Indentation for evaluating cracking and delamination of thin coatings using finite element analysis

Vacuum 122 (2015) 17e30

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Indentation for evaluating cracking and delamination of thin coatings using finite element analysis Yangyi Xiao*, Wankai Shi, Jing Luo The State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 May 2015 Received in revised form 1 September 2015 Accepted 2 September 2015 Available online 5 September 2015

The mechanics of indentation induced coating cracking and interfacial delamination in a typical weakly bonded brittle thin coating-substrate system (diamond-like carbon coating with low adhesion on steel substrate) is investigated by using finite element method. A bilinear cohesive zone model with prescribed cohesive strength and energy is applied to simulate evolutions of the crack and delamination. The effects of the cohesive zones, coating elastic modulus, and coating thickness on the indentation response are evaluated. It is found that increasing the cohesive strength/energy of coating reduces the probabilities of crack generation and propagation, whereas increases the susceptibility to interfacial delamination. At a critical value of the cohesive strength of the bonding layer, the resistance to interfacial delamination reaches its minimum. Nevertheless, the coating cracking is not sensitive to the interface adhesive properties. Moreover, a coating can receive better protection with lower coating elastic modulus. Additionally, increasing the coating thickness generally increases the critical load for coating/ interface failures, but opposite effect occurs when the coating is thinner than a critical thickness. Numerical results have also been compared with other emulational or experimental works, and can establish a theoretical basis for improving the durability of brittle thin coatings. © 2015 Published by Elsevier Ltd.

Keywords: Indentation Coating cracking Interfacial delamination Cohesive zone model

1. Introduction Thin hard coatings have been widely applied in mechanical components, such as gears, bearings, and joints, to reduce the coefficient of friction, improve the wear performance, and achieve excellent loading capacity of the surfaces. Although these coated mechanical components can realize perfect tribology performance, the major failure modes of thin hard coatings (usually brittle) on relatively softer substrates system are coating cracking and interfacial delamination [1e3], which we should highly regard. Physical vapor deposition (PVD) coatings, as a general rule, are hard and brittle, but cannot show good bonding performance with steels under some special conditions. For instance, an oxide layer on the substrate or rough surface of the substrate before deposition can result in an interfacial problem of low adhesive performance [4]. Furthermore, a PVD coating shows inferior adhesive performance at an elevated temperature operating condition [5,6]. Therefore, the selection of a typical weakly bonded brittle thin coating-substrate

* Corresponding author. E-mail address: [email protected] (Y. Xiao). http://dx.doi.org/10.1016/j.vacuum.2015.09.003 0042-207X/© 2015 Published by Elsevier Ltd.

system is critical to the investigation. Indentation test is an important way to assess failures of the coating [1,7e10]. Of course, mechanical behavior of coatingsubstrate system has more immediate concerns. For numerical simulation on indentation, the finite element method is often applied to analyze its mechanical behavior. In previous studies, for example, the stress characteristics of coating surface and bonding layer are analyzed to evaluate the coating failures [3,8e11]. Tilbrook et al. [12] obtained the damage mechanisms of hard TiN coatings based on a model by anisotropic property definitions and nodal coupling. The crack growth in thin elastic coatings bonded on elastic substrates was investigated by using the J-integral method [13,14]. These methods can only predict the failure locations or analyze the preexisting cracks. Moreover, the hard coating cracking under indentation was simulated by applying cohesive zones in the possible crack locations [15]. But Abdul-Baqi et al. [15] studied the circumferential coating cracks only by the method of step-by-step discerning analysis. In their research, the location of the (n þ 1)th crack can be found by stress analysis after the nth crack forms, and all coating cracks are assumed to initiate from the coating surface. Actually, this method has the shortcoming that it cannot simulate the crack behaviors if some coating cracks may form

18

Y. Xiao et al. / Vacuum 122 (2015) 17e30

simultaneously. Also, some of the cracks initiate from the coating surface, and some initiate from the coating side of the interface. In addition to these works, the bonding layer was modeled by a cohesive surface to study the interfacial delamination [16e19]. Zhang et al. [19] obtained the effects of the interface adhesive properties and the thickness of the thin coating for cases of a perfect hard coating on a soft substrate on the onset and growth of interfacial delamination. In summary, there are three cases on the numerical models of indentation in previous works: perfect coating-substrate system, cohesive interface layer between perfect coating and substrate, and cracks on coating with perfect bonding layer. However, it has been shown that coating cracking often follows by interfacial delamination [15,16,20] under indentation, especially in a weakly bonded thin hard coating-substrate system. The method in neither case is going to be the best for evaluation on brittle thin coating failures. There are few studies that consider the interaction of coating cracking and interfacial delamination. Thus, these gaps will be the emphasis of the current paper. Currently, the cohesive zone model (CZM) is one of the most modern evolutions in the area of fracture mechanics to simulate the cracks (between the coating neighbor segments) and delamination (between the coating and substrate) [15e23]. The major features of CZM is that it is able to adequately predict the behavior of uncracked structures, but also can be used to investigate the onset and growth of separations if damage occurs [18]. Additionally, the CZM does not represent any physical material, but describes the cohesive forces which occur when material elements are being pulled apart. In the present paper, both the coating cracking and interfacial delamination are taken into account for brittle thin coatings by applying a cohesive zone finite element model during indentation simulation. A system consisting of a diamond-like carbon (DLC) coating on a steel substrate is investigated in this study. For simplicity, the residual stress is not considered in the coatingsubstrate system, which means the coating is assumed to be stress free prior to indentation. A spherical indenter is used in the indentation test. The radial cracking in the coating is not included in the study, because this type of indenter (spherical) is less likely to induce radial cracking, but rather ring cracking. Here, it is worth noting that the shapes of both the coating cracks (Hertzian ring cracks [8,21]) and the interfacial shear delamination are circumferential. The aim of this study is to offer an improved understanding of coating cracking and interfacial shear delamination during the loading stage of indentation and to improve the durability of brittle thin coatings by varying several parameters. The effects of the cohesive zones, coating elastic modulus, and coating thickness on the indentation response which includes initiations of coating cracking and interfacial delamination and their propagations are studied. Importantly, numerical results will be compared with other emulational or experimental works.

