Finite-plasticity theories often feature nonlocal energetic contributions in the plastic variables. By introducing a length-scale for plastic effects in the picture, these nonlocal terms open the way to existence results (Mainik and Mielke in J Nonlinear Sci 19(3):221–248, 2009). We focus here on a reference example in this direction, where a specific energetic contribution in terms of dislocation-density tensor is considered (Mielke and Müller in ZAMM Z Angew Math Mech 86:233–250, 2006). When external forces are small and dissipative terms are suitably rescaled, the finite-strain elastoplastic problem converges toward its linearized counterpart. We prove a \(\Gamma \)-convergence result making this asymptotics rigorous, both at the incremental level and at the level of quasistatic evolution.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Agostiniani, V., Dal Maso, G., De Simone, A.: Linear elasticity obtained from finite elasticity by \(\Gamma \)-convergence under weak coerciveness conditions. Ann. I. H. Poincaré 29(5), 715–735 (2012)
Agostiniani, V., DeSimone, A.: Gamma-convergence of energies for nematic elastomers in the small strain limit. Contin. Mech. Thermodyn. 23(3), 257–274 (2011)
Ball, J.M.: Minimizers and the Euler-Lagrange equations. In: Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), volume 195 of Lecture Notes in Phys., pp. 1–4. Springer, Berlin (1984)
Ball, J.M.: Some open problems in elasticity. In: Newton, P., et al. (eds.) Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, New York (2002)
Braides, A., Solci, M., Vitali, E.: A derivation of linear elastic energies from pair-interaction atomistic systems. Netw. Heterog. Media 2(3), 551–567 (2007)
Cermelli, P., Gurtin, M.: On the characterization of geometrically necessary dislocations in finite plasticity. J. Mech. Phys. Solids 49, 1539–1568 (2001)
Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Birkhäuser, Boston (1993)
Dal Maso, G., Negri, M., Percivale, D.: Linearized elasticity as \(\Gamma \)-limit of finite elasticity. Set-Valued Anal. 10(2–3), 165–183 (2002)
Davoli, E.: Linearized plastic plate models as \(\Gamma \)-limits of 3D finite elastoplasticity. ESAIM Control Optim. Calc. Var. 20, 725–747 (2014)
Davoli, E.: Quasistatic evolution models for thin plates arising as low energy \(\Gamma \)-limits of finite plasticity. Math. Models Methods Appl. Sci. 24, 2085–2153 (2014)
Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)
Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245–2271 (2001)
Francfort, G., Mielke, A.: Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595, 55–91 (2006)
Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55(11), 1461–1506 (2002)
Giacomini, A., Musesti, A.: Quasi-static evolutions in linear perfect plasticity as a variational limit of finite plasticity: a one-dimensional case. Math. Models Methods Appl. Sci. 23, 1275–1308 (2013)
Gloria, A., Neukamm, S.: Commutability of homogenization and linearization at identity in finite elasticity and applications. Ann. I. H. Poincaré 28(6), 941–964 (2011)
Grandi, D., Stefanelli, U.: Existence and linearization for the Souza-Auricchio model at finite strains. Discrete Contin. Dyn. Syst. Ser. S 10(6), 1257–1280 (2017)
Grandi, D., Stefanelli, U.: Finite plasticity in \(P^\top P\). Part II: quasistatic evolution and linearization. SIAM J. Math. Anal. 49, 1356–1384 (2017)
Gurtin, M.E.: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32 (2002)
Han, W., Reddy, B.D.: Plasticity, Mathematical Theory and Numerical Analysis. Springer, New York (1999)
Hill, R.: The mathematical theory of plasticity. Reprint of the 1950 original. Oxford Classic Texts in the Physical Sciences. Oxford Engineering Science Series, 11. The Clarendon Press, Oxford University Press, New York (1998)
Lee, E.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36, 1–6 (1969)
