put to test in various numerical experiments intended to bring forward possible locking phenomena and to evaluate the accuracy attainable with the considered element formulations. The proposed eight- and nine-node elements that incorporate polynomial approximations of second order in all three directions prove to be advantageous both with respect to the actual performance and with regard to the numerical efficiency when compared to other absolute nodal coordinate formulation plate elements. A comparison with a four-node high-order element corroborates the supposition that the usage of in-plane slopes as nodal coordinates has a negative effect on numerical convergence properties in thin-plate use cases. An additional example showcases the functioning of two of the higher-order elements in a dynamic simulation.