Effect of rock shapes on brittle fracture using Smoothed Particle Hydrodynamics

Breakage of rocks or particulates plays a major role in various industries, such as mineral and ore processing. Many of the processes used for fracturing materials in these industries have the requirement to produce specified size and/or shape of the products. Numerical modelling can assist in understanding and predicting complex fracture processes, and can be used in designing the equipment and setting the process parameters to ensure desired product quality. In this paper, a mesh-free numerical method, called Smoothed Particle Hydrodynamics (SPH), is extended to predict impact fracture of rocks. SPH is a particle based Lagrangian method which is particularly suited to the analysis of fracture due to its capacity to model large deformation and track the free surfaces generated. A continuum damage model is used to predict the fracture of rocks. Evolution of damage is predicted using the strain history of each particle. Damage inhibits the transmission of tensile stress between particles, and once it reaches unity, the particle is unable to transmit tensile stress, resulting in a macro-crack. Connected macro-cracks lead to complete fragmentation.Firstly, an Unconfined Compressive Strength (UCS) test under uniaxial compression of a rock sample is modelled using SPH and compared against experiments to validate the capability of SPH for prediction of fracture in rocks. The SPH prediction matched the well-known experimentally observed diagonal fracture pattern. SPH is subsequently used to simulate brittle fracture of rocks during impact. Rock specimens of different shapes are examined to determine the effects of shape on both the fracture pattern and the energy dissipation during impact fracture. Rock shape is found to have considerable influence on the fracture process, fragment sizes, energy dissipation, and post-fracture motion of the fragments.     

Impact failure of granular materials – Non-equilibrium multiscale simulations and high-speed experiments

We present results of high-speed impact experiments on aluminum oxide (Al₂O₃) and silicon carbide (SiC) ceramics and propose a mesoscopic way to model the fracture behavior of these brittle materials based on discrete particles. The two-dimensional model used here has only three adjustable parameters, but is able of reproducing many salient features of the investigated ceramics under compressive, tensile and shock impact load. We discuss our particle model in detail and then consider strain and shear load simulations. In particular, we model explicitly the macroscopic experimental set-up of the edge-on impact experiment and show that the experimentally observed crack patterns can in principle be explained by the random distribution of particle overlaps and the thereby generated differences in the local strength of the material.     

Analysis of shock wave reflection from fixed and moving boundaries using a stabilized particle method

In the present paper, the efficiency of an enhanced formulation of the stabilized corrective smoothed particle method （CSPM） for simulation of shock wave propagation and reflection from fixed and moving solid boundaries in compressible fluids is investigated. The Lagrangian nature and its accuracy for imposing the boundary conditions are the two main reasons for adoption of CSPM. The governing equations are further modified for imposition of moving solid boundary conditions. In addition to the traditional artificial viscosity, which can remove numerically induced abnormal jumps in the field values, a velocity field smoothing technique is introduced as an efficient method for stabilizing the solution. The method has been implemented for one- and two-dimensional shock wave propagation and reflection from fixed and moving boundaries and the results have been compared with other available solutions. The method has also been adopted for simulation of shock wave propagation and reflection from infinite and finite solid boundaries.     

不同撞击速度下穿燃弹侵彻陶瓷/铝合金复合靶板时弹芯破碎失效特性研究

为了研究12.7 mm穿燃弹以不同速度撞击陶瓷/铝合金复合靶板时弹芯的破碎及失效特性,开展了12.7 mm穿燃弹以434.5～844.6 m/s速度撞击SiC陶瓷/6061T6铝合金复合靶板的弹道试验,分析了弹靶的失效模式。弹芯在侵彻靶板后会产生不同尺寸的碎片,使用回收箱收集弹芯碎片并用不同孔径筛网对其进行筛分、称重,得到了不同撞击速度下弹芯碎片的质量分布,并对不同部位的弹芯碎片断口形貌进行了宏观和微观观测分析。研究结果表明:背板失效模式为碟形变形-剪切穿孔-花瓣形失效,试验后的弹芯碎片累积质量分布符合Rosin-Rammler幂率分布规律,且随着着靶速度的增大,小质量碎片质量增加;弹芯在冲击过程中等效直径较大碎片（大于8 mm）失效模式为拉伸脆性断裂,而等效直径小于2 mm的碎片上存在局部塑性剪切断裂。     

