Inelastic responses of a two-bar system with temperature-dependent elastic modulus under cyclic thermomechanical loadings

Abstract

a b s t r a c tThis paper is concerned with the elastic plastic response of a two-bar system with temperature-dependent elastic coefficients under cyclic thermomechanical loadings. Such materials are characterized by lack of results concerning the asymptotic behaviors and conditions for shakedown occurrence. This study shows that the considered simple structure is sufficiently complex to experience different periodic long-term behaviors as in classical elastoplasticity. In order to understand how Melan-Koiter method works for such materials, the evolution of the structure's response until the stabilization of the plastic strain ('shakedown') or the asymptotic dissipative behavior ('alternating plasticity' or 'ratcheting') is analytically addressed and the Bree diagram is then constructed. The main result of this work is that the residual stress and strain fields are time-dependent even when shakedown occurs. Besides, we proved that Halphen's conjecture (Halphen, 2005) giving a sufficient condition for shakedown occurrence is not a necessary condition. Finally, numerical results performed by an incremental finite element procedure are presented.

Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli

“…Similar remarks hold for the examples studied by Halphen and di Domizio (2005). Therefore, there is no contradiction between Theorem 1 and the results of Halphen and di Domizio (2005); Hasbroucq et al (2010Hasbroucq et al ( , 2012: for all the examples studied by the authors, it turns out that L(t) is of the form (53), in which case condition (ii) is automatically satisfied if condition (i) is.…”

“…Remark 3 : as mentioned in the introduction, incremental analysis has been used by Halphen and di Domizio (2005); Hasbroucq et al (2010Hasbroucq et al ( , 2012 to study the asymptotic behaviour of some specific mechanical systems with time-dependent elastic moduli. For the examples considered by those authors, condition (i) was found to be sufficient for shakedown to occur independently on the initial state.…”

“…It follows from the definition (13) that the corresponding operator L(t) is of the form (53). In the two-bar system considered in Hasbroucq et al (2010), the Young's modulus is not spatially homogeneous but the space H of self-stresses is of dimension 1. Therefore, the linear operator L(t) : H → H is also of the radial form (53).…”

“…Using incremental analysis, the asymptotic behavior of some specific mechanical systems has been studied by Halphen and di Domizio (2005); Hasbroucq et al (2010Hasbroucq et al ( , 2012. The obtained results led to some conjectures about shakedown theorems for time-dependent elastic moduli, but not theoretical result is available.…”

On Shakedown of Elastic-Plastic Bodies With Temperature Dependent Properties

“…The main motivation is to extend the range of applications of Melan's theorem to thermomechanical loading histories with large temperature fluctuations: In such case, the variation of the material properties with the temperature cannot be neglected. Whereas the case of temperature-dependent yield limits is well understood [17], the case of temperature-dependent elastic moduli remains a long standing issue and has been the object of several conjectures [18,19,20,21]. The main difficulty is that the proof used in the original Melan's theorem -as well as in most of its knows extensions -crucially relies on some contraction properties that are lost when the elastic moduli are allowed to vary in time.…”