This chapter examines analytical and empirical relations used to extract mechanical properties from indentation experiments. Finite element simulations are performed for spherical and conical indentation of elastic, elastic-plastic, and coated materials. The material response is characterized by load-displacement curves. Existing relations for contact area, contact stiffness, and hardness are compared to finite element results. The chapter lays groundwork for using indentation to characterize properties and failure modes of coated materials.

1. Failure of Brittle Coatings on Ductile
Metallic Substrates

3. 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.

4. Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. E. van der Giessen
Samenstelling 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 Delft
The work of A.J.J. Abdul-Baqi was supported by the Program for Innovative Research, surface
technology (IOP oppervlakte technologie), under the contract number IOT96005.
Copyright c Shaker Publishing 2002
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or otherwise, without the prior permission of the publishers.
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Tel.: +31 43 3500424
Fax: +31 43 3255090
http://www.shaker.nl

5. To my Family

7. Contents
1 Introduction 1
2 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 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

8. Samenvatting 91
Propositions 93
Stellingen 95
Curriculum Vitae 97
Acknowledgement 99
vi

9. Chapter 1
Introduction
Hard coatings are usually applied to materials to enhance performance and reliability such as
chemical resistance and wear resistance. Ceramic coatings, for example, are used as protective
layers in many mechanical applications such as cutting tools. These coatings are usually brittle
and the enhancement gained by the coating is always accompanied by the risk of its failure
leading to a premature failure of otherwise long lasting systems. Failure may occur in the
coating itself or at the interface with the substrate. Therefore, mechanical characterization of
such systems, including the possible failure modes under various loading circumstances, is
critical for the understanding and the improvement of its performance.
Indentation has become one of the most common methods to determine the mechanical
properties of materials such as elastic properties, plastic properties and strength. In this test, an
indenter is pushed into the surface of a sample under continuous recording of the applied load
and corresponding penetration depth (Weppelmann and Swain, 1996). Indenters have different
geometries including spheres and cones. They are usually made of diamond due to its extreme
properties like hardness and stiffness. For hard coatings, indentation is one of the simplest tests
in terms of sample preparation (Drory and Hutchinson, 1996). However, the interpretation of
indentation results still poses a big challenge. This has motivated extensive experimental as
well as theoretical studies which covers various indenter geometries and constitutive material
models. The material response in an indentation experiment is governed by both its mechanical
properties 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 systems
comprising a strong coating on relatively softer substrate. Qualitative description of the coating
and the interface fracture characteristics is inferred from failure events. In addition, estimation
of the coating and the interface fracture energies from failure events as commonly done in
indentation experiments is also discussed. The analysis is carried out numerically using a finite
strain, finite element method. An overview of the most common methods used to determine
the mechanical properties of materials by indentation is given in Chapter 2. Both the loading
and the unloading are modeled using the finite element method. The emphasis is based on the
load versus displacement data in comparison with the prediction of some existing analytical and
1

10. 2 Chapter 1
empirical relations. The analysis in this chapter assumes that failure events do not occur during
indentation. This assumption holds true if the generated stresses do not reach the material
strength; otherwise, failure is inevitable.
The main failure events discussed in this thesis are interfacial delamination and coating
cracking. Crack initiation and propagation are modeled within a cohesive surface framework
where the fracture characteristics of the material are embedded in a constitutive model for the
cohesive 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 carrying
capability, and a fracture energy. Additional criteria for crack initiation and propagation are not
required. 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 interface
prior to indentation. The coating is assumed to remain intact and failure is only allowed to
occur at the interface. Shear delamination (mode II) is possible during the loading stage of
the indentation process as discussed in Chapter 3. It is found that a ring-shaped portion of the
coating, outside the contact region, is detached from the substrate. On the other hand, normal
delamination (mode I) can occur during the unloading stage as discussed in Chapter 4. In
this case, a circular portion of the coating, directly under the contact region, is lifted off from
the substrate. Delamination is imprinted on the load–displacement curve by a rather sudden
decrease in the indentation stiffness. For relatively strong interfaces, the stiffness might even
become negative. This leads to a kink on the loading curve and a hump on the unloading curve
in the case of shear and normal delamination, respectively. The latter has recently been observed
experimentally 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 in
between all continuum elements in the coating leads to serious numerical problems in addition
to an artificial enhancement of the overall compliance (Xu and Needleman, 1994). In this
study we adopt a procedure in which the number of cohesive zones is minimized and placed
only at precalculated locations. The interface between the coating and the substrate is also
modeled by means of cohesive zones but with interface properties. It is shown that successive
circumferential through-thickness cracking occurs outside the contact region with crack spacing
of the order of the coating thickness. Each cracking event is imprinted on the load–displacement
curve 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 fracture
energy compared to the actual values. This is mainly attributed to the fact that, in such a highly
nonlinear problem, these methods oversimplify the estimation of the energy release associated
with the failure event.

11. Introduction 3
References
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.

12. 4 Chapter 1

13. Chapter 2
Indentation 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 Introduction
In 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 has
motivated extensive experimental as well as theoretical studies which cover various indenter
geometries and material models. The most common indenter geometries are a sphere (Brinell
test), 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 Berlin
Physical Society (Johnson, 1985). The publication of his classic paper On the contact of elastic
solids in 1882 (Hertz, 1882) may be viewed, according to Johnson (1985), to have started the
subject of contact mechanics. However, developments in the Hertz theory did not appear in the
literature until the beginning of the 20th century (Johnson, 1985). The problem of determining
the stress distribution within an elastic half space due to surface tractions and a concentrated
normal force has been considered first by Boussinesq (1885). Based on his solution, partial
numerical 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 of
the 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 depth
and the contact area for punches of different axisymmetric geometries. In the above studies, the
contact is assumed frictionless. Contact involving a sticking indenter has been latter analyzed
by Spence (1968).
5

14. 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 behaviour
of the indented material and different attempts has been carried out to account for the plastic
deformation. An empirical relation was found by Tabor (1951) which correlates between the
hardness, defined as the mean pressure supported by the material under load, and the material’s
plastic properties. Hill et al. (1989) have carried out a theoretical study of indentation of a
power law hardening material. They were able to predict Tabor’s empirical relation and to
study in detail the deformation field beneath the indenter. Indentation of power law creeping
material has been studied by Matthews (1980) and Hill (1992). Proceeding from the study
by Hill, Bower et al. (1993) have also studied indentation of creeping materials and provided
relations 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 the
unloading 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. Doerner
and 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, they
have combined the elastic properties of the coating and substrate linearly in one effective elastic
modulus in a way which fits measured experimental data. King (1987) modified the formula
proposed 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 derive
closed-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 them
with numerical results. The main focus will be on the indentation load versus displacement
curve in the case of purely elastic material, elastic-plastic material and coated material.
2.2 Elastic contact
The normal contact between a spherical indenter and an elastic half space is given by the Hertz
theory. For the geometry shown in Fig. 2.1, Hertz theory provides an analytical solution for the
stress 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)

15. 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
Q
contact 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 effects
including boundary changes and radial displacements of points along the contact surface. The
latter 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 the
spherical and conical indenter profiles. The main intention is to examine the accuracy of the
prediction 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
6
modulus d d D © ¥ GPa and a Poisson’s ratio ad © ¦
. Figure 2.2 shows the pressure distribution
s
at the contact surface. The numerical pressure distribution seems to agree reasonably with
the analytical distribution proposed by Hertz (Eq. 2.1). The load–displacement data for both

16. 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 depth
vxe %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 discussed
previously are not fully satisfied, mainly the small strain assumption.
The main attraction of the Hertz theory is the analytical solution it provides for the contact
problem. However, the validity of the theory and other existing analytical solutions is limited
to infinitesimal deformations. The problem involving finite deformations or nonlinear material
behaviour has no analytical solution. Such problems are generally solved numerically using the
finite element method.
2.3 Elastic-plastic contact
The contact problem involving elastic-plastic materials does not have a complete analytical
solution due to the highly nonlinear material response. However, approximate solutions limited
by simplifying assumptions are available in the literature. In this section we will examine
some of the existing analytical and empirical relations, namely those that relate the response
to indentation and the material’s mechanical properties, and compare their predictions with our
numerical results.
There are several constitutive models in the literature which account for plasticity in the
material. Examples of the most common models used in indentation modeling include elastic-

17. 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 dependent
plasticity models (e.g. Bower et al., 1993; Mesarovic and Fleck, 1999). In this section we will
consider a material with an elastic-perfectly plastic response. Such material is characterized
by its elastic properties ( , ) and a yield stress . In the FEM calculations we have used
¦ ¥  ‚t
sd 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 ,
Q
the indentation load is predicted based on the so-called cavity model (Johnson, 1985). It is

18. 10 Chapter 2
related 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 hemispherical
core, 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 body
which contains a spherical cavity under a pressure equal that of the core. Based on the conical
indenter 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 are
derivable from a single solution by appropriate scaling (Bower et al., 1993; Mesarovic and
Fleck, 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
¤k
where is the strain hardening exponent,
l d ˜
is the yield strain and the constants and are k
functions of the strain hardening exponent, the indenter geometry and the frictional condition
k
between 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 behaviour
occurs at © l ˜ k
. Bower et al. (1993) tabulated the values of the constants and for a range of
hardening exponents and indenter profiles. The elastic-perfectly plastic material corresponds to
taking
t0l D
s 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. Both
the similarity solution, Eq. (2.10), and the cavity model solution, Eq. (2.8), seem to be in close
agreement 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 from
the numerical results as seen clearly in Fig. 2.4(a). This deviation is not surprising in view of
the approximations made.

