韧性材料的动态碎裂特性研究
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摘要
材料在冲击拉伸载荷的作用下常常会断裂成多个碎片,对材料动态碎裂机制的研究是力学、应用物理学、航空航天和兵器工程等多个领域共同关心的重要课题。关于材料的碎裂化现象,一个直观的问题是：在给定材料性能及载荷条件时,能否成功预测产生碎片的几何尺寸?而更进一步的问题则是：产生碎片的尺寸有没有统计特征?
针对韧性金属材料,Grady和Kipp将一个与断裂能相关的内聚力断裂模型引入Mott卸载波传播分析中,推导出了一个预测韧性材料拉伸碎裂过程中产生碎片的平均尺度的公式。为检验该公式的适用性,本文利用ABAQUS/Explicit动态有限元软件模拟了一维应力状态下的弹塑性金属(45号钢杆和TU1无氧铜膨胀环)在高应变率拉伸变形过程中的碎裂现象,并研究了Grady-Kipp公式中的关键参数如应变率、材料断裂能和材料密度对碎裂过程的影响。研究结果表明：Grady-Kipp公式在广泛的材料参数和应变率范围内能较好地预测碎裂过程中产生的碎片的平均尺寸。
为研究韧性碎片的分布规律,通过对具有不同初始膨胀速度的膨胀环断裂产生的碎片尺寸进行统计分析发现：1)无论初始膨胀速度如何,碎片归一化尺寸的分布具有相似性,可以用一个统一的具有初始阈值的Weibull分布来描述,这个分布还可以近似的简化为Rayleigh分布；2)碎片尺寸的累积分布曲线呈现阶梯特性,表现出较为明显的“量子化”特性。建立了一个Monte-Carlo模型,概率模拟结果表明：碎片尺寸分布的阶梯分布特性与初始颈缩间距的分布特性相关,初始颈缩间距分布所服从的Weibull分布的宽度决定了后期碎片尺寸分布的光滑程度,颈缩间距分布的跨度越小,碎片尺寸分布的“量子化”特性越明显。
在材料制造和试件加工过程中不可避免地会引入初始缺陷,这些缺陷的空间分布往往具有一定规律。为研究初始缺陷对材料高应变率碎裂过程的影响,本文采用有限元方法模拟了具有周期性几何缺陷的韧性金属圆杆在高应变率拉伸过程中的碎裂现象。模拟结果表明：1)与无初始缺陷的韧性杆件相比,具有一定幅值的初始缺陷的杆件在同等拉伸速度下发生断(碎)裂的时刻一般提前；2)初始缺陷对碎片的尺寸和大小分布具有明显影响,在一定的应变率范围内,周期性缺陷完全控制了韧性材料碎裂过程中产生碎片的个数,可称这个碎裂过程为“缺陷控制碎裂”；3)改变初始缺陷的空间间距和幅值,出现“缺陷控制碎裂”的应变率窗口将发生明显变化。并进一步讨论了具有双幅值和双周期的复合缺陷对拉伸碎裂过程的影响。
提出了一种基于分离式Hopkinson压杆(SHPB)的冲击膨胀环实验装置,为研究材料的动态拉伸碎裂问题提供了一种简单实用的加载方式。实验装置包括一个液压腔,一侧为驱动活塞,另一侧为圆环试件封闭。对活塞施加轴向冲击,利用液体体积近似不可压缩的特性,通过液压腔截面积的大比例缩小,将较低速度的对活塞冲击转化为高速的圆环试件沿径向膨胀,从而驱动试件发生拉伸变形直至断(碎)裂。利用此冲击膨胀环实验技术,获得了韧性金属环在不同速度下的碎裂结果。断口分析表明韧性金属环断口均是由周向拉伸应力拉伸断裂而成。随着子弹撞击速度增大,韧性金属的表观断裂应变呈增加的趋势,碎裂产生的碎片尺度呈减小趋势。
采用理论方法分析了刚-理想塑性材料中,多个等间距裂纹同时开动时的卸载波(Mott波)传播、以及裂纹之间材料的卸载过程。模拟分析了中间位置包含缺陷的一维弹塑性杆的高应力率拉伸下的碎裂过程,计算结果显示初始长度和碎裂时间之间关系呈U型关系,即存在一个最优裂纹间距,对应于最快的裂纹体卸载过程。而最优裂纹间距的位置均落于Grady-Kipp公式给出的韧性碎片尺寸分布附近,即Grady-Kipp等给出的韧性材料碎片尺寸恰好对应着最快速卸载原理。
数值方法模拟了二维弹塑性平板在高应变率拉伸过程中的碎裂现象,通过对碎裂过程产生碎片的平均尺度的分析来验证Grady二维韧性碎裂尺度公式的适用性。模拟结果表明,Grady推广的二维碎片尺度公式显著高估了碎片尺度,理论碎片面积值约为数值实验结果4-5倍。相对一维情况下,二维碎裂过程中两轴的拉伸载荷使得颈缩发展历程较短,材料提前断裂,从而Mott波传播距离相对较短了,进而碎片尺度更小。
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. For a given set of material, structure and loading parameters, estimating the average fragment size is a crucial issue. The further question remains on how to determine this characteristic size distribution.
