Analytical study of the asymptotic behavior of a thin plate with temperature-dependent elastic modulus under cyclic thermomechanical loadings

In a recent paper (Hasbroucq et al., 2010 [1]), we have analyzed the long-term behavior of a discrete mechanical system with temperature-dependent elastic properties under cyclic thermomechanical loadings. In particular, we have shown that the residual stress and strain fields are time-dependent when shakedown occurs and thus the Halphen's (2005) [2] conjecture is not a necessary shakedown condition. Also, we have shown that there is loss of convexity of elastic and shakedown domains in the Bree diagram. In order to examine if these facts are related to the finite dimensional character of the structure and to the uniaxial path load in Hasbroucq et al. (2010) [1], we consider here a continuum media, namely a thin plate undergoing a biaxial loading. Linear and quadratic dependence of Young's modulus and Poisson's coefficient with respect to the temperature are considered. Closed-form expressions of the asymptotic responses of the plate under a given thermomechanical history loads are presented for shakedown, alternating plasticity and ratcheting regimes. In addition, the present study shows that in the Bree diagram, the elastic domain is convex while the shakedown zone is not so and, contrary to Hasbroucq et al. (2010) [1], the residual (elastic) strain field is time-independent for shakedown. Therefore, the present example falls in the Halphen's shakedown conjecture.