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 theorems 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.
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References
Brassart, L., Zhao, K., Suo, Z.: Cyclic plasticity and shakedown in high-capacity electrodes of lithium-ion batteries. Int. J. Solids. Struct. 49, 1120–1129 (2013)
Frederick, C., Armstrong, P.: Convergent internal stresses and steady cyclic states of stress. J. Strain Anal. Eng. Des. 1, 154–159 (1966)
Halphen, B.: Steady cycles and shakedown in elastic-viscoplastic and plastic structures. In: Structures et matériaux sous chargement cyclique. Association amicale des ingénieurs anciens éleves de l’E.N.P.C., pp. 203–229 (1978)
Johnson, C., Mercier, B., Nedelec, J.C.: Convergence to a periodic solution in perfect plasticity. In: Structures et matériaux sous chargement cyclique. Association amicale des ingénieurs anciens éleves de l’E.N.P.C., pp. 253–255 (1978)
Maitournam, H., Pommier, B., Thomas, J.J.: Détermination de la réponse asymptotique d’une structure anélastique sous chargement cyclique. C.R. Mecanique 330, 703–708 (2002)
Peigney, M., Stolz, C.: Approche par contrôle optimal des structures élastoviscoplastiques sous chargement cyclique. C.R. Acad. Sci. Paris II 329, 643–648 (2001)
Peigney, M., Stolz, C.: An optimal control approach to the analysis of inelastic structures under cyclic loading. J. Mech. Phys. Solids 51, 575–605 (2003)
Spiliopoulos, K.V., Panagiotou, K.D.: A direct method to predict cyclic steady states of elastoplastic structures. Comput. Methods Appl. Mech. Eng. 223, 186–198 (2012)
Klarbring, A., Barber, J.R., Spagnoli, A., Terzano, M.: Shakedown of discrete systems involving plasticity and friction. Eur. J. Mech. A 64, 160–164
Peigney, M.: Recoverable strains in composite shape memory alloys. J. Mech. Phys. Solids 56, 360–375 (2008)
Peigney, M.: Shakedown theorems and asymptotic behaviour of solids in non-smooth mechanics. Eur. J. Mech. A 29, 784–793 (2010)
Peigney, M.: On shakedown of shape memory alloys structures. Ann. Solid Struct. Mech. 6, 17–28 (2014)
Peigney, M.: Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli. J. Mech. Phys. Solids 71, 112–131 (2014)
Pham, D.C.: Consistent limited kinematic hardening plasticity theory and path-independent shakedown theorems. Int. J. Mech. Sci. 130, 11–18 (2017)
Larché, F., Cahn, J.W.: A linear theory of thermochemical equilibrium of solids under stress. Acta Metall. 21, 1051–1063 (1973)
Dang Van, K., Papadopoulos I.V.: Introduction to fatigue analysis in mechanical design by the multiscale approach. In: High-Cycle Metal Fatigue. Springer, Berlin (1999)
Koiter, W.T.: General theorems for elastic-plastic solids. In: Progress in Solid Mechanics (1960)
Melan, E.: Theorie statisch unbestimmter Systeme aus ideal-plastischen Baustoff. Sitz. Berl. Ak. Wiss. 145, 195–218 (1936)
Symonds, P.S.: Shakedown in continuous media. J. Appl. Mech. 18, 85–89 (1951)
Peigney, M.: Cyclic steady states in diffusion-induced plasticity with applications to lithium-ion batteries. J. Mech. Phys. Solids 111, 530–556 (2018)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive map**s. Bull. Am. Math. Soc. 73, 591–597 (1967)
Débordes, O., Nayroles, B., et al.: Sur la théorie et le calcul à l’adaptation des structures élastoplastiques. J. Mecanique 15, 1–53 (1976)
Nguyen, Q.S.: On shakedown analysis in hardening plasticity. J. Mech. Phys. Solids 51, 101–125 (2003)
Peigney, M.: Static and kinematic shakedown theorems in diffusion-induced plasticity. J. Thero. App. Mech. 58(2), 415–424 (2020)
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Peigney, M. (2021). On Cyclic Steady States and Elastic Shakedown in Diffusion-Induced Plasticity. In: Pisano, A., Spiliopoulos, K., Weichert, D. (eds) Direct Methods. Lecture Notes in Applied and Computational Mechanics, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-030-48834-5_9
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