基于连续-非连续单元法的三维脆性颗粒冲击破碎特性分析

通过在连续-非连续单元法(CDEM)中引入考虑应变率效应的断裂能本构以及能量统计算法,实现了球体冲击破碎过程中损伤破裂程度及能量演化的定量分析。计算结果表明,冲击破碎过程分为接触蓄能阶段、损伤破碎阶段和碎块飞散阶段。首先,颗粒的部分动能转化为单元弹性变形能,随后这部分变形能和动能迅速转化为摩擦消耗、阻尼消耗及弹簧断裂能,破碎基本完全后碎块继续飞散。不同冲击速度下,颗粒分别出现了反弹、开裂、破碎和粉碎的现象。随冲击速度的增加,D50的变化速率逐渐放缓,破碎块度逐渐趋于稳定;破裂度、损伤度以及平均损伤因子的变化速率先增加后放缓,颗粒破坏以拉伸破坏为主。以上结论可为脆性材料冲击破碎工艺的优化设计提供依据。     

PMMA膨胀环动态拉伸碎裂实验研究

在强动载作用下, 脆性材料的碎裂问题是一个重要的研究课题, 而脆性材料在冲击拉伸载荷下的力学行为的实验研究相对较匮乏. 提出了一种动态拉伸断(碎)裂的液压膨胀环实验技术, 可用于准脆性/脆性材料的动态拉伸. 利用该技术对有机玻璃(PMMA)圆环试件进行了不同膨胀速度下的动态碎裂实验研究. 从回收碎片的断口形貌和碎片内部残余裂纹观察可知试件的破碎由环向拉伸应力造成, 碎片断口处发出的稀疏波会将周围的拉伸应力卸载, 从而抑制其他裂纹的进一步发展. 利用超高速相机记录了试件的膨胀碎裂过程, 利用DISAR激光速度干涉仪获得了试件外表面粒子的径向膨胀速度历史, 通过试件上的应变片获得了试件的应变历史和断裂应变. 实验结果表明: 在拉伸应变率150\mathrm{~}500s^{-1}范围, 材料的动态断裂应变低于准静态加载下的断裂应变, 体现出“动脆”现象; 随着加载应变率的提高, PMMA 材料的碎片尺寸减小; 无量纲化的PMMA圆环的平均碎片尺寸介于韧性碎裂模型和脆性碎裂模型的预测数值之间, 反映出材料的准脆性特性.      

Effects of non-uniform strains on tensile fracture of fiber-reinforced ceramic composites

Effects of non-uniform strains on tensile fracture of fiber-reinforced ceramic–matrix composites have not been satisfactorily explained by existing mechanics-based models. In this paper, we use an exact model of fiber fragmentation under global load sharing conditions to predict fracture in three model problems in which non-uniform strains occur: (i) an end-constrained plate subject to a linear transverse temperature gradient; (ii) an internally-pressurized cylindrical tube with a linear through-thickness temperature gradient; and (iii) a rectangular beam under combined bending and tension. Fracture is assumed to occur when the global load reaches a maximum value. Approximations to the exact fragmentation model are also assessed, with the goal of decoupling the effects of two important parts of the computed stress–strain response: the rate of post-peak strain softening and the magnitude of the plateau “flow” stress once fiber fragmentation is complete. We find that for cases in which the fiber Weibull modulus is low and hence its plateau strength is high relative to its peak and the loading yields a sufficiently high strain gradient, the failure strain lies in the plateau regime. Consequently, the results can be predicted with good accuracy using a perfectly-plastic representation of the post-peak response. In contrast, for cases in which the fiber Weibull modulus is high, the failure strain lies in the softening portion of the curve. Here a linear-softening model is found to yield accurate results. A preliminary assessment of the model has been made by comparing predicted and measured bending/tension strength and failure strain ratios for one specific composite. The correlations appear good, though additional experiments are required in order to critically assess the model predictions over a range of loading scenarios.     

