Elastic-plastic-creep Cyclic Loading of Thick Pressure Vessels based on the Frederick-Armstrong Kinematic Hardening Model

Transactions of the 17 th International Conference on Structural Mechanics in Reactor Technology (SMiRT 17) Prague, Czech Republic, August 17 –22, 2003 Paper # F05-7 1 Elastic-plastic-creep Cyclic Loading of Thick Pressure Vessels based on the Frederick-Armstrong Kinematic Hardening Model Hossein Mahbadi 1) and Mohamad R. Eslami 2) 1) ME Dept., Islamic Azad University, Tehran Branch, Tehran, Iran 2) ME Dept., Distinguished Solid Mechanics Group, Amirkabir Univ. of Tech., Tehran, Iran ABSTRACT In this article the behavior of thick cylindrical and spherical vessels under mechanical cyclic loads is obtained. Frederick-Armstong kinematic hardening theory is used to evaluate the plastic strain distribution of thick vessels due to the cyclic loadings. The results are compared for two cases, where creep deformation is included and the case where creep is excluded. KEY WORDS: cyclic loading, creep, ratcheting, shakedown, load controlled, deformation controlled, thick sphere, thick cylinder, Frederick-Armstrong, INTRODUCTION According to Mahbadi and Eslami [1,2], when creep is not considered, cyclic loading analysis of structures based on the Prager kinematic hardening model results to reversed plasticity for the load controlled stress and shakedown for the deformation controlled stress. The Frederick-Armstrong kinematic hardening model, on the other hand, predicts ratcheting for the load controlled stress and shakedown or reversed plasticity for the deformation controlled stress [1,2]. The latter model is generally recommended for the cyclic loading analysis of structures due to its more precise results. However, when creep deformation is considered, the cyclic loading analysis of structures based on the Prager and Frederick-Armstrong models qualitatively predict similar results [3,4,5]. Cyclic loading analysis of thick cylinders and spheres of homogeneous and isotropic materials using the isotropic and kinematic hardening theories are given in references [6,7]. It is shown that the isotropic hardening model always predicts shakedown, when creep is not considered. When creep deformation is considered, this model predicts shakedown for the deformation controlled loads and ratcheting for the load controlled cylic loading. In this paper, cyclic loading analysis of thick spherical and cylindrical vessels are obtained using the Frederick- Armstrong kinematic hardening model. Two types of loads, namely the load controlled and deformation controlled loads, are checked to calculate the stresses and strains of thick cylindrical and spherical vessels. Secondary creep is considered to evaluate the creep deformation of the vessels. The results are justified with the given data in the literature. An effective numerical iterative method is proposed for the cyclic loading analysis of structures. The method is effective for the one-dimensional stress state as well as multi dimensional stress cyclic loading analysis. MATHEMATICAL FORMULATION Consider a thick vessel (spherical or cylindrical) of inside radius a and outside radius b under internal and external pressures p I and p o and radial temperature distribution. A) THICK CYLINDER: The axisymmetric and plain strain conditions are assumed for the thick cylinder. The following differential equations together with the stress-strain relations and incompressibility conditions for plastic and creep deformations in the cylindrical coordinates are solved to obtain the thermoelastic-plastic and creep stress and strain distributions in the thick cylinder. The equilibrium equation is: