Thermography spatial-transient-stage mathematical tensor construction and material property variation track Bin Gao a , Aijun Yin b, * , Guiyun Tian a, c , W.L. Woo c a School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, China b The State Key Laboratory of Mechanical Transmission, College of Mechanical Engineering, Chongqing University, Chongqing 400044, China c School of Electrical and Electronic Engineering, Newcastle University, UK article info Article history: Received 17 December 2013 Received in revised form 16 June 2014 Accepted 16 June 2014 Available online Keywords: Thermal analysis Tensor mathematical model Non-destructive testing and evaluation Tucker decomposition Material properties variation tracking Gear fatigue evaluation abstract Characterizing and tracking the properties variation in conductive material such as electrical conduc- tivity, magnetic permeability and thermal conductivity have promising potential for the detection and evaluation of material state undertaken by fatigue or residual stress. This is a challenge task for the research field of non-destructive testing and evaluation. This paper proposes a spatial-transient-stage tensor mathematical model of inductive thermography system and Tucker decomposition algorithm for characterizing and tracking the variation of properties. The inductive thermography has advantages in such as rapid inspection and high sensitivity of defect detection. The links between mathematical and physics models have been discussed. The simulation experiments of tracking physic properties of steel material are investigated and verified. In addition, the real experiment of the measurement for gears with different cycles of fatigue tests is evaluated. The estimation of normalized stage basis by using Tucker decomposition has shown high correlation relationships with different variation of physics properties in material. © 2014 Elsevier Masson SAS. All rights reserved. 1. Introduction Non-destructive testing and evaluation (NDT&E) is a wide group of analysis technique used in science and industry to eval- uate the properties of material, component or system without causing damage [1e4]. Infrared thermography methods have reached a prominent status as an NDT&E method [5e9] with the advantages of being fast, and providing non-contact, non-inter- action, real-time measurements over a large detection area with a long range, security of personnel, relatively easy interpretation of results. Infrared thermography can be used to assess and predict the structure or behavior beneath the surface by measuring the distribution of infrared radiation and converting the measure- ments into a temperature scale. Infrared thermography is gener- ally divided into two main streams: passive infrared thermography (PIT) and active infrared thermography (AIT). Pas- sive Thermography (PIT) is defined as measuring the temperature differences between the target materials and the surroundings under different ambient temperature conditions. AIT [10] was developed to provide more accurate information by considering the amount of thermal radiation and heat transfer. The common thermal stimulation techniques in AIT [11] are: pulsed thermog- raphy (e.g., flash thermography), step heating (long pulse), lock-in thermography, and vibrothermography (e.g., ultrasonic IR ther- mography). Traditional thermal radiation heating is the earliest and the simplest direct technique in AIT. The method employs a lighting source or radiative source to heat the surface of the test object. Flaws or suspicious response can be captured, according to the slow heat transfer, by an infrared (IR) camera inspecting system. The eddy current pulsed thermography (ECPT) is an emerging inductive infrared thermography for conductive material NDT&E method with an increasing span of applications [12]. Comparing with other thermography NDT&E methods, the heat of ECPT is not limited to the sample surface, rather it can reach a certain depth, which governed by the skin depth of eddy current. Furthermore, ECPT focus the heat on the defect due to friction or eddy current distortion, which increases the temperature contrast between the defective region and defect-free areas. During the testing, a high- current electromagnetic pulse induces eddy current on the * Corresponding author. E-mail addresses: bin_gao@uestc.edu.cn (B. Gao), aijun.yin@cqu.edu.cn (A. Yin). Contents lists available at ScienceDirect International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts http://dx.doi.org/10.1016/j.ijthermalsci.2014.06.018 1290-0729/© 2014 Elsevier Masson SAS. All rights reserved. International Journal of Thermal Sciences 85 (2014) 112e122 Author´s Personal Copy