Hertzian Impact Theory Research Articles

The development of sub-surface damage during high energy solid particle erosion of a thermally sprayed WC–Co–Cr coating

The erosion wear performance of a thermally sprayed hard coating has been found to be inferior to that of sintered bulk material of the same composition. The anisotropic microstructure of thermally sprayed WC–Co–Cr coatings, in particular the low fracture toughness in a direction parallel to the substrate, i.e. parallel to the long side of the splats that make up the thermal spray coating has been observed to affect the nature of crack formation both in indentation testing and under high energy solid particle erosion conditions. The present work quantifies the number and length of cracks found, both parallel and transverse to the substrate boundary, in eroded samples of thermally sprayed WC–Co–Cr material. The initiation sites of these cracks are also studied and the importance of voids and other microstructural features (i.e. cobalt lakes, splat boundaries, interfacial inclusions) in the coating as initiation sites is highlighted. Dynamic Hertzian impact theory is used to estimate the elastic contact stresses and sub-surface shear stresses induced by solid particle impacts as well as to infer likely zones of plasticity. These stress fields are related to the location of cracks and to possible mechanisms for crack propagation. The cracks appear in near-surface layers that are likely regions of localised plasticity in the matrix and could result from a mixed mode of ductile fractures driven by plastic strain accumulated after multiple solid impacts as well as fractures driven by elastic stress intensity primarily dominated by sub-surface shear stresses and surface tensile stresses.

Inelastic collision and the Hertz theory of impact

The area, A, and the duration of contact, T, have been measured as a function of impact speed, U, for balls striking a flat surface. The balls lost about 40% of their kinetic energy over the range of speeds studied, but, surprisingly, the results for A(U) and T(U) appear to be consistent with Hertz’s elastic theory of impact. Possible reasons are discussed for this unexpected behavior.

Analysis of Stresses Induced by Dynamic Load Head-Disk Contacts

The dynamic load head-disk contact induced impact stress was studied. A dual channel LDV was used to measure the head-disk relative motion during impact, and an analytical model incorporating the Hertz theory of impact was developed to quantitatively estimate the impact induced contact force and stress based on the LDV-measured results. 70 percent sliders were used in order to compare the results with our previous study. From the estimated maximum contact stresses and the results of our previous study, it was found that when the average maximum stress was 511 MPa, the head-disk interface did not show any damage after 100,000 cycles of repeated head-disk impacts. When the average maximum stress was 880 MPa, however, 100,000 repeated head-disk impacts caused significant wear of the disk’s overcoat even though a single impact did not cause any observable damage. From the analysis it can be seen that a lower head-disk impact velocity and/or a larger radius of curvature at the contacting corner of the slider result in a smaller head-disk impact stress on the disk. Based on the analyses, we estimated the radius of curvature needed for a 50 percent (Nano) slider and a 30 percent (Pico) slider to have at least 100,000 cycles of dynamic load head-disk interface durability. Such radius of curvature can be realized, for example, by edge-blending the sliders. [S0742-4787(00)02901-5]

An impact analysis of a flexible bat using an iterative solver

Although technology has now infiltrated and prompted evolution in most mass participation sports, the advances in bat technology in such sports as baseball and cricket have been relatively minor. In this study, we used a simple finite element modelling approach to try to shed new light upon the underlying mechanics of the bat-ball impact, with a view to the future optimization of bat design. The analysis of a flexible bat showed that the point of impact that produced the maximum post-impact ball velocity was a function of the bat's vibrational properties and was not necessarily at the centre of percussion. The details of the analysis agreed well with traditional Hertzian impact theory, and broadly with empirical data. An inspection of the relative modal contributions to the deformations during impact also showed that the position of the node of the first flexure mode was important. In conclusion, considerable importance should be attached to the bat's vibrational properties in future design and analysis.

Effects of water-mixture film on impact contact in abrasive waterjet machining

The effects of a water-mixture film adhering to solid materials on the impact contact are examined by employing exact solutions for an inhomogeneous contact problem and the Hertz impact theory. The elastic modulus of the water on a surface is assumed to be zero and the modulus of the water-mixture between a water surface and a target material varies from zero to the modulus of the target material. Non-elastic deformation is considered based on an energy balance. A new expression of closed form for impact force is obtained. The film effect on the impact force is very significant for low impact energy and becomes stable and less influential once impact energy increases to a certain value. Experimental results for quasi-static force are in agreement with the theoretical analysis in which two unknown parameters are ascertained by experiments. The minimum effective velocity of garnet abrasives achieved in this work for waterjet machining is consistent with previous results.

