Integration Algorithms for Elastoplastic Constitutive Laws in Large Deformation Problems
Publication: Journal of Engineering Mechanics
Volume 142, Issue 9
Abstract
In most finite-element codes, a Newton-Raphson procedure is used for solving the global equilibrium problem. To preserve quadratic convergence of the aforementioned iterative scheme, this paper presents an analytical full linearization of the principal of virtual work in an updated Lagrangian framework and develops tangent operators consistent with the integration algorithms. Four implementations of the most-used objective rates are described and are shown to be consistent, stable, and objective. The rigid body rotation tensor is obtained by a new computational implementation of polar decomposition scheme in two-dimensional problems. An automatic substepping algorithm is used for integrating elastoplastic constitutive laws in large deformation problems. Numerical examples are presented to test the algorithms, validate the proposed schemes, and compare their performance with other integration algorithms.
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References
Aunay, O., Aunay, S., Chorlay, D., Touzot, G., and Vayssade, M. (1988). “Using artificial intelligence in an open software architecture for modeling in engineering.” Artificial intelligence in computational engineering, Ellis Horwood.
Bathe, K. J. (1996). Finite element procedures, Prentice Hall, Englewood Cliffs, NJ.
Bažant, Z. P., et al. (2000). “Large-strain generalization of microplane model for concrete and application.” J. Eng. Mech., 971–980.
Bažant, Z. P., and Vorel, J. (2014). “Energy conservation error due to the use of Green-Naghdi objective stress rate in commercial finite-element codes and its compensation.” ASME J. Appl. Mech., 81, 021008.
Braudel, H. J., Abouaf, M., and Chenot, J. L. (1986). “An implicit incrementally objective formulation for the solution of elastoplastic problems at finite strain by the F.E.M.” Compt. Struct., 24(6), 825–843.
Bruhns, O. T. (2009). “Eulerian elastoplasticity: Basic issues and recent results.” Theor. Appl. Mech., 36(3), 167–205.
Charlier, R. (1987). “Approche unifie de quelques problèmes non linéaires de mécanique des milieux continues par la méthode des éléments finis.” Thesis, Univ. of Liège, Liège, Belgium.
Çolak, O. U. (2003). “Modeling of large simple shear using a viscoplastic overstress model and classical plasticity model with different objective stress rates.” Turkish J. Eng. Environ. Sci., 27, 95–106.
Crisfield, M. A. (1991). Non-linear finite element analysis of solids and structures, Willey, New York.
Detraux, J. M. (1985). “Formulation et application en grandes déformations des solides.” Thesis, Univ. of Technology of Compiegne, Compiegne,France.
Dienes, J. K. (1979). “On the analyses of rotation and stress rate in deforming bodies.” Acta Mech., 32(4), 217–232.
Garino, C. G., and Oliver, J. (1991). “A numerical model for elastoplastic large strain problems. Fundamentals and applications.” Proc., Int. Conf. Computational Plasticity, CIMNE, Barcelona, Spain.
Green, A. E., and Naghdi, P. M. (1965). “A general theory of an elastic-plastic continuum.” Arch. Ration. Mech. Anal., 18(4), 251–281.
Healy, B. E., and Dodds, R. H. (1992). “A large strain plasticity model for implicit finite element analyses.” Compt. Mech., 9(2), 95–112.
Hibbit, H. D. (1979). “Some follower forces and load stiffness.” Int. J. Numer. Methods Eng., 14(6), 937–941.
Hoger, A., and Carlson, D. E. (1984). “Determination of the stretch and rotation in the polar decomposition of the deformation gradient.” Q. App. Math., 42(1), 113–117.
Hughes, T. J. R., and Winget, J. (1980). “Finite rotation effects in numerical integration of rate constitutive equations arising in large deformation analysis.” Int. J. Numer. Methods Eng., 15(12), 1862–1867.
Lubarda, V. A. (2001). Elastoplasticity theory, CRC Press, Boca Raton, FL.
Meyers, A. (2013). “View on a traditional elastoplasticity model.” Proc. Indian National Sci. Acad., 79(4), 547–552.
Meyers, A., Xiao, H., and Bruhns, O. T. (2006). “Choice of objective rate in single parameter hypoelastic deformation cycles.” Compt. Struct., 84(17), 1134–1140.
Ortiz, M., Pinsky, P. M., and Taylor, R. L. (1983). “Operator split method for the numerical solution of the elastoplastic dynamic problem.” Comput. Methods Appl. Mech. Eng., 39(2), 137–157.
Pinsky, P. M., Ortiz, M., and Pister, K. S. (1983). “Numerical integration of rate constitutive equations in finite deformation analysis.” Compt. Methods Appl. Mech. Eng., 40(2), 137–158.
Pinto, Y. (1990). “Contact et frottement en Grande Déformations Plastiques.” Doctoral Thesis, Univ. of Province Aix-Marseilles I, Marseilles, France.
Rouhauda, E., Panicauda, B., and Kernerb, R. (2013). “Canonical frame-indifferent transport operators with the four-dimensional formalism of differential geometry.” Comput. Mater. Sci., 77, 120–130.
Roy, S., Fossum, A. F., and Dexter, R. J. (1992). “On the use of polar decomposition in the integration of hypoelastic constitutive laws.” Int. J. Eng. Sci., 30(2), 119–133.
Simo, J. C., and Taylor, R. S. (1985). “Consistent tangent operators for rate independent elasto-plasticity.” Int. J. Numer. Methods Eng., 22(3), 649–670.
Touzot, G., and Dabounou, J. (1993). “Intégration numérique de lois de comportement élastoplastique.” Revue Européenne des éléments finis, 2(4), 465–494.
Truesdell, C. (1966). The elements of continuum mechanics, Springer, New York.
Voyiadjis, G. Z., and Abed, F. H. (2006). “Implicit algorithm for finite deformation hypoelastic-viscoplasticity in fcc metals.” Int. J. Numer. Methods Eng., 67(7), 933–959.
Zabaras, N., and Arif, A. F. M. (1992). “A family of integration algorithms for constitutive equations in finite deformation elasto-viscoplasticity.” Int. J. Numer. Methods Eng., 33(1), 59–84.
Zienkiewicz, O. C., and Taylor, R. L. (2000). The finite element method, 5th Ed., Butterworth-Heinemann, Oxford, U.K.
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© 2016 American Society of Civil Engineers.
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Received: Nov 12, 2015
Accepted: Mar 9, 2016
Published online: May 4, 2016
Published in print: Sep 1, 2016
Discussion open until: Oct 4, 2016
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