rus
Issue #5/2026
S.A.Votyakov, S.I.Votyakova, G.Kh.Sultanova, A.S.Useinov
SUBSTRATE EFFECT ON MEASUREMENT OF MECHANICAL PROPERTIES OF THIN FILMS BY A MICROSPHERICAL INDENTER
SUBSTRATE EFFECT ON MEASUREMENT OF MECHANICAL PROPERTIES OF THIN FILMS BY A MICROSPHERICAL INDENTER
INTRODUCTION
When developing and using structural materials, it is vital to assess their physical and mechanical properties. In many cases, a product’s performance is determined by the condition of its surface, as it is the surface layer that is the first to bear contact loads, friction and localised damage. Consequently, to improve wear resistance, corrosion resistance and contact strength, coatings are widely used, the mechanical properties of which must be assessed separately from those of the base material. One method of such assessment is indentation, in which the material resists the penetration of a harder indenter [1]. Instrumental indentation allows a small area of the specimen to be tested with virtually no damage to it, and provides information on the material’s resistance to deformation. During indenter loading, the material undergoes stages of elastic and plastic deformation, as well as a stage of failure; therefore, the indentation diagram can be used to analyse elastic, strength and plastic properties [1]. This paper focuses primarily on the load branch of the ‘load – penetration depth’ diagram (the P–h curve), as it is the shape of this curve that reflects the development of the contact interaction between the indenter and the ‘film – substrate’ system and allows us to trace the emergence of the substrate’s contribution to the measured response.
When studying thin films, the task becomes considerably more complex, as the measured response depends on the film’s properties, its thickness, roughness, the condition of the interface, residual stresses, and the size of the contact area and the properties of the substrate [2]. Consequently, as the penetration depth increases, the experimental plot may reflect not the intrinsic properties of the film, but the response of the entire ‘film – substrate’ system. To minimise the influence of the substrate, the 1/10 rule is often used, according to which the penetration depth should not exceed approximately 10 per cent of the film thickness. However, this rule should not be regarded as universal, as it is primarily intended as a practical guideline for measuring hardness, whereas for the modulus of elasticity, the permissible depth may be less [3].
The use of spherical indenters holds promise for analysing the mechanical response of thin-film systems. Historically, spherical indentation has been associated with the Brinell method, in which hardness is determined from the indentation mark produced when a ball is pressed into the material [1]. Unlike sharp indenters, a spherical indenter ensures a gradual increase in the contact area as the penetration depth increases, which makes it possible to investigate the initial stages of deformation and the transition from the elastic to the elastoplastic state of the material [4]. Spherical indentation is used to reconstruct the local deformation diagram of the material. In a number of methods, the loading region during the indentation of a spherical indenter is converted into an equivalent ‘stress-strain’ diagram. This approach is based on the relationship between the average contact pressure and the characteristic stress, whilst the characteristic strain of the material is related to the contact geometry [5]. This approach was subsequently developed in studies where a more accurate description of the contact geometry and finite element modelling were used to refine the ‘stress-strain’ diagrams [6]. In [7], spherical diamond indenters with radii of 5, 10 and 15 µm were used to reconstruct the stress-strain diagrams of steel based on indentation data. It was found that taking the actual geometry of the indenter into account improves the accuracy of the reconstruction of such diagrams, as the effective contact radius varies with the depth of penetration. A comparison of different geometries shows that spherical and pyramidal indenters involve different volumes of material in the deformation; consequently, their sensitivity to local inhomogeneities and the influence of the substrate differs [8].
Russian studies [9, 10] examine the manufacture and application of small microspherical diamond indenters. It has been shown that such tips enable the investigation of the local mechanical properties of materials. In particular, a microspherical indenter made of a single-crystal diamond with a radius of curvature of 2.5 µm was proposed as a means of obtaining ‘stress-strain’ diagrams from indentation data [10]. However, for the correct application of this approach, it is necessary to take into account the geometry of the indenter, the characteristics of initial contact and the method for calculating the contact area [10, 11].
