no. 4
A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 4, pp. 615-660

The subject of this paper is the rigorous derivation of a quasistatic evolution model for a linearly elastic–perfectly plastic thin plate. As the thickness of the plate tends to zero, we prove via Γ-convergence techniques that solutions to the three-dimensional quasistatic evolution problem of Prandtl–Reuss elastoplasticity converge to a quasistatic evolution of a suitable reduced model. In this limiting model the admissible displacements are of Kirchhoff–Love type and the stretching and bending components of the stress are coupled through a plastic flow rule. Some equivalent formulations of the limiting problem in rate form are derived, together with some two-dimensional characterizations for suitable choices of the data.

DOI : 10.1016/j.anihpc.2012.11.001
Classification : 74C05, 74G65, 74K20, 49J45
Keywords: Quasistatic evolution, Rate-independent processes, Perfect plasticity, Thin plates, Prandtl–Reuss plasticity, Γ-convergence 
@article{AIHPC_2013__30_4_615_0,
     author = {Davoli, Elisa and Mora, Maria Giovanna},
     title = {A quasistatic evolution model for perfectly plastic plates derived by {\protect\emph{\ensuremath{\Gamma}}-convergence}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {615--660},
     year = {2013},
     publisher = {Elsevier},
     volume = {30},
     number = {4},
     doi = {10.1016/j.anihpc.2012.11.001},
     mrnumber = {3082478},
     zbl = {06295435},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2012.11.001/}
}
TY  - JOUR
AU  - Davoli, Elisa
AU  - Mora, Maria Giovanna
TI  - A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 615
EP  - 660
VL  - 30
IS  - 4
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2012.11.001/
DO  - 10.1016/j.anihpc.2012.11.001
LA  - en
ID  - AIHPC_2013__30_4_615_0
ER  - 
%0 Journal Article
%A Davoli, Elisa
%A Mora, Maria Giovanna
%T A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 615-660
%V 30
%N 4
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2012.11.001/
%R 10.1016/j.anihpc.2012.11.001
%G en
%F AIHPC_2013__30_4_615_0
Davoli, Elisa; Mora, Maria Giovanna. A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 4, pp. 615-660. doi: 10.1016/j.anihpc.2012.11.001

[1] H. Abels, M.G. Mora, S. Müller, The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity, Calc. Var. Partial Differential Equations 41 (2011), 241-259 | MR | Zbl

[2] H. Abels, M.G. Mora, S. Müller, Large time existence for thin vibrating plates, Comm. Partial Differential Equations 36 (2011), 2062-2102 | MR | Zbl

[3] E. Acerbi, G. Buttazzo, D. Percivale, A variational definition for the strain energy of an elastic string, J. Elasticity 25 (1991), 137-148 | MR | Zbl

[4] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Elsevier/Academic Press, Amsterdam (2003) | MR | Zbl

[5] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York (2000) | MR | Zbl

[6] G. Anzellotti, S. Baldo, D. Percivale, Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity, Asymptot. Anal. 9 (1994), 61-100 | MR | Zbl

[7] J.-F. Babadjian, Quasistatic evolution of a brittle thin film, Calc. Var. Partial Differential Equations 26 (2006), 69-118 | MR | Zbl

[8] F. Bourquin, P.G. Ciarlet, G. Geymonat, A. Raoult, Γ-convergence et analyse asymptotique des plaques minces, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 1017-1024 | MR | Zbl

[9] M. Brokate, A.M. Khludnev, Existence of solutions in the Prandtl–Reuss theory of elastoplastic plates, Adv. Math. Sci. Appl. 10 (2000), 399-415 | MR | Zbl

[10] G. Dal Maso, A. Desimone, M.G. Mora, Quasistatic evolution problems for linearly elastic–perfectly plastic materials, Arch. Ration. Mech. Anal. 180 (2006), 237-291 | MR | Zbl

[11] F. Demengel, Problèmes variationnels en plasticité parfaite des plaques, Numer. Funct. Anal. Optim. 6 (1983), 73-119 | MR | Zbl

