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Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates

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Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates.

Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates.

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  • 1. Failure of Brittle Coatings on Ductile Metallic Substrates
  • 2. Failure of Brittle Coatings on Ductile Metallic Substrates Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus Prof. dr. ir. J. T. Fokkema, voorzitter van het College voor Promoties,in het openbaar te verdedigen op dinsdag 26 februari 2002 om 16:00 uur door Adnan Jawdat Judeh ABDUL-BAQI, Master of Science, Bergen, Norway geboren te Zawieh, Palestine.
  • 3. Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. E. van der GiessenSamenstelling promotiecommissie: Rector Magnificus, voorzitter Prof. dr. ir. E. van der Giessen, Rijksuniversiteit Groningen, promotor Prof. dr. J.Th.M. de Hosson, Rijksuniversiteit Groningen Prof. dr. ir. M.G.D. Geers, Technische Universiteit Eindhoven Prof. dr. G. de With, Technische Universiteit Eindhoven Prof. dr. ir. F. van Keulen, Technische Universiteit Delft Dr. G.C.A.M. Janssen, Technische Universiteit DelftThe work of A.J.J. Abdul-Baqi was supported by the Program for Innovative Research, surfacetechnology (IOP oppervlakte technologie), under the contract number IOT96005.  Copyright c Shaker Publishing 2002All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,or transmitted, in any form or by any means, electronic, mechanical, photocopying, recordingor otherwise, without the prior permission of the publishers.Printed in The Netherlands.ISBN 90-423-0181-3Shaker Publishing B.V.St. Maartenslaan 266221 AX MaastrichtTel.: +31 43 3500424Fax: +31 43 3255090http://www.shaker.nl
  • 4. To my Family
  • 5. Contents1 Introduction 12 Indentation of bulk and coated materials 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Elastic contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Elastic-plastic contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Coated materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Indentation-induced interface delamination of a strong film on a ductile substrate 19 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Delamination of a strong film from a ductile substrate during indentation unload- ing 41 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 Indentation-induced cracking of brittle coatings on ductile substrates 65 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 Stress distribution in a perfect coating . . . . . . . . . . . . . . . . . . . . . . 70 5.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.5 Effect of geometrical, material and cohesive parameters . . . . . . . . . . . . . 77 5.6 Fracture energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Summary 89 v
  • 6. Samenvatting 91Propositions 93Stellingen 95Curriculum Vitae 97Acknowledgement 99 vi
  • 7. Chapter 1IntroductionHard coatings are usually applied to materials to enhance performance and reliability such aschemical resistance and wear resistance. Ceramic coatings, for example, are used as protectivelayers in many mechanical applications such as cutting tools. These coatings are usually brittleand the enhancement gained by the coating is always accompanied by the risk of its failureleading to a premature failure of otherwise long lasting systems. Failure may occur in thecoating itself or at the interface with the substrate. Therefore, mechanical characterization ofsuch systems, including the possible failure modes under various loading circumstances, iscritical for the understanding and the improvement of its performance. Indentation has become one of the most common methods to determine the mechanicalproperties of materials such as elastic properties, plastic properties and strength. In this test, anindenter is pushed into the surface of a sample under continuous recording of the applied loadand corresponding penetration depth (Weppelmann and Swain, 1996). Indenters have differentgeometries including spheres and cones. They are usually made of diamond due to its extremeproperties like hardness and stiffness. For hard coatings, indentation is one of the simplest testsin terms of sample preparation (Drory and Hutchinson, 1996). However, the interpretation ofindentation results still poses a big challenge. This has motivated extensive experimental aswell as theoretical studies which covers various indenter geometries and constitutive materialmodels. The material response in an indentation experiment is governed by both its mechanicalproperties and the indenter geometry. One of the most common outputs in indentation exper-iments is the indentation force versus the indentation depth data (load–displacement curve),from which material parameters can be extracted. This thesis provides an improved understanding of indentation-induced failure of systemscomprising a strong coating on relatively softer substrate. Qualitative description of the coatingand the interface fracture characteristics is inferred from failure events. In addition, estimationof the coating and the interface fracture energies from failure events as commonly done inindentation experiments is also discussed. The analysis is carried out numerically using a finitestrain, finite element method. An overview of the most common methods used to determinethe mechanical properties of materials by indentation is given in Chapter 2. Both the loadingand the unloading are modeled using the finite element method. The emphasis is based on theload versus displacement data in comparison with the prediction of some existing analytical and 1
  • 8. 2 Chapter 1empirical relations. The analysis in this chapter assumes that failure events do not occur duringindentation. This assumption holds true if the generated stresses do not reach the materialstrength; otherwise, failure is inevitable. The main failure events discussed in this thesis are interfacial delamination and coatingcracking. Crack initiation and propagation are modeled within a cohesive surface frameworkwhere the fracture characteristics of the material are embedded in a constitutive model for thecohesive surfaces. This model is a relation between the traction and the separation of the cohe-sive zone. It is mainly characterized by a peak traction which reflects the material load carryingcapability, and a fracture energy. Additional criteria for crack initiation and propagation are notrequired. The cohesive law we adopt in this study is the one given by Xu and Needleman (1993).The normal response in this law is motivated by the universal binding law of Rose and Ferrante(1981), while the tangential (shear) response is considered as entirely phenomenological. In modeling interfacial delamination, a single cohesive surface is placed along the interfaceprior to indentation. The coating is assumed to remain intact and failure is only allowed tooccur at the interface. Shear delamination (mode II) is possible during the loading stage ofthe indentation process as discussed in Chapter 3. It is found that a ring-shaped portion of thecoating, outside the contact region, is detached from the substrate. On the other hand, normaldelamination (mode I) can occur during the unloading stage as discussed in Chapter 4. Inthis case, a circular portion of the coating, directly under the contact region, is lifted off fromthe substrate. Delamination is imprinted on the load–displacement curve by a rather suddendecrease in the indentation stiffness. For relatively strong interfaces, the stiffness might evenbecome negative. This leads to a kink on the loading curve and a hump on the unloading curvein the case of shear and normal delamination, respectively. The latter has recently been observedexperimentally by Carvalho and De Hosson (2001). Coating cracking is one of the failure events frequently observed in indentation experiments.The simulation of coating cracking is presented in Chapter 5. Embedding cohesive zones inbetween all continuum elements in the coating leads to serious numerical problems in additionto an artificial enhancement of the overall compliance (Xu and Needleman, 1994). In thisstudy we adopt a procedure in which the number of cohesive zones is minimized and placedonly at precalculated locations. The interface between the coating and the substrate is alsomodeled by means of cohesive zones but with interface properties. It is shown that successivecircumferential through-thickness cracking occurs outside the contact region with crack spacingof the order of the coating thickness. Each cracking event is imprinted on the load–displacementcurve as a kink. Estimation of the interface and coating fracture energies from failure events is also investi-gated in Chapters 4 and 5, respectively. It is found that methods used in indentation experiments(Hainsworth et al., 1998; Li et al., 1997) generally result in overestimated values of the fractureenergy compared to the actual values. This is mainly attributed to the fact that, in such a highlynonlinear problem, these methods oversimplify the estimation of the energy release associatedwith the failure event.
  • 9. Introduction 3References Carvalho, N.J.M., De Hosson, J.Th.M., 2001. Characterization of mechanical properties of tungsten carbide/carbon multilayers: Cross-sectional electron microscopy and nanoin- dentation observations. J. Mater. Res. 16, 2213–2222. Drory, M.D., Hutchinson, J.W., 1996. Measurement of the adhesion of a brittle film on a ductile substrate by indentation. Proc. Roy. Soc. Lond. A 452, 2319–2341. Hainsworth, S.V., McGurk, M.R., Page, T.F., 1998. The effect of coating cracking on the indentation response of thin hard-coated systems. Surf. Coat. Technol. 102, 97–107. Li, X., Diao, D., Bhushan, B., 1997. Fracture mechanisms of thin amorphous carbon films in nanoindentation. Acta Mater. 45, 4453–4461. Rose, J.H., Ferrante, J., 1981. Universal binding energy curves for metals and bimetallic interfaces. Phys. Rev. Lett. 47, 675–678. Weppelmann, E., Swain, M.V., 1996. Investigation of the stresses and stress intensity factors responsible for fracture of thin protective films during ultra-micro indentation tests with spherical indenters. Thin Solid Films 286, 111–121. Xu, X.-P., Needleman, A., 1993. Void nucleation by inclusion debonding in a crystal matrix. Model. Simul. Mater. Sci. Eng. 1, 111–132. Xu, X.-P., Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, 1397–1434.
  • 10. 4 Chapter 1
  • 11. Chapter 2Indentation of bulk and coated materials Indentation experiments are widely used to measure mechanical properties of materials. Such properties are extracted from the material response to indentation by means of ana- lytical and empirical relations available in the literature. The material response is usually given in terms of load versus displacement data. In this chapter we will examine some of the existing relations and compare their predictions with our finite-element results. Inden- tation is modeled for two indenter geometries, namely spherical and conical. The response of purely elastic materials, elastic-plastic materials and coated materials is investigated.2.1 IntroductionIn the past few decades, indentation has become a powerful tool to determine the mechani-cal properties of materials such as elastic properties, plastic properties and strength. This hasmotivated extensive experimental as well as theoretical studies which cover various indentergeometries and material models. The most common indenter geometries are a sphere (Brinelltest), a cone (Rockwell test) and a rectangular pyramid (Vickers test). The response in an in-dentation experiment is governed by both the material properties and indenter geometry. The first analysis of the stresses arising from a frictionless contact between two elastic bod-ies was first studied by Heinrich Hertz in 1881 when he presented his theory to the BerlinPhysical Society (Johnson, 1985). The publication of his classic paper On the contact of elasticsolids in 1882 (Hertz, 1882) may be viewed, according to Johnson (1985), to have started thesubject of contact mechanics. However, developments in the Hertz theory did not appear in theliterature until the beginning of the 20th century (Johnson, 1985). The problem of determiningthe stress distribution within an elastic half space due to surface tractions and a concentratednormal force has been considered first by Boussinesq (1885). Based on his solution, partialnumerical results were derived later by Love for a flat-ended cylindrical punch (Love, 1929)and for a conical punch (Love, 1939). Starting in 1945, a more comprehensive treatment ofthe contact problem was followed up by Sneddon in a series of publications listed in (Sneddon,1965). He has derived analytical formulas which relate the applied load, the indentation depthand the contact area for punches of different axisymmetric geometries. In the above studies, thecontact is assumed frictionless. Contact involving a sticking indenter has been latter analyzedby Spence (1968). 5
  • 12. 6 Chapter 2 Materials in general have an elastic limit beyond which they undergo plastic deformation.After the onset of plasticity, the previously mentioned solutions fail to describe the behaviourof the indented material and different attempts has been carried out to account for the plasticdeformation. An empirical relation was found by Tabor (1951) which correlates between thehardness, defined as the mean pressure supported by the material under load, and the material’splastic properties. Hill et al. (1989) have carried out a theoretical study of indentation of apower law hardening material. They were able to predict Tabor’s empirical relation and tostudy in detail the deformation field beneath the indenter. Indentation of power law creepingmaterial has been studied by Matthews (1980) and Hill (1992). Proceeding from the studyby Hill, Bower et al. (1993) have also studied indentation of creeping materials and providedrelations between material parameters and indentation response for several indenter profiles.The Young’s modulus can also be deducted by indentation experiments. Loubet et al. (1984)suggested to infer the Young’s modulus from an elastic analysis of the initial elastic slope of theunloading portion of the load versus displacement curve. Coated materials have also been investigated using the indentation technique. The mechani-cal properties of the coating as well as of the substrate can be deducted by indentation. Doernerand Nix (1986) extended the idea of Loubet et al. (1984) to indentation of thin coatings de-posited on substrates. Due to the lack of elastic contact solutions for layered materials, theyhave combined the elastic properties of the coating and substrate linearly in one effective elasticmodulus in a way which fits measured experimental data. King (1987) modified the formulaproposed by Doerner and Nix (1986) to fit his numerical data. Motivated by the already ex-isting studies, Gao et al. (1992) have used a first-order moduli-perturbation method to deriveclosed-form elastic solutions for the contact compliance of multi-layered materials. In this chapter we will list some of the previously mentioned predictions and compare themwith numerical results. The main focus will be on the indentation load versus displacementcurve in the case of purely elastic material, elastic-plastic material and coated material.2.2 Elastic contactThe normal contact between a spherical indenter and an elastic half space is given by the Hertztheory. For the geometry shown in Fig. 2.1, Hertz theory provides an analytical solution for thestress distribution in the elastic half space and for the relation between the applied force ( ), ¡indentation depth ( ), contact radius ( ), indenter radius ( ) and elastic properties ( , ). The ¢ £ ¤ ¦ ¥pressure distribution as a function of the radial distance between the indenter and the solid is §proposed by Hertz (Johnson, 1985) to be 320)%#"§   ¨ £ 1 § ( &$ £!   ¨ © (2.1)where ¨ is the maximum pressure. The theory results in the following relations &£ © ¢ (2.2) ¤ BA¤ @8¥ 65© ¡ ¢ 97 4 (2.3)
  • 13. Indentation of bulk and coated materials 7 F O R r h a Symmetry axis z L L Figure 2.1: Geometry of the analyzed problem. 6 ¡ #D C¨ E ©  (2.4) & F£where $ & ¦ PI 2¥ H7 ¥   ! G (2.5)The stress field in the material is also given by the theory (Barquins, 1982). For a conical indenter with semiangle , the relations between load, penetration depth and Qcontact radius are given by Sneddon (1965) D & WQ #US7 ¥ E © ¡ ¢ VTR (2.6) E Q R `H£ D © ¢ YX (2.7)These analytical solutions assume frictionless contact and do not account for nonlinear effectsincluding boundary changes and radial displacements of points along the contact surface. Thelatter is only satisfied at small indentation strains; small values of  for a spherical indenter ba£ ¤!and large semiangle (close to ) for a conical indenter (Johnson, 1985). Q d "c In this section we perform finite-element simulations of the indentation process using thespherical and conical indenter profiles. The main intention is to examine the accuracy of theprediction of the analytical solutions in comparison to the numerical results. We have used a  "q©spherical indenter of radius ig"D © ¤ hfe , a conical indenter with a semiangle , a Young’s rp Q 6modulus d "d D © ¥ GPa and a Poisson’s ratio ad © ¦ . Figure 2.2 shows the pressure distribution sat the contact surface. The numerical pressure distribution seems to agree reasonably withthe analytical distribution proposed by Hertz (Eq. 2.1). The load–displacement data for both
  • 14. 8 Chapter 2 25 FEM 20 Analytical: Eq. (2.1) 15 σ zz (GPa) 10 5 0 0 1 2 3 4 5 r (µm)Figure 2.2: (a) The distribution of the normal stress component wvt uu and the Hertzian assumption(Eq. 2.1) at vxe %d © ¢ hf s .spherical and conical indenters is plotted in Fig. 2.3. The analytical solutions, Eqs. (2.3) and(2.6), seem to underestimate the force. However, the error at the maximum indentation depthvxe %d © ¢hf s in the spherical and conical indenter predictions is about and € "y € "d  , respectively.This error is attributed to the fact that some of the analytical solutions assumptions discussedpreviously are not fully satisfied, mainly the small strain assumption. The main attraction of the Hertz theory is the analytical solution it provides for the contactproblem. However, the validity of the theory and other existing analytical solutions is limitedto infinitesimal deformations. The problem involving finite deformations or nonlinear materialbehaviour has no analytical solution. Such problems are generally solved numerically using thefinite element method.2.3 Elastic-plastic contactThe contact problem involving elastic-plastic materials does not have a complete analyticalsolution due to the highly nonlinear material response. However, approximate solutions limitedby simplifying assumptions are available in the literature. In this section we will examinesome of the existing analytical and empirical relations, namely those that relate the responseto indentation and the material’s mechanical properties, and compare their predictions with ournumerical results. There are several constitutive models in the literature which account for plasticity in thematerial. Examples of the most common models used in indentation modeling include elastic-
  • 15. Indentation of bulk and coated materials 9 0.6 0.5 FEM Analytical: Eq. (2.3) 0.4 F (N) 0.3 0.2 0.1 Spherical indenter (a) 0 0 0.1 0.2 0.3 0.4 0.5 0.1 0.08 FEM Analytical: Eq. (2.6) 0.06 0.04 0.02 Conical indenter (b) 0 0 0.1 0.2 0.3 0.4 0.5 h (µm) Figure 2.3: Force versus indentation depth for an elastic material.perfectly plastic, elastoplastic with linear or power-law strain hardening and time dependentplasticity models (e.g. Bower et al., 1993; Mesarovic and Fleck, 1999). In this section we willconsider a material with an elastic-perfectly plastic response. Such material is characterizedby its elastic properties ( , ) and a yield stress . In the FEM calculations we have used ¦ ¥  ‚tsd ƒ©  t  GPa; all other parameter values are the same as in the previous section. For an elastic perfectly-plastic material indented by a conical indenter with a semiangle , Qthe indentation load is predicted based on the so-called cavity model (Johnson, 1985). It is
  • 16. 10 Chapter 2related to the material properties and indenter geometry by (Cheng, 1999) ¦ D P D6 † Q R `X  ¥ V ˆH …& £ E St D6 © ¡ Y ‚¦ ‘I  t p ‰ ‡ † „  (2.8) ”•’“¦ P $  The cavity model assumes that the contact surface of the indenter is encased in a hemisphericalcore, inside which the hydrostatic stress is constant. Outside the core, the stress and displace-ments have a radial symmetry and are the same as in an infinite elastic-perfectly plastic bodywhich contains a spherical cavity under a pressure equal that of the core. Based on the conicalindenter solution, Johnson (1985) suggested an approximate solution for a spherical indenter.The strain imposed by the indenter, ba£ ¤! , is simply replaced by Q R `X Y , i.e. ¦ D  D £  6 † ¥ E D ”—’0¦  – ¤ ‚¦ $ PI St p ‰ V W „ & £ St 6 © ¡    ‡ † (2.9) For a power-law hardening material, Hill et al. (1989) showed that the solution is self-similar, i.e. that the geometry, stress and strain fields throughout the indentation process arederivable from a single solution by appropriate scaling (Bower et al., 1993; Mesarovic andFleck, 1999). For an axisymmetric indenter with a smooth profile, the force is given by ¡ ¢ ‰ ™© ¡ E ˜ (2.10) ijhef’aIdF£ g  SIF£  t&The relation between the contact radius and the indentation depth £ ¢ is given by Q R `H£ © ¢ YX Conical indenter k (2.11) &£D © Spherical indenter ¤&kwhere is the strain hardening exponent, l d ˜ is the yield strain and the constants and are kfunctions of the strain hardening exponent, the indenter geometry and the frictional condition kbetween the indenter and the half space. The constant is the ratio of the true to nominal(geometrical) contact radius. For onk  m , the material sinks-in at the edge of the contact area,whereas for qfk  p6 , the material piles-up. The switch between a sink-in and a pile-up behaviouroccurs at © l ˜ k . Bower et al. (1993) tabulated the values of the constants and for a range ofhardening exponents and indenter profiles. The elastic-perfectly plastic material corresponds totaking t0l Ds r in Eq. (2.10). From the tabulated values, e sd 6 u˜ © for both indenter geometries, p s ƒ–k  © for the conical indenter and D s vk  © for the spherical indenter. Figure 2.4 shows the numerical load versus contact radius data and the prediction of Eqs. (2.8–2.10). The steps in the curve originate from the node-to-node growth of the contact region. Boththe similarity solution, Eq. (2.10), and the cavity model solution, Eq. (2.8), seem to be in closeagreement with the numerical results in the case of the conical indenter as shown in Fig. 2.4(b).In the case of the spherical indenter, the cavity model solution, Eq. (2.9), seems to deviate fromthe numerical results as seen clearly in Fig. 2.4(a). This deviation is not surprising in view ofthe approximations made.
  • 17. Indentation of bulk and coated materials 11 0.25 0.2 FEM Analytical: Eq. (2.10) Analytical: Eq. (2.9) 0.15 F (N) 0.1 0.05 Spherical indenter (a) 0 0 1 2 3 4 5 0.025 0.02 FEM Analytical: Eq. (2.10) Analytical: Eq. (2.8) 0.015 0.01 0.005 Conical indenter (b) 0 0 0.5 1 1.5 a (µm) Figure 2.4: Force versus contact radius for an elastic-perfectly plastic material. Extensive work has been done to extract the plastic properties from the loading portion of theload–displacement curve (Tabor, 1951; Hill et al., 1998; Matthews, 1980; Hill, 1992; Bower etal., 1993). One of the most common parameters in indentation experiments is hardness, definedas w s & £¡ E © (2.12)The extraction of the material’s plastic properties from hardness is not straightforward. Inthe case of strain hardening materials, the hardness depends on the the yield stress, contact
  • 18. 12 Chapter 2radius, strain hardening exponent and indenter geometry (Bower et al., 1993). For rigid-plasticmaterials, for example, hardness is related to the yield stress as (Tabor, 1996) 6 w t © (2.13) w e sd 6 y˜Eq. (2.10) leads to a similar expression for hardness; , where  t x© ˜ . On the ©other hand, Eqs. (2.8) and (2.9) lead to rather complicated expressions for hardness due to thepresence of elasticity. In these equations, hardness continuously increases with the ratio of theapplied strain ( for the cone and Q R `X Y for the sphere) to the yield strain ba£ ¤! . Based on ¥ b!  tEq. (2.13), we have calculated the yield stress from the maximum load and the correspondingmaximum contact radius (Fig. 2.4). We have chosen this data point to ensure a negligible in-fluence of elasticity since at higher indentation depths, the indentation response is dominatedby the plastic flow (Mesarovic and Fleck, 1999). The estimated values in the case of the spher-ical and conical indenters are and "c %d c s e c %d GPa, respectively. This estimate is close to the sactual value z© St   GPa. Eq. (2.10) would results in similar values. Solving Eqs. (2.9) and(2.8) numerically for the yield stress using the same data point resulted in the values and r es s GPa, respectively. The overestimation of the yield stress by Eq. (2.9) is tied to the fact thatthis relation underestimates the force as seen in Fig. 2.4a and explained previously. The unloading portion of the load–displacement curve is also of importance in indentationexperiments. Even though the material has undergone elastic-plastic deformation during load-ing, the initial unloading is an elastic event (Loubet et al., 1984). Therefore, the Young’s mod-ulus can be inferred from an elastic analysis of this portion. For an indenter with axisymmetricsmooth profile, the initial slope is related to the Young’s modulus by ¡ & ¦ –  D © (2.14) €¢ “h)"8¥  ~ ~ } | { £ ˆ‡…†ƒ‚ƒ  „This expression can be derived from the elastic analytical relations discussed in section 2.2. Figure 2.5 shows the load–displacement curves for the two indenter profiles. Making use ofEq. (2.14), we estimate the Young’s modulus to be d "D D GPa from the spherical indenter resultsand r ‰ D GPa from the conical indenter. Compared to the actual value dbd D © ¥ GPa, the erroris about †d  . This error is attributed to the finite-strain effects that are not accounted for in €the elastic analytical analysis as discussed in section 2.2. Cheng et al. (1998) have performedindentation experiments and numerical simulations using a conical indenter. Using a wide rangeof material parameters, they have found that their results agree with the relation ¡ & ¦   y s D © (2.15) €¢ “h)"8¥  ~ ~ } | { £ ˆ‡…†ƒ‚ƒ  „They argued that the deviation from the elastic analysis represented by Eq. (2.14) resultedfrom the nonlinear effects, including large strain and moving contact boundaries. According6 DEq. (2.15), the calculated values of the Young’s modulus are and GPa. It should be d dDnoted that if the Poisson’s ratio is also unknown, Young’s modulus can not be determined by ¦this method. In this case, only the composite modulus $ & ¦ PI 2¥ can be determined.   !
