Research on the Volume and Line Fractal Dimensions of Fragments from the Explosion of Warhead Shell

doi:10.3969/j.issn.1000-1093.2018.08.006

Abstract

Abstract: The fragment characteristics and inherent features of metal shells under instantaneous explosion loading are presented. An experiment is designed to measure the fragmentation of a steel cylinder shell filled with high explosives. The expressions for the volume and line fractal dimensions of the fragments are obtained by using fractal mathematical models. The volume fractal dimension D₃, which is used to describe the mass or size distribution of the fragments, and the line fractal dimension D₁, which is used to describe the projected contour of fragments, are measured by using a MATLAB mathematical model. The results reveal that the characteristic mass distribution and morphology of the fragments are statistically self-similar and can be characterized by the fractal dimension. The line fractal dimension D₁ is 1.298 7, the volume fractal dimension D₃ is 2.385 9, and the resulting volume and line fractal dimensions meet the relationship D₃+2D₁=5. Key

Key words: warheadshell, explosivefragmentation, fractaldimension, massdistribution, projectedcontouroffragment

CLC Number:

TJ410.3⁺3

YANG Yun-chuan, ZHU Jian-jun, ZHENG Yu, LI Wen-bin, WANG Xiao-ming, QIAO Xiang-xin， LI Rui. Research on the Volume and Line Fractal Dimensions of Fragments from the Explosion of Warhead Shell[J]. Acta Armamentarii, 2018, 39(8): 1499-1506.

References

［1］ Ning J G, Yan D, Xu X Z, et al. Velocity characteristics of fragments from prismatic casing under internal explosive loading[J]. International Journal of Impact Engineering, 2017, 109:29-38.
[2］ Grisaro H Y, Dancygier A N. Spatial mass distribution of fragments striking a protective structure[J]. International Journal of Impact Engineering, 2018, 112:1-14.
[3］ Li Y, Li Y H, Wen Y Q. Radial distribution of fragment velocity of asymmetrically initiated warhead[J]. International Journal of Impact Engineering, 2017, 99:39-47.
[4］ Mott N F. A theory of the fragmentation of shells and bombs[M]∥Grady D. Fragmentation of Rings and Shells: the Legacy of N.F. Mott. Heidelberg, Berlin, Germany: Springer, 2006: 243- 294.
[5］ Brown W K, Wohletz K H. Derivation of the Weibull distribution based on physical principles and its connection to the Rosin-Rammler and lognormal distributions[J]. Journal of Applied Physics, 1995, 78(4):2758-2763.
[6］ Gilvarry J J. Fracture of brittle solids. I. Distribution function for fragment size in single fracture[J]. Journal of Applied Physics, 1961, 32(3): 391-399.
[7］ Zhou F, Molinari J F, Ramesh K T. Characteristic fragment size distributions in dynamic fragmentation[J]. Applied Physics Letters, 2006, 88(26): 261918.
[8］ Elek P, Jaramaz S. Size distribution of fragments generated by detonation of fragmenting warheads[C]∥Proceedings of the 23rd International Symposium on Ballistics. Tarragona, Spain: International Ballistics Society, 2007: 153-160.
[9］ Cohen E A. New formulas for predicting the size distribution of warhead fragments[J]. Mathematical Modelling, 1981, 2(1): 19-32.

[10］林鸿溢，李映雪. 分形论——奇异性探索[M]. 北京:北京理工大学出版社, 1992:27.
LIN Hong-yi, LI Ying-xue. Fractal theory—singularity exploration[M]. Beijing: Beijing Institute of Technology Press, 1992:27.(in Chinese)
[11］ Mandelbrot B B. The fractal geometry of nature[M]. San Francisco, CA, US: W. H. Freeman and Company, 1982.
[12］ Huang J Y, Hu S S, Xu S L, et al. Fractal crushing of granular materials under confined compression at different strain rates[J]. International Journal of Impact Engineering, 2017, 106: 259-265.
[13］ Krisnan R, Ruchi J. Association between vascular density and loss of protective RAS during early NPDR by fractal dimension[C］∥2017 ARVO Annual Meeting: Global Connections in Vision Research. Baltimore, MD, US: the Association for Research in Vision and Ophthalmology, 2017.
[14］ Zhou B, Wang J. Three-dimensional sphericity, roundness and fractal dimension of sand particles[J]. Géotechnique, 2017, 112: 207-219.
[15］ Florindo J B, Bruno O M. Closed contour fractal dimension estimation by the Fourier transform[J]. Chaos, Solitons & Fractals, 2011, 44(10): 851-861.
[16］ Logan B E. Environmental transport processes[M]. New York, NY, US: John Wiley & Sons, 1999.
[17］ Helalizadeh A, Müller-Steinhagen H, Jamialahmadi M. Application of fractal theory for characterization of crystalline deposits[J]. Chemical Engineering Science, 2006, 61: 2069-2078.
[18］章冠人. 动力破碎分形和加载的关系[J]. 高压物理学报, 1997, 11(2）:81-83.
ZHANG Guan-ren. A corelation of dynamic fragmental fractal dimension with the loading[J]. Chinese Journal of High Pressure Physics, 1997, 11(2): 81-83.(in Chinese)
[19］章冠人. 动力破碎的几何统计和分形方法[J]. 高压物理学报, 1996, 10(3): 215-219.
ZHANG Guan-ren. Geometrical statistics and fractal method for the fragment distribution of dynamic loading[J]. Chinese Journal of High Pressure Physics, 1996, 10(3): 215-219.(in Chinese)
[20］ Held M. Fragment mass distribution of HE projectile[J]. Propellants, Explosives, Pyrotechnics, 1990, 15(6): 254-260.
[21］ Strmse E, Ingebrigtsen K O. A modification of the Mott formula for prediction of the fragment size distribution[J]. Propellants, Explosives, Pyrotechnics, 1987, 12(5): 175-178.
[22］ Inaoka H, Ohno M. New universality class of impact fragmentation[J]. Fractals, 2003, 11(4): 369-376.
[23］ Yang Y C, Wang L, Feng Y. Study on fractal mathematical models of pulverizing theory for ore[J]. Powder Technology, 2016, 288(1): 354-359.
[24］卢寿慈, 沈志刚, 郑水林, 等. 粉体技术手册[M]. 北京: 化学工业出版社, 2004.
LU Shou-ci, SHEN Zhi-gang, ZHENG Shui-lin, et al. Handbook of powder technology[M]. Beijing: Chemical Industry Press, 2004.(in Chinese)
[25］孙文瑜, 徐成贤,朱德通.最优化方法[M]. 北京:高等教育出版社, 2004:75.
SUN Wen-yu, XU Cheng-xian, ZHU De-tong. Optimization method[M]. Beijing: Higher Education Press, 2004:75.(in Chinese)
[26］ Chapra S C, Canale R P. Numerical methods for engineers[M]. 7th ed. New York, NY, US: McGraw-Hill, 2015:471-478.
[27］杨虎, 刘琼荪, 钟波. 数理统计[M]. 北京:高等教育出版社, 2008:110.
YANG Hu, LIU Qiong-sun, ZHONG Bo. Mathematical statistics[M]. Beijing: Higher Education Press, 2008:110. (in Chinese)

第39卷
第8期2018 年8月兵工学报ACTA
ARMAMENTARIIVol.39No.8Aug.2018

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