19. 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 the
load–displacement curve (Tabor, 1951; Hill et al., 1998; Matthews, 1980; Hill, 1992; Bower et
al., 1993). One of the most common parameters in indentation experiments is hardness, defined
as w
s £¡ E © (2.12)
The extraction of the material’s plastic properties from hardness is not straightforward. In
the case of strain hardening materials, the hardness depends on the the yield stress, contact

20. 12 Chapter 2
radius, strain hardening exponent and indenter geometry (Bower et al., 1993). For rigid-plastic
materials, 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 the
presence of elasticity. In these equations, hardness continuously increases with the ratio of the
applied strain ( for the cone and
Q R `X
Y for the sphere) to the yield strain
ba£
¤! . Based on ¥ b!  t
Eq. (2.13), we have calculated the yield stress from the maximum load and the corresponding
maximum 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 dominated
by 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
s
actual 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 that
this 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 indentation
experiments. 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 axisymmetric
smooth 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 of
Eq. (2.14), we estimate the Young’s modulus to be d D
D
GPa from the spherical indenter results
and r ‰ D
GPa from the conical indenter. Compared to the actual value dbd D © ¥
GPa, the error
is 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 performed
indentation experiments and numerical simulations using a conical indenter. Using a wide range
of 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) resulted
from the nonlinear effects, including large strain and moving contact boundaries. According6 D
Eq. (2.15), the calculated values of the Young’s modulus are and GPa. It should be d dD
noted 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. !

21. 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 lines
illustrate the slope of the initial portion of the unloading curves.
2.4 Coated materials
Indentation of coated materials is far more complicated as compared to bulk materials. In coated
systems, the indentation response is controlled by the mechanical properties of both the coating
and the substrate. In this section we will investigate the indentation of an elastic-perfectly
plastic substrate coated by a relatively stronger elastic coating. The coating is characterized
6
by its thickness
vf ‹© Š
h and elastic properties d d e © ‘¥
Œ GPa, $ %d © b¦
s Π. The substrate is

22. 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 semiangle
Q 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• ” “ b'Š •
£! ©
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

23. 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 with
different 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¦
Ž
the 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 element
analysis. They found that the solution is valid, within an error of , at least for moduli ratio
€ y
up to 2. For larger moduli ratio, the weight functions (Eq. 2.18) fail to accurately represent the

24. 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 plastic
substrate.
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 effective
properties.
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 different
from these of the substrate, the initial unloading stiffness in this case is related to the effective

25. Indentation of bulk and coated materials 17
modulus 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 is
known. 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. These
are 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 value
of 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 mechanical
properties of the system (Wittling et al., 1995; Korsunsky et al., 1998). The main idea is to
introduce weighting functions to interpolate between the two limiting cases where the coating
and 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 by
delamination 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.

26. 18 Chapter 2
Hill, 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.

27. Based on: A. Abdul-Baqi and E. Van der Giessen, Indentation-induced interface delamination of a strong film on
a ductile substrate, Thin Solid Films 381 (2001) 143.
Chapter 3
Indentation-induced interface
delamination of a strong film on a ductile
substrate
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 Introduction
Indentation is one of the traditional methods to quantify the mechanical properties of materials
and during the last decades it has also been advocated as a tool to characterize the properties of
thin 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, many
experimental as well as theoretical studies have been devoted to indentation of coated systems
during 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
c
possibility to extract the interfacial energy from indentation tests. Assuming the use of a small
spherical indenter, they identified five different classes of interfacial failure, depending on the
relative 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 an
important role when indentation is continued until interface failure. As emphasized further by
Bagchi and Evans (1996), this makes the deduction of the interface energy from global inden-
19

28. 20 Chapter 3
tation 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, coating
delamination is coupled to plastic expansion of the film with the driving force for delamination
being delivered via buckling of the film. The key mechanics ingredients of this mechanism have
been presented by Marshall and Evans (1984), and Kriese and Gerberich (1999) have recently
extended 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 through
the film. The mechanics of delamination in such systems has been analyzed by Drory and
Hutchinson (1996) for deep indentation with depths that are two to three orders of magnitude
larger than the coating thickness. The determination of interface toughness in systems that show
coating cracking has been demonstrated recently by e.g. Wang et al. (1998). In both types of
material systems there have been reports of ”fingerprints” on the load–displacement curves in
the 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
c
origin of these kinks remains somewhat unclear, however.
A final class considered in (Swain and Menˇ ik, 1994) is that of hard, strong coatings on
c
ductile substrates, where Swain and Menˇ ik hypothesized that indentation with a spherical in-
c
denter would not lead to cracking of the coating but just to delamination. This class has not
yet received much attention, probably because most deposited coatings, except diamond or
diamond-like carbon, are not sufficiently strong to remain intact until delamination. On the
other hand, it provides a relatively simple system that serves well to gain a deep understanding
of the coupling between interfacial delamination and plasticity in the substrate. An analysis of
this class is the subject of this paper.
In the present study, we perform a numerical simulation of the process of indentation of
thin 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- perfectly
plastic. The interface is modeled by a cohesive surface, which allows to study initiation and
propagation of delamination during the 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 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 the
interface strength needed to avoid delamination. We will also study the ”fingerprint” left on
the load–displacement curve by delamination, and see if delamination itself can lead to kinks
as mentioned above in other systems. It is emphasized that the calculations assume that other
failure events, mainly through-thickness coating cracks, do not occur.

29. 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 formulation
3.2.1 Governing equations
We consider a system comprising an elastic-perfectly plastic material (substrate) coated by an
elastic thin film and indented by a spherical indenter. The indenter is assumed rigid and only
characterized 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 Š
by an interface, which will be specified in the next section. The substrate is taken to have a
height of Š ž and radius , with large enough so that the solution is independent of and
ž ž ž
the substrate can be regarded as a half space.
The analysis is carried out numerically using a finite strain, finite element method. It uses
a Total Lagrangian formulation in which equilibrium is expressed in terms of the principle of
virtual 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 ¦ Š
are the components of displacement and traction vector, respectively;¢£ ¥
are the components of
Second Piola-Kirchhoff stress while are the dual Lagrangian strain components. The latter

30. 22 Chapter 3
are 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 µ
left-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 interface
has components , while the displacement jump is
» v­ ˜
, with being either the local normal
direction (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 are
controlled by the motion of the indenter, i.e.
° °
d © †U#§ Â 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 precise
boundary conditions only have a significant effect very close to the contact area and do not alter
the results for delamination to be presented later. The indentation force is computed from the ¡
tractions in the contact region, Ã ÄŸ
§ § 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
ž
from the precise remote conditions.
The equations (3.1) and (3.2) need to be supplemented with the constitutive equations for
the coating and the substrate, as well as the interface. As the latter are central to the results of
this study, these will be explained in detail in the forthcoming section. The substrate is supposed
to be a standard isotropic elastoplastic material with plastic flow being controlled by the von
Mises stress. For numerical convenience, however, we adopt a rate-sensitive version of this
model, 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 ’ t
for the plastic part of the strain rate, © ¢£ Á. Here, B ÇÈ© Ž t ¢£ £ Ž¢ ¢£ is the von Mises stress,
Á Á
expressed in terms of the deviatoric stress components , ± ± l is the rate sensitivity exponent and
±

31. 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 r
rate-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
Á
stress 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. For
this 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 the
forward gradient version of the viscoplastic law (3.7) due to Peirce et al. (1984).
3.2.2 The cohesive surface model
In the description of the interface as a cohesive surface, a small displacement jump be- ¬ Î ›¬ Í
tween the film and substrate is allowed, with normal and tangential components and ,
© Ω
respectively. The interfacial behaviour is specified in terms of a constitutive equation for the
i
corresponding traction components and at the same location. The constitutive law we
adopt in this study is an elastic one, so that any energy dissipation associated with separation is
i
ignored. 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 use
the 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 Î’
i i a@xÎ i i
ÔÓÒ (3.10)
i Ï#! Ï ÖÕ ©
‰¤ a@Ò ¤
Î ÔÓ
with and the normal and tangential works of separation ( Ï ), and two char-
Ï
acteristics 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 ¬
Õ˜– † 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 ÔÓ†
Î
i s Î ¤ ‰ a@Ò ¤ ‰ ÔÓ ¬ (3.11)
(3.12)
”i ’ § ’i i ’ ’
%@Ò i
ÔÓ i a@Ò i i
ÔÓ

32. 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 Î
law 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 ©

33. 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 4
Figure 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 ¡
i
separation 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
Î
now 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
ÔÓ
Using equation (3.13) together with the relation #! Ï AÕ
Ï © a@Ò
ÔÓ
, we can relate the uncoupled normal
and 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
¬ D
mum 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
i
the most realistic, and the parameter values used in the present study ensure this.
i
i

34. 26 Chapter 3
3.3 Results and discussion
3.3.1 Analysis and parameters
The solution to the problem depends on a number of nondimensional parameters. We have
chosen 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 hSœãiâ'he 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 that
rate-dependent plasticity, Eq. (3.7), is governed by the parameters and only through the Æ Ád Æt
combination . 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
æ0l
s 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 nondimensional
groups, we have chosen to work primarily in terms of real dimensional values in order to sim-
e
plify the interpretation. We have used an indenter of radius
e 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 dd  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 a
constant 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
â
limit. For the cohesive surface we have chosen the same values for and , namely d
mm. Most of the results to be presented are for and
e %d Õ e %d © §
, but we will also briefly © s
¡ i s
investigate 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

