bonding mechanism

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Acta Materialia 51 (2003) 4379–4394 www.actamat-journals.com Bonding mechanism in cold gas spraying Hamid Assadi a , Frank Ga ¨rtner b,, Thorsten Stoltenhoff b , Heinrich Kreye b a Department of Materials Engineering, Tarbiat Modarres University, P.O. Box 14115-143, Tehran, Iran b Department of Mechanical Engineering, University of the Federal Armed Forces Hamburg, Hamburg D-22043 Germany Received 31 July 2002; accepted 5 May 2003 Abstract Cold gas spraying is a relatively new coating process by which coatings can be produced without significant heating of the sprayed powder. In contrast to the well-known thermal spray processes such as flame, arc, and plasma spraying, in cold spraying there is no melting of particles prior to impact on the substrate. The adhesion of particles in this process is due solely to their kinetic energy upon impact. Experimental investigations show that successful bonding is achieved only above a critical particle velocity, whose value depends on the temperature and the thermomechanical properties of the sprayed material. This paper supplies a hypothesis for the bonding of particles in cold gas spraying, by making use of numerical modelling of the deformation during particle impact. The results of modelling are assessed with respect to the experimentally evaluated critical velocities, impact morphologies and strengths of coatings. The analysis demonstrates that bonding can be attributed to adiabatic shear instabilities which occur at the particle surface at or beyond the critical velocity. On the basis of this criterion, critical velocities can be predicted and used to optimise process parameters for various materials. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Cold gas spraying; Particle impact; Modelling; Bonding; Adiabatic shear instability 1. Introduction In thermal spray processes, the quality of coat- ings is influenced by both, the degree of melting and the velocity of the sprayed particles. In high velocity oxy-fuel (HVOF) spraying, for instance, a significant proportion of particles is still solid as they impinge the substrate [1–4]. The formation of dense and tightly bonded coatings in spraying pro- Corresponding author. Tel.: +49-40-6541-2887; fax: +49- 40-6541-2006. E-mail address: [email protected] (F. Ga ¨rtner). 1359-6454/03/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00274-X cesses can thus be attributed not only to the thermal energy but also to the kinetic energy of particles upon impact. In cold gas spraying, very high par- ticle velocities are obtained by the acceleration of an expanding gas stream to velocities in the range of 500–1200 m/s in a converging diverging DeLa- val type nozzle [5–10]. In this process, the gas is heated, without using combustion, only to increase the gas and particle velocity and to facilitate the deformation of particles upon impact. The gas and particle temperatures remain well below the melt- ing temperature of the spray material. The particles are therefore in the solid state as they hit the sub- strate, and formation of coatings occurs only due

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Page 1: Bonding Mechanism

Acta Materialia 51 (2003) 4379–4394www.actamat-journals.com

Bonding mechanism in cold gas spraying

Hamid Assadia, Frank Ga¨rtnerb,∗, Thorsten Stoltenhoffb, Heinrich Kreyeb

a Department of Materials Engineering, Tarbiat Modarres University, P.O. Box 14115-143, Tehran, Iranb Department of Mechanical Engineering, University of the Federal Armed Forces Hamburg, Hamburg D-22043 Germany

Received 31 July 2002; accepted 5 May 2003

Abstract

Cold gas spraying is a relatively new coating process by which coatings can be produced without significant heatingof the sprayed powder. In contrast to the well-known thermal spray processes such as flame, arc, and plasma spraying,in cold spraying there is no melting of particles prior to impact on the substrate. The adhesion of particles in thisprocess is due solely to their kinetic energy upon impact. Experimental investigations show that successful bonding isachieved only above a critical particle velocity, whose value depends on the temperature and the thermomechanicalproperties of the sprayed material. This paper supplies a hypothesis for the bonding of particles in cold gas spraying,by making use of numerical modelling of the deformation during particle impact. The results of modelling are assessedwith respect to the experimentally evaluated critical velocities, impact morphologies and strengths of coatings. Theanalysis demonstrates that bonding can be attributed to adiabatic shear instabilities which occur at the particle surfaceat or beyond the critical velocity. On the basis of this criterion, critical velocities can be predicted and used to optimiseprocess parameters for various materials. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Cold gas spraying; Particle impact; Modelling; Bonding; Adiabatic shear instability

1. Introduction

In thermal spray processes, the quality of coat-ings is influenced by both, the degree of meltingand the velocity of the sprayed particles. In highvelocity oxy-fuel (HVOF) spraying, for instance, asignificant proportion of particles is still solid asthey impinge the substrate[1–4]. The formation ofdense and tightly bonded coatings in spraying pro-

∗ Corresponding author. Tel.:+49-40-6541-2887; fax:+49-40-6541-2006.

E-mail address: [email protected] (F.Gartner).

1359-6454/03/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.doi:10.1016/S1359-6454(03)00274-X

cesses can thus be attributed not only to the thermalenergy but also to the kinetic energy of particlesupon impact. In cold gas spraying, very high par-ticle velocities are obtained by the acceleration ofan expanding gas stream to velocities in the rangeof 500–1200 m/s in a converging diverging DeLa-val type nozzle[5–10]. In this process, the gas isheated, without using combustion, only to increasethe gas and particle velocity and to facilitate thedeformation of particles upon impact. The gas andparticle temperatures remain well below the melt-ing temperature of the spray material. The particlesare therefore in the solid state as they hit the sub-strate, and formation of coatings occurs only due

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to the kinetic energy of the particles. Cold gasspraying is therefore particularly suitable for spray-ing and coating materials which are sensitive toheat or oxidation and would transform or react inconventional spraying processes. It has been dem-onstrated that cold gas sprayed coatings of oxi-dation sensitive materials such as Ti or MCrAlYshow similar, or only slightly higher oxygen con-tents than their respective powder feedstock[11,12].

