Extremum theorem and convergence criterion for an iterative solution to the finite-step problem in elastoplasticity with mixed nonlinear hardening

Claudia Comi; Giulio Maier

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1989)

  • Volume: 83, Issue: 1, page 177-186
  • ISSN: 1120-6330

Abstract

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For a class of elastic-plastic constitutive laws with nonlinear kinematic and isotropic hardening, the problem of determining the response to a finite load step is formulated according to an implicit backward difference scheme (stepwise holonomic formulation), with reference to discrete structural models. This problem is shown to be amenable to a nonlinear mathematical programming problem and a criterion is derived which guarantees monotonie convergence of an iterative algorithm for the solution of the finite-step analysis problem. This communication anticipates in an abbreviated form results to be presented elsewhere in an extended form: here proofs and various comments are omitted.

How to cite

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Comi, Claudia, and Maier, Giulio. "Extremum theorem and convergence criterion for an iterative solution to the finite-step problem in elastoplasticity with mixed nonlinear hardening." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 83.1 (1989): 177-186. <http://eudml.org/doc/287496>.

@article{Comi1989,
abstract = {For a class of elastic-plastic constitutive laws with nonlinear kinematic and isotropic hardening, the problem of determining the response to a finite load step is formulated according to an implicit backward difference scheme (stepwise holonomic formulation), with reference to discrete structural models. This problem is shown to be amenable to a nonlinear mathematical programming problem and a criterion is derived which guarantees monotonie convergence of an iterative algorithm for the solution of the finite-step analysis problem. This communication anticipates in an abbreviated form results to be presented elsewhere in an extended form: here proofs and various comments are omitted.},
author = {Comi, Claudia, Maier, Giulio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Elastoplasticity; Finite-step; Extremum theorem; Convergence; elastic-plastic constitutive laws; nonlinear kinematic and isotropic hardening; finite load step; implicit backward difference scheme; stepwise holonomic formulation; nonlinear mathematical programming; monotonic convergence; iterative algorithm; finite-step analysis problem},
language = {eng},
month = {12},
number = {1},
pages = {177-186},
publisher = {Accademia Nazionale dei Lincei},
title = {Extremum theorem and convergence criterion for an iterative solution to the finite-step problem in elastoplasticity with mixed nonlinear hardening},
url = {http://eudml.org/doc/287496},
volume = {83},
year = {1989},
}

TY - JOUR
AU - Comi, Claudia
AU - Maier, Giulio
TI - Extremum theorem and convergence criterion for an iterative solution to the finite-step problem in elastoplasticity with mixed nonlinear hardening
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1989/12//
PB - Accademia Nazionale dei Lincei
VL - 83
IS - 1
SP - 177
EP - 186
AB - For a class of elastic-plastic constitutive laws with nonlinear kinematic and isotropic hardening, the problem of determining the response to a finite load step is formulated according to an implicit backward difference scheme (stepwise holonomic formulation), with reference to discrete structural models. This problem is shown to be amenable to a nonlinear mathematical programming problem and a criterion is derived which guarantees monotonie convergence of an iterative algorithm for the solution of the finite-step analysis problem. This communication anticipates in an abbreviated form results to be presented elsewhere in an extended form: here proofs and various comments are omitted.
LA - eng
KW - Elastoplasticity; Finite-step; Extremum theorem; Convergence; elastic-plastic constitutive laws; nonlinear kinematic and isotropic hardening; finite load step; implicit backward difference scheme; stepwise holonomic formulation; nonlinear mathematical programming; monotonic convergence; iterative algorithm; finite-step analysis problem
UR - http://eudml.org/doc/287496
ER -

References

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  1. CASCIARO, R. - MANCUSO, M., 1988. Un approccio numerico al problema dell'elastoplasticità incrementale. Omaggio a Giulio Ceradini: 177-187. 
  2. CORRADI, L., 1983. A Displacement formulation for the Finite Element Elastic-Plastic Problem. Meccanica, 18: 77-91. Zbl0519.73072
  3. DE DONATO, O. - MAIER, G., 1979. Mathematical Programming Methods for the Inelastic Analysis of Reinforced Concrete Frames Allowing for Limited Rotation Capacity. Int. J. for Num. Meth. in Eng., 4: 307-329. Zbl0359.73063
  4. DRUCKER, D.C., 1964. On the Postulate of Stability of Material in Mechanics of Continua. Journal de Mécanique, 3: 235-249. MR167053
  5. FRANCHI, A. - GENNA, F., 1984. Minimum Principles and Initial Stress Method in Elastic-Plastic Analysis. Engineering Structures: 65-69. 
  6. MAIER, G., 1969. Some Theorems for Plastic Strain Rates and Plastic Strains. Journal de Mécanique, 8: 5-19. Zbl0176.25901
  7. MATER, G., 1970. A Matrix Structural Theory of Piecewise-Linear Plasticity with Interacting Yield Planes. Meccanica, 5: 55-66. Zbl0197.23303
  8. MAIER, G. - NAPPI, A., 1989. Backward Difference Time Integration, Nonlinear Programming and Extremum Theorems in Elastoplastic Analysis. Solid Mechanics Archives, 14(1): 37-64. Zbl0679.73021MR994147
  9. MAIER, G. - NOVATI, G., 1988. Externum Theorems for Finite-Step Backward-Difference Analysis of Elastic-Plastic Nonlinearly Hardening Solids. Rend. Acc. Naz. dei Lincei, Cl. Sci., to appear. Zbl0737.73049MR1139818
  10. MARTIN, J.B. - NAPPI, A., An Internal Variable Formulation of Perfectly Plastic and Linear Kinematic and Isotropic Hardening Relations with a Von Mises Yield Condition. European Journal of Mechanics, to appear. Zbl0706.73035
  11. MARTIN, J.B. - REDDY, B.D., 1988. Variational Principles and Solution Algorithms for Internal Variable Formulations of Problems in Plasticity. Omaggio a Giulio Ceradini: 465-477. 
  12. ORTIZ, M. - POPOV, E.P., 1985. Accuracy and Stability of Integration Algorithms for Elastoplastic Constitutive Relations. Int. J. Num. Methods in Engineering, 21: 1561-1576. Zbl0585.73057MR810515DOI10.1002/nme.1620210902
  13. ORTIZ, M. - SIMO, J.C., 1986. An Analysis of a New Class of Integration Algorithms for Elastoplastic Constitutive Relations. Int. J. Numer. Methods Eng., 23: 353-366. Zbl0585.73058MR833184DOI10.1002/nme.1620230303
  14. PEREGO, U., 1988. Explicit Backward Difference Operators and Consistent Predictors for Linear Hardening Elastic-Plastic Constitutive Laws. Sol. Mech. Arch., 13: 65-102. Zbl0711.73270
  15. RESENDE, L. - MARTIN, J.B., 1985. Formulation of Drucker-Prager Cap Model. ASCE J. of Eng. Mech., 111, 7: 855-881. 
  16. SIMO, J.C. - TAYLOR, R.L., 1985. Consistent Tangent Operators for Rate-Independent Elastoplasticity. Comp. Methods Appl. Mech. Eng., 48: 101-118. Zbl0535.73025
  17. SIMO, J.C. - KENNEDY, J.G. - GOVINDJEE, S., 1988. Non-Smooth Multisurface Plasticity and Viscoplasticity Loading/Unloading Conditions and Numerical Algorithms. Int. Jour, for Num. Methods in Eng., 26: 2161-2185. Zbl0661.73058MR960023DOI10.1002/nme.1620261003

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