35. Indentation-induced interface delamination of a strong film on a ductile substrate 27
d ‰
p
J/m , which are realistic values for the interface toughnesses for well-adhering deposited
films (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-
ž
infinite medium. The mesh is an arrangement of quadrilateral elements, and
d bd p c
nodes. cDD
The elements are built up of four linear strain triangles in a cross arrangement to minimize
numerical problems due to plastic incompressibility. To account properly for the high stress
gradients under the indenter and for an accurate detection of the contact nodes, the mesh is
made 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 ª
location develops when exceeds . The critical instant is here taken to be when .
D ª¤ © 6
Larger critical values may be used as well (Xu and Needleman, 1994), but using or times
does 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 and
the 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, which
would 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-
Î
vance here, is the stress distribution that develops at the coating-substrate interface. Figure 3.4
shows the normal ( ) and shear ( ) stress components at different indentation depths for the
case of
D d %ts d © Pb! Æ t
e d . Other values of
ç ¥
t
would give the same qualitative behaviour.
‘#! Æ t
ç ¥
i
Note 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-
vҤ
£ 1
pression (Fig. 3.4.a). The reason for this stress being the same for all indentation depths shown
is that the contact region is contained within the plastic zone in the substrate, which exhibits
6
no 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 a
local maximum (in absolute value): near the edge of the contact radius, at about
6 and in a £D
region between £e
to . The highest shear stress, roughly
£ «t p ad
, is attained in this outer re-
Æ s
gion. Closer examination of the results reveals that this shear stress is caused by the outward
radial plastic flow of the substrate under the indenter relative to the coating. The latter is being
pulled inwards due to the indentation. This relative motion is opposite to that observed in the
calculations of Narasimhan and Biswas (1997) who considered a plastically deforming film on
a rigid substrate.
The most noteworthy part of the stress distributions in Fig. 3.4 is that there is an annular
6
region around £ p §
where the shear stresses are large and the normal stress is positive. This

36. 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⁄a
Figure 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 normal
stress, corresponding to d p
, significantly reduces the maximal shear stress before shear
6
failure, as shown in Fig. 3.3, so that this region
i £ p §
is a potential location for delamination.

37. Indentation-induced interface delamination of a strong film on a ductile substrate 29
3.3.3 Initiation of delamination
ª ¬
In the presence of a cohesive surface along the interface, a distribution of normal and tangential
separations 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¬ Î
parameter combinations investigated (cf. Sec. 3.3.1) we indeed find that delamination occurs
by 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 t
e 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 t
the 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 will
be 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 Š
radius 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 the
contact radius , this factor depending on both interface strength and substrate yield strength.
For
ebD dd 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 determining
whether or not delamination takes place (we will return to this later).
£6 £ £ £ 6 eç b‰t
¥!Æ
†c ë 64 %îDbBc yp e6 %d r Srp ep %d D6 †eydr 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).

38. 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 ⁄ R
Figure 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

39. 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
¥!Æ
combined to give ¡ e ë
† IŠIì¢ £ © $ £ † 'Š#I¢ £ $ Æ #! hØ{
t }| (3.21)
( £ $ ! B $ !ì
ë e

40. 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 e
to 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 the
interface strength so that extreme care must be taken by extrapolation of (3.19)–(3.22) outside
the 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 this
coupling, the shear stress along the interface is limited by the plastic flow in the substrate. In
the time-independent limit, in (3.7), we have
ðl
s r so that the maximum value that
qA«t 6
Æt 1 ñ
the shear stress can reach at the interface is ; when plasticity is slightly rate-dependent
6 9!Æt
as 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 interfacial
region 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 reduces
the local shear strength
h){ Î
}| ¡
. Figure 3.7 demonstrates this coupling for the case © ‘b! Æ t
ç ¥
D d %d
e 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
Ω
zone gradually engulfs this point. As this occurs, increases until a certain indentation depth Ω
Î
and the effective shear strength
j}){
|
decreases. However, just before would coincide
j}){
|
i
with and delamination would initiate, the normal opening starts to decrease and delamination
t
becomes 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 to
be a rather good linear correlation over the range considered, with the deviation from the line
6
9!Æt Î Æt
increasing with . The figure also shows the two values for interfaces characterized 6
by 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.
Õ
Hence, 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

41. 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 depth
corresponding peak tangential traction
D s v© § $ ¢ ¬ j}){ Î ©
¤4
of shear stress
at
| .
t , normal opening and
i
i
3.3.5 Propagation of delamination
After 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 tip
of the zone that propagates towards the indenter gets arrested after a short time because of the
compressive normal traction close to the contact area as seen in Fig. 3.4.a. The leading tip of the
delaminated zone keeps propagating with continued indentation. Figure 3.9.a shows the leading
tip 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§
¤! Ì
Fig. 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-state
speed after traveling a certain distance from the initiation point. The reason for this is that the
dominant driving force for initial propagation is the elastic energy stored in the system; once
this energy gets released, it gives rise to more or less instantaneous growth which is just limited
here by the rate at which plastic deformation can evolve in the substrate. On the other hand, the
steady state propagation is predominantly driven by the indentation process which is performed

42. 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 «t
line 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 fÕ
s © 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. We
attribute this to the stored elastic energy in these cases being so large that ellipticity of the
problem is locally lost when this energy is released. More advanced solution procedures are
necessary 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 included
as well (Fig. 3.10.a). The region in the substrate where the value of is close to signifies
ñt Æt
the region of substantial plastic deformation (note that we are using viscoplasticity so that ñ it
is not limited to ). The delaminated zone in Fig. 3.10.b is identified by a white line, and
ö Æ t x¢
ö
extends 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!ì
and a critical radius of d %d Ϥ
s ì mm. Comparison of the two plots shows, first of all, that
a plastic region has developed near the leading tip of this zone with a size on the order of
the coating thickness. This plastic zone moves with the propagating tip of the delamination
zone. Apparently, there is a stress concentration around the tip of the delaminated zone which
induces local plastic flow in the substrate. The second difference between Fig. 3.10.a and b

43. 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⁄t
Figure 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 somewhat
larger than in the case of a finite-strength interface. This is attributed to the fact that interfacial
delamination serves as another mechanism for stress relaxation in the system. Except for the
near-tip region of the delaminated zone, this tends to reduce the necessity for stress relaxation
by plastic deformation, thereby reducing the overall dimension of the plastic zone.

44. 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 indentation
One of the most common outputs of indentation experiments is the indentation force versus
indentation depth curve (load–displacement curve). Figure 3.11 shows the predictions for some

45. 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⁄t
Figure 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 results
differ very slightly from those with a perfectly bonded coating, which is just due to the finite
stiffness 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 and
10% 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 ¥!Æ
have 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 here
is 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 stored
elastic energy is higher for higher interfacial strength and higher substrate yield stress, the kink
will be sharper accordingly.
3.4 Conclusions
Numerical simulations have been carried out of the indentation process of a coated material
by a spherical indenter. The interface between the film and the substrate was modeled by a

46. 38 Chapter 3
cohesive 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 in
the constitutive model and does not require any additional criteria.
A parametric study has been carried out to investigate the influence of interfacial strength
and substrate yield strength on delamination. Delamination is found to be driven by the shear
stress at the interface associated with the plastic deformation in the substrate, and consequently
occurs in a tangential mode. Nevertheless, interfacial normal stresses have a significant effect
on delamination, due to the coupling between normal and tangential response. Here we have
focused on the coupling whereby the interfacial resistance to tangential delamination is reduced
by tensile stresses along the interface. We find, however, that the results are quite sensitive
to variations in this coupling so that quantitative predictions require an accurate description of
this 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
6
tangential response, this limiting strength is simply 9 ! Æ t © h){
}|, but it is significantly higher
due 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 the
contact radius, and propagates in two directions; towards and away from the indenter, thus
generating a ring-shaped delaminated zone. The front which travels towards the indenter gets
arrested after a short distance because of the compressive normal stress in that region. After
a high initial speed of propagation, the other top settles at a steady propagation at a constant
velocity. 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 process
itself.
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 findings
both 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 under
consideration will be limited to plastic zones immediately under the indenter, which are often
smaller than that in the substrate. In such cases, the precise instant of delamination and the rate
of propagation are expected to differ somewhat quantitatively, but to leave the phenomenology
essentially unchanged.
For simplicity, the present calculations have assumed perfect but rate-dependent plasticity
for the substrate. Rate dependency was never significantly affecting the results here because the
time scale for the indentation process was always much larger. Strain hardening of the substrate
will, obviously, change the quantitative results, especially for the indentation force vs. depth
response. However, we do not expect a qualitatively significant effect on delamination, since
the leading front of the delaminated zone seems to propagate with the front of the plastic zone
in which hardening has not yet taken place.
Again for simplicity, this study has not accounted for the presence of residual stress in the
coating, due for example to thermal mismatch relative to the substrate. This as well as the

47. Indentation-induced interface delamination of a strong film on a ductile substrate 39
influence 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.

48. 40 Chapter 3
Xu, X.-P., Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids.
J. Mech. Phys. Solids 42, 1397–1434.