The bonding of particles in cold gas sprayingis presumed to be the result of extensive plasticdeformation and related phenomena at the interface[9,13–15]. The bonding mechanisms in cold gasspraying can thus be compared to those in pro-cesses such as explosive cladding or shock wavepowder compaction. These latter processes havebeen explored widely with respect to mechanismsof bonding as well as to possible applications [16–18]. In explosive cladding, successful bondingoccurs within a certain range of impact angles,impact velocities and materials properties, and it isoften manifested by the formation of an out-flow-ing jet of material at the contact zone [19]. Withina distance of a few millimetres from the interface,there is a sequence of regions with severe defor-mation, highly elongated grains, recrystallisedgrains, and sometimes resolidified microstructures,though melting and resolidification are often lim-ited to a thickness of less than a micrometer at theinterface [20]. In explosive powder compaction,dense solids of materials ranging from metals suchas aluminium, steels and nickel based superalloys,to ceramics can be produced by appropriate selec-tion of shock pressure and duration [21–26]. Inanalogy with explosive cladding, successful bond-ing in powder compaction, also, has been relatedto critical conditions for extensive plastic defor-mation at the particle/particle interface [23,27].Microstructural studies of compacted powder ofsome nickel-base alloys, indicate the formation ofamorphous phases due to rapid cooling [28]. Thereis also evidence for the formation of a fast travel-ling jet at the particle/particle interface in powdercompaction studies [18,22,25].

Despite the existing analogies among these pro-cesses, it has not yet become clear as to what extentthe concepts of explosive cladding or shock wave

powder compaction can be applied to cold gasspraying. There have been a number of studiesfocusing on the relationships between the meas-ured deposition efficiencies and the particle velo-cities as evaluated by semi-empirical methods andnumerical modelling [8,12,29,30]. These investi-gations propose the existence of a critical particlevelocity for successful bonding, whose valuedepends most significantly on the thermomechan-ical properties of the spray and also the substratematerials. Below this velocity, impacting particleswould only cause densification and abrasion of thesubstrate, in a way similar to that in grit blasting orshot peening. Moreover, scanning and transmissionelectron microscopy of cold gas sprayed coatingsshow features on a very localised scale which arecomparable to those observed in explosive clad-ding or explosive powder compaction [13–15].Previous studies also include numerical analyses ofparticle impact in cold gas spraying, providing adescription of the deformation pattern and tem-perature distribution in the particle and substrate[13].

The aim of this paper is to explore mechanismsof bonding and to explain the observed criticalvelocities in cold gas spraying, by means ofnumerical analysis of particle impact. The presentstudy includes a description of the mechanical stateand thermal history of the individual particles dur-ing impact. The results of the analyses are thendiscussed in view of possible correlations with themeasured critical velocities and other experimentalobservations. In this way, possible criteria for par-ticle bonding in cold gas spraying are assessed.Based on these criteria, correlations between thecritical velocity and key parameters of the processand material properties are evaluated. These corre-lations provide a better understanding of the coldspray process and contribute to its optimisation.

2. Methods

2.1. The cold gas spraying process

Principles of cold gas spraying and the experi-mental set-up for the preparation of coatings havebeen described elsewhere [12] and are summarised

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schematically in Fig. 1. Process parameters suchas gas type, pressure and temperature wereoptimised to yield maximum deposition efficiencyand minimal porosity, with respect to size distri-bution, morphology and oxygen content of the par-ticles of the spraying powder. In case of cold gasspraying of copper, nitrogen or helium, preheatedto 320 °C, were used as process gases. To studythe impact of single particles on the substrate, lessthan a monolayer of particles was deposited bywipe tests, in which the substrate was moved rap-idly perpendicular to the stream of particles.

2.2. Semi-empirical determination of criticalvelocities

Details of the procedure used to determine thecritical velocities are reported in detail in [12].Initially, particle velocity distributions for a givenset of processing conditions were calculated byusing computational fluid dynamics (CFD). Thiswas then verified against the corresponding meanvelocity as measured by laser doppler anemometry(LDA). Subsequently, the velocity distribution wascombined with the measured deposition efficiencyand the particle size distribution, to give the sizeand velocity of the largest particle which wouldbond successfully to the substrate. The correspond-ing calculated velocity for this largest, and henceslowest, bonded particle was taken as the criticalvelocity for bonding.

Fig. 1. Schematic diagram of the set-up used for cold gasspraying in the present study. The spray gun is mounted on arobot system to give the required raster speed.

2.3. Microstructural investigations

Impact morphologies of copper particles on cop-per substrates were studied by scanning electronmicroscopy. Cross sections of the coated sampleswere prepared to study the particle/substrate inter-faces. In case of wipe tests, the samples were alsostudied from a top or a perspective view.

2.4. Measurement of strength

The adhesive strength of cold gas sprayed coat-ings was determined according to the Europeanstandard EN 582. Top surfaces of cylindricalsamples were coated and then glued to respectivecounter-bodies of the same size. The specimenswere then exposed to tensile forces along their axesin a tensile-testing machine. The ruptured speci-mens were screened to determine the location offailure, which could be at the interfaces between(i) coating and substrate, (ii) neighbouring particleswithin the coating, or (iii) coating and glue.

2.5. Modelling of deformation

Deformation of the particles upon impact wasanalysed by using the finite element programABAQUS/Explicit version 6.2-1 [31]. The analysisaccounted for strain hardening, strain-rate harden-ing, thermal softening, and heating due to fric-tional, plastic and viscous dissipation. In mostcases, the heating was assumed to be adiabatic, i.e.,heat transfer was not considered. In modelling ofhigh strain rate phenomena, the validity of thisassumption can be assessed with respect to thevalue of the dimensionless parameter x 2 /D tht, inwhich x is a characteristic system dimension, Dth

is thermal diffusivity and t is the process time [32].A value of x 2 /D tht equal or above unity is con-sidered to justify the assumption of adiabatic heat-ing. The value of this parameter can be shown tobe well above unity for the case of particle impactin cold gas spraying, by taking typical values of10�6 m2/s for Dth, 10 ns for t and 10�6 m for theelement size in a particle of 10�5 m in diameter.It should nevertheless be noted that the above argu-ment is based on the solution of the diffusive heatequation. At very small length scales, heat conduc-

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tion would be dominated by a wave propagationmechanism rather than by diffusion [33,34]. Thisimplies that with decreasing particle size, the speedof heat propagation would approach that of plasticwaves, and would in any case be limited by thespeed of sound in the particle. In other words, forhigh speed impacts of much smaller particles, heatconduction could be even slower than what is pre-dicted by the diffusive heat equation. Therefore, itcould still be reasonable to assume adiabatic heat-ing even for the impact of very small particles forwhich x 2 /D tht would fall below unity. It has alsobeen suggested that the dissipation of kineticenergy into heat is strain rate dependent, i.e., thefraction of plastic work dissipated into heat wouldbe larger for higher strain rates [35]. This wouldfurther support the assumption of adiabatic heatingin cold gas sprayed particles. Nevertheless, in orderto leave a margin for heat conduction and storedenergy, only 90% of the kinetic energy wasassumed to dissipate into heat.