sys and 1 mm for convenience. Fig. 1 describes the geometry of 1 GPa the simplified indentation model and its boundary conditions. The coating-substrate system has the shape of a cylinder and that the indentation is made just in the center of the coating surface. A twodimensional finite element model is considered in the radialeaxial (rez) coordinate due to its symmetry along the axis, and the contact radius a can be obtained during indentation. As seen in Fig. 1, the parameters are defined as radius of spherical indenter of R ¼ 50 D, coating thickness of hc ¼ 2D, substrate thickness of hs ¼ 200D, bonding layer thickness of hb ¼ 0.1D, and system radius of L ¼ 200D. Roller-boundary conditions are applied along the axis of symmetry and fixed-boundary conditions are applied to the substrate base. In this finite element analysis, the indenter is considered as a rigid body for simplicity, because it is made of diamond which is much harder than the specimen. Then, a reference node located on the center of the spherical indenter is used as the reference point of the rigid indenter, and a specified displacement is applied to this reference node. In order to get accurate results and improve the convergent rate, different mesh densities are applied to the coating-substrate system. There is the highest mesh density near the contact and bonding areas. The contact behavior between indenter and coating is divided into two parts: normal and tangential behaviors. The normal behavior applied to the model is “hard” contact. For tangential behavior, a penalty-type contact algorithm is carried out that places imaginary springs between the “master” surface and the “slave” surface. The coefficient of friction for contact between the indenter and coating is specified as 0.1. Cohesive zone models (CZMs) are applied on cracks and the bonding layer that can be found in the next section. In continuum mechanics, the substrate is defined as homogeneous, isotropic, and elasticeplastic materials. It accords with the following uniaxial stressestrain law:

s¼

8 > < > : sy

Eε 

ε 

sy E

n

   ε < sy Es 

  ε  sy Es

(1)

where, s and ε are the stress and strain of the material, respectively.

2. Indentation model description 2.1. Finite element model During a loading step of the indentation test, the reaction force of indenter is continuously recorded as a function of the indenter displacement. Then the loadedisplacement (Peh) curves can be obtained. The quasi-static structural analysis (rate-dependent, but not involving inertia) is performed on the step. The indenter moves along the axis of symmetry and penetrates the coating-substrate system up to a prescribed depth (maximum value is equal to the coating thickness in the present paper) in loading stage. In the simulation, the length and stress are respectively scaled by and D(substrate yield stress), which are respectively chosen as

Fig. 1. Geometry of the simplified indentation model and boundary conditions.

Y. Xiao et al. / Vacuum 122 (2015) 17e30

E is the elastic modulus. n is the work-hardening exponent. Here, it is assumed that the substrate material is an elastic-perfectly plastic solid (n ¼ 0). Es is the elastic modulus of substrate and is taken as 200sys. The Poisson's ratio of substrate is ns ¼ 0:3. For a DLC coating, the great variety of its structures and compositions (in particular sp3 content) leads to a wide range of mechanical property. Indeed, its elastic modulus varies from 60 GPa up to 650 GPa [24]. Unlike the substrate, the coating is supposed to be fully elastic, which is a fairly good approximation. Its material properties dthe elastic modulus and Poisson's ratiod are Ec ¼ 100sys and nc ¼ 0:3, respectively.

2.2. Cohesive zone model The constitutive equation of the cohesive zone is based on the bilinear tractioneseparation (TeS) law, which is very suitable to describe the material separation damage [23,25,26]. The cohesive relation in the mixed mode of the total TeS law (coupling the tension and shear modes) is shown in Fig. 2. The bilinear constitutive response shown in Fig. 2 can be described by the following equations:

t¼

8 > > > < > > > :

Kd ðI  DÞKd 0

0

ðd < d 0

 f

ðd  d < d  ðd  df



(2)

tT T is the separating stress matrix, d ¼ ½ dN dT T  T is the separating displacement matrix, d0 ¼ d0N d0T is the separating displacement matrix for damage initiation within the iT h is the separating displacement cohesive zone, df ¼ dfN dfT

 KN 0 matrix for failure, and K ¼ is the undamaged surface 0 KT stiffness matrix. The cohesive zone in this analysis is assumed to be isotropic (KN ¼ KT ¼ K). High values of K avoid interpenetration of the separating faces but can lead to numerical problems, and several stiffness values of K (107, 5.7  107, and 108 MPa/mm, etc.) have been proposed by previous researchers [27]. Here, high stiffness values of 109 and 108 MPa/mm are individually chosen for potential coating cracks and bonding layer in this study, but it can also avoid potential convergence problems during the nonlinear where, t ¼ ½ tN

Fig. 2. Bilinear constitutive model in the mixed mode.

19





DN 0 is the diagonal matrix representing the 0 DT damage accumulated in the cohesive zone, I is the identity matrix, and subscript N and T, respectively, denote the normal and tangent directions. The above all normal separating stress is effective only under the condition of dN > 0, but the tangent separating stress is not limited in that way. This means the softening response illustrated in Fig. 2 is representative of the tension or the shear response but not compression. It is assumed that compression loads neither cause delamination nor have the effect on damage of the surface in the present paper [27]. In a bilinear CZM, the critical cohesive energy for a surface is given by: procedure. D ¼

1 GC ¼ t 0 df 2

(3)

 T where, GC ¼ GCN GCT is the critical cohesive energy matrix and  T 0 t 0 ¼ tN tT0 is the cohesive strength matrix. The detection of damage is maximum stress criterion which is shown as follow [22]:

( ) htN i tT max ; ¼1 0 t0 tN T

(4)

where, 〈tN〉 represents the Macaulay bracket that can be described as:

htN i ¼

tN ðtN  0Þ 0 ðtN < 0Þ

(5)

The most widely used criterion to predict damage propagation under mixed mode loading is the power law criterion. It can be described as follow:

