New variational principles for the asymptotic elastoplastic response under cyclic loads
Nestor Zouain1; Ivaldo Pontes2
1 Mechanical Engineering Department, COPPE-Poli / UFRJ Federal University of Rio de Janeiro; 2 Civil Engineering Department, UFPE Federal University of Prnambuco
doi:10.20906/CPS/CILAMCE2015-0141
Resumo
The subject of the paper is the direct computation of the asymptotic steady state produced in an elastoplastic body submitted to cyclic loadings exceeding the shakedown limits (Polizzotto, 2003; Ponter and Chen, 2001; Zouain, 2004). In summary, we present here: (i) two new mixed variational principles for the continuum model of the body, whose stationary conditions properly represent known equations characterizing the asymptotic response of stress and plastic strain rate fields (ii) two different time-space discretizations of the mentioned principles giving optimization problems amenable to practical application and (iii) some examples of application. The basic tool for the attainment of the present variational formulations for the continuum is a mixed minimization principle, in terms of stress and plastic strain rates, for the flow law of the elastic-ideally plastic material. Both variational principles proposed here are strictly equivalent in the context of the continuum formulation where the corresponding constraints are supposed to be valid at any material point as well as at any instant, for exact time rates. However, it is shown in the paper that one of the formulations leads directly to a consistent discrete principle while the other loses crucial properties when discretized under usual assumptions. We discuss in the article the differences between the present formulation and (the few) existing ones. References Polizzotto, C. (2003). Variational methods for the steady state response of elastic-plastic solids subjected to cyclic loads. Int. J. Solids Struct., 40, 2673-2697. Ponter, A. R. S., and Chen, H. (2001). A minimum theorem for cyclic load in excess of shakedown, with application to the evaluation of a ratchet limit. Eur. J. Mech. A/Solids, 20, 539-553. Zouain, N. (2004). Shakedown and safety assessment. In E. Stein, R. de Borst, and T. J. R. Hughes (Eds.), Encyclopedia of Computational Mechanics (p. 291-334). John Wiley.
Palavras-chave: cyclic plasticity; variational principles; shakedown