49. Based on: A. Abdul-Baqi and E. Van der Giessen, Delamination of a strong film from a ductile substrate during
indentation unloading, Journal of Materials Research 16 (2001) 1396.
Chapter 4
Delamination of a strong film from a
ductile substrate during indentation
unloading
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 Introduction
Industrial application of thin hard-film-coated systems continuously progresses. Coatings are
commonly used to enhance reliability, such as chemical resistance, wear resistance, corrosion
resistance and thermal barriers. Adhesion between the film and the substrate determines, to a
great 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 the
interface between the coating and the substrate may lead to premature failure of otherwise long
lasting 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 characterize
the properties of thin films or coatings. At the same time, for example for hard wear-resistant
coatings, indentation can be viewed as an elementary step of concentrated loading. For these
reasons, many experimental as well as theoretical studies have been devoted to indentation of
coated systems during recent years.
41

50. 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 (Kriese
and Gerberich, 1999; Li and Bhushan, 1998). The corresponding load vs displacement curves
show a reduction in the stiffness or even a sudden discontinuity which is usually attributed to
the coating cracking. Delamination without any accompanying through-thickness cracks has
been observed by Li and Bhushan (1998) in their nanoindenation experiments on single and
multilayer coatings. There is no evidence in the literature, to the authors’ knowledge, whether
delamination 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 motivated
by residual stress. The emphasis in their work is on the role of the interface debond energy
and the methods of its quantitative measurement. They argue that most thin film adhesion
tests do not measure the interface debond energy because the strain energy release rate cannot
be deconvoluted from the work done by the external load. Viable procedures to extract the
interfacial energy from indentation experiments will depend strongly on the precise mechanisms
involved. The relative contribution of each mechanism to the overall observed behaviour and
failure mode depends on the material properties and loading conditions in a complex manner. In
the case of ductile films on a hard substrate, coating delamination is coupled to plastic expansion
of the film with the driving force for delamination being delivered via buckling of the film (see
also Marshall and Evans, 1984). On the other hand, coatings on relatively ductile substrates
often 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 than
the coating thickness. They have also reviewed briefly the commonly used test methods for
evaluating 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. In
this model, the elastic energy of the indented coating is approximated by the elastic energy of a
centrally loaded disc. The idea has also been used in cross-sectional indentation by S´ nchez et
a
al. (1999) as a new technique to characterize interfacial adhesion. The proportionality between
the delamination area and the film lateral deflection predicted by the model was confirmed by
the 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 this
purpose, we perform a numerical simulation of the process of indentation of thin elastic film on
a relatively soft substrate with a small spherical indenter. The complete cycle of the indentation
process, both loading and unloading, is simulated. The indenter is assumed to be rigid, the
film is elastic and strong, and the substrate is elastic-perfectly plastic. The interface is modeled
by a cohesive surface, which allows to study initiation and propagation of delamination during
the 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 interfacial
delamination with emphasis on the unloading part of the indentation process and the associated
normal delamination. The interfacial failure during the loading part has been studied by the

51. 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 found
to occur in a tangential mode driven by the shear stress at the interface. It is initiated at a radial
distance which is two or three times the contact radius resulting in a ring-shaped delaminated
area 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 for
the occurrence/suppression this mode of failure, its fingerprint on the load–displacement curve
and provide some quantitative measures about the interfacial strength. The effect of residual
stress in the film and waviness of the interface on delamination will also be investigated. It
is emphasized that the calculations assume that other failure events, mainly through-thickness
coating cracks, do not occur.
4.2 Problem formulation
The 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 and
tangential components and , respectively. The interfacial behaviour is specified in terms © Ω
of a constitutive equation for the corresponding traction components and at the same
i
location. The constitutive law we adopt in this study is an elastic one, so that any energy
dissipation associated with separation is ignored. Thus, it can be specified through a potential,
i
i.e.
2¬ ´ ƒ© «©
ª‚ ª
Ï sƒIŠ·xl ÑИ
$ ( © (4.1)
´

52. 44 Chapter 4
The potential reflects the physics of the adhesion between coating and substrate. Here, we use
the 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
ÔÓÒ
with and the normal and tangential works of separation (
Ï Ï #! Ï © ÙÕ ), and and two i i Ï
characteristics 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 separation
separation of (
Î
with 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 Î
now 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)
Î
Using these along with the definition
a@Ò
ÔÓ b! Ï i —Õ
Ï © a@Ò i
ÔÓ
, we can relate the normal and shear strengths
through ¡ 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-Baqi
and 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 flow
being controlled by the von Mises stress. For numerical convenience, however, we adopt a
rate-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
Æ
for the plastic part of the strain rate, ¢£ ¡ © ¢£ Á ¥ ¢£ £ Ž¢ ¢£
. Here, are the deviatoric components of the
¢£ Á Á
Piola-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 reference
strain rate. In the limit of
± ±
l
, this constitutive model reduces to the rate-independent von
øul
s r
@ÁÆ d
Mises plasticity with yield stress . Values of on the order of
Æt are frequently used for
l dd
metals (see e.g. Becker et al., 1988), so that the value of at yield is within a few percent of
Æ t
Ž
for the strain rates that are encountered in our analysis. The elastic part of the strain rate, ,
«t £ Ž¢ ¥
Á
is 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).
¦

53. 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 Î
Figure 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 their
respective 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 to
have a spherical tip characterized by its radius . The film is characterized by its thickness
¤ Š
and is bonded to a half-infinite substrate by an interface specified above. Assuming both coating
and 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 ž
and thus approaches the half-infinite substrate solution.

54. 46 Chapter 4
The analysis is carried out numerically using a finite strain, finite element method. It uses
a total Lagrangian formulation in which equilibrium is expressed in terms of the principle of
virtual work as
¢£ #¤ ¢£ ¡ Ÿ
¥
†
ª b¤ ª © @§ Ÿ
¬ ¨ ¢ # ¢¤ ƒ® Ÿ ©
°
(4.7) Š s
Here, 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 strains
correspond 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 µ
left-hand side of Eq. (4.7) is the contribution of the interface, which is here measured in the
deformed 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 process
is 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 are
specified with respect to a rotated local frame of reference þ v°
as shown in Fig. 4.1. In the ·SU( üû
$¶( ý
normal direction, the displacement rate is controlled by the motion of the indenter, while in
Á
the 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 precise
boundary conditions only have a significant effect very close to the contact area and do not alter
the 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 indentation
depth 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 on
the indenter; if the normal component of the nodal force is smaller than a critical value, and
the vertical distance between the node and the indenter is positive, the node is released from
contact. 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 significantly
affect 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)
~ ”

55. Delamination of a strong film from a ductile substrate during indentation unloading 47
The 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 Â
$( for 3• —d
ž 1 1 . (4.12)
However, the size will be chosen large enough that the solution is independent from the
ž
precise remote conditions.
4.3 Model parameters
There 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 ‚t
subsequently 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 d66 d © l %d ©
s
elastic 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 s
in 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 i
and 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 normal
failure 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 Õ
implies 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
vgD © ¤
hfe
and most of the results are for a film
thickness
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 e
be large enough that the results are independent of and therefore identical to those for a
ž
coated 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
4
to minimize numerical problems due to plastic incompressibility. To resolve properly the high
stress gradients under the indenter and for an accurate detection of the contact nodes, the mesh
is 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 crack
has 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 then
and the present model is no longer applicable.

56. 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 max
Figure 4.3: Normal stress variations along a perfect interface at the beginning of unloading
hØ)¡ © ¡
(
}| { ) until complete retrieval of the indenter ( d © ¡
).
4.4 Results and discussion
4.4.1 Perfect interface
For the purpose of reference, we first consider a system with a perfect interface, i.e. its strength
is sufficiently higher than the stresses induced by the particular loading. This can be achieved
by 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 interface
during the unloading stage, and in particular the component normal to the interface . Fromt
this, 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
i
indentation 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 is
of course compressive, and almost uniform over the current contact area due to plastic flow in
ö
the substrate. The compressive stress attains a peak value of 4 GPa just outside the contact
region of radius
hØ{ £ 6 ö
}| . Relatively low tensile normal stresses are found beyond the compressive
region, at j}){ £ §
| . This is a result of the resistance of the substrate to the film bending in this
region. It was demonstrated by the authors (Abdul-Baqi and Van der Giessen, 2001) that the
normal 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

57. Delamination of a strong film from a ductile substrate during indentation unloading 49
tends to seek its original flat shape. For the material parameters here this peeling tendency
induces reversed plastic flow in the substrate under the indenter. As this proceeds, the initially
compressive stress evolves into a tensile stress in the interface directly under the initial contact
region (Fig. 4.3). The figure also shows that the tensile area increases slowly in size during the
process 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-
i
ued unloading, the peeling effect causes interfacial tension to develop rapidly, Fig. 4.4a, with
the 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 of
the unloading and located at the symmetry axis.
i
Based on these results, interfacial failure leading to normal delamination may be expected
during the unloading stage when the interfacial strength j}){ t
| is lower than the maximum tensile
stress
j}){ t
| reached at any moment. In the present case, normal delamination is avoided on the
other 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 of
a 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. Several
techniques have been reported in the literature to extract the mechanical properties of both
homogeneous and composite or coated materials from indentation experiments (Doerner and
Nix, 1986; Bhattacharya and Nix, 1988; Gao et al., 1992; King, 1987; Lim et al., 1999). In the
forthcoming section, we will therefore study the interfacial failure process in more detail and
provide some qualitative measures of the interfacial strength.
4.4.2 Finite-strength interface
In this section, and throughout the rest of this paper, we will study interfaces with finite strengths
to allow for interfacial delamination to develop. To demonstrate the effect of the interfacial
failure on the load–displacement data, Fig. 4.5 shows the predicted curves for different values
of interfacial strength j}){ t
| . The rest of the material and geometrical parameters are the same
as 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 delamination
is 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 displacement
curve as a hump where the stiffness becomes negative. This phenomena will be explained in
more detail later in this section. Another characteristic of delamination that can be observed
in the load–displacement curve is the negligible residual indentation depth at the end of the
unloading. In the absence of delamination (case e), the residual indentation depth is more than
half 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 shear
delamination at that stage, as discussed in detail in (Abdul-Baqi and Van der Giessen, 2001).