GN GCN

!a þ

GT GCT

!a ¼1

(6)

where, GN and GT are the works done by the traction in the normal and shear directions, respectively. In this work, the power exponential a is defined as 1 [25,28,29] which is the most commonly used value and corresponds to a linear fracture criterion. Two important parameters (the cohesive energy and strength) will be taken into consideration, because the predictions of damage by other CZM parameters appear to be insensitive to the details of the TeS law [28,30]. There are many test methods to assess the fracture and adhesion of a system composed of a DLC coating on a steel substrate, like indentation and tensile tests, etc. Table 1 lists the fracture toughness and strength of DLC coating from previous literature. The adhesive toughness and strength of DLC coating on steel are presented in Table 2. In general, the DLC coating bonds well with steel due to its large enough bond toughness and strength. However, sometimes a DLC coating shows low values of bond toughness and strength, which are attributed to the presence of thick and friable oxide layers on the substrate [4]. Also, its bonding performance will decrease at an elevated temperature condition in which DLC coating has the poor stability of its microstructure and properties [5]. Thus, weakly bonded DLC coating should be of concern. It also needs to be pointed out that the cohesive element properties are affected by the deposition process and substrate conditions, which may markedly vary the model parameters [21]. After referring to the data from Tables 1 and 2, reasonable values about cohesive zones, which represent a typical weakly bonded

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Y. Xiao et al. / Vacuum 122 (2015) 17e30

Table 1 Values for DLC coating (on steel substrate) fracture toughness and strength from literature. Testing assessment

Fracture toughness

Multistrain & indentation Indentation Tensile test

Fracture strength

Reference

[J/m2]

Dimensionless

[GPa]

Dimensionless

30 ~20 e

0.03sysD ~0.02sysD

1.44 e 1.5e8.75

1.44sys

[31] [32] [33]

(1.5e8.75)sys

Table 2 Values for DLC coating (on steel substrate) adhesive toughness and strength from literature. Empiric assessment

Adhesive toughness

Multistrain & indentation Deposition & cooling Deposition & cooling Pull-out Tensile test

Adhesive strength

[J/m2]

Dimensionless

150 7 200 e e

0.15sysD 0.007sysD 0.2sysD

6

4

[31] [4] [4] [34] [33]

(0.0087e0.0379)sys (0.285e1.94)sys

0 ¼ 2s , GC ¼ 0:03s D). For the on the normal direction (tTc ys ys Tc 0 ¼ bonding layer (interface) between the coating and substrate, tNb C C 0 ¼ 0:2s tTb and G ¼ G ¼ 0:008s D are respectively selected. ys ys Nb Tb 0 ¼ t 0 , as well as Note that t0 in the following sections means t 0 ¼ tN T

(a)

h/hc=0.1 h/hc=0.4 h/hc=0.7 h/hc=1

5

Dimensionless

e e e 8.7e37.9 285e1940

brittle thin coating-substrate model, are chosen as follows. For the 0 ¼ 2s cohesive zones between the coating neighbor segments, tNc ys and GCNc ¼ 0:03sys D are respectively chosen. It is assumed that the shear cohesive strength and energy have the same values as those

(a)

Reference

[MPa]

1

0

2

-1

σ Ν /σys

σ r/σys

3

1 0

-2

-1

h/hc=0.1 h/hc=0.4 h/hc=0.7 h/hc=1

-3

-2 -3 0

(b)

1

2

3

r/a

4

5

-4

6

4

0

1

2

r/a

3

4

(b) 0.6

h/hc=0.1 h/hc=0.4 h/hc=0.7 h/hc=1

3 2

h/hc=0.1 h/hc=0.4 h/hc=0.7 h/hc=1

0.4

0.2

0

σ T/σys

σ r/σys

1

-1

0.0

-2 -0.2

-3 -4 -5

-0.4

0

1

2

3

r/a

4

5

6

Fig. 3. Radial stress distribution under different indentation depths for the coating: (a) along the coating surface, (b) along the interface on the coating side.

0

1

2

r/a

3

4

Fig. 4. Interfacial tractions with (a) normal stress and (b) shear stress along the perfect bonding layer under different indentation depths.

Y. Xiao et al. / Vacuum 122 (2015) 17e30

21

Fig. 5. Contour of the maximum principle stress at indentation depth of h/hc ¼ 1 for perfect coating-substrate system.

Fig. 6. Detail finite element coating-substrate system mesh around the axis of symmetry, where the red color meshes indicate cohesive elements for potential coating cracks and bonding layer. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

GC and other matrixes in bold. Elastic modulus and coating thickness are principal parameters affecting the magnitude of stresses in the coating-substrate system. Thus, it is feasible to obtain the effects of the cohesive zone, coating elastic modulus, and coating thickness on the indentation response by varying their values. Furthermore, it is worth noting that if the parameters of geometry and material as well as boundary conditions are not specifically illustrated in the following numerical model, the above definitions will be applied remaining unchanged.

3. Indentation simulation Before having the analysis on cracks between the coating neighbor segments, it is significant to take the perfect coating (no cohesive zones) into consideration. Therefore, the initial coatingsubstrate system is composed of perfect coating, perfect bonding

0 ¼ t 0 /∞), and substrate. layer (no failures on cohesive zone tNb Tb The stress distributions of the coating surface and bonding layer in this ideal model are analyzed to predict the locations where the failure may occur. The radial stress distribution along the coating surface under different indentation depths are shown in Fig. 3(a). The maximum tensile stress is found outside the contact region at r z 1.2a. The compressive stress is shown in the region of r < a. Fig. 3(b) shows the variation of radial stress along the coating side of the interface. Contrary to the radial stress along the coating surface, the radial stress along the coating side of the interface is found to be tensile in the region of r < a. Its maximum compressive stress is found in the region of r z (1.2  1.5)a. Through above analysis, it can be summarized that these regions of r < a (interface) and r z 1.2a (coating surface) are possible locations from where the circumferential crack initiates because of the maximum tensile radial stresses at

Fig. 7. Schematic illustration of coating surface cracking and interfacial delamination.

