Dynamic buckling and fragmentation in brittle rods

We present experiments on the dynamic buckling and fragmentation of slender rods axially impacted by a projectile. By combining the results of Saint-Venant and elastic beam theory, we derive a preferred wavelength lambda for the buckling instability, and experimentally verify the resulting scaling law for a range of materials including teflon, dry pasta, glass, and steel. For brittle materials, buckling leads to the fragmentation of the rod. Measured fragment length distributions show two peaks near lambda/2 and lambda/4. The nonmonotonic nature of the distributions reflect the influence of the deterministic buckling process on the more random fragmentation processes.     

Fragmentation of rods by cascading cracks: why spaghetti does not break in half

When thin brittle rods such as dry spaghetti pasta are bent beyond their limit curvature, they often break into more than two pieces, typically three or four. With the aim of understanding these multiple breakings, we study the dynamics of a bent rod that is suddenly released at one end. We find that the sudden relaxation of the curvature at this end leads to a burst of flexural waves, whose dynamics are described by a self-similar solution with no adjustable parameters. These flexural waves locally increase the curvature in the rod, and we argue that this counterintuitive mechanism is responsible for the fragmentation of brittle rods under bending. A simple experiment supporting the claim is presented.     

Models of fragmentation with composite power laws

Some models for binary fragmentation are introduced in which a time dependent transition size produces two regions of fragment sizes above and below the transition size. In the first model we assume a fixed rate of fragmentation for the largest fragment and two different rates of fragmentation in the two regions of sizes above and below the transition size. The model is solved exactly in the long time limit to reveal stable time-invariant solutions for the fragment size and mass distributions. These solutions exhibit composite power law behaviours; power laws with two different exponents for fragments in smaller and larger regions. A special case of the model with no fragmentation in the smaller size region is also examined. Another model is also introduced which have three regions of fragment sizes with different rates of fragmentation. The similarities between the stable distributions in our models and composite power law distributions from experimental work on shock fragmentation of long thin glass rods and thick clay plates are discussed.     

Fragmentation of shells

We present a theoretical and experimental study of the fragmentation of closed thin shells made of a disordered brittle material. Experiments were performed on brown and white hen egg shells under two different loading conditions: impact with a hard wall and explosion by a combustible mixture. Both give rise to power law fragment size distributions. A three-dimensional discrete element model of shells is worked out. Based on simulations of the model, we give evidence that power law fragment mass distributions arise due to an underlying phase transition which proved to be abrupt for explosion and continuous for impact. We demonstrate that the fragmentation of closed shells defines a new universality class of fragmentation phenomena.     

无序性对脆性材料冲击破碎的影响

脆性材料受冲击荷载作用产生损伤开裂是一个连续介质离散化的过程.采用连续离散耦合方法模拟了一个脆性圆球以不同初始速度与刚性板的冲击,重点研究了无序性对脆性材料冲击破碎的影响,并对其内在机理进行了分析.本文不考虑材料细观结构的无序性,材料的无序仅体现在细观断裂参数的非均质性.数值实验同样揭示了脆性材料在冲击破坏中存在两种破坏模式,即低速时接触区域的局部损伤和高速时的整体碎裂.研究表明,材料无序性对临界冲击速度、破碎模式、碎片形态影响显著.随着无序度增加,材料的临界速度增大,损伤开裂由少量贯穿性裂纹主导转变为全域性的分叉裂纹.高无序度圆球冲击产生的碎片表面更粗糙,体型更为扁平细长.这与细观断裂的主导机制有关,无序度较高时剪切导致开裂的比重更大,碎片内部损伤裂纹面更多.     

Buckling of coupled elastic rods

A simple physical system is studied which demonstrates the unfolding of a multiple bifurcation point. Two coupled elastic rods are subjected to a compressive load. The rods are joined such that their buckling directions are orthogonal. As the load is quasistatically increased from zero one of the two rods will buckle independently of the other. The “preferred” buckling mode is determined by two parameters, the length ratio and the stiffness ratio. The experimental system consists of two spring steel strips joined through a Teflon connector such that the buckling planes are orthogonal. The theoretical model consists of an analogous rigid rod and spring model which we assume retains the essential features of the continous system. Secondary bifurcation is essential to a continuous exchange of priority between the two buckling modes, and its role is examined for both imperfection-free and perturbed rod-spring systems. Our computational results are then compared with some experiments.     

