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Finite element analysis of the contact stresses in diamond coatings subjected to a uniform normal load

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Finite element analysis of the contact stresses in diamond coatings subjected to a uniform normal load
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  Diamond and Related Materials 9 (2000) 26–36www.elsevier.com /  locate /  diamond Finite element analysis of the contact stresses in diamond coatingssubjected to a uniform normal load M.J. Fagan *, S.J. Park, L. Wang School of Engineering, University of Hull, Cottingham Road, Hull HU6 7RX, UK  Accepted 7 September 1999 Abstract Finite element analysis is used to predict the stress distribution in a diamond-coated substrate. The model examines the e ff  ectof coating thickness in detail, and compares and validates the finite element predictions with experimental results obtained fromthe soft impressor method applied to a series of diamond coatings on a silicon substrate. The results demonstrate that theperformance of a coating can be related to the ratio of coating thickness to contact radius ( t /  a ), so for example, for the siliconsubstrate most of the protection is achieved when the  t /  a  ratio is about 0.1, and for the diamond coating there is no advantage inusing coatings with  t /  a  ratios grater than 1.3. © 2000 Elsevier Science S.A. All rights reserved. Keywords:  Contact load; Diamond coating; Finite element analysis; Modelling; Stress 1. Introduction  The life of a diamond coating will probably not belimited by wear, but by failure of the coating system,The use of a coating to protect a substrate is now a i.e. failure of either the coating, the interface or thewell-established technique to improve the mechanical substrate. Hence, to select and use these coatings success-and tribological properties of components in many fully it is essential that the development and distributionengineering applications. There has been interest for of the internal stress distribution in the coating systemsome time in the use of diamond coatings, because of  is thoroughly understood.diamond’s excellent properties, such as high hardness, The work described in this paper is concerned withlow friction and chemical inertness, making it attractive a detailed study of these stresses by finite elementfor a wide range of application. However, while there analysis. It focuses in particular on diamond coatingshas been and continues to be considerable fundamental applied to a silicon substrate, which might not be aresearch undertaken on the development of diamond particularly practical combination of materials, but iscoatings on a wide range of di ff  erent substrates, the convenient in this research for a number of reasons.number of genuine applications of the coatings taking Firstly, diamond can be grown relatively easily, reliablyadvantage of its superior mechanical properties is rela- and accurately on silicon and the residual stresses duetively modest. Certainly the flood of products using to the mismatch in thermal expansions coe ffi cient arediamond coatings that was promised when the first not as large as in most other combinations. Also, thecoatings were produced, has not materialised. The combination is well suited to the soft impressor tech-reason for this is due in part to the complexities of the nique for evaluating the e ff  ectiveness of coatings, whichdeposition process and the relatively slow growth rates. is used here to validate the finite element model, and aHowever, it is also a result of a general lack of under- large set of experimental data was available for thisstanding of the real capabilities and operation of dia- system. Finally, it was found that a number of di ff  erentmond coatings and undefined procedures for selecting failure modes were observed with this configuration,the optimum coating for a given application. which should all be predicted by the finite elementmodel, allowing the accuracy and validity of the modelto be tested to the full. * Corresponding author. Tel: + 44-1482-465058; Thus, the purpose of this present work is to start the fax: + 44-1482-466664. E-mail address:  m.j.fagan@eng.hull.ac.uk (M.J. Fagan)  development of guidelines for the selection of the opti- 0925-9635 /  00 /  $ - see front matter © 2000 Elsevier Science S.A. All rights reserved.PII: S0925-9635(99)00190-9  27 M.J. Fagan et al.  /   Diamond and Related Materials 9 (2000) 26–36  mum coating configuration for a given loading condi- stresses for contact of arbitrary surface shapes [5]. Theypresented numerical results for the case when the layertion, an understanding of the failure mechanisms of diamond coatings, and in the process, validation of the to substrate shear modulus was 3, simulating a steellayer on an aluminium substrate.soft impressor technique as a method for testing theintegrity of coatings. The e ff  ect of thin hard coatings on the stressesinduced in brittle materials by a rigid spherical indenterwas analysed using finite element analysis by van der 1.1. Modelling and analysis of contract stresses in coated and uncoated materials  Zwaag and Field [6]. They considered coatings of di ff  erent sti ff  ness and thickness and found that themaximum tensile stress in the substrate was reducedLove analysed the distribution of stress in a semi-infinite elastic and isotropic material, subjected to a with increasing Young’s modulus and thickness, but thiswas accompanied by an increase in the tensile stress inuniform pressure over a circular area, using potentialfunctions in 1929 [1]. He showed the stresses along the the coating. However, the maximum coating thicknessto contact radius ratio,  t /  a  they analysed was 0.2, i.e.surface of the solid were:only thin coatings were considered.Djabella and Arnell considered the cases of Hertzian s r = P 2 p ( S  1 − 2 n S  2 ),  s h = P 2 p (2 n S  1 − S  2 ),spherical and cylindrical contact on single layer [7] anddouble layer systems [8]. They did not consider particu-lar coating materials, but instead the ratio of Young’s s z = P 2 p ( S  1 − S  2 ),  t rz = 0,moduli of the coating(s) and substrate, where the sub-strate had a modulus of 200 GPa, i.e. was notionallywheresteel. They found that the stresses in the systems showeda complex relationship between the ratio of material S  1 = P  02 p  ( r − cos  w ) cos  wr 2 − 1 − 2 r  cos  w d w properties and ratio of coating thickness to contactradius, but that a double layer coating could be used toreduce the stresses.and  S  2 = P  02 p  sin 2 wr 2 − 1 − 2 r  cos  w d w , 1.2. The soft impressor method  and  P  is the applied pressure,  a  the contact radius,  v  isPoisson’s ratio,  r ,  h  and  z  the polar coordinate directions, The soft impressor method uses a blunted softimpressor to apply a uniform pressure to a surfaceand  r = r /  a .Several analytical methods of determining the contact coating on a substrate until failure is detected, asillustrated in Fig. 1, and can be used to quantify thepressures, deformations and stresses in normally loadedcoated materials have been developed. Most consideredHertzian contact, where a ball or cylinder is in contactwith a plane coated surface. The solutions allow thee ff  ects of varying parameters such as coating thickness,geometry, loading and material properties to be investi-gated. However, the reported models do not ofteninclude experimental comparisons and hence agreementbetween theory and real contacts is di ffi cult to make.For example, Hannah considered the problem of athin elastic layer on a rigid substrate contacted by arigid indenter and obtained results for a limited numberof cases [2]. Meijers extended the work to consider anyvalue of contact radius to coating thickness ratio [3]. Ageneralised solution for an elastic indenter contacting alayered elastic solid was attempted by Gupta andWalowit who presented numerical results for variousratios of elastic modulus of layer and substrate for arigid indenter and including the case where the elasticmodulus of the indenter is the same as that of thesubstrate [4].Chui and Harnett calculated the surface deformationof a layered half space under a uniform rectangular load  Fig. 1. The soft impressor method uses a blunted soft impressor toapply a uniform pressure to a surface coating until failure is detected. and presented an analytical solution of the subsurface  28  M.J. Fagan et al.  /   Diamond and Related Materials 9 (2000) 26–36  Table 1 ~ 900 ° C, tantalum filament temperature ~ 2000 ° C, pro- Failure modes of di ff  erent diamond coating thicknesses tested by the cess pressure of 20 Torr and total gas flow 200 sccm.) soft impressor method (from Ref. [10])Coating Applied Failure mode Integritythickness ( m m) pressure (MPa) (see Fig. 2) ratio 2. Finite element analysis 0 1600 Substrate crack 1.02 1630 Diamond ring crack 1.02 Finite element analysis allows a quantitative assess- 3.5 2740 Diamond ring crack 1.71 ment of the e ff  ect of coating thickness, contact radius 12 3470 Substrate crack /  ring 2.17 and material properties of the coating and substrate. delamination The models used here were parameterised to allow the 22 6160 Substrate crack /  ring 3.85delamination  above properties to be varied with the minimum of  48 8000 Substrate crack /  ring 5.0 additional work and consisted of three areas as shown delamination in Fig. 3, namely the coating, substrate and interface. 90 8000 Complete 5.0 The interface area was present in the model to allow delamination the input of a third material property to take accountof any mixed layer between coating and substrate. Themodel was meshed using second order axisymmetricprotection given by the coating to the substrate [9]. Thelevel of protection achieved is referred to as the ‘integrity elements, the uniform pressure,  P , thus being appliedover a circle of radius  a . This therefore represents simpleratio’ and is defined as the ratio of the mean contactpressure to cause failure of the coated system to the static contact or sliding with negligible friction. Themodels were constrained in the  z -direction along themean contact pressure to cause failure of the uncoatedsubstrate material. lower edge of the substrate, and the material propertiesused for the diamond coatings and silicon are summar-The technique has been used in particular to measurethe e ff  ectiveness of diamond coatings of di ff  erent thick- ised in Table 2. The interface was assumed to have thesame properties as the coating, i.e. the coating wasnesses on a silicon substrate [10], and sample results areincluded in Table 1. Diamond coatings are of course perfectly bonded to the substrate.Model convergence is essential if finite element resultswell suited to this test because they can be transparentallowing the state of the coating, substrate and interface are to be reliable and was tested in this case by systemati-cally increasing the mesh density and plotting sampleto be monitored visually. A number of modes of failuremay be observed with diamond and this material combi- output. The models were assumed to have convergedwhen an increase in mesh density did not result in anation, and although these will not be discussed in detailhere the primary indications of the failures are illustrated significant change in output. It was found that at least10 elements were required through the thickness of thein Fig. 2. It is worth noting, however, that substratecracking was not necessarily seen with the diamond ring coating, so that the final models consisted of approxi-mately 8000 elements. Fig. 3 shows a detail of the verycrack failures, but it was invariably observed with thering delaminations. fine graduated mesh near the edge of the contact region.A series of analyses was conducted where the coating(The diamond coatings discussed here were all depos-ited by HFCVD using a 1 %  methane in hydrogen gas thickness,  t , was varied between 2 and 90  m m (and0.02 ≤ t /  a ≥ 2.3), with di ff  erent applied uniform pres-mixture. The conditions were: substrate temperature Fig. 2. Primary modes of failure of a diamond coating on a silicon substrate.  29 M.J. Fagan et al.  /   Diamond and Related Materials 9 (2000) 26–36  Fig. 3. Finite element model of the coated substrate, with the detail of the mesh at the edge of the contact region. sures, up to 8 GPa, and corresponding contact radii. level in the material is very sensitive to the value of Poisson’s ratio used in the analysis, as illustrated inThe uncoated case was also analysed, allowing a com-parison with analytical and experimental results, allow- Fig. 4 which shows the variation of maximum resolvedtensile stress with Poisson’s ratio. The graph shows thating further validation of the model.As discussed later, the results of the finite element using a value of 0.23 for the Poisson’s ratio (rather thanthe 0.2 in Table 2) in the finite element model wouldmodels presented here, in common with other publishedfinite element studies, do not include the e ff  ects of have resulted in a maximum resolved shear stress of 0.27 GPa.residual stresses in the system, which can be significant.This is to be the subject of a forthcoming paper. A detailed verification of the model’s results wereundertaken by comparison with Love’s analytical solu-tion for the problem [1], and the values predicted wereproven to be accurate. Thus, Fig. 