Model for collisions in granular gases.

We propose a model for collisions between particles of a granular material and calculate the restitution coefficients for the normal and tangential motion as functions of the impact velocity from considerations of dissipative viscoelastic collisions. Existing models of impact with dissipation as well as the classical Hertz impact theory are included in the present model as special cases. We find that the type of collision (smooth, reflecting or sticky) is determined by the impact velocity and by the surface properties of the colliding grains. We observe a rather nontrivial dependence of the tangential restitution coefficient on the impact velocity.

Dynamic loading impact-induced demagnetization in thin film media

In order to eliminate the contact stress and deformation at the instant of head-disk impact during dynamic loading, the Hertz's impact theory is implemented in the dynamic loading numerical simulator. On the basis of the calculated stress field, inverse magnetostriction effects in longitudinal recording thin film media are considered and demagnetization due to the head-disk impact is simulated. >

Sound radiated from the impact of two cylinders

The sound radiated from the impact of two cylinders is the basis of the study on noise from many mechanical structures, such as the noise from the engagement of gears. Insofar as the authors know, the sound radiated from the impact of two cylinders side-to-side, has not been reported. In this paper, the Hertzian impact theory and the acoustic theory are used to study the sound radiated from the impact of two cylinders side-to-side, and the theoretical result is confirmed experimentally.

Impact of fruit on flexible beams to sense modulus of elasticity

The increasing use of nonselective mechanical harvesters of fruit requires sorting the fruit on firmness. A model and computer program were developed based upon Hertzian impact theory which can predict the impact force function for a spherical fruit hitting a simply supported beam. The model was exercised to observe the effects of beam dimensions and fruit characteristics, including firmness, on the force function. Experimental work showed that measurements from a strain gage on the impacted beam can be used to determine the force function. The measured force function approximated the function predicted by the impact theory model. This work indicated that by the proper design of a sensing beam, fruit firmness may be sensed.

Particle Sizing by Quantitative Acoustic Emission

Abstract Two methods of particle sizing using an acoustic method are presented. The technique relies on the detection and measurement of elastic waves arising from impact of the particles with a small target plate. The acoustic impact signal, as measured from a high fidelity piezoelectric transducer, is characterised by the first wave arrival (compression wave) amplitude and risetime. A theoretical approach, based upon Hertzian impact theory and elastic wave propagation theory, is developed and used to determine the relationship between the impact dynamics of the particle and the acoustic signal. The first method of particle sizing relies upon amplitude measurements alone but requires accurate system calibration and full knowledge of the particle impact velocity, incident angle and coefficient of restitution. The velocity measurements were determined using a laser Doppler technique. The second approach for particle sizing incorporates risetime measurement and only depends very weakly upon the compression wave amplitude, thus minimizing the need for accurate absolute calibration. Knowledge of the impact velocity, incident angle, and coefficient of restitution were not required. Spherical particles of glass in the size range 35 to 140 μm diameter were dropped onto an aluminium target plate at an incident velocity of 8.1 m/s and angle to the surface normal of 0°, 40°, and 61°. The two acoustic methods were used to size the particles, and the results were compared to the particle size distributions obtained from video microscope measurements. The first sizing method undersized the smaller particles by 20% to 30% and oversized the larger ones by 5% to 10%. However, the second approach gave much smaller errors and consistently undersized all sizes of particle by just a few percent. This undersizing was understood in terms of some plasticity in the target that occurred during particle impact.

Particle impact damage in silicon carbide.

The impact damage that occurred in silicon carbide plates was investigated by projectile impact tests. Target plates of silicon carbide with two different thicknesses were impacted by three kinds of projectiles with various speeds ranging from 30 m/s to 500 m/s. Projectiles were 5 mmφ and 3 mmφ steel spheres and 5mmφ silicon carbide spheres. Various modes of damage, such as indentation concave, ring and cone cracks, radial and median cracks, lateral cracks, spalling and formation of a percussion cone were observed. The relationship between the mode of damage and the speed or the kinetic energy of the projectile was investigated. The damage caused by the impact load of the present study was very similar to that caused by quasi-static point indentation load. It was also observed that the outermost diameter of ring cracks formulated in the impacted SiC plate was well calculated by the elastic Hertz's impact theory or a modification of it including the effect of the plastic deformation of projectiles.