Modelling of particle impact was performedinitially for copper as a reference material. Thechoice of copper was made in view of both feasi-bility of coating formation by cold gas sprayingand availability of relevant material data in theliterature (e.g., [36–39]). In most cases, the plasticresponse of copper was assumed to comply withthe Johnson–Cook plasticity model as follows

s � [A � Ben][1 � Clne∗][1�T∗m] (1)

where s is the flow stress, e is the equivalent plas-tic strain, e∗ is the equivalent plastic strain rate nor-malised with respect to a reference strain rate, andT∗m = (T�Tref)/(Tm�Tref) in which Tref is the tem-perature above which thermal softening can occur,and Tm is the melting temperature. The constantsA, B, n, and m are taken from [36] and are listedin Table 1. The elastic response of the material wasassumed to follow a linear elasticity model. Thisassumption was found to be adequate for mostcases of simulations, especially for low and moder-ate particle impact velocities. A linear Mie–Gru-neisen equation of state (EOS) was employed alter-natively for comparison with the linear elasticitymodel. In ABAQUS, this has the following form

p�pH � r�(Em�EH) (2)

Table 1Material data for copper

Properties Parameter Unit Value

General Density kg/m3 8960Specific heat J/kg K 383Melting temperature K 1356Thermal expansion 1/K 5 × 10�5

Elastic Young’s modulus GPa 124Poison’s ratio – 0.34

Plastic A MPa 90(Johnson– B MPa 292Cook model) n – 0.31

C – 0.025m – 1.09Reference K 298temperatureReference strain rate 1/s 1

where pH and EH are the Hugoniot pressure andspecific energy, respectively, and � is the Grun-iesen ratio [31]. The material parameters requiredfor these calculations were taken from [37].

Analyses were performed by using axisym-metric, and three-dimensional models of particleimpacts, with various combinations of calculationsettings concerning element type, initial and adapt-ive meshing, contact interaction, etc. For axisym-metric analyses, 4-node elements with hourglasscontrol were used. Initial meshing was performedby using the built-in free-meshing algorithm ofABAQUS/CAE. The initial element size variedfrom region to region, but for most cases of analy-sis it was kept about 1/50 of the particle size nearthe contact area. Adaptive meshing was usedinitially to cope with large deformations near thecontact surfaces. This was done with the objectiveof preserving initial mesh grading, combined withthe built-in second-order advection and half-indexshift momentum advection methods ofABAQUS/Explicit [31]. Nevertheless, for moder-ate and high particle impact velocities, frequentremeshing resulted in non-conserving energy vari-ation of the output set, and in unphysical shape ofthe out-flowing jet of material at the interface. Toalleviate this problem, remeshing was performedonly as frequent as necessary; i.e., in most casesthe frequency of remeshing was kept lower thanone in every 50 increments. Three-dimensional

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analyses were performed without adaptive mesh-ing, by using 4-node linear tetrahedron elements.Both types of analysis were performed mainly foridentical particle and substrate materials and forparticle sizes in the range from 5 to 100 µm. Theinitial impact velocity was taken as the prime inputvariable. The deformation pattern and the key fieldvariables were examined generally throughout theparticle and substrate, but also particularly alongvarious paths within the particle. The temporalevolution of these variables at specified locationsat the surface were also studied. Fig. 2 shows anexample of the initial configuration, the paths, andthe boundary conditions assumed for an axisym-metric analysis. The substrate has no degree offreedom at the bottom, and can only slide in thevertical directions at the lateral edges.

Fig. 2. Initial configuration and boundary conditions for anaxisymmetric model. The substrate is fixed at the bottom, andhorizontal displacement is not allowed at the axis of symmetryand at the outer edge. The profiles of field variables are investi-gated along meridian and radial paths as shown in the figure.The development of these variables with time is studied atselected nodes at the surface, which are subject to highestamount of deformation.

3. Experimental results

3.1. Critical velocities

By using the method described in Section 2.2, acritical velocity of 570 m/s was obtained for inert-gas-atomised 99.8% copper powder with particlesize ranging from 5 to 22 µm. The critical velocitywas determined to be 660 m/s for inert-gas-atom-ised 99.93% aluminium powder with particle sizebelow 45 µm. These values are consistent withthose reported previously by other researchers[5,8]. The mean particle velocities, as measured byLDA, agreed well with the calculated mean velo-cities. However, this experimental method couldnot give an unambiguous correlation between par-ticle sizes and velocities

3.2. Deformation morphologies

Fig. 3 shows micrographs of impacted particlesof copper on a copper substrate. The impact velo-

Fig. 3. Scanning electron micrographs (secondary electronmode) of wipe test samples of copper particles on a coppersubstrate, showing (a) an overview and (b) a close-up image.

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cities were calculated to be in the range from 550to 670 m/s for particle sizes between 5 and 25 µm.Overviews at low magnifications demonstrate thatmore than 80% of the impacting particles adhereto the substrate (Fig. 3a). The close-up view in Fig.3b illustrates a bonded particle which isaccompanied by the formation of a ring of a jet-type morphology around the impact zone. Themicrographs also show that the deformation of sec-ondary particles is influenced by the morphologycreated by primary impacts. The non-adhering par-ticles leave craters in the substrate without anyindication of a jet-type feature. The height anddiameter of the adhering particles in a N2-sprayedsample were determined from cross-section micro-graphs. The mean flattening ratio, defined as theratio of the diameter of splat to that of a sphericalparticle of the same volume, was calculated to be1.31 for this sample. It should be noted that thisvalue is inevitably associated with an error, whicharises from the underestimation of the particle vol-ume. This is due to the fact that cross sections donot normally pass through the centre of splats. Fora He-sprayed sample a larger flattening ratio wasobtained by, alternatively, comparing the diametersof the adhering particles—as obtained from top-view SEM micrographs—to the particle size distri-bution of the initial powder feedstock—as determ-ined by sieving analysis. More results will be givenand compared to the results of modelling later inthis paper.

Fig. 4 shows a cross-section of a 1 mm thickcopper coating. The particles appear to be tightlybonded throughout their entire surfaces and thereis only a very small amount of porosity within thecoating. The micrograph reveals jet-type featuresand elongated morphologies.

3.3. Strength of coatings

The strength tests of cold gas sprayed coatingsof copper on copper or steel substrates reveal thatin most cases failure occurs at the coating/substrateinterface. For the employed spraying conditions,the adhesive strengths were typically in the rangefrom 30 to 40 MPa. Higher values were obtainedwith higher gas and particle velocities. This meansthat the strength of cold gas sprayed coatings of

Fig. 4. Scanning electron micrographs (back-scattering mode)of a cross-section of a cold gas sprayed copper coating on analuminium substrate. The sample was etched to reveal theparticle/particle boundaries. The particles appear to be tightlybonded, and there is very little porosity within the coating.

copper can be as low as 20% of the yield strengthof work hardened copper, when typical velocitiesfor building up dense coatings are used. In contrastto the tight appearance of the coatings (Fig. 4), therelatively low strength of these coatings suggestsimperfect bonding at the particle/particle inter-faces.

4. Results of numerical modelling

4.1. General aspects of particle impact

Fig. 5 shows the deformation pattern in a copperparticle during impact onto a flat copper substrate,for two initial velocities of 500 and 600 m/s. Theparticles and substrates are initially at room tem-perature. The contours indicate temperature and thearrows represent the magnitude of velocity at therespective surfaces. The figure shows very local-ised heating near the particle/substrate interface forboth impact velocities, though this is more severefor the higher velocity impact. The figure alsoshows the formation of a jet-type flow of materialat the interface at an early stage of impact. To ver-ify parameter settings, flattening ratios obtained bymodelling were compared to results frominvestigating cross sections of single impacts. Asshown in Fig. 6, the modelled flattening ratio ofCu-particles impacting on Cu-substrates, is slightly

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Fig. 5. Simulated impact of a copper particle on a copper sub-strate, for the initial velocities of 500 and 600 m/s. The arrowsrepresent the velocities of nodes at the respective surfaces ofparticle and substrate, and the contours indicate temperature dis-tribution.

Fig. 6. Flattening ratios as obtained from impact modellingand experimental investigations. Cross-sections are from anitrogen-sprayed copper powder. The data for copper particleson steel are taken from Ref. [13].

increasing with velocity. Profilometric investi-gations of single impacts of Cu-particles on steelshow larger flattening ratios which can be attri-buted to the higher hardness of the substratematerial [13]. For N2-sprayed powder, a mean par-ticle velocity can be calculated by CFD. For simi-lar velocities, the experimental flattening ratio isslightly smaller than that obtained by modelling.This difference can be attributed to the underestim-ation of particle diameters in cross sections andtherefore the overestimation of impact velocities.A better statistical basis can be obtained by com-paring diameters of adhering particles in top viewwith that of the initial powder feedstock (Fig. 7).For the different sizes of the initial powder popu-lation as calculated on the basis of sieving analy-ses, particle velocities were determined by CFD.With the velocity-dependent flattening ratioobtained from impact modelling, the top view sizedistribution of impacted particles of that particularpowder can be calculated. As shown in Fig. 7, cal-culated and measured top view size distributionsmatch reasonably. The slightly larger size of theexperimentally determined population can be attri-buted to the fact that over-hanging material or jetscontribute to an overestimation of splat sizes asobtained by SEM-analysis.

Fig. 7. Size distributions of a copper powder, before and afterspraying with helium. The splat diameters are also calculatedbased on the corresponding initial sizes and the CFD/impactmodelling. As a guide for the eye, Gaussian fits to the respectivepopulations are indicated.

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As shown in Fig. 8, the velocity of a node at theparticle surface can reach twice as much the impactvelocity. The figure also shows that the flow ofsubstrate material can take place at a speed greaterthan the initial impact velocity. This rapid increaseof velocity at the particle and substrate surfacescan be considered as a measure to indicate jet for-mation.

In order to obtain a better understanding of thelocalisation of plastic deformation, the strain andstress profiles along a radial path (as indicated inFig. 2) can be examined. Fig. 9a shows an exampleof such a profile for a 10 µm particle, 5 ns afterthe beginning of the impact, for two initial velo-cities of 300 and 900 m/s. The corresponding vari-ation of the equivalent (von Mises) flow stressalong the same path is shown in Fig. 9b. For theimpact velocity of 300 m/s, there is a monotonicincrease in flow stress by approaching the inter-face. This is a result of hardening effects due tohigh strains and strain rates at the interface. Thehigh plastic strain at the interface can on the otherhand result in softening due to adiabatic heating.This softening effect can dominate against thehardening effects at higher velocities and lead toan adiabatic shear instability at the interface. Thiseffect is observed clearly for an impact velocity of900 m/s. For this velocity, despite the high magni-tude of plastic strain at the interface, the shear

Fig. 8. Calculated velocity magnitudes at the critical nodes (asindicated in Fig. 2) on the respective surfaces of particle andsubstrate. The velocity of the monitored nodes at the particleand substrate surfaces exceeds the initial particle velocity, fora short period after the initial contact.

Fig. 9. Calculated profiles of (a) plastic strain and (b) flowstress along the radial path (as indicated in Fig. 2) for impactvelocities of 300 and 900 m/s. Both, strain and stress increasemonotonically by approaching the interface for the smallerimpact velocity. For the higher velocity, however, the shearstress falls to zero at the interface, indicating an adiabaticshear instability.

strength of material falls to values near zero dueto dominance of thermal softening over harden-ing effects.