58. 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 during
unloading. (b) The corresponding location at the interface at which the stress is maximum.
i
In 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 must
be 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 compliance
results 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

59. 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 the
normal 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 loading
stage ( j}){ ¡ © ¡
| ) for the case (c) in Fig. 4.5 with
e s ó© hØ{ t
}|
GPa. The size of the plastic
i
zone at this depth of ŠD © ¢
is about 5 times the maximum contact radius. To illustrate the
delamination process, Fig. 4.6b shows a contour plot of the vertical stress component at the u hu t
end of the unloading process ( d © ¡
). First thing to observe is that the radius of the delaminated
zone, , 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 was
initiated under the retrieving indenter, expanded in the radial direction and was arrested in this
compressive interfacial stress region.
The progressive development of delamination with continued unloading is shown in Fig. 4.7
for 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 the
symmetry 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 relatively
high 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¢

60. 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⁄t
Figure 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.

61. 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 several
values of h){ t
}| (or equivalently ). Ï
i
compressive stress (Fig. 4.6). The final delamination radius is about
es
times the maximum
contact 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 interfacial
strength, 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, where
the 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 figure
that the plastic energy is constant at the initial stage of the unloading, i.e., the initial stage for
the unloading is almost purely elastic. This is in agreement with what is commonly observed
in indentation experiments (Doerner and Nix, 1986). Limited reverse plasticity is seen to have
contributed 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 substrate
to the elastic energy is demonstrated in Fig. 4.8b. The elastic energy of the substrate is seen
to decrease more rapidly compared to the elastic energy of the film at the initial stage of the
unloading. This is in agreement with what is reported in the literature that the initial stiffness

62. 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 than
the 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 a
constant value, whereas the film elastic energy decreases as the film unbends. This indicates
that the main contribution to the energy release, and hence the advance of delamination, comes
from the film. It is also interesting to notice that at the end of the unloading, there still exists

63. 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 opening
of the interface at the initial stage of delamination (Fig. 4.7). As explained in the previous
paragraph and shown in Fig. 4.8, the processes that control the system during delamination are
the unflexing of the coating and the interfacial delamination. The coating evidently provides
a positive contribution to the overall stiffness, whereas the energy release from the interface
gives a negative contribution. This can be seen in Fig. 4.8, where the stiffness provided by each
energy source is the curvature of the corresponding curve. For relatively strong interfaces, the
energy release from the interface dominates during the first stage of delamination when the rate
of 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 (note
that the coating response is constrained by the indenter which is withdrawn at a given rate). It
is this complex interplay between these two terms which shapes the overall behaviour of the
system, including the load–displacement curve.
4.4.3 Comparison with a simple estimate
Deduction of quantitative information about the interfacial strength from indentation experi-
ments, in particular from load–displacement curves and delamination areas, is hindered by the
rather complicated interplay between the film elastic energy and the interfacial energy. A simple
estimate for the work of interfacial debonding from final delamination results has been given by
Hainsworth et al. (1998). This estimate is based on an energy balance involving the interfacial
energy 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$ ¦
ë
in terms of directly measurable quantities.
i
As the model shows a strong dependence on the coating thickness and the maximum Š
indentation depthj})¢
|{ , we have chosen to vary these two parameters over a certain range and
compare the model predictions with our FEM findings. A set of calculations using a conical
p
indenter with a r
semi-angle is also performed to examine the model’s sensitivity to the
indenter’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). The
results of a series computations for two different strengths are summarized in Fig. 4.9a, and
are seen to be consistent with this scaling. The conical indenter results presented in the figure

64. 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 ¤§ vs
coating thickness . Š g 'ë
show the same trend. S´ nchez et al. (1999) have used Eq. (4.13) and a modified version of
a
it on their cross-sectional indentation data, and they have also confirmed the linear relation
between the delamination area and the maximum deflection of the coating. Secondly, according
to (4.13),B ¤§ is proportional to , with all other quantities being the same. Our results, shown
Š 6
in Fig. 4.9b are consistent with this as well. Finally, over the range of of
g 'ë – GPa, dd p d e © ¥
Œ
the proportionality between and is also found to be consistent with the prediction of
§ ¤ Œ ¥
Eq. (4.13). ë

65. 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 for
ige s D © Š
hf ÷
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 . hآ
}| {
÷ ÷
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 for
vge © 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 the
delamination area vs hآ
}| { curve for higher values of interfacial strength, whereas the FEM
results presented in Fig. 4.9a show the opposite tendency.
The more serious limitation of Eq. (4.13) is that the interfacial energy estimated from the
numerical 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

66. 58 Chapter 4
the higher the maximum indentation depth, the better the estimate. This can be understood by
recalling that the model is based on the expression for the deflection of a clamped disc loaded
at the center (Timoshenko and Woinowsky-Krieger, 1959), where the deformation is assumed
to be pure bending. The contribution of the stretching is ignored; this is reasonable when the
radius of the disc is large compared to its thickness. In the case of indentation, this condition
is 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 is
also observed for the conical indenter in Table 4.2, but the quality of the estimate here is even
worse. 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 the
estimate according to (4.13). The same explanation holds here too. Evidently, the assumption
that the disc is clamped at its boundary is questionable. If it is assumed that the disc is simply
6
supported, the expression for
Ž Ï
in Eq. (4.13) must be multiplied by Ib¦ † II†¦ I
$Œ !$Œ †
. This
will 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 the
accuracy 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
to
dd 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 the
substrate, but not the permanent deformation immediately below the indenter; the latter is what
is controlling the delamination radius. However, it should be noted, as will be discussed in the
next section, that the yield stress plays a major role in determining whether delamination will
take place at all.
4.4.4 Critical value of interfacial strength for delamination
Whether 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 the
interface 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 elastic
modulus, the substrate yield stress, the maximum indentation depth and the coating thickness.
For each parameter combination, delamination is suppressed if the interfacial strength is higher
than a critical value hØì{ t
}| .
As an example, Fig. 4.10 shows load–displacement curves for different values of maximum
indentation 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. Lower
indentation 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 yield
stress. Higher values of the coating Young’s modulus , the coating thickness , or the max-
Œ ¥ BŠ

67. 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 of
e 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 the
limitations of the circular disc model pointed out before, these trends are roughly consistent
with 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 in
the coating. The bending moment is proportional to the curvature of the coating which increases
rapidly with the indentation depth until the coating takes the shape of the indenter. After that
point, the curvature does not change much but the bent region propagates outwards and this
corresponds 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 increase
of the bending moment with . For thicker coatings, the critical strength decreases due to the
BŠ
decrease in the coating curvature because the substrate becomes relatively softer. Figure 4.12a
shows an almost linear increase of the critical strength with the coating Young’s modulus. The
increase of the critical strength with the substrate yield stress is shown in Fig. 4.12b. This
t
increase is caused by the reverse plasticity that takes place prior to delamination (Fig. 4.8). The
higher the yield stress, the higher the stresses which can be reached at the substrate. Since the
normal stress is continuous across the interface, higher tensile normal stress can be reached with
increasing , thus making it possible to delaminate stronger interfaces.
t

68. 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).
Š
4.4.5 Residual stresses and interfacial waviness
Coated systems generally contain residual stresses. These are due to the deposition process
itself, 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 introduced
uniform 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 from
d Ö
d
GPa (compressive) to GPa (tensile). Subsequently, we perform the indentation calcula-
tions as before.

69. 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 even
prevent delamination, whereas the opposite happens with tensile stresses. This is explained by
the fact that residual stress will have an out-of-plane component after the deformation of the
coating. In the case of tensile stress, this component will tend to enhance the unbending of
coating during the unloading, and thus will assist delamination. As a consequence, the critical
strength to prevent delamination will increase with residual tension in the coating. Compressive
stress has the opposite effect. For example, a coating of the default thickness of with
ige s D © Š
hf
a interfacial strength of
vge © 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

70. 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 a
sinusoidal wave with an amplitude of
ige
hf 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-free
coating (cf. Tab. 4.1).
Roughness of the interface is commonly simplified by a sinusoidal wave (e.g. Clarke and
Pompe, 1999). To study the effect of roughness on delamination, a wave of an amplitude up to
Š D %d
s ŠD
and a wavelength up to were introduced along the interface, see Fig. 4.13. Delamination
is 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 next
crest/valley. Even though the precise evolution of delamination depends on the waviness of
the interface, for all cases considered here we did not find a significant effect on the critical
indentation depth at which delamination starts, nor on the final delamination radius.
4.5 Conclusions
For the purpose of studying interfacial delamination, numerical simulations have been carried
out 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

71. Delamination of a strong film from a ductile substrate during indentation unloading 63
cohesive 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 in
the 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, driven
by the bending moment in the coating. Normal delamination is recognized by the imprint left
on the load vs displacement curve and the negligible residual indentation depth. For any given
indentation 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 normal
delamination takes place. Compared to normal delamination, shear delamination can occur for
relatively low interfacial strength. Conversely, if the interface strength is high enough to prevent
normal 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 elastic
energy, 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 separation
can be derived iteratively.
The disc model estimate (Hainsworth et al., 1998) has been compared with our numerical
findings 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 parameters
are easily interpreted, whereas the details of this variation are governed by the nonlinear nature
of the problem.
Compressive residual stress in the film delays delamination and, if high enough, it might
even prevent delamination, whereas, tensile residual stress has an opposite effect. Waviness
of 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.