粉末粒子の静電加速による特殊加工法の研究 (第4報)

In order to develop the unconventional machining method, the stock-removing and machined-surface properties of Si wafer was investigated by making collide obliquely submicron-sized tungsten and its carbide particles with the velocities of about 100 m/s. The results obtained are as follows : (1) The volume removal per hard carbide particle measured is well approximated by an elastic penetration volume introduced from Hertzian impact theory. This suggests that the removal may be caused by cutting action based on ductile shear fracturing of Si. But the depth of cut estimated with relatively soft tungsten particle became of atomic scale smaller than elastic penetration value, and which may be due to severe plastic flow in the particle. (2) The RHEED observation of the machined Si surface whose roughness is of the order of 10 nm suggests the existence of slightly distorted lattice layer. (3) The phenomenological change from Si removal to particle deposition obviously proceeds at particle incident angle beyond 45°. Such a phenomenon is briefly discussed from a viewpoint of particle-energy consumption by measuring the reflection spectra of post-impact particles.

Elastic impact of spheres on thin shallow spherical shells

This investigation was concerned with the elastic impact of spheres on thin shallow spherical shells. A non-linear integro-differential equation of the impact process was developed on the basis of Reissner's approximate theory for transverse vibrations of shallow shells and the quasistatic Hertzian impact theory. This equation was numerically integrated and its main results were experimentally verified. Special attention was given to the apparently new observations that the rebound height of a sphere dropped onto a given shell has a minimum value for a critical size of the sphere, and that the contact time differs substantially for sphere sizes greater or less than this critical value.

Loose parts signal theory

A loose parts signal model was developed using Hertz impact theory and equations for plate wave propagation, accelerometer response, and signal filtering. The detected signal frequency content, amplitude, and dispersion were calculated for impacts representative of loose part size and energy on typical reactor structures. Effects of accelerometer mounting, component dimensions, multiple path transmission, and filtering were investigated. The results of this study provide insight to loose part signal properties and methods for more quantitative monitoring and diagnostics.

Encapsulated Toners in Xerography and their Shell Fracture Mechanics

The effect of various elastic and physical parameters on the strength of the shell in encapsulated toner is examined. The contact problem of a toner particle and a pressure roller is discussed, and the mechanism of shell fracture from the viewpoint of Hertz's impact theory is reviewed. The magnitudes of the impact force, the fracturing force, and the paper damaging stress are estimated. Within the framework of linear elasticity theory, general expressions for the stresses induced in the shell and core have been derived for both solid and liquid cores. Particular cases have been computed and presented in normalized graphical form on a unit loading basis. The shell strength depends strongly on the geometrical and elastic properties of the core and shell materials, specifically on the shell thickness, the ratio of the core and shell shear moduli, and the tensile strength of the shell material. The modes of toner fracture and the significant effects of the elastic properties of the paper on the crushing force of the particles is discussed. It is shown that the breaking of the particles is facilitated by a harder surfaced paper.

Transient sound radiated by spheres undergoing an elastic collision

An experimental and theoretical analysis of the sound field radiated by the collision of two elastic spheres is presented. The sound pressure measured at any point in space is the proper sum in time of the sound radiated independently by each sphere. It is shown that the combination of classical Hertzian impact theory and acoustical theory is sufficient to define the pressure time waveforms as measured in the experiments.

