Model formulations

A schematic of the system of a substrate, a coating, and an indenter is shown in Fig. 1(h) where a Cartesian coordinate system is assigned. The x-axis is parallel to the scratch direction, the y-axis is pointing to the negative direction of the indentation (i.e., the out-of-coating direction), and the z-axis is pointing out of this figure. The z = 0 plane splits the structures into two symmetrical halves.

Figure 1.

Distribution of the normal stress σ _zzin the thin films at the indentation depth of 0.1 µm. (a) The mesh distribution in half a film, and the green box region in (a) is the same region as (b). (b), (d), and (f) With prestress σ _xx = −σ _{y 1}, and (c), (e), and (g) do not have prestress applied. The film thickness is 3 µm (b, c), 6 µm (d, e), and 18 µm (f, g), respectively. (h) Schematic of the scratch process on a prestressed film coated on a substrate.

The loadings are applied in three continuous steps. First, a homogeneous prestress at the coating is applied, and then the coating is attached to a substrate. Second, a vertical external force F _n is continuously exerted along the y-axis on the spherical indenter to compress it into the coating surface, which is called the indentation step. Third, a horizontal external force F _t along the x-axis and a moment M _z are applied to move the indenter in the x-axis direction at a fixed indentation depth without the indenter rotation, which is called the scratch step. The scratch under a constant indentation depth is widely employed in both experiments and FEM simulations on scratch tests.^{[Reference Feng and Chen10–Reference Feng12]}

In Fig. 1(h), the initial thickness of the coating in the undeformed configuration is considered 3, 6, and 18 µm, respectively, and the coating width along the z-axis and the length along the x-axis is 160 and 300 µm, respectively. The radius of the spherical indenter r is 200 µm which is a common geometry for the indenter as defined in European Standard EN 10109-2,^{[Reference Holmberg, Laukkanen, Ronkainen, Wallin and Varjus20]} and the indenter is indented and scratches at the middle of the thin film width. This system geometry is consistent with the previous study.^{[Reference Feng and Chen10–Reference Feng12]} The coating thickness in a few to several tens of micrometers is also commonly used in both FEM and experiments for the scratch tests.^{[Reference Bucaille, Gauthier, Felder and Schirrer18,Reference Holmberg, Laukkanen, Ronkainen, Wallin and Varjus20,Reference Li and Beres21]} The following material properties were used for the metallic coatings^{[Reference Feng11,Reference Ott, Blue, Santella and Blau27]} : yield strength $\sigma _{{\rm y}1} = 234{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm MPa}$, Young's modulus E ₁ = 74 GPa, and v ₁ = 0.3. For the high-speed steel substrate^{[Reference Holmberg, Laukkanen, Ronkainen, Wallin and Varjus20]}: $\sigma _{{\rm y}2} = 4.1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm GPa}$, v ₂ = 0.29, and $E_2 = 200{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm GPa}$. Coulomb friction coefficient is μ = 0.3. The elastic modulus and yield strength are much larger in the substrate than those in the soft coating, and under the loading used in this paper, the substrate is under extremely small elastic deformation without plasticity, which does not affect the deformation of soft thin film.^{[Reference Feng and Chen10–Reference Feng12]} Therefore, the substrate can be considered a rigid body. The indentation or scratch depth is 0.1 µm. All results discussed in this paper are at a depth of 0.1 µm, and the ratio between indentation/scratch depth and the indenter radius is 1/2000. The size of the thin film is described in micrometers above. If the size of the scratch system and indentation depth are multiplied by n, the patterns of stress fields will be the same as in this paper and the applied force will be changed from F _n (or F _t) to n ² · F _n (or n ² · F _t). Because of the symmetry with respect to the z = 0 plane, only half of the structure will be analyzed.

