## 1 Introduction  
Pressure vessels and piping systems operating under various extreme cyclic bending or thermal loading situations exhibit different cyclic plasticity behaviors. The applicability domains of these structures, particularly functionally graded material (FGM) pipes under repeated start-stop loading conditions, arise in many engineering areas such as nuclear reactors, aircrafts, space shuttles, etc. Thus, understanding the responses of FGM pipes under a combination of sustained and cyclic loadings, particularly cyclic accumulated plastic deformation (ratcheting), contributes to a higher safety-in-design and hinders premature structural failures. In order to assess the material behaviors under different cyclic loading conditions, the interaction diagram referred to as [PERSON]'s diagram is applied in design criteria in engineering structures. However, achieving precise trend lines of boundaries between cyclic stress regimes in [PERSON]'s diagram necessitates the application of proper and efficient plasticity models. Over the past decades, a large number of constitutive models in the field of plastic mechanics such as [PERSON] and [PERSON] (A-F) [1], [PERSON] [2], [PERSON] (O-W) [3], [PERSON] [4], [PERSON] (J-S) [5], [PERSON] and [PERSON] [6] and [PERSON] (C-J-K) [7] have been proposed to predict cyclic responses of material. In addition, a number of reported studies used these cyclic plasticity models to investigate various elastic-plastic problems with different types of structures, which are briefly presented in the following. [PERSON] et al. [8] examined the robustness of different plasticity models on the prediction of uniaxial and biaxial cyclic plasticity responses of straight pressurized homogeneous pipe through comparison with experimental data. [PERSON] et al. [9] experimentally investigated cyclic elastic-plastic behaviors of homogeneous pipes submitted to internal pressure, bending and transverse load controls. [PERSON] and [PERSON] [10] used the yield surface distortion rule of Baltov combined with the Chaboche nonlinear kinematic hardening model to predict cyclic behaviors of metals subjected to multiaxial and uniaxial loading conditions. [PERSON] et al. [11] experimentally investigated the cyclic behavior of short steel pipes filled with three different concrete mixtures. [PERSON] and [PERSON] [12] utilized different steel pipes to illustrate the effects of loading paths and directions on cyclic behaviors through the newly developed kinematic hardening model of Ahmadzadeh-Varvani (A-V). Elastic-plastic responses of pressurized 90\({}^{\circ}\) pipe bend submitted to in-plane cyclic bending and cyclic thermal loading was investigated by [PERSON] et al. [13]. Elastic-plastic behaviors of homogeneous pipes under cyclic axial loading and constant internal pressure were investigated by [PERSON] et al. [14]. However, limited published research papers regarding the elastic-plastic analysis of FG composite materials are available which will be described in the following paragraph. [PERSON] et al. [15] presented a coupled elastoplastic damage model for an FG plate with material properties varied through an in-plane direction. [PERSON] and [PERSON] [16] developed analytical solutions for the elastic-plastic problem of FG pressurized pipe under the case of plane strain. The analytical solution for the both elastic and elastic-plastic problems of a pressurized pipe by means of a meshless radial basis function (RBF) collocation method was presented by [PERSON] and [PERSON] [17]. [PERSON] et al. [18] investigated the static and elastic-plastic responses of Bree plate made of FGM involved with two simultaneous loading conditions of constant mechanical load and cyclic temperature changes. Moreover, the elastic solutions for different FG components under thermomechanical loadings can be found in many works [19; 20; 21; 22; 23]. In addition, [PERSON] et al. [24] investigated the yield onset of thick-walled FG cylindrical tubes under thermomechanical loading conditions.