  • 19. Indentation of bulk and coated materials 13 0.25 (a) Spherical indenter 0.2 0.15 F (N) 0.1 0.05 dF dh 0 0 0.1 0.2 0.3 0.4 0.5 0.02 (b) Conical indenter 0.015 0.01 0.005 0 0 0.1 0.2 0.3 0.4 0.5 h (µm)Figure 2.5: Force versus indentation depth for an elastic-perfectly plastic material. Dashed linesillustrate the slope of the initial portion of the unloading curves.2.4 Coated materialsIndentation of coated materials is far more complicated as compared to bulk materials. In coatedsystems, the indentation response is controlled by the mechanical properties of both the coatingand the substrate. In this section we will investigate the indentation of an elastic-perfectlyplastic substrate coated by a relatively stronger elastic coating. The coating is characterized 6by its thickness vf ‹© Š h  and elastic properties d "d e © ‘¥  Œ GPa, $ %d © b¦ s Œ . The substrate is
  • 20. 14 Chapter 2 6 characterized by its elastic properties d D ©  GPa, idg"D ‘¥ ¤ h "f e  © $ %d © b¦ and a yield stress s  GPa. ƒ© t   The spherical indenter has a radius  bq© , while the conical indenter has a semiangleQ rp . The subscripts c and s refer to the coating and substrate, respectively. The deduction of the elastic properties of the coating or the substrate from the initial unload- ing stiffness is not as straightforward as in the case of bulk material. In coated materials, the unloading stiffness is a function of the elastic properties of both the coating and the substrate. However, there are two limiting cases. For indentation depths that are very small compared to the coating thickness, the initial stiffness is dominated by the coating elastic properties, whereas for large depths, the stiffness is dominated by the substrate’s elastic properties (King, 1987; Gao et al., 1992). Between these two limiting cases, an empirical relation for the initial stiffness as a function of the elastic properties of the coating and substrate was introduced by King (1987). His relation uses a numerical constant which depends on the ratio of the contact radius to the coating thickness and on the indenter geometry. This constant has to be extracted from a set of curves. Motivated by previous work, Gao et al. (1992) derived a closed-form solution of the effective modulus for a multi-layered material. They assumed that the indentation response ˆP¥ Ž of a multi-layered elastic half space can be obtained from the existing elastic solutions for bulk materials (e.g. the Hertzian solution). In such solutions, the Young’s modulus and Poisson’s ratio have to be replaced by an effective Young’s modulus and an effective Poisson’s ra- ˆ¥ Ž tio , respectively. These parameters are functions of the elastic properties of elastic layers ˆb¦ Ž and the contact conditions. For an elastic coating on an elastic substrate, the effective Young’s modulus and Poisson’s ratio are are given by (Gao et al., 1992) j–$ ˆ• `ƒb¦ –¥H  b¦ –¥W ‰ † b¦ –¥W ‘$ ˆŽ ¦ ‘I © ˆŽ ¥  ”“ ’ † Œ † Œ   † ’ †  (2.16) $ ˆ• “ $ ¦  Œ ¦ q†  ¦ © ˆb¦  Ž (2.17) where . and e $ ˆ• “ $ ˆ• ” “ bŠ • £! © are weight functions that reflect the substrate effect and given by e D •  & ˜‘ …• D –I • † ‡  – & •ˆH † & • V ‚¦ $   ’ ‚¦ PI D ‘• #UR X #nE © $ ˆ• ” “ $   E † VT —T (2.18) & ˆH V ‡ • E ‘• #UR X #T D E © $ ˆ• • † † VT — &• “ where the Poisson’s ratio can be taken as coating or substrate value since its effect on and ¦ e is negligible (Gao et al., 1992). Both of these functions approach unity at small indentation ”“ “ depths ( ) and the effective elastic properties are equal to those of the coating. On the €bIŠ ™ £! e other hand, at large indentation depths (  ›bIŠ ), both and approach zero and the effective š £! “ œ“” elastic properties are equal to those of the substrate. e To investigate the accuracy of this solution, we have performed a calculation with an elastic substrate (without plasticity). The corresponding load–displacement curve is shown in Fig. 2.6. The analytical solution shown for comparison, is obtained from Eqs. (2.3) and (2.6) by using the
  • 21. Indentation of bulk and coated materials 15 0.7 0.6 FEM Analytical: Eq. (2.3) 0.5 0.4 F (N) 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.2 FEM 0.15 Analytical: Eq. (2.6) 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 h (µm)Figure 2.6: Force versus indentation depth for an elastic coating on an elastic substrate withdifferent Young’s modulus. The analytical results in (a) and (b) are obtained from Eqs. (2.3)and (2.6), respectively. The effective properties (Eq. 2.16) and ˆ¥ Ž (Eq. 2.17) are used in ˆb¦ Žthe definition of 7 ¥(Eq. 2.5).effective properties and ˆP¥ Ž in the definition of ˆb¦ Ž (Eq. 2.5). It is seen that the analytical 7 p¥solution overestimates the force by a maximum of and € €by  in (a) and (b), respectively.Gao et al. (1992) also investigated the range of validity of this solution through finite elementanalysis. They found that the solution is valid, within an error of , at least for moduli ratio € "yup to 2. For larger moduli ratio, the weight functions (Eq. 2.18) fail to accurately represent the
  • 22. 16 Chapter 2 0.25 (a) Spherical indenter 0.2 0.15 F (N) 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.07 0.06 (b) Conical indenter 0.05 0.04 0.03 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 h (µm)Figure 2.7: Force versus indentation depth for an elastic coating on an elastic-perfectly plasticsubstrate.relative influence of the coating and the substrate. In the current calculations where ©  b! Œ ¥ ¥es D , these weight functions have apparently exaggerated the coating contribution to the effectiveproperties. In the case of an elastic-perfectly plastic substrate with a yield stress v© ‚t   GPa, the load–displacement curve is shown in Fig. 2.7. Since the elastic properties of the coating are differentfrom these of the substrate, the initial unloading stiffness in this case is related to the effective
  • 23. Indentation of bulk and coated materials 17modulus by ˆ& Ž ¦ – ¡  D © (2.19) “¢ ~ }h)"£ ˆŽ ¥  ~ | {Based on the calculated value of from the numerical results, the Young’s modulus of the ˆ‡…†ƒƒ  „ ‚ ˆŽ ¥coating or the substrate can be calculated by Eq. (2.16) provided that the other modulus isknown. The load–displacement curve is shown in Fig. 2.7. From the unloading stiffness in (a)and (b), the calculated values of the Young’s modulus are and D GPa, respectively. Theseare reasonable estimates compared to the actual value GPa. e y © 4 ‘¥ d"d Œ d "c 4 Hardness of coated systems is also defined by Eq. (2.12). The measured or apparent valueof hardness depends on the mechanical properties of each of the constituents and on the con-tact conditions. Various models have been proposed to relate the hardness to the mechanicalproperties of the system (Wittling et al., 1995; Korsunsky et al., 1998). The main idea is tointroduce weighting functions to interpolate between the two limiting cases where the coatingand substrate properties are dominant at small and large indentation depths, respectively. The previous analysis assumes that failure events do not occur during indentation. This as-sumption holds true if the stresses generated by the indenter do not reach the material strength;otherwise, failure is inevitable. The possible failure mechanisms are discussed in the forth-coming chapters including the failure of the interface between the coating and the substrate bydelamination and the failure of the coating itself by cracking.References Barquins, M., Maugis, D., 1982. Adhesive contact of axisymmetric punches on an elastic half- space: the modified Hertz-Huber stress tensor for contacting spheres. J. Mec. Theori. Appl. 1, 331–357. ` ´ ´ Boussinesq, J., Applications des Potentiels a l’Etude de l’Equilibre et du Mouvement des ´ Solides Elastiques (Gauthier-Villars, Paris, 1885). Bower, A.F., Fleck, N.A., Needleman, A., Ogbonna, N., 1993. Indentation of power law creeping solids. Proc. Roy. Soc. Lond. A 441, 97–124. Cheng, Y.-T., Cheng, C.-M., 1998. Scaling approach to conical indentation in elastic-plastic solids with work hardening. J. Appl. Phys. 84, 1284–1291. Cheng, Y.-T., Cheng, C.-M., 1999. Scaling relationships in conical indentation of elastic- perfectly plastic solids. Int. J. Solids Struct. 36, 1231–1243. Doerner, M.F., Nix, W.D., 1986. A method for interpreting the data from depth-sensing inden- tation instruments. J. Mater. Res. 4, 601–609. Gao, H., Chiu, C.-H., Lee, J., 1992. Elastic contact versus indentation modeling of multi- layered materials. Int. J. Solids Struct. 29, 2471–2492. ¨ Hertz, H., 1882. Uber die Ber¨ hrung fester elastischer K¨ rper (On the contact of elastic u o solids). J. reine und angewandte Mathematik 92, 156–171.
  • 24. 18 Chapter 2Hill, R., 1992. Similarity analysis of creep indentation tests. Proc. Roy. Soc. Lond. A 436, 617–630.Hill, R., Stor˚ kers, B., Zdunek, A.B., 1989. A theoretical study of the Brinell hardness test. a Proc. Roy. Soc. Lond. A 423, 301–330.Johnson, K.L., Contact Mechanics (Cambridge University Press, Cambridge, United King- dom, 1985).King, R.B., 1987. Elastic analysis of some punch problems for a layered medium. Int. J. Solids Struct. 23, 1657–1664.Korsunsky, A.M., McGurk, M.R., Bull, S.J., Page, T.F., 1997. On the hardness of coated systems. Surf. Coat. Technol. 99, 171–183.Loubet, J., Georges, J., Marchesini, J., Meille, G., 1984. Vickers indentation curves of mag- nesium oxide (MgO). J. Tribology 106, 43–48.Love, A.E.H., 1929. Stress produced in a semi-infinite solid by pressure on part of the bound- ary. Phil. Trans. A. 228, 377.Love, A.E.H., 1939. Boussinesq’s problem for a rigid cone. Quart. J. Math. 10, 161.Matthews, J.R., 1980. Indentation hardness and hot pressing. Acta Metall. 28, 311.Mesarovic, S.Dj., Fleck, N.A., 1999. Spherical Indentation of elastic-plastic solids. Proc. Roy. Soc. Lond. A 455, 2707–2728.Sneddon, I.N., 1965. The relation between load and penetration in the axisymmetric Boussi- nesq problem for a punch of arbitrary profile. Int. J. Engng. Sci. 3, 47–57.Spence, D.A., 1968. Self-similar solutions to adhesive contact problems with incremental loading. Proc. Roy. Soc. Lond. A 305, 55.Tabor, D., The Hardness of Metals (Clarendon Press, Oxford, 1951).Tunvisut, K., O’Dowd, N.P., Busso, E.P., 2001. Use of scaling functions to determine me- chanical properties of thin coatings from microindentation tests. Int. J. Solids Struct. 38, 335–351.Wittling, M., Bendavid, A., Martin, P.J., Swain, M.V., 1995. Influence of thickness and sub- strate on the hardness and deformation of TiN films. Thin Solid Films 270, 283–288.
  • 25. Based on: A. Abdul-Baqi and E. Van der Giessen, Indentation-induced interface delamination of a strong film ona ductile substrate, Thin Solid Films 381 (2001) 143.Chapter 3Indentation-induced interfacedelamination of a strong film on a ductilesubstrate The objective of this work is to study indentation-induced delamination of a strong film from a ductile substrate. To this end, spherical indentation of an elastic-perfectly plas- tic substrate coated by an elastic thin film is simulated, with the interface being modeled by means of a cohesive surface. The constitutive law of the cohesive surface includes a coupled description of normal and tangential failure. Cracking of the coating itself is not included and residual stresses are ignored. Delamination initiation and growth are analyzed for several interfacial strengths and properties of the substrate. It is found that delamination occurs in a tangential mode rather than a normal one and is initiated at two to three times the contact radius. It is also demonstrated that the higher the interfacial strength, the higher the initial speed of propagation of the delamination and the lower the steady state speed. Indentation load vs depth curves are obtained where, for relatively strong interfaces, the delamination initiation is imprinted on this curve as a kink.3.1 IntroductionIndentation is one of the traditional methods to quantify the mechanical properties of materialsand during the last decades it has also been advocated as a tool to characterize the properties ofthin films or coatings. At the same time, for example for hard wear-resistant coatings, inden-tation can be viewed as an elementary step of concentrated loading. For these reasons, manyexperimental as well as theoretical studies have been devoted to indentation of coated systemsduring recent years. Proceeding from a review by Page and Hainsworth (1993) on the ability of using indenta-tion to determine the properties of thin films, Swain and Menˇ ik (1994) have considered the cpossibility to extract the interfacial energy from indentation tests. Assuming the use of a smallspherical indenter, they identified five different classes of interfacial failure, depending on therelative properties of film and substrate (hard/brittle versus ductile), and the quality of the ad-hesion. Except for elastic complaint films, they envisioned that plastic deformation plays animportant role when indentation is continued until interface failure. As emphasized further byBagchi and Evans (1996), this makes the deduction of the interface energy from global inden- 19
  • 26. 20 Chapter 3tation load versus depth curves a complex matter. Viable procedures to extract the interfacial energy will depend strongly on the precise mech-anisms involved during indentation. In the case of ductile films on a hard substrate, coatingdelamination is coupled to plastic expansion of the film with the driving force for delaminationbeing delivered via buckling of the film. The key mechanics ingredients of this mechanism havebeen presented by Marshall and Evans (1984), and Kriese and Gerberich (1999) have recentlyextended the analysis to multilayer films. On the other hand, coatings on relatively ductile sub-strates often fail during indentation by radial and in some cases circumferential cracks throughthe film. The mechanics of delamination in such systems has been analyzed by Drory andHutchinson (1996) for deep indentation with depths that are two to three orders of magnitudelarger than the coating thickness. The determination of interface toughness in systems that showcoating cracking has been demonstrated recently by e.g. Wang et al. (1998). In both types ofmaterial systems there have been reports of ”fingerprints” on the load–displacement curves inthe form of kinks (Kriese and Gerberich, 1999; Hainsworth et al., 1997; Li and Bhushan, 1997),in addition to the reduction of hardness (softening) envisaged in (Swain and Menˇ ik, 1994). The corigin of these kinks remains somewhat unclear, however. A final class considered in (Swain and Menˇ ik, 1994) is that of hard, strong coatings on cductile substrates, where Swain and Menˇ ik hypothesized that indentation with a spherical in- cdenter would not lead to cracking of the coating but just to delamination. This class has notyet received much attention, probably because most deposited coatings, except diamond ordiamond-like carbon, are not sufficiently strong to remain intact until delamination. On theother hand, it provides a relatively simple system that serves well to gain a deep understandingof the coupling between interfacial delamination and plasticity in the substrate. An analysis ofthis class is the subject of this paper. In the present study, we perform a numerical simulation of the process of indentation ofthin elastic film on a relatively softer substrate with a small spherical indenter. The inden-ter is assumed to be rigid, the film is elastic and strong, and the substrate is elastic- perfectlyplastic. The interface is modeled by a cohesive surface, which allows to study initiation andpropagation of delamination during the indentation process. Separate criteria for delaminationgrowth are not needed in this way. The aim of this study is to investigate the possibility andthe phenomenology of interfacial delamination. Once we have established the critical condi-tions for delamination to occur, we can address more design-like questions, such as what is theinterface strength needed to avoid delamination. We will also study the ”fingerprint” left onthe load–displacement curve by delamination, and see if delamination itself can lead to kinksas mentioned above in other systems. It is emphasized that the calculations assume that otherfailure events, mainly through-thickness coating cracks, do not occur.
  • 27. Indentation-induced interface delamination of a strong film on a ductile substrate 21 ˙ h R O a r h Film t Interface z Substrate Symmetry axis L L Figure 3.1: Illustration of the boundary value problem analyzed in this study.3.2 Problem formulation3.2.1 Governing equationsWe consider a system comprising an elastic-perfectly plastic material (substrate) coated by anelastic thin film and indented by a spherical indenter. The indenter is assumed rigid and onlycharacterized by its radius . Assuming both coating and substrate to be isotropic, the problem ¤is axisymmetric, with radial coordinate and axial coordinate in the indentation direction, as § illustrated in Fig. 3.1. The film is characterized by its thickness and is bonded to the substrate Šby an interface, which will be specified in the next section. The substrate is taken to have aheight of Š  ž and radius , with large enough so that the solution is independent of and ž ž žthe substrate can be regarded as a half space. The analysis is carried out numerically using a finite strain, finite element method. It usesa Total Lagrangian formulation in which equilibrium is expressed in terms of the principle ofvirtual work as ¢£ #ƒ£ %vSŸ ¥¤¢¡  † 2b†«`@SŸ ª ¬¤ª ©¨§ ¢ #¯¤¢ ƒ%Ÿ © °  ® (3.1) Š s ¦ †~ ­ a~ ± %~Here, is the total region analyzed and § is its boundary, both in the undeformed ž ²¢ ³ƒž ‹© µ ´ ¢£ ¡ ¢° ¢configuration. With ¦ the coordinates in the undeformed configuration, $†·(SU(# ¶  and ¦ Šare the components of displacement and traction vector, respectively;¢£ ¥ are the components ofSecond Piola-Kirchhoff stress while are the dual Lagrangian strain components. The latter
  • 28. 22 Chapter 3are expressed in terms of the displacement fields in the standard manner, £ ¸ º ° ¸¢ ‚° † ¢ h£ ° † £ ¹¸¢ °   © ¢£ ¥ º ¸ $ D (3.2) ¢where a comma denotes (covariant) differentiation with respect to . The second term in the µleft-hand side of Eq. (3.1) is the contribution of the interface, which is here measured in the ª©deformed configuration ( ¾ ·Š ©  ½© ¼ ª ¬ ). The (true) traction transmitted across the interfacehas components , while the displacement jump is » v­ ˜ , with being either the local normaldirection (l ¿˜ © ) or the tangential direction ( ) in the Š À˜ © -plane. Here, and in the $(  ¢ ¥ © B ¢ ¡ © B F©U#§ °remainder, the axisymmetry of the problem is exploited, so that . d © B Š B The precise boundary conditions are illustrated in Fig. 3.1. The indentation process is per-formed incrementally with a constant indentation rate . Outside the contact area with radius Á¢ £in the reference configuration, the film surface is stress free, d © †U#§  u Š © †U§  wŠ $d( $d(  for €ž —3•£ s 1 § 1 (3.3)Inside the contact area we assume perfect sticking conditions so that the displacement rates arecontrolled by the motion of the indenter, i.e. ° ° d © †U#§   Á fÁ ¢ © "U#§  u Á $d( ( $d( for –£ —3—d s 1 § 1 (3.4)Numerical experiments using perfect sliding conditions instead have shown that the preciseboundary conditions only have a significant effect very close to the contact area and do not alterthe results for delamination to be presented later. The indentation force is computed from the ¡tractions in the contact region, à ğ § § D †U#§  u Š s E $d( © ¡ (3.5) ~ ”The substrate is simply supported at the bottom, so that the remaining boundary conditions read ° ° d © ‘U§  u $ ž( for d © FU%d   ž —3—d $( ; 1 § 1 for 3• —d ž 1 1 . (3.6)As mentioned previously, the size will be chosen large enough that the solution is independent žfrom the precise remote conditions. The equations (3.1) and (3.2) need to be supplemented with the constitutive equations forthe coating and the substrate, as well as the interface. As the latter are central to the results ofthis study, these will be explained in detail in the forthcoming section. The substrate is supposedto be a standard isotropic elastoplastic material with plastic flow being controlled by the vonMises stress. For numerical convenience, however, we adopt a rate-sensitive version of thismodel, expressed by 6 Å a¥ ¢£ D © ¢£ Á t ‰ ·d © Å d ( Å d Ž Æ (3.7) t ± %¥ Å Æ Á Á ¥ Á¥ Ž ¢£ ¢£ i ’ tfor the plastic part of the strain rate, © ¢£ Á. Here, B& ÇÈ© Ž t ¢£ £ Ž¢  ¢£ is the von Mises stress, Á Áexpressed in terms of the deviatoric stress components , ± ± l is the rate sensitivity exponent and ±
  • 29. Indentation-induced interface delamination of a strong film on a ductile substrate 23 is a reference strain rate. In the limit of·ÁÆ d , this constitutive model reduces to the Ƀl s rrate-independent von Mises plasticity with yield stress . Values of on the order of Æt are l d"d a few percent of . The elastic part of the strain rate, , is given in terms of the Jaumann Æt £ Ž¢ ¥frequently used for metals (see e.g. Becker et al., 1998), so that the value of at yield is within Ž vt Ástress rate as ËwŽº ¥ wº ¢£ ¤ © ¢£ Ê ¡ Ë (3.8) Á Ë wº ¢£with the elastic modulus tensor being determined by the Young’s modulus ¤ and Pois-  ¥son’s ration (subscript s for substrate). ¦ The coating is assumed to be a strong, perfectly elastic material with Young’s modulus Ì ‘¥and Poisson’s ration (subscript f for film). ̦ The above equations, supplemented with the constitutive law for the interface to be dis-cussed presently, form a nonlinear problem that is solved in a linear incremental manner. Forthis purpose, the incremental virtual work statement is furnished with an equilibrium correc-tion to avoid drifting from the true equilibrium path. Time integration is performed using theforward gradient version of the viscoplastic law (3.7) due to Peirce et al. (1984).3.2.2 The cohesive surface modelIn the description of the interface as a cohesive surface, a small displacement jump be- ¬ Î ›¬ Ítween the film and substrate is allowed, with normal and tangential components and , © Ωrespectively. The interfacial behaviour is specified in terms of a constitutive equation for the icorresponding traction components and at the same location. The constitutive law weadopt in this study is an elastic one, so that any energy dissipation associated with separation is iignored. Thus, it can be specified through a potential, i.e. 2¬ ´ ƒ© ª © ª‚  Ï sƒIŠ·xl ÑИ $ ( © (3.9) ´The potential reflects the physics of the adhesion between coating and substrate. Here, we usethe potential that was given by Xu and Needleman (1993), i.e. Ï ¬ ¬ ¬ Î ¬  ˆ – ¤ † § P ’ „ ¤  ‰ Õ   §  Ï † Ï © Ï ¤ Õ ˆ § † s ”—’ & & Î ¤  ‰   § ‰ Õ ’ – i ’i i Β i i a@xÎ i i ÔÓÒ (3.10) i Ï#! Ï ÖÕ © ‰¤ a@Ò ¤ Î ÔÓwith and the normal and tangential works of separation ( Ï ), and two char- Ïacteristics lengths, and a parameter that governs the coupling between normal and tangential §separation. The corresponding traction–separation laws from (3.9) read i i i Î ¬ ¬ ¬ ¬ Î ¬ ՘– † & Î ¤   ¤ „ ¤ ‰ ¤Ï © © ¤  § ’u– & Î ¤  ‰  P ’   § ’& ‰ ¬ i ›¬ ’ i ¤ a@Ò i ‚© i Î × ™” – i ’¬ & ¤ ˆ § ‰a@Ò Õi „ Î ¤ i Î ¤ ‰ Ô ¤ Ï Ó D i© Î Õ Ôӆ Î i s & Î ¤  ‰ a@Ò ¤  ‰ ÔÓ ¬ (3.11) (3.12) ”i ’  § ’i i ’& ’ %@Ò i ÔÓ i a@Ò i i ÔÓ
  • 30. 24 Chapter 3 1.5 1 0.5 T n ⁄ σ max 0 −0.5 −1 −1.5 −2 (a) −2.5 −1 0 1 2 3 4 5 6 ∆n ⁄ δn 1.5 1 0.5 T t ⁄ τ max 0 −0.5 −1 (b) −1.5 −3 −2 −1 0 1 2 3 ∆t ⁄ δt © ¬ Î ¬Figure 3.2: The uncoupled normal and tangential responses according to the cohesive surface Î ¬ Ω ¬law (3.11)–(3.12). (a) Normal response with $  . (b) Tangential response d © ¡ $ with d © . Both are normalized by their respective peak values i and i . j})‚t |{ hØ{ }| i © Λ¬ © The form of the normal response, © ©  is motivated by the universal binding $ †d Ù¬ Ú Îlaw of Rose and Ferrante (1981). In the presence of tangential separation, i i, the expres- d ©sion (3.11) is a phenomenological extension of this law, while the tangential response (3.12) Î ¬should be considered as entirely phenomenological. The uncoupled responses, i.e. with d ©
