Fragmentations of solids under impact tension

ZHENG Yuxuan; ZHOU Fenghua; HU Shisheng; YU T.X.

doi:10.6052/1000-0992-16-004

Volume 46 Issue 1

May 2016

Turn off MathJax

Article Contents

Article Navigation > Advances in Mechanics > 2016 > 46(1): 201612

ZHENG Yuxuan, ZHOU Fenghua, HU Shisheng, YU T.X.. Fragmentations of solids under impact tension[J]. Advances in Mechanics, 2016, 46(1): 201612. doi: 10.6052/1000-0992-16-004

Citation:

ZHENG Yuxuan, ZHOU Fenghua, HU Shisheng, YU T.X.. Fragmentations of solids under impact tension[J]. Advances in Mechanics, 2016, 46(1): 201612. doi: 10.6052/1000-0992-16-004

ZHENG Yuxuan, ZHOU Fenghua, HU Shisheng, YU T.X.. Fragmentations of solids under impact tension[J]. Advances in Mechanics, 2016, 46(1): 201612. doi: 10.6052/1000-0992-16-004

Citation:

ZHENG Yuxuan, ZHOU Fenghua, HU Shisheng, YU T.X.. Fragmentations of solids under impact tension[J]. Advances in Mechanics, 2016, 46(1): 201612. doi: 10.6052/1000-0992-16-004

PDF( 6752 KB)

Fragmentations of solids under impact tension

doi: 10.6052/1000-0992-16-004

1.
MOE Key Laboratory of Impact and Safety Engineering, Ningbo University, Ningbo 315211, China
2. CAS Key Laboratory for Mechanical Behavior and Design of Materials, The University of Science and Technology of China, Hefei 230027, China
3. Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, 999077, Hong Kong, China

More Information

Corresponding author: ZHOU Fenghua
Received Date: 2016-01-26
Rev Recd Date: 2016-05-03
Publish Date: 2016-05-20

Abstract

Abstract

Materials usually break into many pieces (fragments) under high strain-rate tension. Understanding the fragmentation properties of solids is important to the researchers in fields of physics, mechanics, aerospace and defense engineering. In this paper, we reviewed the researches on tensile fragmentation undertaken since 1940s. Recent advances in the fragmentation studies from theoretical analysis, experimental investigation, and numerical simulation are addressed. Some suggestions were provided for the future development in this field.
- impact tension,
- fragmentation,
- brittle and ductile fracture,
- fragment size,
- frag-ment size distribution

FullText(HTML)

References(129)