22

Y. Xiao et al. / Vacuum 122 (2015) 17e30

Fig. 8. Contour of the maximum principle stress at indentation depth of h/hc ¼ 1 for different coating and bonding cases: (a) brittle coating and perfect bonding layer, (b) perfect coating and weak bonding layer, and (c) brittle coating and weak bonding layer.

these regions [15]. Fig. 4(a) and (b) shows the distributions of normal stress sN and shear stress sT along the bonding layer under different indentation depths, respectively. We do not care much about the compressive stress (negative normal stress). What is relevant for interfacial delamination is that the normal stress along the bonding layer is positive in the region of around r > 1.2a where the maximum shear stress can also be found. After analysis, it is seen that an interfacial delamination is likely to initiate in the region of r > 1.2a [16], and the delamination behavior remains a mode II dominated defect [17]. The maximum principal stress is a common parameter in fracture mechanics. Fig. 5 shows that after the indentation depth of h/ hc ¼ 1, the substrate yielded, and the maximum principal stress reaches a maximum around the contact edge or on the subsurface near the bonding layer. It further evidences that the failure locations of coating cracking and delamination are also around the contact edge and interface if the maximum principal stress criterion is used here, which is the same as the above result by analysis of radial stress and interfacial traction. After analysis of the potential failure locations of the system, an overall model which can study the cracking and delamination is established. As shown in Fig. 6, four-node interface cohesive elements are added into the coating-substrate system to simulate the coating cracking and interfacial delamination. According to the value of contact radius calculated before as well as the balance between solution efficiency and accuracy, 30 columns of cohesive zone vertically extend through the thickness of the coating and are

evenly distributed in lateral direction (r-coordinate) around the axis of symmetry to represent the potential cracks. The distance between each column of cohesive zone elements is 0.615D (relatively small). The cohesive elements are also used to stand for the bonding layer along the interface. Then, the indentation simulation is performed on this developed model. Furthermore, viscous regularization options are used for the cohesive elements in order to improve the convergence performance due to a softening behavior in the cohesive zone [25,28]. The influence of various parameters on the initiation of the first circumferential crack and the spacing between successive cracks, as well as the onset of the first delamination and delamination length is investigated. The coating surface cracking and interfacial delamination can be schematically illustrated in Fig. 7. The overall length of cracking for circumferential cracks is described by P Lc ¼ m i¼1 Lci ; where m and Lci are the number of cracks and the length of one crack, respectively. The average crack spacing Sc beP tween successive cracks is m i¼1 Sci =m; where Sci denotes the distance between two consecutive cracks. The overall length of P delamination can be described as Ld ¼ ni¼1 Ldj ; where n and Ldj respectively denote the number of delaminations and the size of one delamination. 4. Results and discussion 4.1. Cracking and delamination If the cohesive strength between a coating and a substrate is

Y. Xiao et al. / Vacuum 122 (2015) 17e30

strong enough, the model above without failure on the bonding layer (strong enough) can be used. As shown in Fig. 8(a), 6 cracks can be found on the brittle coating and perfect bonding layer system after the indentation depth of h/hc ¼ 1. The 4th (white one) and 5th cracks initiate from the interface, and others initiate from the coating surface. Fig. 8(b) describes the case of a perfect coating and

(a)

1.1 1.0 0.9

Perfect coating & perfect interface Cracking & perfect interface 4th Crack

0.8 0.7

P/Pperfect

5th Crack

3rd Crack

0.6

2nd Crack

0.5 st

1 Crack

0.4 0.3 0.2 0.1 0.0 0.0

(b)

0.2

0.4

h/hc

0.6

0.8

1.0

23

weak bonding layer model (delamination failure occurred). The interaction of coating cracking and interfacial delamination is considered as shown in Fig. 8(c). By comparing Figs. 5 and 8, it is observed that the maximum principal stress attains a maximum on the perfect coating-substrate system, and gets a minimum on the brittle coating and perfect bonding layer model. It means that the coating cracking and interfacial delamination occur with the stress releasing. The corresponding loadedisplacement (Peh) curves for different coating and bonding cases are also studied. As shown in Fig. 9, the above four coating-substrate systems have almost the same Peh curves although there are several failures on some of them, because the stiffness of the cohesive zone is finite [16] and there is not much difference between elastic modulus of coating (100sys) and substrate (200sys) for this case. Pperfect is acquired from the simulation for the case of perfect coating-substrate system, which is the maximum reaction force of indenter during the loading stage. The Peh curve of perfect coating-substrate is smooth, but there are small fluctuations (kinks) around it in some regions of the other three models. So, the corresponding indentation loads and depths of initial cracking and delamination can be obtained from these fluctuant regions. But it is not obvious to find kinks in some first cracking locations (the 1st and 3rd cracks in Fig. 9(c) initiating from the coating side of the interface) on Peh curve, because there are some additional stresses to overcome the delamination in these areas. Combined with Figs. 5, 8 and 9, we can

1.1 1.0

Perfect coating & perfect interface Perfect coating & delamination

0.9 0.8

P/Pperfect

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

(c)

0.2 1 Delamination

0.4

h/hc

0.6

0.8 1.0 2 Delamination

1.1 1.0

Perfect coating & perfect interface Cracking & delamination

6th Crack

0.9 0.8

5th Crack

P/Pperfect

0.7

4th Crack 3 Crack

0.6

rd

0.5

2nd Crack

0.4

1st Crack

0.3 0.2 0.1 0.0 0.0

0.2 Delamination

0.4

h/hc

0.6

0.8

1.0

Fig. 9. Indentation load vs. indentation depth for different coating and bonding situations: (a) cracking and perfect interface, (b) perfect coating and delamination, and (c) cracking and delamination.

Fig. 10. (a) The first cracking location and the corresponding indentation depth, and (b) the overall length of cracks and average crack spacing under different cohesive strengths of coating.

24

Y. Xiao et al. / Vacuum 122 (2015) 17e30

conjectures will be studied in the following sections. In general, it indicates that the cracking and delamination appear to be linked in some way from Figs. 5, 8 and 9. Some interactions between them will be investigated in the following sections, including the first cracking location and the corresponding indentation depth, the overall length of cracking Lc, the average crack spacing between successive cracks Sc, as well as the first delamination location and the corresponding indentation depth, and the overall length of delamination Ld. 4.2. Effect of cohesive zone 4.2.1. For coating

Fig. 11. The length of delamination and the corresponding indentation depth under different cohesive strengths of coating.

find that it may be harder to delaminate on the interface if the coating cracking happened. Also, it may be more likely to cause cracking if there is an interfacial delamination in the system. These

Fig. 12. (a) The first cracking location and the corresponding indentation depth, and (b) the overall length of cracks and average crack spacing under different cohesive energies of coating.