脆性材料破坏模拟的网格生成算法

脆性材料的破坏过程具有随机性,当前的网格生成算法没有充分考虑脆性材料破坏时裂纹扩展和碎块生成的随机性。在Persson网格生成算法与Delaunay随机网格剖分理论基础上,提出了一种可根据模拟需要动态控制网格品质的网格生成算法。通过对随机分布点的Delauna三角化,生成初始网格,然后将网格体系比拟为桁架结构,网格节点即为桁架节点。桁架节点在虚拟力作用下可动态调整位置,并最终达到整个体系受力平衡。对Persson 算法中的尺寸分布函数和收敛条件进行了修正,从而提高了收敛速度,并适用于任意形状对象的网格剖分。 基于VC++平台开发了算法程序。通过实例对算法进行了验证,表明算法能够满足脆性材料破碎模拟的需要。     

Analysis of fracture initiation angle in some cracked ceramics using the generalized maximum tangential stress criterion

Brittle fracture in ceramics sometimes occurs under combined opening-sliding (or mixed mode I/II) crack deformation. In this paper, a generalized maximum tangential stress criterion is employed for predicting the fracture initiation angle under mixed mode I/II loading in some brittle ceramics including alumina, zirconia, soda lime glass and three silicon based ceramics. The experimental results reported for the fracture angles in these ceramics have been obtained from fracture tests on the centrally cracked circular disc (often called the Brazilian disc). Very good agreement is shown to exist between the experimental results and the theoretical predictions. According to the fracture model, the mixed mode fracture angle is strongly dependent on the elastic T-stress in the tested ceramics. The negative T-stress that exists in the Brazilian disc specimen can be the main influencing parameter for decreasing the fracture initiation angle in the investigated ceramics.     

Search for conditions of compressive fracture of hard brittle ceramics at impact loading

In this paper we discuss three different experimental configurations to diagnosing the modes of inelastic deformation and to evaluating the failure thresholds at shock compression of hard brittle solids. One of the manifestations of brittle material response is the failure wave phenomenon, which has been previously observed in shock-compressed glasses. However, based on the measurements from our “theory critical” experiments, both alumina and boron carbide did not exhibit this phenomenon. In experiments with free and pre-stressed ceramics, while the Hugoniot elastic limit (HEL) in high-density B₄C ceramic was found to be very sensitive to the transverse stress, it was found relatively less sensitive in Al₂O₃, implying brittle response of the boron carbide and ductile behavior of alumina. To further investigate the effects of stress states on the shock response of brittle materials, a “divergent flow or spherical shock wave” based plate impact experimental technique was employed to vary the ratio of longitudinal and transversal stresses and to probe conditions for compressive fracture thresholds. Two different experimental approaches were considered to generate both longitudinal and shear waves in the target through the impact of convex flyer plates. In the ceramic target plates, the shear wave separates a region of highly divergent flow behind the decaying spherical longitudinal shock wave and a region of low-divergent flow. Experiments with divergent shock loading of alumina and boron carbide ceramic plates coupled with computer simulations demonstrated the validity of these experimental approaches to develop a better understanding of fracture phenomena.     

冲击压缩下氧化铝陶瓷中破坏阵面的传播

进行了平面冲击波压缩下氧化铝陶瓷中破坏阵面的实验测试和理论探索. 通过氧 化铝陶瓷的平板碰撞实验,借助VISAR测试系统测量了试件自由面的质点速度历程,并对回 收试件进行了电镜扫描观察. 质点自由面速度历程曲线表明,氧化铝陶瓷材料中存在破坏阵 面的传播. 考察了破坏阵面的传播特性,给出了陶瓷材料的动态破坏模型,并对破坏阵面的 传播进行了数值分析.     

