On estimation of internal state by an optimal control approach for elastoplastic material

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April 2016, 9(2): 599-611. doi: 10.3934/dcdss.2016014

On estimation of internal state by an optimal control approach for elastoplastic material

1.	GeM, UMR CNRS 6183,1 rue de la Noe, F-44321 Nantes, EdF-CEA-ENSTA UMR CNRS 8193, 1 avenue General Leclerc, F- 92141 Clamart, France

Received October 2014 Revised November 2015 Published March 2016

Abstract
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After a general formulation of the evolution of an elastoplastic body using duality based on the constitutive behaviour, some classes of inverse problems (estimation of the internal state, determination of an unknown history, ...) for such materials are investigated. A general formulation based on optimal control is proposed, the control variables are related to the internal state. In each class of inverse problem, the solution is obtained by introducing a adjoin state and a suitable cost function.

Keywords: adjoin state., elastoplasticity, Inverse problem, optimal control.

Mathematics Subject Classification: 74CXX, 49-XX, 49-N4.

Citation: Claude Stolz. On estimation of internal state by an optimal control approach for elastoplastic material. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 599-611. doi: 10.3934/dcdss.2016014

References:

[1]	P. Ballard and A. Constantinescu, On the inversion of subsurface residual stresses from surface stress measurements,, J. Mech. Phys. Solids, 42 (1994), 1767. doi: 10.1016/0022-5096(94)90071-X.
[2]	H. D. Bui, Introduction Aux Problèmes Inverses en Mécanique des Matériaux,, Eyrolles, (1993).
[3]	B. Halphen, Stress accommodation in elastic perfectly plastic and viscoplastic structures,, Mech. Res. Comm., 2 (1975), 273. doi: 10.1016/0093-6413(75)90057-9.
[4]	J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations Aux Dérivées Partielles,, Avant propos de P. Lelong Dunod, (1968).
[5]	Q. S. Nguyen, Bifurcation et stabilité des systèmes irréversibles obéissant au principe de dissipation maximale,, J. Mécanique Théorique et appliquée, 3 (1984), 41.
[6]	M. Peigney and C. Stolz, An optimal control approach to the analysis of inelastic structures under cyclic loading,, J. Mech. Phys. Solids, 51 (2003), 575. doi: 10.1016/S0022-5096(02)00104-7.
[7]	M. Peigney and C. Stolz, Approche par contrôle optimal des structures élastoviscoplastique sous chargement cyclique,, C. R. Mécanique, 339 (2001), 643.
[8]	C. Stolz, Optimal control approach in non linear mechanics,, C. R. Mécanique, 336 (2008), 238.
[9]	C. Stolz, Some applications of optimal control to inverse problems in elastoplasticity,, J. of Mechanics of Materials and Structures, 20 (2015), 411. doi: 10.2140/jomms.2015.10.411.

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References:

[1]	P. Ballard and A. Constantinescu, On the inversion of subsurface residual stresses from surface stress measurements,, J. Mech. Phys. Solids, 42 (1994), 1767. doi: 10.1016/0022-5096(94)90071-X.
[2]	H. D. Bui, Introduction Aux Problèmes Inverses en Mécanique des Matériaux,, Eyrolles, (1993).
[3]	B. Halphen, Stress accommodation in elastic perfectly plastic and viscoplastic structures,, Mech. Res. Comm., 2 (1975), 273. doi: 10.1016/0093-6413(75)90057-9.
[4]	J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations Aux Dérivées Partielles,, Avant propos de P. Lelong Dunod, (1968).
[5]	Q. S. Nguyen, Bifurcation et stabilité des systèmes irréversibles obéissant au principe de dissipation maximale,, J. Mécanique Théorique et appliquée, 3 (1984), 41.
[6]	M. Peigney and C. Stolz, An optimal control approach to the analysis of inelastic structures under cyclic loading,, J. Mech. Phys. Solids, 51 (2003), 575. doi: 10.1016/S0022-5096(02)00104-7.
[7]	M. Peigney and C. Stolz, Approche par contrôle optimal des structures élastoviscoplastique sous chargement cyclique,, C. R. Mécanique, 339 (2001), 643.
[8]	C. Stolz, Optimal control approach in non linear mechanics,, C. R. Mécanique, 336 (2008), 238.
[9]	C. Stolz, Some applications of optimal control to inverse problems in elastoplasticity,, J. of Mechanics of Materials and Structures, 20 (2015), 411. doi: 10.2140/jomms.2015.10.411.

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