Deformation And Fracture Analysis Of Coating-Substrate Systems

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A R C H I V E S O F M E T A L L U R G Y A N D M A T E R I A L S

Volume 60

DOI: 10.1515/amm-2015-0378

M. KOT* #

2015 Issue 3

DEFORMATION AND FRACTURE ANALYSIS OF COATING-SUBSTRATE SYSTEMS

ANALIZA ODKSZTAŁCENIA I PĘKANIA W UKŁADZIE POWŁOKA-PODŁOŻE

The paper presents the deformation and fracture analysis of coating-substrate systems during spherical indentation. CrN and
TiN ceramic coatings with a thickness of 1-5 μm were tested using 10 to 200 μm tip radius spherical indenters. The typical results
of indentation tests i.e. force-penetration depth curves were transformed into stress-strain curves using an algorithm developed by
the author. The test results are compared with the results of numerical analysis conducted using FEM modelling. Such a complex
analysis allows users to determine the level of tensile stress leading to the formation of cracks observed using SEM and TEM
microscopy, and to define the failure maps for the coating substrate-systems.

_Keywords:_ coatings, deformation, stress, fracture

W pracy przedstawiono analizę deformacji i pękania układów powłoka-podłoże przy użyciu testów indentacji i sferycznych
wgłębników. Badano powłoki CrN i TiN o grubościach z zakresu 1-5 μm stosując wgłębniki o promieniach zaokrąglenia z zakresu
10 do 200 μm. Typowe wyniki testów w postaci krzywych siła-głębokość penetracji transformowano na krzywe naprężenie-odkształcenie przy użyciu algorytmu opracowanego przez autora. Wyniki porównywano z wynikami modelowania przy użyciu metody
elementów skończonych MES. Taka procedura pozwoliła określić stan naprężeń rozciągających prowadzących do powstawania
pęknięć obserwowanych przy użyciu mikroskopii skaningowej SEM i transmisyjnej TEM. Opracowano także mapy deformacji
badanych układów powłoka-podłoże.

1. Introduction

Thin, hard surface layers and coatings are increasingly used
in mechanical engineering, mainly in order to reduce wear and
minimize friction. Many of the coatings are deposited by PVD
and CVD techniques, and typical materials are carbides or nitrides
of transition metals or carbon with different _sp_ 2 to _sp_ 3 bonds
ratio. Ceramic coatings are characterized by high hardness, high
elasticity modulus and in many cases, high wear and corrosion
resistance. The great interest in carbon coatings derives from
their excellent tribological properties. Their low coefficients
of friction in contact with a number of engineering materials is
primarily a result of the formation of a thin graphite tribolayer
on the surface, which significantly reduces friction [1]. However,
the main problem of both these groups of coatings is their low
fracture resistance, limiting their application especially under
dynamic or impact loading. The value of their fracture toughness

_KIC_ = 1-10 MPa · m 1/2 is tens of times lower than for metal alloys. Their fracture resistance can be improved by modifying
the architecture and microstructure, as is the case for multilayers [2,3], and nanocomposite coatings [4,5], characterized by
significantly better mechanical properties over single coatings.
Multilayer coatings can consist of various types of materials,
while for machine parts mainly ceramic/metal ones, such as
Ti / TiN [6] and Cr/CrN [2,7], are proposed. The hard ceramic
layers are responsible for the tribological properties and metal
layers reduce stress and improve adhesion to the substrate.
Microscopic examination showed that such multilayer coatings
exhibited improved fracture resistance than single ceramic coatings. This is due to the interaction between the layers that leads
to the specific failure mechanism. Microcracks appear on the
surface of the ceramic layers and propagate through their whole
thickness towards the interface with soft layers, where they are
stopped in locally yielded metal layers [8]. Crack initiation in the

- AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF MECHANICAL ENGINEERING AND ROBOTICS, AL. A.MICKIEWICZA 30, 30-059 KRAKOW, POLAND