References

[1]	陈磊, 周风华, 汤铁钢. 2011. 韧性金属圆环高速膨胀碎裂过程的有限元模拟. 力学学报, 43: 861-870 (Chen L, Zhou F H, Tang T G. 2011. Finite element simulations of the high velocity expansion and fragmentation of ductile metallic rings. Acta Mechanica Sinica, 43: 861-870).
[2]	桂毓林, 孙承纬, 李强, 张光升. 2006. 实现金属环动态拉伸的电磁加载技术研究. 爆炸与冲击, 26: 481-485 (Gui Y L, Sun C W, Li Q, Zhang G S. 2006. Experimental studies on dynamic tension of metal ring by electromagnetic loading. Explosion and Shock Waves, 26: 481-485).
[3]	桂毓林, 孙承纬, 路中华, 李强, 张光升. 2007. 一维快速拉伸下无氧铜的动态断裂与破碎. 爆炸与冲击, 27: 40-44 (Gui Y L, Sun C W, Lu Z H, Li Q, Zhang G S. 2007. The dynamic fracture and fragmentation of OFHC Cu under 1-D fast tension. Explosion and Shock Waves, 27: 40-44).
[4]	郑宇轩, 周风华, 胡时胜, 余同希: 固体的冲击拉伸碎裂533
[5]	桂毓林. 2007. 电磁加载下金属膨胀环的动态断裂与碎裂研究.[博士论文]. 绵阳:中国工程物理研究院(Gui Y L. 2007. Study on dynamic fracture and fragmentation of metal expanding ring under Electromagnetic Loading. [PhD Thesis]. Mianyang: China Academy of Engineering Physics).
[6]	胡八一, 董庆东, 韩长生, 胡海波, 王德生, 高新华. 1992. 爆炸金属管的绝热剪切断裂宏观研究. 爆炸与冲击, 12: 319-325 ( Hu B Y, Dong Q D, Han C S, Hu H B, Wang D S, Gao X H. 1992. The macroscopic study of adiabatic shear fracture of metal tubes under explosive loading. Explosion and Shock Waves, 12: 319-325).
[7]	胡八一, 董庆东, 韩长生, 胡海波, 王德生. 1993. 爆炸金属管绝热剪切断裂的细观研究. 爆炸与冲击, 13: 305-312 (Hu B Y, Dong Q D, Han C S, Hu H B, Wang D S. 1993. Microscopic study of adiabatic shear fracture of metal tubes under internal explosive loading. Explosion and Shock Waves, 13: 305-312).
[8]	胡八一, 董庆东, 韩长生, 王德生, 胡海波. 1993. 内部爆轰加载下的钢管膨胀断裂研究. 爆炸与冲击, 13: 49-54 (Hu B Y, Dong Q D, Han C S, Wang D S, Hu H B. 1993. Studies of explosion and fracture of explosive-filled steel cylinders. Explosion and Shock Waves, 13: 49-54).
[9]	胡时胜, 邓德涛, 任小彬. 1998. 材料冲击拉伸实验的若干问题探讨. 实验力学, 13: 9-14 (Hu S S, Deng D T, Ren X B. 1998. A study on impact tensile test of materials. Journal of Experimental Mechanics, 13: 9-14).
[10]	汤铁钢, 谷岩, 李庆忠, 华劲松, 孙学林. 2003. 爆轰加载下金属柱壳膨胀破裂过程研究. 爆炸与冲击, 23: 529-533 (Tang T G, Gu Y, Li Q Z, Hua J S, Sun X L. 2003. Expanding fracture of steel cylinder shell by detonation driving. Explosion and Shock Waves, 23: 529-533).
[11]	汤铁钢, 桂毓林, 李庆忠, 陈永涛, 童慧峰, 刘仓理. 2010. 爆炸膨胀环实验数据处理方法讨论. 爆炸与冲击, 30: 505-510 (Tang T G, Gui Y L, Li Q Z, Cheng Y T, Tong H F, Liu C L. 2010. A discussion of data processing techniques for expanding ring tests. Explosion and Shock Waves, 30: 505-510).
[12]	汤铁钢, 李庆忠, 陈永涛, 童慧峰, 刘仓理. 2010. 爆炸膨胀环一维应力假定的分析与讨论. 爆炸与冲击, 30: 577-582 (Tang T G, Li Q Z, Cheng Y T, Tong H F, Liu C L. 2010. Discussion about one-dimensional stress presume for explosion expanding ring test. Explosion and Shock Waves, 30: 577-582).
[13]	汤铁钢, 李庆忠, 刘仓理, 桂毓林. 2010. 爆炸膨胀环的截面尺寸效应. 爆炸与冲击, 30: 39-54 (Tang T G, Li Q Z, Liu C L, Gui Y L. 2010. Size effects of expanding ring by explosive loading. Explosion and Shock Waves, 30: 39-54).
[14]	汤铁钢, 李庆忠, 孙学林, 孙占峰, 金山, 谷岩. 2006. 钢柱壳膨胀断裂的应变率效应. 爆炸与冲击, 26: 129-133 (Tang T G, Li Q Z, Sun X L, Sun Z F, Jin S, Gu Y. 2006. Strain-rate effects of expanding fracture of 45 steel cylinder shells driven by detonation. Explosion and Shock Waves, 26: 129-133).
[15]	王永刚, 周风华. 2008. 径向膨胀Al2O3 陶瓷环动态拉伸破碎的实验研究. 固体力学学报, 29: 245-249 (Wang Y G, Zhou F H. 2008. Experimental study on the dynamic tensile fragmentation of Al2O3 rings under radial expansion. Acta Mechanica Solida Sinica, 29: 245-249).
[16]	郑宇轩, 陈磊, 胡时胜, 周风华. 2013. 韧性材料冲击拉伸碎裂中的碎片尺寸分布规律. 力学学报, 45: 580-587 (Zheng Y X, Chen L, Hu S S, Zhou F H. 2013. Characteristics of fragment size distribution during the expansion of a ductile metallic ring under high strain-rate tension. Acta Mechanica Sinica, 45: 580-587).
