On Cyclic Steady States and Elastic Shakedown in Diffusion-Induced Plasticity

Abstract

This chapter is devoted to media in which plasticity and diffusion are coupled, such as electrode materials in lithium ion batteries. We present some recent results on the large time behavior of such media when they are submitted to cyclic chemo-mechanical loadings. Under suitable technical assumptions, we notably show that there is convergence towards a cyclic steady state in which the stress, the plastic strain rate, the chemical potential and the concentration of guest atoms are all periodic in time (with the same period as the applied loading). A special case of interest is that of elastic shakedown, which corresponds to the situation where the medium behaves elastically in the large time limit. We present general theorem that allow one to construct both lower and upper bounds of the set of loadings for which elastic shakedown occurs, in the spirit of Melan and Koiter theorems in classical plasticity. An illustrative example-for which all the relevant calculations can be done in closed-form-is presented.

On the elastic–plastic behaviors of centrifugal steam compressor impeller under cyclic thermomechanical loading

“…However, generating unambiguous design limits for practical structures, such as Bree‐type diagrams, 14 using this approach can be computationally expensive. Hence, direct methods based on the upper and lower bounding theorems have been extensively explored 15–17 . These methods transform the determination of shakedown and ratchet boundaries into mathematical programming problems that involve finding solutions for maximum or minimum values while considering equality and inequality constraints, and this approach effectively addresses the requirements for shakedown and ratchet limits in the design phase.…”