# Corresponding author: kotmarc@agh.edu.pl

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next ceramic layer when stress reaches a critical level requires
a rise in load. The total effect of fracture toughness enhancement
of a ceramic/metal multilayer could be significant considering
that these metal layers may number even several dozen. A similar
effect was also observed for the multilayers with hydrogenated
carbon and thin titanium layers TiN / Ti / a-C:H [9]. However, in
the nanocomposite coatings, such a barrier to easy crack propagation in a usually amorphous a-C:H, a-C, a-Si3N4 matrix are
carbides or nitride nanoparticles WC, CrC, TiN [5,10,11]. The
optimization of nanocomposite microstructure was presented in
a previous paper [4]. However, even the best coating will not fulfil
its functions if the substrate does not provide adequate support.
The large contact stresses, being a result of external loads, can
lead to local plastic deformation of the substrate. The development of a plastically deformed zone is responsible for the large
deformation of the whole coating-substrate system and, in the
case of stiff and low fracture resistant coatings, for their fracture,
chipping and premature wear [12]. To prevent this phenomenon,
additional treatments are applied for substrate strengthening
i.e. nitriding or hardening [13,14] and for surfaces with further
coating deposition these technologies are called duplex. Unfortunately, despite the rapid development of deposition technology
and design of the newest groups of sophisticated coatings, the
problem of the coating selection to specific applications has still
not been solved. This is due to the lack of mathematical equations
for determining the load bearing capacity of coated surfaces. For
homogeneous materials, engineers use the Hertz theory [15,16],
which allows them to calculate the stress field, deformation and
predict the yield strength when load and contact geometry are
known. However, there are many assumptions in this theory that
preclude it for the coating-substrate systems where the coating
has different properties than the core material. In addition, local
plastic deformation may occur in metal substrates at high contact
loads. There are some studies where the authors developed Hertz
equations for coated components [17]. Their essence is the introduction of additional correction factors that depend on the contact
geometry, coating and substrate properties and coating thickness
to well known equations valid for homogeneous materials.

The mechanical testing of coating-substrate systems is very
commonly conducted using an instrumental indentation method,
where load and penetration depth are continuously measured
while pressing an indenter with a defined geometry into the
surface. The obtained indentation load-penetration depth curves
allow users to calculate the hardness and elasticity modulus of the
tested samples [18]. The obtained results correspond to coating
properties at small deformations and loads usually in a range of
several milinewtons.

But still the problem is how to determine the fracture toughness of coatings, therefore this parameter for brittle coatings is
extremely important. In the literature, there are some techniques
for determination of a coating’s fracture resistance, like threepoint bending [19], buckling [20], and uniaxial stretching [21].
Each of them has its own limitations, but the techniques most
often used for calculating the _KIC_ parameter are indentation and
scratch tests. Many equations are given for _KIC_ determined from

where _E_ is Young’s modulus, _ν_ - Poisson number, 2 _πCR_ crack
length and _t_ is a coating thickness.

However, for thin coatings, it is difficult to exclude the
influence of the substrate on the measurement results because
the deformation of the system to some extent are restricted by
the substrate and the cracked coating is still supported on the
substrate. On the other hand, from the designer’s point of view, it
is necessary to know how the whole system, not the coating itself,
deformations are acceptable and what the load is at which a specific form of the coating-substrate system destruction appears.
In the face of many problems encountered in the construction
of analytical models for the analysis of deformation and stress
distribution in coating-substrate systems, the numerical method
based on finite element modelling (FEM) is a very effective and
commonly used tool. Most frequently indentations of coatings
deposited on elastic or elastic-plastic substrates [23,24], and
even coatings with a complex structure as multilayers [25,26]
are modelled. The aim of these studies is to find the critical
loads leading to the plastic deformation of the substrate, coating
fracture or delamination.

The author of the paper proposes to analyse the deformation
of coating-substrate systems method based on complex analysis
of indentation curves, their transformation into stress-strain
_σ_ curves and comparison with modelling results (algorithm in
Fig. 1). These _σ – ε_ curves are plotted by measuring the contact
radius _a_ C corresponding to the actual load _F_ and calculated from
the following equations [27]:

- - _pm_ - �� _FaC_ 2, - _i_ - 0,2 _Riac_, _aC_ - 2 _hC_ _Ri_ - _hC_ 2 (2)

where: _Ri_ - indenter tip radius, _hC_ - contact depth.

