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Ferromagnetic structures such as pipes or vessels are widely used in petroleum, chemical and power generation industries. Periodical nondestructive testing (NDT) is vital for continued safe operation. As a NDT technology, pulsed eddy current testing (PECT) technology which is excited by a square-wave pulse rather than a sinusoidal waveform has been widely used for its advantages of non-contact and acquisition of information at various depths in one excitation process. In PECT, the analytical modeling is important because it gives a better understanding of the signal and benefits the inverse process of PECT in extracting information of structures. The foundation of theoretical model of PECT is the Dodd-Deeds model presented by Dodd and Deeds in 1968, Theodoulidis and Kriezis represented the integral solution of Dodd-Deeds model in the form of series by using the truncated region eigenfunction expansion (TREE) method. Using the Dodd-Deeds model and the TREE method, other analytical modelings have been solved. However, most modelings assume that the wall thinning of the specimen is uniform, and the analytical solution only contains the variables in the z direction (the direction perpendicular to the surface of the specimen), such as the thickness of the specimen. With the rapid development of PECT, problems such as the footprint of the probe, the quantitative analysis of local wall thinning also need to be solved. These problems are related to the variable in the r direction (the direction parallel to the surface of the specimen), so the analytical modelings mentioned above are not available any more. To solve these problems, the analytical modeling of the plate with a flat-bottom hole is proposed. Considering the fact that the boundary condition in the analytical modeling of the plate with a flat-bottom hole is complicated, the assumption that the transverse wave number and the longitudinal wave number in the layer where the flat-bottom hole located are the same is made in this paper, and the transverse wave number is set to be only related to the structure in the r direction. Firstly, the expressions of magnetic vector potential in all the layers are obtained by using the reflection and refraction theory of electromagnetic wave. Then the analytical solution is solved based on the extended Cheng's matrix method by introducing the construction coefficient Wn. Finally, the 16MnR specimen with the flat bottom holes is conducted as an example, and experiments are carried out. The good agreement between results calculated by the analytical model and the experimental results measured verifies the developed analytical model.
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Keywords:
- pulsed eddy current testing /
- reflection and transmission theory /
- flat-bottom hole /
- analytical modeling
[1] Fan M B, Yin Y D, Cao B H 2012 Acta Phys. Sin. 61 088105 (in Chinese)[范孟豹, 尹亚丹, 曹丙花2012物理学报61 088105]
[2] Fu J J, Lei Y Z 2016 J. Sci. Instrum. 37 617 (in Chinese)[付剑津, 雷银照2016仪器仪表学报37 617]
[3] Yang L J, Su J M, Gao S W, Liu B 2016 NDT 40 10 (in Chinese)[杨理践, 孙靖萌, 高松巍, 刘斌2016无损探伤40 10]
[4] Kang X W, Fu Y W 2011 Nondestr. Test. 33 40 (in Chinese)[康小伟, 付跃文2011无损检测33 40]
[5] Wu X J, Zhang Q, Shen G T 2016 J. Sci. Instrum. 37 1698 (in Chinese)[武新军, 张卿, 沈功田2016仪器仪表学报37 1698]
[6] Dodd C V, Deeds W E 1968 J. Appl. Phys. 39 2829
[7] Theodoulidis T P, Kriezis E E 2005 J. Mater. Process. Technol. 161 343
[8] Theodoulidis T P, Kriezis E E 2006 Eddy Current Canonical Problems (with applications to nondestructive evaluation) (Forsyth:Tech Science Press) pp93-135
[9] Fan M B, Huang P J, Ye B, Hou D B, Zhang G X, Zhou Z K 2009 Acta Phys. Sin. 58 5950 (in Chinese)[范孟豹, 黄平捷, 叶波, 侯迪波, 张光新, 周泽魁2009物理学报58 5950]
[10] Chen X L, Lei Y Z 2015 Chin. Phys. B 24 030301
[11] Xu Z Y, Wu X J, Li J, Kang Y H 2012 NDT & E Int. 51 24
[12] Tian G Y, Li Y, Mandache C 2009 IEEE Trans. Magn. 45 184
[13] Li J, Wu X J, Zhang Q, Sun P F 2015 NDT & E Int. 75 57
[14] Fu F, Bowler J 2006 IEEE Trans. Magn. 42 2029
[15] Xu Z Y 2012 Ph. D. Dissertation (Wuhan:Huazhong University of Science and Technology) (in Chinese)[徐志远2012博士学位论文(武汉:华中科技大学)]