By introducing a cohesive fracture model into the Mott momentum diffusion analysis, Grady and Kipp deduced a formula for predicting the average size of the fragments during a ductile fragmentation process. To quantitatively evaluate the accuracy of the Grady-Kipp formula, in this paper, we numerically simulated the fragmentation processes of an elastic-plastic bar undergoing initially uniform high strainrate tensile deformation. The key material parameters, including the fracture energy, the material density, yield stress A, and the strain-rate sensitivity C, were varied intentionally for evaluating their effects on the fragmentation process. The average fragment sizes were calculated for a wide range of the prescribed strainrates and the material parameters. It was concluded that the Grady-Kipp model provides reasonably close predictions of the lower limit of the ductile fragment sizes, though slight deviations exist in the cases when the fundamental assumptions in the Grady-Kipp analysis do not apply.
The numerical fragments obtained from the FEM simulations were collected for statistical analysis. It is found that:1) The cumulative distributions of the normalized fragment sizes at different initial expansion velocities are similar, and collectively the fragment size distributions are modeled as a Weibull distribution with an initial threshold. Approximately, this distribution can be further simplified as a Rayleigh distribution, which is the special case with the Weibull parameter to be2;2) The cumulative distribution of the fragment sizes exhibits a step-like nature, which means that the fragment sizes may be "quantized". A Monte-Carlo model is established to describe the origination of such quantization. In the model, the fractures occur at the sites where the tensioned material necks. The spacing of the necking sites follows a narrow Weibull distributions. As the fragment size is the sum of several (a random integer) necking spacing, the distributions of the fragment sizes automatically inherit the quantum properties of the random integers as long as the spacing distributions are not so wide.
Actually during the fabrication and the machining process, metallic components are inevitably brought with defects or inhomogeneities. Generally such defects or inhomogeneities have fixed geometric distributions. In this paper, we use the explicit FEM code to simulate the dynamic fragmentation processes of a thin elastic-plastic bar undergoing uniform high strain-rate tensile deformations. The thin bar is prescribed with periodical geometrical defects. Through numerical experiments, it was found that the bar with the initial defects usually broke into pieces earlier than the bar without defects. For periodically distributed defects, there exists a strain-rate region in which the fragmentation process is completely controlled by the defects. This is called the "defect controlled fragmentation" process. The spacing and the size of the defect also affect the fragmentation process, by moving the strain-rate region of the "defect controlled fragmentation". The effects of the combined defect distribution on the ductile fragmentation process are also investigated.
A new loading experimental technology was developed for conducting expanding ring tests, basing on the Split Hopkinson Pressure Bar (SHPB). The tests are useful for the studies on the dynamic tensile deformation and the fracture (fragmentation) properties of materials. The loading fixture includes a hydraulic cylinder full of incompressible fluid, which is pushed by a piston connected to the input bar. As the liquid is driven, it compresses and expands the metallic ring specimen in the radial direction. The approximately incompressible property of the liquid makes it possible to transfer a low piston-velocity to a very high radial velocity of the specimen, as the sectional areas of the cylinder narrows extremely. Using this experimental technology the ring specimens made of ductile metals were dynamically expanded and fragmentized. Results show apparent increases of the fragment numbers and the fracture strain of the specimen with the increase of the impact velocity.