边缘冲击（EOI）作用下透明陶瓷破坏特性研究

相较于传统透明材料,相同面密度下透明陶瓷具有更优异的抗冲击性能,使其成为极具应用前景的透明装甲防护材料。研究透明陶瓷在冲击下的破坏响应及损伤演化规律,对透明陶瓷装甲的结构设计及防护能力的提高起到至关重要的作用。为了比较传统透明材料与典型透明陶瓷材料在冲击过程中的破坏特性差异,利用9 mm弹道枪发射平台进行了浮法玻璃、YAG透明陶瓷及镁铝尖晶石透明陶瓷3种透明材料的边缘冲击试验,破片发射速度为200~300 m/s。通过高速摄影捕捉破片的撞击过程,分析了粉碎区及主裂纹扩展距离随时间的变化规律。结果表明,3种材料在不同速度破片的冲击作用下,粉碎区面积与材料强度呈负相关性。对同种材料,在200~300 m/s速度范围内,破片撞击速度对主裂纹的扩展速度没有影响。同时比较了玻璃与透明陶瓷在宏观尺度上的损伤演化特征差异:玻璃在粉碎区两侧产生三角形的次裂纹区域,陶瓷材料则会产生细长的次裂纹簇,并会产生较明显的裂纹“分叉”现象。利用扫描电子显微镜对回收到的陶瓷碎片进行观测,并分析了2种透明陶瓷材料在细观尺度破坏特征的异同。2种透明陶瓷的径向裂纹断面上会出现从沿晶断裂到穿晶断裂的过渡变化,而环向断裂面上几乎都是沿晶和穿晶混合断裂。2种透明陶瓷中,仅YAG透明陶瓷在沿晶断裂时会出现晶体“剥落”现象。     

On the importance of thermo-elastic cooling in the fracture of glassy polymers at high rates

In a previous thermo-mechanical analysis [Estevez, R., Basu, S., van der Giessen, E., 2005. Analysis of temperature effects near mode I cracks in glassy polymers. Int. J. Fract. 132, 249–273] in which shear yielding of the bulk and failure by crazing were accounted for, we examined which of these two viscoplastic processes contributed to heat in mode I fracture. The present study completes this work by investigating the conditions for thermo-elastic cooling prior to crack propagation as reported experimentally by Rittel [Rittel, D., 1998. Experimental investigation of transient thermo-elastic effects in dynamic fracture. Int. J. Solids Struct. 35, 2959–2973] and Bougaut and Rittel [Bougaut, O., Rittel, D., 2001. On crack tip cooling during dynamic crack propagation. Int. J. Solids Struct. 38, 2517–2532] on high strain rate loading of PMMA. To this end, coupled thermo-mechanical finite element simulations are carried out by accounting for the thermo-elastic source, in addition to the heat sources related to shear yielding and crazing. The bulk as well as cohesive zone parameters for crazing realistically describe PMMA as they are obtained from detailed calibration experiments. Our results show that if significant thermo-elastic cooling has to be observed in the vicinity of the crack tip of a polymeric material, suppression of shear yielding as well as suppression of crazing is necessary. It seems that at these high strain rates a brittle fracture mechanism activated at very high stresses takes over from crazing, or at least that craze initiation occurs for stress levels very different to those for quasi-static conditions.     

Effects of static electric field on the fracture behavior of piezoelectric ceramics

The paper gives an overview on experimental observations of the failure behavior of electrically insulating and conducting cracks in piezoelectric ceramics. The experiments include the indentation fracture test, the bending test on smooth samples, and the fracture test on pre-notched (or pre-cracked) compact tension samples. For electrically insulating cracks, the experimental results show a complicated fracture behavior under electrical and mechanical loading. Fracture data are much scattered when a static electric field is applied. A statistically based fracture criterion is required. For electrically conducting cracks, the experimental results demonstrate that static electric fields can fracture poled and depoled lead zirconate titanate ceramics and that the concepts of fracture mechanics can be used to measure the electrical fracture toughness. Furthermore, the electrical fracture toughness is much higher than the mechanical fracture toughness. The highly electrical fracture toughness arises from the greater energy dissipation around the conductive crack tip under purely electric loading, which is impossible under mechanical loading in the brittle ceramics. The project supported by an RGC grant from the Research Grant Council of the Hong Kong Special Administrative Region, China     

