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

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.
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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