4.2.1.1. Cohesive strength. To study the effect of cohesive strength of coating on the response of indentation, the simulation here is respectively performed for tc0 =sys ¼ 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 with GCc =ðsys DÞ ¼ 0.03, as well as parameters of the bonding layer: tb0 =sys ¼ 0.1 and GCb =ðsys DÞ ¼ 0.008. Fig. 10(a) shows the first cracking location and the corresponding indentation depth under different cohesive strengths of coating. It can be seen that the radius of the first crack increases with the increase in its cohesive strength. For cohesive strength tc0 under 2sys, the first crack initiates from the interface. The coating cracking initiates from the coating surface whose cohesive strength is more than 2.5sys. So there exists a watershed cohesive strength that describes the first crack initiating from the interface to the coating surface. Moreover, the larger the cohesive strength, the larger the indentation depth required to form the first crack. It can be seen from Fig. 10(b) that with the increase in cohesive strength, the overall length of cracks decreases and the average crack spacing increases. This phenomenon accords with the above bilinear TeS law. It means that an increase in cohesive strength of coating can effectively prevent the initiation and growth of cracks. In addition, the influence of coating cracking on interfacial delamination is investigated. The response of delamination length and indentation depth under different cohesive strengths of coating is described in Fig. 11. It can be seen that the radius of the first interfacial delamination decreases with increasing cohesive strength of coating. The larger the cohesive strength of coating, the smaller the indentation depth needed to trigger delamination and the larger the overall length of delamination. It can be explained that if the coating is easier to crack, more strain energy will be

Fig. 13. The length of delamination and the corresponding indentation depth under different cohesive energies of coating.

Y. Xiao et al. / Vacuum 122 (2015) 17e30

taken on coating cracks and less energy will be consumed on interfacial delamination. It indicates that the coating cracks consume more damage energy from the indenter to protect the bonding layer. 4.2.1.2. Cohesive energy. In order to evaluate the effect of cohesive energy of coating on indentation, the present simulation is respectively carried out for GCc =ðsys DÞ ¼ 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, and 0.08 with tc0 =sys ¼ 2.0, and including parameters of

25

the bonding layer: tb0 =sys ¼ 0.1 and GCb =ðsys DÞ ¼ 0.008. The first cracking location and the corresponding indentation depth under different cohesive energies of coating can be seen in Fig. 12(a). The radius of the first crack increases with the increase in its cohesive energy. The coating cracking respectively initiates from the interface and coating surface when cohesive energies of coating GCc are less than 0.03sysD and more than 0.04sysD. Also, the greater the cohesive energy, the larger the indentation depth required to form the first crack. It can be seen from Fig. 12(b) that the overall length of cracks decreases and the average crack spacing increases as the cohesive energy increases. It can be illustrated by the bilinear TeS law that both the damage accumulated stiffness and separating f displacement of the coating (Dc and dc ) increase with the increasing C of its cohesive energy Gc . It also can be found that the influence of cohesive energy on coating cracking is almost the same as that of cohesive strength. Fig.13 shows the response of delamination length and indentation depth under condition of different cohesive energies of coating. Also, the radius of the first interfacial delamination decreases with the increment of cohesive energy of coating. The indentation depth needed to form delamination gets smaller and the overall length of delamination becomes larger as the cohesive energy of coating increases. Obviously, these results can be explained by the above strain energy consumption theory which is also used to illustrate the effect of cohesive strength of coating on

(a)

10 9

0.45

First cracking location Corresponding indentation depth

8

0.40

Perfect bonding layer

7

0.35

h/hc

r/Δ

6 5

0.30

4 3 2 1 0 0.0

0.25 Increasingly hard to delaminate

Increasingly easy to delaminate 0.1

0.2

0.3

0.4

t b/σ

0.5

0.6

0.7

0.20 0.8

0

(b)

2.50

4.4 4.3

Overall length of cracks Average crack spacing

2.45 2.40

4.2 2.35 2.30

4.0

Perfect bonding layer

2.25

Sc/Δ

Lc/hc

4.1

2.20

3.9

2.15 3.8 2.10 3.7 Increasingly easy to delaminate 3.6 0.0

Fig. 14. (a) The first delamination location and its corresponding indentation depth, (b) the overall length of delamination, and (c) the length of delamination and the corresponding indentation depth under different cohesive strengths of bonding layer.

0.1

0.2

0.3

0.4

Increasingly hard to delaminate

t0b/σ

0.5

0.6

0.7

2.05 2.00 0.8

Fig. 15. (a) The first cracking location and the corresponding indentation depth, and (b) the overall length of cracks and average crack spacing under different cohesive strength of bonding layer.

26

Y. Xiao et al. / Vacuum 122 (2015) 17e30

condition of a larger cohesive strength (more than 0.5sys). For the corresponding indentation depth, it also has the tendency to decrease at first and then increase (the watershed value of ~0.4sys) with the increase in cohesive strength. This phenomenon (having watershed in cohesive strength) about the coating propagation can also be seen in Fig. 15(b). The theory used to state the effect of cohesive strength of bonding layer on interfacial delamination can also explain this phenomenon about coating cracking. It indicates that if the bonding layer is damaged (weakly bonded), the cracks in the coating will be weakly supported by the substrate. Then more elastic energy is transferred to the coating along the horizontal direction, which will be easier to generate cracks.

