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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

Claude Stolz 1,

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

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.  Google Scholar

[2]

H. D. Bui, Introduction Aux Problèmes Inverses en Mécanique des Matériaux,, Eyrolles, (1993).   Google Scholar

[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.  Google Scholar

[4]

J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations Aux Dérivées Partielles,, Avant propos de P. Lelong Dunod, (1968).   Google Scholar

[5]

Q. S. Nguyen, Bifurcation et stabilité des systèmes irréversibles obéissant au principe de dissipation maximale,, J. Mécanique Théorique et appliquée, 3 (1984), 41.   Google Scholar

[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.  Google Scholar

[7]

M. Peigney and C. Stolz, Approche par contrôle optimal des structures élastoviscoplastique sous chargement cyclique,, C. R. Mécanique, 339 (2001), 643.   Google Scholar

[8]

C. Stolz, Optimal control approach in non linear mechanics,, C. R. Mécanique, 336 (2008), 238.   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

H. D. Bui, Introduction Aux Problèmes Inverses en Mécanique des Matériaux,, Eyrolles, (1993).   Google Scholar

[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.  Google Scholar

[4]

J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations Aux Dérivées Partielles,, Avant propos de P. Lelong Dunod, (1968).   Google Scholar

[5]

Q. S. Nguyen, Bifurcation et stabilité des systèmes irréversibles obéissant au principe de dissipation maximale,, J. Mécanique Théorique et appliquée, 3 (1984), 41.   Google Scholar

[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.  Google Scholar

[7]

M. Peigney and C. Stolz, Approche par contrôle optimal des structures élastoviscoplastique sous chargement cyclique,, C. R. Mécanique, 339 (2001), 643.   Google Scholar

[8]

C. Stolz, Optimal control approach in non linear mechanics,, C. R. Mécanique, 336 (2008), 238.   Google Scholar

[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.  Google Scholar

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