-
Previous Article
Thermodynamics of perfect plasticity
- DCDS-S Home
- This Issue
-
Next Article
Crack propagation by a regularization of the principle of local symmetry
A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems
1. | Dipartimento di Matematica, Università di Brescia, Via Valotti 9, I-25133 Brescia, Italy |
2. | Dipartimento di Matematica “F. Casorati", Università di Pavia, Via Ferrata 1, I- 27100 Pavia, Italy |
In this paper we study both notions in the one-dimensional setting and we obtain a full characterization of BV and energetic solutions for a broad family of energy functionals. In the case of monotone loadings we provide a simple and explicit characterization of such solutions, which allows for a direct comparison of the two concepts.
References:
[1] |
Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. |
[2] |
J. Convex Analysis, 13 (2006), 151-167. |
[3] |
Dunod, Gauthier-Villars, Paris, 1974. |
[4] |
Calc. Var. Partial Differential Equations, 22 (2005), 73-99.
doi: 10.1007/s00526-004-0267-8. |
[5] |
in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam II (2005), 461-559. |
[6] |
in "Nonlinear PDE's and Applications'' (eds: L. Ambrosio and G. Savaré), Lecture Notes, C. I. M. E. Summer School, Cetraro, Italy, (2008), 87-171, Springer, 2011. |
[7] |
Discrete Contin. Dyn. Syst., 25 (2009), 585-615.
doi: 10.3934/dcds.2009.25.585. |
[8] |
ESAIM Control Optim. Calc. Var, Calc., 18 (2012), 36-80.
doi: 10.1051/cocv/2010054.Published. |
[9] |
in "Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering'' (eds. H.-D. Alber, R. M. Balean and R. Farwig), Aachen, Shaker-Verlag, (1999), 117-129. Google Scholar |
[10] |
NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.
doi: 10.1007/s00030-003-1052-7. |
[11] |
SIAM J. Math. Anal., 41 (2009), 1340-1365.
doi: 10.1137/090750809. |
[12] |
Adv. Calc. Var., 3 (2010), 149-212. |
[13] |
Math. Nachr., 282 (2009), 1492-1512.
doi: 10.1002/mana.200810803. |
[14] |
Applied Mathematical Sciences, 111, Springer-Verlag, Berlin, 1994. |
show all references
References:
[1] |
Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. |
[2] |
J. Convex Analysis, 13 (2006), 151-167. |
[3] |
Dunod, Gauthier-Villars, Paris, 1974. |
[4] |
Calc. Var. Partial Differential Equations, 22 (2005), 73-99.
doi: 10.1007/s00526-004-0267-8. |
[5] |
in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam II (2005), 461-559. |
[6] |
in "Nonlinear PDE's and Applications'' (eds: L. Ambrosio and G. Savaré), Lecture Notes, C. I. M. E. Summer School, Cetraro, Italy, (2008), 87-171, Springer, 2011. |
[7] |
Discrete Contin. Dyn. Syst., 25 (2009), 585-615.
doi: 10.3934/dcds.2009.25.585. |
[8] |
ESAIM Control Optim. Calc. Var, Calc., 18 (2012), 36-80.
doi: 10.1051/cocv/2010054.Published. |
[9] |
in "Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering'' (eds. H.-D. Alber, R. M. Balean and R. Farwig), Aachen, Shaker-Verlag, (1999), 117-129. Google Scholar |
[10] |
NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.
doi: 10.1007/s00030-003-1052-7. |
[11] |
SIAM J. Math. Anal., 41 (2009), 1340-1365.
doi: 10.1137/090750809. |
[12] |
Adv. Calc. Var., 3 (2010), 149-212. |
[13] |
Math. Nachr., 282 (2009), 1492-1512.
doi: 10.1002/mana.200810803. |
[14] |
Applied Mathematical Sciences, 111, Springer-Verlag, Berlin, 1994. |
[1] |
Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332 |
[2] |
Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 |
[3] |
Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256 |
[4] |
Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585 |
[5] |
Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i |
[6] |
Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145 |
[7] |
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 |
[8] |
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 |
[9] |
Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411 |
[10] |
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 |
[11] |
Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257 |
[12] |
Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649 |
[13] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331 |
[17] |
Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801 |
[18] |
Hernán R. Henríquez, Claudio Cuevas, Juan C. Pozo, Herme Soto. Existence of solutions for a class of abstract neutral differential equations. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2455-2482. doi: 10.3934/dcds.2017106 |
[19] |
Nobuyuki Kenmochi, Jürgen Sprekels. Phase-field systems with vectorial order parameters including diffusional hysteresis effects. Communications on Pure & Applied Analysis, 2002, 1 (4) : 495-511. doi: 10.3934/cpaa.2002.1.495 |
[20] |
Robert Hesse, Alexandra Neamţu. Global solutions and random dynamical systems for rough evolution equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2723-2748. doi: 10.3934/dcdsb.2020029 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]