In traditional Coulomb friction, the slippage on the contact surface starts only when the magnitude of the friction stress τ, τ = |τ| reaches the critical friction stress τ _crit = μσ _n where σ _n is the normal contact stress and μ is the traditional Coulomb friction coefficient. In addition, once friction stress τ reaches the shear yield strength of material $\tau _{\rm y}\, = \,\sigma _{\rm y}/\sqrt 3 $ (von Mises yield condition is utilized and σ _y is the yield strength) in the plastic state, slippage can also initiate even if τ < μσ _n. Therefore, the traditional Coulomb friction can be modified by defining the critical friction stress as $\tau _{{\rm crit}}\, = \,\min \,(\mu \sigma _{\rm n},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \tau _{\rm y})$, and when the friction stress τ reaches τ _crit the slippage can occur. A total system of friction equations can be found in previous models,^{[Reference Feng and Chen10,Reference Feng and Levitas28]} and it is used for the contact surface between the indenter and thin film during the indentation and scratch steps. The cohesion condition between the contact surfaces of the film and substrate is used, and these two surfaces are always tied together and there is no sliding between them.

The rigid body assumption for the indenter is accepted in all pioneering FEM models in scratch tests in the literature, which is also used in this paper. The elastoplastic models introduced below will be used for both the substrate and coating, while the high-speed steel substrate is almost rigid under the loading applied in this paper. The position vector of a particle in the current deformed configuration ${\bi r} = {\bi r}({\bi r}_0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm t})$ is described by a function of the current time instant t and the initial position vector r ₀ in the reference undeformed configuration at the initial time t₀. As with Ref. Reference Feng, Levitas and Hemley29, the deformation gradient F = ∂r/∂r ₀ can be multiplicatively decomposed into a plastic F _p contribution and a symmetric elastic stretch tensor V _e as F = V _e · F _p. The small elastic deformation assumption ε _e = V _e − I (I is the second-rank unit tensor) is used here, but the material rotations and plastic strains are allowed to be large and considered in the deformation gradient F _p. A complete system of equations is summarized below.

The velocity gradient ${\bi D}= {\rm (}\dot{{\bi F}}\cdot {\bi F}^{-1}{\rm )}_{\rm s}$ is additively decomposed into elastic (subscript e) and plastic (subscript p) contributions:

(1)$${\bi D} = {\bi D}_p + \mathop {{\bi \varepsilon} _e} \limits^\nabla. $$

Subscript s stands for symmetrization of a tensor, and $\mathop {{\bi \varepsilon} _{\rm e}}\limits^{\rm \nabla} $ is the Jaumann objective time derivative of the elastic strain ε _e.

Hooke's law for deviatorics and volumetric (negative pressure p) components of the Cauchy stress T:

(2)$${\bi s} = 2G{\kern 1pt} {\kern 1pt} dev{\kern 1pt} {\bi \varepsilon} _e ;\quad -p = {\kern 1pt} \displaystyle{{\sigma _{xx} + \sigma _{yy} + \sigma _{zz}} \over 3} = K\varepsilon _{v {\kern 1pt}}, {\kern 1pt} $$

where ${\bi s} = dev{\bi T}$; $\varepsilon _{{\rm v}{\kern 1pt}} $ is the volumetric strain; G and K are the shear and bulk moduli, respectively.

Von Mises yield condition obeys

(3)$$\sigma _i = \left( {\displaystyle{3 \over 2}{\rm {\bf s}}:{\rm {\bf s}}} \right)^{0.5} \le \sigma _y, $$

in which σ _y is the material yield strength; σ _i is the von Mises effective stress: $\sigma _i = \sqrt {3/2{\bi s}:{\bi s}} $.

In the plastic region, plastic flow rule follows:

(4)$$\sigma _i\, = \sigma _{\rm y}\to {\bi D}_{\rm p} = \lambda {\bi s}\quad {\rm and}\quad \lambda \, \ge \,0.$$

The parameter λ is iteratively updated by satisfying the von Mises yield condition in Eq. (3), and the Newton–Raphson method and radial return method are frequently used for solving λ.

