Thermal and mechanical cyclic loading of thick spherical vessels made of transversely isotropic materials M. Komijani a , H. Mahbadi b , M.R. Eslami a, * a Mechanical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran b Mechanical Engineering Department, Islamic Azad University, Central Tehran Branch, Tehran, Iran article info Article history: Received 30 September 2012 Received in revised form 29 December 2012 Accepted 21 January 2013 Keywords: Kinematic hardening model Ratcheting Shakedown Reversed plasticity Stress categories abstract The aim of this paper is to obtain the dependency of the ratcheting, reversed plasticity, or shakedown behavior of spherical vessels made of some anisotropic materials to the stress category of imposed cyclic loading. The Hill anisotropic yield criterion with the kinematic hardening theories of plasticity based on the Prager and ArmstrongeFrederick models are used to predict the yield of the vessel and obtain the plastic strains. An iterative numerical method is used to simulate the cyclic loading behavior of the structure. The effect of mean and amplitude of the mechanical and thermal loads on cyclic behavior and ratcheting rate of the vessel is investigated respectively. The ratcheting rate for the vessels made of transversely isotropic material is evaluated for the various ratios of anisotropy. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Cyclic loading of structures results in structural shakedown, ratcheting, or reversed plasticity. The anisotropy of metal during the manufacturing process may affect the cyclic loading results of the structure. Extensive efforts have been made to improve the accuracy of Hill’s [1,2] quadratic classical anisotropic theory of flow by many researchers such as, Hill [3e5], Caddell and Hosford [6,7], Karafillis and Bocye [8], Barlat et al. [9e12], and others. In most of these works the non-quadratic yield functions are introduced to improve the anisotropic behavior of sheet metals during the plastic works. While, the non-quadratic associated flow rules increase the complexity of the plastic analysis of the structures due to their large number of material parameters, the non-associated flow rule pre- sented by researchers such as Stoughton [13], Staughton and Yoon [14], Aretz [15] are presented which benefits from the convenient description of anisotropic yielding and flow. Cyclic loading effects due to imposed cyclic mechanical and thermal loads cannot be predicted by the common isotropic hardening theories, and a suitable kinematic hardening theory is needed to model the Bauschinger effect. The associated kinematic hardening proposed by Wu [16], and Hahm and Kim [17], are among the models which are capable to simulate the anisotropic properties of cyclic deformations. The kinematic hardening models proposed in these researches are based on the Prager [18] and ArmstrongeFrederick [19] models. It is possible to relate the structural behavior to the stress category in cyclic loading condition. Two stress categories that may be applied to the structure are the load and strain controlled types. According to Eslami and Shariyat [20], a load controlled stress is a normal or shear stress developed by the imposed loading which is necessary to satisfy the simple laws of equilibrium of external and internal forces and moments. It is a state of constant stress with limited final strain associated with linear elastic strain. The basic characteristic of load controlled stress is that it is not self-limiting. A deformation controlled stress is a normal or shear stress developed by the constraint of the adjacent structure. It is a state of stress with constant total strain where unlimited strain can occur causing stress reduction. The basic characteristic of the deformation controlled stress is that it is self- limiting and self-equilibrating [20]. Mahbadi and Eslami [21] inves- tigated the cyclic loading analysis of thick vessels made of isotropic materials under load and deformation controlled conditions based on the Prager and ArmstrongeFrederick kinematic hardening models. Their investigations show that the response of a structure to a cyclic loading depends on the stress category of the load applied to the structure and the hardening model used in the plastic analysis. In this paper, cyclic loading behavior of thick spherical vessels made of some anisotropic materials under different types of loading such as thermal, mechanical, and their combinations are investigated. The materials are assumed to be anisotropice homogeneous and obey a non-linear strain hardening law in the * Corresponding author. Tel.: þ98 21 6640 5844. E-mail address: eslami@aut.ac.ir (M.R. Eslami). Contents lists available at SciVerse ScienceDirect International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp 0308-0161/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijpvp.2013.01.006 International Journal of Pressure Vessels and Piping 107 (2013) 1e11