The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of Γ-convergence, in the framework of finite plasticity. Denoting by ε the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy per unit volume is of order ε2α-2, with α ≥ 3. According to the value of α, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von Kármán plate theory and the linearized plate theory.
Mots-clés : finite plasticity, thin plates, Γ-convergence
@article{COCV_2014__20_3_725_0,
author = {Davoli, Elisa},
title = {Linearized plastic plate models as $\Gamma $-limits of {3D} finite elastoplasticity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {725--747},
publisher = {EDP-Sciences},
volume = {20},
number = {3},
year = {2014},
doi = {10.1051/cocv/2013081},
zbl = {1298.74145},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2013081/}
}
TY - JOUR AU - Davoli, Elisa TI - Linearized plastic plate models as $\Gamma $-limits of 3D finite elastoplasticity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 725 EP - 747 VL - 20 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2013081/ DO - 10.1051/cocv/2013081 LA - en ID - COCV_2014__20_3_725_0 ER -
%0 Journal Article %A Davoli, Elisa %T Linearized plastic plate models as $\Gamma $-limits of 3D finite elastoplasticity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 725-747 %V 20 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2013081/ %R 10.1051/cocv/2013081 %G en %F COCV_2014__20_3_725_0
Davoli, Elisa. Linearized plastic plate models as $\Gamma $-limits of 3D finite elastoplasticity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 725-747. doi : 10.1051/cocv/2013081. http://archive.numdam.org/articles/10.1051/cocv/2013081/
[1] , and , A variational definition for the strain energy of an elastic string. J. Elasticity 25 (1991) 137-148. | MR | Zbl
[2] , An alternative approach to finite plasticity based on material isomorphisms. Int. J. Plasticity 15 (1999) 353-374. | Zbl
[3] , and , Non-convex potentials and microstructures in finite-strain plasticity. R. Soc. London Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002) 299-317. | MR | Zbl
[4] , An introduction to Γ-convergence. Boston, Birkhäuser (1993). | MR | Zbl
[5] and , Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27 (2010) 257-290. | Numdam | MR | Zbl
[6] , Quasistatic evolution models for thin plates arising as low energy Γ-limits of finite plasticity. Preprint SISSA (2012), Trieste. | MR
[7] and , Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 55-91. | MR | Zbl
[8] , and , A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55 (2002) 1461-1506. | MR | Zbl
[9] , and , A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Rational Mech. Anal. 180 (2006) 183-236. | MR | Zbl
[10] and , Stability of slender bodies under compression and validity of the Von Kármán theory. Arch. Ration. Mech. Anal. 193 (2009) 255-310. | MR | Zbl
[11] and , The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549-578. | MR | Zbl
[12] , Elastic-plastic deformation at finite strains. J. Appl. Mech. 36 (1969) 1-6. | Zbl
[13] and , An evolutionary elastoplastic plate model derived via Γ-convergence. Math. Models Methods Appl. Sci. 21 (2011) 1961-1986. | MR | Zbl
[14] and , Rigorous derivation of a plate theory in linear elastoplasticity via Γ-convergence. NoDEA Nonlinear Differ. Eqs. Appl. 19 (2012) 437-457. | MR | Zbl
[15] and , Global existence for rate-independent gradient plasticity at finite strain. J. Nonlinear Sci. 19 (2009) 221-248. | MR | Zbl
[16] , Equations constitutive et directeur dans les milieux plastiques et viscoplastique. Int. J. Sol. Struct. 9 (1973) 725-740. | Zbl
[17] , Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15 (2003) 351-382. | MR | Zbl
[18] , Finite elastoplasticity, Lie groups and geodesics on SL(d), Geometry, Dynamics, and Mechanics. Springer, New York (2002) 61-90. | MR | Zbl
[19] and , Linearized plasticity is the evolutionary Gamma-limit of finite plasticity. J. Eur. Math. Soc. 15 (2013) 923-948. | MR
[20] , A critical review of the state of finite plasticity. Z. Angew. Math. Phys. 41 (1990) 315-394. | MR | Zbl
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