The impact strength of diamond under different rates of strain

Diamond spheres were fractured by impacts of durations varying from 10 +3·5 to 10 –6 s. Except at the shortest time of contact, the spheres were loaded so that the maximum stress attained during a free impact of the prescribed duration was sufficient to give an even chance of breaking at least one sphere. The maximum ‘nominal’ stresses were calculated from the Hertz theory of impact. As the duration of impact decreased from 10 +3·5 to 10 –4 s, the nominal stress at fracture increased from 6·3 x 10 11 dyn/cm 2 to 1·3 x 10 12 dyn/cm 2 . At the shortest times of impact the nominal stress fell again, reaching 7·0 x 10 11 dyn/cm 2 at a duration of 10 -6 s. The results are explained by assuming that under impacts of duration more than 10 –4 s, fracture occurs by the slow formation of ring and cone cracks, followed by crystallographic fracture. The formation of the ring and cone cracks reduces the true maximum stress below the ‘nominal’ stress. In impacts of short duration there is no time for the rings and cones to form, and cones are never observed on the broken spheres. Crystallographic fracture then occurs directly under a stress which has not been reduced by the ring and cone cracks.

Photoelastic study of elastic impact on the edge of a plate

The Hertz impact theory assumes a quasi-static stress distribution adjacent to the point of contact. This assumption has been tested by a plane photoelastic investigation. The impacting bodies were made from glass plate, which is well suited to impact tests because at high rates of loading it is purely elastic up to fracture and shows no relaxation. Its small photoelastic sensitivity is an advantage for the investigation of high local stresses. A disadvantage is its large elastic modulus giving high propagation velocities. An apparatus based on single-shot exposure has been developed allowing us to follow quasi-cinematographically the wave propagation in the impacted plate at time intervals of micro-seconds with an exposure time determined by a spark of 0·6 μsec half-value duration. The fringes at the point of impact could be photographed very sharply with this arrangement because of their low velocities. The isoclinics were also photographed at various instants during the impact time which was between 45 and 100 μsec according to the impacting mass. Comparison of photographs with others taken under static compression of the same bodies in similar conditions confirms the similarity of the stress distribution near the impact or contact point. Hence the impact force can be determined photoelastically by static calibration. The maximum impact forces found in this way for the various impact masses and velocities coincide well with those calculated from the elementary theory. As an additional check, the area of the surface of contact was measured under static and dynamic loading. A comparison with the values calculated from the Hertz theory would be inaccurate because the impact duration was of the same order of magnitude as the natural period of the impinging bodies and hence smaller than the impact duration according to Hertz. The quasi-cinematographical series of photographs shows clearly the vibrations excited in the impacting bodies. The coefficient of restitution, which relates mainly to this part, could be determined from the impact force—time curve and the impulse. The fracture strength of glass proved appreciably higher under impact than under static loading.

Erratum: On the Pressure Estimation in Impact Sensitivity Experiments on Explosives and the Problem of Initiation

Pressures generated on the explosive layer under conditions of experiments on the impact sensitivity of explosives have been calculated making use of the basic assumptions embodied in the Hertz theory of impact, taking data from certain experiments on explosion. In the absence of any knowledge of the elastic constants of explosives under dynamic conditions of loading, calculations have been made for two sets of elastic constants, one for steel to steel impact and the other for steel to a solid having elastic constants one‐tenth that of steel. The experimental pressures lie between these two values. A reinterpretation of the results leads to a relation between a mass m and its height of h for fifty percent explosion efficiency. The conduction at the solid air interface due to an entrapped gas bubble compressed by these pressures has been considered. The amount of the explosive heated to the optimum temperature required for explosion, within times comparable to impact explosion delays, has been shown to be ...

On the Pressure Estimation in Impact Sensitivity Experiments on Explosives and the Problem of Initiation

Pressures generated on the explosive layer under conditions of experiments on the impact sensitivity of explosives have been calculated making use of the basic assumptions embodied in the Hertz theory of impact, taking data from certain experiments on explosion. In the absence of any knowledge of the elastic constants of explosives under dynamic conditions of loading, calculations have been made for two sets of elastic constants, one for steel to steel impact and the other for steel to a solid having elastic constants one-tenth that of steel. The experimental pressures lie between these two values. A reinterpretation of the results leads to a relation between a mass m and its height of h for fifty percent explosion efficiency. The conduction at the solid air interface due to an entrapped gas bubble compressed by these pressures has been considered. The amount of the explosive heated to the optimum temperature required for explosion, within times comparable to impact explosion delays, has been shown to be of the order of the hot spot size found from other considerations. The variation of the height h and the pressure p of the gas surrounding the explosive layer for fifty percent efficiency is shown to be governed by the relation h/p2 = constant.