损伤演化对韧性金属碎裂过程的影响

固体在冲击拉伸载荷作用下会断裂成多个碎片,基于线性内聚力断裂假设的Mott-Grady模型能较好地预测碎裂过程所产生的平均碎片尺度的下限。然而实际上,韧性金属的损伤演化是多元化的,为此通过数值模拟方法研究了不同损伤演化规律对韧性碎裂过程的影响。利用ABAQUS/Explicit动态有限元软件数值再现了韧性金属杆（45钢）在高应变率下拉伸碎裂的过程,分析了线性和非线性损伤演化对韧性碎裂过程的影响规律。结果表明:损伤演化规律对韧性金属的碎裂过程具有显著影响,非线性指标α越大,碎裂过程产生的碎片数越少;Grady-Kipp碎裂公式仍能在一定范围内预测韧性碎裂过程中产生的碎片尺寸;当非线性指标α远大于零时,在较低冲击拉伸载荷作用下,数值模拟结果和Grady-Kipp模型预测值偏差较大,随着应变率增大,数值模拟结果与Grady-Kipp模型预测值吻合较好。     

改进巴西试验:从平台巴西圆盘到切口巴西圆盘

巴西试验是测试混凝土、陶瓷、岩石等脆性或准脆性材料拉伸强度的标准方法,在材料科学和土木工程中有广泛的应用,研究该方法的改进在科学和工程界都受到日益增长的关注.我们对巴西试验的原始试样即巴西圆盘(Brazilian disc,BD)提出一种新的构型:切口巴西圆盘(grooved Brazilian disc,GBD),利用GBD加载直径两端的窄而浅的切口,消除完整巴西圆盘对径压缩加载时可能非中心起裂的缺点,这一点与我们过去提出的平台巴西圆盘(flattened Brazilian disc,FBD)相同.GBD的构型更普适,因为它涵盖了巴西圆盘和平台巴西圆盘.对于求解GBD切口顶点及其正前方近邻的应力,提出两个近似解析模型,分别求出对应的压应力和拉应力,用叠加原理最后推出的公式能够定性地预测应力的变化趋势,以及试样几何参数的影响.近似解析公式,有限元数值计算,对比实验的结果都证实,对径压缩圆盘的几何与表面边界条件的一个微小的改变,对巴西试验试样的几何稳定性和断裂过程产生显著和有利的影响.利用推出的切口应力公式进行试样几何参数的优化选择,可以使GBD比BD和FBD更具优越性.     

A method for calculating the stress–strain state in the general boundary value problem of metal forming—part 2. Impact of a bar against a rigid obstacle

The method for calculating stress–strain state and fracture proposed by Kolmogorov, 1995and in Part 1 of this present paper is illustrated by the simple problem of a thin bar impacting a rigid obstacle. Known exact solutions are used to test the method. On the basis of the stability theory, the one-dimensional solution has been shown to be legitimate. Mathematical simulation of bar fragmentation resulting from impact has been carried out.     

固体结构损伤破坏统一相场理论、算法和应用

固体开裂引起的损伤和断裂是工程材料和结构最为普遍的破坏形式.为了防止这种破坏,结构设计首先必须了解裂缝在固体内如何萌生、扩展、分叉、汇聚甚至破碎;更重要的是,还需要准确量化这些裂缝演化过程对于结构完整性和安全性降低的不利影响.针对上述固体结构损伤破坏问题,本工作系统地介绍了笔者提出的统一相场理论、算法及其应用.作为一种...     

脆性无序介质冲击破碎过程及其非线性动力学特性的细观模拟

为了模拟脆性无序介质的冲击破碎过程以及揭示其所伴随的非线性动力学特性，提出了梁－颗粒细观数值模型。在该模型中，介质被随机地离散成Ｖｏｒｏｎｏｉ多边形单元，相邻多边形采用弹脆性梁单元联结，介质的连续损伤和破坏由梁单元的断裂来模拟。算例表明，梁－颗粒细观数值模型不仅能实时给出介质的破碎图象及其分形维数，而且还能模拟介质内部应力和质点运动速度随时间的演化过程。