[16] Cheng W, Komura I 2012 Proceedings of the 9th International Conference on NDE in Relation to Structural Integrity for Nuclear and Pressurized Components Seattle, USA, May 22-24, 2012 p336
[17] Wang J, Teng Y P, Fu Y G, Sun M X, Liu Z B, Fan Z Y, Shi K 2013 Nondestr. Test. 35 54 (in Chinese)[王健, 滕永平, 傅迎光, 孙明璇, 刘再斌, 范智勇, 石坤2013无损检测35 54]
[18] Xie S, Chen Z, Takagi T, Uchimoto T 2012 NDT & E Int. 51 45
[19] Bowler J R, Theodoulidis T P 2006 J. Phys. D:Appl. Phys. 39 2862
[20] Theodoulidis T P, Bowler J R 2010 IEEE Trans. Magn. 46 1034
[21] Theodoulidis T P, Bowler J R 2005 Rev. Prog. Quantit. Nondestr. Eval. 24 403
[22] Zhang Q, Wu X J, Li J, Sun P F 2014 Proceedings of the 19th International Workshop on Electromagnetic Nondestructive Evaluation Xi'an, China, June 23-28, 2014 p95
[23] Cheng C C, Dodd C V, Deeds W E 1971 Int. J. Nondestr. Test. 3 109
[24] Feng C Z, Ma X K 2000 An Introduction to Engineering Electromagnetic Fields (Beijing:Higher Education Press) p228(in Chinese)[冯慈璋, 马西奎2000工程电磁场导论(北京:高等教育出版社)第228页]
[25] Yang Z 2009 Ph. D. Dissertation (Shandong:China University of Petroleum) (in Chinese)[杨震2009博士学位论文(山东:中国石油大学)]
[26] Xu Z Y, Wu X J, Huang C, Kang Y H 2011 J. Huazhong Univ. Sci. Techn (Nat. Sci. Ed.) 39 91(in Chinese)[徐志远, 武新军, 黄琛, 康宜华2011华中科技大学学报(自然科学版) 39 91]
[27] Xie M X, Guo J Z, Zhang H B, Chen K 2010 Computer Eng. Sci. 32 92(in Chinese)[谢明霞, 郭建忠, 张海波, 陈科2010计算机工程与科学32 92]
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[1] Fan M B, Yin Y D, Cao B H 2012 Acta Phys. Sin. 61 088105 (in Chinese)[范孟豹, 尹亚丹, 曹丙花2012物理学报61 088105]
[2] Fu J J, Lei Y Z 2016 J. Sci. Instrum. 37 617 (in Chinese)[付剑津, 雷银照2016仪器仪表学报37 617]
[3] Yang L J, Su J M, Gao S W, Liu B 2016 NDT 40 10 (in Chinese)[杨理践, 孙靖萌, 高松巍, 刘斌2016无损探伤40 10]
[4] Kang X W, Fu Y W 2011 Nondestr. Test. 33 40 (in Chinese)[康小伟, 付跃文2011无损检测33 40]
[5] Wu X J, Zhang Q, Shen G T 2016 J. Sci. Instrum. 37 1698 (in Chinese)[武新军, 张卿, 沈功田2016仪器仪表学报37 1698]
[6] Dodd C V, Deeds W E 1968 J. Appl. Phys. 39 2829
[7] Theodoulidis T P, Kriezis E E 2005 J. Mater. Process. Technol. 161 343
[8] Theodoulidis T P, Kriezis E E 2006 Eddy Current Canonical Problems (with applications to nondestructive evaluation) (Forsyth:Tech Science Press) pp93-135
[9] Fan M B, Huang P J, Ye B, Hou D B, Zhang G X, Zhou Z K 2009 Acta Phys. Sin. 58 5950 (in Chinese)[范孟豹, 黄平捷, 叶波, 侯迪波, 张光新, 周泽魁2009物理学报58 5950]
[10] Chen X L, Lei Y Z 2015 Chin. Phys. B 24 030301
[11] Xu Z Y, Wu X J, Li J, Kang Y H 2012 NDT & E Int. 51 24
[12] Tian G Y, Li Y, Mandache C 2009 IEEE Trans. Magn. 45 184
[13] Li J, Wu X J, Zhang Q, Sun P F 2015 NDT & E Int. 75 57
[14] Fu F, Bowler J 2006 IEEE Trans. Magn. 42 2029
[15] Xu Z Y 2012 Ph. D. Dissertation (Wuhan:Huazhong University of Science and Technology) (in Chinese)[徐志远2012博士学位论文(武汉:华中科技大学)]
[16] Cheng W, Komura I 2012 Proceedings of the 9th International Conference on NDE in Relation to Structural Integrity for Nuclear and Pressurized Components Seattle, USA, May 22-24, 2012 p336
[17] Wang J, Teng Y P, Fu Y G, Sun M X, Liu Z B, Fan Z Y, Shi K 2013 Nondestr. Test. 35 54 (in Chinese)[王健, 滕永平, 傅迎光, 孙明璇, 刘再斌, 范智勇, 石坤2013无损检测35 54]
[18] Xie S, Chen Z, Takagi T, Uchimoto T 2012 NDT & E Int. 51 45
[19] Bowler J R, Theodoulidis T P 2006 J. Phys. D:Appl. Phys. 39 2862
[20] Theodoulidis T P, Bowler J R 2010 IEEE Trans. Magn. 46 1034
[21] Theodoulidis T P, Bowler J R 2005 Rev. Prog. Quantit. Nondestr. Eval. 24 403
[22] Zhang Q, Wu X J, Li J, Sun P F 2014 Proceedings of the 19th International Workshop on Electromagnetic Nondestructive Evaluation Xi'an, China, June 23-28, 2014 p95
[23] Cheng C C, Dodd C V, Deeds W E 1971 Int. J. Nondestr. Test. 3 109
[24] Feng C Z, Ma X K 2000 An Introduction to Engineering Electromagnetic Fields (Beijing:Higher Education Press) p228(in Chinese)[冯慈璋, 马西奎2000工程电磁场导论(北京:高等教育出版社)第228页]
[25] Yang Z 2009 Ph. D. Dissertation (Shandong:China University of Petroleum) (in Chinese)[杨震2009博士学位论文(山东:中国石油大学)]
[26] Xu Z Y, Wu X J, Huang C, Kang Y H 2011 J. Huazhong Univ. Sci. Techn (Nat. Sci. Ed.) 39 91(in Chinese)[徐志远, 武新军, 黄琛, 康宜华2011华中科技大学学报(自然科学版) 39 91]
[27] Xie M X, Guo J Z, Zhang H B, Chen K 2010 Computer Eng. Sci. 32 92(in Chinese)[谢明霞, 郭建忠, 张海波, 陈科2010计算机工程与科学32 92]
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