Comparing the mean pressure, stress distribution and
concentration areas, the maximum tensile stress level leading
to brittle failure of the coating can be determined. It should be
noted that residual stresses, usually compressive in PVD deposition processes for ceramic and carbon coatings (up to a few
GPa [28,29]), must be added to the stress field derived from
external loading. The presented procedure for analysing the
mechanical properties of coated surfaces, including analysis of
the experimental and modelling results, allows to create failure
maps for the coating-substrate systems, and can be helpful in the
selection of the coating and its thickness optimal for a particular
application. To break free from a particular contact geometry,
such maps are created in the relative thickness _t_ / _Ri_ ( _t_ - thickness

indentation experiments but all of them link material properties,
load and length of cracks on the coating surface. Analysis of
indentation curves makes it possible to determine the amount
of energy dissipated during the crack formation _U_, which can
be seen in the curves as a sudden increase in penetration depth
called pop-in. Li et al [22] proposed the relation for fracture
toughness _KIC_ of thin films as:

1/2

�� �� _E_ ��� _U_ ��

- ������ �1� _v_ 2 - 2� _CR_ ����� _t_ �����

_K_ _IC_ - ����� �1� _v_ 2 _E_ - 2� _CR_ ������� _Ut_

_IC_ - ��� 1� _v_ 2 2

_R_

(1)

1� _v_ 2

of the coating, _Ri_ - the radius of the indenter) and the relative
load _F_ / _Ri_ 2 ( _F_ - critical load leading to the characteristic forms

of destruction) coordinates.

2. Experimental part

TiN and CrN coatings were deposited by unbalanced DC
pulsed magnetron sputtering. Prior to deposition, 25×25×1 mm
X10CrNi18-8 austenitic steel substrates were cleaned in an industrial washing machine (Miele, Guetersloh, Germany) with
tenside washing agents, dried and mounted on a 3D rotable planetary in the industrial-like R&D deposition equipment (Leybold
Vakuum, Cologne, Germany). After pumping to high vacuum
conditions (2×10 –5 mbar), ion plasma treatment was performed
without prior heating of the chamber and substrates at room tem

2309

perature. Titanium nitride and chromium nitride were deposited
using pure titanium and chromium targets (RHP Technologies,
Seibersdorf, Austria) in an Ar-N2 gas flow. A suitable thickness
of the coatings was obtained by changing the time of deposition,
while the growth rate was previously found for thicker coatings.
CrN coatings 1 and 5.2 μm and TiN 1 and 2.4 μm thick were
tested by instrumented indentation techniques to analyse the
deformation and fracture of coating-substrate systems. Tests
were conducted with different spherical indenter radii _Ri_ = 10,
20, 50 and 200 μm. These indenters allowed users to obtain the
various states of deformation and stress distribution, dependent
mainly on the relative film thickness _t_ / _Ri_ as shown in previous
works [12,27]. Within the analysed indenters and coating thickness, this parameter varied inside the range 0.005 ≤ _t_ / _Ri_ ≤ 0.5.
Indentation curves were analysed using the algorithm shown in
Figure 1 and described above.

Fig. 1. Diagram of a complex analysis of spherical indentation using the experiment and FEM modelling results

2310

Numerical experiments were also conducted using FEM
and the ANSYS 15 software. A 2D axisymmetric model of the
coating-substrate system and indenter with a fixed finite element
mesh is shown in Figure 2.

Fig. 2. Part of finite element model of indenter pressed into coated
substrate, with the refined meshes in the region around contact zone

The boundary conditions were specified so that the model
matches the real indentation experiments on the coating-substrate
system. Nodes in the bottom plane of the model were fixed assuming the deformation _uy_ and _uz_ = 0, while nodes on the symmetry axis of the indenter (left side) cannot move in the radial
direction _uy_ = 0. The size of the model means that the elastic
deformation in the substrate did not reach the bottom and right
plane of the model, hence it may be considered as infinitely
large. Both dimensions of the model were over 40 times higher
than the maximal radius of the contact area. A very fine mesh,
an 8-node element “solid structural 2D PLANE183” [30] with
additional nodes on the sides of the element, were used in the
coating and the substrate near the contact zone for accurate
analysis of strain and stress fields where the stress concentration
was expected (Fig. 2). Even the thinnest coating had 10 nodes in
the z-direction, while for thicker coatings the number of nodes
was higher. Normal load was gradually applied on the top of the
indenter with at least 50 steps during loading and unloading up
to a maximum selected value and down to zero. FEM analysis
was conducted to determine the effect of the contact geometry
and material properties on:

- stress distributions at different ranges of coating-substrate
system deformation,

- areas subjected to maximum radial stress concentration
leading to coating fracture.

3. FEM modelling results

The action of the indenter on a coating-substrate system is
different for thin and thick coatings. Thick coatings substantially
reduce the deformations of the whole system, as is shown in the

indentation curves for 1 and 5 μm thick coatings and a 20 μm
indenter tip radius (Fig. 3).