1.0 0.9 0.8

6.1

h/hc

0.7 0.6

2.6

0.5 0.4

Γ Cb=0.008σ Δ Γ Cb=0.012σ Δ Γ Cb=0.014σ Δ

0.3 0.2 0.1

2

4

6

8

r/Δ

10

12

14

Fig. 16. The length of delamination and the corresponding indentation depth under different cohesive energies of bonding layer.

interfacial delamination. Combined with the descriptions about effects of cohesive strength and energy of coating on interfacial delamination, it can be concluded that the stronger the coating, the smaller the indentation depth needed to trigger delamination and the larger the overall length of delamination. 4.2.2. For bonding layer

(a)

0.45 9

Increasingly hard to delaminate

8 0.40

Perfect bonding layer

7 6 5

0.35

h/hc

4 3 0.30

2

First cracking location Corresponding indentation depth

1 0 0.002

0.004

0.006

0.008

0.010

Γ b/(σ Δ)

0.012

0.014

0.016

0.25 0.018

C

(b) 4.00

2.50

Overall length of cracks Average crack spacing

3.95

2.45 2.40

3.90

2.35 2.30

3.85

Perfect bonding layer 3.80

2.25

Sc/Δ

4.2.2.1. Cohesive strength. This section is proposed to investigate the effect of cohesive strength of the bonding layer on indentation. The numerical analysis is respectively performed based on the parameters of the bonding layer: tb0 =sys ¼ 0.1, 0.2, 0.3, 0.4, 0.5, and 0.55 with GCb =ðsys DÞ ¼ 0.008, as well as parameters of the coating: tc0 =sys ¼ 2 and GCc =ðsys DÞ ¼ 0.03. Fig. 14(a) shows that with the increase in cohesive strength of the bonding layer, both the first delamination radius and the corresponding indentation depth decrease firstly followed by an increase. But as seen in Fig. 14(b) the overall length of delamination has an opposing tendency. Fig. 14(c) describes the length of delamination and the corresponding indentation depth under three typical cohesive strengths of bonding layer. A previous study made by Xia et al. [35] also concluded that a minimum in resistance to shear crack nucleation (shear interfacial delamination) occurs when the cohesive strength of the interface reaches a critical value. It is not difficult to explain these results by the bilinear TeS law. No other parameters changed except cohesive strength of the bonding layer. If the cohesive strength of the bonding layer tb0 improves from 0.1sys to 0.4sys, both the damage accumulated stiffness Db and separating displacement of the bonding layer dfb will decrease. Thus, it is easier to cause damage on the bonding layer. However, for the case of coating, its cohesive strength is much larger than the cohesive strength of coating, e.g., tc0 ¼ 2sys and tb0 ¼ 0:1sys . As shown in Fig. 2 about the bilinear constitutive curve, although the damage accumulated stiffness and separating f displacement of the coating (Dc and dc ) also decrease with the in0 crease of its cohesive strength tc , the cracking on the coating is just accumulated and delayed without damage. Furthermore, if the cohesive strength of the bonding layer tb0 remains elevated after 0.4sys, the interfacial delamination will also be delayed and its damage will decrease. As seen in Fig. 15(a), the coating cracking initiates from the interface for the case of cohesive strength tb0 less than 0.5sys, and the coating cracking initiates from the coating surface under the

4.2.2.2. Cohesive energy. The numerical analysis is respectively executed for GCb =ðsys DÞ ¼ 0.004, 0.006, 0.008, 0.010, 0.012, 0.014, and 0.016 with tb0 =sys ¼ 0.2, and containing parameters of the coating: tc0 =sys ¼ 2 and GCc =ðsys DÞ ¼ 0.03. Fig. 16 shows the relationship between the length of delamination and the corresponding indentation depth under different cohesive energies of bonding layer. The results indicate that with the increase in cohesive energy of bonding layer, the values of all the three variables (the radius of first interfacial delamination, the indentation depth needed to cause delamination, and the overall length of delamination) decrease. It can be easily explained by the bilinear TeS law that both of the damage accumulated stiffness and separating displacement of the bonding layer (Db and dfb ) increase with

r/Δ

11.2

Lc/hc

1.1

2.20 2.15

3.75

2.10 3.70

2.05 Increasingly hard to delaminate

3.65 0.002

0.004

0.006

0.008

0.010

Γ Cb/(σ Δ)

0.012

2.00 0.014

0.016

0.018

Fig. 17. (a) The first cracking location and the corresponding indentation depth, and (b) the overall length of cracks and average crack spacing under different cohesive energies of bonding layer.

Y. Xiao et al. / Vacuum 122 (2015) 17e30

(a)

14

Effect of coating CZM on coating cracking Effect of bonding layer CZM on coating cracking

(b)

27

1.0 0.9

12

Effect of coating CZM on coating cracking Effect of bonding layer CZM on coating cracking

0.8

10

0.7

8

r/Δ

h/hc

0.6

6

0.5

4

0.4

2

0.3

0

(c)

7

Fig. 10(a)

Fig. 12(a)

Fig. 15(a)

0.2

Fig. 17(a)

Effect of coating CZM on coating cracking Effect of bonding layer CZM on coating cracking

6

(d)

14 12 10

4

8

Sc/Δ

Lc/hc

5

3

6

2

4

1

2

Fig. 10(b)

Fig. 12(b)

Fig. 15(b)

Fig. 17(b)

Fig. 10(a)

Fig. 12(a)

Fig. 15(a)

Fig. 17(a)

Effect of coating CZM on coating cracking Effect of bonding layer CZM on coating cracking

Fig. 10(b)

Fig. 12(b)

Fig. 15(b)

Fig. 17(b)

Fig. 18. Comparisons of effects of bonding layer adhesive properties and coating fracture properties on coating cracking by summarizing the following variables from the previous Figs.: (a) the first cracking location and (b) the corresponding indentation depth, as well as (c) the overall length of cracks and (d) the average crack spacing.

increasing of the cohesive energy. Beyond that, there are some effects of interfacial delamination on coating cracking. Fig. 17(a) shows that both of the radius of the first coating crack and its corresponding indentation depth increase with the increase in cohesive energy of the bonding layer. It can be seen from Fig. 17(b) that the overall length of cracks decreases and the average crack spacing increases as the cohesive energy increases. Combined with the explanations about effects of cohesive strength and energy of the bonding layer on coating cracking, it is more likely to cause coating cracking if the bonding layer is easier to delaminate due to the energy transmission theory. From the above analyses, results on sensitivity of coating cracking to interface adhesive properties are summarized in Fig. 18. It does not appear to be much difference between effects of bonding layer adhesive properties and coating fracture properties on the first cracking location r/D from Fig. 18(a). Nevertheless, compared to the effect of coating fracture properties on coating cracking (Fig. 18(bed)), it is found that the variations (fluctuations) of h/hc, Lc/hc, and Sc/D scaling the effect of bonding layer adhesive properties on coating cracking are very small. Thus, the conclusion can be drawn that the coating cracking is not sensitive to the interface adhesive properties, which corresponds well with another work by Hu et al. [21].