  • 31. Indentation-induced interface delamination of a strong film on a ductile substrate 25 3 3 max T ⁄ τ max (a) (a) q = 0.3 (b) (b) q = 0.5 2 2 (c) q = 0.7 r ≥ 0, q = 1 (d) r = 0, q > 0 (c) 1 1 (d) 0 0 −1 0 1 2 3 4 ∆n ⁄ δn −1 0 Ω 1 2 ¡ 3 4Figure 3.3: The maximum shear traction , normalized by hØ{ }| (see Fig. 3.2), as a function hØ{ }|of the normal separation for different combinations of the coefficients and . In (a)-(c), § Õ © §e %d s . ¬( d © ) for the normal (tangential) response, are shown in Fig. 3.2. Both are highly nonlinear ¡ iseparation of ( D 9 !Τ © Î ¬ ¤ © ¬with a distinction maximum of the normal (tangential) traction of ( ) which occurs at a ). The normal (tangential) work of separation, ¡ ( ), can j}){ h)‚t | }| { Ï Ï Înow be expressed in terms of the corresponding strengths ( ) as hØ{ hØ{ t }| }| ¤ i i i © Ï  ƒ© Î Î`¤ ¡ ( j}|){ w$ I t  D Û Ï s h){ $ I }|  (3.13) Î i a@Ò i ÔÓUsing equation (3.13) together with the relation #! Ï AÕ Ï © a@Ò ÔÓ , we can relate the uncoupled normaland shear strengths through Î `¤ © j})St i¡ ¤  |{ j}|){ $ I D  Õ  (3.14) ¬ ¤ `¤ 2¬ Î Î i © a@Ò The coupling parameter can be interpreted as the value of the normal separation § ÔÓ !after complete shear separation ( ) with ! d © . Some insight into the coupling s Ür i Î i¤ Î ¬ Î ©between normal and shear response can be obtained from Fig. 3.3, which shows the maxi- i ¬ Ω ¬ Dmum shear traction as a function of the normal displacement, i.e. © !  G $  j}){ |$ ( 9! . It is seen that this is quite sensitive to the values of and . The maximum i Õ §( id p ¬shear traction that can be transmitted decreases when there is opening in the normal direction ) for all parameter combinations shown. However, in normal compression ( d m ), ¬ ¬the maximum shear stress can either increase or decrease with . An increase appears to be  ithe most realistic, and the parameter values used in the present study ensure this. i i
  • 32. 26 Chapter 33.3 Results and discussion3.3.1 Analysis and parametersThe solution to the problem depends on a number of nondimensional parameters. We havechosen the following nondimensional groups: geometry ¤ (3.15) Š Ý material ¦ ¦ Æ ŒbÞ( Ì Þ( –t¥ ( P¥¥ Ý Ì (3.16) ¡ ¤ ¤ interface hØ{ ( `¤ ( }| Î Õ (3.17) §ß( Æ t i Ši ( Ý  ¥ÆdŠ ¡ ¢ loading hSœãiâhe á³ÁvÆ Á t 0Æ t & ¤ ( Š Ý åe ä g ¢ à (3.18)Note that the rather complex form of the last loading parameter is dictated by the fact thatrate-dependent plasticity, Eq. (3.7), is governed by the parameters and only through the Æ Ád Ætcombination . This nondimensional parameter immediately shows that the indentation S#‰UÁÆ d Æt!force depends on the indentation rate in a rate-dependent material. However, in the limit that i æ0ls r , the bracketed factor becomes , so that the parameter reduces to  . $ Æ t & ¤  ¡ ! In the results to be presented we focus mainly on the effect of the normalized substrate yield¡strength , which is simply the yield strain, and the normalized interfacial shear strength P#œt  ¥!Æ on the initiation and advance of interfacial delamination. The relation between normal #`hØ{ Æ t !}|and tangential interfacial strengths is given by equation (3.14). Three values of e bd %d "D "d %d were Pb! Æ ¡ t ç ¥chosen, namely , and d s e d s . For each value of  d ad , several values of s are Pb! Æ t ç ¥ Æ t#! h){ }|chosen. Even though the solution is formally governed by the above-mentioned nondimensionalgroups, we have chosen to work primarily in terms of real dimensional values in order to sim- eplify the interpretation. We have used an indenter of radiuse bd %d © Š mm and a film thickness 6 s © D © P¥ "6 %d d a© d Ì ¦ ¤ 6 "d 6 s © mm. The elastic properties are GPa, , d "d e © Ì ¥ GPa and d"d  s s . The yield stress of the substrate is varied, as discussed above, and the reference %d  ¦strain rate is taken to be s with  %d © Æ Á d s "d è© l d  . The indentation is performed under aconstant rate  è© Á ¢ mm/sec. For a typical value of e of Sä , the value of the factor e bd %d Pb! Æ t ç ¥ d s é Á ¢ Æ #!  ¥ Æ Á d Š ê t in (3.18) is b†c ad yy s ; this is less than ¤ € es D smaller than the rate-independent Τ jåeSäi Ijge âlimit. For the cohesive surface we have chosen the same values for and , namely dmm. Most of the results to be presented are for and e %d Õ e %d © § , but we will also briefly © s¡ i sinvestigate the sensitivity of the results to these values. As mentioned above, various values of ë Fä hØ{ }| ¡ Õ will be considered; it should be noted that by using a constant ratio , the normal strength ¡}hØSt |{ varies with according to (3.14). The values of j}){ | to be investigated vary between hØ{ }|0.3 and 1.4 GPa. This corresponds to interfacial energies for shear failure ranging from 35 to
  • 33. Indentation-induced interface delamination of a strong film on a ductile substrate 27d ‰ p & J/m , which are realistic values for the interface toughnesses for well-adhering depositedfilms (Bagchi and Evans, 1996). The size of the system analyzed (Fig. 3.1) is taken to be ž . This proved to be large ¤ "d enough that the results are independent of and therefore identical to those for a coated half- žinfinite medium. The mesh is an arrangement of quadrilateral elements, and d bd p c  nodes. cDDThe elements are built up of four linear strain triangles in a cross arrangement to minimizenumerical problems due to plastic incompressibility. To account properly for the high stressgradients under the indenter and for an accurate detection of the contact nodes, the mesh ismade very fine locally near the contact area with an element size of . d  IŠ ! Consistent with the type of elements in the coating and the substrate, linear two-noded ele-ments are used along the interface. Integration of the cohesive surface contribution in Eq. (3.1) ª —¬ ª ‰¤is carried out using two-point Gauss integration. Failure, or delamination, of the interface at any ‰¤ Ù¬ ª ªlocation develops when exceeds . The critical instant is here taken to be when . D ª¤ © 6Larger critical values may be used as well (Xu and Needleman, 1994), but using or timesdoes not significantly change the critical indentation depth or force. The maximum indentation depth applied in all calculations is . Further indentation Š D © h){ ¢ }|can be done, but was not considered relevant since real coatings will have cracked by then andthe present model is no longer applicable.3.3.2 Perfect interface ¡For the purpose of reference, we first consider a coated system with a perfect interface, whichwould correspond to taking . We have analyzed this simply by rigidly connecting ær Æ #! hØ{ s t }|the coating to the substrate (cf. also Begley et al., 1999; Sen et al., 1998). Of particular rele- Îvance here, is the stress distribution that develops at the coating-substrate interface. Figure 3.4shows the normal ( ) and shear ( ) stress components at different indentation depths for thecase of "D "d %ts d © Pb! Æ t e d . Other values of ç ¥ t would give the same qualitative behaviour. ‘#! Æ t ç ¥ iNote that the radial position is normalized, for each , by the current contact radius . § ¢ £ In the contact area below the indenter ( ) the interface is, of course, under high com- v“§ £ 1pression (Fig. 3.4.a). The reason for this stress being the same for all indentation depths shownis that the contact region is contained within the plastic zone in the substrate, which exhibits 6no hardening. For £ p § roughly, there is an annular region with an opening normal stress.According to Fig. 3.4.b there are three regions in which the interfacial shear stress exhibits alocal maximum (in absolute value): near the edge of the contact radius, at about 6 and in a £Dregion between £e to . The highest shear stress, roughly £ «t p ad , is attained in this outer re- Æ sgion. Closer examination of the results reveals that this shear stress is caused by the outwardradial plastic flow of the substrate under the indenter relative to the coating. The latter is beingpulled inwards due to the indentation. This relative motion is opposite to that observed in thecalculations of Narasimhan and Biswas (1997) who considered a plastically deforming film ona rigid substrate. The most noteworthy part of the stress distributions in Fig. 3.4 is that there is an annular 6region around £ p § where the shear stresses are large and the normal stress is positive. This
  • 34. 28 Chapter 3 0.5 0 coating −0.5 σn −1 substrate −1.5 σn ⁄ σy −2 −2.5 h ⁄ t = 0.5 h ⁄ t = 1.0 −3 h ⁄ t = 2.0 −3.5 (a) −4 0 1 2 3 4 5 6 r⁄a 0.8 coating 0.6 σt substrate 0.4 0.2 σt ⁄ σy 0 h ⁄ t = 0.5 −0.2 h ⁄ t = 1.0 h ⁄ t = 2.0 −0.4 (b) −0.6 0 1 2 3 4 5 6 r⁄aFigure 3.4: Interfacial tractions along a perfectly bonded interface at three indentation depths( is the instantaneous contact radius) for£ "D "d %d © ‘#! Æ t e d s ç ¥ . (a) Normal stress. (b) Shear stress.is relevant for delamination because of the coupling between normal and tangential response of ¬the interface. Rephrased in terms of the cohesive surface model used here, the opening normalstress, corresponding to d p , significantly reduces the maximal shear stress before shear 6failure, as shown in Fig. 3.3, so that this region i £ p § is a potential location for delamination.
  • 35. Indentation-induced interface delamination of a strong film on a ductile substrate 293.3.3 Initiation of delamination ª ¬In the presence of a cohesive surface along the interface, a distribution of normal and tangentialseparations develops, b§  ¬ ¤ Î ›¬ Î `¤ , with progressive indentation . The actual initiation of delamina- $ ¢tion is identified when or © $b§  © b§  for any . The indentation depth at this instant $ §is denoted by , the corresponding contact radius is and the critical position is . For all ì ‚¢ ì a£ ì œ§ i Î`¤ i 2¬ Îparameter combinations investigated (cf. Sec. 3.3.1) we indeed find that delamination occursby tangential failure, . –b§  í $ A parameter study has been carried out to determine the dependence of , and on the ‰§ «¢ ì ì ì F£¡two key material characteristics: the substrate yield strength and the interfacial shear strength Æt ¡ . This is done by scanning, for three values of "D "d %d © Pb! Æ te d s ç ¥ hØ{ }| ¡ , several values of the ratio ç bœ«t ¥!Æ . Æ#!`h|Ø{ t }In the case of , the values of are 0.6, 0.7, 0.75, 0.8, 0.85 and 0.86, in Æ #! h){ t }| e "d ad © Pb! Æ tthe case of d s ç ¥ the values are 0.3, 0.4, 0.5, 0.6, 0.7, 0.75 and 0.8, and in the case of  d %d © ‘b! Æ t¡ the values are 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7. For each s ç ¥ , higher values of ‘#! Æ t ç ¥ were considered but above a certain value delamination was not found. The fact that #`hØ{ Æ t !}|there is a limiting value for the interfacial strength above which delamination is prevented willbe discussed in the next subsection. Figure 3.5 summarizes the obtained values of , and for the various cases. Figure 3.5.a ‰§ «¢ ì ì ì F£¡shows that for each of the three substrate yield strengths, increases with the interface strength ì§ t }| in proportion with the indentation depth . It should be noted that the direct pro- Æ #! hØ{ ì «¢portionality between and is valid for the range of interface stresses considered here, but ì œ§ ì ‚¢breaks down at much smaller strengths. The strengths considered here are such that the in-dentation depth has to reach roughly the coating thickness before delamination occurs. The Šradius at which delamination then initiates is seen to be on the order of the indenter radius ¤which here is Š bd  . Figure 3.5.b shows that this delamination radius is between 2 and 4 times thecontact radius , this factor depending on both interface strength and substrate yield strength.For ebD d"d ì %£ d © ç b`t s , we find ¥!Æ 6 #! ì £ p b! ì § , which is consistent with the expectation in the ¤ ¤previous section. The same data is re-plotted in Fig. 3.6 but now as a function of the ratio between interface ¡and substrate yield strength. We see that both and increase nonlinearly with ì¢ . ì§ ¡ #`hØ{ Æ t !}|Contrary to , which appears to be depending practically only on ì§ (Fig. 3.6.b), the #‰h){ Æ t !}|indentation depth is quite a strong function of the substrate yield strain (Fig. 3.6.a). The ì ¢latter strongly indicates that plastic deformation in the substrate is a key factor in determiningwhether or not delamination takes place (we will return to this later). £6 £ £ £ 6 eç b‰t ¥!Æ †c ë 64 %îD"bBc yp e6 %d r S"&rp ep %d  D6 †ey"d"r s  bDe "d %d y s d c s D s d s "ye 4 %d r y "D %ï` 6 %d  6 p r e %d  "d %d y s s d pd s s d s 4 c 4 %d c D %d s s s4 %d d 4 %d s d %d s Table 3.1: Coefficients for fitting relations (3.19)–(3.22).
  • 36. 30 Chapter 3 1.4 1.3 σ y ⁄ E s = 0.0025 σ y ⁄ E s = 0.005 σ y ⁄ E s = 0.01 1.2 1.1 τ max rc ⁄ R 1 ---------- σy 0.9 0.8 (a) 0.7 0.8 1 1.2 1.4 1.6 1.8 2 hc ⁄ t 1.4 1.3 σ y ⁄ E s = 0.0025 σ y ⁄ E s = 0.005 σ y ⁄ E s = 0.01 1.2 1.1 τ max rc ⁄ R 1 ---------- σy 0.9 0.8 (b) 0.7 0.25 0.3 0.35 0.4 0.45 0.5 ac ⁄ RFigure 3.5: (a) Location of delamination initiation vs critical indentation depth . (b) ì § ì ‚¢ ì ‰§vs contact radius at delamination initiation. Discrete points are results from the numerical ¡computations; the lines are (a) linear (see Eq. (3.19)) and (b) second degree polynomial fits.The arrows indicate the direction of increasing . œhØ{ Æ t ! }| The results presented in Fig. 3.5 can be fitted quite accurately by the following relationships: ( & £ † I! ì ¢  £ © #! ì § ¡ $Š ¤ (3.19) † II¢  £ © & efbI‰§  $ Æ #! hØ{  ( £ $Š!ì B $ ¤ !ì t }| (3.20) ë
  • 37. Indentation-induced interface delamination of a strong film on a ductile substrate 31 2 1.8 σ y ⁄ E s = 0.0025 σ y ⁄ E s = 0.005 σ y ⁄ E s = 0.01 1.6 1.4 hc ⁄ t 1.2 1 0.8 (a) 0.6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 τ max ⁄ σ y 1.4 1.3 σ y ⁄ E s = 0.0025 σ y ⁄ E s = 0.005 σ y ⁄ E s = 0.01 1.2 1.1 rc ⁄ R 1 0.9 0.8 (b) 0.7 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 τ max ⁄ σ y ¡Figure 3.6: (a) Critical indentation depth Š and (b) critical delamination radius #! ì ¢ versus ¤ b! ì §relative interfacial strength . Discrete points are results from the numerical computa- œhØ{ Æ t ! }|tions; the lines are according to fits: (a) Eq. (3.21), (b) Eq. (3.22).The least–squares method has been used to obtain the coefficients through for the best fit £ £for each of the three values of (see Table 3.1). The relationships (3.19) and (3.20) can be ç #‰‚t ¥!Æcombined to give ¡ e ë † IŠIì¢  £ © & $ £ † Š#I¢  £  $ Æ #! hØ{  t }| (3.21) ( £ $ ! B & $ !ì ë e
  • 38. 32 Chapter 3 ¡ ( £ £ † $ & £  #! ì §  B £ © & fb! ì §  `#‰h){  £ ¤ $ ¤ $Æ t !}| (3.22)which have then been plotted in Fig. 3.6. It is seen that they give a reasonable representation ë e eto the numerical results over the range of parameters considered here. As mentioned before,the critical indentation depth differs from the trend in Fig. 3.5 for much smaller values of theinterface strength so that extreme care must be taken by extrapolation of (3.19)–(3.22) outsidethe considered parameter range.3.3.4 Strength limit to delamination ¡As mentioned before, there is value of beyond which delamination does not take place. œhØ{ Æ t ! }|This value is seen in Fig. 3.6 to be dependent on and, upon closer examination, is related Pb! Æ t ç ¥to the coupling between normal and tangential behaviour of the interface. In the absence of thiscoupling, the shear stress along the interface is limited by the plastic flow in the substrate. Inthe time-independent limit, in (3.7), we have ðl s r so that the maximum value that qA«t 6 Æt 1 ñthe shear stress can reach at the interface is ; when plasticity is slightly rate-dependent 6 9!Ætas here ( "d ò© l d  ) this value can be somewhat larger but is still a very good estimate. 9!Æt ¡Hence, if the interface response is not coupled, delamination is not possible for strengths 6 ¡ j}){ |exceeding . The reason for which we find delamination in Fig. 3.6 at higher 9 ‰‚t !Æ #`hØ{ Æ t !}|must be attributed to the coupling in the interface behaviour as described by (3.11)–(3.12). Ω In the presence of coupling, the maximal tangential traction ¬ at any point along the h){ }|interface depends on the local normal displacement 6 , as shown in Fig. 3.3. The interfacialregion that is of interest then is £ p § or so, where a tensile interface stress develops during i ©indentation (Fig. 3.4.a), since the corresponding normal opening of the cohesive surface reducesthe local shear strength h){ Î }| ¡ . Figure 3.7 demonstrates this coupling for the case © ‘b! Æ t ç ¥ "D "d %de d s and ¬ y r Î %© d © #‰hØ{ ¬ ÆÎ t ! } | , for which delamination did not occur. This figure shows the s{evolution of , $  h) and }| with indentation depth at t ¡ (this is the location ¤ 4 D s ó© § where delamination does initiate when i i 6 "r %d © Æ #! j}){ p s ). We see that the local shear stress at t |the interface gradually increases to a value of around 9!Æ t ¬ , which evidences that the plastic Ωzone gradually engulfs this point. As this occurs, increases until a certain indentation depth Ω Îand the effective shear strength j}){ | decreases. However, just before would coincide j}){ | iwith and delamination would initiate, the normal opening starts to decrease and delamination tbecomes excluded. ¡ Figure 3.8 summarizes the limiting values of the interfacial shear strength for the vari- hØ{ }|ous cases analyzed and plots them directly versus the substrate yield strength. There appears tobe a rather good linear correlation over the range considered, with the deviation from the line 69!Æt Î Æt increasing with . The figure also shows the two values for interfaces characterized 6by different values of Ï#! Ï nÕ © than 0.5, namely ad nÕ s © and 0.7. According to Fig. 3.3, a Õ ¡smaller value of , for example, implies a stronger reduction of the interfacial shear strength. ÕHence, the smaller , the larger the value of i that suppresses shear delamination. It is seen hØ{ }|from Fig. 3.8 that the result is quite sensitive to the precise value of . Õ
  • 39. Indentation-induced interface delamination of a strong film on a ductile substrate 33 1 max Tt ( ∆n ⁄ δn) ⁄ σy 0.8 0.6 1⁄ 3 0.4 σt ⁄ σy 0.2 ∆n ⁄ δn 0 −0.2 0 0.5 1 1.5 2 h⁄t Î ¬Figure 3.7: Evolution with indentation depthcorresponding peak tangential traction D s v© § $ ¢ ¬  j}){ Î © ¤4  of shear stress at | . t , normal opening and i i3.3.5 Propagation of delaminationAfter delamination initiates at the location and at a corresponding indentation depth , it ì § ì «¢propagates as a shear crack in both directions: towards and away from the indenter. The tipof the zone that propagates towards the indenter gets arrested after a short time because of thecompressive normal traction close to the contact area as seen in Fig. 3.4.a. The leading tip of thedelaminated zone keeps propagating with continued indentation. Figure 3.9.a shows the leadingtip location,¤ , as a function of the indentation depth, b! Ì § , for the case of #«¢ Š! "D bd %d © ç b‰vt e d s ¥!Æ(other values of show the same qualitative behaviour), and for four different values of ‘b! Æ t ç ¥interfacial strength. The width of the ring-shaped delaminated area, , is shown in bI$ §  œ§  ¤! ÌFig. 3.9.b as a function of the indentation depth. Since the indentation rate is constant, the Á ¢»propagation speed can be expressed as given by ÌÁ§ f#! Ì §  ´ ¤ Á ¢ © @§ $ ¤ I#«¢  ´ Š Ì Á $Š!so that it is simply I#f¤  Á ¢ times the slope of the curves in Fig. 3.9.a. In each of the four cases $Š!shown, delamination starts off with a relatively high speed and reaches a constant steady-statespeed after traveling a certain distance from the initiation point. The reason for this is that thedominant driving force for initial propagation is the elastic energy stored in the system; oncethis energy gets released, it gives rise to more or less instantaneous growth which is just limitedhere by the rate at which plastic deformation can evolve in the substrate. On the other hand, thesteady state propagation is predominantly driven by the indentation process which is performed
  • 40. 34 Chapter 3 1.6 1.4 τ max = 0.7374σ y + 0.073 1.2 τ max (GPa) 1 no delamination 0.8 0.6 σy ------ - 0.4 3 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 σ y (GPa) ¡Figure 3.8: The interfacial shear strength above which delamination is excluded as a func- j}){ |tion of the substrate yield stress . Open circles are the numerical results, while the solid Æ «tline is a linear fit (the dash-dotted line is the limit in the absence of coupling between normal 6 †and tangential separation in the interface). The symbols and represent isolated results for ô %d fÕs © and y %d fÕ s ©, respectively.at a constant rate, thus explaining the constant steady-state propagation speed. ¡ It is of interest to mention here that for some parameter combinations, such as  d %d © Pb! Æ t s ç ¥and y %d © Æ ! hØ{ s t }| , we encounter numerical instabilities after the onset of delamination. Weattribute this to the stored elastic energy in these cases being so large that ellipticity of theproblem is locally lost when this energy is released. More advanced solution procedures arenecessary to handle this. To further demonstrate the interaction between delamination and plasticity in the substrate,e s õ© #«¢ ¡Fig. 3.10 shows the distribution of the Von Mises effective stress in the indented region for "D "d ad © ‘#! Æ t vt e d s ñ  Š! . The particular case shown is for r %d © Æ ! j}){ sand t | . For the ç ¥purpose of comparison, the distribution for the case of a perfectly bonded interface is includedas well (Fig. 3.10.a). The region in the substrate where the value of is close to signifies ñt Ætthe region of substantial plastic deformation (note that we are using viscoplasticity so that ñ itis not limited to ). The delaminated zone in Fig. 3.10.b is identified by a white line, and ö Æ t x¢ öextends from mm to "d %d e @§ ö 4 d %d § y s Ì s mm. It initiated at an indentation depth ‰ s y© #I‚¢ p  Š!ìand a critical radius of d %d œ§ s ì mm. Comparison of the two plots shows, first of all, thata plastic region has developed near the leading tip of this zone with a size on the order ofthe coating thickness. This plastic zone moves with the propagating tip of the delaminationzone. Apparently, there is a stress concentration around the tip of the delaminated zone whichinduces local plastic flow in the substrate. The second difference between Fig. 3.10.a and b
  • 41. Indentation-induced interface delamination of a strong film on a ductile substrate 35 2.2 2 τ max ⁄ σ y = 0.3 τ max ⁄ σ y = 0.5 1.8 τ max ⁄ σ y = 0.7 τ max ⁄ σ y = 0.8 1.6 rf ⁄ R 1.4 1.2 1 0.8 (a) 0.6 0.8 1 1.2 1.4 1.6 1.8 2 h⁄t 1.4 τ max ⁄ σ y = 0.3 1.2 τ max ⁄ σ y = 0.5 τ max ⁄ σ y = 0.7 1 τ max ⁄ σ y = 0.8 ( rf – ri) ⁄ R 0.8 0.6 0.4 0.2 (b) 0 0.8 1 1.2 1.4 1.6 1.8 2 h⁄tFigure 3.9: (a) Location of the leading tip of the delaminated zone, , vs indentation depth . Ì œ§ ¢(b) The width of the ring-shaped delaminated zone, "D "d %d © ç b‰t e d s §  ̧ , vs indentation depth . This figure ¢is for the case of , and several values of interfacial strengths. ¥!Æ »is that a perfectly bonded interface induces a plastic zone in the substrate that is somewhatlarger than in the case of a finite-strength interface. This is attributed to the fact that interfacialdelamination serves as another mechanism for stress relaxation in the system. Except for thenear-tip region of the delaminated zone, this tends to reduce the necessity for stress relaxationby plastic deformation, thereby reducing the overall dimension of the plastic zone.