Universal dynamic fragmentation in D dimensions

A generic model is introduced for brittle fragmentation in D dimensions, and this model is shown to lead to a fragment-size distribution with two distinct components. In the small fragment-size limit a scale-invariant size distribution results from a crack branching-merging process. At larger sizes the distribution becomes exponential as a result of a Poisson process, which introduces a large-scale cutoff. Numerical simulations are used to demonstrate the validity of the distribution for D=2. Data from laboratory-scale experiments and large-scale quarry blastings of granitic gneiss confirm its validity for D=3. In the experiments the nonzero grain size of rock causes deviation from the ideal model distribution in the small-size limit. The size of the cutoff seems to diverge at the minimum energy sufficient for fragmentation to occur, but the scaling exponent is not universal.     

Explosive fragmentation of a thin ceramic tube using pulsed power

This study experimentally examined the explosive fragmentation of thin ceramic tubes using pulsed power. A thin ceramic tube was threaded on a thin copper wire, and high voltage was applied to the wire using a pulsed power generator. This melted the wire and the resulting vapor put pressure on the ceramic tube, causing it to fragment. We examined the statistical properties of the fragment mass distribution. The cumulative fragment mass distribution obeyed the double exponential or power law with exponential decay. Both distributions agreed well with the experimental data. Finally, we obtained universal scaling for fragmentation, which is applicable to both impact and explosive fragmentation.     

Statistical models of brittle fragmentation

Recent developments in statistical models for fragmentation of brittle material are reviewed. The generic objective of these models is understanding the origin of the fragment size distributions (FSDs) that result from fracturing brittle material. Brittle fragmentation can be divided into two categories: (1) Instantaneous fragmentation for which breakup generations are not distinguishable and (2) continuous fragmentation for which generations of chronological fragment breakups can be identified. This categorization becomes obvious in mining industry applications where instantaneous fragmentation refers to blasting of rock and continuous fragmentation to the consequent crushing and grinding of the blasted rock fragments. A model of unstable cracks and crack-branch merging contains both of the FSDs usually related to instantaneous fragmentation: the scale invariant FSD with the power exponent (2?1/D) and the double exponential FSD which relates to Poisson process fragmentation. The FSDs commonly related to continuous fragmentation are: the lognormal FSD originating from uncorrelated breakup and the power-law FSD which can be modeled as a cascade of breakups. Various solutions to the generic rate equation of continuous fragmentation are briefly listed. Simulations of crushing experiments reveal that both cascade and uncorrelated fragmentations are possible, but that also a mechanism of maximizing packing density related to Apollonian packing may be relevant for slow compressive crushing.

Statistical models of brittle fragmentation

All authors

J. A. Åström

https://doi.org/10.1080/00018730600731907

Published online:

28 November 2010

Table     

Patterns of fragmentation for polymer coatings

We investigated the fragmentation of thin coatings on substrates under uniaxial tension. Theoretical models based on linear elasticity predict a power-law dependence of the mean fragment length < L> on the strain e to which the substrate is subjected: namely, one has <L> ∞?^?α with α ≤ 1/2. Experimentally, such a relation is obeyed in the fragmentation of thin SiO _x coatings on polyethylene terephthalate (PET) substrates. In our experiments, we used brittle paints to coat rubber substrates: the ensuing mean fragment lengths also obey power laws in ?: we found, however, α = 0.71 ± 0.03. Such values of α can be reproduced theoretically by taking into account the nonlinearity of the coupling between coating and substrate.     

Impact fragmentation of polyurethane and polypropylene cylinder

The impact fragmentation of a bulk polyurethane elastomer (PU) and polypropylene (PP) cylinder have been investigated using a Cu plate projectile launched by a propellant gun at a velocity of 0.53–1.4 km/s. A projectile drills into a PU sample and forms a cavity in the sample. A small number of tiny fragments are formed. When the projectile smashes in at 1.4 km/s, the PU cylinder bursts and PU fragments form. On the other hand, a brittle fracture occurs on the PP cylinder. The mass of fragments from the PU sample generated at a lower impact velocity is distributed in the lognormal form, whereas the mass of fragments from the PU sample generated by a 1.4 km/s impact follows a power-law distribution. The fragment mass distribution of the PP sample generated at a lower impact velocity obeys the power-law form, whereas that generated at a higher impact velocity follows the lognormal form.     

Nuclear multifragmentation: Basic concepts

We present a brief overview of nuclear multifragmentation reaction. Basic formalism of canonical thermodynamical model based on equilibrium statistical mechanics is described. This model is used to calculate basic observables of nuclear multifragmentation like mass distribution, fragment multiplicity, isotopic distribution and isoscaling. Extension of canonical thermodynamical model to a projectile fragmentation model is outlined. Application of the projectile fragmentation model for calculating average number of intermediate mass fragments and the average size of the largest cluster at different Z _bound, differential charge distribution and cross-section of neutron-rich nuclei of different projectile fragmentation reactions at different energies are described. Application of nuclear multifragmentation reaction in basic research as well as in other domains is outlined.     