make obvious changes on follow-up analysis results. Here, the simulation is respectively implemented for coating elastic moduli Ec =Es ¼ 0.5, 1.0, 1.5, 2.0, and 2.5 with the parameters of coating: tc0 =sys ¼ 4 and GCc =ðsys DÞ ¼ 0.08, as well as the parameters of the bonding layer: tb0 =sys ¼ 0.55 and GCb =ðsys DÞ ¼ 0.009. It can be seen from Fig. 19 (a) that increasing coating elastic modulus lowers both the first cracking radius and its corresponding depth. As shown in Fig. 19(b), the overall length of cracks increases with the increase of the coating elastic modulus, while the opposite trend holds for the average crack spacing. For the case of interfacial delamination (Fig. 20), the first delamination radius and its corresponding indentation depth decrease with the increase of the coating elastic modulus, but the opposite trend is for the case of the overall length of delamination. In a word, lowering the coating elastic modulus can reduce the risk of coating cracking and interfacial delamination. To a great extent, the radial stress in the coating sr and interfacial tractions (normal stress sN and shear stress sT) in the bonding layer depend on the coating elastic modulus of coating Ec. Accordingly, Hutchinson et al. [36] proposed a formula to evaluate the stress in the coating sc:

4.3. Effect of coating elastic modulus

where Z is the dimensionless coefficient. It can be seen that decreasing the coating elastic modulus Ec would lead to decreasing the stress in the coating sc under the conditions of other same calculation parameters. Thus, coatings of lower elastic modulus require more bending or equivalently larger indentation depths to

In general, the stress in a loaded body increases with the increase in its elastic modulus. Hence, relatively large parameter values of the coating and bonding layer are applied in this model to

sffiffiffiffiffiffiffiffiffiffiffi GC Ec sc ¼ Zhc

(7)

28

Y. Xiao et al. / Vacuum 122 (2015) 17e30 0.9

(a) First cracking location Corresponding indentation depth

12

10

1.0

(a)

First delamination location Corresponding indentation depth

12 0.8

0.8

11

0.7

0.9

r/Δ

6

0.5

10 0.6 9

0.5

0.4 4

0.4

8 0.3

2

(b)

0.5

1.0

1.5

Ec/Es

2.0

h/hc

0.6

h/hc

r/Δ

0.7 8

0.3 7

2.5

1.0

1.5

2.0

Ec/Es

2.5

8

(b)

3.5

0.5

7 3.0

7

6 6

5

Ld/Δ

Overall length of cracks Average crack spacing

2.0

S c/ Δ

Lc/hc

2.5

5

1.5

4 4

1.0

3

Overall length of delamination

0.5 1.0

1.5

Ec/Es

2.0

2.5

3

2

Fig. 19. (a) The first cracking location and the corresponding indentation depth, and (b) the overall length of cracks and average crack spacing under different coating elastic moduli.

reach the cohesive strength and subsequently crack or delamination [15]. 4.4. Effect of coating thickness

1.0

1.5

2.0

Ec/Es

2.5

As described in Fig. 22, the first interfacial delamination occurs farther to the axis of symmetry for thinner coatings (hc/D  3). Moreover, the thinner the coating, the larger the indentation depth required to generate the first delamination. It indicates that the interfacial delamination is less likely to produce if the coating is sufficiently thin. Liu et al. [18] also found the similar result. It results

1.0

12

First cracking location Corresponding indentation depth

10

0.9

8

0.8

6

0.7

4

0.6

2

h/Δ

The research subject in this study is thin coating, so coating thicknesses hc/D ¼ 0.5, 1, 1.5, 2, 2.5, 3, 4, 5, 6, 7, 8, 9, and 10 are respectively applied to the present model with parameters of the coating: tc0 =sys ¼ 2 and GCc =ðsys DÞ ¼ 0.03, and with parameters of the bonding layer: tb0 =sys ¼ 0.2 and GCb =ðsys DÞ ¼ 0.01. The cohesive energy of the bonding layer is artificially slightly enhanced to not allow the delamination damage easy occurring. The cases of the first cracking and the first delamination (including the critical indentation depths of cracking and delamination) are investigated. Fig. 21 shows that the critical indentation depth to cause the crack increases with the coating thickness over a critical value hc/D ¼ 3. The reason is that the stress in the coating sc decreases with the increase of coating thickness hc from Eq. (7). However, such a trend is reversed for thinner coatings (hc/D  3). Also, similar results have been observed by Madsen et al. [37]. On the other hand, the first crack initiates from the coating surface for thinner coatings (hc/ D  1.5), while the first crack is more inclined to generate from the interface for relatively thicker coatings (Fig. 21). For sufficiently thin coatings, the tensile stress along the interface on the coating side is not large enough to initiate crack, while the tensile stress along the coating surface is adequately large to generate crack.

0.5

Fig. 20. (a) The first delamination location and the corresponding indentation depth, and (b) the overall length of delamination under different coating elastic moduli.

r/Δ

0.5

0.5

0 0

1

2

3

4

5

hc/Δ

6

7

8

9

10

0.4

Fig. 21. The first cracking location and the corresponding indentation depth under different coating thicknesses.

Y. Xiao et al. / Vacuum 122 (2015) 17e30

According to the numerical results, the following conclusions can be drawn.

