March  2010, 5(1): 97-132. doi: 10.3934/nhm.2010.5.97

Quasistatic evolution for Cam-Clay plasticity: The spatially homogeneous case

1. 

SISSA, via Bonomea 265, 34136 Trieste, Italy

2. 

SISSA, Via Beirut 2-4, 34151 Trieste, Italy

Received  July 2009 Revised  December 2009 Published  February 2010

We study the spatially uniform case of the quasistatic evolution in Cam-Clay plasticity, a relevant example of small strain nonassociative elastoplasticity. Introducing a viscous approximation, the problem reduces to determine the limit behavior of the solutions of a singularly perturbed system of ODE's in a finite dimensional Banach space. Depending on the sign of two explicit scalar indicators, we see that the limit dynamics presents, under quite generic assumptions, the alternation of three possible regimes: the elastic regime, when the limit equation is just the equation of linearized elasticity; the slow dynamics, when the stress evolves smoothly on the yield surface and plastic flow is produced; the fast dynamics, which may happen only in the softening regime, when viscous solutions exhibit a jump determined by the heteroclinic orbit of an auxiliary system. We give an iterative procedure to construct a viscous solution.
Citation: 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
[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]

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

[3]

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

[4]

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

[5]

Tomáš Roubíček. Thermodynamics of perfect plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 193-214. doi: 10.3934/dcdss.2013.6.193

[6]

Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309

[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]

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

[9]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015

[10]

Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure & Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479

[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]

Monica Conti, Vittorino Pata, M. Squassina. Singular limit of dissipative hyperbolic equations with memory. Conference Publications, 2005, 2005 (Special) : 200-208. doi: 10.3934/proc.2005.2005.200

[13]

Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179

[14]

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

[15]

M. Bulíček, P. Kaplický. Incompressible fluids with shear rate and pressure dependent viscosity: Regularity of steady planar flows. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 41-50. doi: 10.3934/dcdss.2008.1.41

[16]

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

[17]

Felipe Alvarez, Alexandre Cabot. Asymptotic selection of viscosity equilibria of semilinear evolution equations by the introduction of a slowly vanishing term. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 921-938. doi: 10.3934/dcds.2006.15.921

[18]

Stefano Bianchini, Alberto Bressan. A case study in vanishing viscosity. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 449-476. doi: 10.3934/dcds.2001.7.449

[19]

Umberto Mosco, Maria Agostina Vivaldi. Vanishing viscosity for fractal sets. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1207-1235. doi: 10.3934/dcds.2010.28.1207

[20]

Wenjun Wang, Lei Yao. Vanishing viscosity limit to rarefaction waves for the full compressible fluid models of Korteweg type. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2331-2350. doi: 10.3934/cpaa.2014.13.2331

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]