# Study on the Residual Strength of Nonlinear Fatigue-Damaged Pipeline Structures
## 1 Introduction  
The pipeline is the only channel for underground oil reservoirs and surface oil and gas exploitation and transportation. During the service process of the pipeline, it is often accompanied by cyclic plastic deformation, which is characterized by alternating elastic-plastic behavior and is a typical low-cycle fatigue process [1]. Low-cycle fatigue runs through the entire service life cycle of materials. Low-strength, soft metals tend to cyclic harden in low-cycle fatigue; however, high-strength metals generally tend to cyclic soften during low-cycle fatigue, resulting in continuous deterioration of material properties. The evolution of material properties during service plays a decisive role in the service safety of pipelines [2]. During the service of circulating steam huff and puff thermal recovery well casing, the casing material is under the service conditions of accumulated plastic deformation and strain fatigue, and necking and fracture failures often occur [3; 4; 5]. The traditional design method for casing strings primarily focuses on strength-based checks based on the safety factor method, ensuring that the load applied to the casing string remains within its strength limits and within the elastic range [6; 7; 8]. The evolution process of the residual strength of the material under fatigue damage is not considered, and the failure risk is significantly increased. The reason for the failure of the strength check is that the prediction or calculation of the residual strength of the material is not accurate. Therefore, the study of residual strength is of great significance for the safe design of pipeline services. The nonlinear damage to pipelines is caused by the nonuniform deformation of the cross-section [9]. During a deformation process, different deformations occur at different positions in the vertical direction of the pipe section, resulting in different damage degrees for each part. According to the assumption of the neutral layer during deformation [10], during the repeated bending process of the pipe, there is no deformation in the neutral layer, the degree of deformation increases from the neutral layer to the outermost edge, and the deformation of the outermost edge is the largest. Therefore, there is no damage to the neutral layer. The further the distance between the other layers of the cross-section and the neutral layer, the greater the damage, and the outermost edge damage of the pipe is the most serious. After structural damage, the residual strength of the neutral layer is still the initial strength of the material, and the residual strength of the outermost edge is the lowest. As the number of alternate deformations increases, the outermost edge of the pipe first begins to yield and initiate plastic deformation to initiate cracks, and then gradually expands to the neutral layer until the pipe body completely enters the plastic stage and finally breaks. To solve the problem of difficult prediction of nonlinear fatigue accumulation, a number of theories and methods have been proposed to model the fatigue damage process of engineering structures/components [11; 12; 13; 14; 15; 16; 17; 18]. Linear damage theory is commonly used in engineering to describe the fatigue of metal materials [19], for example, [PERSON]'s rule. However, it is only an approximate and empirical one, and its use can result in significant errors, especially the inability to reflect the influence of the order of load action, thus having significant limitations [20; 21]. The Corten-Dolan nonlinear damage accumulation model can well describe the fatigue behavior of engineering materials and structures under variable amplitude loads, while the determination of parameter d is difficult and still open [22; 23; 24]. [PERSON] and [PERSON] [25] used a phenomenological technique to establish a reversal-by-reversal cumulative damage rule for predicting the strain life under variable amplitude loading (VAL).