4.2. Condition for adiabatic shear instability

Thermal softening and adiabatic shear instabilityat the interface could play a significant role in theset of microscopic processes leading to successfulbonding of particles in cold gas spraying. Theresults of the above analyses (Fig. 9) outline a win-

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dow of conditions for this to take place. The nextstep would be to obtain a more precise evaluationof the particle impact velocity corresponding to thethreshold of adiabatic shear instability. For suchevaluations, it should be considered that the vari-ation of the degree of deformation with distance isstrongly non-linear near the particle/substrate inter-face. On the other hand, the calculated field vari-ables, such as strain or temperature, would onlyrepresent the average respective values within aparticular element. For an element at the interface,where the plastic deformation is highly localised,such average values may therefore deviate substan-tially from the real values at the interface. This isso because the element size scales with the shortestdistance from the interface. A larger element sizewould consequently result in a larger deviation.This causes a part of the results from modelling todepend strongly on the element size. Fig. 10 showsthe calculated maximum temperature at the inter-face for various element sizes and impact velo-cities. As indicated in the figure, all calculated tem-peratures fall below melting temperature of copper.However, these vary with the element size, in thesense that higher temperatures are obtained whena finer mesh is chosen. As an initial solution tofilter out the element size effect, the “ real”maximum temperature has been worked out by lin-ear extrapolation to “zero element size” . This real

Fig. 10. Calculated maximum temperatures for variouselement sizes and impact velocities, showing an increase oftemperature with decreasing element size. Extrapolation ofmaximum temperature to zero element size coincides with themelting temperature of copper for an impact velocity of 545m/s.

maximum temperature coincides with the meltingtemperature of copper for an impact velocity of545 m/s. In a first approximation, therefore, thisvalue could be taken as corresponding to the thres-hold of an adiabatic instability.

In subsequent analyses, the problem of elementsize dependence was alleviated by refining themesh structure near the interface, down to elementsizes of about 0.2 µm. Fig. 11 shows the develop-ment of plastic strain, temperature and flow stressin a so-called critical element at the particle surfacewhich undergoes the highest amount of defor-mation within the particle. As shown in Fig. 11a,the plastic strain increases very rapidly at a rate ofup to 0.5 × 109 s�1, before reaching its final valueof about 4 for three velocities of 450, 500 and 550m/s. For the velocity of 580 m/s the trend is never-theless different, showing a further increase ofstrain up to a final value of about 10. The increaseof strain at this velocity could be the result of achange in mechanism from plastic to viscous flow.This will be discussed below in more detail. Thetemporal development of temperature in the criticalelement (Fig. 11b) is similar to that of strain, butwith small differences which are due to differencesin frictional and viscous dissipations. The heatingrates for all impact velocities are around 109 K/sin the first step. For the impact velocity of 580 m/s,the temperature approaches the melting tempera-ture of copper, whereas in other cases it remainswell below it. Fig. 11c shows the temporal devel-opment of the equivalent stress. For lower ormedium velocities of up to 550 m/s, there is a dropin stress after 0.05 µs, which can be attributed tothe loading conditions from the substrate. But forthe impact velocity of 580 m/s, there is a signifi-cant change in trend of temporal development ofstress at an earlier time of 0.03 µs. Beyond thistime, the stress variation is accompanied with largefluctuations, as well as a decrease in the overallvalue. The change in stress variation coincides withthe increase of strain and temperature in the criticalelement, as shown in Fig. 11. The fluctuations intemporal development of stress at a high velocity(580 m/s) could be explained with respect to achange in deformation mechanism from plastic toviscous flow. Near the conditions for thermalsoftening, the resistance of material to shear flow

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Fig. 11. Calculated temporal development of (a) plastic strain(b) temperature and (c) flow stress at the critical node of asprayed particle (as indicated in Fig. 2) for various impact velo-cities. There is a change in trend of variation of these variableswith time, as the initial particle velocity is increased from 550to 580 m/s, indicating a shear instability.

is generally low. This means that by approachingthe melting temperature, the material loses its shearstrength and undergoes excessive deformation forany amount of imposed shear stress. On the otherhand, this excessive deformation would create aviscous-type resistance which would hinder furtherdeformation, particularly under high pressure. Inthe present analysis, this effect will be due to theincrease in the second term of the Johnson-Cookmodel which accounts for strain-rate hardening.Additionally, the loading conditions at the particlesurface may change as a result of changes instrength, thus contributing to further fluctuations instress. A successive interplay of these three effectsresults in large-amplitude fluctuations as observedin the stress plots. It should be noted that thesefluctuations may not accurately represent the realstate of stress at the interface, as they are alsoinfluenced by the selected calculations parameterssuch as mesh size. However, they are stillimportant features in the results of modelling, asthey signify the beginning of a shear instability.The overall level of stress decreases with timebecause of accumulating viscous dissipation whichresults in further increase of temperature anddecrease of strength. The rapid changes of fieldvariables such as strain, temperature and stress fora critical element show that transitions from plasticflow to viscous flow can occur within a narrowrange of velocity between 550 and 580 m/s. Takinginto account the effect of element size, whichwould lead to an overestimation of the velocity foradiabatic shear instability, the latter result can beconsidered to be in agreement with the transitionvelocity of 545 m/s as estimated earlier (Fig. 10).

4.3. Non-uniformities at the particle–substrateinterface

The interpretation of the results of modelling hasso far been based on the development of field vari-ables at selected nodes at the interface. In order toobtain a better insight to the bonding mechanismin cold gas spraying, several other factors have tobe taken into account. It would for instance be ofprime importance to know not only the thresholdof transition to thermal softening but also theextent of plastic deformation and viscous flow at

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the interface. This can be illustrated by plotting thefield variables such as the equivalent plastic strainalong a meridian path (Fig. 2) on the particle sur-face. Fig. 12 shows examples of such plots corre-sponding to various times and impact velocities.The plots also indicate that the contact areabetween particle and substrate grows with the pro-gress of impact and with increasing the impactvelocity. For short and medium times after the

Fig. 12. Calculated profile of plastic strain along the meridianpath on the particle surface, for various times after impact, forimpact velocity of (a) 550 and (b) 580 m/s. The limit of thecontact zone is indicated by vertical lines on plot (a). The strainincreases always monotonically from the initial point of contact(south pole) up to a maximum at this limit. The extent of regionwhich undergoes adiabatic shear instability (b) is limited onlyto a small fraction of the overall contact area.

initial contact, there is an increase of plastic strainwith distance from the south pole up to the edgeof the contact zone. The increase of plastic strainwith time at specific distances along the path ishowever small. The condition for shear instabilityis manifested here by a sharp rise in the plasticstrain near the edge of the contact zone for theimpact velocity of 580 m/s. The temperature distri-bution along a meridian path is similar to that ofstrain, though this has not been shown here. Again,the start of adiabatic instability is manifested by asharp increase of temperature to values near themelting temperature of copper. The only differenceis that the rise in temperature near shear instabilityconditions is less pronounced than that in strain.Similarly, this could be attributed to the changefrom plastic to viscous dissipation of kineticenergy into heat.