  • 42. 36 Chapter 3 0.00 σe/σy 0.04 1.4 1.2 0.08 1 z (mm) 0.8 0.6 0.12 0.4 0.2 0.16 (a) 0.20 0.00 0.04 0.08 0.12 0.16 0.20 r (mm) 0.00 σe/σy 0.04 1.4 1.2 0.08 1 z (mm) 0.8 0.6 0.12 0.4 0.2 0.16 (b) 0.20 0.00 0.04 0.08 0.12 0.16 0.20 r (mm)Figure 3.10: Contours of the Von Mises effective stress at "D "d %d © Pb! Æ t e s y© #¡Š «¢ e d s ç ¥ for  . (a)perfectly bonded interface; (b) finite-strength interface with r %d © S#‰j}){ ! . The white line in s Æt! |(b) denotes the delaminated zone, the arrows indicating the direction of interfacial shear.3.3.6 Load versus depth of indentationOne of the most common outputs of indentation experiments is the indentation force versusindentation depth curve (load–displacement curve). Figure 3.11 shows the predictions for some
  • 43. Indentation-induced interface delamination of a strong film on a ductile substrate 37 2.5 perfect interface 2 finite-strength interface 0.005 1.5 0.01 F ⁄ (πR σ y ) σ y ⁄ E s = 0.0025 2 1 0.5 0 0 0.5 1 1.5 2 2.5 h⁄tFigure 3.11: Normalized load versus indentation depth curves for different values of . ç b`it ¥!Æ ¡Dashed lines represent cases with perfect interface and solid lines represent cases with a finite-strength interface with p %d © Æ ! hØ{ . The dash-dotted lines identify the initiation values s t }|ì¢ © ¢ (cf. Fig. 3.6.a). ¡of the cases considered here with an interface strength of p %d © #`hØ{ , in comparison with s Æ t !}|the corresponding ones for the perfect interface. Prior to initiation of delamination, the resultsdiffer very slightly from those with a perfectly bonded coating, which is just due to the finitestiffness of the cohesive surface. After initiation of the shear delamination, , the system ì¢ p ¢is noticeably more compliant, yielding an indentation force that is reduced by between 5 and10% compared to the same case with the perfect interface. For the smaller yield strengths, the effect of delamination in the – curve evolves grad- ¢ ¡ually, but a distinct kink at ì¢ © ¢ is observed in Fig. 3.11 for  d %d © ç bœ‚t . Similar kinks s ¥!Æhave been observed experimentally by Hainsworth et al. (1997) and Li and Bhuchan (1997),but they have been associated with through-thickness ring or radial cracks. The kink found hereis a result of the high energy release rate at the moment of crack initiation, that gives rise to al-most instantaneous growth of the delaminated zone (see Fig. 3.9.a). Since the amount of storedelastic energy is higher for higher interfacial strength and higher substrate yield stress, the kinkwill be sharper accordingly.3.4 ConclusionsNumerical simulations have been carried out of the indentation process of a coated materialby a spherical indenter. The interface between the film and the substrate was modeled by a
  • 44. 38 Chapter 3cohesive surface, with a coupled constitutive law for the normal and the tangential response.Failure of the interface by normal or tangential separation, or a combination, is embedded inthe constitutive model and does not require any additional criteria. A parametric study has been carried out to investigate the influence of interfacial strengthand substrate yield strength on delamination. Delamination is found to be driven by the shearstress at the interface associated with the plastic deformation in the substrate, and consequentlyoccurs in a tangential mode. Nevertheless, interfacial normal stresses have a significant effecton delamination, due to the coupling between normal and tangential response. Here we havefocused on the coupling whereby the interfacial resistance to tangential delamination is reducedby tensile stresses along the interface. We find, however, that the results are quite sensitiveto variations in this coupling so that quantitative predictions require an accurate description ofthis interface coupling. This pertains, for instance, to the value of the interfacial shear strength ¡above which delamination never occurs. In the absence of any coupling between normal and 6tangential response, this limiting strength is simply 9 ! Æ t © h){ }|, but it is significantly higherdue to the coupling. This indicates that especially this coupling in the adopted interface model(3.11)–(3.12) needs closer examination and comparison with dedicated experiments. In all cases analyzed, delamination initiates at a radial location that is more than twice thecontact radius, and propagates in two directions; towards and away from the indenter, thusgenerating a ring-shaped delaminated zone. The front which travels towards the indenter getsarrested after a short distance because of the compressive normal stress in that region. Aftera high initial speed of propagation, the other top settles at a steady propagation at a constantvelocity. The initial propagation speed is governed by the high elastic energy stored in the sys-tem prior to delamination, while the steady-state speed is controlled by the indentation processitself. It bears emphasis at this point to recall that the coating is assumed to be elastic and strong.Deviations from this, such as plasticity of the coating or cracking, may affect our findingsboth for the initiation and the propagation of delamination. As mentioned in the Introduction,cracking is a potential mechanism for hard coatings and will change the picture dramatically;this requires a totally different analysis. However, plastic deformation in the hard coatings underconsideration will be limited to plastic zones immediately under the indenter, which are oftensmaller than that in the substrate. In such cases, the precise instant of delamination and the rateof propagation are expected to differ somewhat quantitatively, but to leave the phenomenologyessentially unchanged. For simplicity, the present calculations have assumed perfect but rate-dependent plasticityfor the substrate. Rate dependency was never significantly affecting the results here because thetime scale for the indentation process was always much larger. Strain hardening of the substratewill, obviously, change the quantitative results, especially for the indentation force vs. depthresponse. However, we do not expect a qualitatively significant effect on delamination, sincethe leading front of the delaminated zone seems to propagate with the front of the plastic zonein which hardening has not yet taken place. Again for simplicity, this study has not accounted for the presence of residual stress in thecoating, due for example to thermal mismatch relative to the substrate. This as well as the
  • 45. Indentation-induced interface delamination of a strong film on a ductile substrate 39influence of wavy interfaces will be examined in a forthcoming paper.References Bagchi, A., Evans, A.G., 1996. The mechanics and physics of thin film decohesion and its measurement. Interface Science 3, 169–193. Becker, R., Needleman, A., Richmond, O., Tvergaard, V., 1988. Void growth and failure in notched bars. J. Mech. Phys. Solids 36, 317–351. Begley, M.R., Evans, A.G., Hutchinson, J.W., 1999. Spherical impression of thin elastic films on elastic-plastic substrates. Int. J. Solids Struct. 36, 2773–2788. Drory, M.D., Hutchinson, J.W., 1996. Measurement of the adhesion of a brittle film on a ductile substrate by indentation. Proc. Roy. Soc. Lond. A 452, 2319–2341. Hainsworth, S.V., McGurk, M.R., Page, T.F., 1998. The effect of coating cracking on the indentation response of thin hard-coated systems. Surf. Coat. Technol. 102, 97–107. Kriese, M.D., Gerberich, W.W., 1999. Quantitative adhesion measures of multilayer films. Part II. Indentation of W/Cu, W/W, Cr/W. J. Mater. Res. 14, 3019–3026. Li, X., Bhushan, B., 1998. Measurement of fracture toughness of ultra-thin amorphous carbon films. Thin Solid Films 315, 214–221. Marshall, B.D., Evans, A.G., 1984. Measurement of adherence of residually stressed thin films by indentation: I. Mechanics of interface delamination. J. Appl. Phys. 56, 2632–2638. Narasimhan, R., Biswas, S.K., 1998. A finite element study of the indentation mechanics of an adhesively bonded layered solid. Int. J. Mech. Sci. 40, 357–370. Page, T.F., Hainsworth, S.V., 1993. Using nanoindentation techniques for the characterization of coated systems: a critique. Surf. Coat. Technol. 61, 201–208. Peirce, D., Shih, C.F., Needleman, A., 1984. A tangent modulus method for rate dependent solids. Computers and Structures 18, 875–887. Rose, J.H., Ferrante, J., 1981. Universal binding energy curves for metals and bimetallic interfaces. Phys. Rev. Lett. 47, 675–678. Sen, S., Aksakal, B., Ozel, A., 1998. A finite-element analysis of the indentation of an elastic- work hardening layered half-space by an elastic sphere. Int. J. Mech. Sci. 40, 1281–1293. Swain, M.V., Menˇ ik, J., 1994. Mechanical property characterization of thin films using spher- c ical tipped indenters. Thin Solid Films 253, 204–211. Wang, J.S., Sugimura, Y., Evans, A.G., Tredway, W.K., 1998. The mechanical performance of DLC films on steel substrate. Thin Solid Films 325, 163–174. Xu, X.-P., Needleman, A., 1993. Void nucleation by inclusion debonding in a crystal matrix. Model. Simul. Mater. Sci. Eng. 1, 111–132.
  • 46. 40 Chapter 3Xu, X.-P., Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, 1397–1434.
  • 47. Based on: A. Abdul-Baqi and E. Van der Giessen, Delamination of a strong film from a ductile substrate duringindentation unloading, Journal of Materials Research 16 (2001) 1396.Chapter 4Delamination of a strong film from aductile substrate during indentationunloading In this work, a finite element method is performed to simulate the spherical indentation of a ductile substrate coated by a strong thin film. Our objective is to study indentation-induced delamination of the film from the substrate. The film is assumed to be linear elastic, the substrate is elastic-perfectly plastic and the indenter is rigid. The interface is modeled by means of a cohesive surface. The constitutive law of the cohesive surface includes a coupled description of normal and tangential failure. Cracking of the coating itself is not included. During loading, it is found that delamination occurs in a tangential mode rather than a normal one and is initiated at two to three times the contact radius. Normal delamination occurs during the unloading stage, where a circular part of the coating, directly under the contact area is lifted off from the substrate. Normal delamination is imprinted on the load vs displacement curve as a hump. There is critical value of the interfacial strength above which delamination is prevented for a given material system and a given indentation depth. The energy consumption by the delamination process is calculated and separated from the part dissipated by the substrate. The effect of residual stress in the film and waviness of the interface on delamination will be discussed.4.1 IntroductionIndustrial application of thin hard-film-coated systems continuously progresses. Coatings arecommonly used to enhance reliability, such as chemical resistance, wear resistance, corrosionresistance and thermal barriers. Adhesion between the film and the substrate determines, to agreat deal, the durability of that system. The enhancement gained by the coating may be ac-companied by the risk of poor adhesion between the coating and the substrate. Failure of theinterface between the coating and the substrate may lead to premature failure of otherwise longlasting systems. Indentation is one of the traditional methods to quantify the mechanical prop-erties of materials and during the last decades it has also been advocated as a tool to characterizethe properties of thin films or coatings. At the same time, for example for hard wear-resistantcoatings, indentation can be viewed as an elementary step of concentrated loading. For thesereasons, many experimental as well as theoretical studies have been devoted to indentation ofcoated systems during recent years. 41
  • 48. 42 Chapter 4 Interfacial delamination is commonly observed in indentation experiments to be accom-panied by other failure phenomena, such as coating cracking and subsequent spalling (Krieseand Gerberich, 1999; Li and Bhushan, 1998). The corresponding load vs displacement curvesshow a reduction in the stiffness or even a sudden discontinuity which is usually attributed tothe coating cracking. Delamination without any accompanying through-thickness cracks hasbeen observed by Li and Bhushan (1998) in their nanoindenation experiments on single andmultilayer coatings. There is no evidence in the literature, to the authors’ knowledge, whetherdelamination can give rise to any characteristic fingerprint on the load vs displacement curve. Bagchi and Evans (1996) have reviewed the mechanics of thin film decohesion motivatedby residual stress. The emphasis in their work is on the role of the interface debond energyand the methods of its quantitative measurement. They argue that most thin film adhesiontests do not measure the interface debond energy because the strain energy release rate cannotbe deconvoluted from the work done by the external load. Viable procedures to extract theinterfacial energy from indentation experiments will depend strongly on the precise mechanismsinvolved. The relative contribution of each mechanism to the overall observed behaviour andfailure mode depends on the material properties and loading conditions in a complex manner. Inthe case of ductile films on a hard substrate, coating delamination is coupled to plastic expansionof the film with the driving force for delamination being delivered via buckling of the film (seealso Marshall and Evans, 1984). On the other hand, coatings on relatively ductile substratesoften fail during indentation by radial and in some cases circumferential cracks through the film.The mechanics of delamination in such systems has been analysed by Drory and Hutchinson(1996) for deep indentation with depths that are two to three orders of magnitude larger thanthe coating thickness. They have also reviewed briefly the commonly used test methods forevaluating adhesion. Hainsworth et al. (1998) have suggested a simple model for estimating the work of inter-facial debonding from the maximum indentation depth and the final delamination radius. Inthis model, the elastic energy of the indented coating is approximated by the elastic energy of acentrally loaded disc. The idea has also been used in cross-sectional indentation by S´ nchez et aal. (1999) as a new technique to characterize interfacial adhesion. The proportionality betweenthe delamination area and the film lateral deflection predicted by the model was confirmed bythe experimental results. The objective of the present paper is to offer an improved understanding of indentation-induced delamination and to test the validity of the above-mentioned simple estimates. For thispurpose, we perform a numerical simulation of the process of indentation of thin elastic film ona relatively soft substrate with a small spherical indenter. The complete cycle of the indentationprocess, both loading and unloading, is simulated. The indenter is assumed to be rigid, thefilm is elastic and strong, and the substrate is elastic-perfectly plastic. The interface is modeledby a cohesive surface, which allows to study initiation and propagation of delamination duringthe indentation process. Separate criteria for delamination growth are not needed in this way.The aim of this study is to investigate the possibility and the phenomenology of interfacialdelamination with emphasis on the unloading part of the indentation process and the associatednormal delamination. The interfacial failure during the loading part has been studied by the
  • 49. Delamination of a strong film from a ductile substrate during indentation unloading 43 ˙ h O R r h a ρ Film t Interface φ ξ z Substrate Symmetry axis L L Figure 4.1: Geometry of the analyzed problem.authors in a previous work (Abdul-Baqi and Van der Giessen, 2001). Delamination was foundto occur in a tangential mode driven by the shear stress at the interface. It is initiated at a radialdistance which is two or three times the contact radius resulting in a ring-shaped delaminatedarea and imprinted on the load–displacement curve as a kink (Abdul-Baqi and Van der Giessen,2001). In this paper we will study the characteristics of normal delamination; conditions forthe occurrence/suppression this mode of failure, its fingerprint on the load–displacement curveand provide some quantitative measures about the interfacial strength. The effect of residualstress in the film and waviness of the interface on delamination will also be investigated. Itis emphasized that the calculations assume that other failure events, mainly through-thicknesscoating cracks, do not occur.4.2 Problem formulationThe interface between the coating and the substrate is modeled by means of a cohesive surface, ¬ Î ›¬where a small displacement jump between the film and substrate is allowed, with normal andtangential components and , respectively. The interfacial behaviour is specified in terms © Ωof a constitutive equation for the corresponding traction components and at the same ilocation. The constitutive law we adopt in this study is an elastic one, so that any energydissipation associated with separation is ignored. Thus, it can be specified through a potential, ii.e. 2¬ ´ ƒ© «© ª‚  ª Ï sƒIŠ·xl ÑИ $ ( © (4.1) ´
  • 50. 44 Chapter 4The potential reflects the physics of the adhesion between coating and substrate. Here, we usethe potential that was given by Xu and Needleman (1993), Ï Î ¬ ¬ ¬ ¬ Õ ˆ § Õ  ˆ ¤ † P Õ  Ï © Ï s ”—’ & & Î ¤  ‰ – ¤ ’  §‰ † ’   § – §  ’ „ ’ ¤ ‰ Ï † Î ‰¤ ¤ a@Ò i Î ÔÓ i i (4.2) i a@xÎ i i ÔÓÒwith and the normal and tangential works of separation ( Ï Ï #! Ï © ÙÕ ), and and two i i Ïcharacteristics lengths. The parameter governs the coupling between normal and tangential § i i i ¡responses. As shown in Fig. 4.2, both tractions are highly nonlinear functions of separationseparation of ( Îwith a distinct maximum of the normal (tangential) traction of Ï Ï hØ{ hØ{ t }| }| ¡ ( ) which occurs at a ). The normal (tangential) work of separation, ( ), can D 9 ! `¤ © ›¬ ¤ © ¬ Î Înow be expressed in terms of the corresponding strengths ( hØ{ hØ«t }| }| { ) as i ¤ i i Î `¤ ¡  ƒ© Î © Ï s h){ $ I D Û Ï ( j}){ w$ I }|  | t  (4.3) ÎUsing these along with the definition a@Ò ÔÓ b! Ï i —Õ Ï © a@Ò i ÔÓ , we can relate the normal and shear strengthsthrough ¡ Τ i xh){ ¤ $ I D  Õ © h)%t s }|  }| { (4.4) i a@Ò 2¬ Õ ÷ ›¬ ÔÓ ÷The coupling parameters and are chosen such that the shear peak traction decreases with §positive and increases with negative (Fig. 4.2b). More details are given in (Abdul-Baqiand Van der Giessen, 2001). The coating is assumed to be a strong, perfectly elastic material with Young’s modulus Œ A¥and Poisson’s ration (subscript c for coating). Œ¦ The substrate is supposed to be a standard isotropic elastoplastic material with plastic flowbeing controlled by the von Mises stress. For numerical convenience, however, we adopt arate-sensitive version of this model, expressed by Å 6 a¥ ¢£ © ¢£ Á Ž t ‰ ·d © Å d ( Å d (4.5) t ± D a¥ Å i ’Ft Æ Á Á ¥ Á ¥ Ž Æfor the plastic part of the strain rate, ¢£ ¡ © ¢£ Á ¥ ¢£ £ Ž¢  ¢£ . Here, are the deviatoric components of the ¢£ Á ÁPiola-Kirchhoff stress ¢£ ¢£ and are the dual Lagrangean strain-rate components. Furthermore, Á ± Ç ò© Ž t &B is the von Mises stress, is the rate sensitivity exponent and is a referencestrain rate. In the limit of± ± l , this constitutive model reduces to the rate-independent von øul s r @ÁÆ dMises plasticity with yield stress . Values of on the order of Æt are frequently used for l d"d metals (see e.g. Becker et al., 1988), so that the value of at yield is within a few percent ofÆ t Ž for the strain rates that are encountered in our analysis. The elastic part of the strain rate, , «t £ Ž¢ ¥ Áis given in terms of the Jaumann stress rate as wŽË º ¥ wº ¢£ ¤ © ¢£ Ê ¡ Ë (4.6) Á Ë wº ¢£with the elastic modulus tensor being determined by the Young’s modulus ¤  ¥ and Pois-son’s ration (subscript s for substrate). ¦
  • 51. Delamination of a strong film from a ductile substrate during indentation unloading 45 1.5 1 ∆t = 0 0.5 0 T n ⁄ σ max increasing ∆ t −0.5 −1 −1.5 −2 (a) −2.5 −1 0 1 2 3 4 5 6 ∆n ⁄ δn 3 2 ∆n = 0 1 T t ⁄ τ max 0 increasing ∆ n −1 −2 (b) −3 −3 −2 −1 0 1 2 3 ∆t ⁄ δt ¬ © ›¬ v© Î ÎFigure 4.2: The normal and tangential responses according to the interfacial potential (Eq. 4.1).(a) Normal response ¡ $  . (b) Tangential response $  . Both are normalized by theirrespective peak values and } . h|Ø{ i j})Si t |{ The problem actually solved is illustrated in Fig. 4.1. The indenter is assumed rigid and tohave a spherical tip characterized by its radius . The film is characterized by its thickness ¤ Šand is bonded to a half-infinite substrate by an interface specified above. Assuming both coatingand substrate to be isotropic, the problem is axisymmetric, with radial coordinate and axial §coordinate in the indentation direction. The actual calculation is carried out for a substrate of height Š  ž and radius , but is taken large enough so that the solution is independent of ž ž žand thus approaches the half-infinite substrate solution.