5 shows the variation 3. Results of the axial, radial and maximum shear stresses downthe centreline of the model predicted by the finite elementThe model was first used to simulate the testing of uncoated silicon by the soft impressor method. The model; the results are essentially identical to the analyti-cal solution when presented graphically. For example,pressure required to initiate failure was found experi-mentally to be 1.6 GPa over a contact radius of 88.4  m m.Failure is observed to occur on the {111} cleavage planewhen the loading is applied to the (001) surface, that isat 54 ° 44 ∞ to the sample’s surface. The respective maxi-mum (resolved) tensile stress predicted by the finiteelement model was found to be 0.30 GPa which issimilar to the cleavage stress for silicon of 0.27 GPa[11], and occurs just outside the contact radius asexpected. However, it should be noted that the stress Table 2Material properties used in the finite element modelsDiamond SiliconYoung’s modulus (GPa) 1050 110Fig. 4. Variation of the maximum resolved tensile stress in uncoatedPoisson’s ratio 0.1 0.2silicon as a function of Poisson’s ratio.  30  M.J. Fagan et al.  /   Diamond and Related Materials 9 (2000) 26–36  occur on the surface of the diamond near the edge of the contact. For the thickest coating the maximum stressoccurs at the diamond /  silicon interface beneath thecentre of contact.More specifically, Fig. 7 presents the variation inmaximum tensile stress in the diamond with coatingthickness and contact radius. At the  t /  a  ratio increasesfrom zero, the stress decreases until a local minimum isreached when it starts to increase again to a localmaximum and then decreases again. This reflects thethin and thick behaviour of the coating, with the verythin coatings only able to support membrane-type loadsand the thickest coatings dominated by bending behavi-our. The variation of the maximum tensile  surface stresses in the diamond as a function of the  t /  a  ratio isshown in Fig. 8. This stress decreases as the ratioincreases, until a value of approximately 1.3 is reached,when the surface stress starts gradually to increase again.The reason for this behaviour is revealed in Fig. 9, which Fig. 5. Variation of the axial, radial and maximum shear stresses down shows the distribution of the maximum surface stress in the centreline of an uncoated silicon specimen (as predicted analyticallyand by the finite element model).  the diamond for the di ff  erent coating thicknesses. Forthin coatings, the maximum stress occurs  near  the edgeof the contact, but as the thickness increases, the magni-the maximum shear stress predicted by the theory istude of the stress decreases and the location of the570.5 MPa occurring at a  z /  a  ratio of 0.603, while themaximum moves away from the contact. However, forfinite element model predicts a value of 577.6 MPathicker coatings, the stress  at  the edge of the contactestimated to occur at  z /  a = 0.595. The error in both thebecomes significant, and increases to dominate thevalue and position is just 1.2 % .stress field.For coated substrates it is convenient to discuss theFor the substrate, Fig. 10 shows how the maximumgeneral performance of di ff  erent coating configurationstensile stress decreases as the coating thickness increases.in terms of the thickness to contact radius ration,  t /  a ,The stress appears to reach a plateau, and then startsand to normalise the induced stresses with respect toto decrease again coinciding with the minimum observedthe applied pressure. For the diamond-coated siliconin the coating stress distribution (in Fig. 8). Somemodels the results show di ff  erent behaviours for ‘thin’quantification of the true e ff  ectiveness of the di ff  erentand ‘thick’ coatings. For example, contour plots forcoating configurations is presented in Fig. 11, whichcoating thickness of 3.5, 12.0 and 22.0  m m subjected tocompares the maximum tensile stress in coated sub-the same loading conditions are shown in Fig. 6, corre-strates with that predicted in an uncoated sample againsponding to  t /  a  ratios of 0.09, 0.30 and 0.56, respectively.For the thinnest coating the maximum stress is seen to as a function of coating thickness and contact radius. It Fig. 6. Contour plots of maximum principal stress for coatings of di ff  erent thickness with the same applied load ( t /  a  ratios = 0.09, 0.30 and 0.56respectively).
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