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
Abstract
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.
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