  • 52. 46 Chapter 4 The analysis is carried out numerically using a finite strain, finite element method. It usesa total Lagrangian formulation in which equilibrium is expressed in terms of the principle ofvirtual work as ¢£ #¤ ¢£ ¡   Ÿ ¥ † ª b¤ ª © @§ Ÿ ¬ ¨ ¢ # ¢¤ ƒ® Ÿ © °   (4.7) Š sHere, is the total region analyzed and ž ²¢ ³ƒž ¦ †~ ­ is its boundary, both in the undeformed a~ ´ ± %~ ° ¢configuration. With ¦ $†·(SU(# ¶  § ‹© µ the coordinates in the undeformed configuration, and ¦ ¢£ Š ¥b¤ ¢ ¢ #¤ °are the components of displacement and traction vector, respectively. The virtual strainscorrespond to the virtual displacement field via the strain definition, £ ¸ º ° º ° ¢ h£ ° £ ¹¸¢ °  ¢£ ¥ ¸ $ ¸¢ † †  D © ¢ (4.8)where a comma denotes (covariant) differentiation with respect to . The second term in the µleft-hand side of Eq. (4.7) is the contribution of the interface, which is here measured in thedeformed configuration ( ª v© ·Š ©  § À© ù ¼ ). The (true) traction transmitted across the interface ¾ ª —¬has components , while the displacement jump is »­ , with being either the local normal ˜direction ( l ¿˜ © ) or the tangential direction ( ) in the Š À˜ © -plane. Here, and in the ú FU#v§ú( ° $ ú¢ ¡ ú¢ ¥remainder, the axisymmetry of the problem is exploited, so that . © Š © © d © The precise boundary conditions are also illustrated in Fig. 4.1. The indentation processis performed incrementally with a constant indentation rate . Outside the contact area with Á¢radius in the reference configuration, the film surface is traction free, £ d © †U#§  u Š © †U§  wŠ $d( $d(  for €ž —3•£ s 1 § 1 (4.9)Inside the contact area we assume perfect sliding conditions. The boundary conditions arespecified with respect to a rotated local frame of reference þ v° as shown in Fig. 4.1. In the "·SU( üû $¶( ýnormal direction, the displacement rate is controlled by the motion of the indenter, while in Áthe tangential direction the traction is set to zero, i.e. ÿ wŠ þ° © $FU(#§  ÿ ŠÞ(8Ï%¡`Á ¢ © FU§  Á d    Y X $( for P£ —3—d s 1 § 1 (4.10)Numerical experiments using perfect sticking conditions instead have shown that the preciseboundary conditions only have a significant effect very close to the contact area and do not alterthe results for delamination to be presented later. During the loading part, contact nodes are ¬identified by their spatial location with respect to the indenter; simply, at a certain indentationdepth and displacement increment ¢ , the node is considered to be in contact if the vertical ¢ ¬distance between the node and the indenter is not greater than . During the unloading part, ¢a node is released from contact based on both its spatial location and the force it exerts onthe indenter; if the normal component of the nodal force is smaller than a critical value, andthe vertical distance between the node and the indenter is positive, the node is released fromcontact. The critical value for the nodal force is taken to be of the average current nodal €force. It should be noted that using a value one order of magnitude smaller did not significantlyaffect the results. The indentation force is computed from the tractions in the contact region, ¡ à Ÿ § § D †U#§  u Š s E $d( © ¡ (4.11) ~ ”
  • 53. Delamination of a strong film from a ductile substrate during indentation unloading 47The substrate is simply supported at the bottom, so that the remaining boundary conditions read ° ° d © ‘U§  u $ ž( for ž —3—d 1 § 1 ; d © FU%d   $( for 3• —d ž 1 1 . (4.12)However, the size will be chosen large enough that the solution is independent from the žprecise remote conditions.4.3 Model parametersThere are various material parameters that enter the problem, but the main ones are the interfa-cial normal strength , the coating thickness , the coating Young’s modulus , the maxi- j})t |{ Š Œ ¥mum indentation depth h)¢ }| { and the substrate yield strength . In the results to be presented Æ ‚tsubsequently we focus mainly on the effect of the interfacial normal strength , keeping the h){ t }|same value of sd o© Æ t  GPa (with a reference strain rate of Æd 6and ). The d6""6 d  © l    %d © selastic properties are taken to be GPa, d bd e © ‘¥ Œ , © Á –¥ "6 %d © b¦ GPa and  s Œ . s © b¦ `¤ ä "d D ¤ ad  Î% d e For the cohesive surface we have chosen the same values for and , namely m. As f  ad sin the previous study (Abdul-Baqi and Van der Giessen, 2001), the coupling parameters and §Õ are both taken equal to e %d which give rise to qualitatively realistic coupling between normal s iand tangential responses of the interface. The values of that have been investigated vary j})‚t |{approximately between 0.5 and 2.0 GPa. These correspond to interfacial energies for normalfailure ranging from 150 to & h ! ¢ bd p , which are realistic values for the interface toughnesses d ¡of well-adhering deposited films (Wei and Hutchinson, 1999). Note that a constant value of Õimplies that the shear strength always scales with the normal strength hØ{ }| according to h؂t }| {Eq. (4.4). We have used an indenter of radius vg"D © ¤ hfe and most of the results are for a filmthickness ige s D © Š hf . Indentation as well as retraction are performed at a constant rate © Á¢ £  mm/sec. The size of the system analyzed (Fig. 4.1) is taken to be ž . This proved to Šbd ebe large enough that the results are independent of and therefore identical to those for a žcoated half-infinite medium. The mesh is an arrangement of D 6D ""d D  quadrilateral elements, and dd nodes. The elements are built up of four linear strain triangles in a cross arrangement 4to minimize numerical problems due to plastic incompressibility. To resolve properly the highstress gradients under the indenter and for an accurate detection of the contact nodes, the meshis made very fine locally near the contact area with an element size of . d  Š ! Consistent with the type of elements in the coating and the substrate, linear two-noded ele-ments are used along the interface. Integration of the cohesive surface contribution in Eq. (4.7) ª ¬ ª¤is carried out using two-point Gauss integration. Failure, or delamination, of the interface at ª ¬any location develops when ª¤Dexceeds . A practical definition of when a complete crackhas formed, is © (Xu and Needleman, 1994). The maximum indentation depth applied in all calculations is . Further indenta- Š D —j})‚¢ 1 |{tion can be done, but was not considered relevant since real coatings will have cracked by thenand the present model is no longer applicable.
  • 54. 48 Chapter 4 3 F ⁄ F max = 0.00 2 0.02 1 0.12 σn ⁄ σy 0.35 0 0.54 −1 −2 unloading −3 1.00 −4 0 1 2 3 4 r ⁄ a maxFigure 4.3: Normal stress variations along a perfect interface at the beginning of unloadinghØ)¡ © ¡(}| { ) until complete retrieval of the indenter ( d © ¡ ).4.4 Results and discussion4.4.1 Perfect interfaceFor the purpose of reference, we first consider a system with a perfect interface, i.e. its strengthis sufficiently higher than the stresses induced by the particular loading. This can be achievedby rigidly connecting the coating to the substrate, which corresponds to taking s r Æ #! h){ t t }| .Of particular relevance here, is the development of the stress distribution along the interfaceduring the unloading stage, and in particular the component normal to the interface . Fromtthis, we can already get qualitative insight into when and where delamination may occur. Figure 4.3 shows the normal stress at the interface at different instants between maximum iindentation depth and complete retraction of the indenter, as specified through the load rela- ¡tive to the maximum indenter load. At the maximum indentation depth, the interface stress isof course compressive, and almost uniform over the current contact area due to plastic flow in öthe substrate. The compressive stress attains a peak value of 4 GPa just outside the contactregion of radius hØ{ £ 6 ö }| . Relatively low tensile normal stresses are found beyond the compressiveregion, at j}){ £ § | . This is a result of the resistance of the substrate to the film bending in thisregion. It was demonstrated by the authors (Abdul-Baqi and Van der Giessen, 2001) that thenormal displacement induced by this stress will reduce the interfacial shear strength (Fig. 4.2b),which in turn may lead to shear delamination. As the indenter is withdrawn, at the same rate as during loading, the elastically bent coating
  • 55. Delamination of a strong film from a ductile substrate during indentation unloading 49tends to seek its original flat shape. For the material parameters here this peeling tendencyinduces reversed plastic flow in the substrate under the indenter. As this proceeds, the initiallycompressive stress evolves into a tensile stress in the interface directly under the initial contactregion (Fig. 4.3). The figure also shows that the tensile area increases slowly in size during theprocess of unloading, and its final size is roughly the same as the maximum contact radius . j})j}| %£ { {) To study the evolution of the tensile normal stress at the interface, its maximum value is t |recorded together with its position along the interface, as shown in Fig. 4.4. In the initial stages §of unloading, tension is found only in the ring outside the contact area (Fig. 4.3). Upon contin- iued unloading, the peeling effect causes interfacial tension to develop rapidly, Fig. 4.4a, withthe location of the maximum closely following the instantaneous contact radius (Fig.4.4b). £The largest value of sy D © j}){ t | GPa obtained in this particular case, is reached at the end ofthe unloading and located at the symmetry axis. i Based on these results, interfacial failure leading to normal delamination may be expectedduring the unloading stage when the interfacial strength j}){ t | is lower than the maximum tensilestress j}){ t | reached at any moment. In the present case, normal delamination is avoided on theother hand if the interfacial strength i exceeds h){ t }| sy D GPa. Figure 4.5, curve (e), shows the indentation load versus displacement curve for this case ofa perfect interface. Such a curve is one of the most common outputs of indentation experiments.Its importance stems from the fact that it is a signature of the indented material system. Severaltechniques have been reported in the literature to extract the mechanical properties of bothhomogeneous and composite or coated materials from indentation experiments (Doerner andNix, 1986; Bhattacharya and Nix, 1988; Gao et al., 1992; King, 1987; Lim et al., 1999). In theforthcoming section, we will therefore study the interfacial failure process in more detail andprovide some qualitative measures of the interfacial strength.4.4.2 Finite-strength interfaceIn this section, and throughout the rest of this paper, we will study interfaces with finite strengthsto allow for interfacial delamination to develop. To demonstrate the effect of the interfacialfailure on the load–displacement data, Fig. 4.5 shows the predicted curves for different valuesof interfacial strength j}){ t | . The rest of the material and geometrical parameters are the sameas before. Interfacial delamination during unloading was found in all cases shown in Fig. 4.5(except case e). Compared to the perfect interface case (curve e), the initiation of delaminationis seen to result in a rather sudden reduction of the unloading stiffness at sufficiently small¡ . For higher interfacial strengths, delamination is imprinted on the load versus displacementcurve as a hump where the stiffness becomes negative. This phenomena will be explained inmore detail later in this section. Another characteristic of delamination that can be observedin the load–displacement curve is the negligible residual indentation depth at the end of theunloading. In the absence of delamination (case e), the residual indentation depth is more thanhalf the maximum indentation depth. Curve a, which corresponds to the lowest , shows j})«t |{a little decrease in the stiffness at the end of the loading stage. This reduction is due to sheardelamination at that stage, as discussed in detail in (Abdul-Baqi and Van der Giessen, 2001).
  • 56. 50 Chapter 4 3 unloading 2 ⁄ σy max σn 1 (a) 0 3.5 4 4.5 5 h (µm) 1.5 r⁄R a⁄R 1 unloading 0.5 (b) 0 3.5 4 4.5 5 h (µm)Figure 4.4: (a) Evolution of the maximum normal stress h){ t }| with indentation depth duringunloading. (b) The corresponding location at the interface at which the stress is maximum. iIn all other cases shown, the interface strength was large enough to prevent shear delamination,but not normal delamination. The interfacial strength above which delamination is prevented is found to be © h)«t }| { Ds DGPa (curve d in Fig. 4.5). From the results discussed above for a perfect interface, how-ever, we expected delamination at even higher strengths, up to sy D GPa. The difference mustbe attributed to the fact that the cohesive surface description for the finite-strength interface ¬ ¤provides additional compliance to the system even before failure. This additional complianceresults from the limited normal opening at the interface  m , whereas a perfect interface, $by definition, does not allow such opening. Although the energy consumed at the interface in i i
  • 57. Delamination of a strong film from a ductile substrate during indentation unloading 51 2 0.5 a b 0.25 1.5 c d e F ⁄ ( πσ y R ) 2 0 3 3.5 4 1 0.5 0 0 1 2 3 4 5 h (µm)Figure 4.5: Load vs displacement curves for several values of interfacial strength hØ{ t }| : (a) 0.55;(b) 1.1; (c) 1.5 and (d) 2.2 GPa. Curve (e) is for a perfect interface.this state is extremely small, the extra compliance does give rise to a small redistribution of thenormal stress over the interface and a reduction of the maximum normal stress . j}){ t | Figure 4.6a shows a contour plot of the von Mises effective stress at the end of the loadingstage ( j}){ ¡ © ¡ | ) for the case (c) in Fig. 4.5 with e s ó© hØ{ t  }| GPa. The size of the plastic izone at this depth of ŠD © ¢ is about 5 times the maximum contact radius. To illustrate thedelamination process, Fig. 4.6b shows a contour plot of the vertical stress component at the u hu tend of the unloading process ( d © ¡ ). First thing to observe is that the radius of the delaminatedzone, , is about § ¤ ¥ € †d e larger than the maximum contact radius reached during indentation. j}){ £ |Secondly, we observe a region with compressive normal stress in front of the delamination tip.This region is the remainder of the compressive region generated during the loading stage,which has apparently hardly changed during unloading. It thus seemed that delamination wasinitiated under the retrieving indenter, expanded in the radial direction and was arrested in thiscompressive interfacial stress region. The progressive development of delamination with continued unloading is shown in Fig. 4.7for several values of . It should be noted that, except for h){ t }| GPa, delamination D s D © hØ{ t }|starts at a distance from the symmetry axis. For these cases represents the location of the § ¤delamination tip which is traveling away from the symmetry axis. Since the other tip reaches thesymmetry axis almost immediately, can be considered to a good approximation as the radius § ¤ ¥of the delaminated circular area. In all cases shown in Fig. 4.7 delamination starts at a relativelyhigh initial propagation velocity compared to the indentation rate , and then reaches a lower Á¢velocity on the order of . The crack is stopped when it reaches the region with sufficiently high Á¢
  • 58. 52 Chapter 4 a max ⁄ t 0 σe ⁄ σy 1.60 5 1.40 1.20 1.00 10 0.80 0.60 z⁄t 15 0.40 0.20 0.00 20 25 (a) 30 0 5 10 15 20 25 30 r⁄t rd ⁄ t 0 σ zz ⁄ σ max 1.00 5 0.75 0.50 0.25 10 0.00 -0.25 z⁄t 15 -0.50 20 25 (b) 30 0 5 10 15 20 25 30 r⁄tFigure 4.6: (a) Contour plot of the von Mises stress at the end of loading ( }| { © ). (b) e s  © hØ)¡{ t ¡ d © ¡Contour plot of the stress component w«t uu at the end of the unloading ( ) for} h|GPa (curve c in Fig. 4.5). The plot also shows the delaminated region.
  • 59. Delamination of a strong film from a ductile substrate during indentation unloading 53 2 1.5 σ max = 0.55 GPa r d ⁄ a max 1 1.1 unloading 1.5 0.5 1.85 2.2 0 0 1 2 3 4 5 h (µm)Figure 4.7: Evolution of the delamination radius during unloading for ige © h){ ¢ h f }| and severalvalues of h){ t }| (or equivalently ). Ï icompressive stress (Fig. 4.6). The final delamination radius is about es  times the maximumcontact area for all values of . It is clear in the figure that for lower interfacial strengths, h)St }| {delamination starts earlier in the unloading process. On the other hand, the lower the interfacialstrength, the lower the residual indentation depth (permanent indentation depth left at the ¢ ¦ §end of the unloading). Figure 4.7 reveals that residual indentation depth for several values of ¢ ¦hØSt}| { . Lower interfacial strengths even lead to small negative residual indentation depths, wherethe coating bulges upwards at the end of the unloading. The observations indicate that delamination is the outcome of a complex interaction be-tween various mechanisms. To get further insight into this competition, Fig. 4.8a shows the ÷decomposition of the total energy of the system into interfacial energy Å , elastic energy ¨ © Ž  ¨(in the film and substrate) and dissipated, plastic energy for the case of ¨ © GPa e s ó© j}){ t »  |(curve c in Fig. 4.5). Other values of interfacial strength show the same qualitative behavior.In this particular case, delamination initiated at hvfxe sy 6 © Š e s  © ¢ . It is clear in the figurethat the plastic energy is constant at the initial stage of the unloading, i.e., the initial stage forthe unloading is almost purely elastic. This is in agreement with what is commonly observedin indentation experiments (Doerner and Nix, 1986). Limited reverse plasticity is seen to havecontributed to a little increase (less than € †d  ) in the plastic energy. At the onset of delamina-tion, the plastic energy reaches a constant value. The contribution of the film and the substrateto the elastic energy is demonstrated in Fig. 4.8b. The elastic energy of the substrate is seento decrease more rapidly compared to the elastic energy of the film at the initial stage of theunloading. This is in agreement with what is reported in the literature that the initial stiffness
  • 60. 54 Chapter 4 1 U tot unloading 0.8 0.6 U ⁄ U max U pl 0.4 loading 0.2 U in U el 0 (a) 0 1 2 3 4 5 h (µm) 0.25 0.2 U el ⁄ U max 0.15 film 0.1 0.05 substrate 0 (b) 0 1 2 3 4 5 h (µm) ÷Figure 4.8: (a) Decomposition of total energy into interfacial energy ( ), elastic energy ( ) Å ¨ Ž © …  ¨and plastic energy ( ). (b) Contribution of the film and substrate to the elastic energy. In » (a) and (b) e s ¨ ¿© hØ{ t  }| © GPa, the normalization constant is ¨ ¢ ¡ © hØ{ and the vertical }|dashed lines identify the initiation of delamination. ~ ‡ˆ†„  ” …of the unloading is predominantly controlled by the substrate for indentation depths larger thanthe film thickness (Doerner and Nix, 1986; Bhattacharya and Nix, 1988; Gao et al., 1992; King,1987; Lim et al., 1999). At the onset of delamination, the substrate elastic energy reaches aconstant value, whereas the film elastic energy decreases as the film unbends. This indicatesthat the main contribution to the energy release, and hence the advance of delamination, comesfrom the film. It is also interesting to notice that at the end of the unloading, there still exists
  • 61. Delamination of a strong film from a ductile substrate during indentation unloading 55 ÷some elastic energy in the system. This energy is small compared to the dissipated energy (plas-tic energy), but, when compared with the interfacial energy, , it seems to have a significant ¨value. » On the basis of the above observations, the unstable part of the load–displacement curves,with negative stiffness, shown in Fig. 4.7, is now readily attributed to the spontaneous openingof the interface at the initial stage of delamination (Fig. 4.7). As explained in the previousparagraph and shown in Fig. 4.8, the processes that control the system during delamination arethe unflexing of the coating and the interfacial delamination. The coating evidently providesa positive contribution to the overall stiffness, whereas the energy release from the interfacegives a negative contribution. This can be seen in Fig. 4.8, where the stiffness provided by eachenergy source is the curvature of the corresponding curve. For relatively strong interfaces, theenergy release from the interface dominates during the first stage of delamination when the rateof propagation, relative to the indentation rate , is high. During the second stage, the process Á¢is governed by the unflexing of the coating, thus giving rise to a positive overall stiffness (notethat the coating response is constrained by the indenter which is withdrawn at a given rate). Itis this complex interplay between these two terms which shapes the overall behaviour of thesystem, including the load–displacement curve.4.4.3 Comparison with a simple estimateDeduction of quantitative information about the interfacial strength from indentation experi-ments, in particular from load–displacement curves and delamination areas, is hindered by therather complicated interplay between the film elastic energy and the interfacial energy. A simpleestimate for the work of interfacial debonding from final delamination results has been given byHainsworth et al. (1998). This estimate is based on an energy balance involving the interfacialenergy and the elastic energy in the coating (the elastic energy in the substrate is neglected).The latter is approximated by the elastic energy of a centrally loaded disc of radius with § ¤ clamped edges. Based on this model, the interfacial work of separation is estimated by $ &¦ ¢  hØ&{ ¢  B Š 6 Œ ¥ D ©  Ž Ï }|   (4.13) § & Œ PI ¤ w$ ¦ ëin terms of directly measurable quantities. i As the model shows a strong dependence on the coating thickness and the maximum Šindentation depthj})¢ |{ , we have chosen to vary these two parameters over a certain range andcompare the model predictions with our FEM findings. A set of calculations using a conical  "pindenter with a r semi-angle is also performed to examine the model’s sensitivity to theindenter’s geometry which is not captured by Eq. (4.13). Despite its very approximate nature, Eq. (4.13) does capture some of the qualitative trends,as shown in Fig. 4.9. For instance, one expects from (4.13) that for a given interfacial &§  ¤ hØ{ ¢ }|strength (or energy) and coating properties (and neglecting the residual indentation depth). Theresults of a series computations for two different strengths are summarized in Fig. 4.9a, andare seen to be consistent with this scaling. The conical indenter results presented in the figure
  • 62. 56 Chapter 4 1000 delamination area πr d (µm ) 2 800 2 σ max = 0.92 GPa 600 400 1.84 200 1.84 (conical indenter) 0 (a) 0 1 2 3 4 5 h max (µm) 60 h max = 5.0 µm 50 σ max = 1.84 GPa ) 4⁄3 (µm 40 4⁄3 rd h max = 2.5 µm 30 σ max = 0.92 GPa 20 (b) 1 2 3 4 5 t (µm)Figure 4.9: (a) Delamination area &¤ § E vs the maximum indentation depth hØ{ ¢ }| . (b) B ¤§ vscoating thickness . Š g ëshow the same trend. S´ nchez et al. (1999) have used Eq. (4.13) and a modified version of ait on their cross-sectional indentation data, and they have also confirmed the linear relationbetween the delamination area and the maximum deflection of the coating. Secondly, accordingto (4.13),B ¤§ is proportional to , with all other quantities being the same. Our results, shown Š 6in Fig. 4.9b are consistent with this as well. Finally, over the range of of g ë – GPa, d"d p d e © ¥ Œthe proportionality between and is also found to be consistent with the prediction of § ¤ Œ ¥Eq. (4.13). ë
  • 63. Delamination of a strong film from a ductile substrate during indentation unloading 57 ÷ ÷ Ï #!  Ž Ï $ if  § $ if  ¢ $ if  j}){ ¢ h ¤ h ¦ h | D sd e   r byp %d c s e s D6 p "r s D 4 p %ss d  e  6 ayp e "D s D6  y p %d s e sd s 6 D s se  Sp s  4 p %d s D rs D rs  6 64 p %d sp s %d e sd s 4 p 4 s 44 e4 4 4 sc  s %d sd e ÷Table 4.1: Estimates forige s D © Šhf ÷ from (4.13) on the basis of the computed values Ï ¢ ¦ and § ¤ for & h ! ¢ "d e © Ï and several values of . The actual value is d . hØ¢ }| { ÷ ÷ Ï #!  Ž Ï $ if  § $ if  ¢ $ if  j}){ ¢ h ¤ h ¦ h | 6 6 e sy 6 e s D6 4 Dac r e  s 4p D6 4 %dd s D4 c say 6 sr D Sr4 say  c6 s 4% s4 s %d e sd s 6 e %d D4 6 ay D s sr ‰ e4 " D %%dssd c  y 4 %d 4s s e sd s 4 4 p "s p 4s  c 4 %d s sd e Table 4.2: Same as in Table 4.1 but for conical indenter. ÷ ÷ Ïb! Ž Ï $ vf  § $ vf  ¢ $ vf  Š h ¤ h ¦ h   e 6p e s D6 p 4s 4 r es p 4 e sc `p r e %dd s bpd d %"cs  c6 D s  sr ay  p e %d se s % e sd s 6 Dc p ss  64 ay` sr s4  4 e %d se s %d e sd s 4 e y ay4  bd %c 4 4 s d s 4 %d s sd e ÷Table 4.3: Estimates forvge © j})S¢hf Ï from (4.13) on the basis of the computed values of ÷ ¢ ¦ and § ¤ for |{ and several values of . The actual value is Š . & h ! ¢ bd e © Ï d However, not all trends are correct. For example, Eq. (4.13) predicts a lower slope for thedelamination area vs hØ¢ }| { curve for higher values of interfacial strength, whereas the FEMresults presented in Fig. 4.9a show the opposite tendency. The more serious limitation of Eq. (4.13) is that the interfacial energy estimated from thenumerical results do not agree quantitatively with the actual energies. As demonstrated in Ta-bles 4.1 to 4.3, the interfacial energy are severely overestimated. In Table 4.1 we notice that
  • 64. 58 Chapter 4the higher the maximum indentation depth, the better the estimate. This can be understood byrecalling that the model is based on the expression for the deflection of a clamped disc loadedat the center (Timoshenko and Woinowsky-Krieger, 1959), where the deformation is assumedto be pure bending. The contribution of the stretching is ignored; this is reasonable when theradius of the disc is large compared to its thickness. In the case of indentation, this conditionis analogous to contact radius (or maximum indentation depth) being larger than the film thick-ness. This explains the better estimation at larger maximum indentation depths. This trend isalso observed for the conical indenter in Table 4.2, but the quality of the estimate here is evenworse. The reason is that the cone produces more stretching of the film than the sphere, result-ing in less accuracy of the model. In Table 4.3, the smaller the coating thickness, the better theestimate according to (4.13). The same explanation holds here too. Evidently, the assumptionthat the disc is clamped at its boundary is questionable. If it is assumed that the disc is simply 6supported, the expression for Ž Ï  in Eq. (4.13) must be multiplied by Ib¦ †  II†¦ I $Œ !$Œ †  . Thiswill give better estimates, but large errors are still possible. i Note that Eq. (4.13) does not incorporate the influence of the substrate. In order to see theaccuracy of this approximation, we have investigated the dependence of the delamination radius on the substrate properties e ¤§ and . Varying the substrate Young’s modulus  ‘¥  St from  ¥ d "d tod"d GPa, the resulting delamination radius increases with D y %d by € "D e . On the other hand,  ¥an increase of the yield stress from t to sd D GPa gives values of that decrease by only s ¤ ¥§€p . The reason for this is that the yield stress determines the size of the plastic zone in thesubstrate, but not the permanent deformation immediately below the indenter; the latter is whatis controlling the delamination radius. However, it should be noted, as will be discussed in thenext section, that the yield stress plays a major role in determining whether delamination willtake place at all.4.4.4 Critical value of interfacial strength for delaminationWhether or not delamination takes place, depends on the tensile normal stress that can be gener-ated at the interface during the unloading process. The ultimate value of this stress relative to theinterface strength j}){ t | depends on almost all parameters involved in the boundary value prob-lem in a rather complex way. We have performed a parameter study involving the coating elasticmodulus, the substrate yield stress, the maximum indentation depth and the coating thickness.For each parameter combination, delamination is suppressed if the interfacial strength is higherthan a critical value hØì{ t }| . As an example, Fig. 4.10 shows load–displacement curves for different values of maximumindentation depths. Delamination is seen to occur if is above a certain critical value, and it h)‚¢ }| {is recognized by the hump left on the curve and the negligible residual indentation depth. Lowerindentation depths do not create normal stresses that exceed the interfacial strength , and j}){ t |therefore do not lead to delamination. Figures 4.11 and 4.12 shows the variation of the critical strength j})ì { t with the maximum |indentation depth, the coating thickness, the coating Young’s modulus and the substrate yieldstress. Higher values of the coating Young’s modulus , the coating thickness , or the max- Œ ¥ BŠ
  • 65. Delamination of a strong film from a ductile substrate during indentation unloading 59 2 delamination 1.5 no delamination F ⁄ ( πσ y R ) 2 1 0.5 0 0 1 2 3 4 5 h (µm)Figure 4.10: Load–displacement curves for several values of h){ ¢ }| , for a coating strength ofe r s ƒ© hØSt  }| {GPa.imum indentation depth lead to delamination of stronger interfaces. These are explained j})¢ |{by the fact that the driving force for delamination is the unbending of the coating. Despite thelimitations of the circular disc model pointed out before, these trends are roughly consistentwith Eq. (4.13), but not when looked at in more detail. For values of less than the coating thickness ( j})¢ |{ ), ige s D © Š hf hØì{ t shows a relatively }|rapid increase, Fig. 4.11a. This increase is attributed to the increase in the bending moment inthe coating. The bending moment is proportional to the curvature of the coating which increasesrapidly with the indentation depth until the coating takes the shape of the indenter. After thatpoint, the curvature does not change much but the bent region propagates outwards and thiscorresponds to the slower increase in hØì{ t for higher indentation depths. Fig. 4.11b shows also }|an initial rapid increase in the critical strength with the coating thickness due to the increaseof the bending moment with . For thicker coatings, the critical strength decreases due to the BŠdecrease in the coating curvature because the substrate becomes relatively softer. Figure 4.12ashows an almost linear increase of the critical strength with the coating Young’s modulus. Theincrease of the critical strength with the substrate yield stress is shown in Fig. 4.12b. This tincrease is caused by the reverse plasticity that takes place prior to delamination (Fig. 4.8). Thehigher the yield stress, the higher the stresses which can be reached at the substrate. Since thenormal stress is continuous across the interface, higher tensile normal stress can be reached withincreasing , thus making it possible to delaminate stronger interfaces. t
  • 66. 60 Chapter 4 2.5 no delamination 2 σ max (GPa) 1.5 delamination c 1 R = 25 µm t = 2.5 µm (a) 0.5 0 1 2 3 4 5 h max (µm) 2.5 2 σ max (GPa) 1.5 c 1 R = 25 µm h max = 2.5 µm (b) 0.5 1 2 3 4 5 t (µm) Figure 4.11: Critical value of the interfacial strength hØì{ t }| versus hØ{ ¢ }| (a) and (b). Š4.4.5 Residual stresses and interfacial wavinessCoated systems generally contain residual stresses. These are due to the deposition processitself, to the thermal expansion mismatch between the coating and the substrate, or a combina-tion of the two. To study the influence of residual stresses on delamination, we have introduceduniform in-plane stress in the film prior to indentation. This has been achieved, for numeri-cal convenience, by assigning different thermal expansion coefficients to coating and substrate,and by subjecting the system to various temperature changes to generate stresses ranging fromd Ö  d GPa (compressive) to GPa (tensile). Subsequently, we perform the indentation calcula-tions as before.