72. 64 Chapter 4
Becker, 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.

73. Based on: A. Abdul-Baqi and E. Van der Giessen, Numerical analysis of indentation-induced cracking of brittle
coatings on ductile substrates, International Journal of Solids and Structures, accepted (2002).
Chapter 5
Indentation-induced cracking of brittle
coatings 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 Introduction
Hard coatings on relatively soft substrates and their industrial applications have been receiving
more 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 of
applications 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 used
as protective layers in many mechanical applications such as cutting tools. Such coatings are
usually 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 such
failure.
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 properties
of 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 characterize
the properties of thin films or coatings. At the same time, for example for hard wear-resistant
65

74. 66 Chapter 5
coatings, indentation can be viewed as an elementary step of concentrated loading. For these
reasons, many experimental as well as theoretical studies have been devoted to indentation of
coated 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 der
Varst and de With, 2001; Malzbender et al., 2000; Wang et al., 1998). The main emphasis
in such investigations has been to extract quantitative data about the coating and interfacial
fracture energies and strengths. Diverse results have been obtained by different studies due to
the complex nature of the problem and the different estimation procedures (van der Varst and de
With, 2001). The complexity is mainly attributed to the fact that such system is a combination of
at 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 of
the 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 was
found to occur in mode II driven by the shear stress at the interface outside the contact region
during the loading stage. During the unloading stage, delamination was found to occur in mode
I 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 the
coating side of the interface and may grow into a through-thickness crack. Li et al. (1997) have
proposed a method to estimate the fracture toughness from indentation tests. From the load
drop or plateau caused by the first circumferential crack, they estimate a strain energy release
to create the crack. The fracture energy is then calculated using the energy release, the crack
area and the elastic properties of the coating. This approach has been also used by others (Li
and Bhushan, 1998; van der Varst and de With, 2001; Malzbender et al., 2000). Van der Varst
and de With (2001) have performed Vickers indentation experiments on TiN coatings on tool
steel. Arguing that the energy release which is estimated from the load–displacement curve is
spent 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
4
used to estimate the energy release from the load–displacement curve is different from the one
suggested 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 and
the 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 without
any 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 circumferential
cracks which initiate from the coating surface, its imprint on the load–displacement curve and
the spacing between successive cracks. Using our numerical findings, the above-mentioned
estimates of the fracture energy are tested. The effect of the material, geometric and cohesive
parameters is also investigated.

75. 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
L
Figure 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 formulation
The system considered in this study is illustrated in Fig. 5.1. It comprises an elastic-perfectly
plastic substrate coated by an elastic thin coating and indented by a spherical indenter. The
indenter is assumed rigid and only characterized by its radius . Assuming both
vxbD © ¤
hfe
coating 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© Š
h and elastic properties ( GPa, d e © Œ ¥ %d © Œ ¦
) and is bonded to the substrate by an
d s
interface. The substrate is supposed to be a standard isotropic elastoplastic material with plastic
flow being controlled by the von Mises effective stress . It is taken to have a height and Žt Š ž
a radius , with large enough compared to film thickness so that the solution is independent
ž ž
of and the substrate can be regarded as a half space; thus, it is only characterized by its elastic
ž 6 6
properties ( 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 is
performed 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§ Â Š
$d( $d( for €ž —3•£
s 1 § 1 (5.1)
Inside the contact area we assume perfect sliding conditions. The boundary conditions are
specified with respect to a local frame of reference that is rotated over an angle such FU( üû
$ý Ï
that is always perpendicular to the coating surface as shown in Fig. 5.1. The definition of this
ý
frame for points inside the material is also shown in Fig. 5.1. The angle is calculated for each Ï

76. 68 Chapter 5
material element in the deformed state (solid line) based on the rotation of the element with
þ°
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 ÿŠ
set 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. The
mesh is an arrangement of four-noded quadrilateral elements. The elements are built up of four
linear strain triangles in a cross arrangement to minimize numerical problems due to plastic
incompressibility. To resolve properly the high stress gradients under the indenter and for an
accurate detection of the contact nodes, the mesh is made very fine locally near the contact area
with 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 indenter
is 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 have
used a threshold value for tensile nodal forces equal to the average current nodal force. A value
that is an order of magnitude smaller did not show any significant effect on the results. After a
node is released from contact, the distance between the node and the indenter is checked each
increment 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
i
components 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 Ï
traction components are determined by two potentials and $ ( Ï
according to
i s Î ‚i¬Ï ´ © «© ( ‚´ © ©
ΠϬ i (5.5)
´ i ´ i
The resulting traction-separation relations in normal and tangential direction are illustrated in
i
Fig. 5.2. In both directions, the peak traction represents the cohesive strength and the area under

77. 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 Î
Figure 5.2: The normal and shear cohesive tractions. (a) Normal traction $ ¡ . (b) Shear
traction $
. Both are normalized by their respective peak values and hi} |){ i hØ{ t
}| .
Î
the curve corresponds to the work of fracture. The work of normal (tangential) separation, ¡ Ï
( ), can be expressed in terms of the corresponding strengths
Ï ( ) as j}){ h){ t
| }|
¤ Î Î `¤ ¡ i
Û © © Ï
D Ï ( hØ{ w$ I
}| t j}){ $ I
| (5.6)
¤ Î ‰¤ Î
where and are two characteristics lengths andÔ%b@!ÓÏ Ò Ï Õ © i a@Ò i
ÔÓ
is a coupling parameter. From
the previous equations, the normal and tangential strengths are related by
i i ¡ `¤ Î
j}|){ ¤ $ I D Õ © j})St
|{ (5.7)
i a@ÒÔÓ

78. 70 Chapter 5
More details of this constitutive model are found in (Xu and Needleman, 1993; Abdul-Baqi and
Van der Giessen, 2001a). The cohesive zones are incorporated in the finite element calculation
using linear two-noded elements with two Gaussian integration points which is consistent with
the 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
i
comparable 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 and
coating were chosen such that, when combined with the work of separation, reasonable values
for 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). The
ratio 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 coating
thickness, the coating Young’s modulus, the yield stress and the coating cohesive parameters
will 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 the
cohesive parameters since, unless otherwise specified, they all will refer to the coating.
5.3 Stress distribution in a perfect coating
Before performing the cracking analysis, it is instructive to have a brief look at the stresses that
generate 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 e
is 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 the
ratio . 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 within
the contact area £ m §
. It is interesting to note in Fig. 5.3(a) the decrease of the compressive
stress in the contact region with increasing indentation depth. This is a consequence of the
plastic flow of the substrate material in the indented region which induces an additional positive
radial straining in the coating, leading to a decrease in the compressive stress at the contact
region. 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 suggests
that 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
ÿt
Fig. 5.3(c). The highest value of this stress is found under the contact edge at . The £ © §

79. 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⁄a
Figure 5.3: Stress distribution at several indentation depths for
D ! ¿© bIŠ
e ¤! . (a) Radial stress
along the coating surface. (b) Radial stress along the coating side of the interface. (c) Shear
stress along the middle plane in the coating (
D IŠ © 
! ).

80. 72 Chapter 5
shear stress has a smaller maximum value than the radial stress and furthermore is located at
the 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 radial
stresses along the surface with those on the coating side of the interface, we see that each tensile
stress is faced by a compressive stress on the opposite side of the coating due to the bending-like
loading. This indicates that a crack, whether initiated at the surface or the interface, will not be
able to propagate through the whole coating thickness. Tensile radial stresses at the surface and
interface are seen to be of the same order of magnitude. But crack initiation and propagation
from the coating side of the interface suffers an extra resistance as this requires some interfacial
shearing (see inset in Fig. 5.1) and hence more energy. Therefore, the main emphasis in this
study 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 on
surface cracks as will be discussed later in this paper.
5.4 Analysis
A 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 to
their 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 cohesive
zones reach the peak traction h)St
}| { and start softening almost simultaneously, which breaks the
uniqueness of the solution. Mesh refinement does not solve the problem. When fine meshes
are used, the region of more or less homogeneous, high tensile radial stresses comprises several
elements so that there is an even larger probability of running into a loss of uniqueness. On
the 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 and
the coating. In addition, embedding of cohesive elements leads to an artificial enhancement of
the overall compliance (Xu and Needleman, 1994). This increase will tend to underestimate the
stresses in the coating and therefore the occurrence of failure. Since the compliance increase is
proportional to the number of cohesive zones, we adopt a procedure in this study in which the
number of cohesive zones is minimized.
The location of the necessary cohesive zones is determined in the following way. We first
conduct a calculation without any cohesive zone and we trace the evolution of the maximum
tensile 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-node
growth of the contact region. After an initial transient, both the maximum stress and its location
seem 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 6t
h 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 placed
at this location prior to indentation. The first crack will then indeed form at the expected inden-