[17]	郑宇轩, 胡时胜, 周风华. 2012. 韧性材料高应变率拉伸碎裂过程及材料参数影响. 固体力学学报, 33: 358-369 (Zheng Y X, Hu S S, Zhou F H. 2012. Effect of material properties on the fragmentation of ductile materials under high strainrate tension. Acta Mechanica Solida Sinica, 33: 358-369).
[18]	郑宇轩, 周风华, 胡时胜. 2013. 周期分布缺陷对韧性材料高应变率拉伸碎裂过程的影响. 爆炸与冲击, 33: 113-119 (Zheng Y X, Zhou F H, Hu S S. 2013. Effects of periodically-distributed defects on ductile fragmentation process of materials under high strain-rate tension. Explosion and Shock Waves, 33: 113-119).
[19]	郑宇轩, 周风华, 余同希. 2016. 动态碎裂过程中的最快速卸载现象. 中国科学(技术科学), 46: 332-338 (Zheng Y X, Zhou F H, Yu T X. 2016. The rapidest unloading in dynamic fragmentation events. Science China (Technological Sciences), 46: 332-338).
[20]	周风华, 郭丽娜, 王礼立. 2010. 脆性固体碎裂过程中的最快卸载特性. 固体力学学报, 31: 286-295 (Zhou F H, Guo L N,Wang L L. 2010. The rapidest unloading characteristics in the fragmentation process of brittle solids. Acta Mechanica Solida Sinica, 31: 286-295).
[21]	周风华, 王礼立. 2010. 脆性固体中内聚断裂点阵列的扩张行为及间隔影响. 力学学报, 42: 691-701 (Zhou F H, Wang L L. 2010. Growth of an array of cohesive crack points in brtiile solids and the influence of crack spacing. Chinese Journal of Theoretical and Applied Mechanics, 42: 691-701).
[22]	Altynova M, Hu X, Daehn G S. 1996. Increased ductility in high velocity electromagnetic ring expansion. Metallurgical and Materials Science, 27: 1837-1844.
[23]	Amery B T. 1976. Wide range velocity interferometer//Sixth Symposium on Detonation (Office of Naval Research, Dept. of the Navy, Arlington, VA, 1976), 673-681.
[24]	Barker L M, Hollenbach R E. 1972. Laser interferometer for measuring high velocities of any reflecting surface. Journal Applied Physics, 43: 4669-4675.
[25]	Barton D C. 2004. Determination of the high strain rate fracture properties of ductile materials using a combined experimental/numerical approach. International Journal of Impact Engineering, 30: 1147-1159.
[26]	Ba?ant Z P, Li Y N. 1997. Cohesive crack with rate-dependent opening and viscoelasticity: I. Mathematical model and scaling. International Journal of Fracture, 86: 247-265.
[27]	Brown W K, Wohletz K H. 1995. Derivation of the Weibull distribution based on physical principles and its connection to the Rosin-Rammler and lognormal distributions. Journal of Applied Physics, 78: 2758-2763.
[28]	Camacho G T, Ortiz M. 1996. Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures, 33: 2899-2938.
[29]	Cohen R D. 1993. Second law limitations on particle size distribution functions. Proc. R. Soc. London, 440: 611-620.
[30]	Curran D R, Seaman L, Cooper T, Shockey D A. 1993. Micromechanical model for comminution and granular flow of brittle material under high strain rate application to penetration of ceramic targets. International Journal of Impact Engineering, 13: 53-83.
[31]	Drugan W J. 2001. Dynamic fragmentation of brittle materials. Journal of the Mechanics and Physics of Solids, 49: 1181-1208.
[32]	Englman R, Rivier N, Jaeger Z. 1987. Fragment size distribution in disintegration by maximun entropy formalism. Phil. Mag. B, 56: 751-759.
[33]	Epstein B. 1947. The mathmatical description of certain breakage mechanisms leading to the logarithmico-normal distribution. J. Franklin Inst, 244: 471-477.
[34]	Espinosa H D, Zavattieri P D, Dwivedi S K. 1998. A finite deformation continuum/discrete model for the description of fragmentation and damage in brittle materials. Journal of the Mechanics and Physics of Solids, 46: 1909-1942.
[35]	Fressengeas C, Molinari A. 1985. Inertia and thermal effects on the localization of plastic flow. Acta Metallurgica, 33: 387-396.