空间弹性细杆在扭矩作用下的屈曲

用有限元方法研究了三维弹性细杆在扭矩作用下的屈曲.利用自然坐标形式的细长空间曲杆的能量方程和2节点12个自由度的自然坐标形式的三维曲梁单元,采用特征值分析方法,研究分析了同时受有轴力和扭矩作用时的空间弹性细杆的屈曲问题.数值结果与存在的理论解极为吻合.具有一定曲率和挠率的空间细长曲杆,其临界扭矩值与扭矩的指向有着极其明显的关系.     

Projectile fragment emission in the fragmentation of ~(56)Fe on C,Al and CH_2 targets at 471 A MeV

The emission angle and the transverse momentum distributions of projectile fragments produced in the fragmentation of ⁵⁶Fe on CH₂, C and Al targets at 471 A MeV are measured. It is found that for the same target, the average value and width of the angular distribution decrease with an increase of the projectile fragment charge; for the same projectile fragment, the average value of the distribution increases and the width of the distribution decreases with increasing the target charge number. The transverse momentum distribution of a projectile fragment can be explained by a single Gaussian distribution and the averaged transverse momentum per nucleon decreases with the increase of the charge of projectile fragment. The cumulated squared transverse momentum distribution of a projectile fragment can be explained well by a single Rayleigh distribution. The temperature parameter of the emission source of the projectile fragment, calculated from the cumulated squared transverse momentum distribution, decreases with the increase of the size of the projectile fragment.     

Space-time scale invariance in dynamically fragmented quasi-brittle materials

The paper reports on a study to investigate the influence of load intensity and material structure on statistical regularities of fragmentation in ZrO₂-based ceramics differing in porosity. The study was performed on a modified split Hopkinson pressure bar system and allowed a comparative analysis of dynamic stress-strain curves and statistical characteristics of fragmentation such as distributions of emitted light pulses (fractoluminescence) and fragment sizes. It is shown that increasing the ceramic porosity changes both the form of stress-strain curves and the pulse distribution. In ceramic specimens with up to 45% porosity, the pulse distribution is described by a bimodal power law; in ceramic specimens with 60% porosity, by a power law. The fragment size distribution in the material corresponds to a power law with the exponent dependent on porosity and load intensity.     

Cold and “hot” fragmentation in thermal neutron induced fission of 245Cm

In thermal neutron induced fission of ²⁴⁵Cm a fine structure in mass distribution is observed in cold and “hot” fragmentation regions for light as well as for heavy fragments. The complementarity of the light and heavy fragment masses in the fine structures is discussed.     

Flexible polymers and thin rods far from equilibrium: buckling dynamics

We investigate the dynamics of the classical Euler buckling instability of compressed objects such as flexible molecular chains and thin rods moving in a viscous medium. We find that flexible chains undergo a coarsening process self-similar in time. They develop a wavelike pattern whose amplitude and wavelength grow in time. We relate the buckling dynamics to phase ordering phenomena. The role of the order parameter here is played by the chain slope with respect to the straight initial chain configuration.     

Scale invariance and lack of self-averaging in fragmentation

We derive exact statistical properties of a recursive fragmentation process. We show that introducing a fragmentation probability 0相似文献   

Polymerization force driven buckling of microtubule bundles determines the wavelength of patterns formed in tubulin solutions

We present a model for the spontaneous formation of a striated pattern in polymerizing microtubule solutions. It describes the buckling of a single microtubule (MT) bundle within an elastic network formed by other similarly aligned and buckling bundles and unaligned MTs. Phase contrast and polarization microscopy studies of the temporal evolution of the pattern imply that the polymerization of MTs within the bundles creates the driving compressional force. Using the measured rate of buckling, the established MT force-velocity curve and the pattern wavelength, we obtain reasonable estimates for the MT bundle bending rigidity and the elastic constant of the network. The analysis implies that the bundles buckle as solid rods.     

Scale invariance in dynamic fragmentation of quartz

In the work, we studied statistical regularities in fragmentation of cylindrical quartz specimens under dynamic loading. The original equipment used in the study ensured integrity of the specimens after impact for the determination of fragment size distribution (spatial scaling) and allowed recording fractoluminescence pulses at newly formed fracture surfaces for the determination of pulse spacing distribution (temporal scaling). The results of experimental data processing show that the size distributions for both the spatial parameter (fragment size) and the temporal parameter (fractoluminescence pulse spacing) are described by a power function. This enables us to refer dynamic fragmentation of quartz to phenomena exhibiting self-organized criticality.