In the elastic region:

(5)$$\sigma _i\, \lt \sigma _{\rm y}\to {\bi D}_{\rm p} = 0\quad {\rm and}\quad \lambda \, = \,0.$$

Equilibrium equation yields

(6)$$\nabla \cdot {{\bi T}} = {\bf 0}.$$

In this paper the material is considered isotropic. When the size of the thin film is reduced to at or below several tens of nanometers the material may become single crystal. Anisotropy of single crystal theory can be found in Ref. Reference Feng, Bronkhorst, Addessio, Morrow, Cerreta, Lookman, Lebensohn and Low30.

Numerical results and discussion

In this section three-dimensional indentation and scratch models with prestress proposed in the previous section are simulated and discussed by using finite element code ABAQUS 6.14. Similar to our previous models,^{[Reference Feng and Chen10–Reference Feng12]} an 8-node linear brick element (C3D8R) is used in the simulations of this paper. Due to symmetry on the z = 0 plane only half of the structure is considered in simulations and discussion. As seen in Fig. 1(a), the front surface is the z = 0 plane. The number of elements is around 100,000 for the thin film, with 6–10 elements in the thickness direction. As shown in Fig. 1(a), the distance of the indentation location and the left boundary of film is 100 µm, which can exclude the boundary effects. The numerical results for the scratch test are obtained at the location when the indenter scratches 100 µm from the indentation location to exclude effects from the indentation location [see Fig. 1(a)]. Compared with the thin film strength (e.g., yield strength and Young's modulus) the substrate is a very hard material in this paper, and therefore the deformation in the substrate is completely elastic and extremely small. Therefore, we will only discuss the results for the soft thin film as in Refs. Reference Feng and Chen10–Reference Feng12. For Figs. 1–3 the homogeneous uniaxial compressive prestress in the scratch direction (i.e., the x-axis) σ _xx = −σ _{y 1} is applied, where σ _{y 1} is the yield strength of the films. For Fig. 4, in addition to the compressive prestress σ _xx = −σ _{y 1}, we also apply the compressive prestress σ _zz from 0 to − σ _{y 1}, so the prestress is biaxial. This helps us to understand the effects of the magnitudes of prestress in the z axis in a range (i.e., when it varies from zero to the maximum compressive prestress). In all cases the prestress is homogeneously distributed in the surface coating, and no plastic deformation occurs before the indentation step starts. The results may be very different if both plastic deformation and prestress are applied simultaneously, but this is not the focus of this paper. In addition, the results discussed in this paper are under an indentation or scratch depth of 0.1 µm for both the indentation and scratch tests.

Figure 2.

Distributions of the normal stress σ _yy(a, b) and σ _xx (c, d) of the thin films in the indentation test. (a) and (c) have the prestress σ _xx = −σ _{y 1}, and (b) and (d) are without prestress applied. The thickness of the films is 3, 6, and 18 µm.

Figure 3.

(a) Indentation force F _n applied on the indenter along the y-axis versus the film thickness for prestressed (σ _xx = −σ _{y 1}) and non-prestressed films in the indentation test. Normal force F _n (b) and transverse force F _t (c) applied on the indenter for the scratch distance 100 µm for prestressed (σ _xx = −σ _{y 1}) and non-prestressed films at the scratch depth of 0.1 µm. (d) The scratch friction coefficient μ _s :μ _s = F _t/F _n.

Figure 4.

Normal F _n (a) and transverse F _t (b) force applied on the indenter when the scratch distance is 100 µm and scratch depth is 0.1 µm for prestressed σ _xx = −σ _{y 1} and prestressed σ _zz varying from 0 to $\sigma _{y 1}{\kern 1pt} \;(compressive)$. In (a) the normal force F _n for the indentation test is also plotted. This film thickness is 18  µm.

