###### Abstract  
Special geometric features and complex working environments render the internal defects of small-diameter thick-walled steel pipes \"easy to expand, difficult to detect\". In this paper, a magnetic permeability perturbation testing (MPPT) method is proposed to assess the internal axial cracks of small-diameter thick-walled steel pipes. The mechanism of the MPPT method and its corresponding probe and magnetizer are introduced, and its feasibility is verified through a series of simulations and experiments. Experiments conducted using different sizes of small-diameter thick-walled pipes show that this method offers good performance with respect to the detection of internal axial cracks. Additionally, both diameter and wall thickness significantly affect the MPPT signal. To a certain extent, a greater wall thickness or a smaller diameter brings about a weaker signal. This method does not benefit from the lift-off effect, nor is it limited by the skin effect, which has great practical value as a supplement to the evaluation of thick-walled steel pipes.  
magnetic permeability perturbation testing; small-diameter; thick-walled steel pipes; internal axial cracks +
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affected by the \"magnetic shielding effect\" of the pipe wall, and its detection sensitivity gradually decreases with the increase in pipe wall thickness [17; 18; 19; 20; 21; 22; 23]. Electromagnetic acoustic transducers (EMATs) are suitable for detecting wall-thinning defects, but they are limited by a low efficiency of energy transfer [24; 25]. TTBT detects leaks in pipelines through pressure changes and offers high sensitivity, but it is susceptible to environmental changes such as temperature changes and changes in the medium inside the pipeline [26; 27].  
Ferromagnetic materials exhibit significant non-linear magnetization characteristics under an external magnetic field. In our previous study, we found that ferromagnetic materials inevitably generate permeability perturbation in the DC magnetization state [28]. Based on the mechanism of magnetization or permeability change of ferromagnetic materials, [PERSON] achieved the detection of internal wall defects through the co-excitation of DC and AC for differential permeability testing [29]. [PERSON] designed an improved magnetizer consisting of a DC coil and a U-shaped permanent magnet [30]. [PERSON] used DC-magnetization-based eddy current array testing to quantify the inner surface defects of steel plates [31]. [PERSON] proposed a method of scanning induction thermal imaging systems using a DC bias, which relies on effective permeability changes [32].  
This paper proposes an MPPT method for the detection of axial cracks in the internal walls of small-diameter thick-walled pipes. Section 1 introduces the MPPT mechanism and the method for detecting axial cracks in the internal wall of the steel pipe. Section 2 investigates the effect of the diameter of a pipe on the distribution characteristics of a steel pipe's magnetic permeability perturbation (MPP) and the signal amplitude. Section 3 verifies the feasibility of the MPPT method through a series of experiments and analyzes the effects of pipe diameter on the detection signal. Section 4 discusses some details of the detection method. The conclusions and prospects for future work are summarized in Section 5.