[36]	Fressengeas C, Molinari A. 1994. Fragmentation of rapidly stretching sheets. European Journal of Mechan-ics. A. Solids, 13: 251-268.
[37]	Glenn L A, Chudnovsky A. 1986. Strain-energy effects on dynamic fragmentation. Journal Applied Physics, 59: 1379-1380.
[38]	Goto D M, Becker R, Orzechowski T J, Springer H K, Sunwoo A J, Syn C K. 2008. Investigation of the fracture and fragmentation of explosively driven rings and cylinders. International Journal of Impact Engineering, 35: 1547-1556.
[39]	GourdinWH,Weinland S L, Boling R M. 1989. Development of the electromagnetically launched expanding ring as a high-strain-rate test technique. Review of Scientific Instruments, 60: 427-432.
[40]	Gourdin W H. 1989. Analysis and assessment of electromagnetic ring expansion as a high-strain-rate test. Journal Applied Physics, 65: 411-422.
[41]	Grady D E, Benson D A. 1983. Fragmentation of metal rings by electromagnetic loading. Experimental Mechanics, 23: 393-400.
[42]	Grady D E, Hightower M M. 1992. Natural fragmentation of exploding cylinders//Meyers M A, Murr L E, Staudhammer K P, eds. Shock Wave and High-Strain Rate Phenomena in Materials. New York: Marcel Dekker, Inc. 713-721.
[43]	Grady D E, Kipp M E. 1980. Continuum modelling of explosive fracture in oil shale. International Journal of Rock Mechanics and Mining Sciences, 17: 147-157.
[44]	Grady D E, Kipp M E. 1985a. Geometric statistics and dynamic fragmentation. Journal Applied Physics, 58: 1210-1222.
[45]	Grady D E, Kipp M E. 1985b. Mechanisms of dynamic fragmentation: Factors governing fragment size. Mechanics of Materials, 4: 311-320.
[46]	Grady D E, Kipp M E. 1987. The growth of unstable thermoplastic shear with application to steady-wave shock compression in solids. Journal of the Mechanics and Physics of Solids, 35: 95-119.
[47]	Grady D E, Kipp M E. 1995. Experimental measurement of dynamic failure and fragmentation properties of metals. International Journal of Solids and Structures, 32: 2779-2781, 2783-2791.
[48]	Grady D E, Kipp M E. 1997. Fragmentation properties of metals. International Journal of Impact Engi-neering, 20: 293-308.
[49]	Grady D E, Olsen M L. 2003. A statistics and energy based theory of dynamic fragmentation. International Journal of Impact Engineering, 29: 293-306.
[50]	Grady D E. 1981. Fragmentation of solids under impulsive stress loading. Journal of Geophysical Research, 86: 1047-1054.
[51]	Grady D E. 1982. Local inertial effects in dynamic fragmentation. Journal Applied Physics, 53: 322-325.
[52]	Grady D E. 2006a. Comparison of hypervelocity fragmentation and spall experiments with Tuler-Butcher spall and fragment size criteria. International Journal of Impact Engineering, 33: 305-315.
[53]	Grady D E. 2006b. Fragmentation of Rings and Shells: The Legacy of N.F. Mott. Spring.
[54]	Grady D E. 2008. Fragment size distributions from the dynamic fragmentation of brittle solids. International
[55]	Journal of Impact Engineering, 35: 1557-1562.
[56]	Grady D E. 2010. Length scales and size distributions in dynamic fragmentation. International Journal of Fracture, 163: 85-99.
[57]	Griffith A A. 1921. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London, A221: 163-198.
[58]	Guduru P R, Freund L B. 2002. The dynamics of multiple neck formation and fragmentation in high rate extension of ductile materials. International Journal of Solids and Structures, 39: 5615-5632.
[59]	Gurney R W. 1943. The initial velocities of fragment from bombs, shell, and grenades. Army Ballistic Research Laboratory Report BRL 405.