When using a microspherical diamond indenter with a radius of 2.5 µm to study thin metal films, the key challenge is to separate the film response from the substrate contribution. Research into spherical indentation of film systems shows that the load curve depends on the ratio of the film’s and substrate’s moduli of elasticity, the film thickness and the indenter radius [12, 13]. Theoretical models of film stiffness also indicate that the contribution of the substrate must be considered in conjunction with the indentation geometry and the film thickness [14]. To correctly separate the contributions of the film and the substrate, it is necessary to take into account the evolution of the stress-strain state in the contact zone, as the influence of the substrate may manifest itself both through the propagation of the elastic field to the interface and through the onset of plastic deformation of the substrate [15]. It is therefore important, when interpreting P–h curves, to analyse the stresses distribution, strains and contact pressure within the ‘film-substrate’ system. The finite element method makes it possible to track the development of the deformation zone and to determine the conditions under which the measured diagram begins to depend significantly on the substrate [16–18].
In this study, nickel films on a 9XS steel substrate have been selected as the model system. This choice is methodologically justified, as nickel is a ductile metallic material with a pronounced elastic-plastic response, whilst 9XS steel is a widely used structural material in manufacturing, employed as a substrate for coating application. The "nickel film-steel" system makes it possible to observe how changes in film thickness affect the shape of the P–h curve. The experiment examines films with thicknesses of 9.2 and 3.7 µm, which allows a comparison of cases where the P–h curve exhibits different sensitivities to the influence of the substrate when using a microspherical indenter with a radius of 2.5 µm. Thus, despite the widespread application of the 1/10 rule, a separate assessment of acceptable measurement conditions for thin films is required for small-radius microspherical diamond indenters. The experimental data are compared with the results of finite-element modelling for film thicknesses ranging from 0.1 to 9.2 µm. This approach makes it possible to determine the depth range within which the measured response corresponds predominantly to the nickel film, and to formulate recommendations on the selection of penetration depth when investigating thin metallic films using a microspherical indenter.
RESEARCH METHODS
Experiment
Samples of nickel film on steel (9XS) with thicknesses of 3.7 and 9.2 µm were studied. The films were deposited by electroplating. The arithmetic mean roughness of the original steel surface was 40 nm, and that of the films was 70 nm.
Instrumental indentation tests were carried out using a spherical indenter with a radius of 2.5 µm on a "TESTURION T2" nanoindenter (Russia). The tests were carried out with a maximum loading force of 40 mN, as the effective working length of the tip is 800 nm, which limits the load range.
The measured mechanical characteristics are influenced by factors such as micro-irregularities on the contacting surfaces, noise, and changes in the probe’s speed when the criterion for its approach to the surface is triggered. To eliminate the last of these factors, a loading mode without feedback is used, in which the probe is moved at a set speed both before and after it comes into contact with the surface. The loading rate set for the experiment was 60 nm/s. The curves are smoothed by applying an averaging mask and decimating the data, which helps to eliminate spikes and outliers. An important step in data post-processing is determining the point of contact between the indenter and the surface, which defines the position of the curve and the strain at which plastic flow of the material begins.
Numerical modelling
Simulation of indentation using a spherical diamond tip was carried out using the finite element method in the COMSOL Multiphysics software package, in a two-dimensional axisymmetric configuration for a "nickel film/steel substrate" system. The indenter radius was 2.5 µm. An experimental validation of the computational model was carried out for a nickel film 9.2 µm thick at a maximum load of approximately 40 mN and a penetration depth of approximately 530 nm. Following the comparison of the computational and experimental "load-depth" curves (P–h), a series of calculations was carried out for film thicknesses ranging from 0.1 to 9.2 µm, and bulk nickel and steel were also investigated as limiting cases; the 9.2 µm film matches bulk nickel in terms of mechanical properties.
The nickel film and the steel substrate were modelled using an elastic-plastic model with isotropic hardening according to the Voise law. For nickel, the following parameters were used: E = 200 GPa, ν = 0.31, ρ = 8900 kg/m3, σ0 = 1.30 GPa, σsat = 2.0 GPa, β = 3.2; for 9XS steel: E = 210 GPa, ν = 0.30, ρ = 7750 kg/m3, σ0 = 0.34 GPa, σsat = 0.55 GPa, β = 5. The diamond indenter was modelled as a linearly elastic material with E = 1140 GPa and ν = 0.07. The contact was simulated without taking friction into account, and the lower boundary of the substrate was fixed. A local mesh refinement was applied in the contact zone, with a minimum element size of 35 nm on the sample surface and 70 nm on the indenter surface (Fig.1).
