# A Point Crack Source Location Method without Velocity Information in Anisotropic Plates
## 1 Introduction  
Acoustic emission (AE) is a phenomenon in which an elastic wave is generated when microcracks start forming before a material is destroyed, during the process of small deformations, or when a material is destroyed. The method for detecting and evaluating this elastic wave with sensors is called AE testing [1]. The purpose of AE testing is to diagnose the tendency of the occurrence of AE, to find the location of the defect in the material, and to know what is happening at the source. AE testing can be applied to various materials, such as concrete [2, 3], still [4, 5], and fiber-reinforced plastics [6], depending on several application areas [7]. Finding the crack location or acoustic source location (ASL) of a subject using AE is valuable both for locating defects during safety monitoring and for fundamental laboratory studies of fractures. To locate a crack, multiple AE sensors are used to measure the AE signals from a single acoustic source. There are two ways to find a crack's location: using a measured signal waveform and using a signal parameter such as the signal arrival time, or the so-called time of arrival (TOA). Time reverse modeling (TRM) is a location method [8, 9] which uses the entire measured AE signal waveform. The TRM method uses the measured AE signal as an input source to solve the elastic equation and provide the location of the crack and the characteristics of the crack. This method has disadvantages in that it is necessary to know the properties of the object and to solve the elastic wave equation. To find a crack's location using the TOA, many researchers have developed ASL methods for isotropic and anisotropic objects. The ASL methods widely used in isotropic media include the triangular position representation method [10] and the simplex method [11], and several other methods can be found in [12, 13]. In anisotropic materials, since the elastic wave speed is different depending on the propagation direction [14], it is necessary to develop an ASL method in consideration of this [6, 15]. For the anisotropic materials, the ASL techniques are studied based on the optimization technique [16; 17; 18]. These methods estimate the locations of cracks using the time measured by the sensor and the propagation speed for each angle of the anisotropic material. There have been many studies to develop an ASL method for the case where the propagation speed of an anisotropic medium is unknown [19; 20; 21; 22]. In order to avoid using the propagation speed in anisotropic media, the prior research's approaches require special sensor placements, namely the L-shaped sensor clusters in [19; 20; 23], Z-shaped sensor clusters in [21], and square-shaped sensor clusters in [22]. In this paper, we propose an ASL method using only the TOA in anisotropic plates. This method allows for arbitrary placement of the sensor and includes the TOA errors. The proposed ASL method has an iterative scheme to minimize the nonlinear least squares. The objective function of the least squares is generated by assuming that the propagation velocity of an elastic wave in an anisotropic material is constant and based on the ellipse equation [16; 17], which is modified so that the unknown velocity term is erased. In addition, we add the TOA error correction constant to the objective function and apply a well-known Tikhonov regularization method [24] for stable solving of the optimization problem. We call the proposed ASL method the speed deviation minimization method (SDMM), because it is designed to minimize the difference in speed measured by each sensor. The result of applying the SDMM is the estimated crack position of the material, the crack start time, and the wave propagation velocity, of which the estimated crack position is used as the final result. Many experiments were conducted to evaluate the accuracy and robustness of the crack location by the SDMM. As a result of the experiment, it has been confirmed that it has an accuracy within 5% and works stably against noise. This paper is organized as follows.