Figure 1 presents the distribution of the normal stress σ _zzof the thin film in the indentation test. Figure 1(b) is the zoomed-in region of the green rectangle in Fig. 1(a). This nonzero stress region is quite small compared with the total volume of the thin film, and thus the boundary effects are excluded from our simulations. It is found from Fig. 1 that the patterns of the normal stress σ _zz distribution in the high-stress (large absolute values) region between the prestressed and non-prestressed films are very similar. However, the stress σ _zz is higher in the prestressed film, while there is no prestress applied in the z-axis. In the low-stress region the patterns of σ _zz are very different between with prestress and without prestress. In particular with a large thickness (e.g., 18 µm) the patterns between them are completely different. One of the important reasons why prestress applied in one direction affects the stress in another direction is that the von Mises effective stress σ _i is limited by the yield strength in shear, and thus the difference between two normal stresses cannot increase without a limit. In Ref. Reference Zhang, Cheng, Peng, Ma, Jiang, Hu and Chen14 the prestress effects in the bulk material are discussed with the Coulomb friction conditions. In this paper we extend the models not only for bulk material but also for thin films, so that combined Coulomb friction and plastic friction are considered. In Fig. 1, with an increase in film thickness, the patterns of the σ _zz distribution vary significantly. There are two obvious features: first, with an increase in thickness, the stress concentration reduces and the thinner film has the larger stress at a fixed indentation depth of 0.1 µm; second, the normal stress gradient along the thickness direction increases with the increase of film thickness. Specifically, σ _zz only varies very slightly in the film having 3 µm thickness, while in the film with 18 µm thickness σ _zz is very high at the contact surface but close to zero at the bottom of the film.

Figures 2(a) and 2(b) presents the distribution of the normal stress σ _yy of the thin film in the indentation test. In comparison with σ _xx and σ _zz, σ _yy is the largest normal stress in the absolute value due to the compression of the indenter into the film. The patterns of all three normal stress distributions are quite different. Similar to σ _zz, the stress concentration of σ _yy is larger with prestress than that without prestress. With increasing coating thickness the stress σ _yy concentration reduces, and the stress gradient increases in the thickness direction. The patterns of the σ _yy distribution with prestress and without prestress are almost the same in both the high- and low-stress zones while the patterns of σ _xx (or σ _zz) are very different between prestressed and non-prestressed films in the low-stress zones. As shown in Figs. 2(c) and 2(d), the minimum normal stress σ _xx in absolute value is equal to the yield strength σ _{y 1} of the prestressed film, while it is zero for the non-prestressed film. The stress concentration for σ _xx is larger in the prestressed film than that in the non-prestressed film, which is consistent with σ _yy and σ _zz. The stress σ _xx gradient along the thickness direction in the film with 18 µm thickness is larger than that in the 3 and 6 µm films. For the prestressed film the number of different colors in the σ _xx is less than the number in the non-prestressed film and the stress σ _xx gradient in the x and y directions is smaller in the prestressed film since there are less colors, which is different from the cases about σ _yy and σ _zz.

Figure 3(a) plots the vertical force F _n applied on the indenter along the y-axis for prestressed and non-prestressed films at the indentation depth of 0.1 µm. F _n is equal to the integration of the contact stress at the film along the direction normal to the undeformed film surface (i.e., along the y-axis):

(7)$$F_{\rm n} = \int {{\bi \tau} _y} da + \int {\sigma _{{\rm n}y}} da.$$

In Eq. (7), subscript y represents the components of the friction stress τand the normal contact reaction σ _n along the y-axis, and $\int {da} $ is the area of the contact surface. It is found in Fig. 3(a) that to reach the same indentation depth, the normal force F _n is much larger in the prestressed film than that in the non-prestressed film. Therefore, to cause the same amount of deformation of the thin film, a larger force should be applied on the prestressed film, and the material will become harder to deform for the prestressed film. For example, for a film thickness from 3 to 18 µm, the increase in the normal force F _n for the prestressed film is 10%–12% under the indentation depth 0.1 µm, in comparison with the non-prestressed film. In addition, with an increase in the film thickness the normal force F _n reduces drastically for very thin films; then the rate of the F _n reduction becomes slow and then becomes a constant for very thick films.