The accuracy of the model was assessed by comparing the calculated and experimental P–h curves. To analyse the mechanical response, the distributions of equivalent stress (Mises) and equivalent plastic strain were examined, enabling an assessment of the contact stress zone, the localisation of plastic flow and the extent of substrate involvement.
RESULTS
The influence of the substrate when measuring thin films is determined by the size of the contact area and the deformed region. Let us introduce a parameter, defined as the relative thickness of the coating (as a percentage of the film thickness), which determines the point at which the substrate begins to influence the measured mechanical properties of the coating:
η = 100 · hs / t,
where hs is critical depth at which the substrate begins to exert an influence; t – coating thickness.
Deviation of the load curves for films on a substrate from the curve of a bulk material makes it possible to assess the point at which the measured response ceases to correspond solely to the film and the substrate’s contribution begins to be detected, indicating the substrate’s involvement in the deformation process. Thicker coatings are most suitable for measurements, as even at a great indenter penetration depth, the substrate may not influence the measured properties of the coating, which will then correspond to the bulk material curve. Figure 2 shows the P–h curves (load-displacement curves) of nickel coatings, measured at a maximum indentation depth of hm ≈ 530 nm. Films of two thicknesses are shown: a thick film t1 = 9.2 µm and a thin film t2 = 3.7 µm relative to hm, at which the substrate is expected to influence the measured data. The curve for film t2 begins to diverge from that of film t1 at a depth of 130 nm, which corresponds to a parameter η = 3.5%. Although the standard recommendation for measuring thin coatings at hm/t ≈ 0.1 was followed, the value of η is significantly lower than expected.
A finite-element model was constructed to explain the interaction between the microspherical indenter and the ‘film-substrate’ system, as well as to explain the deviation of η from the expected values. The von Mises stress distribution for films t1 and t2 is shown in Figs.3a and 3b respectively; hm is similar to the experimental value. The region of highest stress in the film is concentrated beneath the indenter tip. As expected, interaction with a thick film does not lead to interaction with the substrate: the entire stress field is concentrated within the film. For a film with a thickness of t2 µm, the stress state spreads noticeably towards the interface and partially penetrates into the substrate. For this film, a noticeable interaction with the substrate is observed, rather than the onset of its influence; as can be seen in Fig.2, the influence of the substrate manifests itself before the depth hm.
A series of load curves was calculated for various film thicknesses: from 0.1 to 9.2 µm. The reference load curve is the curve for the 9.2 µm film, for which no substrate contribution is observed, as shown in Fig.3a. The calculations are shown in Fig.4. It is shown that for films of 5 µm thick and thicker, with a penetration depth of up to 530 nm, no significant deviation from the reference curve is observed. For thinner films of less than 5 µm, the curves begin to diverge at depths of hs; as expected, this depth decreases as the coating thickness decreases, and the curve approaches that of steel, i.e. the measured mechanical properties approach those of the substrate.
Table 2 shows the values of the parameter η for films of various thicknesses up to 4.5 µm, where the influence of the substrate is observed at a maximum depth of hm ≈ 530 nm. This parameter is difficult to determine for films thinner than 0.5 µm, as the load branch of such a film is barely distinguishable from that of the bare substrate. A value of η close to the experimental result corresponds to a 0.4 µm film. The significant discrepancies can be explained by the fact that the computational model is idealised: it does not take into account surface roughness, local thickness inhomogeneities, film defects, residual stresses, possible adhesion characteristics, and actual deviations in the indenter’s shape from an ideal sphere. These factors require further theoretical investigation. Thus, the model reveals the fundamental cause of the variation in the parameter η, which is linked to the spherical shape of the microindenter.
DISCUSSION
The characteristic contact size of a spherical indenter, unlike that of Berkovich’s self-similar ideal indenter, increases non-linearly with increasing contact depth:
as ≈ √(2Rhc),
where R is the radius of indenter.
When indenting thin films on a substrate with a spherical indenter, the value of η decreases as the film thickness decreases, in contrast to ideal self-similar indenters, where this value remains constant. This is due to the geometry of the spherical indenter, for which the characteristic contact size increases non-linearly. Consequently, for the same relative penetration depth, a larger relative contact area (as/t) is formed on thinner films. As a result, the stress-strain state reaches the interface with the substrate more rapidly, leading to a systematic decrease in the parameter η for thinner coatings. In particular, work [19] shows that even for self-similar Berkovich tips, different tip sharpness and tip-tip rounding alter the localisation of deformation and, consequently, the degree of substrate influence. This is consistent with the assumption that η changes with changes in film thickness for both spherical and rounded tips.