Fig. 3. Modelled indentation curves of coating-substrate systems with
different _t_ / _Ri_ ratio

For a 1 μm coating ( _t_ / _Ri_ = 0.05) the maximum penetration
depth is 2450 nm, while for 5 μm ( _t_ / _Ri_ = 0.25) it is reduced to
1000 nm. In comparison, the penetration depth for an uncoated
substrate is 2700 nm. Initially, in the elastic state of deformation,
the radial stresses depend mainly on the elasticity modulus of the
coatings. The rise in coating stiffness results in a smaller contact
radius and contact area between the coating and indenter and
hence higher contact pressure. Figure 4a shows the change in
radial stress on the coating surface at a 1 mN load. An increase
in the coating elasticity modulus _Ec_ also leads to an increase in
maximum tensile stress at the surface of the coating from 280
to 440 MPa for _Ec_ = 210 and 630 GPa, respectively. At small
loads the increase in coating thickness from 1 to 5 μm does not
greatly affect the stress distribution in the contact area (Fig. 4b).
Meanwhile tensile stress grows from 340 to 410 MPa (detail
Fig. 4b).

Significant differences in the contact mechanics of coatingsubstrate systems appear at higher loads when the substrate is
locally plastically deformed below the contact zone [31]. The
analyses presented in the previous work have shown that the
substrate yield occurred at _FPL_ load [27]:

2 _Y_ 3

_FRPli_ 2 - _A_ - _YS_ - ��� _EECS_ - _B_ ������ _Rti_ ��� - 21,08� _EY_ 2 _S_

- _A_ - _YS_ - ��� _EECS_ - _B_ ������ _Rti_ ��� - 21,08� _EYind_ 2 _S_ 3 (3)

_Pl_ 2 - _A_ - _YS_ - �� _EEC_ - _B_ ���� _Rt_ �� - 21,08� _EY_ 2 _S_

_i_ - _ES_ �� _Ri_ - _Eind_

where: _YS_ - substrate yield strength, _Ri_ - indenter radius, _E_ elasticity modulus, _A_ and _B_ - coefficients determined from the
FEM modelling. Meanwhile, the reduced elasticity modulus

1
_Eind_ is calculated from the equation _Eind_ - 1 2 1 2

_E_

- . The

1�� _D_ 2 - 1�� _S_ 2

_ind_ - 1�� _D_ 2 1

_D_ 1�� _S_

_ED_ _E_

_D_ _ES_

subscripts _c_, _s_ and _d_ denote coating, substrate and diamond indenter. For an uncoated substrate the yield can be predicted from
the equation which is the second part of the sum in equation (3):

Fig. 4. Radial stress distributions in elastic deformation regime
for: a) 2 μm coatings with different elasticity modulus, b) coatings
_E_ = 420GPa with different thickness _t_

_Fpl_ _YS_

_pl_ - 21,08� _YS_
2 _E_ 2

_pl_ - 21,08� _S_ (4)

_Ri_ 2 _Eind_ 2

2311

Fig. 5. The surface of the coating-substrate systems profiles under 1N
load

This large substrate deformation induces high tensile stresses at the surface and above the coating strength to its fracture.

Fig. 6. Analogy of the coating-substrate system deformation b), to the
deformation of circular plate with constant load acting on fixed radius _a_ ).

_i_ _Eind_

The values of the _A_ and _B_ coefficients vary from 0.85 to 1.1
and from 0.5 to 1.2 respectively [27]. Higher values correspond
to smaller values of _Ec_ / _Es_ (the ratio of the coating and substrate
elasticity modulus) and higher yield strength. Initially, for
_t_ / _Ri_ < 0.01 coatings, it does not increase the load bearing capacity
of the surface. FEM results indicate that the stress distribution in
the system with such a thin coating is practically the same as for
the uncoated substrate. Above this relative thickness, the coating
starts to play a greater role and the substrate no longer has to carry
the whole load. For all the analysed coating-substrate systems,
substrate yield occurred at _h_ max / _t_ = 0.004-0.04. Equation (3) indicates that _Fpl_ increases with the square of coating thickness. The
plastic deformation of metallic substrates at higher loads leads
to the formation of pile-up around the contact area. This pile-up
is greater for lower relative coating thickness _t_ / _Ri_ . In Figure 5
there is a large pile-up for the thick coating _t_ / _Ri_ = 1/20, and its
lack for the thin coating _t_ / _Ri_ = 1/200 at the 1N load.