[60]	Harding J, Wood E O, Campbell J D. 1960. Tensile testing of materials at impact rates of strain. Journal of Mechanical Engineering Science, 2: 88-96.
[61]	Hemsing W F. 1979. Velocity sensing interferometer (VISAR) modification. Review of Scientific Instru-ments, 50: 73-78.
[62]	Hild F, Denoual C, Forquin P, Brajer X. 2003. On the probabilistic-deterministic transition involved in a fragmentation process of brittle materials. Computers and Structures, 81: 1241-1253.
[63]	Hiroe T, Fujiwara K, Hata H, Takahashi H. 2008. Deformation and fragmentation behaviour of exploded metal cylinders and the effects of wall materials, configuration, explosive energy and initiated locations. International Journal of Impact Engineering, 35: 1578-1586.
[64]	Hiroe T, Fujiwara K, Hata H, Yamauchi M, Tsutsumi K, Igawa T. 2010. Explosively driven expansion and fragmentation behavior for cylinders, spheres and rings of 304 stainless steel. Materials Science Forum, 638: 1035-1040.
[65]	Hoggatt C R, Recht R F. 1968. Fracture behavior of tubular bombs. Journal of Applied Physics, 39: 1856-1862.
[66]	Hoggatt C R, Recht R F. 1969. Stress-strain data obtained at high rates using en expanding ring. Experi-mental Mechanics, 9: 441-448.
[67]	Hopkinson J. 1872. Collected Scientific Papers. Cambridge: Cambridge University Press.
[68]	Hu J W, Jin Y H, Chen D N, Wu S X, Wang H R, Ma D F. 2008. Measurement of critical impact velocity of copper in tension. Chinese Physics Letters, 25: 1049-1051. Inaoka H, Toyosawa E, Takayasu H. 1997. Aspect ratio dependence of impact fragmentation. Physical Review Letters, 78: 3455-3458.
[69]	Irwin G. 1957. Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics, 24: 361-364.
[70]	Ishii T, Matsushita M. 1992. Fragmentation of long thin glass rods. Journal of the Physical Society of Japan, 61: 3474-3477.
[71]	Johnson P C, Stein B A, Davh R S. 1963. Measurement of dynamic plastic flow properties under uniform stress//Symposium on the Dynamic Behavior of Materials: ASTM Special Publication.
[72]	Kadono T. 1997. Fragment mass distribution of platelike objects. Physical Review Letters, 78: 1444-1447.
[73]	Karman T V, Duwez P. 1950. The propagation of plastic deformation in solids. Journal of Applied Physics, 21: 987-994.
[74]	Kipp M E, Grady D E, Chen E P. 1980. Strain-rate dependent fracture initiation. International Journal of Fracture, 16: 471-478.
[75]	Kipp M E, Grady D E. 1985. Dynamic fracture growth and interaction in one dimension. Journal of the Mechanics and Physics of Solids, 33: 399-415.
[76]	Knoche P, Needleman A. 1993. The effect of size on the ductility of dynamically loaded tensile bars. Eur. J. Mech. A/Solids, 12: 585-601.
[77]	Lankford J, Blanchard C R. 1991. Fragmentation of brittle materials at high rates of loading. Journal of Materials Science, 26: 3067-3072.
[78]	Larsson M, Needleman A, Tvergaard V, Storåkers B. 1982. Instability and failure of internally pressurized ductile metal cylinders. Journal of the Mechanics and Physics of Solids, 30: 121-154.
[79]	Lienau C C. 1936. Random fracture of a brittle solid. J. Franklin Inst, 221: 485-494, 674-686, 769-787.
[80]	Maiti S, Rangaswamy K, Geubelle P H. 2005. Mesoscale analysis of dynamic fragmentation of ceramics under tension. Acta Materialia, 53: 823-834.
[81]	Marsili M, Zhang Y C. 1996. Probabilistic fragmentation and effective power law. Physical Review Letters, 77: 3577-3580.
[82]	Meibom A, Balslev I. 1996. Composite power laws in shock fragmentation. Physical Review Letters, 76: 2492-2494.
[83]	Mercier S, Granier N, Molinari A, Llorcab F, Buyb F. 2010. Multiple necking during the dynamic expansion of hemispherical metallic shells, from experiments to modelling. Journal of the Mechanics and Physics of Solids, 58: 955-982.