One-dimensional theoretical models are established to study the unloading processes of the ductile materials due to an array of internal equally-spaced defects. By symmetry, a unit region containing one defect is considered. The Mott wave propagations and the interactions in the region were analyzed theoretically or numerically, leading to the historical curves of the average stress across the region. The critical time of fracture, defined as the time at which the average stress dropped to zero, is determined from these curves. It appears that for a prescribed strainrate, there always exists an optimum defect spacing corresponding to the rapidest unloading process. Fortunately, the optimum defect spacings for different strainrates approximate the fragment sizes given by Grady-Kipp formula.
To quantitatively evaluate the accuracy of the Grady-Kipp formula for2D fragmentation, we numerically simulated the fragmentation processes of an elastic-plastic plate undergoing initially uniform high strainrate tensile deformation. Through numerical experiments, it was found that the calculated fragment size is significantly different from Grady's theoretic estimate, which is four or five times of the numerical value.
针对韧性金属材料,Grady和Kipp将一个与断裂能相关的内聚力断裂模型引入Mott卸载波传播分析中,推导出了一个预测韧性材料拉伸碎裂过程中产生碎片的平均尺度的公式。为检验该公式的适用性,本文利用ABAQUS/Explicit动态有限元软件模拟了一维应力状态下的弹塑性金属(45号钢杆和TU1无氧铜膨胀环)在高应变率拉伸变形过程中的碎裂现象,并研究了Grady-Kipp公式中的关键参数如应变率、材料断裂能和材料密度对碎裂过程的影响。研究结果表明：Grady-Kipp公式在广泛的材料参数和应变率范围内能较好地预测碎裂过程中产生的碎片的平均尺寸。
为研究韧性碎片的分布规律,通过对具有不同初始膨胀速度的膨胀环断裂产生的碎片尺寸进行统计分析发现：1)无论初始膨胀速度如何,碎片归一化尺寸的分布具有相似性,可以用一个统一的具有初始阈值的Weibull分布来描述,这个分布还可以近似的简化为Rayleigh分布；2)碎片尺寸的累积分布曲线呈现阶梯特性,表现出较为明显的“量子化”特性。建立了一个Monte-Carlo模型,概率模拟结果表明：碎片尺寸分布的阶梯分布特性与初始颈缩间距的分布特性相关,初始颈缩间距分布所服从的Weibull分布的宽度决定了后期碎片尺寸分布的光滑程度,颈缩间距分布的跨度越小,碎片尺寸分布的“量子化”特性越明显。
在材料制造和试件加工过程中不可避免地会引入初始缺陷,这些缺陷的空间分布往往具有一定规律。为研究初始缺陷对材料高应变率碎裂过程的影响,本文采用有限元方法模拟了具有周期性几何缺陷的韧性金属圆杆在高应变率拉伸过程中的碎裂现象。模拟结果表明：1)与无初始缺陷的韧性杆件相比,具有一定幅值的初始缺陷的杆件在同等拉伸速度下发生断(碎)裂的时刻一般提前；2)初始缺陷对碎片的尺寸和大小分布具有明显影响,在一定的应变率范围内,周期性缺陷完全控制了韧性材料碎裂过程中产生碎片的个数,可称这个碎裂过程为“缺陷控制碎裂”；3)改变初始缺陷的空间间距和幅值,出现“缺陷控制碎裂”的应变率窗口将发生明显变化。并进一步讨论了具有双幅值和双周期的复合缺陷对拉伸碎裂过程的影响。
提出了一种基于分离式Hopkinson压杆(SHPB)的冲击膨胀环实验装置,为研究材料的动态拉伸碎裂问题提供了一种简单实用的加载方式。实验装置包括一个液压腔,一侧为驱动活塞,另一侧为圆环试件封闭。对活塞施加轴向冲击,利用液体体积近似不可压缩的特性,通过液压腔截面积的大比例缩小,将较低速度的对活塞冲击转化为高速的圆环试件沿径向膨胀,从而驱动试件发生拉伸变形直至断(碎)裂。利用此冲击膨胀环实验技术,获得了韧性金属环在不同速度下的碎裂结果。断口分析表明韧性金属环断口均是由周向拉伸应力拉伸断裂而成。随着子弹撞击速度增大,韧性金属的表观断裂应变呈增加的趋势,碎裂产生的碎片尺度呈减小趋势。
采用理论方法分析了刚-理想塑性材料中,多个等间距裂纹同时开动时的卸载波(Mott波)传播、以及裂纹之间材料的卸载过程。模拟分析了中间位置包含缺陷的一维弹塑性杆的高应力率拉伸下的碎裂过程,计算结果显示初始长度和碎裂时间之间关系呈U型关系,即存在一个最优裂纹间距,对应于最快的裂纹体卸载过程。而最优裂纹间距的位置均落于Grady-Kipp公式给出的韧性碎片尺寸分布附近,即Grady-Kipp等给出的韧性材料碎片尺寸恰好对应着最快速卸载原理。
数值方法模拟了二维弹塑性平板在高应变率拉伸过程中的碎裂现象,通过对碎裂过程产生碎片的平均尺度的分析来验证Grady二维韧性碎裂尺度公式的适用性。模拟结果表明,Grady推广的二维碎片尺度公式显著高估了碎片尺度,理论碎片面积值约为数值实验结果4-5倍。相对一维情况下,二维碎裂过程中两轴的拉伸载荷使得颈缩发展历程较短,材料提前断裂,从而Mott波传播距离相对较短了,进而碎片尺度更小。
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. For a given set of material, structure and loading parameters, estimating the average fragment size is a crucial issue. The further question remains on how to determine this characteristic size distribution.