  • 67. Delamination of a strong film from a ductile substrate during indentation unloading 61 2.25 2 σ max (GPa) 1.75 c 1.5 (a) 1.25 200 300 400 500 E c (GPa) 3 2.5 σ max (GPa) 2 c 1.5 (b) 1 0.5 1 1.5 2 σ y (GPa) Figure 4.12: Critical value of the interfacial strength j})ì { t | versus Œ P¥ (a) and  St (b). Compressive stress in the coating is found to delay the delamination process, or to evenprevent delamination, whereas the opposite happens with tensile stresses. This is explained bythe fact that residual stress will have an out-of-plane component after the deformation of thecoating. In the case of tensile stress, this component will tend to enhance the unbending ofcoating during the unloading, and thus will assist delamination. As a consequence, the criticalstrength to prevent delamination will increase with residual tension in the coating. Compressivestress has the opposite effect. For example, a coating of the default thickness of with ige s D © Š hfa interfacial strength ofvge © j})S¢hf 4 r s  © hØt GPa was found earlier to delaminate after indentation to }| { |{ (see Fig. 4.11a), but delamination is prevented under a residual stress of GPa. d Ö  e ay The delamination radius is relatively insensitive to the residual stress: over a range of ¤ ¥§ s
  • 68. 62 Chapter 4 0 5 10 z (µm) 15 20 25 0 5 10 15 20 25 r (µm)Figure 4.13: Example of normal delamination for a case with a rough interface, modeled as asinusoidal wave with an amplitude ofigehf e r s v© hØ{ t  }| Š D  %d and a wave length equal to . In this case, s Š © hØ{ ¢ }| and GPa. d if sy ‰ h p h vf sc e 5© ¥§to GPa, varies between § ¤  and4s 4  compared to 4  ¤ for the stress-freecoating (cf. Tab. 4.1). Roughness of the interface is commonly simplified by a sinusoidal wave (e.g. Clarke andPompe, 1999). To study the effect of roughness on delamination, a wave of an amplitude up toŠ D %d s ŠD and a wavelength up to were introduced along the interface, see Fig. 4.13. Delaminationis found to start at valleys and crests where the normal stress component has a local maximum.Neighboring delaminated areas link up before the delamination front propagates to the nextcrest/valley. Even though the precise evolution of delamination depends on the waviness ofthe interface, for all cases considered here we did not find a significant effect on the criticalindentation depth at which delamination starts, nor on the final delamination radius.4.5 ConclusionsFor the purpose of studying interfacial delamination, numerical simulations have been carriedout of the indentation process of a coated material by a spherical indenter. To describe in-terfacial failure, the interface between the film and the substrate was modeled by means of a
  • 69. Delamination of a strong film from a ductile substrate during indentation unloading 63cohesive surface, with a coupled constitutive law for the normal and the tangential response.Failure of the interface by normal or tangential separation, or a combination, is embedded inthe constitutive model and does not require any additional criteria. Normal delamination occurs during the unloading stage of the indentation process. A cir-cular part of the coating, directly under the contact area, is lifted off from the substrate, drivenby the bending moment in the coating. Normal delamination is recognized by the imprint lefton the load vs displacement curve and the negligible residual indentation depth. For any givenindentation depth, the normal stress that can be attained at the interface is larger for thicker coat-ings, for coatings with a higher Young’s modulus or for substrates with a higher yield strength.To prevent delamination of such coatings, stronger interfaces are necessary. It should be noted that shear delamination can occur during indentation, before normaldelamination takes place. Compared to normal delamination, shear delamination can occur forrelatively low interfacial strength. Conversely, if the interface strength is high enough to preventnormal delamination, shear delamination will also be avoided. The energy consumed by the delamination process has been explicitly calculated and sepa-rated from the part dissipated by plastic deformation in the substrate. A small amount of elasticenergy, but still comparable with the total interfacial energy, is left in the system after unload-ing. Delamination is driven by the coating energy as it unflexes to retain its initial configuration.Deduction of quantitative information about the interfacial work of separation or strength is hin-dered by the complex interplay between the coating elastic energy and the interfacial energy.However, the present model does allow for an inverse approach by which the work of separationcan be derived iteratively. The disc model estimate (Hainsworth et al., 1998) has been compared with our numericalfindings for a range of parameters. It does capture some of the qualitative aspects of delamina-tion. But, it tends to strongly overestimate the interfacial strength or energy of separation. Critical values of the interfacial strength were calculated for several parameter combina-tions. The general trends of the variation of these critical values with the involved parametersare easily interpreted, whereas the details of this variation are governed by the nonlinear natureof the problem. Compressive residual stress in the film delays delamination and, if high enough, it mighteven prevent delamination, whereas, tensile residual stress has an opposite effect. Wavinessof the interface was not found to have a significant effect on delamination. Both conclusions,however, are intimately tied to the assumption that the coating remains intact during indentation.References Abdul-Baqi, A., Van der Giessen, E., 2001. Indentation-induced interface delamination of a strong film on a ductile substrate. Thin Solid Films 381, 143–154. Bagchi, A., Evans, A.G., 1996. The mechanics and physics of thin film decohesion and its measurement. Interface Science 3, 169–193.
  • 70. 64 Chapter 4Becker, R., Needleman, A., Richmond, O., Tvergaard, V., 1988. Void growth and failure in notched bars. J. Mech. Phys. Solids 36, 317–351.Bhattacharya, A.K., Nix, W.D., 1988. Analysis of elastic and plastic deformations associated with indentation testing of thin films on substrates. Int. J. Solids Struct. 24, 1287–1298.Clarke, D.R., Pompe, W., 1999. Critical radius for interface separation of a compressively stressed film from a rough surface. Acta Mater. 47, 1749–1756.Doerner, M.F., Nix, W.D., 1986. A method for interpreting the data from depth-sensing inden- tation instruments. J. Mater. Res. 4, 601–609.Drory, M.D., Hutchinson, J.W., 1996. Measurement of the adhesion of a brittle film on a ductile substrate by indentation. Proc. Roy. Soc. Lond. A 452, 2319–2341.Gao, H., Chiu, C.-H., Lee, J., 1992. Elastic contact versus indentation modeling of multi- layered materials. Int. J. Solids Struct. 29, 2471–2492.Hainsworth, S.V., McGurk, M.R., Page, T.F., 1998. The effect of coating cracking on the indentation response of thin hard-coated systems. Surf. Coat. Technol. 102, 97–107.King, R.B., 1987. Elastic analysis of some punch problems for a layered medium. Int. J. Solids Struct. 23, 1657–1664.Kriese, M.D., Gerberich, W.W., 1999. Quantitative adhesion measures of multilayer films. Part II. Indentation of W/Cu, W/W, Cr/W. J. Mater. Res. 14, 3019–3026.Li, X., Bhushan, B., 1998. Measurement of fracture toughness of ultra-thin amorphous carbon films. Thin Solid Films 315, 214–221.Lim, Y.Y., Chaudhri, M.M., Enomoto, Y., 1999. Accurate determination of the mechanical properties of thin aluminum films deposited on sapphire flats using nanoindentations. J. Mater. Res. 14, 2314–2327.Marshall, B.D., Evans, A.G., 1984. Measurement of adherence of residually stressed thin films by indentation: I. Mechanics of interface delamination. J. Appl. Phys. 56, 2632–2638.S´ nchez, J.M., El-Mansy, S., Sun, B., Scherban, T., Fang, N., Pantuso, D., Ford, W., Elizalde, a M.R., Mart´ inez-Esnaola, J.M., Mart´ in-Meizoso, A., Gil-Sevillano, J., Fuentes, M., Maiz, J., 1999. Cross-sectional nanoindentation: a new technique for thin film interfacial adhe- sion characterization. Acta Mater. 47, 4405–4413.Timoshenko, S., Woinowsky-Krieger, S., Theory of Plates and Shells (McGraw-Hill, New Yourk, 1959), 2nd Edn.Xu, X.-P., Needleman, A., 1993. Void nucleation by inclusion debonding in a crystal matrix. Model. Simul. Mater. Sci. Eng. 1, 111–132.Xu, X.-P., Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, 1397–1434.Wei, Y., Hutchinson, J.W., 1999. Models of interface separation accompanied by plastic dissi- pation at multiple scales. Int. J. Fract. 95, 1–17.
  • 71. Based on: A. Abdul-Baqi and E. Van der Giessen, Numerical analysis of indentation-induced cracking of brittlecoatings on ductile substrates, International Journal of Solids and Structures, accepted (2002).Chapter 5Indentation-induced cracking of brittlecoatings on ductile substrates Cracking of hard coatings during indentation is studied using the finite element method. The coating is assumed to be linear elastic, the substrate is elastic-perfectly plastic and the indenter is spherical and rigid. Through-thickness cracks are modeled using cohesive surfaces, with a finite strength and fracture energy. The interface between the coating and the substrate is also modeled by means of cohesive zones but with interface properties. The primary potential locations for the initiation of coating cracks are the coating surface close to the contact edge and the coating side of the interface in the contact region, where high values of tensile radial stress are found. Circumferential cracks are found to initiate from the coating surface and to propagate towards the interface. The initiation and advance of a crack is imprinted on the load–displacement curve as a kink. The spacing between successive cracks is found to be of the order of the coating thickness. The influence of other material and cohesive parameters on the initiation of the first crack and spacing between successive cracks is also investigated.5.1 IntroductionHard coatings on relatively soft substrates and their industrial applications have been receivingmore and more attention during the past few decades. The coating industry has advanced con-siderably in the production of various kinds of coatings to cope with the increasing number ofapplications of hard-coated systems. Hard coatings are usually applied to relatively soft sub-strates to enhance reliability and performance. Hard ceramic coatings, for example, are usedas protective layers in many mechanical applications such as cutting tools. Such coatings areusually brittle and subject to fracture of the coating or failure of the interface with the substrate.Therefore, the enhancement gained by the coatings is always accompanied by the risk of suchfailure. Indentation is one of the traditional methods to quantify the mechanical properties of mate-rials. Several techniques have been reported in the literature to extract the mechanical propertiesof both homogeneous and composite or coated materials from indentation experiments (Bhat-tacharya and Nix, 1988; Doerner and Nix, 1986; Gao et al., 1992; King, 1987; Lim et al.,1999; Oliver and Pharr, 1992). Indentation has also been advocated as a tool to characterizethe properties of thin films or coatings. At the same time, for example for hard wear-resistant 65
  • 72. 66 Chapter 5coatings, indentation can be viewed as an elementary step of concentrated loading. For thesereasons, many experimental as well as theoretical studies have been devoted to indentation ofcoated systems during recent years. Contact-induced failure of coated systems has also been investigated during recent years(Abdul-Baqi and Van der Giessen, 2001a, b; Li et al., 1997; Li and Bhushan, 1998; van derVarst and de With, 2001; Malzbender et al., 2000; Wang et al., 1998). The main emphasisin such investigations has been to extract quantitative data about the coating and interfacialfracture energies and strengths. Diverse results have been obtained by different studies due tothe complex nature of the problem and the different estimation procedures (van der Varst and deWith, 2001). The complexity is mainly attributed to the fact that such system is a combination ofat least two materials with different mechanical properties. Failure of such systems may include,for example, the plastic deformation of the substrate, the cracking of the coating or the failure ofthe interface. Interfacial delamination has been studied by the authors in previous work (Abdul-Baqi and Van der Giessen, 2001a, b). In the absence of coating cracking, delamination wasfound to occur in mode II driven by the shear stress at the interface outside the contact regionduring the loading stage. During the unloading stage, delamination was found to occur in modeI driven by the tensile stress at the interface in the contact region. Coating cracking is one of the common failure events usually observed in indentation ex-periments. Radial or circumferential cracks might initiate from the coating surface or from thecoating side of the interface and may grow into a through-thickness crack. Li et al. (1997) haveproposed a method to estimate the fracture toughness from indentation tests. From the loaddrop or plateau caused by the first circumferential crack, they estimate a strain energy releaseto create the crack. The fracture energy is then calculated using the energy release, the crackarea and the elastic properties of the coating. This approach has been also used by others (Liand Bhushan, 1998; van der Varst and de With, 2001; Malzbender et al., 2000). Van der Varstand de With (2001) have performed Vickers indentation experiments on TiN coatings on toolsteel. Arguing that the energy release which is estimated from the load–displacement curve isspent in both creating the crack and some plastic dissipation in the substrate, they estimate an e &upper bound of J/m for the fracture energy of the coating. However, the method they have 4used to estimate the energy release from the load–displacement curve is different from the onesuggested by Li et al. (1997). The objective of the present paper is to provide an improved understanding of coating crack-ing during indentation. To this end, numerical simulation of the indentation process is per-formed. The coating is assumed elastic and strong, the substrate is elastic-perfectly plastic andthe indenter is spherical and rigid. Coating cracking is modeled by means of cohesive zones.The cohesive zone methodology allows the study of crack initiation and propagation withoutany separate criteria since the fracture characteristics of the material are embedded in a con-stitutive model for the cohesive zone. The emphasis in this study will be on circumferentialcracks which initiate from the coating surface, its imprint on the load–displacement curve andthe spacing between successive cracks. Using our numerical findings, the above-mentionedestimates of the fracture energy are tested. The effect of the material, geometric and cohesiveparameters is also investigated.
  • 73. Indentation-induced cracking of brittle coatings on ductile substrates 67 ˙ h O R r a ρ t h Coating cohesive surface φ ς Symmetry axis z Interface L ρ φ ς Substrate LFigure 5.1: Geometry of the analyzed problem, including definition of local frame of reference,FU( üû$ý , and detail of the layout of continuum elements and cohesive elements along the interface.5.2 Problem formulationThe system considered in this study is illustrated in Fig. 5.1. It comprises an elastic-perfectlyplastic substrate coated by an elastic thin coating and indented by a spherical indenter. Theindenter is assumed rigid and only characterized by its radius . Assuming both vxbD © ¤ hfecoating and substrate to be isotropic, the problem is axisymmetric, with radial coordinate and §axial coordinate in the indentation direction. The coating is characterized by its thickness  6 "6 vf ƒ© Šh  and elastic properties ( GPa, "d e © Œ ¥ %d © Œ ¦ ) and is bonded to the substrate by an d sinterface. The substrate is supposed to be a standard isotropic elastoplastic material with plasticflow being controlled by the von Mises effective stress . It is taken to have a height and Žt Š  ža radius , with large enough compared to film thickness so that the solution is independent ž žof and the substrate can be regarded as a half space; thus, it is only characterized by its elastic ž 6 "6properties ( d bd D ©  ¥ GPa, ) and the yield stresss %d ©  ¦ GPa. sd ƒ©  t  The precise boundary conditions are also illustrated in Fig. 5.1. The indentation process isperformed incrementally with a constant indentation rate mm/sec. Outside the contact  ò© Á ¢area, with radius measured in the reference configuration, the film surface is traction free, £ d © †U#§  u Š © †U§   Š $d( $d( for €ž —3•£ s 1 § 1 (5.1)Inside the contact area we assume perfect sliding conditions. The boundary conditions arespecified with respect to a local frame of reference that is rotated over an angle such FU( üû $ý Ïthat is always perpendicular to the coating surface as shown in Fig. 5.1. The definition of this ýframe for points inside the material is also shown in Fig. 5.1. The angle is calculated for each Ï
  • 74. 68 Chapter 5material element in the deformed state (solid line) based on the rotation of the element withþ°respect to its original configuration (dashed line). In the normal direction, the displacement rate Á is controlled by the motion of the indenter, while in the tangential direction the traction is ÿŠset to zero, i.e. þ ‚° d © FU#§  ÿ Þ8%¡`Á ¢ © FU§  Á $( Š ( Ï  YX $( for P£ —3—d s 1 § 1 (5.2)The substrate is simply supported at the bottom, so that the remaining boundary conditions read ° ° d © $FU%d   ž —3—d ( for1 § 1 d © ‘U§  u ; $ ž( for 3• —d ž 1 1 . (5.3)The indentation force is computed from the tractions in the contact region, ¡ à ğ s§ § D †U#§  u Š E $d( © ¡ (5.4) ~ ” The analysis is carried out numerically using a finite strain, finite element method. Themesh is an arrangement of four-noded quadrilateral elements. The elements are built up of fourlinear strain triangles in a cross arrangement to minimize numerical problems due to plasticincompressibility. To resolve properly the high stress gradients under the indenter and for anaccurate detection of the contact nodes, the mesh is made very fine locally near the contact areawith an element size of . d  IŠ ! Since the indenter is rigid, contact nodes are identified simply by their spatial location with ¬respect to the indenter. At a certain indentation depth and displacement increment ¢ , the ¢ ¬node is considered to be in contact if the vertical distance between the node and the indenteris not greater than . Local loss of contact may occur if an open crack falls into the contact ¢region. In the calculations, contact is released when the nodal force becomes tensile. We haveused a threshold value for tensile nodal forces equal to the average current nodal force. A valuethat is an order of magnitude smaller did not show any significant effect on the results. After anode is released from contact, the distance between the node and the indenter is checked eachincrement for possible re-formation of contact. Coating cracking is represented by inserting cohesive zones into the coating, perpendicu-lar to the surface. A cohesive zone is also used to model the interface between coating and ¬substrate. A cohesive zone is a surface along which a small displacement jump between the Î ¬two sides is allowed with normal and tangential components and , respectively. The co- © Î «©hesive behavior is specified in terms of a constitutive equation for the corresponding traction icomponents and at the same location. ¬ Î ¬ The cohesive law we adopt in this study is the one given by Xu and Needleman (1993). The i ¬ ÎÎ ¬ $ ( Î Ïtraction components are determined by two potentials and $ (  Ï according to i s Î ‚i¬Ï ´ © «© ( ‚´ © © ΠϬ i (5.5) ´ i ´ iThe resulting traction-separation relations in normal and tangential direction are illustrated in iFig. 5.2. In both directions, the peak traction represents the cohesive strength and the area under
  • 75. Indentation-induced cracking of brittle coatings on ductile substrates 69 1.5 1 ∆t = 0 0.5 0 T n ⁄ σ max increasing ∆ t −0.5 −1 −1.5 −2 (a) −2.5 −1 0 1 2 3 4 5 6 ∆n ⁄ δn 3 2 ∆n = 0 1 T t ⁄ τ max 0 increasing ∆ n −1 −2 (b) −3 −3 −2 −1 0 1 2 3 ∆t ⁄ δt ¬ © ›¬ «© Î ÎFigure 5.2: The normal and shear cohesive tractions. (a) Normal traction $ ¡ . (b) Sheartraction $  . Both are normalized by their respective peak values and hi} |){ i hØ{ t }| . Îthe curve corresponds to the work of fracture. The work of normal (tangential) separation, ¡ Ï( ), can be expressed in terms of the corresponding strengthsÏ ( ) as j}){ h){ t | }| ¤ Î Î `¤ ¡ i  Û © © Ï D Ï ( hØ{ w$ I }| t  j}){ $ I |  (5.6) ¤ Î ‰¤ Îwhere and are two characteristics lengths andÔ%b@!ÓÏ Ò Ï Õ © i a@Ò i ÔÓ is a coupling parameter. Fromthe previous equations, the normal and tangential strengths are related by i i ¡ `¤ Î  j}|){ ¤ $ I D  Õ © j})St  |{ (5.7) i a@ÒÔÓ
  • 76. 70 Chapter 5More details of this constitutive model are found in (Xu and Needleman, 1993; Abdul-Baqi andVan der Giessen, 2001a). The cohesive zones are incorporated in the finite element calculationusing linear two-noded elements with two Gaussian integration points which is consistent withthe type of continuum elements in the coating and the substrate. Τ ¤ For the cohesive zones at the interface we have chosen , %d 2Õ vf  %d © © s4 © h s and r "r ¡ D©6 Ï d "d D & ¤ Î `¤ J/m . Making use of Eqs. (5.6) and (5.7), these values correspond to © hØ{ t }| » 6 © j}|){p ay¡ s GPa. For the coating nm, © sd  © and J/m so that & d © Ï i "r D 4 %d “Õ r s © © hØ» Œ{ t }|hØiŒ {}| "v© GPa. The values assigned to the work of separation for both coating and interface are icomparable to these obtained by Wang et al. (1998) for diamond-like carbon (DLC) coatings on Î `¤ i ¤steel substrates. The values assigned to the characteristic lengths and for both interface andcoating were chosen such that, when combined with the work of separation, reasonable valuesfor the strength are obtained. In terms of the coating material and the cohesive parameters, i ÷the critical stress intensity factors for mode I and II fracture along a cohesive zone, i.e and #! $"%" !$" , are and $ &Œ ¦ e –I !  Ï Œ ¥  $ & Œ ¦ –I ! Ï Œ ¥      , respectively (Xu and Needleman, 1994). Theratio of to ! being p %d is about s Õ 9 . This value is reasonable compared to the values $ " ! %" $"obtained by different experimental estimates as listed in (Malzbender et al., 2000). The effect of the variation of some of previously given parameters, such as the coatingthickness, the coating Young’s modulus, the yield stress and the coating cohesive parameterswill be investigated. The values given in this section will serve as a reference case in this study.In the remainder of this study we will drop the superscripts from quantities referring to thecohesive parameters since, unless otherwise specified, they all will refer to the coating.5.3 Stress distribution in a perfect coatingBefore performing the cracking analysis, it is instructive to have a brief look at the stresses thatgenerate in a perfect coating during indentation (i.e. without cohesive zones). Figure 5.3(a)shows the radial stress distribution along the coating surface, b§  wÿ t "D ö . A tensile radial stress $ ÿ eis found outside the contact area with a maximum at , where is the instantaneous £ £ s §contact radius. It should be noted that the location of this maximum is strongly dependent on theratio . For the same indentation depth, the smaller the ratio bIŠ ¤! , the closer the location of the #IŠ ¤!maximum tensile stress to the contact edge. The radial stress is found to be compressive withinthe contact area £ m § . It is interesting to note in Fig. 5.3(a) the decrease of the compressivestress in the contact region with increasing indentation depth. This is a consequence of theplastic flow of the substrate material in the indented region which induces an additional positiveradial straining in the coating, leading to a decrease in the compressive stress at the contactregion. Tensile stress is also seen along the coating side of the interface as shown in Fig. 5.3(b).It is located at the contact region £ m § with a maximum at the symmetry axis. This suggeststhat a circumferential crack is likely to initiate from the interface at ö or the coating surface £ m §at § £ "D s  e due to the maximum tensile radial stresses at these regions. þ The distribution of the shear stress along the mid-plane of the coating is shown in ÿtFig. 5.3(c). The highest value of this stress is found under the contact edge at . The £ © §
  • 77. Indentation-induced cracking of brittle coatings on ductile substrates 71 20 h (µm) 15 0.25 10 0.50 0.75 5 σ ρρ (GPa) 1.0 0 -5 -10 (a) -15 0 1 2 3 4 5 20 h (µm) 15 0.25 10 0.50 σ ρρ (GPa) 0.75 5 1.0 0 -5 -10 (b) -15 0 1 2 3 4 5 6 5 h (µm) 0.25 4 0.50 3 0.75 σ ρζ (GPa) 1.0 2 1 0 -1 (c) -2 0 1 2 3 4 5 r⁄aFigure 5.3: Stress distribution at several indentation depths for "D ! ¿© bIŠ e  ¤! . (a) Radial stressalong the coating surface. (b) Radial stress along the coating side of the interface. (c) Shearstress along the middle plane in the coating ( D IŠ ©  ! ).
