Observations on Return Mapping Algorithms for Piecewise Linear Yield Criteria
Publication: International Journal of Geomechanics
Volume 8, Issue 4
Abstract
This paper shows that for perfect plasticity, the closest point projection method (CPPM) and the cutting plane algorithm (CPA) for return mapping are exactly equivalent for piecewise linear yield criteria under both associated and nonassociated plastic flow. The paper demonstrates this by presenting closed-form expressions for returned stresses in terms of predicted stresses. A consequence of this exact approach is that the final stresses can be obtained in a single iteration. The equivalence of CPPM and CPA is further demonstrated numerically by comparing five previously published algorithms for return mapping to Mohr–Coulomb in a finite-element analysis of bearing capacity. The analyses also highlight issues relating to singularities that occur at the corners of the Mohr–Coulomb surface. It is shown that many of these problems can be avoided if the return mapping is performed in principal stress space as opposed to general stress space.
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Acknowledgments
The writers gratefully acknowledge the support of the National Science Foundation under Grant No. CMS-0408150.
References
Abbo, A. J., and Sloan, S. W. (1995). “A smooth hyperbolic approximation to the Mohr–Coulomb yield criterion.” Comput. Struct., 54(3), 427–441.
Asensio, G., and Moreno, C. (2003). “Linearization and return mapping algorithms for elastoplasticity models.” Int. J. Numer. Methods Eng., 57(7), 991–1014.
Bathe, K. J. (1996). Finite-element procedures, Prentice-Hall, Upper Saddle River, N.J.
Bathe, K. J., Chaudhary, A., Dvorkin, E., and Kojic, M. (1984). “On the solution of nonlinear finite-element equations.” Proc., Int. Conf. on Computer-Aided Analysis and Design of Concrete Structures, Split, Yugoslavia.
Bathe, K. J., Ramm, E., and Wilson, E. L. (1975). “Finite-element formulations for large deformation dynamic analysis.” Int. J. Numer. Methods Eng., 9(2), 353–386.
Bićanić, N., and Pearce, C. J. (1996). “Computational aspects of a softening plasticity model for plain concrete.” Mech. Cohesive-Frict. Mater., 1(1), 75–94.
Borja, R. I., Sama, K. M., and Sanz, P. F. (2003). “On the numerical integration of three-invariant elastoplastic constitutive models.” Comput. Methods Appl. Mech. Eng., 192(9-10), 1227–1258.
Caddemi, S., and Martin, J. B. (1991). “Convergence of the Newton–Raphson algorithm in elastic–plastic incremental analysis.” Int. J. Numer. Methods Eng., 31(1), 177–191.
Clausen, J. (2006). “Efficient nonlinear finite-element implementation of elasto–plasticity for geotechnical problem.” Ph.D. thesis, Aalborg Univ.
Clausen, J., Damkilde, L., and Andersen, L. (2004). “One-step direct return method for Mohr–Coulomb plasticity.” Proc., 17th Nordic Seminar of Computational Mechanics, A. Eriksson, J. Månsson, and G. Tibert, eds., KTH Mechanics, Stockholm, 156–159.
Clausen, J., Damkilde, L., and Andersen, L. (2005). “An efficient return algorithm for nonassociated Mohr–Coulomb Plasticity.” Proc., 10th Int. Conf. on Civil, Structural, and Environmental Engineering Computing, Rome, 144.
Clausen, J., Damkilde, L., and Andersen, L. (2006). “Efficient return algorithms for associated plasticity with multiple yield planes.” Int. J. Numer. Methods Eng., 66(6), 1036–1059.
Crisfield, M. A. (1987). “Plasticity computations using the Mohr–Coulomb yield criterion.” Eng. Comput., 4(6), 300–308.
Crisfield, M. A. (1991). Nonlinear finite-element analysis of solids and structures: Essential, Vol. 1, Wiley, New York.
Crisfield, M. A. (1997). Nonlinear finite-element analysis of solids and structures: Advanced topics, Vol. 2, Wiley, New York.
de Borst, R. (1987). “Integration of plasticity equations for singular yield functions.” Comput. Struct., 26(5), 823–829.
de Souza Neto, E. A., Perić, D., and Owen, D. R. J. (1994). “A model for elastoplastic damage at finite strains: Algorithmic issues and applications.” Eng. Comput., 11(3), 257–281.
Griffiths, D. V., and Lane, P. A. (1999). “Slope stability analysis by finite elements.” Geotechnique, 49(3), 387–403.
Hiriart-Urruty, J. B., and Lemarechal, C. (1993). Convex analysis and minimization algorithms, Vol. I., Springer, Berlin.
Koiter, W. T. (1953). “Stress–strain relations, uniqueness, and variational theorems for elastic–plastic materials with a singular yield surface.” Q. Appl. Math., 11, 350–354.
Kojić, M., and Bathe, K. J. (1987). “The ‘effective-stress-function’ algorithm for thermoelastoplasticity and creep.” Int. J. Numer. Methods Eng., 24(8), 1509–1532.
Kojić, M., and Bathe, K. J. (2005). Inelastic analysis of solids and structures, Springer, Berlin.
Krieg, R. D., and Krieg, D. B. (1977). “Accuracies of numerical solution methods for the elastic–perfectly plastic model.” ASME J. Pressure Vessel Technol., 99(4), 510–515.
Larsson, R., and Runesson, K. (1996). “Implicit integration and consistent linearization for yield criteria of the Mohr–Coulomb type.” Mech. Cohesive-Frict. Mater., 1(4), 367–383.
Ortiz, M., and Popov, E. P. (1985). “Accuracy and stability of integration algorithms for elastoplastic constitutive relations.” Int. J. Numer. Methods Eng., 21(9), 1561–1576.
Ortiz, M., and Simo, J. C. (1986). “An analysis of a new class of integration algorithms for elastoplastic constitutive relations.” Int. J. Numer. Methods Eng., 23(3), 353–366.
Pankaj, and Bićanić, N. (1997). “Detection of multiple active yield conditions for Mohr–Coulomb elastoplasticity.” Comput. Struct., 62(1), 51–61.
Péréz-Foguet, A., Rodríguez-Ferran, A., and Huerta, A. (2001). “Consistent tangent matrices for substepping schemes.” Comput. Methods Appl. Mech. Eng., 190(35-36), 4627–4647.
Prandtl, L. (1921). “über die eindringungsfestigkeit (härte) plastischer baustoffe und die festigkeit von schneiden.” Z. Angew. Math. Mech., 1, 15–20.
Simo, J. C. (1998). “Numerical analysis of classical plasticity.” Handbook for numerical analysis, P. G. Ciarlet and J. J. Lions, eds., Vol. VI, Elsevier, Amsterdam.
Simo, J. C., and Hughes, T. J. R. (1998). Computational inelasticity, Springer, New York.
Simo, J. C., and Ortiz, M. A. (1985). “A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations.” Comput. Methods Appl. Mech. Eng., 49(2), 221–245.
Simo, J. C., and Taylor, R. L. (1985). “Consistent tangent operators for rate-independent elastoplasticity.” Comput. Methods Appl. Mech. Eng., 48(3), 101–118.
Smith, I. M., and Griffiths, D. V. (1988). Programming the finite-element method, 2nd Ed., Wiley, New York.
Smith, I. M., and Griffiths, D. V. (2004). Programming the finite-element method, 4th Ed., Wiley, New York.
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Received: Jul 5, 2007
Accepted: Nov 20, 2007
Published online: Jul 1, 2008
Published in print: Jul 2008
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