CONCLUSIONS
When indenting thin films of varying thicknesses with a microspherical indenter, it is to be expected that the influence of the substrate will manifest itself at different depths; under certain indentation conditions, this causes discrepancies in the P–h diagrams and, consequently, leads to discrepancies in the calculated mechanical properties of coatings made from the same materials. These circumstances explain the discrepancies in the measured mechanical properties of nickel coatings of different thicknesses. It has been found that, for microspherical indenters, the value of η during the indentation of coatings of varying thicknesses is not a constant, unlike in the case of self-similar indenters, where η is approximately constant and averages 10% of the coating thickness. This imposes new constraints on the process of measuring thin coatings using microspherical indenters.
Finite element modelling confirms this trend; moreover, as the model of the indenter and specimens is idealised, this effect can be attributed solely to the geometry of the spherical tip, for which the characteristic contact size increases non-linearly. Consequently, for the same relative penetration depth, a larger relative contact area is formed on thinner films. As a result, the stress-strain state reaches the interface with the substrate more rapidly.
The influence of the substrate in the experiment at depths shallower than those in the idealised model has important implications for the measurement of thin coatings using spherical microindenters. For a spherical microindenter with a radius of 2.5 µm, it is important to know the actual shape of the indenter and the surface quality of the coatings (roughness), particularly for films thinner than 4.5 µm, in which, even in the idealised case, the substrate will have an influence when η is less than 10%. Thus, a microspherical indenter with a radius of 2.5 µm is an effective tool for analysing the mechanical response of thin films; however, for coatings that are too thin, the contribution of the substrate becomes apparent earlier. Therefore, for measuring films whose roughness value is comparable to η (hs), it is recommended to use an indenter with a smaller radius.
ACKNOWLEDGMENTS
This work was carried out with financial support from the Russian Science Foundation (Grant No. 25-29-00291).
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
When developing and using structural materials, it is vital to assess their physical and mechanical properties. In many cases, a product’s performance is determined by the condition of its surface, as it is the surface layer that is the first to bear contact loads, friction and localised damage. Consequently, to improve wear resistance, corrosion resistance and contact strength, coatings are widely used, the mechanical properties of which must be assessed separately from those of the base material. One method of such assessment is indentation, in which the material resists the penetration of a harder indenter [1]. Instrumental indentation allows a small area of the specimen to be tested with virtually no damage to it, and provides information on the material’s resistance to deformation. During indenter loading, the material undergoes stages of elastic and plastic deformation, as well as a stage of failure; therefore, the indentation diagram can be used to analyse elastic, strength and plastic properties [1]. This paper focuses primarily on the load branch of the ‘load – penetration depth’ diagram (the P–h curve), as it is the shape of this curve that reflects the development of the contact interaction between the indenter and the ‘film – substrate’ system and allows us to trace the emergence of the substrate’s contribution to the measured response.
When studying thin films, the task becomes considerably more complex, as the measured response depends on the film’s properties, its thickness, roughness, the condition of the interface, residual stresses, and the size of the contact area and the properties of the substrate [2]. Consequently, as the penetration depth increases, the experimental plot may reflect not the intrinsic properties of the film, but the response of the entire ‘film – substrate’ system. To minimise the influence of the substrate, the 1/10 rule is often used, according to which the penetration depth should not exceed approximately 10 per cent of the film thickness. However, this rule should not be regarded as universal, as it is primarily intended as a practical guideline for measuring hardness, whereas for the modulus of elasticity, the permissible depth may be less [3].
