A generalization to nonlinear hardening of the first shakedown theorem for discrete elastic-plastic structural models

Giulio Maier

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1987)

  • Volume: 81, Issue: 2, page 161-174
  • ISSN: 1120-6330

Abstract

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In the plastic constitutive laws the yield functions are assumed to be linear in the stresses, but generally non-linear in the internal variables which are non-decreasing measures of the contribution to plastic strains by each face of the yield surface. The structural models referred to for simplicity are aggregates of constant-strain finite elements. Influence of geometry changes on equilibrium are allowed for in a linearized way (the equilibrium equation contains a bilinear term in the displacements and pre-existing stresses). It is shown that shakedown (which means plastic work bounded in time) is guaranteed under variable-repeated quasi-static external actions, when the hardening behaviour exhibits reciprocal interaction, a suitably defined energy function of the internal variables is convex and the yield conditions can be satisfied at any time by some constant internal variable vector and by the linear elastic stress response. Some interpretations and extensions of this result are envisaged. By specialization to linear hardening, earlier results are recovered, which reduce to Melan's classical theorem for non-hardening (perfectly plastic) cases.

How to cite

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Maier, Giulio. "A generalization to nonlinear hardening of the first shakedown theorem for discrete elastic-plastic structural models." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 81.2 (1987): 161-174. <http://eudml.org/doc/287464>.

@article{Maier1987,
abstract = {In the plastic constitutive laws the yield functions are assumed to be linear in the stresses, but generally non-linear in the internal variables which are non-decreasing measures of the contribution to plastic strains by each face of the yield surface. The structural models referred to for simplicity are aggregates of constant-strain finite elements. Influence of geometry changes on equilibrium are allowed for in a linearized way (the equilibrium equation contains a bilinear term in the displacements and pre-existing stresses). It is shown that shakedown (which means plastic work bounded in time) is guaranteed under variable-repeated quasi-static external actions, when the hardening behaviour exhibits reciprocal interaction, a suitably defined energy function of the internal variables is convex and the yield conditions can be satisfied at any time by some constant internal variable vector and by the linear elastic stress response. Some interpretations and extensions of this result are envisaged. By specialization to linear hardening, earlier results are recovered, which reduce to Melan's classical theorem for non-hardening (perfectly plastic) cases.},
author = {Maier, Giulio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Plasticity; Shakedown; Hardening; safety factor; convex energy function; reciprocal interaction of hardening behaviour; constant-strain finite elements; geometry changes; equilibrium; linearized; variable-repeated quasi-static external actions},
language = {eng},
month = {6},
number = {2},
pages = {161-174},
publisher = {Accademia Nazionale dei Lincei},
title = {A generalization to nonlinear hardening of the first shakedown theorem for discrete elastic-plastic structural models},
url = {http://eudml.org/doc/287464},
volume = {81},
year = {1987},
}

TY - JOUR
AU - Maier, Giulio
TI - A generalization to nonlinear hardening of the first shakedown theorem for discrete elastic-plastic structural models
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1987/6//
PB - Accademia Nazionale dei Lincei
VL - 81
IS - 2
SP - 161
EP - 174
AB - In the plastic constitutive laws the yield functions are assumed to be linear in the stresses, but generally non-linear in the internal variables which are non-decreasing measures of the contribution to plastic strains by each face of the yield surface. The structural models referred to for simplicity are aggregates of constant-strain finite elements. Influence of geometry changes on equilibrium are allowed for in a linearized way (the equilibrium equation contains a bilinear term in the displacements and pre-existing stresses). It is shown that shakedown (which means plastic work bounded in time) is guaranteed under variable-repeated quasi-static external actions, when the hardening behaviour exhibits reciprocal interaction, a suitably defined energy function of the internal variables is convex and the yield conditions can be satisfied at any time by some constant internal variable vector and by the linear elastic stress response. Some interpretations and extensions of this result are envisaged. By specialization to linear hardening, earlier results are recovered, which reduce to Melan's classical theorem for non-hardening (perfectly plastic) cases.
LA - eng
KW - Plasticity; Shakedown; Hardening; safety factor; convex energy function; reciprocal interaction of hardening behaviour; constant-strain finite elements; geometry changes; equilibrium; linearized; variable-repeated quasi-static external actions
UR - http://eudml.org/doc/287464
ER -

References

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  1. GOKHFELD, D.A. and CHERNIAVSKY, O.F. (1980) - Limit Analysis of Structures at Thermal Cycling, Sijthoff and Nordhoff, Leyden, 1980. MR661252
  2. KÖNIG, J.A. and MAIER, G. (1981) - Shakedown analysis of elastoplastic structures: a review of recent developments, «Nuclear Engineering and Design», 66, 81-95. 
  3. KÖNIG, J.A. (1986) - Shakedown of Elastic-Plastic Structures, Elsevier, Amsterdam. 
  4. POLIZZOTTO, C. (1982) - A unified treatment of shakedown theory and related bounding techniques, «Solid Mechanics Archives», 7 (1), 19-76. Zbl0487.73035
  5. CERADINI, G. (1980) - Dynamic shakedown in elastic plastic bodies, «J. Eng. Mech. Div., Proc. ASCE», 106, n. EM3, 481-499. 
  6. MAIER, G. (1970) - A matrix structural theory of piecewise-linear plasticity with interacting yield planes, «Meccanica», 5 (1), 55-66. Zbl0197.23303
  7. MAIER, G. (1973) - A shakedown matrix theory allowing for work-hardening and second-order geometric effects, Int. Symp. on Foundation of Plasticity, Warsaw, Aug. 30 Sept. 2, 1972; Vol. 1; A. Sawkcuz Ed., Noordhoff, Leyden, 417-433. 
  8. CORRADI, L. (1978) - On compatible finite element models for elastoplastic analysis, «Meccanica», 13, 133-140. Zbl0417.73073
  9. MARTIN, J.B. (1975) - Plasticity: Fundamentals and General Results, MIT Press. 
  10. MARTIN, J.B. (1980) - An internal variable approach to the formulation of finite element problems in plasticity, «Physical Nonlinearity in Structural Analysis», J. Huit, J. Lemaitre Eds., Springer-Verlag, pp. 165-176. 
  11. CYRAS, A.A. (1983) - Mathematical Models for the Analysis and Optimization of Elastoplastic Structures, Ellis Horwood, J. Wiley, Chichester. Zbl0616.73079MR743751
  12. MAIER, G. (1969) - Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: a finite element, linear programming approach, «Meccanica», 4 (3), 250-260. Zbl0219.73039

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