[84]	Miller O, Freund L B, Needleman A. 1999. Modeling and simulation of dynamic fragmentation in brittle materials. International Journal of Fracture, 96:101-125.
[85]	Mock W, Holt W H. 1983. Fragmentation behavior of armco iron and HF-1 steel explosive filled cylinders. Journal of Applied Physics, 54: 2344-2351.
[86]	Morales S A, Albrecht A B, Zhang H, Liechti K M, Ravi-Chandar K. 2011. On the dynamics of localization and fragmentation: V. Response of polymer coated Al 6061-O tubes. International Journal of Fracture, 172: 161-185.
[87]	Mott N F, Linfoot E H. 1943. A theory of fragmentation. Ministry of Supply, A.C.3348.
[88]	Mott N F. 1943. A theory of the fragmentation of shells and bombs. Ministry of Supply, A.C.4035.
[89]	Mott N F. 1947. Fragmentation of shell cases. Proc. R. Soc. London, Ser. A, 189: 300-308
[90]	Nicholas T. 1981. Tensile testing of materials at high rates of strain. Experimental Mechanics, 21: 177-185.
[91]	Nilsson K. 2001. Effects of inertia on dynamic neck formation in tensile bars. European Journal of Mechan-ics, A/Solids, 20: 713-729.
[92]	Niordson F I. 1965. A unit for testing materials at high strain rates. Experimental Mechanics, 5: 29-32.
[93]	Oddershede L, Dimon P, Bohr J. 1993. Self-organized criticality in fragmenting. Physical Review Letters, 71: 3107-3110.
[94]	Pandolfi A, Krysl P, Ortiz M. 1999. Finite element simulation of ring expansion and fragmentation. Inter-national Journal of Fracture, 95: 279-297.
[95]	Rusinek A, Zaera R. 2007. Finite element simulation of steel ring fragmentation under radial expansion. International Journal of Impact Engineering, 34: 799-822.
[96]	Seaman L, Curran D R, Shockey D A. 1976. Computational models for ductile and brittle fracture. Journal of Applied Physics, 47: 4814-4826.
[97]	Shenoy V B, Freund L B. 1999. Necking bifurcations during high strain rate extension. Journal of the Mechanics and Physics of Solids, 47: 2209-2233.
[98]	Shenoy V B, Kim K S. 2003. Disorder effects in dynamic fragmentation of brittle materials. Journal of the Mechanics and Physics of Solids, 51: 2023-2035.
[99]	Sørensen N J, Freund L B. 2000. Unstable neck formation in a ductile ring subjected to impulsive radial loading. International Journal of Solids and Structures, 37: 2265-2283.
[100]	Sturges J L, Cole B N. 2001. Flying wedge: A method for high-strain-rate tensile testing. Part 1. Reasons for its development and general description. International Journal of Impact Engineering, 25: 251-264.
[101]	Tamhane A A, Altynova M M, Daehn G S. 1996. Effect of sample size on ductility in electromagnetic ring expansion. Scripta Materialia, 34: 1345-1350.
[102]	Taylor G I. 1963. Fragmentation of tubular bombs//Scientific Papers of G. I. Taylor, Cambridge University Press.
[103]	Tvergaard V. 1990. Bifurcation in elastic-plastic tubes under internal pressure. Eur. J. Mech. A/Solids, 9: 21-35
[104]	Vogler T J, Thornhill T F, Reinhart W D, Chhabildas L C, Grady D E, Wilson L T, Hurricane O A, Sunwoo A. 2003. Fragmentation of materials in expanding tube experiments. International Journal of Impact Engineering, 29: 735-746.
[105]	Walling H C, Forrestal M J. 1973. Elastic-plastic expansion of 6061-T6 aluminum rings. AIAA Journal, 11: 1196-1197.
[106]	Walsh J M. 1984. Plastic instability and particulation in stretching metal jets. Journal Applied Physics, 56: 1997-2006.
[107]	Wang H, Ramesh K T. 2004. Dynamic strength and fragmentation of hot-pressed silicon carbide under uniaxial compression. Acta Materialia, 52: 355-367.