2312

The stress maxima correspond to the minima of radius at
which the coating is bent. From a mechanical point of view, the
stress state is similar to the deformation of the circular plate fixed
at its ends – radius _b_ and loaded by _pm_ pressure in the radius _a_
(Fig. 6a). Stress on the upper and lower surface of the plate can
be calculated as:

2
2
_r_ - _a_

2
2 -
- _R_ - _bottom_ - - 3� 4 _pm_ - _t_ 2� _ac_ ���1�� _c_ - - (2 _ln_ _raplc_ - ���� _raplc_ ���� �1 ��

2 -
3� 4 _pm_ - _t_ 2� _ac_ ���1�� _c_ - - (2 _ln_ _raplc_ - ���� _raplc_ ���� �1 ��

_R_ - _bottom_ - - 3� _pm_ 2� _ac_ ��1�� _c_ - - (2 _ln_ _rapl_ - ��� _ac_

- _bottom_ - - � _m_ 2� _c_ ���1�� _c_

_c_ - _rpl_

2 2 - 2

_a_ - _a_

2 2 - 2
- _R_ - _top_ - - 3� 2 _pm_ - _t_ 2� _ac_ ��� _atc_ �����1����� _raplc_ ���� ��

(5)

_pl_

_R_ - _top_ - - 3� _pm_ 2� _ac_ �� _ac_ ���1��� _ac_

- _top_

2 2
3� _pm_ � _ac_ - _ac_ ��1
2 - _t_ 2 �� _t_ ��

For the coating-substrate system it can be assumed (Fig. 6b)
that _b_ corresponds to the radius of the plastically deformed zone
in the substrate _rpl_ and _a_ corresponds to the radius of the contact
zone _ac_ . The larger stress is mainly determined by the _ac_ / _t_ and
_ac_ / _rpl_ ratios. The radial stress distribution at a 1 N load is quite
different for thin and thick coatings (Fig. 7).

Fig. 7. Radial stress distribution in: a) thin _t_ / _Ri_ = 0,005, b) thick
_t_ / _Ri_ = 0,25 coatings

The maximum of the tensile stresses is just outside the
contact area on the surface of the thin coating. In the coatingsubstrate interface, the highest stress is in the indenter symmetry
axis, but its level is lower than in the surface. For thick coatings the maximum stress is initially on the coating surface, like
in the case of thin coatings. However, with the load increase,
maximum stress concentration moves to the interface with the
relative penetration depth _h_ max/ _t_ - 0.03 (Fig. 8).

Fig. 8. Maximal radial stress on the top (top) and in the coating substrate
interface (bot) for coatings two times stiffer than substrate _E_ c/ _E_ s = 2
with thickness _t_ = 1 and 5 μm, during spherical indentation with indenter
tip radius _Ri_ = 20 μm

FEM modelling results show that this deformation corresponds to the moment when the radius of the yielded zone _rpl_
in the substrate is about 2 times greater than the radius of the
contact area _ac_ . Therefore, the range of substrate plastic deformation seems to be crucial for areas subjected to fracture. For
thick coatings, the maxima of the tensile stresses in the coating
surface appears at a large distance from the contact surface.
This is confirmed by SEM observations where the circumferential cracks appear at even 3-5 times greater distance than the
contact radius. Such cracks are presented in SEM images of
a 2.4 μm thick TiN coating after indentation with _Ri_ = 200 μm
( _t_ / _Ri_ = 0.012), and _Ri_ = 20 μm ( _t / Ri_ = 0.12) indenters – Figures 9a and 9b respectively.

Fig. 9. Indents images after spherical indentation: a) SEM TiN = 2,4
_Ri_ = 200 μm, b) SEM TiN = 2,4 _Ri_ = 20 μm, c) TEM TiN = 1,4 _Ri_ = 20 μm

Figure 9c shows the TEM image of the cross-section in
the indent axis performed on a 1.4 μm TiN coated sample with
clearly visible cracks in two characteristic areas – outside the
contact zone in the surface and in the interface close to the symmetry axis. The changes in the stress concentration coefficient
_σR_ / _pm_ for coatings with varying thickness and the _EC_ / _ES_ = 2 ratio,
which corresponds to many ceramic coatings on steel substrates,
are presented in Figure 10. On the surface of the coatings, at low
deformation, the coefficient _σR_ / _pm_ is in the range 0.14-0.2. Load
increase leads to a significant stress concentration in the surface
(Fig. 10a) and in the interface (Fig. 10b).