[12] F. Demengel, Fonctions à hessien borné, Ann. Inst. Fourier (Grénoble) 34 (1984), 155-190 | MR | EuDML | Zbl | Numdam

[13] A. Demyanov, Quasistatic evolution in the theory of perfectly elasto-plastic plates. I. Existence of a weak solution, Math. Models Methods Appl. Sci. 19 (2009), 229-256 | MR | Zbl

[14] A. Demyanov, Quasistatic evolution in the theory of perfect elasto-plastic plates. II. Regularity of bending moments, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 2137-2163 | MR | EuDML | Zbl | Numdam

[15] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, Classics Appl. Math. vol. 28, SIAM, Philadelphia, PA (1999) | MR | Zbl

[16] G.A. Francfort, A. Giacomini, Small strain heterogeneous elasto-plasticity revisited, Comm. Pure Appl. Math. 65 (2012), 1185-1241 | MR | Zbl

[17] L. Freddi, R. Paroni, C. Zanini, Dimension reduction of a crack evolution problem in a linearly elastic plate, Asymptot. Anal. 70 (2010), 101-123 | MR | Zbl

[18] G. Friesecke, R.D. James, M.G. Mora, S. Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Math. Acad. Sci. Paris 336 (2003), 697-702 | Zbl

[19] G. Friesecke, R.D. James, S. Müller, 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

[20] G. Friesecke, R.D. James, S. Müller, A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Ration. Mech. Anal. 180 (2006), 183-236 | MR | Zbl

[21] C. Goffman, J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J. 31 (1964), 159-178 | MR | Zbl

[22] R.V. Kohn, R. Temam, Dual spaces of stresses and strains, with applications to Hencky plasticity, Appl. Math. Optim. 10 (1983), 1-35 | MR | Zbl

[23] H. Le Dret, A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. 74 (1995), 549-578 | MR | Zbl

[24] M. Lewicka, M.G. Mora, M.R. Pakzad, Shell theories arising as low-energy Γ-limit of 3d nonlinear elasticity, Ann. Sc. Norm. Super. Pisa Cl. Sci. 9 (2010), 253-295 | MR | Zbl | Numdam

[25] M. Lewicka, M.G. Mora, M.R. Pakzad, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Ration. Mech. Anal. 200 (2011), 1023-1050 | MR | Zbl

[26] M. Liero, A. Mielke, An evolutionary elasto-plastic plate model derived via Γ-convergence, Math. Models Methods Appl. Sci. 21 (2011), 1961-1986 | MR | Zbl

[27] M. Liero, T. Roche, Rigorous derivation of a plate theory in linear elasto-plasticity via Γ-convergence, NoDEA Nonlinear Differential Equations Appl. 19 (2012), 437-457 | MR | Zbl

[28] A. Mainik, A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations 22 (2005), 73-99 | MR | Zbl

[29] A. Mielke, T. Roubíček, U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations 31 (2008), 387-416 | MR | Zbl

[30] A. Mielke, T. Roubíček, M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, Preprint WIAS 1542, Berlin, 2010. | MR

[31] M.G. Mora, S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-convergence, Calc. Var. 18 (2003), 287-305 | MR | Zbl

[32] M.G. Mora, S. Müller, A nonlinear model for inextensible rods as low energy Γ-limit of three-dimensional nonlinear elasticity, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 271-293 | MR | EuDML | Zbl | Numdam

[33] D. Percivale, Perfectly plastic plates: a variational definition, J. Reine Angew. Math. 411 (1990), 39-50 | MR | EuDML

[34] L. Scardia, The nonlinear bending-torsion theory for curved rods as Γ-limit of three-dimensional elasticity, Asymptot. Anal. 47 (2006), 317-343 | MR | Zbl

[35] L. Scardia, Asymptotic models for curved rods derived from nonlinear elasticity by Gamma-convergence, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), 1037-1070 | MR | Zbl

[36] P.-M. Suquet, Sur les équations de la plasticité: existence et regularité des solutions, J. Mécanique 20 (1981), 3-39 | MR | Zbl

[37] R. Temam, Mathematical Problems in Plasticity, Gauthier–Villars, Paris (1985) | Zbl

Cité par Sources :