American Institute of Mathematical Sciences

September  2008, 3(3): 567-614. doi: 10.3934/nhm.2008.3.567

Globally stable quasistatic evolution in plasticity with softening

 1 SISSA-International School for Advanced Studies, Via Beirut 2-4, 34014, Trieste, Italy 2 SISSA-International School for Advanced Studies, Via Beirut 2-4, 34014 Trieste 3 SISSA-International School for Advanced Studies, Via Beirut 2-4,, 34014, Trieste, Italy

Received  February 2008 Published  June 2008

We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.
Citation: G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Globally stable quasistatic evolution in plasticity with softening. Networks & Heterogeneous Media, 2008, 3 (3) : 567-614. doi: 10.3934/nhm.2008.3.567
 [1] Francesco Solombrino. Quasistatic evolution for plasticity with softening: The spatially homogeneous case. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1189-1217. doi: 10.3934/dcds.2010.27.1189 [2] Gianni Dal Maso, Francesco Solombrino. Quasistatic evolution for Cam-Clay plasticity: The spatially homogeneous case. Networks & Heterogeneous Media, 2010, 5 (1) : 97-132. doi: 10.3934/nhm.2010.5.97 [3] Tomáš Roubíček. Thermodynamics of perfect plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 193-214. doi: 10.3934/dcdss.2013.6.193 [4] M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure & Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495 [5] T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215 [6] Sergio Conti, Georg Dolzmann, Carolin Kreisbeck. Relaxation and microstructure in a model for finite crystal plasticity with one slip system in three dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 1-16. doi: 10.3934/dcdss.2013.6.1 [7] Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070 [8] Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks & Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257 [9] Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076 [10] Virginia Agostiniani. Second order approximations of quasistatic evolution problems in finite dimension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1125-1167. doi: 10.3934/dcds.2012.32.1125 [11] Gilles A. Francfort, Alessandro Giacomini, Alessandro Musesti. On the Fleck and Willis homogenization procedure in strain gradient plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 43-62. doi: 10.3934/dcdss.2013.6.43 [12] Tomáš Roubíček. On certain convex compactifications for relaxation in evolution problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 467-482. doi: 10.3934/dcdss.2011.4.467 [13] Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591 [14] Alessandro Giacomini. On the energetic formulation of the Gurtin and Anand model in strain gradient plasticity. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 527-552. doi: 10.3934/dcdsb.2012.17.527 [15] Steffen Arnrich. Modelling phase transitions via Young measures. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 29-48. doi: 10.3934/dcdss.2012.5.29 [16] G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1 [17] Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256 [18] Gianni Dal Maso, Alexander Mielke, Ulisse Stefanelli. Preface: Rate-independent evolutions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : i-ii. doi: 10.3934/dcdss.2013.6.1i [19] Marita Thomas. Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 235-255. doi: 10.3934/dcdss.2013.6.235 [20] Martin Kružík, Ulisse Stefanelli, Chiara Zanini. Quasistatic evolution of magnetoelastic plates via dimension reduction. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5999-6013. doi: 10.3934/dcds.2015.35.5999

2018 Impact Factor: 0.871