2313

For both these areas, _σR_ / _pm_ rises with coating thickness. For
the thinnest coating _t_ / _Ri_ = 0.05 ( _t_ = 1 μm, _Ri_ = 20 μm), within the
analysed load range, tensile stress at the surface can be up to five
times higher than the mean pressure in the contact zone. Meanwhile for a thicker coating _t_ / _Ri_ = 0.25 ( _t_ = 5 μm, _Ri_ = 20 μm),
_σR_ / _pm_ grows only up to 0.6. The same phenomena was found
for stress concentration in the coating-substrate interface where
_σR_ / _pm_ reaches 5 and 1.5 for _t_ / _Ri_ = 0.05 and 0.25 respectively. In
Figure 10, the dashed curves correspond to the same loads 50,
200, 500 and 1000 mN. For example, a 500 mN load induces the
relative penetration depth _h_ max / _t_ = 1.2 and 0.2 for the thinnest

Fig. 10. Changes of the ratio of maximum radial stress to contact pressure _σR_ / _pm_ as a function of relative penetration depth _h_ max / _t_ for modulus
ratio _E_ c / _E_ s = 1, 2, 3 and relative thickness range _t_ / _Ri_ = 0.05-0.25 in: a) coating surface, b) coating-substrate interface

and thickest coatings, while the stress concentration coefficient
reaches 4 and 0.25 respectively. Although the stress concentration
for thicker coatings is lower the radial stress, _σR_ could be higher
than in thinner ones due to the substantially higher pressure _pm_
in the contact zone to which they are referenced. Previous studies presented in [27] allowed the mathematical relation binding
the stress concentration coefficient _C = σR_ / _pm_ with the relative
penetration depth _h_ max / _t_ on the surface (C _Top_ ) and interface (C _Bott_ )
to be determined:

1
_Top_ - 0,14 - ��ln �� _t_ �� �1,15�� - ��0,37 ��� _hmax_ ��2 ��2 _hmax_ �� - �� _Ec_ �� 3

- 0,14 - ��ln �� _t_ �� �1,15�� - ��0,37 ��� _hmax_ ��2 ��2 _hmax_ �� - �� _Ec_ ��
�� - _Ri_ - ���� - _t_ - _t_ �� - _Es_

_CTop_ - 0,14 - ��ln �� _t_ �� �1,15�� - ��0,37 ��� _hmax_ ��2 ��2 _hmax_ �� - �� _Ec_

��               - _Ri_               -               -               -               - _t_               - _t_               -               - _Es_

_i_ - ���� - _t_ - _t_ �� - _Es_

2 2
0,47 - 0,55 _Bott_ - 0,13 - �� _Ec_ �� - �� _t_ �� 3 - ln �� _hmax_ �� - 0,4 - �� _Ec_ �� - �� _t_ �� 3

Figure 11a shows the indentation curves of 1 and 5.2 μm
thick CrN coatings obtained using a 10 μm indenter radius. The
penetration depth of the indenter at the 200 mN load decreases
from 1470 to 520 nm with increasing coating thickness from 1 to
5.2 μm. Subsequent curves for the larger indenter radii _Ri_ = 20,
50 and 200 μm are presented in Figures 12a, 13a and 14a, respectively.

Comparing consecutive figures, the declining influence of
the coating thickness (ratio _t_ / _Ri_ ) on the deformation of the whole
system is clearly visible. The penetration depth ratio values
_Pd_ ( _t_ = 1 ρm) / _Pd_ ( _t_ = 5.2 ρm) at 200 mN are 2.65, 1.57 and 1.35
for _Ri_ = 20, 50 and 200 μm indenters, while for the smallest radius
_Ri_ = 10 μm this parameter is 2.82. Hence, one can conclude that
the smaller the relative coating thickness, the lower the impact of
the coating on the system deformation. The stress-strain curves
calculated from the presented indentation curves on the following figures are shown in Figures 11b, 12b, 13b and 14b. The
significant effect of the _t_ / _Ri_ ratio on the mean pressure _pm_ in the
contact zone is presented in Fig. 11b where _pm_ reaches 3.5 and
7.3 GPa for _t_ / _Ri_ = 0.1 and 0.52 respectively (the results for all
indenters are summarized in Table 1). For both coatings, pressure
initially rises linearly up to the strain _ε_ = 0.02-0.05. The systems
are elastically deformed, which was confirmed by the results of
tests carried out with a smaller load when loading and unloading

0,47 - 0,55
- 0,13 - �� _Ec_ �� - �� _t_ �� 3 - ln �� _hmax_ �� - 0,4 - �� _Ec_ �� - �� _t_ ��

- _Es_     -     - _Ri_     -     - _t_     -     - _Es_     -     - _Ri_

_CBott_ - 0,13 - �� _Ec_ �� - �� _t_ �� 3 - ln �� _hmax_ �� - 0,4 - �� _Ec_ �� - �� _t_

- _Es_        -        - _Ri_        -        - _t_        -        - _Es_        -        - _Ri_

_s_ - - _Ri_ - - _t_ - - _Es_ - - _Ri_

(6)

4. Indentation results

Indentation tests using different tip radius spherical indenters with the algorithm shown in Figure 11 allow the calculation
of the mean pressure changes in the contact zone.