81. 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 'ÿ
maximum.
tation depth and we continue the computation to determine the location of the second crack in
the 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 the
fourth step in which three coating cracks have developed. Figure 5.5(a) shows the distribution
of radial stress at an indentation depth of
vxe y %d
hf s
. The first crack at
h vf sd e © §
is visible
ÿ 'ÿ t
in the figure and a tensile stress is seen to develop in the coating outside the crack radius. At
vxD s o© ¢
hfe e
an indentation depth of
vf c ay
h s , three cracks have already occurred at locations
vf s v© §
h p , 4 s p sd
and , see Figure 5.5(b). The crack at is not visible in the figure since it has
4

82. 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
vgD s
hfe
(b).
ÿ 'ÿ t
closed up. The first crack has also closed from the surface side as a result of the compressive
stress in the contact region. Note that local loss of contact has occurred above the closed-up
crack as explained in section 5.2. The third crack did not reach the interface yet at this stage of
loading due to the compressive stress in this region.
The procedure is further illustrated in Fig. 5.6(a), showing the evolution of the maximum
radial stress along the coating surface. The maximum stress is seen to increase with indentation
depth until the moment of failure where a sudden drop occurs due to stress relaxation and
redistribution. After initiation, the crack propagates through the coating thickness and stops
just before reaching the interface due to the compressive stress at that region. About of € †d

83. 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 'ÿ
dentation depth . The curves belong to four different calculations with the number of cohesive
¢ 6
surfaces (and later cracks) increasing from to . (b) Load–displacement curve for the case
d
shown 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 in
mode I resulting in two small drops in the maximum stress. The location of the fourth crack is
predicted to be
vf %c © §
h s after
vge s © ¢
hfe . The four successive cracks thus found are seen
4 hige s
f
to have an almost uniform spacing of . The dependence of crack spacing on some of

84. 76 Chapter 5
the model parameters will be discussed in section 5.5.
The corresponding load–displacement curve shown in Fig. 5.6(b) reflects these cracking
events by a sudden drop of the load. Each of three major drops on the curve is associated
with the initiation of a crack. Since the fracture energy is proportional to the crack radius, the
larger the crack radius, the larger the drop in the load. With further indentation, the contact
radius increases, thus expanding the tensile region at the interface. The growth of the freshly
nucleated crack to the interface gives rise to small load drops in between the major ones. The
third 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 on
the load–displacement curve at this depth.
These results demonstrate that the procedure described above is indeed capable of predicting
the initiation and growth of coating cracks within the cohesive zone framework. It should be
pointed out, however, that even with this small number of cohesive zones, some numerical
problems were faced at first. Careful examination has shown that these were caused by the
fact that the cohesive zone parameters chosen here approach atomic separation properties. The
coating 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, and
negative, which would lead to local snap back in the finite element system. This can probably
be 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 stiffness
of the cohesive zone as soon as the normal strength is reached, , but to update the
í
tractions directly from (5.5). Of course, this leads to an error in the solution, but this error is
i i
corrected by way of the equilibrium correction in the next time step. Practically speaking, the
effect 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 between
a cohesive zone element and a continuum element is smaller than a few percent, even when the
instantaneous 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 maximum
tensile 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. The
radial stress distribution at an indentation depth of
h if sd
and the evolution of the maximum
radial stress on the coating surface are shown in Fig. 5.7. Cracks at the three specified locations
are seen to have occurred. None of these cracks has reached the surface since at the moment
of initiation of each crack, the opposite side of the coating is already in contact and hence the
stress state is compressive. The stress at the coating surface outside the contact region is mainly
driven by the contact between the indenter and the coating at the contact edge £ © §
. Since all
cracks initiating from the coating side of the interface lie in the region £ m §
, their effect on the
stress outside the contact region is insignificant as shown in Fig. 5.7(b).

85. 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
h
trates the three cracks initiating from the coating side of the interface at locations
vxe s
hf , and sd s j}){
| t
4 . (b) The corresponding maximum radial stress at the coating surface (solid line)
compared with the stress of the non cracked coating (dashed line).
ÿ 'ÿ
5.5 Effect of geometrical, material and cohesive parameters
The 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

86. 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⁄R
Figure 5.8: Normalized crack spacing f#v
$ ¤! versus normalized coating thickness . Single fb'Š
$ ¤!
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.
t
coating 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
¤! ö
variation of the crack spacing with the coating thickness. The relation provides
b«
¤! b'Š 4 s
¤!
the best linear fit. It should be noted that spacing in this figure is taken to be distance between
ö
the first two cracks. From the average spacing we get b«
¤! b'Š 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 and
corresponding 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 the
coating 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 e
any effect on crack spacing. However, the location of the first crack and the corresponding
indentation depth strongly depend on . The radial stress in the coating
Œ ¥ is proportional to
the degree of bending and to Young’s modulus. Therefore, coatings of lower modulus require
ÿ 'ÿ t
more 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 with
indentation depth (Fig. 5.4b), coatings of lower Young’s modulus crack at larger radii as seen in
Fig. 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

87. 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 influence
on 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

88. 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
6
depth. 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 d
are 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 s
cumferential cracks of radii of , and , respectively. The corresponding indentation
4
depths did not deviate significantly from the reference case, where . The crack
vxe %d © ¢
hf s
spacing increased to
h
for
vf sc e %d © Æ t
GPa, whereas it decreased to
s in the case of
vxD s
hf
sd 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 the
coating outside the contact region. Therefore, lower yield stresses tend to lead to larger radial
location of the maximum stress on the coating surface and therefore a larger crack radius. The
corresponding 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 crack
and the crack spacing is also investigated. In the case of perfect sticking contact conditions, the
first crack occurred at a radius
ige s e © §
hf
and a corresponding indentation depth ,
if c e %d © ¢
h s
instead 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-induced
cracks in titanium nitride coatings on stainless steel substrates. They have used a Rockwell
B indenter (
ige by © ¤
hf c ) and coating thicknesses of , and s D 4 ad
. They have found
s igD s
hf
4
that cracks are almost uniformly spaced which is consistent with our numerical results. The

89. 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 i
crack spacings observed for the various thicknesses were found to be
6 , and sd e e s
,
vf d %d
h s
respectively, 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 several
reasons for this discrepancy. One reason is the presence of residual compressive stresses of – ãy%d
s
es 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 friction
between the indenter and the coating which is usually present in indentation experiments. In
the presence friction, the crack spacing increases as mentioned previously. In addition, the
material parameters chosen in this study are not precisely those of titanium nitride coatings on
stainless steel substrates. Variation of these parameters has been shown above to influence the
crack spacing.
The coupling and reversibility in the cohesive law are also investigated. In the original
Xu-Needleman formulation (Xu and Needleman, 1993), the constitutive law of the cohesive
zone is assumed to be reversible, so that the loading and unloading paths are the same. In this
case, if a crack closes it heals and the system recovers all the energy consumed. Even though
this energy is very small in comparison with the plastic dissipation, the effect of taking it into
account is studied by using an irreversible version of the tractions. Therefore, we modify the
constitutive law by introducing a secant unloading branch as illustrated in Fig. 5.11. If the
opening velocity changes sign (unloading), the traction is assumed to decrease linearly to zero
as the opening diminishes to zero (Camacho and Ortiz, 1996). The same method is applied to
the shear traction. It should be noted that in the presence of coupling between the tractions, the
formulation for irreversibility would not be clear. Therefore, the effect of reversibility is studied
by comparing the results achieved when using reversible and irreversible uncoupled tractions.

90. 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 cohesive
zones compared to the coupled ones.
On the other hand, the effect of coupling is investigated by studying the difference between
using coupled and uncoupled reversible tractions.
¬ © © Î ›¬
The results of coupled reversible tractions which has been used so far are compared to the Î ‚©
¬ ›¬ ‚©
Î Î
results 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
h
predicted as explained in section 5.4. The crack spacing is found to be , which is less than
hvfge s i
the value of found previously in the presence of coupling. Moreover, additional cracks
occur at smaller indentation depths compared to the corresponding ones in the presence of
coupling. This is demonstrated in Fig. 5.12. When a crack opens, the shear strength is reduced
in the presence of coupling as demonstrated in Fig. 5.2(b). The moment the crack reaches the
interface, 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 the
uncoupled tractions where shearing off does not occur, indenting to larger depth is required to
arrive to the same degree of bending and form an additional crack. Moreover, larger indentation
depth 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 tractions
does not seem to have any significant effect on the results, even though we have noticed crack
closure taking place (e.g. Fig. 5.5b). Uncoupled reversible and irreversible tractions lead to
almost indistinguishable load–displacement curves and to the same crack spacing, .
vf 6 s v©
h

91. 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 the
energy release associated with a cracking event.
e
5.6 Fracture energy estimates
Several researchers have tried to estimate the fracture energy from indentation experiments. One
of the approaches which has been used is to estimate the energy consumed to create the first
circumferential crack from the corresponding load drop or plateau on the load–displacement
curve (Li et al., 1997; Li and Bhushan, 1998; van der Varst and de With, 2001; Malzbender et
al., 2000). In this section we will compare the prediction of these methods with our numerical
findings.
¬
Li et al. (1997) have suggested an approach which is based on the idea that all the energy
released 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 i
literature 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 curve
and 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
¨ B
fracture mechanics for cracking under constant load and constant displacement (indentation

92. 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
e
single points are numerical results and lines are linear fits.
i
depth), respectively (Kanninen and Popelar, 1985).
We now use our numerical simulations to test the accuracy of the estimates obtained from
Eq. (5.8). Obviously, the values of the energy release estimated in the three methods are com-