4.4. Three-dimensional simulation

Axisymmetric models can be applied only toperpendicular impact of a single particle. In orderto simulate interaction of particles during depo-sition, three-dimensional models should be used.Fig. 13 shows the result of a three-dimensionalsimulation for the overlapping impact of two cop-per particles of 5 µm diameter on a copper sub-strate. With respect to realistic impact conditions,initial velocities and temperatures were assumed tobe 600 m/s and 200 °C, respectively. The initialdistance between the two particles allows embed-ding of the first particle before the second particlehits the substrate (Fig. 13a). The formation of ajet-type ring of material around the impact zone isevident for both particles (Fig. 13b). However, theformation of the jet for the second particle isinfluenced by the change in morphology andproperties of the underlying substrate as a result ofthe first impact (Fig. 13c). It should be noted thatthe relatively shorter time scale in three-dimen-sional analysis is due to the relatively smaller par-ticle size as compared to those assumed in the axi-symmetric models.

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Fig. 13. Three-dimensional simulation of impact of two 5 µmparticles with initial impact velocity of 600 m/s, at 5, 10, and 15ns after the initial contact of the first particle with the substrate.

5. Discussion

5.1. Criteria for bonding

Bonding of particles in cold gas spraying can beinfluenced by various factors. These factors rangefrom geometrical parameters such as the contactsurface area, the depth and width of crater [13] tothermomechanical parameters such as plasticstrain, flow stress, pressure, and temperature at theinterface. These parameters are in turn influencedby the particle impact velocity. It would thereforebe reasonable to assume that at velocities near thecritical velocity, these parameters also would reacha critical value, or would show a change in trendin their variation with velocity. Postulation of anycriteria for bonding would require determination ofthese critical values and conditions.

In explosive welding, the formation of a jet atthe interface is often considered as a criterion forsuccessful bonding [19]. This concept might seemto be applicable to the bonding of particles in coldgas spraying. Modelling of particle impact on arigid substrate illustrates clearly the formation ofa jet-type flow of material at the interface, but forimpact on substrates of the same material, the jetis morphologically less prominent. In addition,depending on the calculation parameters such asthe mesh size, a jet-type flow of material couldoccur over a wide range of impact velocities, thusmaking the criterion ineffective.

The results of the modelling show a change intrend of almost all key parameters near the particleinterface with the beginning of adiabatic shearinstability at the interface. Adiabatic shear insta-bility is characteristically associated with highstrain rate deformation. This phenomenon can bea cause for failure of mechanical componentswhich are subject to high strain rate deformation.In cold spraying of copper, on the other hand, thevelocity at the threshold of this instability is closeto the experimentally evaluated critical velocity forcopper [12]. This suggests that adiabatic shearinstability could be the cause for bonding of par-ticles in cold gas spraying. Based on this hypoth-esis, a criterion for bonding can be formulated bytaking the critical velocity as being the same asthat required for reaching adiabatic shear insta-

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bility. This criterion allows the prediction of thecritical velocity for a given material and particletemperature. However, it should be noted that theabove criterion describes only one of several con-ditions required for bonding of particles. Forinstance, if the contact time is too short, or theapplied tensile stress at the interface is too high, aparticle may bounce back before the conditions forbonding are achieved. A particle impinging thesubstrate at an angle is an example of this case. Inan angular impact, an additional temperature risecan occur at the interface due to frictional dissi-pation, which could promote shear instability.However, the tangential component of momentumof the particle in an angular impact would createa tensile force at the interface, which could be largeenough to detach the particle from the substrate,despite the occurrence of shear instability at theinterface.

5.2. Influence of process and materialsparameters on critical velocities

By using the experimental method describedpreviously, a critical velocity of 570 m/s wasdetermined for cold gas spraying of a gas-atomisedcopper powder. This result is in good agreementwith the transition velocities of 545 and 580 m/sas obtained from the modelling. To extend theresults of the modelling to other materials andspraying conditions, the effect of various processparameters and material properties on the calcu-lated critical velocity can be estimated by using themethod described in Appendix A. This method canbe used to estimate the effect of small changes inmaterial and process parameters on the criticalvelocity. Examples of these parameters and theirrespective effects on the critical velocity are givenin Table 2. The sign of the calculated derivativesindicates that the critical velocity increases withincreasing yield stress and melting temperature,whereas it decreases with increasing density andparticle temperature. For a typical range of materialproperties and process parameters, yield stress andmelting temperature have a smaller effect on thecritical velocity as compared to density and particletemperature. For instance, for each 1 g/cm3

increase in density, there will be 14 m/s decrease

Table 2Rate of change of critical velocity (m/s) with respect to materialand process parameters

Property (ai) Unit ∂vcr

∂ai

Yield stress MPa 0.1Density g/cm3 �14Elastic modulus GPa 0.008Melting temperature K 0.08Particle temperature K �0.4

in the critical velocity. The overall effect of theseparameters can be summarised into the followingsimple formula

vcr � 667 � 14r � 0.08Tm � 0.1su � 0.4Ti (3)

where r is the density in g/cm3, Tm is the meltingtemperature in °C, su is the ultimate strength inMPa and Ti is the initial particle temperature in °C.By using respective values of density and meltingtemperature for aluminium, for instance, the criti-cal velocity for this material can be estimatedroughly to be about 620 m/s. Fig. 14 shows thecritical velocities for a number of other materialswhich are estimated by using this approach forqualitative comparison. It should be stressed again

Fig. 14. The critical velocities of various materials, calculated(FEA) and measured (Exp.) for copper and aluminium, and esti-mated (Eq. (3)) for other materials by the approach describedin Appendix A, by taking copper as reference material and con-sidering only the differences in density and melting tempera-tures.

