Extremum theorems for finite-step back-ward-difference analysis of elastic-plastic nonlinearly hardening solids

Giulio Maier; Giorgio Novati

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1988)

  • Volume: 82, Issue: 4, page 711-715
  • ISSN: 1120-6330

Abstract

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For the finite-step, backward-difference analysis of elastic-plastic solids in small strains, a kinematic (potential energy) and a static (complementary energy) extremum property of the step solution are given under the following hypotheses: each yield function is the sum of an equivalent stress and a yield limit; the former is a positively homogeneous function of order one of stresses, the latter a nonlinear function of nondecreasing internal variables; suitable conditions of "material stability" are assumed. This communication anticipates results to be presented elsewhere in an extended version. Therefore, proofs of the statements and various comments are omitted.

How to cite

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Maier, Giulio, and Novati, Giorgio. "Extremum theorems for finite-step back-ward-difference analysis of elastic-plastic nonlinearly hardening solids." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 82.4 (1988): 711-715. <http://eudml.org/doc/287244>.

@article{Maier1988,
abstract = {For the finite-step, backward-difference analysis of elastic-plastic solids in small strains, a kinematic (potential energy) and a static (complementary energy) extremum property of the step solution are given under the following hypotheses: each yield function is the sum of an equivalent stress and a yield limit; the former is a positively homogeneous function of order one of stresses, the latter a nonlinear function of nondecreasing internal variables; suitable conditions of "material stability" are assumed. This communication anticipates results to be presented elsewhere in an extended version. Therefore, proofs of the statements and various comments are omitted.},
author = {Maier, Giulio, Novati, Giorgio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Computational Plasticity; Extremum theorems; Hardening; small strains; potential energy; complementary energy; yield function; sum of equivalent stress and yield limit; internal variables; material stability; hardening},
language = {eng},
month = {12},
number = {4},
pages = {711-715},
publisher = {Accademia Nazionale dei Lincei},
title = {Extremum theorems for finite-step back-ward-difference analysis of elastic-plastic nonlinearly hardening solids},
url = {http://eudml.org/doc/287244},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Maier, Giulio
AU - Novati, Giorgio
TI - Extremum theorems for finite-step back-ward-difference analysis of elastic-plastic nonlinearly hardening solids
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/12//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 4
SP - 711
EP - 715
AB - For the finite-step, backward-difference analysis of elastic-plastic solids in small strains, a kinematic (potential energy) and a static (complementary energy) extremum property of the step solution are given under the following hypotheses: each yield function is the sum of an equivalent stress and a yield limit; the former is a positively homogeneous function of order one of stresses, the latter a nonlinear function of nondecreasing internal variables; suitable conditions of "material stability" are assumed. This communication anticipates results to be presented elsewhere in an extended version. Therefore, proofs of the statements and various comments are omitted.
LA - eng
KW - Computational Plasticity; Extremum theorems; Hardening; small strains; potential energy; complementary energy; yield function; sum of equivalent stress and yield limit; internal variables; material stability; hardening
UR - http://eudml.org/doc/287244
ER -

References

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  1. MARTIN, J.B., REDDY, B.D., GRIFFIN, T.B. and BIRD, W.W. (1987) - Applications of Mathematical Programming Concepts to Incremental Elastic-Plastic Analysis, Engineering Structures, 9, 171- 180. 
  2. ORTIZ, M., POPOV, E.P. (1985) - Accuracy and Stability of Integration Algorithms for Elastoplastic Constitutive Relations, Int. J. Num. Meth. Engng., 21, 1561-1576. Zbl0585.73057MR810515DOI10.1002/nme.1620210902
  3. SIMO, J.C. and TAYLOR, R.L. (1986) - A Return Mapping Algorithm for Plane Stress Elastoplasticity, Int J. Num. Meth. Engng., 22, 649-670. Zbl0585.73059MR839299DOI10.1002/nme.1620220310
  4. REDDY, B.D. and GRIFFIN, T.B. (1986) - Variational Principles and Convergence of Finite Element Approximations of Holonomic Elastic-Plastic Problems, UCT/CSIR Applied Mechanics Research Unit, Technical Report, N. 83. Zbl0618.73034
  5. MAIER, G., DE DONATO, O. and CORRADI, L. (1972) - Inelastic Analysis of Reinforced Concrete Frames by Quadratic Programming, In: Inelasticity and Non-Linearity in Structural Concrete, University of Waterloo Press, Study N. 8, paper 10, 265-288. 
  6. DE DONATO, O. and MAIER, G. (1973) - Finite Element Elastoplastic Analysis by Quadratic Programming: the Multistage Method, Proc. 2nd Int. Conf. on Structural Mechanics in Reactor Technology (SMIRT), Berlin, Vol. V, Part M. 
  7. FRANCHI, A. and GENNA, F. (1987) - A Numerical Scheme for Integrating the Rate Plasticity Equations with an "A Priori" Error Control, Comput. Meth. Appl. Mech. Engng., 60 (3), 317-342. Zbl0611.73038MR878836DOI10.1016/0045-7825(87)90138-1
  8. MAIER, G. and MUNRO, J. (1982) - Mathematical Programming Methods in Engineering Plastic Analysis, Appl. Mech. Rev., 35 (12), 1631-1643. 
  9. MAIER, G. (1968) - Quadratic Programming and Theory of Elastic-Perfectly Plastic Structures, Meccanica, 3 (4), 265-273. Zbl0181.53704MR266487
  10. MAIER, G. (1969) - Teoremi di Minimo in Termini Finiti per Continui Elastoplastici con Leggi Costitutive Linearizzate a Tratti, Rendiconti dell'Istituto Lombardo di Scienze e Lettere, Vol. 103, 1066-1080. Zbl0213.28002MR263297
  11. MAIER, G. (1969) - Complementary Plastic Work Theorems in Piecewise Linear Elastoplasticity, Int. J. Solids Struct., 5, 261-270. Zbl0164.27101
  12. RESENDE, L. and MARTIN, J.B. (1985) - Formulation of Drucker-Prager Cap Model, ASCE-J. of Eng. Mech., 111 (7), 855-881. 
  13. CAPURSO, M. (1969) - Principi di Minimo per la Soluzione Incrementale dei Problemi Elastoplastici, Rend. Acc. Naz. Lincei, Cl. Sci. Zbl0187.48003
  14. CAPURSO, M. and MAIER, G. (1970) - Incremental Elastoplastic Analysis and Quadratic Optimization, Meccanica, 4 (1), 107-116. Zbl0198.58301
  15. FRANCHI, A. and GENNA, F. (1984) - Minimum Principles and Initial Stress Method in Elastic-Plastic Analysis, Engng. Struct.6 (1), 65-69. 
  16. PONTER, A.R.S. and MARTIN, J.B. (1979) - Some Extremal Properties and Energy Theorems for Inelastic Materials and their Relationship to the Deformation Theory of Plasticity, J. Mech. Phys. Solids, 20, 281-300. Zbl0241.73003MR349113
  17. MARTIN, J.B. and PONTER, A.R.S. (1972) - On Dual Energy Theorems for a Class of Elastic-Plastic Problems Due to G. Maier, J. Mech. Phys. Solids, 20, 301-306. Zbl0241.73008MR349114

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