Abstract
In the general theory of continuum mechanics, the state of rotation and deformation of material points can be uniquely defined from the displacement field by using the nine independent components of the displacement gradients. For this reason, the use of the absolute rotation parameters as nodal coordinates, without relating them to the displacement gradients, leads to coordinate redundancy that leads to numerical and fundamental problems in many existing large rotation finite element formulations. Because of this fundamental problem, special measures that require modifications of the numerical integration methods were proposed in the literature in order to satisfy the principle of work and energy. As demonstrated in this paper, no such measures need to be taken when the finite element absolute nodal coordinate formulation is used since the principle of work and energy are automatically satisfied. This formulation does not suffer from the problem of coordinate redundancy and ensures the continuity of stresses and strains at the nodal points. In this study, the use of the implicit integration methods with the consistent Lagrangian elasto-plastic tangent moduli is examined when the absolute nodal coordinate formulation is used. The performance of different numerical integration methods in the dynamic analysis of large elasto-plastic deformation problems is investigated. It is shown that all these methods, in the case of convergence, yield a solution that satisfies the principle of work and energy without the need of taking any special measures. Semi-implicit integration methods, however, can lead to numerical difficulties in the case of very stiff problems due to the linearization made in these methods in order to avoid the iterative Newton--Raphson procedure. It is also demonstrated that the use of the consistent Lagrangian-plastic tangent moduli derived in this investigation using the absolute nodal coordinate formulation leads to better convergence of the iterative Newton--Raphson procedure used in the implicit integration methods.
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References
Shabana, A. A., 'Finite element incremental approach and exact rigid body inertia', ASME Journal of Mechanical Design 118, 1996, 171–178.
Simo, J. C. and Vu-Quoc, L., 'On the dynamics of flexible beams under large overall motions', ASME Journal of Applied Mechanics 53, 1986, 849–863.
Simo, J. C. and Vu-Quoc, L., 'Three-dimensional finite-strain rod model. Part II: Computational aspects', Computer Methods in Applied Mechanics and Engineering 58, 1986, 79–116.
Ibrahimbegovic, A., 'On finite element implementation of geometrically nonlinear reissner's beam theory: Three-dimensional curved beam elements', Computer Methods in Applied Mechanics and Engineering 122, 1995, 11–26.
Bonet, J. and Wood, R. D., Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, Cambridge, 1997.
Shabana, A. A., Dynamics of Multibody Systems, 2nd edn., Cambridge University Press, Cambridge, 1998.
Shabana, A. A. and Yakoub, R. Y., 'Three dimensional absolute nodal coordinate formulation for beam elements', ASME Journal of Mechanical Design 123, 2001, 606–621.
Von Dombrowski, S., 'Analysis of large flexible body deformation in multibody systems using absolute coordinates', Multibody System Dynamics 8, 2002, 409–432.
Shabana, A. A. and Mikkola, A. M., 'Use of the finite element absolute nodal coordinate formulation in modeling slope discontinuity', ASME Journal of Mechanical Design 125, 2003, 342–350.
Mikkola, A. M. and Shabana, A. A., 'A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications', Multibody System Dynamics 9, 2003, 283–309.
Dmitrochenko, O. N. and Pogorelov, D.Y., 'Generalization of plate finite elements for absolute nodal coordinate formulation', Multibody System Dynamics 10, 2003, 17–43.
Sugiyama, H., Escalona, J. L., and Shabana, A. A., 'Formulation of three-dimensional joint constraints using the absolute nodal coordinates', Nonlinear Dynamics 31, 2003, 167–195.
Sopanen, J. T. and Mikkola, A. M., 'Description of elastic forces in absolute nodal coordinate formulation', Nonlinear Dynamics 34, 2003, 53–74.
Garcia Vallejo, D., Escalona, J. L., Mayo, J., Alvarez, A., and Dominguez, J., 'Describing rigid-flexible multibody systems using absolute coordinates', Nonlinear Dynamics 34, 2003, 75–94.
Gerstmayr, J., 'The absolute nodal coordinate formulation with elasto-plastic deformations', in Proceedings of Multibody Dynamics 2003, Lisbon, Portugal, 2003.
Sugiyama, H. and Shabana, A. A., 'Use of plasticity theory in flexible multibody system dynamics', in Proceedings of ASME International Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Chicago, Illinois, 2003.
Yoo, W.-S., Park, S.-J., Lee, J.-H., Sohn, J.-H., Pogorelov, D. Y., and Dmitrochenko, O. N., 'Large deflection analysis of a thin plate: Computer simulations and experiments', Multibody System Dynamics 11, 2004, 185–208.
Sugiyama, H. and Shabana, A. A., 'Application of plasticity theory and absolute nodal coordinate formulation to flexible multibody system dynamics', ASME Journal of Mechanical Design 126, 2004, 478–487.
Rankin, C. C. and Brogan, F. A., 'Element independent corotational procedure for the treatment of large rotations', ASME Journal of Pressure Vessel Technology 108, 1986, 165–174.
Crisfield, M. A., 'A consistent corotational formulation for nonlinear three-dimensional beam-elements', Computer Methods in Applied Mechanics and Engineering 81, 1990, 131–150.
Simo, J. C. and Hughes, T. J. R., Computational Inelasticity, Springer-Verlag, New York, 1998.
Simo, J. C. and Taylor, R. L., 'Consistent tangent operators for rate-independent elastoplasticity', Computer Methods in Applied Mechanics and Engineering 48, 1985, 101–118.
Yakoub, R. Y. and Shabana, A. A., 'Use of Cholesky coordinates and the absolute nodal coordinate formulation in the computer simulation of flexible multibody systems', Nonlinear Dynamics 20, 1999, 267–282.
Hairer, E., Nørsett, S. P., and Wanner, G., Solving Ordinary Differential Equations: Nonstiff Problems, Springer-Verlag, New York, 1993.
Hairer, E. and Wanner, G., Solving Ordinary Differential Equations: Stiff and Differential-Algebraic Problems, Springer-Verlag, New York, 1996.
Shampine, L. F., 'Implementation of Rosenbrock methods', ACM Transaction of Mathematical Software 8, 1982, 93–113.
Deuflhard, P., 'Recent progress in extrapolation methods for ordinary differential equations', SIAM Review 27, 1985, 505–535.
Bader, G. and Deuflhard, P., 'A semi-implicit midpoint rule for stiff systems of ordinary differential equations', Numerische Mathematik 41, 1983, 373–398.
Stoer, J. and Bulirsch, R., Introduction to Numerical Analysis, Springer-Verlag, New York, 1980.
Deuflhard, P., 'Order and stepsize control in extrapolation methods', Numerische Mathematik 41, 1983, 399–422.
Shampine, L. F. and Gordon, M. K., Computer Solution of Ordinary Differential Equations, W. H. Freeman, San Francisco, California, 1975.
Shampine, L. F., Watts, H. A., and Davenport, S. M., 'Solving nonstiff ordinary differential equations-the state of the art', SIAM Review 18, 1976, 376–411.
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Sugiyama, H., Shabana, A.A. On the Use of Implicit Integration Methods and the Absolute Nodal Coordinate Formulation in the Analysis of Elasto-Plastic Deformation Problems. Nonlinear Dynamics 37, 245–270 (2004). https://doi.org/10.1023/B:NODY.0000044644.53684.5b
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DOI: https://doi.org/10.1023/B:NODY.0000044644.53684.5b