  • 78. 72 Chapter 5shear stress has a smaller maximum value than the radial stress and furthermore is located atthe edge of the contact where only relatively small tensile radial stresses are found. This indi-cates that shear stress is likely to play a minor role in the fracture process. Comparing the radialstresses along the surface with those on the coating side of the interface, we see that each tensilestress is faced by a compressive stress on the opposite side of the coating due to the bending-likeloading. This indicates that a crack, whether initiated at the surface or the interface, will not beable to propagate through the whole coating thickness. Tensile radial stresses at the surface andinterface are seen to be of the same order of magnitude. But crack initiation and propagationfrom the coating side of the interface suffers an extra resistance as this requires some interfacialshearing (see inset in Fig. 5.1) and hence more energy. Therefore, the main emphasis in thisstudy will be on cracks that initiate from the coating surface and advance towards the interface.Cracks which initiate from the coating side of the interface do not have any significant effect onsurface cracks as will be discussed later in this paper.5.4 AnalysisA first approach to simulate coating failure was to place cohesive zones in between all contin-uum elements in the coating, as pioneered by Xu and Needleman (1993, 1994). Contrary totheir work, however, the present computation is a quasi-static one and this led to serious nu-merical problems. The numerical problems originate from the fact that some adjacent cohesivezones reach the peak traction h)St }| { and start softening almost simultaneously, which breaks theuniqueness of the solution. Mesh refinement does not solve the problem. When fine meshesare used, the region of more or less homogeneous, high tensile radial stresses comprises severalelements so that there is an even larger probability of running into a loss of uniqueness. Onthe other hand, using coarser meshes to overcome this problem decreases the accuracy in repre-senting the high stress gradients in the coating and the precise contact between the indenter andthe coating. In addition, embedding of cohesive elements leads to an artificial enhancement ofthe overall compliance (Xu and Needleman, 1994). This increase will tend to underestimate thestresses in the coating and therefore the occurrence of failure. Since the compliance increase isproportional to the number of cohesive zones, we adopt a procedure in this study in which thenumber of cohesive zones is minimized. The location of the necessary cohesive zones is determined in the following way. We firstconduct a calculation without any cohesive zone and we trace the evolution of the maximumtensile radial stress along the coating surface along with its location at each indentation incre-ment. This is shown in Fig. 5.4, with the steps in the curves originating from the node-to-nodegrowth of the contact region. After an initial transient, both the maximum stress and its locationseem to increase linearly with the contact radius . The first crack is predicted to occur when £ © h){ }| . Figure 5.4 shows that "‹© |  GPa is reached when the contact radius is vxe %d hØ© { t ¢ vf ÿsyÿ 6th f s }| ( h h if sd e © § j}){ t ) and at the radial location . £ In the second step, a new calculation is carried out with a single cohesive zone being placedat this location prior to indentation. The first crack will then indeed form at the expected inden-
  • 79. Indentation-induced cracking of brittle coatings on ductile substrates 73 20 15 σ ρρ (GPa) 10 max 5 (b) 0 0 1 2 3 4 5 6 7 6 5 r ( µm ) 4 3 r = a 2 1 (a) 0 0 1 2 3 4 5 6 a ( µm )Figure 5.4: (a) Evolution of the maximum radial stress at the coating surface with the j}){ t |contact radius . (b) The corresponding location at the coating surface at which the stress is £ § ÿ ÿmaximum.tation depth and we continue the computation to determine the location of the second crack inthe same way as for the first crack. A third computation is then started with two cohesive zones,etc. Continuing this procedure, more cracks are simulated. Figure 5.5 gives the result after thefourth step in which three coating cracks have developed. Figure 5.5(a) shows the distributionof radial stress at an indentation depth of vxe y %d hf s . The first crack at h vf sd e © § is visible ÿ ÿ tin the figure and a tensile stress is seen to develop in the coating outside the crack radius. At vx"D s o© ¢ hfe  ean indentation depth of vf c ay h s , three cracks have already occurred at locations vf s v© § h p , 4 s p sdand , see Figure 5.5(b). The crack at is not visible in the figure since it has 4
  • 80. 74 Chapter 5 0 σ ρρ ⁄ σ max 1.10 2 0.90 1st crack 0.70 0.50 4 0.30 z ( µm ) 0.10 -0.10 6 -0.30 -0.50 -0.70 8 -0.90 (a) 10 0 2 4 6 8 10 12 14 0 σ ρρ ⁄ σ max 1.20 2 1.00 3rd crack 0.80 1st crack 0.60 4 0.40 z ( µm ) 0.20 0.00 6 -0.20 -0.40 -0.60 8 -0.80 (b) 10 0 2 4 6 8 10 12 14 r ( µm )Figure 5.5: Distribution of the radial stress at indentation depths of vxe y %d hf s (a) and vg"D s  hfe(b). ÿ ÿ tclosed up. The first crack has also closed from the surface side as a result of the compressivestress in the contact region. Note that local loss of contact has occurred above the closed-upcrack as explained in section 5.2. The third crack did not reach the interface yet at this stage ofloading due to the compressive stress in this region. The procedure is further illustrated in Fig. 5.6(a), showing the evolution of the maximumradial stress along the coating surface. The maximum stress is seen to increase with indentationdepth until the moment of failure where a sudden drop occurs due to stress relaxation andredistribution. After initiation, the crack propagates through the coating thickness and stopsjust before reaching the interface due to the compressive stress at that region. About of € †d 
  • 81. Indentation-induced cracking of brittle coatings on ductile substrates 75 2.5 2 1.5 (0) (1) (2) (3) σ ρρ ⁄ σ max 1 0.5 (a) 0 0 0.5 1 1.5 2 1.2 1 0.8 3rd F (N) 0.6 2nd 0.4 1st 0.2 (b) 0 0 0.5 1 1.5 2 h ( µm )Figure 5.6: (a) Evolution of the maximum radial stress at the coating surface hØ{ t }| with the in- ÿ ÿdentation depth . The curves belong to four different calculations with the number of cohesive ¢ 6surfaces (and later cracks) increasing from to . (b) Load–displacement curve for the case dshown in Fig. 5.5. Arrows point to sudden load drops caused by cracking events.the coating thickness remains intact for a while until it shears off (mode II) and later opens inmode I resulting in two small drops in the maximum stress. The location of the fourth crack ispredicted to be vf %c © § h s after vg"e s  © ¢ hfe . The four successive cracks thus found are seen 4 hige s  & fto have an almost uniform spacing of . The dependence of crack spacing on some of
  • 82. 76 Chapter 5the model parameters will be discussed in section 5.5. The corresponding load–displacement curve shown in Fig. 5.6(b) reflects these crackingevents by a sudden drop of the load. Each of three major drops on the curve is associatedwith the initiation of a crack. Since the fracture energy is proportional to the crack radius, thelarger the crack radius, the larger the drop in the load. With further indentation, the contactradius increases, thus expanding the tensile region at the interface. The growth of the freshlynucleated crack to the interface gives rise to small load drops in between the major ones. Thethird crack shown in Fig. 5.5(b) for example reaches the interface forming a complete through-thickness crack at an indentation depth of h if 6 s  . This gives rise to the small load drop seen onthe load–displacement curve at this depth. These results demonstrate that the procedure described above is indeed capable of predictingthe initiation and growth of coating cracks within the cohesive zone framework. It should bepointed out, however, that even with this small number of cohesive zones, some numericalproblems were faced at first. Careful examination has shown that these were caused by thefact that the cohesive zone parameters chosen here approach atomic separation properties. Thecoating strength is of the order of tens of GPa’s and the critical opening is only a nanometer.The consequence of this is that the cohesive stiffness beyond the peak strength is very large, andnegative, which would lead to local snap back in the finite element system. This can probablybe dealt with using, for instance, indirect displacement control (de Borst, 1987), but we have ¬ ¤adopted a simpler method to circumvent this. The idea is to reduce the instantaneous stiffnessof the cohesive zone as soon as the normal strength is reached, , but to update the ítractions directly from (5.5). Of course, this leads to an error in the solution, but this error is i icorrected by way of the equilibrium correction in the next time step. Practically speaking, theeffect of this procedure is that snap back instabilities are avoided and stability is maintained.Extensive testing has shown that the final error in, for example, the traction continuity betweena cohesive zone element and a continuum element is smaller than a few percent, even when theinstantaneous cohesive zone stiffness is reduced down to . €e Cracks initiating from the coating side of the interface have been excluded from the simula-tions despite the high tensile stress found there. The main reason is the fact that the maximumtensile stress is always found at the symmetry axis (Fig. 5.3b). A cohesive surface at this lo-cation will have a zero area in an axisymmetric formulation making it impossible to simulate.To study the effect of coating cracking starting from the interface, three cohesive surfaces are 6 e placed at radial locations of , s and sd ige s hf . Being all located at distances less than h vf sd e 4(the expected location of the first surface crack), surface cracks are not expected to occur. Theradial stress distribution at an indentation depth of h if sd  and the evolution of the maximumradial stress on the coating surface are shown in Fig. 5.7. Cracks at the three specified locationsare seen to have occurred. None of these cracks has reached the surface since at the momentof initiation of each crack, the opposite side of the coating is already in contact and hence thestress state is compressive. The stress at the coating surface outside the contact region is mainlydriven by the contact between the indenter and the coating at the contact edge £ © § . Since allcracks initiating from the coating side of the interface lie in the region £ m § , their effect on thestress outside the contact region is insignificant as shown in Fig. 5.7(b).
  • 83. Indentation-induced cracking of brittle coatings on ductile substrates 77 0 σ ρρ ⁄ σ max 1.40 2 1.15 0.90 0.65 z ( µm ) 4 0.40 0.15 -0.10 6 -0.35 -0.60 -0.85 8 -1.10 (a) 10 0 2 4 6 8 10 12 14 r ( µm ) 2 1.5 σ ρρ ⁄ σ max 1 0.5 (b) 0 0 0.2 0.4 0.6 0.8 1 h ( µm )Figure 5.7: (a) Distribution of the radial stress at an indentation depth of ÿ ÿ t . It illus- 6 e vf sd  htrates the three cracks initiating from the coating side of the interface at locations vxe s hf , and sd s  j}){ | t4 . (b) The corresponding maximum radial stress at the coating surface (solid line)compared with the stress of the non cracked coating (dashed line). ÿ ÿ5.5 Effect of geometrical, material and cohesive parametersThe effect of various parameters on the initiation of the first circumferential crack and the spac-ing between successive cracks is studied. The chosen parameters are: the coating thickness , & Š
  • 84. 78 Chapter 5 0.2 ( λ ⁄ R ) ≈ 1.4 ( t ⁄ R ) 0.15 λ⁄R 0.1 0.05 0 0 0.05 0.1 0.15 t⁄RFigure 5.8: Normalized crack spacing f#v&  $ ¤! versus normalized coating thickness . Single fbŠ  $ ¤!points are the numerical results and solid line is a linear fit. The error in is half the finite-& $ £element size.the coating Young’s modulus , the substrate yield stress Œ ¥ and the cohesive properties, i.e. tcoating strength j}){ t | as well as the coupling and reversibility of the cohesive tractions. The coating thickness and indenter radius are the main length scales in the system and Š ¤their effect is studied by varying the nondimensional parameter . Figure 5.8 shows the #IŠ ¤! övariation of the crack spacing with the coating thickness. The relation provides b«& ¤! bŠ 4 s  ¤!the best linear fit. It should be noted that spacing in this figure is taken to be distance between öthe first two cracks. From the average spacing we get b«& ¤! bŠ e s  . In the remainder of this ¤!study, will refer to the average crack spacing. & Figure 5.9 shows the location of the first crack as a function of the coating thickness andcorresponding indentation depth for two values of . It can be seen that first crack occurs j}){ t |closer to the contact edge for thinner coatings and higher strengths. Moreover, the thicker thecoating the larger the indentation depth required to form the first crack. A variation of the coating’s Young’s modulus ranging from Œ ¥ to bd D GPa did not have d d "d eany effect on crack spacing. However, the location of the first crack and the correspondingindentation depth strongly depend on . The radial stress in the coating Œ ¥ is proportional tothe degree of bending and to Young’s modulus. Therefore, coatings of lower modulus require ÿ ÿ tmore bending or equivalently larger indentation depths to reach the fracture strength and subse-quently crack. Since the location of the maximum stress on the coating surface increases withindentation depth (Fig. 5.4b), coatings of lower Young’s modulus crack at larger radii as seen inFig. 5.10. An increase in the coating strength seems to be equivalent to a decrease in the j}){ t |Young’s modulus in terms of location of the first crack and the corresponding indentation depth
  • 85. Indentation-induced cracking of brittle coatings on ductile substrates 79 3 2.5 σ max = 5.5 GPa 2 1.5 σ max = 11 GPa r⁄a 1 0.5 (a) 0 0 0.5 1 1.5 2 2.5 3 3.5 t (µm) 12 increasing t 10 σ max = 5.5 GPa 8 r (µm) 6 σ max = 11 GPa 4 2 (b) 0 0 0.25 0.5 0.75 1 1.25 1.5 h (µm)Figure 5.9: The location of the first circumferential crack as a function of the coating thickness(a) and the corresponding indentation depth (b) for two values of . Discrete points are hØ{ t }|numerical results, lines are linear fits.as shown in Fig. 5.10. This suggests that the first crack radius and the corresponding indenta-tion depth depend on the ratio which can be considered a measure of the critical strain Pb! h){ t ì ¥ }|for cracking. Contrary to the Young’s modulus, the coating strength does have some influenceon crack spacing. Strengths of e s e © h)t and }| { e s ` p GPa resulted in crack spacings of and Ds h vf r s  , respectively. The effect of the yield stress of the substrate is studied by comparing the results obtained Æt
  • 86. 80 Chapter 5 8 decreasing E c 7 6 r (µm) 5 4 increasing σ max 3 2 0 0.25 0.5 0.75 1 1.25 1.5 h (µm)Figure 5.10: The location of the first circumferential crack versus the corresponding indentation 6depth. Open circles are the numerical results using " of , , "d 4 e "d ‰ "d D b ì bD¥ r e e and d dp d  eGPa, "d e © de © h)St }| { GPa. Stars correspond to values of of , , and hØ{ t }| GPa, ìP¥ s GPa. Lines s s d "dare quadratic fits.using three different values, namely, , sd D vf sd  e %d e e and GPa. These values resulted in first cir- h ãys s sd p scumferential cracks of radii of , and , respectively. The corresponding indentation 4depths did not deviate significantly from the reference case, where . The crack vxe %d © ¢ hf sspacing increased to h for vf sc  e %d © Æ t GPa, whereas it decreased to s in the case of vxD s  hfsd D © t Æ GPa. The effect of the yield stress is explained in terms of the size of the plastic zone.At the same indentation depth , the lower the yield stress, the larger the plastic zone size. The ¢expansion of the plastic zone towards the surface is partially accommodated by bending of thecoating outside the contact region. Therefore, lower yield stresses tend to lead to larger radiallocation of the maximum stress on the coating surface and therefore a larger crack radius. Thecorresponding larger crack spacing is also explained by the same argument. The effect of friction between the indenter and the coating on the location of the first crackand the crack spacing is also investigated. In the case of perfect sticking contact conditions, thefirst crack occurred at a radius ige s e © § hf and a corresponding indentation depth , if c e %d © ¢ h sinstead of h vf sd e © § and vge %d © ¢ hf s obtained previously in the case of perfect sliding contact.The crack spacing also increased from hvfxe s  to vxD hf when perfect sticking was assumed. Souza et al. (2001) have experimentally investigated the spacing of indentation-inducedcracks in titanium nitride coatings on stainless steel substrates. They have used a RockwellB indenter ( ige by © ¤ hf c ) and coating thicknesses of , and  s D 4 ad . They have found s igD s hf 4that cracks are almost uniformly spaced which is consistent with our numerical results. The
  • 87. Indentation-induced cracking of brittle coatings on ductile substrates 81 1.2 0.9 loading 0.6 T n ⁄ σ max unloading 0.3 reloading 0 −0.3 −0.6 0 1 2 3 4 5 6 ∆n ⁄ δn ¬ © Î ›¬ Figure 5.11: Irreversible normal cohesive traction $  at d © . i icrack spacings observed for the various thicknesses were found to be 6 , and sd e e s  , vf d %d  h srespectively, so that varies between b«& ¤! bIŠ 4 s D ¤! and bIŠ e ãys . However, these experimental ¤!values are seen to be larger than those obtained in our calculations. There may be severalreasons for this discrepancy. One reason is the presence of residual compressive stresses of – ãy%d ses GPa in the coatings investigated by Souza et al. (2001), whereas in the current simulations,the coating is assumed to be stress-free prior to indentation. Another reason is the frictionbetween the indenter and the coating which is usually present in indentation experiments. Inthe presence friction, the crack spacing increases as mentioned previously. In addition, thematerial parameters chosen in this study are not precisely those of titanium nitride coatings onstainless steel substrates. Variation of these parameters has been shown above to influence thecrack spacing. The coupling and reversibility in the cohesive law are also investigated. In the originalXu-Needleman formulation (Xu and Needleman, 1993), the constitutive law of the cohesivezone is assumed to be reversible, so that the loading and unloading paths are the same. In thiscase, if a crack closes it heals and the system recovers all the energy consumed. Even thoughthis energy is very small in comparison with the plastic dissipation, the effect of taking it intoaccount is studied by using an irreversible version of the tractions. Therefore, we modify theconstitutive law by introducing a secant unloading branch as illustrated in Fig. 5.11. If theopening velocity changes sign (unloading), the traction is assumed to decrease linearly to zeroas the opening diminishes to zero (Camacho and Ortiz, 1996). The same method is applied tothe shear traction. It should be noted that in the presence of coupling between the tractions, theformulation for irreversibility would not be clear. Therefore, the effect of reversibility is studiedby comparing the results achieved when using reversible and irreversible uncoupled tractions.
