Abstract
In this paper we prove a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures \(\alpha _k\,{\mathrm {E}}u_k\) with respect to the weak convergence \(\alpha _k\rightharpoonup \alpha \) in \(W^{1,n}(\Omega )\) and the weak\(^*\) convergence \(u_k{\mathop {\rightharpoonup }\limits ^{*}}u\) in \(BD(\Omega )\), \({\mathrm {E}}\) denoting the symmetrised gradient. A concentration compactness argument shows that the limit has the form \(\alpha \,{\mathrm {E}}u+\eta \), with \(\eta \) supported on an at most countable set.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Avoid common mistakes on your manuscript.
References
Alessi, R., Marigo, J.-J., Maurini, C., Vidoli, S.: Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: one-dimensional examples. J. Mech. Sci. Int. (2017). https://doi.org/10.1016/j.ijmecsci.2017.05.047
Alessi, R., Marigo, J.-J., Vidoli, S.: Gradient damage models coupled with plasticity and nucleation of cohesive cracks. Arch. Ration. Mech. Anal. 214, 575–615 (2014)
Alessi, R., Marigo, J.-J., Vidoli, S.: Gradient damage models coupled with plasticity: variational formulation and main properties. Mech. Mater. 80, 351–367 (2015)
Ambati, M., Kruse, R., De Lorenzis, L.: A phase-field model for ductile fracture at finite strains and its experimental verification. Comput. Mech. 57, 149–167 (2016)
Ambrosio, L., Coscia, A., Dal Maso, G.: Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal. 139, 201–238 (1997)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, New York (2000)
Babadjian, J.-F., Francfort, G.A., Mora, M.G.: Quasistatic evolution in non-associative plasticity the cap model. SIAM J. Math. Anal 44, 245–292 (2012)
Crismale, V.: Globally stable quasistatic evolution for a coupled elastoplastic-damage model. ESAIM Control Optim. Calc. Var. 22, 883–912 (2016)
Crismale, V.: Globally stable quasistatic evolution for strain gradient plasticity coupled with damage. Ann. Mat. Pura Appl. 196, 641–685 (2017)
Crismale, V.: Some results on quasistatic evolution problems for unidirectional processes. Ph.D. Thesis (2016)
Crismale, V., Lazzaroni, G.: Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model. Calc. Var. Partial Differ. Equ. 55, 17 (2016)
Dal Maso, G.: On the integral representation of certain local functionals. Ricerche Mat. 32, 85–113 (1983)
Dal Maso, G., DeSimone, A., Mora, M.G.: Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180, 237–291 (2006)
Dal Maso, G., DeSimone, A., Solombrino, F.: Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling. Calc. Var. Partial Differ. Equ. 40, 125–181 (2011)
Dal Maso, G., Orlando, G., Toader, R.: Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case. Calc. Var. Partial Differ. Equ. 55, 45 (2016)
Dal Maso, G., Orlando, G., Toader, R.: Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation. Adv. Calc. Var. 10, 183–207 (2017)
Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS 74. American Mathematical Society, Providence (1990)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Francfort, G.A., Mora, M.G.: Quasistatic evolution in non-associative plasticity revisited. Calc. Var. Partial Differ. Equ. 57, 11 (2018)
Francfort, G.A., Stefanelli, U.: Quasi-static evolution for the Armstrong–Frederick hardening-plasticity model. Appl. Math. Res. Express AMRX 2013, 297–344 (2013)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover Publications Inc, Mineola, NY (2006). (unabridged republication of the 1993 original)
Knees, D., Rossi, R., Zanini, C.: A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23, 565–616 (2013)
Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case, Part I. Rev. Mat. Iberoam. 1(1), 145–201 (1985)
Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case, Part II. Rev. Mat. Iberoam. 1(2), 45–121 (1985)
Miehe, C., Aldakheel, F., Raina, A.: Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory. Int. J. Plast. 84, 1–32 (2016)
Miehe, C., Hofacker, M., Schänzel, L., Aldakheel, F.: Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic-plastic solids. Comput. Methods Appl. Mech. Eng. 294, 486–522 (2015)
Mielke, A.: Evolution of rate-independent systems. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations, Evolutionary Equations, vol. 2, pp. 461–559. Elsevier B.V, Amsterdam (2005)
Mielke, A., Roubíček, T.: Rate Independent Systems: Theory and Application. Springer, New York (2015)
Pham, K., Marigo, J.-J.: Approche variationnelle de lendommagement: I. Les concepts fondamentaux. C. R. Méc. 338, 191–198 (2010)
Pham, K., Marigo, J.-J.: Approche variationnelle de lendommagement: II. Les modèles à gradient. C. R. Méc. 338, 199–206 (2010)
Reshetnyak, Y.G.: Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9, 1039–1045 (1968)
Rossi, R.: Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity. Discrete Contin. Dyn. Syst. Ser. S 10, 1413–1466 (2017)
Roubíček, T., Valdman, J.: Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation. SIAM J. Appl. Math. 75, 314–340 (2016)
Roubíček, T., Valdman, J.: Stress-driven solution to rate-independent elasto-plasticity with damage at small strains and its computer implementation. Math. Mech. Solids 22, 1267–1287 (2017)
Temam, R.: Problèmes mathématiques en plasticité. Méthodes Mathématiques de l’Informatique [Mathematical Methods of Information Science], vol. 12. Gauthier-Villars, Montrouge (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Crismale, V., Orlando, G. A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in \(W^{1,n}\). Nonlinear Differ. Equ. Appl. 25, 16 (2018). https://doi.org/10.1007/s00030-018-0507-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-018-0507-9