The discussion on Figs. 3(b)–3(d) and 4 is focused on the scratch test, and the results are obtained when the scratch distance from the initial indentation location is 100 µm as shown in Fig. 1(a). For the scratch test the indenter scratches on the film with a constant scratch depth of 0.1 µm. The scratch normal force is also determined by Eq. (7). While the indenter displacement in the y-axis is 0.1 µm for both indentation and scratch tests, Figs. 3(a) and 3(b) show that the normal force F _n in the scratch test is higher than F _n in the indentation test. Due to the indenter displacement in the y-axis, the film material in the front of the scratch is extruded out and film thickness is higher than the undeformed initial thickness. When the scratch continues the lump due to extrusion has to be scratched down to the indenter, and it requires an additional normal force, compared with compressing a flat surface in the indentation test. This is the reason why the normal force F _n in the scratch is higher than F _n in the indentation. For the film thicknesses varying from 3 to 18 µm, the increasing ratio of F _n for scratch from the non-prestressed to prestressed film is about 4%–23%, but it is close to a constant 11% for the indentation test. This difference between scratch and indentation is also caused by severe scratch deformations. For a very thin coating the difference in F _n between a prestressed and non-prestressed film during scratch is small.

Figure 3(c) plots the scratch transverse force F _t for the prestressed and non-prestressed films during the scratch test. F _t is the resistance force for the indenter moving along the scratch direction (i.e., along the x-axis), and it can be defined by

(8)$$F_{\rm t} = \int {{\bi \tau} _x} da + \int {\sigma _{{\rm n}x}} da,$$

where subscript x stands for the components of the friction stress τ and the normal contact stress σ _n along the x-axis. The scratch coefficient μ _s in Fig. 3(d) is defined by μ _s = F _t/F _n.

For the indentation test, F _t is zero while τ_x and σ _{n x} are both nonzero at the contact surface due to deformation of contact surface. For the scratch test the variation of F _t with the increasing film thickness has the same trend with the variation of F _n, therefore, the variation of μ _s = F _t/F _n is small during thickness changes as shown in Fig. 3(d). The difference in F _t between the prestressed and non-prestressed films is small for a thin film but it can be larger when the thickness reaches 9 µm. The resistance of the prestressed film for scratch becomes larger, and the indenter is more difficult to move on the prestressed film, in comparison with the case for the non-prestressed film. For thicknesses varying from 3 to 18 µm the increasing ratio of F _t for the prestressed film during scratch is about 5%–25% from the non-prestressed film. The scratch friction coefficient also represents the sliding capacity at the contact surface, and Fig. 3(d) shows that it is more difficult to slide on a prestressed film than a non-prestressed one due to a larger scratch friction coefficient. It is noted that μ _s in Fig. 3(d) can be lower than the Coulomb friction coefficient 0.3 in this paper, while the plastic deformation of contact surface in many cases increases the scratch friction coefficient. The scratch friction coefficient being lower than Coulomb friction for a soft coating is due to plastic slippage, in which the friction stress cannot surpass the yield strength in shear but the normal contact pressure on the contact surface can continually increase. This is similar to the experiment^{[Reference Sinha and Lim6]} and simulation^{[Reference Feng and Chen10]} findings that the scratch friction coefficient can decrease with an increase of indentation depth (or plastic deformation). For Figs. 1–3 the prestress is uniaxial along the scratch direction. Figure 4 shows the normal force F _n in the indentation and scratch, and the scratch transverse force F _t, for the biaxial prestress in which the compressive prestress in the x-axis is σ _{y 1} and the compressive prestress in the z-axis σ _zz varying from 0 to σ _{y 1}. It is found from Fig. 4 that indentation normal force F _n is significantly smaller than scratch normal force F _n. With an increase of prestress σ _zz the film deformation for indentation as in Fig. 4(a) and for scratch as in Fig. 4(b) becomes more difficult, and higher external forces (e.g., F _n and F _t) are required to deform the material and slide at the contact surface.