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Networks and Heterogeneous Media

2010, Volume 5, Issue 1: 97-132. doi: 10.3934/nhm.2010.5.97
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Quasistatic evolution for Cam-Clay plasticity: The spatially homogeneous case

  • 1.
    SISSA, via Bonomea 265, 34136 Trieste
  • 2.
    SISSA, Via Beirut 2-4, 34151 Trieste
  • Received: 01 July 2009 Revised: 01 December 2009

  • 74C05, 74L10, 49L25, 34E10.

  • 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 spatiallyhomogeneous case[J]. Networks and Heterogeneous Media, 2010, 5(1): 97-132. doi: 10.3934/nhm.2010.5.97

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  • 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.


  • This article has been cited by:

    1. R. V. Goldstein, A. V. Dudchenko, S. V. Kuznetsov, The modified Cam-Clay (MCC) model: cyclic kinematic deviatoric loading, 2016, 86, 0939-1533, 2021, 10.1007/s00419-016-1169-x
    2. Virginia Agostiniani, Second order approximations of quasistatic evolution problems in finite dimension, 2012, 32, 1553-5231, 1125, 10.3934/dcds.2012.32.1125
    3. R. Conti, C. Tamagnini, A. DeSimone, Critical softening in Cam-Clay plasticity: Adaptive viscous regularization, dilated time and numerical integration across stress–strain jump discontinuities, 2013, 258, 00457825, 118, 10.1016/j.cma.2013.02.002
    4. G. A. Francfort, M. G. Mora, Quasistatic evolution in non-associative plasticity revisited, 2018, 57, 0944-2669, 10.1007/s00526-017-1284-8
    5. Gianni Dal Maso, Antonio DeSimone, Francesco Solombrino, Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution, 2012, 44, 0944-2669, 495, 10.1007/s00526-011-0443-6
    6. Bruno Després, Frédéric Lagoutière, Nicolas Seguin, Weak solutions to Friedrichs systems with convex constraints, 2011, 24, 0951-7715, 3055, 10.1088/0951-7715/24/11/003
    7. G. A. Francfort, U. Stefanelli, Quasi-Static Evolution for the Armstrong-Frederick Hardening-Plasticity Model, 2013, 1687-1200, 10.1093/amrx/abt001
    8. R.V. Goldstein, S.V. Kuznetsov, Modified Сam-clay model. Theoretical foundations and numerical analysis, 2016, 9, 1999-6691, 162, 10.7242/1999-6691/2016.9.2.14
    9. Gianni Dal Maso, Antonio DeSimone, Francesco Solombrino, Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling, 2011, 40, 0944-2669, 125, 10.1007/s00526-010-0336-0
    10. S. V. Kuznetsov, Critical State Models in Non-Cohesive Media Mechanics (Review), 2020, 55, 0025-6544, 1423, 10.3103/S0025654420080142
    11. J.-F. Babadjian, G. A. Francfort, M. G. Mora, Quasi-static Evolution in Nonassociative Plasticity: The Cap Model, 2012, 44, 0036-1410, 245, 10.1137/110823511
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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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