93. Indentation-induced cracking of brittle coatings on ductile substrates 85
pletely 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 and
Š. At low values of Ï or , the first method underestimates the fracture energy, whereas the
Š
second and third methods give reasonable estimates. On the other hand, at large values of
i Ï i
or , the second and third methods overestimate the fracture energy, whereas the estimation by
Š
the first method seems reasonable. Therefore, estimations of the coating fracture energy from
i
indentation experiments might lead to values different from the actual ones by as much as one
order 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 the
plastic zone in the substrate. Nevertheless, it is interesting to note the proportionality between
the normalized estimated values and the actual ones and the coating thickness.
5.7 Concluding remarks
In 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 simulated
by 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 and
modeled by means of cohesive zones. The interface between the coating and the substrate was
also modeled by means of cohesive zones but with interface properties.
The emphasis was on circumferential cracks which initiate from the coating surface outside
the contact edge and advance towards the interface. The initiation and advance of such cracks
is imprinted on the load–displacement curve as a sudden load drop (kink). The larger the crack
radius, the larger the load drop, since the energy released by the crack is proportional to the crack
area. After initiation, the crack advances almost spontaneously across the coating thickness and
stops just before reaching the interface as a result of a compressive stress at that side. With
further indentation, the tensile region under the indenter expands and when it encompasses the
crack 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. For
several coating thicknesses, the ratio of the crack radius to the instantaneous contact radius was
found to increase linearly with the coating thickness. The smaller the coating thickness, the
closer this ratio is to unity.
With further indentation, successive cracking occurs at larger radii, which each additional
crack also imprinted on the load–displacement curve as a kink. The crack spacing was found
to be predominantly controlled by the ratio of the coating thickness to the indenter radius. For
a fixed indenter radius, the crack spacing is of the order of the coating thickness. Variation of
the coating Young’s modulus did not show any influence on crack spacing, whereas spacing
increased with the coating strength and decreased with the substrate yield stress. Coupling and
irreversibility of the cohesive tractions were also investigated. The presence of coupling was
found to slightly increase crack spacing. Moreover, additional cracking occurs at higher inden-
tation depths compared to uncoupled tractions. Reversible and irreversible cohesive tractions

94. 86 Chapter 5
have led to almost indistinguishable results in terms of crack spacing and load–displacement
data. The predicted crack spacings are consistent with those obtained in recent experiments by
Souza et al. (2001).
Finally, it has been shown that estimation of the coating fracture energy from the first crack
area 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 is
that a small portion of the energy release is consumed by the crack, while the rest of the energy
is consumed by the accompanied plastic dissipation in the substrate.
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96. 88

97. Summary
The response of coated materials to external loading such as indentation is far more complicated
compared to the response of bulk materials. For coated materials, the overall response of the
system 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 a
key parameter in determining the influence of each of the constituents. Initially, when this ratio
is 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 overall
response is dominated by both constituents in a complex manner. Consequently, extraction of
material parameters from indentation experiments in hindered by such complexity. For more
accurate interpretation of indentation experiments, the main emphasis in this study, however,
was to numerically investigate some of the possible indentation-induced failure modes using
an axisymmetric finite element formulation. Failure by interfacial delamination and coating
cracking 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 this
type of failure is very sensitive to the degree of coupling between the interface response in the
normal and tangential directions. During the unloading stage, delamination was found to occur
in a normal mode, where a circular portion of the coating, directly under the contact region, is
lifted off from the substrate. Normal delamination is driven by the coating energy as it unflexes
to retain its original configuration and the mismatch in the mechanical properties of the coating
and substrate. The presence of compressive residual stress in the coating was found to delay
normal delamination and, if high enough, it might even prevent it, whereas tensile residual stress
has an opposite effect. Delamination is imprinted on the load–displacement curve by a rather
sudden 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 the
interface forming through-thickness circumferential cracks. The initiation and propagation of
each 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 thickness
and the indenter radius. For a fixed indenter radius, the crack spacing scales with the coating
thickness. Coupling and irreversibility of the cohesive tractions were also investigated. The
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98. 90
presence 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 spacing
and load–displacement data.
Some of the common methods which are used to estimate the interfacial energy and the
coating fracture energy from indentation experiments were also investigated. It has been found
that estimation of the interfacial energy from the radius of the delaminated area and estimation
of the coating fracture energy from the first crack area and the corresponding load drop, lead to
values that differ from the actual ones by as much as one order of magnitude. The difference
is mainly attributed to the fact that, in a highly nonlinear problem, these methods oversimplify
the estimation of the energy release associated with the failure event. In addition, such simple
models do not take into account the plastic dissipation in the substrate which has been found to
accompany failure.
The indentation method has proven to be a powerful tool for characterizing the mechanical
properties of materials and in the case of coated materials, it maybe considered as one of the
best available methods. This is mainly owed to two reasons; firstly, the relatively easy sample
preparation and secondly, the various types of failure which may occur, and consequently be
investigated, as a result of the heterogeneous stress field it generates. Moreover, besides being a
test method, indentation can also be viewed as one of the possible external loadings the system
maybe subjected to during service. However, the interpretation of the outcome of this method
to extract more accurate quantitative data about the coating, the interface and the substrate, still
requires more theoretical as well as experimental investigations.

99. Samenvatting
Het 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 veel
kleiner 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 tot
gevolg dat het afleiden van materiaal parameters uit een indentatie experiment gehinderd wordt
door 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 door
het 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 deze
manier van bezwijken erg gevoelig is voor de mate van koppeling tussen het gedrag van de
interface in normale en tangentiele richting. Tijdens de ontlastingsfase vindt delaminatie plaats
in een loodrechte toestand, waar een circelvormig deel van de coating recht onder het contact
gebied los komt van het substraat. Loodrechte delaminatie wordt gestuurd door de energie in de
coating, wanneer deze relaxeert naar de oorspronkelijke toestand en door de mispassing in de
mechanische eigenschappen van de coating en het substraat. Het blijkt, dat een achter gebleven
compressieve spanning in de coating vertragend werkt op loodrechte delaminatie en wanneer
het hoog genoeg is, delaminatie zelfs voorkomt. Terwijl achtergebleven trekspanningen een
tegenover gestelde werking hebben. Delaminatie is herkenbaar in een spannings-rek kromme
door een vrij plotselinge afname van de indentatie stijfheid.
Er is ook aangestoond dat het breken van de coating voorkomt tijdens de belastings fase
van indentatie. Scheurtjes beginnen aan de oppervlakte buiten het contact gebied en groeien
naar de interface, waarbij ze in de dikte richting ringvormige scheuren vormen. Het ontstaan
en de voortbeweging van elke scheur staat in de spannings-rek kromme als een kink. Er is
gevonden, 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 de
scheuren met de coating dikte. Koppeling en onomkeerbaarheid van cohesieve tracties zijn ook
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100. 92
onderzocht. Aanwezigheid van koppeling blijkt een lichte toename van de ruimte tussen de
scheuren te geven. Omkeerbare en onomkeerbare cohesive tracties hebben geleid tot bijna niet
te 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 de
breuk energie van de coating te bepalen. Er is gevonden dat bepaling van de interface energie
uit de straal van het delaminatie gebied en de bepaling van de breukenergie van de coating
uit het gebied van de eerste scheur en de bijbehorende daling in de belasting, resultaten geven
die 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 van
de energie die vrij komt bij zo’n proef. Daar komt bij, dat deze methodes geen rekening houden
met 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 eenvoudige
preparatie 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 dat
gegenereerd 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 verkrijgen
over de coating, de interface en het substraat, vereist nog meer theoretisch en experimenteel
onderzoek.

101. 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.
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102. 94

103. 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.
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104. 96

105. Curriculum Vitae
The 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 Mechanical
Engineering Faculty at Birzeit University, Palestine. The study came to an end after less than
two months when the university was closed.
In 1990 the author was admitted to the Yarmouk University in Jordan where he obtained
his Bachelor degree in Physics in the year 1993. In 1994 the author received a scholarship
from Bergen University in Norway where he worked under the supervision of Prof. Ladislav
Kocbach in the theoretical atomic physics group. After the defense of his thesis Atomic physics
aspects of radiation physics and passing his final exam, the author was awarded a Master degree
in physics in the year 1996.
The author started his PhD work in August 1997 with Prof. Erik van der Giessen in the
micromechanics of materials group at Delft University of Technology. The work was on the
project Adhesion of brittle coatings on ductile metallic substrates. Coming from a different
field, the author had initially to get familiar with the concepts of mechanics, the finite-element
method and the related numerical tools. This has been accomplished successfully with the
guidance 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 at
Eindhoven University of Technology in September 2001 to work on fatigue damage modeling
in lead-free solder joints.
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106. 98

107. Acknowledgement
I would like to acknowledge the supervision of Erik van der Giessen in the course of preparation
of this manuscript. Your decision to offer me a doctoral position at the micromechanics of
materials group, despite my different background, is really appreciated. It was a pleasure and a
privilege to work with you.
I thank all my colleagues at the micromechanics of materials group for providing me a
perfect working atmosphere. I especially name Harko, Martin, Sabine, and Sumit. Thanks for
the scientific and social interesting discussions. I acknowledge Jan Booij for his assistance in
computer-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 and
conversations about almost every thing in life has always made your company very interesting
for me.
Last but certainly not least, I wish to express my sincere gratitude to my family and most
notably to my mother for the devotion of her life to her children until her early death. I will
never forget the support of my father which was, and still is, of vital importance for my personal
and academic advancement.
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