1.8

9

First delamination location Corresponding indentation depth

8

1.6 1.4 1.2 1.0

6

h/Δ

r/Δ

7

0.8

5

0.6 4

3

0.4

0

1

2

3

4

5

hc/Δ

6

7

8

9

10

29

0.2

Fig. 22. The first delamination location and the corresponding indentation depth under different coating thicknesses.

from saturation of the stress which occurs at lower stress levels for thinner films (Fig. 4). Nevertheless, it presents an opposite tendency once the coating thickness exceeds 3D, which can also be illustrated by the above Eq. (7). 4.5. Implications of the present analysis This study provides insight into the coating and bonding layer damages during the loading stage of indentation. It is by no means easy to observe and quantify the first damage location and its propagation on coating or interface by experiments. Hence, the results of the present analysis have direct implications in the accurate prediction of coating cracking and interfacial delamination during the indentation test, and can also build up a connection to an experimental assessment of the durability of brittle thin coatings (e.g., DLC coatings) by indentation. To characterize the fracture performance of coating and mechanical adhesion quality of bonding layer, cohesive energy alone is not enough, and the cohesive strength is equally important, especially for the bonding layer. Importantly, although the coating cracking is not sensitive to interface CZM properties, the interface CZM properties have some influences on coating cracking. Considering the relationship between the coating cracking and interfacial delamination, the weakly bonded brittle thin coatingsubstrate system can receive better protection by appropriately reducing the coating fracture performance. The weakly bonded feature can be observed in some special working conditions, like high temperature. This conclusion can be reasoned from the particular to the general. Therefore, if the bonding is weak on the interface, it is essential to improve it instead of just using a coating with high fracture performance. Additionally, this study can also help a great deal with defining rational coating elastic modulus and thickness to improve the durability of brittle thin coatings. 5. Conclusions A cohesive zone finite element model is built to simulate the response of a typical weakly bonded brittle thin coating-substrate (DLC coating with low adhesion on steel substrate) to indentation. The effects of the cohesive zones, coating elastic modulus, and coating thickness on the indentation response are studied. The interaction of coating cracking and interfacial delamination is also considered. Several illustrative numerical examples are presented.

1 In the respect of coating CZM (cohesive strength of 1.5e4 GPa and cohesive energy of 10e70 J/m2), increasing the cohesive strength or energy of the coating can effectively reduce the probabilities of crack initiation and growth, whereas it also increases the susceptibility to interfacial delamination failure. 2 For the case of weak bonding layer CZM (cohesive strength of 100e500 MPa and cohesive energy of 4e16 J/m2), a minimum in resistance to interfacial delamination occurs when the cohesive strength of the bonding layer reaches a critical value (~400 MPa). On the other hand, the larger its cohesive energy is, the firmer the interface becomes. Nevertheless, the coating cracking is not sensitive to the interface adhesive properties. 3 On the aspect of coating elastic modulus (100e500 GPa), it is more difficult to cause cracking and delamination with lower coating elastic modulus in a thin coating-substrate system. 4 For relatively thick coatings (hc > 3 mm), the critical load for coating/interface failure (cracking and delamination) increases with increasing coating thickness. However, such a trend is reversed for thinner coatings (hc  3 mm). Acknowledgments The authors would like to acknowledge the support from the China Scholarship Council (CSC), State Key Laboratory of Mechanical Transmission at the Chongqing University, and NSERC/General Motors of Canada Industrial Research Chair in the Department of Mechanical, Automotive and Materials Engineering at the University of Windsor. The authors also wish to express their gratitude to Prof. Ahmet T. Alpas for his helpful discussion and his generosity in sharing his scientific knowledge. References [1] S. Hainsworth, M. McGurk, T. Page, The effect of coating cracking on the indentation response of thin hard-coated systems, Surf. Coat. Technol. 102 (1998) 97e107. [2] S.A. Oliveira, A.F. Bower, An analysis of fracture and delamination in thin coatings subjected to contact loading, Wear 198 (1996) 15e32. [3] K. Holmberg, H. Ronkainen, A. Laukkanen, K. Wallin, Friction and wear of coated surfacesdscales, modelling and simulation of tribomechanisms, Surf. Coat. Technol. 202 (2007) 1034e1049. [4] X. Peng, T. Clyne, Mechanical stability of DLC films on metallic substrates part IIdinterfacial toughness, debonding and blistering, Thin Solid Films 312 (1998) 219e227. [5] J. Michler, M. Tobler, E. Blank, Thermal annealing behaviour of alloyed DLC films on steel: determination and modelling of mechanical properties, Diam. Relat. Mater. 8 (1999) 510e516. [6] A.A. Gharam, M. Lukitsch, M. Balogh, N. Irish, A. Alpas, High temperature tribological behavior of W-DLC against aluminum, Surf. Coat. Technol. 206 (2011) 1905e1912. [7] M. Swain, J. Men cik, Mechanical property characterization of thin films using spherical tipped indenters, Thin Solid Films 253 (1994) 204e211. [8] J. Michler, E. Blank, Analysis of coating fracture and substrate plasticity induced by spherical indentors: diamond and diamond-like carbon layers on steel substrates, Thin Solid Films 381 (2001) 119e134. [9] Y. Sun, A. Bloyce, T. Bell, Finite element analysis of plastic deformation of various TiN coating/substrate systems under normal contact with a rigid sphere, Thin Solid Films 271 (1995) 122e131. [10] K. Holmberg, A. Laukkanen, H. Ronkainen, K. Wallin, S. Varjus, A model for stresses, crack generation and fracture toughness calculation in scratched TiNcoated steel surfaces, Wear 254 (2003) 278e291. [11] X. Zhao, Z. Xie, P. Munroe, Nanoindentation of hard multilayer coatings: finite element modelling, Mater. Sci. Eng. A 528 (2011) 1111e1116. [12] M.T. Tilbrook, D.J. Paton, Z. Xie, M. Hoffman, Microstructural effects on indentation failure mechanisms in TiN coatings: finite element simulations, Acta Mater. 55 (2007) 2489e2501. [13] Z.C. Xia, J.W. Hutchinson, Crack patterns in thin films, J. Mech. Phys. Solids 48 (2000) 1107e1131. [14] S. Steffensen, H.M. Jensen, Circular channel cracks during indentation in thin films on ductile substrates, Comput. Mater. Sci. 98 (2015) 263e270. [15] A. Abdul-Baqi, E. Van der Giessen, Numerical analysis of indentation-induced

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