Lubliner, J.: Plasticity Theory. Dover, New York (2008)
Kružík, M., Melching, D., Stefanelli, U.: Quasistatic evolution for disclocation-free finite plasticity. ESAIM Control Optim. Calc. Var., to appear (2020)
Mainik, A., Mielke, A.: Global existence for rate-independent gradient plasticity at finite strain. J. Nonlinear Sci. 19(3), 221–248 (2009)
Melching, D., Scala, R., Zeman, J.: Damage model for plastic materials at finite strain. ZAMM Z. Angew. Math. Mech. 99(9), e201800032 (2019). 28 pp
Melching, D., Stefanelli, U.: Well-posedness of a one-dimensional nonlinear kinematic hardening model. Discrete Cont. Dyn. Syst. Ser. S, to appear (2019)
Mielke, A.: Existence of minimizers in incremental elasto-plasticity with finite strains. SIAM J. Math. Anal. 36(2), 384–404 (2004)
Mielke, A.: On evolutionary -convergence for gradient systems. In: Muntean, A., Rademacher, J., Zagaris, A. (eds.) Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. ceedings of Summer School in Twente University (June 2012), Lecture Notes in Appl. Math. Mech., pp. 187–249. Springer, Berlin (2016)
Mielke, A., Müller, S.: Lower semicontinuity and existence of minimizers in incremental finite-strain elastoplasticity. ZAMM Z. Angew. Math. Mech. 86, 233–250 (2006)
Mielke, A., Roubíček, T.: Rate-independent elastoplasticity at finite strains and its numerical approximation. Math. Models Methods Appl. Sci. 26(12), 2203–2236 (2016)
Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Applied Mathematical Sciences, vol. 193. Springer, New York (2015)
Mielke, A., Roubicek, T., Stefanelli, U.: \(\Gamma \)-limit and relaxation of rate-independent evolutionary problems. Calc. Var. Par. Differ. Equ. 31(3), 387–416 (2008)
Mielke, A., Stefanelli, U.: Linearized plasticity is the evolutionary \(\Gamma \)-limit of finite plasticity. J. Eur. Math. Soc. (JEMS) 15(3), 923–948 (2013)
Mielke, A., Theil, F.: On rate-independent hysteresis models. NoDEA Nonlinear Differ. Equ. Appl. 11, 151–189 (2004)
Mühlhaus, H.-B., Aifantis, E.: A variational principle for gradient plasticity. Int. J. Solids Struct. 28, 845–857 (1991)
Müller, S., Neukamm, S.: On the commutability of homogenization and linearization in finite elasticity. Arch. Ration. Mech. Anal. 201(2), 465–500 (2011)
Naghdi, P.M.: A critical review of the state of finite plasticity. J. Appl. Math. Phys. 41, 315–394 (1990)
Paroni, R., Tomassetti, G.: A variational justification of linear elasticity with residual stress. J. Elast. 97, 189–206 (2009)
Paroni, R., Tomassetti, G.: From nonlinear elasticity to linear elasticity with initial stress via \(\Gamma \)-convergence. Contin. Mech. Thermodyn. 23(4), 347–361 (2011)
Scardia, L., Zeppieri, C.I.: Line-tension model for plasticity as the \(\Gamma \)-limit of a nonlinear dislocation energy. SIAM J. Math. Anal. 44(4), 2372–2400 (2012)
Schmidt, B.: Linear \(\Gamma \)-limits of multiwell energies in nonlinear elasticity theory. Contin Mech. Thermodyn. 20(6), 375–396 (2008)
Stefanelli, U.: Existence for dislocation-free finite plasticity. ESAIM Control Optim. Calc. Var. 25, 21 (2019). Art. 21, 20 pp
Svendsen, B.: Continuum thermodynamic models for crystal plasticity including the effects of geometrically necessary dislocations. J. Mech. Phys. Solids 50, 1297–1329 (2002)
US gratefully acknowledges the support of the Vienna Science and Technology Fund (WWTF) through under Project MA14-009 and of the Austrian Science Fund (FWF) under Project F 65. RS gratefully acknowledges the Erwin Schrödinger Institute for financial support.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Andreas Öchsner.
About this article
Cite this article
Scala, R., Stefanelli, U. Linearization for finite plasticity under dislocation-density tensor regularization. Continuum Mech. Thermodyn. (2020). https://doi.org/10.1007/s00161-020-00898-w
- Finite plasticity
- \(\Gamma \)-convergence
- Incremental problem