2314

Fig. 11. Results of spherical indentation _Ri_ = 10 μm of CrN coatings: a) indentation curves, b) stress – strain curves

Fig. 12. Results of spherical indentation _Ri_ = 20 μm of CrN coatings: a) indentation curves, b) stress-strain curves

Fig. 13. Results of spherical indentation _Ri_ = 50 μm of CrN coatings: a) indentation curves, b) stress-strain curves

2315

Fig. 14. Results of spherical indentation _Ri_ = 200 μm of CrN coatings: a) indentation curves, b) stress-strain curves

curve overlaps and indents were not found after the test. Further
load increase leads to substrate yield, which facilitates the easier
deformation of the systems. This plastic deformation occurs at
a lower load for a thinner coating, hence the stress-strain curve
reaches its maximum at a lower pressure. For a thicker coating,
the _σ_ - _ε_ curve changes its slope at 5 GPa pressure, which may
indicate the initial development of a plastically deformed zone.
When the plastic deformation is highly extended, the substrate
does not provide required support for the coating, and pressure
starts to diminish as for the thin coating. The larger maximum
pressure in the contact zone for thicker coatings is the result
of its high stiffness and low contact area. For the _Ri_ = 20 μm
indenter radius, the mean pressure in the contact zone is smaller
and reaches 3.1 and 6.7 GPa for 1 and 5.2 μm coating thickness,
respectively (Fig. 12b). A further increase in indenter radius
reduces the pressure to 2.7 and 5.2 GPa for _Ri_ = 50 μm and 1

and 1.4 for _Ri_ = 200 μm on the surface of the thinner and thicker
coatings, respectively.

Large deformations of coating-substrate systems result in
the tensile stress concentration on the surface of the coating and
its fracture, which is evident on the indentation curves as a popin (sudden rise of penetration depth). The formation of the first
circumferential cracks was observed at the _FFr_ = 100 mN load for
a 1 μm coating while for the thicker one _t_ = 5.2 μm at 350 mN
(Fig. 12a). The summary of relative critical loads leads to the
coating fracture _FFr_ / _Ri_ 2, the corresponding relative penetration

depth _Pd_ / _t_, for all contact geometries and coating thickness are
summarized in Table 1 and in Figure 15. The relative values of
the apparent fracture toughness _Kc_ ’· _Ri_ / _t_ 1/2 (onset of coating
fracture, not the material parameter of coating), calculated as
the integral of the stress-strain curve until the first crack, are
also given in Table 1.

TABLE 1

Summary of the indentation results

Coating
material

CrN

TiN

_t_

[μm]

1 0.1 3.5 1143 0.80 1.76
10
5.2 0.52 7.3 2457 0.18 1.96
1 0.05 3.1 456 0.82 2.18
20
5.2 0.26 6.7 1163 0.15 1.64
1 0.02 2.3 164 0.93 4.46
50
5.2 0.104 3.5 516 0.29 2.29
1 0.005 0.8 96 1.6 4.56
200
5.2 0.026 1.1 113 0.35 2.95

0.7 0.035 2.7 225 0.5 1.84
20
2.4 0.12 5.2 525 0.21 2.01
0.7 0.0035 1.0 9 0.6 2.05
200
2.4 0.012 1.4 22.5 0.37 2.11

_Ri_ _pm_ -max
_t_ / _R_ i _FFr_ / _Ri_

[μm] [GPa]

2 _K_ c’· _Ri_ / _t_ 1/2
_Pd_ / _t_

[MPa·m 1/2 ]

2316

The relative increase in the fracture load _FFr_ / _Ri_ 2 with rising

relative film thickness _t_ / _Ri_ for CrN coatings is presented in Figure 15. However, it occurs for _t_ / _Ri_ - 0.02. The thinner coatings
crack at a relatively constant load of _FFr_ / _Ri_ = 90-130 N/mm 2,
which indicates that such coatings cannot improve the load bearing capacity of the surface. For thicker coatings, the parameter
_FFr_ / _Ri_ 2 increases and reaches a value of about 2200-2400 N/

