Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-09T19:35:23.739Z Has data issue: false hasContentIssue false
  • English
  • Français

Linearized plastic plate models as Γ-limits of 3D finite elastoplasticity

Published online by Cambridge University Press:  27 May 2014

Elisa Davoli
Elisa Davoli*
Affiliation:
Department of Mathematical Sciences, Carnegie-Mellon University, Pittsburgh, PA 15213, USA. [email protected]
Get access

Abstract

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.

Keywords

Finite plasticitythin platesΓ-convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 725 - 747
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E., Buttazzo, G. and Percivale, D., A variational definition for the strain energy of an elastic string. J. Elasticity 25 (1991) 137–148. Google Scholar
Bertram, A., An alternative approach to finite plasticity based on material isomorphisms. Int. J. Plasticity 15 (1999) 353–374. Google Scholar
Carstensen, C., Hackl, K. and Mielke, A., Non-convex potentials and microstructures in finite-strain plasticity. R. Soc. London Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002) 299–317. Google Scholar
G. Dal Maso, An introduction to Γ-convergence. Boston, Birkhäuser (1993).
Dal Maso, G. and Lazzaroni, G., Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27 (2010) 257–290. Google Scholar
E. Davoli, Quasistatic evolution models for thin plates arising as low energy Γ-limits of finite plasticity. Preprint SISSA (2012), Trieste.
Francfort, G.A. and Mielke, A., Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 55–91. Google Scholar
Friesecke, G., James, R.D. and Müller, S., A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55 (2002) 1461–1506. Google Scholar
Friesecke, G., James, R.D. and Müller, S., A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Rational Mech. Anal. 180 (2006) 183–236. Google Scholar
Lecumberry, M. and Müller, S., Stability of slender bodies under compression and validity of the Von Kármán theory. Arch. Ration. Mech. Anal. 193 (2009) 255–310. Google Scholar
Le Dret, H. and Raoult, A., The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549–578. Google Scholar
Lee, E.H., Elastic-plastic deformation at finite strains. J. Appl. Mech. 36 (1969) 1–6. Google Scholar
Liero, M. and Mielke, A., An evolutionary elastoplastic plate model derived via Γ-convergence. Math. Models Methods Appl. Sci. 21 (2011) 1961–1986. Google Scholar
Liero, M. and Roche, T., Rigorous derivation of a plate theory in linear elastoplasticity via Γ-convergence. NoDEA Nonlinear Differ. Eqs. Appl. 19 (2012) 437–457. Google Scholar
Mainik, A. and Mielke, A., Global existence for rate-independent gradient plasticity at finite strain. J. Nonlinear Sci. 19 (2009) 221–248. Google Scholar
Mandel, J., Equations constitutive et directeur dans les milieux plastiques et viscoplastique. Int. J. Sol. Struct. 9 (1973) 725–740. Google Scholar
Mielke, A., Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15 (2003) 351–382. Google Scholar
A. Mielke, Finite elastoplasticity, Lie groups and geodesics on SL(d), Geometry, Dynamics, and Mechanics. Springer, New York (2002) 61–90.
Mielke, A. and Stefanelli, U., Linearized plasticity is the evolutionary Gamma-limit of finite plasticity. J. Eur. Math. Soc. 15 (2013) 923–948. Google Scholar
Naghdi, P.M., A critical review of the state of finite plasticity. Z. Angew. Math. Phys. 41 (1990) 315–394. Google Scholar