Nonlinear Constitutive Models and the Finite Element Absolute Nodal Coordinate Formulation

2007 ◽  
Author(s):  
Luis G. Maqueda ◽  
Ahmed A. Shabana
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Constitutive Models ◽  
Biological Tissues ◽  
Constitutive Law ◽  
Elastic Behavior ◽  
Deformation Analysis ◽  
Absolute Nodal Coordinate ◽  
The Absolute ◽  
Nonlinear Constitutive Models ◽  
Nonlinear Elastic

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the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid.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.

A New Constitutive Model for the Compressibility of Elastomers at Finite Deformations

2001 ◽  
Vol 74 (4) ◽  
pp. 541-559 ◽  
Author(s):  
Jeffrey E. Bischoff ◽  
Ellen M. Arruda ◽  
Karl Grosh
Keyword(s):  
Uniaxial Tension ◽  
Volume Change ◽  
Constitutive Models ◽  
Hydrostatic Compression ◽  
Constitutive Law ◽  
Elastic Behavior ◽  
Compression Tests ◽  
Volume Response ◽  
Using Data ◽  
Nonlinear Elastic

Abstract Although traditional constitutive models for rubbery elastic materials are incompressible, many materials that demonstrate nonlinear elastic behavior are somewhat compressible. Clearly important in hydrostatic deformations, compressibility can also significantly affect the response of elastomers in applications for which several boundaries are rigidly fixed, such as bushings, or triaxial states of stress are realized. Compressibility is also important for convergence of finite element simulations in which a rubbery elastic constitutive law is in use. Volume changes that reflect compressibility have been observed historically in both uniaxial tension and hydrostatic compression tests; however, there appear to be no data obtained from both types of tests on the same material by which to validate a compressible hyperelastic law. In this paper, we propose a new compressible hyperelastic constitutive law for elastomers and other rubbery materials in which entropy and internal energy changes contribute to the volume change. Using data from the literature, we show that this law is capable of reproducing both the pressure—volume response of elastomers in hydrostatic compression, as well as the stress—stretch and volume change—stretch data of elastomers in uniaxial tension.

A New Plate Element Based on the Absolute Nodal Coordinate Formulation

2001 ◽  
Author(s):  
Aki M. Mikkola ◽  
Ahmed A. Shabana
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Body Motion ◽  
Deformation Analysis ◽  
Plate Element ◽  
Large Rotation ◽  
Shell Elements ◽  
Absolute Nodal Coordinate ◽  
Plates And Shells ◽  
The Absolute

Abstract In this investigation, a method for the finite rotation and large deformation analysis of plates is presented. The method, which is based on the absolute nodal coordinate formulation, leads to a plate element capable of representing exact rigid body motion. In this method, continuity conditions on all the displacement gradients are imposed. Therefore, non-smoothness of the plate mid-surface at the nodal points is avoided. By developing such a plate element, a constant mass matrix is obtained, and as a consequence, the centrifugal and Coriolis forces are equal to zero. Generalization of the formulation to the case of shell elements is discussed. Numerical results are presented in order to demonstrate the use of the proposed method in the large rotation and deformation analysis of plates and shells.

Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems

2013 ◽  
Vol 8 (3) ◽  
Author(s):  
Johannes Gerstmayr ◽  
Hiroyuki Sugiyama ◽  
Aki Mikkola
Keyword(s):  
Large Deformation ◽  
Multibody Systems ◽  
Absolute Nodal Coordinate Formulation ◽  
Efficient Solutions ◽  
Rotor Blades ◽  
Deformation Analysis ◽  
Computer Algorithms ◽  
Absolute Nodal Coordinate ◽  
Elasto Plasticity ◽  
The Absolute

The aim of this study is to provide a comprehensive review of the finite element absolute nodal coordinate formulation, which can be used to obtain efficient solutions to large deformation problems of constrained multibody systems. In particular, important features of different types of beam and plate elements that have been proposed since 1996 are reviewed. These elements are categorized by parameterization of the elements (i.e., fully parameterized and gradient deficient elements), strain measures used, and remedies for locking effects. Material nonlinearities and the integration of the absolute nodal coordinate formulation to general multibody dynamics computer algorithms are addressed with particular emphasis on visco-elasticity, elasto-plasticity, and joint constraint formulations. Furthermore, it is shown that the absolute nodal coordinate formulation has been applied to a wide variety of challenging nonlinear dynamics problems that include belt drives, rotor blades, elastic cables, leaf springs, and tires. Unresolved issues and future perspectives of the study of the absolute nodal coordinate formulation are also addressed in this investigation.

Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems

1999 ◽  
Vol 122 (4) ◽  
pp. 498-507 ◽  
Author(s):  
Marcello Campanelli ◽  
Marcello Berzeri ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Multibody Systems ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Systems ◽  
Large Displacement ◽  
Body Motion ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Finite Element Formulations

Many flexible multibody applications are characterized by high inertia forces and motion discontinuities. Because of these characteristics, problems can be encountered when large displacement finite element formulations are used in the simulation of flexible multibody systems. In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed in an earlier article (Rankin, C. C., and Brogan, F. A., 1986, ASME J. Pressure Vessel Technol., 108, pp. 165–174) and the non-incremental absolute nodal coordinate formulation recently proposed (Shabana, A. A., 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge). It is demonstrated in this investigation that the limitation resulting from the use of the infinitesmal nodal rotations in the incremental corotational procedure can lead to simulation problems even when simple flexible multibody applications are considered. The absolute nodal coordinate formulation, on the other hand, does not employ infinitesimal or finite rotation coordinates and leads to a constant mass matrix. Despite the fact that the absolute nodal coordinate formulation leads to a non-linear expression for the elastic forces, the results presented in this study, surprisingly, demonstrate that such a formulation is efficient in static problems as compared to the incremental corotational procedure. The excellent performance of the absolute nodal coordinate formulation in static and dynamic problems can be attributed to the fact that such a formulation does not employ rotations and leads to exact representation of the rigid body motion of the finite element. [S1050-0472(00)00604-8]

Elastic Forces and Coordinate Systems in the Finite Element Formulations

1999 ◽  
Author(s):  
Marcello Berzeri ◽  
Marcello Campanelli ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Frame Of Reference ◽  
Coordinate Systems ◽  
Absolute Nodal Coordinate ◽  
Floating Frame Of Reference ◽  
Elastic Forces ◽  
Floating Frame ◽  
The Absolute ◽  
Finite Element Formulations

Abstract The equivalence of the elastic forces of finite element formulations used in flexible multibody dynamics is the focus of this investigation. Two conceptually different finite element formulations that lead to exact modeling of the rigid body dynamics will be used. These are the floating frame of reference formulation and the absolute nodal coordinate formulation. It is demonstrated in this study that different element coordinate systems, which are used for the convenience of describing the element deformations in the absolute nodal coordinate formulation, lead to similar results as the element size is reduced. The equivalence of the elastic forces in the absolute nodal coordinate and the floating frame of reference formulations is shown. The result of this analysis clearly demonstrates that the instability observed in high speed rotor analytical models due to the neglect of the geometric centrifugal stiffening is not a problem inherent to a particular finite element formulation but only depends on the beam model that is used. Fourier analysis of the solutions obtained in this investigation also sheds new light on the fundamental problem of the choice of the deformable body coordinate system in the floating frame of reference formulation. A new method is presented and used to obtain a simple expression for the elastic forces in the absolute nodal coordinate formulation. This method, which employs a nonlinear elastic strain-displacement relationship, does not result in an unstable solution when the angular velocity is increased.

A Numerical Approach to Solving Flexible Multibody Systems Using the Absolute Nodal Coordinate Formulation

1999 ◽  
Author(s):  
R. Y. Yakoub ◽  
A. A. Shabana
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Sparse Matrix ◽  
Mass Matrix ◽  
Numerical Approach ◽  
Absolute Nodal Coordinates ◽  
Absolute Nodal Coordinate ◽  
Velocity Transformation ◽  
Flexible Multibody ◽  
Nodal Coordinates ◽  
The Absolute

Abstract By utilizing the fact that the absolute nodal coordinate formulation leads to a constant mass matrix, a Cholesky decomposition of the mass matrix can be used to obtain a constant velocity transformation matrix. This velocity transformation can be used to express the absolute nodal coordinates in terms of the generalized Cholesky coordinates. In this case, the inertia matrix associated with the Cholesky coordinates is the identity matrix, and therefore, an optimum sparse matrix structure can be obtained for the augmented multibody equations of motions. The implementation of a computer procedure based on the absolute nodal coordinate formulation and Cholesky coordinates is discussed in this paper. A flexible four-bar linkage is presented in this paper in order to demonstrate the use of Cholesky coordinates in the simulation of the small and large deformations in flexible multibody applications. The results obtained from the absolute nodal coordinate formulation are compared to those obtained from the floating frame of reference formulation.

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