By introducing a cohesive fracture model into the Mott momentum diffusion analysis, Grady and Kipp deduced a formula for predicting the average size of the fragments during a ductile fragmentation process. To quantitatively evaluate the accuracy of the Grady-Kipp formula, in this paper, we numerically simulated the fragmentation processes of an elastic-plastic bar undergoing initially uniform high strainrate tensile deformation. The key material parameters, including the fracture energy, the material density, yield stress A, and the strain-rate sensitivity C, were varied intentionally for evaluating their effects on the fragmentation process. The average fragment sizes were calculated for a wide range of the prescribed strainrates and the material parameters. It was concluded that the Grady-Kipp model provides reasonably close predictions of the lower limit of the ductile fragment sizes, though slight deviations exist in the cases when the fundamental assumptions in the Grady-Kipp analysis do not apply.
The numerical fragments obtained from the FEM simulations were collected for statistical analysis. It is found that:1) The cumulative distributions of the normalized fragment sizes at different initial expansion velocities are similar, and collectively the fragment size distributions are modeled as a Weibull distribution with an initial threshold. Approximately, this distribution can be further simplified as a Rayleigh distribution, which is the special case with the Weibull parameter to be2;2) The cumulative distribution of the fragment sizes exhibits a step-like nature, which means that the fragment sizes may be "quantized". A Monte-Carlo model is established to describe the origination of such quantization. In the model, the fractures occur at the sites where the tensioned material necks. The spacing of the necking sites follows a narrow Weibull distributions. As the fragment size is the sum of several (a random integer) necking spacing, the distributions of the fragment sizes automatically inherit the quantum properties of the random integers as long as the spacing distributions are not so wide.
Actually during the fabrication and the machining process, metallic components are inevitably brought with defects or inhomogeneities. Generally such defects or inhomogeneities have fixed geometric distributions. In this paper, we use the explicit FEM code to simulate the dynamic fragmentation processes of a thin elastic-plastic bar undergoing uniform high strain-rate tensile deformations. The thin bar is prescribed with periodical geometrical defects. Through numerical experiments, it was found that the bar with the initial defects usually broke into pieces earlier than the bar without defects. For periodically distributed defects, there exists a strain-rate region in which the fragmentation process is completely controlled by the defects. This is called the "defect controlled fragmentation" process. The spacing and the size of the defect also affect the fragmentation process, by moving the strain-rate region of the "defect controlled fragmentation". The effects of the combined defect distribution on the ductile fragmentation process are also investigated.
A new loading experimental technology was developed for conducting expanding ring tests, basing on the Split Hopkinson Pressure Bar (SHPB). The tests are useful for the studies on the dynamic tensile deformation and the fracture (fragmentation) properties of materials. The loading fixture includes a hydraulic cylinder full of incompressible fluid, which is pushed by a piston connected to the input bar. As the liquid is driven, it compresses and expands the metallic ring specimen in the radial direction. The approximately incompressible property of the liquid makes it possible to transfer a low piston-velocity to a very high radial velocity of the specimen, as the sectional areas of the cylinder narrows extremely. Using this experimental technology the ring specimens made of ductile metals were dynamically expanded and fragmentized. Results show apparent increases of the fragment numbers and the fracture strain of the specimen with the increase of the impact velocity.