  • 88. 82 Chapter 5 1.2 coupled tractions 1 uncoupled tractions 0.8 F (N) 0.6 0.4 0.2 0 0 0.5 1 1.5 2 h ( µm )Figure 5.12: Load–displacement curve for uncoupled traction-separation laws in the cohesivezones compared to the coupled ones.On the other hand, the effect of coupling is investigated by studying the difference betweenusing coupled and uncoupled reversible tractions. ¬ © © Î ›¬ The results of coupled reversible tractions which has been used so far are compared to the Î ‚© ¬ ›¬ ‚© Î Îresults obtained when using uncoupled reversible tractions; i.e., (  © and $ "d © © $ †d © (  (cf. Fig. 5.2). Three cracks were simulated and the location of the fourth one is i vf 6 i s  i hpredicted as explained in section 5.4. The crack spacing is found to be , which is less thanhvfge s  ithe value of found previously in the presence of coupling. Moreover, additional cracksoccur at smaller indentation depths compared to the corresponding ones in the presence ofcoupling. This is demonstrated in Fig. 5.12. When a crack opens, the shear strength is reducedin the presence of coupling as demonstrated in Fig. 5.2(b). The moment the crack reaches theinterface, it shears off by the shear stress which is present around the contact edge (Fig. 5.3c).The shearing leads to some relaxation of the coating in the form of unbending. Compared to theuncoupled tractions where shearing off does not occur, indenting to larger depth is required toarrive to the same degree of bending and form an additional crack. Moreover, larger indentationdepth means that the maximum tensile stress on the coating surface occurs at a larger radius.This explains the larger crack spacing in the case of coupling. Reversibility of the tractionsdoes not seem to have any significant effect on the results, even though we have noticed crackclosure taking place (e.g. Fig. 5.5b). Uncoupled reversible and irreversible tractions lead toalmost indistinguishable load–displacement curves and to the same crack spacing, & . vf 6 s v© h 
  • 89. Indentation-induced cracking of brittle coatings on ductile substrates 83 F (N) E B A ∆U 1 = area ABCD Fc C ∆U 2 = area OAC ∆F ∆U 3 = area OAD D ∆h O hc h (µm) ¬ ¬Figure 5.13: Schematic load–displacement curve. ¨ to ¨ B are different estimates of theenergy release associated with a cracking event. e5.6 Fracture energy estimatesSeveral researchers have tried to estimate the fracture energy from indentation experiments. Oneof the approaches which has been used is to estimate the energy consumed to create the firstcircumferential crack from the corresponding load drop or plateau on the load–displacementcurve (Li et al., 1997; Li and Bhushan, 1998; van der Varst and de With, 2001; Malzbender etal., 2000). In this section we will compare the prediction of these methods with our numericalfindings. ¬ Li et al. (1997) have suggested an approach which is based on the idea that all the energyreleased upon the formation of a crack, ¬ , is consumed in creating the crack with fracture ¨energy Ï per unit area. By determining , the fracture energy can be estimated as ¬ ¨ i ( #§ ¨ E#D ©  Ž Ï Š (5.8) ¬with the crack radius and the coating thickness. Two methods have been suggested in the § Š iliterature to estimate from the load–displacement curve, as illustrated in Fig. 5.13. The first ¨method (Li et al., 1997) is to extrapolate the load in a tangential direction from the beginning ¬to the end of the discontinuity. The difference between the areas under the extrapolated curveand the discontinuous one ( in Fig. 5.13) is taken to be the estimate of the energy release. ¨ ¬The second method (van der Varst and de With, 2001) estimates the energy release as half the e ¬cracking load times the displacement jump ì x¡ ( in Fig. 5.13). We here suggest a third ¢ ¨ & ¬method which estimates the energy release as half the cracking indentation depth times the ì v¢load jump ( in Fig. 5.13). The second and third methods are borrowed from linear elastic ¨ Bfracture mechanics for cracking under constant load and constant displacement (indentation
  • 90. 84 Chapter 5 10 8 est3 φn ⁄ φn 6 est2 φn ⁄ φn 4 est1 2 φn ⁄ φn (a) 0 10 15 20 25 30 35 40 45 50 2 φ n (J/m ) 14 12 10 est3 φn ⁄ φn 8 est2 φn ⁄ φn 6 est1 4 φn ⁄ φn 2 (b) 0 0 0.5 1 1.5 2 2.5 3 3.5 t (µm)¬ ¬ ¬Figure 5.14: Estimates for from Eq. (5.8). Ï , and &  Ž Ï  Ž Ï are based on the energies B  Ž Ï , and& v¨ f sd  ¨ © Š (cf. Fig. 5.13), respectively. (a) Estimated values versus the actual ones for ¨ 6 h B i . (b) Estimated values versus the coating thickness for i e i J/m . In (a) and (b) d © Ïi & esingle points are numerical results and lines are linear fits. idepth), respectively (Kanninen and Popelar, 1985). We now use our numerical simulations to test the accuracy of the estimates obtained fromEq. (5.8). Obviously, the values of the energy release estimated in the three methods are com-
  • 91. Indentation-induced cracking of brittle coatings on ductile substrates 85pletely different and so will be the estimated fracture energies. Based on our numerical results,Fig. 5.14 shows the estimated values of the coating fracture energy for several values of Ï andŠ. At low values of Ï or , the first method underestimates the fracture energy, whereas the Šsecond and third methods give reasonable estimates. On the other hand, at large values of i Ï ior , the second and third methods overestimate the fracture energy, whereas the estimation by Šthe first method seems reasonable. Therefore, estimations of the coating fracture energy from iindentation experiments might lead to values different from the actual ones by as much as oneorder of magnitude. The origin for the discrepancy is the fact that the estimates assume linear-ity of the problem, while in fact it is quite nonlinear because of the energy dissipation in theplastic zone in the substrate. Nevertheless, it is interesting to note the proportionality betweenthe normalized estimated values and the actual ones and the coating thickness.5.7 Concluding remarksIn this paper, cracking of hard coatings on elastic-perfectly plastic substrates during indenta-tion has been studied using the finite element method. Cracking of the coating was simulatedby assuming single or multiple planes of potential cracks across the coating thickness at pre-determined radial locations prior to indentation. These planes are assigned a finite strength andmodeled by means of cohesive zones. The interface between the coating and the substrate wasalso modeled by means of cohesive zones but with interface properties. The emphasis was on circumferential cracks which initiate from the coating surface outsidethe contact edge and advance towards the interface. The initiation and advance of such cracksis imprinted on the load–displacement curve as a sudden load drop (kink). The larger the crackradius, the larger the load drop, since the energy released by the crack is proportional to the crackarea. After initiation, the crack advances almost spontaneously across the coating thickness andstops just before reaching the interface as a result of a compressive stress at that side. Withfurther indentation, the tensile region under the indenter expands and when it encompasses thecrack tip, the crack advances until the interface creating a complete through-thickness crack.This is also imprinted on the load–displacement curve by a relatively smaller load drop. Forseveral coating thicknesses, the ratio of the crack radius to the instantaneous contact radius wasfound to increase linearly with the coating thickness. The smaller the coating thickness, thecloser this ratio is to unity. With further indentation, successive cracking occurs at larger radii, which each additionalcrack also imprinted on the load–displacement curve as a kink. The crack spacing was foundto be predominantly controlled by the ratio of the coating thickness to the indenter radius. Fora fixed indenter radius, the crack spacing is of the order of the coating thickness. Variation ofthe coating Young’s modulus did not show any influence on crack spacing, whereas spacingincreased with the coating strength and decreased with the substrate yield stress. Coupling andirreversibility of the cohesive tractions were also investigated. The presence of coupling wasfound to slightly increase crack spacing. Moreover, additional cracking occurs at higher inden-tation depths compared to uncoupled tractions. Reversible and irreversible cohesive tractions
  • 92. 86 Chapter 5have led to almost indistinguishable results in terms of crack spacing and load–displacementdata. The predicted crack spacings are consistent with those obtained in recent experiments bySouza et al. (2001). Finally, it has been shown that estimation of the coating fracture energy from the first crackarea and corresponding load drop, as commonly done in indentation experiments, leads to val-ues different from the actual ones by as much as one order of magnitude. The reason for this isthat a small portion of the energy release is consumed by the crack, while the rest of the energyis consumed by the accompanied plastic dissipation in the substrate.References Abdul-Baqi, A., Van der Giessen, E., 2001a. Indentation-induced interface delamination of a strong film on a ductile substrate. Thin Solid Films 381, 143–154. Abdul-Baqi, A., Van der Giessen, E., 2001b. Delamination of a strong film from a ductile substrate during indentation unloading. J. Mater. Res. 16, 1396–1407. Bhattacharya, A.K., Nix, W.D., 1988. Analysis of elastic and plastic deformation associated with indentation testing of thin films on substrates. Int. J. Solids Struct. 24, 1287–1298. Camacho, G.T., Ortiz, M., 1996. Computational modelling of impact damage in brittle mate- rials. Int. J. Solids Struct. 33, 2899–2938. de Borst, R., 1987. Computation of post-bifurcation and post-failure behaviour of strain- softening solids. Computers and Structures 25, 211–224. Doerner, M., Nix, W., 1986. A method for interpreting the data from depth-sensing indentation instruments. J. Mater. Res. 1, 601–609. Gao, H., Chiu, C.-H., Lee, J., 1992. Elastic contact versus indentation modeling of multi- layered materials. Int. J. Solids Struct. 29, 2471–2492. Kanninen, M.F., Popelar, C.H., Advanced Fracture Mechanics, (Oxford University Press, New York, USA, 1985) King, R.B., 1987. Elastic analysis of some punch problems for a layered medium. Int. J. Solids Struct. 23, 1657–1664. Li, X., Bhushan, B., 1998. Measurement of fracture toughness of ultra-thin amorphous carbon films. Thin Solid Films 315, 214–221. Li, X., Diao, D., Bhushan, B., 1997. Fracture mechanisms of thin amorphous carbon films in nanoindentation. Acta Mater. 45, 4453–4461. Lim, Y.Y., Chaudhri, M.M., Enomoto, Y., 1999. Accurate determination of the mechanical properties of thin aluminum films deposited on sapphire flats using nanoindentations. J. Mater. Res. 14, 2314–2327. Malzbender, J., de With, G., den Toonder, J.M.J., 2000. Elastic modulus, indentation pressure and fracture toughness of hybrid coatings on glass. Thin Solid Films 366, 139–149.
  • 93. Indentation-induced cracking of brittle coatings on ductile substrates 87 Oliver, W.C., Pharr, G.M., 1992. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564–1583. Souza, R.M., Sinatora, A., Mustoe, G.G.W., Moore, J.J., 2001. Numerical and experimental study of the circular cracks observed at the contact edges of the indentations of coated systems with soft substrates. Wear 251, 1337–1346. van der Varst, P.G.Th., de With, G., 2001. Energy base approach to the failure of brittle coatings on metallic substrates. Thin Solid Films 384, 85–89. Wang, J.S., Sugimura, Y., Evans, A.G., Tredway, W.K., 1998. The mechanical performance of DLC films on steel substrate. Thin Solid Films 325, 163–174. Xu, X.-P., Needleman, A., 1993. Void nucleation by inclusion debonding in a crystal matrix. Model. Simul. Mater. Sci. Eng. 1, 111–132. Xu, X.-P., Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, 1397–1434.
  • 94. 88
  • 95. SummaryThe response of coated materials to external loading such as indentation is far more complicatedcompared to the response of bulk materials. For coated materials, the overall response of thesystem to indentation is controlled by the mechanical properties of both the coating and the sub-strate as well as the interface. The ratio of the indentation depth and the coating thickness is akey parameter in determining the influence of each of the constituents. Initially, when this ratiois very small compared to unity, the response is dominated almost by the coating. Conversely,when this ratio is very large compared to unity, the coatings influence diminishes and the re-sponse is dominated almost by the substrate. Between these two limiting extremes, the overallresponse is dominated by both constituents in a complex manner. Consequently, extraction ofmaterial parameters from indentation experiments in hindered by such complexity. For moreaccurate interpretation of indentation experiments, the main emphasis in this study, however,was to numerically investigate some of the possible indentation-induced failure modes usingan axisymmetric finite element formulation. Failure by interfacial delamination and coatingcracking has been simulated using the cohesive zone methodology. It has been found that shear delamination may occur during the loading stage of indentation.Driven by the plastic zone expansion in the substrate, a ring-shaped portion of the coating,outside the contact region, was found to detach from the substrate. It was also shown that thistype of failure is very sensitive to the degree of coupling between the interface response in thenormal and tangential directions. During the unloading stage, delamination was found to occurin a normal mode, where a circular portion of the coating, directly under the contact region, islifted off from the substrate. Normal delamination is driven by the coating energy as it unflexesto retain its original configuration and the mismatch in the mechanical properties of the coatingand substrate. The presence of compressive residual stress in the coating was found to delaynormal delamination and, if high enough, it might even prevent it, whereas tensile residual stresshas an opposite effect. Delamination is imprinted on the load–displacement curve by a rathersudden decrease in the indentation stiffness. Coating cracking was also found to occur during the the loading stage of indentation. Crack-ing initiates from the coating surface outside the contact region and propagates towards theinterface forming through-thickness circumferential cracks. The initiation and propagation ofeach crack is imprinted on the load–displacement curve as a kink. The spacing between suc-cessive cracks was found to be predominantly controlled by the ratio of the coating thicknessand the indenter radius. For a fixed indenter radius, the crack spacing scales with the coatingthickness. Coupling and irreversibility of the cohesive tractions were also investigated. The 89
  • 96. 90presence of coupling was found to slightly increase the crack spacing. Reversible and irre-versible cohesive tractions have led to almost indistinguishable results in terms of crack spacingand load–displacement data. Some of the common methods which are used to estimate the interfacial energy and thecoating fracture energy from indentation experiments were also investigated. It has been foundthat estimation of the interfacial energy from the radius of the delaminated area and estimationof the coating fracture energy from the first crack area and the corresponding load drop, lead tovalues that differ from the actual ones by as much as one order of magnitude. The differenceis mainly attributed to the fact that, in a highly nonlinear problem, these methods oversimplifythe estimation of the energy release associated with the failure event. In addition, such simplemodels do not take into account the plastic dissipation in the substrate which has been found toaccompany failure. The indentation method has proven to be a powerful tool for characterizing the mechanicalproperties of materials and in the case of coated materials, it maybe considered as one of thebest available methods. This is mainly owed to two reasons; firstly, the relatively easy samplepreparation and secondly, the various types of failure which may occur, and consequently beinvestigated, as a result of the heterogeneous stress field it generates. Moreover, besides being atest method, indentation can also be viewed as one of the possible external loadings the systemmaybe subjected to during service. However, the interpretation of the outcome of this methodto extract more accurate quantitative data about the coating, the interface and the substrate, stillrequires more theoretical as well as experimental investigations.
  • 97. SamenvattingHet gedrag van gecoate materialen op een uitwendige belasting, zoals indentatie, is veel gecom-pliceerder dan dat van bulk materialen. Het totale gedrag van gecoate materialen onder inden-tatie wordt gestuurd door de mechanische eigenschappen van, zowel de coating en het substraat,als van de interface. De verhouding van indentatie diepte en coating dikte is een belangrijke pa-rameter voor het vast stellen van de invloed van deze onderdelen. Voor een verhouding veelkleiner dan een wordt het gedrag beheerst door de coating terwijl voor een verhouding veel ´´groter dan een het gedrag beheerst wordt door het substraat. Tussen deze limiet gevallen wordt ´´het gedrag door zowel coating als het substraat bepaald op een complexe manier. Dit heeft totgevolg dat het afleiden van materiaal parameters uit een indentatie experiment gehinderd wordtdoor deze complexiteit. Om deze indentatie experimenten beter te begrijpen worden in dit proef-schrift numerieke experimenten beschreven, die bezwijkings modes van indentatie proeven on-derzoeken met behulp van een axisymmetrich eindige elementen model. Het bezwijken doorhet loslaten of breken van de coating is gesimuleerd met de cohesive zone methodologie. Er is gevonden dat schuif delaminatie kan voorkomen tijdens de belastingsfase van inden-tatie. Een ringvormig deel van de coating buiten de contact zone late los van het substraat,wanneer de plastische zone zich uitbreidt in het substraat. Er is ook aangetoond, dat dezemanier van bezwijken erg gevoelig is voor de mate van koppeling tussen het gedrag van deinterface in normale en tangentiele richting. Tijdens de ontlastingsfase vindt delaminatie plaatsin een loodrechte toestand, waar een circelvormig deel van de coating recht onder het contactgebied los komt van het substraat. Loodrechte delaminatie wordt gestuurd door de energie in decoating, wanneer deze relaxeert naar de oorspronkelijke toestand en door de mispassing in demechanische eigenschappen van de coating en het substraat. Het blijkt, dat een achter geblevencompressieve spanning in de coating vertragend werkt op loodrechte delaminatie en wanneerhet hoog genoeg is, delaminatie zelfs voorkomt. Terwijl achtergebleven trekspanningen eentegenover gestelde werking hebben. Delaminatie is herkenbaar in een spannings-rek krommedoor een vrij plotselinge afname van de indentatie stijfheid. Er is ook aangestoond dat het breken van de coating voorkomt tijdens de belastings fasevan indentatie. Scheurtjes beginnen aan de oppervlakte buiten het contact gebied en groeiennaar de interface, waarbij ze in de dikte richting ringvormige scheuren vormen. Het ontstaanen de voortbeweging van elke scheur staat in de spannings-rek kromme als een kink. Er isgevonden, dat de ruimte tussen opeenvolgende scheuren voornamelijk afhangt van de verhoud-ing coating dikte en indentor straal. Voor een vaste indentor straal schaalt de ruimte tussen descheuren met de coating dikte. Koppeling en onomkeerbaarheid van cohesieve tracties zijn ook 91
  • 98. 92onderzocht. Aanwezigheid van koppeling blijkt een lichte toename van de ruimte tussen descheuren te geven. Omkeerbare en onomkeerbare cohesive tracties hebben geleid tot bijna niette onderscheiden resultaten in de ruimte tussen de scheuren en in de spannings-rek data. Een aantal van de gebruikelijke methodes zijn onderzocht om de interface energie en debreuk energie van de coating te bepalen. Er is gevonden dat bepaling van de interface energieuit de straal van het delaminatie gebied en de bepaling van de breukenergie van de coatinguit het gebied van de eerste scheur en de bijbehorende daling in de belasting, resultaten gevendie meer dan een orde afwijken van de daadwerkelijke waardes. Het verschil komt met name,doordat het een sterk niet linear probleem is. De methodes oversimplificeren de bepaling vande energie die vrij komt bij zo’n proef. Daar komt bij, dat deze methodes geen rekening houdenmet de plastische dissipatie die hierbij gepaard gaan. De indentatie methode heeft bewezen een krachtig gereedschap te zijn voor het karakteriz-eren van mechanische eigenschappen van materialen en mag beschouwd worden als een van de ´´beste methodes. Dit komt met name door de volgende twee redenen; ten eerste de eenvoudigepreparatie van proefstukken en ten tweede de verschillende typen van bezwijken die plaats kun-nen vinden en dus onderzocht kunnen worden, als gevolg van het heterogene spanningsveld datgegenereerd wordt. Naast een test methode kan indentatie ook beschouwd worden als een van ´´de mogelijke belastings vormen, dat een systeem kan ondergaan tijdens het gebruik. Echter,de interpretatie van de resultaten van deze methode om betere kwantitatieve data te verkrijgenover de coating, de interface en het substraat, vereist nog meer theoretisch en experimenteelonderzoek.
  • 99. Propositions accompanying the thesis Failure of Brittle Coatings on Ductile Metallic Substrates Adnan Abdul-Baqi 1. The performance enhancement gained by a hard, brittle coating is always accompanied by the risk of its failure. The latter, however, is poorly understood and not thoroughly accounted for in design. this thesis 2. For coated materials, the interpretation of indentation results to extract accurate material and interfacial data still poses a big challenge. This is attributed to the fact that the overall response of such systems is controlled by the coating, the substrate as well as the interface in a complex manner. this thesis 3. The faster the computer, the earlier to find out that a simulation does not work. 4. For foreign employees staying one year or more in the Netherlands, learning the Dutch language should be made compulsory. 5. The prevention and combat of international ’terrorism’ must be proceeded by a unique definition of the term. 6. A substantial progress in solving the increasing world’s problems can be achieved by decoupling them from our own political and economic interests. 7. “ There is a major contradiction between the U.S. role as chief mediator of the Israeli- Palestinian conflict and its role as the principal economic, diplomatic, and military sup- porter of Israeli occupation policies.” Therefore, the leading role in solving this conflict has to be assigned to a neutral party like the U.N. Stephen Zunes, Foreign Policy in Focus, Vol. 6, No. 4, 2001. 8. In the process of seeking the better, one has to make the best out of the available. 93
  • 100. 94
  • 101. Stellingen behorende bij het proefschrift Failure of Brittle Coatings on Ductile Metallic Substrates Adnan Abdul-Baqi 1. Verbetering van de performance verkregen door een harde brosse coating gaat altijd gepaard met het risico van breuk. Over het laatste is nog weinig bekend en er wordt niet grondig rekening mee gehouden tijdens het ontwerp. dit proefschrift 2. Voor gecoate materialen is de interpretatie van indentatieresultaten om accurate materiaal- en interfacegegevens te verkrijgen nog steeds een grote uitdaging. Dit is te wijten aan het feit dat het overall gedrag van zo’n systeem gestuurd wordt door een complex samenspel van de coating, het substraat en de interface. dit proefschrift 3. Hoe sneller de computer, hoe eerder je ondekt dat een simulatie niet werkt. 4. Voor buitenlandse werknemers die een jaar of langer in Nederland blijven, zou het leren van de Nederlandse taal verplicht moeten worden. 5. Het voorkomen en bestrijden van internationaal ’terrorisme’ moet vooraf worden gegaan door een eenduidige definitie van deze term. 6. Een substanti¨ le vooruitgang in het oplossen van het toenemend aantal problemen in de e wereld kan slechts bereikt worden door deze te ontkoppelen van onze eigen politieke en economische belangen. 7. “ Er is een grote contradictie tussen de rol van de V.S. als onderhandelaar in het Israelisch- Palestijns conflict en zijn rol als hoofd ondersteuner in economisch, diplomatiek en mili- taire zin van het Israelische bezettings beleid.” Om deze reden zou de leidende rol in het oplossen van dit conflict toe gewezen moeten worden aan een neutrale partij zoals de V.N. Stephen Zunes, Foreign Policy in Focus, Vol. 6, No. 4, 2001. 8. In de zoektocht naar het betere, moet men het beste maken van het beschikbare. 95
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  • 103. Curriculum VitaeThe author was born on the 9th of May 1969 in the town of Zawieh, West-Bank, Palestine.In 1987 he passed the high school examination and started his education in the MechanicalEngineering Faculty at Birzeit University, Palestine. The study came to an end after less thantwo months when the university was closed. In 1990 the author was admitted to the Yarmouk University in Jordan where he obtainedhis Bachelor degree in Physics in the year 1993. In 1994 the author received a scholarshipfrom Bergen University in Norway where he worked under the supervision of Prof. LadislavKocbach in the theoretical atomic physics group. After the defense of his thesis Atomic physicsaspects of radiation physics and passing his final exam, the author was awarded a Master degreein physics in the year 1996. The author started his PhD work in August 1997 with Prof. Erik van der Giessen in themicromechanics of materials group at Delft University of Technology. The work was on theproject Adhesion of brittle coatings on ductile metallic substrates. Coming from a differentfield, the author had initially to get familiar with the concepts of mechanics, the finite-elementmethod and the related numerical tools. This has been accomplished successfully with theguidance of Prof. Erik van der Giessen and the work finally resulted in the current manuscript. Having decided to continue in research, the author joined the group of Prof. Marc Geers atEindhoven University of Technology in September 2001 to work on fatigue damage modelingin lead-free solder joints. 97
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  • 105. AcknowledgementI would like to acknowledge the supervision of Erik van der Giessen in the course of preparationof this manuscript. Your decision to offer me a doctoral position at the micromechanics ofmaterials group, despite my different background, is really appreciated. It was a pleasure and aprivilege to work with you. I thank all my colleagues at the micromechanics of materials group for providing me aperfect working atmosphere. I especially name Harko, Martin, Sabine, and Sumit. Thanks forthe scientific and social interesting discussions. I acknowledge Jan Booij for his assistance incomputer-related issues and Marianne for her help, including the Dutch to English translations.I also like to thank my friends for their support outside the university. Your sense of humor andconversations about almost every thing in life has always made your company very interestingfor me. Last but certainly not least, I wish to express my sincere gratitude to my family and mostnotably to my mother for the devotion of her life to her children until her early death. I willnever forget the support of my father which was, and still is, of vital importance for my personaland academic advancement. 99