## 1 Introduction  
Life prediction of industrial facilities after many years of service is an important issue related to the stability, economy, and replacement of an entire facility. Since power plants require a huge cost when replacing typical parts or an entire facility, this problem is especially important in the field. The stable operation and reduction of the costs of life evaluation of facilities subjected to mechanical and thermal loads have been continually studied, and it is necessary to develop new approaches in assessing the residual life. Among the key components of power plants, steam pipes, which are designed to carry pressurized steam and are exposed to high temperatures, are in an extreme environment during operation. Thus, periodic maintenance through life prediction of steam pipes is essential. It is known that steam pipes are mainly damaged by creep and fatigue, which are the main damage mechanisms that can be confirmed by metallurgical examinations. Creep and fatigue are already well-established failure modes for such components at room temperature. However, the failure mode becomes more complicated when the components are exposed to high temperatures during operation. Creep-fatigue interaction is the complex mode produced by cyclic and persistent mechanical loads, and it is depicted as the accumulated rate of damage under complex loadings. Fatigue failure is a mechanism involving crack propagation, while creep is manifested by the mobility of voids on the interior grain boundaries. Therefore, in the circumstances of high temperature with cyclic loadings, creep-fatigue interaction, which shows creep cavitation and surface-fatigue damage, may be found within the material and should be considered when evaluating life prediction. From a microscopic viewpoint, crack behaviors have been determined according to operating environments such as cyclic loadings or creep. For low cyclic loadings, the transgranular fracture has been frequently observed because cracks propagate through the grains after they are initiated in a material. At low and intermediate stress, the creep failures occur due to the initiation and growth of creep cavitation or a void along the grain boundaries, which is called intergranular fracture, and the grain boundaries are weakened by the formation of the creep cavitation or a void. The intergranular creep failure follows the process of nucleation, growth, interlinkage of cavities, crack propagation, and fractures. During the creep-fatigue interaction, the creep failure within the material and fatigue damage on the surface are both present. The crack path in the creep-fatigue interactions consists of both the transgranular and intergranular fractures. Once the crack is initiated by the creep-fatigue cycling, it propagates through the interior grains, and the crack path evolves from a mixed mode to intergranular as the hold time is increased. Therefore, the life prediction of creep-fatigue interaction is quite complicated due to the mixed mode of transgranular and intergranular fractures. The life-prediction methods in the creep-fatigue interaction can be categorized into four main models: the linear-damage summation model [1; 2; 3; 4], the ductility-exhaustion model [5; 6; 7], strain-range partitioning [8], and the hysteresis-energy-related damage parameter [9; 10; 11; 12; 13]. The first approach to estimate the total life in creep-fatigue interaction is the linear-damage summation method, which is an empirical model. [PERSON] [1] proposed the linear-damage summation method based on works by [PERSON] [3] and [PERSON] [4]. It is assumed that the fracture occurs when the addition of time-dependent (creep) and time-independent (fatigue) damages reaches unity. The linear-summation rule leads to conservative estimates of life, and has been widely used because it is the procedure advocated by the ASME Boiler and Pressure Vessel Code [14]. In the ductility-exhaustion model, the fatigue and creep damaging processes compete to cause failures [6].