One-dimensional theoretical models are established to study the unloading processes of the ductile materials due to an array of internal equally-spaced defects. By symmetry, a unit region containing one defect is considered. The Mott wave propagations and the interactions in the region were analyzed theoretically or numerically, leading to the historical curves of the average stress across the region. The critical time of fracture, defined as the time at which the average stress dropped to zero, is determined from these curves. It appears that for a prescribed strainrate, there always exists an optimum defect spacing corresponding to the rapidest unloading process. Fortunately, the optimum defect spacings for different strainrates approximate the fragment sizes given by Grady-Kipp formula.
To quantitatively evaluate the accuracy of the Grady-Kipp formula for2D fragmentation, we numerically simulated the fragmentation processes of an elastic-plastic plate undergoing initially uniform high strainrate tensile deformation. Through numerical experiments, it was found that the calculated fragment size is significantly different from Grady's theoretic estimate, which is four or five times of the numerical value.
引文
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[6]Grady D E, Benson D A. Fragmentation of metal rings by electromagnetic loading. Experimental Mechanics,1983,23(4):393-400
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[10]Zhou F, Molinari J-F, Ramesh K T. A cohesive model based fragmentation analysis: effects of strain rate and initial defects distribution. International Journal of Solids and Structures,2005,42(18-19):5181-5207
[11]周风华；,王礼立.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,2010,42(4)：691-701
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[5]桂毓林.电磁加载下金属膨胀环的动态断裂与碎裂研究[博上论文].绵阳中国工程物理研究院2007
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[9]Rusinek A, Zaera R. Finite element simulation of steel ring fragmentation under radial expansion. International Journal of Impact Engineering,2007,34(4):799-822
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[1]Mott N F. A theory of the fragmentation of shells and bombs. Ministry of Supply, AC4035, 1943,May
[2]Grady D E. Local inertial effects in dynamic fragmentation. Journal Applied Physics, 1982,53:322-325
[3]Grady D E, Benson D A. Fragmentation of metal rings by electromagnetic loading. Experimental Mechanics,1983,23(4):393-400
[4]Grady D E, Kipp M E. Geometric statistics and dynamic fragmentation. Journal Applied Physics,1985,58(3):1210-1222
[5]Grady D E, Olsen M L.A statistics and energy based theory of dynamic fragmentation. International Journal of Impact Engineering,2003,29(1-10):293-306
[6]Zhang H, Ravi-Chandar K. On the dynamics of necking and fragmentation-I. Real-time and post-mortem observations in Al 6061-O. International Journal of Fracture,2006, 142(3-4):183-217
[7]Miller O, Freund L B, Needleman A. Modeling and simulation of dynamic fragmentation in brittle materials. International Journal of Fracture,1999,96:101-125
[8]Zhou F, Molinari J-F, Ramesh K T. A cohesive model based fragmentation analysis: effects of strain rate and initial defects distribution. International Journal of Solids and Structures,2005,42(18-19):5181-5207
[9]Ishii T, Matsushita M. Fragmentation of long thin glass rods. Journal of the Physical Society of Japan,1992,61(10):3474-3477
[10]Meibom A, Balslev I. Composite power laws in shock fragmentation. Physical Review Letters,1996,76(14):2492-2494
[11]Kadono T. Fragment mass distribution of platelike objects. Physical Review Letters,1997, 78(8):1444-1447
[12]Inaoka H, Toyosawa E, Takayasu H. Aspect ratio dependence of impact fragmentation. Physical Review Letters,1997,78(18):3455-3458
[13]Oddershede L, Dimon P, Bohr J. Self-organized criticality in fragmenting. Physical Review Letters,1993,71(19):3107-3110
[14]Marsili M, Zhang Y C. Probabilistic fragmentation and effective power law. Physical Review Letters,1996,77(17):3577-3580
[15]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. Journal of Applied Physics,1995,78(4):2758-2763
[16]Mott N F. Fragmentation of shell cases. Proc. R. Soc. London, Ser. A,1947, 189(1018):300-308
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[22]陈磊,周风华,汤铁钢.韧性金属圆环高速膨胀碎裂过程的有限元模拟.力学学报,2011,43(5)：861-870
[23]Zhou F, Molinari J-F, Ramesh K T. Characteristic fragment size distributions in dynamic fragmentation. Applied Physics Letters,2006,88(26):261918
[24]郑宇轩,胡时胜,周风华.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,2012,33(4)：358-369
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