Failure of Brittle Coatings on Ductile         Metallic Substrates
Failure of Brittle Coatings on Ductile          Metallic Substrates                             Proefschrift              ...
Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. E. van der GiessenSamenstelling promotiecommissie: Rector M...
To my Family
Contents1   Introduction                                                                                                  ...
Samenvatting            91Propositions            93Stellingen              95Curriculum Vitae        97Acknowledgement   ...
Chapter 1IntroductionHard coatings are usually applied to materials to enhance performance and reliability such aschemical...
2                                                                                       Chapter 1empirical relations. The ...
Introduction                                                                                3References Carvalho, N.J.M., ...
4   Chapter 1
Chapter 2Indentation of bulk and coated materials      Indentation experiments are widely used to measure mechanical prope...
6                                                                                       Chapter 2     Materials in general...
Indentation of bulk and coated materials                                                                       7          ...
8                                                                                           Chapter 2                     ...
Indentation of bulk and coated materials                                                        9                        0...
10                                                                                               Chapter 2related to the m...
Indentation of bulk and coated materials                                                        11                  0.25  ...
12                                                                                                                   Chapt...
Indentation of bulk and coated materials                                                       13                   0.25  ...
14                                                                                                                Chapter ...
Indentation of bulk and coated materials                                                             15                   ...
16                                                                                        Chapter 2                    0.2...
Indentation of bulk and coated materials                                                         17modulus by             ...
18                                                                                    Chapter 2Hill, R., 1992. Similarity ...
Based on: A. Abdul-Baqi and E. Van der Giessen, Indentation-induced interface delamination of a strong film ona ductile sub...
20                                                                                     Chapter 3tation load versus depth c...
Indentation-induced interface delamination of a strong film on a ductile substrate                                         ...
22                                                                                                                        ...
Indentation-induced interface delamination of a strong film on a ductile substrate                                         ...
24                                                                                                                        ...
Indentation-induced interface delamination of a strong film on a ductile substrate                                         ...
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates
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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Ph.D. Thesis - Failure of Brittle Coatings on Ductile Metallic Substrates

  1. 1. Failure of Brittle Coatings on Ductile Metallic Substrates
  2. 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. 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. 4. To my Family
  5. 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. 6. Samenvatting 91Propositions 93Stellingen 95Curriculum Vitae 97Acknowledgement 99 vi
  7. 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. 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. 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. 10. 4 Chapter 1
  11. 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. 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. 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 igD © ¤ 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. 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. 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. 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£  tThe 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. 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. 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. 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. 20. 14 Chapter 2 6 characterized by its elastic properties d D © GPa, idgD ‘¥ ¤ 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. 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. 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. 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 ‘¥ dd 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. 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. 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. 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. 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. 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. 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 dd 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. 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. 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

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