[108]	Warnes R H, Duffey T A, Karpp R R, Carden A E. 1980. Improved technique for determining dynamic material properties using the expanding ring. International Conference on the Meteallurgical Effects of High Strain Rate Deformation and Fabrication, Albuquerque, NM, USA.
[109]	Weibull W. 1951. A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18: 293-297.
[110]	Weimer R J, Rogers H C. 1979. Dynamic fracture phenomena in high-strength steels. Journal of Applied Physics, 50: 8025-8030.
[111]	Wesenberg D L, Sagartz M J. 1977. Dynamic fracture of 6061-T6 aluminum cylinders. Journal of Applied Mechanics, 44: 643-646.
[112]	Winter R E. 1979. Measurement of fracture strain at high strain rates. Inst. Phys. Conf. Ser., 47: 81-89.
[113]	Xu X P, Needleman A. 1994. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids, 42: 1397-1434.
[114]	Zavattieri P D, Espinosa H D. 2001. Grain level analysis of crack initiation and propagation in brittle materials. Acta Materialia, 49: 4291-4311.
[115]	Zhang H T, Liechti K M, Ravi-Chandar K. 2009. On the dynamics of localization and fragmentation-III. Effect of cladding with a polymer. International Journal of Fracture, 155: 101-118.
[116]	Zhang H T, Ravi-Chandar K. 2006. On the dynamics of necking and fragmentation-I. Real-time and post-mortem observations in Al 6061-O. International Journal of Fracture, 142: 183-217.
[117]	Zhang H T, Ravi-Chandar K. 2008. On the dynamics of necking and fragmentation-II. Effect of material properties, geometrical constraints and absolute size. International Journal of Fracture, 150: 3-36.
[118]	Zhang H T, Ravi-Chandar K. 2009. Dynamic fragmentation of ductile materials. Journal of Physics D: Applied Physics, 42: 214010.
[119]	Zhang H T, Ravi-Chandar K. 2010. On the dynamics of localization and fragmentation-IV. Expansion of Al 6061-O tubes. International Journal of Fracture, 163: 41-65.
[120]	Zheng Y X, Zhou F H, Hu S S. 2014. Dynamic fragmentation tests of LY12 aluminum alloy using a shpb based expanding ring technique. Applied Mechanics and Materials, 566: 281-285.
[121]	Zhou F H, Molinari J F, Ramesh K T. 2005a. A cohesive model based fragmentation analysis: effects of strain rate and initial defects distribution. International Journal of Solids and Structures, 42: 5181-5207.
[122]	Zhou F H, Molinari J F, Ramesh K T. 2005b. A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials. Engineering Fracture Mechanics, 72: 1383-1410.
[123]	Zhou F H, Molinari J F, Ramesh K T. 2006a. An elastic-visco-plastic analysis of ductile expanding ring. International Journal of Impact Engineering, 33: 880-891.
[124]	Zhou F H, Molinari J F, Ramesh K T. 2006b. Analysis of the brittle fragmentation of an expanding ring. Computational Materials Science, 37: 74-85.
[125]	Zhou F H, Molinari J F, Ramesh K T. 2006c. Characteristic fragment size distributions in dynamic frag-mentation. Applied Physics Letters, 88: 261918.
[126]	Zhou F H, Molinari J F, Ramesh K T. Effects of material properties on the fragmentation of brittle materials. International Journal of Fracture, 2006d, 139: 169-196.
[127]	Zhou F H, Molinari J F. 2004. Dynamic crack propagation with cohesive elements: a methodology to address mesh dependency. International Journal for Numerical Methods in Engineering, 59: 1-24.
[128]	Zhou F H, Molinari J F. 2004. Stochastic fracture of ceramics under dynamic tensile loading. International Journal of Solids and Structures, 41: 6573-6596.
[129]	Zhou F H, Wang L L. 2010. Energy transform in a brittle fragmentation process and the estimation of fragment size. Strength, Fracture and Complexity, 6: 3-16.