## 1 Introduction  
The austenitic stainless steel 304 is suitable for large field applications, such as heat exchangers, power plants, oil and gas industry, chemical engineering and especially in nuclear power plants, because of its useful characteristics, such as high temperature service and environment, corrosion resistance, weldability, formability and mechanical properties [1, 2]. Pipes are exposed to a variety of environmental influences, and high temperature and high pressure that cause severe corrosive and environmental deterioration as results in fatigue cracks and flaws in pipes that can appear and grow. A pipe failure can lead to serious ecological disasters, human casualties and financial loss. To predict and avoid such threats and maintain the safety and integrity of pipes, periodic nondestructive testing inspections are necessary [3, 4, 5]. Eddy-current testing (ECT) is adapted to solve such problems. Applied to inspect conductive and ferromagnetic devices to examine their structural integrity [6, 7, 8, 9], it has certain advantages in terms of safety as a testing tool--rapid inspection, high sensitivity, and minimizing contact with the specimen. It is efficiently associated with automatic detection in various work [10, 11]. The ECT technique is frequently used for nondestructive defect inspection of tubes welders and circumferential welds [5, 12, 13]. Many welding processes are presented in several papers; they require a lot of effort and experiments to better understand the parameters [14, 15, 16]. The finite element method associated with experimental investigations is a powerful tool to identify and analyze welding parameters and to obtain an optimal solution in a short time [17]. Usually, in simulation work, the mathematical model of the problem to be treated contains all the necessary information. In this work, a study of weld circumferential thin cracks in AISI 304 pipe was carried out. The used AISI 304 pipe was joined by gas tungsten arc welding (GTAW). Several previous studies focused on crack detection and characterization of cracks in austenitic stainless steel, for instance, detection and evaluation of weld defects using 3-dimension tunnel, and show excellent inspection results for a weld in stainless steel [18]. [PERSON] et al. [19] described weld defects and evaluation of weld quality and how to achieve weld quality improvement. [PERSON] et al. [20] investigated cracking behavior of AISI 304 exposed to high temperature and revealed that strain-induced martensitic transformation in stainless steel has a negative effect on stress corrosion effect. [PERSON] et al. [21] studied microstructure residual stress and corrosion cracking of repair welding on 304 AISI by experiment and simulation, and found that repair welding in 304 stainless steel is recommended no more than two times. All these studies are related to the evaluation of defects in using different methods. However, there are few considerations of delta-ferrite structures, which show ferromagnetic properties, inspite of that the austenitic stainless steel AISI 304 around the welding for analyzing numerical and experimental studies. AISI 304 is classified as paramagnetic material with \(\mu_{r}=1+\mathcal{X}_{m}\), where relative magnetic permeability \(\mu_{r}\) and the magnetic susceptibility \(\mathcal{X}_{m}\), are 1 and 0, respectively. However, after heat treatment (heat welding), the property of AISI 304 as paramagnetic material will disappear and it will become a partial ferromagnetic material named delta-ferrite structure, which results in the change of the relative magnetic permeability at each region \(\mu_{r}\
eq 1\),with high permeability in the weld area, low permeability in the raw material \(\mu_{r}=1\) and decreasing permeability in the HAZ \(\mu_{r}=3\sim 1\)[22; 23; 24]. Therefore, it can be considered that the assembled pipe has three regions with three magnetic permeabilities.