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that the formula in Appendix A is valid only forsmall changes of involved parameters. For largerdeviations from the reference condition, such as inthe case of aluminium, the critical velocity shouldbe calculated through a complete numerical analy-sis for the given set of spray parameters andmaterials properties. Performing such an analysisfor aluminium, for instance, results in a criticalvelocity of 675 m/s, which is in a better agreementwith the experimentally evaluated value of 660m/s.

5.3. Deformation morphologies and the extent ofbonded interfaces

The calculated deformation patterns as obtainedby axisymmetric and three-dimensional modelling,are in reasonable agreement with the observedmorphologies. This includes, in particular, the for-mation of a jet-type flow of material at the inter-face. There is also an agreement between the calcu-lated and measured flattening ratios in this study.As shown in Fig. 7, the calculated and measuredtop view size distributions match also reasonablywell. Differences between the results of modellingand experiments can be attributed partly to theerrors arising in estimating flattening ratios fromcross sections, and also in measuring top viewdiameters in the presence of jet-type features.Nevertheless, profilometric investigation of singleimpacts of copper particles on steel shows signifi-cantly larger flattening ratios [13]. This latter dif-ference can be attributed to the difference in hard-ness of the substrate materials. The results ofmodelling also show that extensive plastic defor-mation at the critical velocity is confined to anouter region within the particle/substrate interface.This means that softening of the entire particle–substrate interface would require higher kineticenergies of impacting particles. On the other hand,the good correlation between the modelled tran-sition velocity and the experimentally observedcritical velocities suggests that attaining high strainrates due to shear instabilities is required only fora localised area to achieve successful bonding. Arough estimation shows that for impact velocitiesof 580 m/s, only about 15–25% of the entire inter-face is subject to shear instability. So far, the mod-

elling has not been able to explain if this amountof surface area would be sufficient for a successfulbonding. On the other hand, this could explain therelatively low bond strength of cold gas sprayedcoatings as compared to bulk material or thermallysprayed coatings. Therefore, tensile tests could beused to supply additional information for assessingthe extent of critical conditions at the interface.The bond strength of cold gas sprayed coatings,with particle velocities just exceeding critical velo-city, are only about 20% of the ultimate tensilestrength of bulk copper. This suggests that the ratioof surface area of the bonded interfaces to the totalarea should be about the same fraction. Taking intoaccount the effect of stress concentrations at theedge of un-bonded areas, and contributions byshear due to impact morphologies, the results oftensile tests can be considered to be in good agree-ment with that of the modelling.

5.4. Open questions

The material properties used for the currentanalysis have been based on the data reported forbulk materials, and for strain rates which are byorders of magnitude smaller than those in cold gasspraying, in which strain rates can be in the rangeof 0.5 × 109/s per nanosecond. In addition, the cur-rent analysis does not take into account any phasechanges in the material. Phase change could inparticular influence the results of modelling forsystems with allotropic transformations, such as foriron. Moreover, the properties of the feedstockpowder may deviate substantially from those ofbulk materials due to effects from high quenchingrates and oxidation of powder during atomisation.

6. Conclusions

The present work demonstrates that plasticdeformation phenomena in cold gas spraying canbe modelled appropriately through finite elementanalysis. The results of the modelling provide abasis for understanding the bonding process. Basedon these results, the bonding of particles can beattributed to adiabatic shear instabilities whichoccurs at the particle/substrate or particle/particle

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interfaces at high velocities. The modelling alsoshows a very non-uniform development of strainand temperature at the interface, suggesting thatthis bonding is confined to a fraction of the inter-acting surfaces. This result is consistent with therelatively low strength of the copper coatings asdetermined experimentally. The analysis also sug-gests that density and particle temperature havesignificant effects on the critical velocity and arethus two of the most influential parameters in coldgas spraying. Overall, the analysis supplies a toolto predict critical velocities and to optimise sprayparameters for different materials.

Acknowledgements

A part of this work was supported by DeutscheForschungsgemeinschaft under grant No.KR1109/3-1. HA is grateful to GKSS and to theUniversity of the Federal Armed Forces Hamburgfor support throughout a visiting professorship.

Appendix A. Influences on critical velocities

The critical velocity, and its total differential,can be expressed as functions of process andmaterial parameters a1,…,an as follows

vcr � f(a1,a2,…,an) (A.1)

dvcr �∂f∂a1

da1 �∂f∂a2

da2 � % �∂f∂an

dan (A.2)

Here, the maximum temperature at the interface,Tmax, is taken as a key intermediate parameter.Assuming that a unique relationship exists betweenTmax and the process parameters, a translation for-mula can be used to give

∂f∂ai

� �(∂Tmax /∂ai)(∂Tmax /∂v)

(A.3)

where v is the impact velocity. In a first approxi-mation, these partial derivatives can be assumed tobe process-independent constants. For smallchanges of parameters with respect to a referencecondition, therefore, the critical velocity for thenew condition can be obtained as

vcr�vrefcr �� ∂v

∂Tmax��∂Tmax

∂a1�a1 � % �

∂Tmax

∂an

�an�(A.4)

in which vrefcr is the critical velocity for the reference

condition. By taking cold gas spraying of copper asthe reference condition, for instance, the referencevelocity shall take the value of 570 m/s. The partialderivatives of Tmax on the right hand side of theabove equation can be worked out numerically bycalculating changes in Tmax for small changes inthe respective parameters. For instance, the changein Tmax for 1 m/s increase in impact velocity hasbeen found to be 1.3 K, over a wide range ofimpact velocity. The other derivatives were calcu-lated in a similar manner with respect to para-meters such as yield stress, density, melting tem-perature, and particle temperature.