The relative increase in the fracture load _FFr_ / _Ri_

_FFr_ / _Ri_ 2 increases and reaches a value of about 2200-2400 N/

mm 2 for _t_ / _Ri_ = 0.52. Within the range 0.02 < _t_ / _Ri_ < 0.52 coatings crack under increasing load, due to the rising rigidity of the
system and force necessary to reach the required deformation
to its destruction. As derived from Figure 10, the increase in
_t_ / _Ri_ results also in the reduction of stress concentration both
on the surface of the coating as well as in the coating-substrate
interface. The lower stress concentration on the coating surface
is due to a smaller bending of the thicker coating, especially just
outside the contact zone. On the other hand, more flexible thin
coatings allow greater deformation to fracture, which proves
higher values of the relative penetration depth _Pd_ / _t_ at the first
crack event for a 1 μm coating than for 5.2 μm indented by the
same diamond (Table 1). For very thick coatings of _t_ / _Ri_ - 0.52
the value of _FFr_ / _Ri_ 2 remains constant. These coatings with such

Fig. 16. Changes of relative fracture resistance vs. relative coating
thickness of CrN coatings

5. Summary

the value of _FFr_ / _Ri_ 2 remains constant. These coatings with such

a high relative thickness start to crack at penetration depths
below 20% of its thickness, which indicates a small effect of
the substrate on the deformation, hence the coating fracture
is limited by fracture resistance as in the case of an infinitely
thick coating. The application of very thick coatings from the
mechanical point of view is aimless, furthermore thick coatings
generally exhibit lower adhesion to the substrates. The changes
in the relative fracture toughness _Kc_ ’· _Ri_ / _t_ 1/2 within the range of
the relative thickness _t_ / _Ri_ are shown in Figures 15, 16.

The analysis of the deformation and fracture of coatingsubstrate systems given in the paper is a very important issue in
the coatings tribology. The wear of such systems is substantially
lower at loads below the critical level leading to coating fracture

[12]. The presented studies allowed these critical loads for typical failure modes, like substrate yield and coating fracture by
spherical indentation, to be determined. Tests performed with
different radii of indenters enable the analysis of the load bearing capacity of coating-substrate systems at different states of
deformation. By transformation of the typical load - the penetration depth curves into stress-strain curves, the analysis of mean
stress changes in the contact area and coating fracture resistance
is possible. The complete procedure for such analyses combined
with the results of numerical experiments are presented in this
paper. The results show the different character of the system
deformation with thin and thick coatings. Coatings enhance the
relative fracture load _FFr_ / _Ri_ 2 but only within 0.02 < _t_ / _Ri_ < 0.5,

relative fracture load _FFr_ / _Ri_ 2 but only within 0.02 < _t_ / _Ri_ < 0.5,

which is clearly seen in the failure map of CrN coatings deposited
on austenitic stainless steel substrates (Fig. 15). The thinner and
thicker coatings have a constant value of the _FFr_ / _Ri_ 2 parameter.

Fig. 15. Changes of relative fracture load vs. relative coating thickness
of CrN coatings

Coatings thicker than _t_ / _Ri_ - 0.2 have this parameter at
a relatively constant level of 1.6-2.3 MPa · m 1/2 . Significantly
higher values of _Kc_ ’· _Ri_ / _t_ 1/2 = 4.5 MPa ·m 1/2 were found for thin
coatings _t_ / _Ri_ < 0.2, which confirms the higher fracture toughness
of thin coatings presented in the literature [34].

thicker coatings have a constant value of the _FFr_ / _Ri_ 2 parameter.

Despite the higher values of the stress concentration coefficient
_σR_ / _pm_ for thin coatings, the greater susceptibility of the whole
system and lower mean pressure in the contact zone ensure its
higher fracture toughness as presented in Figure 16. Systematic
studies on the contact mechanics of coating-substrate systems
and determination of their failure maps could be helpful in predicting the critical failure load during known contact geometry
and optimizing coating thickness.

Acknowledgements

the author would like to thank Dr. habil. Juergen.M. Lackner from Joan
neum Research Forschungsges. mbH, Institute for Surface Technologies and

Photonics, Functional Surfaces, Niklasdorf, Austria for coatings deposition

and Dr. habil. Lukasz Major from Polish Academy of Sciences, Institute

of Metallurgy and Materials Sciences, Krakow, Poland for TEM analysis.

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