## 1 _Introduction_  
Welded joints always work in high temperature, high pressure and corrosive environments, which are prone to cause cracks, fractures and explosions after long-time operation [1]. For this reason, the testing and evaluation of welded joints are very important for pressure vessels, piping and other equipment in service. Principally, it is necessary to identify the early critical state of hidden cracks because accidents caused by hidden cracks may occur at any time [2]. However, the stress concentration zones (SCZs) and early hidden cracks can not be found by traditional nondestructive testing (NDT), such as ultrasonic testing, X-ray inspect and eddy current testing, until the macroscopic crack has been formed [3, 4]. A new NDT technology, called metal magnetic memory (MMM) testing, can not only detect macroscopic cracks but also find SCZs and early hidden cracks without any magnetization devices or special treatment for the tested surface [5, 6]. Based on the self-magnetization phenomenon of ferromagnetic materials, the MMM magnetic-field intensity \(H_{p}\) is characterized by the polarity varying along the normal component \(H_{p}(y)\) and maximum value along the tangential component \(H_{p}(x)\) on SCZs [7, 8]. The appearance of MMM testing brings a great improvement in SCZs and early defect detection. [PERSON] et al. demonstrated that the natural magnetization can be applied to inspect structural inhomogeneity and residual stresses which formed during the fabrication of ferromagnetic material by the MMM method [9]. [PERSON] et al., established a magnetomechanical model based on a nonlinear constitutive relation, and analyzed the model using the finite element method quantitatively [10]. [PERSON] et al. presented a constitutive modelling for ferromagnetic materials under multiaxial multifield loading with different boundary conditions, which can predict magnetization, strain and stress [11]. [PERSON] et al. established a damage model represented by MMM characteristics through fatigue experiments of three point bending and developed a fatigue life prediction method based on the relationship of damage parameter and normalized life [12]. [PERSON] evaluated the contact damage of ferromagnetic materials under nonferromagnetic and ferromagnetic indenters by measuring the signals of the magnetic flux leakage [13]. [PERSON] et al. studied the MMM characteristics of buried defects under different load stages and found the strength of the magnetic field and its gradient decrease with the increasing of the burial depth [14]. [PERSON] proposed the contour plot method to visually analyze the size of defects and established a mathematical model of non-uniform magnetic charge for pipeline leak detection. The peak-to-valley spacing and peak-to-valley values in the contour plot of MFL signal can directly reflect the location and size of defects, and the signal eigenvalues follow the trend of first-order decreasing exponential function, and the first-order derivatives of signal eigenvalues show a trend of first decreasing and then increasing with the decrease of mesh size, and the extreme point of the derivative curve is the best mesh size [15; 16]. By comparing with the Standard JB4730-2005 of X-ray detection, the authors set up the MMM reliability model and investigated MMM signal characteristics of SCZs for welded joints [17; 18; 19]. However, the MMM signals of welded joints often take on fuzzy and uncertainty due to the interruption of welding residual stress [20; 21]. This brings the bottleneck of quantitative identification of weld defect levels with MMM technology, especially between the weld residual stress concentration and the early hidden damage. In order to overcome the bottleneck, a modified MLE MMM quantitative identifying model is proposed based on D-S theory from the perspective of mathematical statistics and information fusion. First, the welded specimens made of Q235B steel are tested with MMM method and synchronous X-ray detection for comparison under the fatigue tension loading.