References

[1] Herman H, Sampath S, McCune R. MRS Bull 2000;25:17.[2] Fincke JR, Neiser RA. MRS Bull 2000;25:26.[3] Kreye H, Gartner F, Kirsten A, Schwetzke RH. High velo-

city oxy-fuel flame spraying state of the art, prospects andalternatives. In: Heinrich P, editor. Proceedings of theFifth HVOF Colloquium. Unterschleißheim (Germany):Gemeinschaft Thermisches Spritzen; 2000. p. 5.

[4] Richter HJ. Fundamental thermodynamic and fluiddynamic concepts related to high velocity oxy-fuel spray-ing. In: Heinrich P, editor. Proceedings of the Fifth HVOFColloquium. Unterschleißheim (Germany): GemeinschaftThermisches Spritzen; 2000. p. 4.

[5] Alkhimov AP, Kosareve VF, Papyrin AN. Dokl AkadNauk SSSR 1990;315:1062.

[6] Alkhimov AP, Papyrin AN, Kosarev VF, Nesterovich NI,Shushpanov MM. Gas-dynamic spray method for applyinga coating. US Patent 5,302,414; April 12, 1994.

[7] Alkhimov AP, Papyrin AN, Kosarev VF, Nesterovich NI,Shushpanov MM. Method and device for coating. Euro-pean Patent 0 484 533 B1; January 25, 1995.

[8] McCune RC, Papyrin AN, Hall JN, Riggs II WL,Zajchowski PH. An exploration of the cold-gas-dynamicspray method for several materials systems. In: BerndtCC, Sampath S, editors. Advances in thermal sprayscience and technology. Materials Park (OH): ASM Inter-national; 1995. p. 1.

[9] McCune RC, Donlon WT, Cartwright EL, Papyrin AN,Rybicki EF, Shadley JR. Characterization of copper andsteel coatings made by the cold-gas-dynamic spraymethod. In: Berndt CC, editor. Thermal spray: practical

Page 16: Bonding Mechanism

4394 H. Assadi et al. / Acta Materialia 51 (2003) 4379–4394

solutions for engineering problems. Materials Park (OH):ASM International; 1996. p. 397.

[10] Dykhuizen RC, Smith MF. J Thermal Spray Tech1998;7:205.

[11] Segall AE, Papyrin AN, Conway JC, Shapiro D. JOM1998;9:52.

[12] Stoltenhoff T, Kreye H, Richter H. J Thermal SprayTech 2002;11:542.

[13] Dykhuizen RC, Smith MF, Gilmore DL, Neiser RA, JiangX, Sampath S. J Thermal Spray Tech 1999;8:559.

[14] Borchers C, Stoltenhoff T, Gartner F, Kreye H, Assadi H.MRS Symp Proc 2001;673:P7.

[15] Papyrin AN, Kosarev VF, Klinkow SV, Alkhimov AP. Onthe interaction of high speed particles with a substrateunder the cold spraying. In: Lugscheider EF, Berndt CC,editors. Proceedings of the International Thermal SprayConference 2002. Dusseldorf (Germany): DVS-Verlag;2002. p. 380.

[16] Brasher DG, Butler DJ. Adv Mater Process 1995;147:37.[17] Murr LE, editor. Shock waves for industrial applications.

New Jersey: Noyes Publications; 1988.[18] Prummer R. Explosivverdichtung pulvriger Substanzen:

Grundlagen, Verfahren, Ergebnisse. Berlin: Springer,1987.

[19] Deribas AA, Zakharenko ID. Fizika Goreniya i Vzryva1974;10:409.

[20] Hammerschmidt M, Kreye H. Microstructure and bondingmechanism in explosive welding. In: Meyers MA, MurrLE, editors. Shock waves and high strain-rate phenomenain metals. New York: Plenum Press; 1981. p. 961.

[21] Zlobin SB, Pai VV, Yakovlev IV, Kuz’min GE. CombustExplosion Shock Waves 2000;36:256.

[22] Meyers MA, Thadhani NN, Yu LH. Explosive shock waveconsolidation of metal and ceramic powder. In: Murr LE,editor. Shock waves for industrial applications. New Jer-sey: Noyes Publications; 1988. p. 265.

[23] Nesterenko VF. Dynamic loading of porous materials:potential and restrictions for novel materials applications.In: Murr LE, Staudhammer KP, Meyers MA, editors. Met-allurgical and materials application of shock wave andhigh strain rate phenomena. Amsterdam: Elsevier; 1995.p. 3.

[24] Kasiray P, Vreeland T, Schwarz RB, Ahrens TJ. ActaMetall 1984;32:1235.

[25] Raybould D. J Mater Sci 1981;16:589.[26] Staudhammer KP, Murr LE. Principles and applications

of shock wave compaction and consolidation of powderedmaterials. In: Murr LE, editor. Shock waves for industrialapplications. New Jersey: Noyes Publications; 1988. p.237.

[27] Schwarz RB, Kasiraj P, Vreeland T, Ahrens TJ. Acta Met-all 1984;32:1243.

[28] Rosato AD, Vreland T, Prinz FB. Int Mater Rev1991;36:45.

[29] Gilmore DL, Dykhuizen RC, Neiser RA, Roemer TJ,Smith MF. J Thermal Spray Tech 1999;8:576.

[30] Papyrin A. Adv Mater Process 2001;159:49.[31] ABAQUS 6.2-1. user manual. Hibbitt, Karlsson & Soer-

ensen. Pawtucket (RI), 2001.[32] Olson GB, Mescall JF, Azrin M. Adiabatic deformation

and strain localization. In: Meyers MA, Murr LE, editors.Shock waves and high-strain-rate phenomena in metals.New York: Plenum Press; 1981. p. 22-1.

[33] Tzou DY. ASME J Heat Transfer 1995;117:8.[34] Prakash GS, Reddy SS, Das SK, Sundararajan T, Seethar-

amu KN. Num Heat Transfer A 2000;38:513.[35] Kapoor R, Nemat-Nasser S. Mech Mater 1998;28:1.[36] Johnson GR. Prog Astronaut Aeronaut 1993;155:165.[37] Karpp RR. Prog Astronaut Aeronaut 1993;155:223.[38] Kamler F, Niessen P, Pick RJ. Can J Phys 1995;73:295.[39] Nemat-Nasser S, Li Y. Acta Mater 1998;46:565.