The use of spherical indenters holds promise for analysing the mechanical response of thin-film systems. Historically, spherical indentation has been associated with the Brinell method, in which hardness is determined from the indentation mark produced when a ball is pressed into the material [1]. Unlike sharp indenters, a spherical indenter ensures a gradual increase in the contact area as the penetration depth increases, which makes it possible to investigate the initial stages of deformation and the transition from the elastic to the elastoplastic state of the material [4]. Spherical indentation is used to reconstruct the local deformation diagram of the material. In a number of methods, the loading region during the indentation of a spherical indenter is converted into an equivalent ‘stress-strain’ diagram. This approach is based on the relationship between the average contact pressure and the characteristic stress, whilst the characteristic strain of the material is related to the contact geometry [5]. This approach was subsequently developed in studies where a more accurate description of the contact geometry and finite element modelling were used to refine the ‘stress-strain’ diagrams [6]. In [7], spherical diamond indenters with radii of 5, 10 and 15 µm were used to reconstruct the stress-strain diagrams of steel based on indentation data. It was found that taking the actual geometry of the indenter into account improves the accuracy of the reconstruction of such diagrams, as the effective contact radius varies with the depth of penetration. A comparison of different geometries shows that spherical and pyramidal indenters involve different volumes of material in the deformation; consequently, their sensitivity to local inhomogeneities and the influence of the substrate differs [8].
Russian studies [9, 10] examine the manufacture and application of small microspherical diamond indenters. It has been shown that such tips enable the investigation of the local mechanical properties of materials. In particular, a microspherical indenter made of a single-crystal diamond with a radius of curvature of 2.5 µm was proposed as a means of obtaining ‘stress-strain’ diagrams from indentation data [10]. However, for the correct application of this approach, it is necessary to take into account the geometry of the indenter, the characteristics of initial contact and the method for calculating the contact area [10, 11].
When using a microspherical diamond indenter with a radius of 2.5 µm to study thin metal films, the key challenge is to separate the film response from the substrate contribution. Research into spherical indentation of film systems shows that the load curve depends on the ratio of the film’s and substrate’s moduli of elasticity, the film thickness and the indenter radius [12, 13]. Theoretical models of film stiffness also indicate that the contribution of the substrate must be considered in conjunction with the indentation geometry and the film thickness [14]. To correctly separate the contributions of the film and the substrate, it is necessary to take into account the evolution of the stress-strain state in the contact zone, as the influence of the substrate may manifest itself both through the propagation of the elastic field to the interface and through the onset of plastic deformation of the substrate [15]. It is therefore important, when interpreting P–h curves, to analyse the stresses distribution, strains and contact pressure within the ‘film-substrate’ system. The finite element method makes it possible to track the development of the deformation zone and to determine the conditions under which the measured diagram begins to depend significantly on the substrate [16–18].
In this study, nickel films on a 9XS steel substrate have been selected as the model system. This choice is methodologically justified, as nickel is a ductile metallic material with a pronounced elastic-plastic response, whilst 9XS steel is a widely used structural material in manufacturing, employed as a substrate for coating application. The "nickel film-steel" system makes it possible to observe how changes in film thickness affect the shape of the P–h curve. The experiment examines films with thicknesses of 9.2 and 3.7 µm, which allows a comparison of cases where the P–h curve exhibits different sensitivities to the influence of the substrate when using a microspherical indenter with a radius of 2.5 µm. Thus, despite the widespread application of the 1/10 rule, a separate assessment of acceptable measurement conditions for thin films is required for small-radius microspherical diamond indenters. The experimental data are compared with the results of finite-element modelling for film thicknesses ranging from 0.1 to 9.2 µm. This approach makes it possible to determine the depth range within which the measured response corresponds predominantly to the nickel film, and to formulate recommendations on the selection of penetration depth when investigating thin metallic films using a microspherical indenter.
RESEARCH METHODS
Experiment
Samples of nickel film on steel (9XS) with thicknesses of 3.7 and 9.2 µm were studied. The films were deposited by electroplating. The arithmetic mean roughness of the original steel surface was 40 nm, and that of the films was 70 nm.
Instrumental indentation tests were carried out using a spherical indenter with a radius of 2.5 µm on a "TESTURION T2" nanoindenter (Russia). The tests were carried out with a maximum loading force of 40 mN, as the effective working length of the tip is 800 nm, which limits the load range.
The measured mechanical characteristics are influenced by factors such as micro-irregularities on the contacting surfaces, noise, and changes in the probe’s speed when the criterion for its approach to the surface is triggered. To eliminate the last of these factors, a loading mode without feedback is used, in which the probe is moved at a set speed both before and after it comes into contact with the surface. The loading rate set for the experiment was 60 nm/s. The curves are smoothed by applying an averaging mask and decimating the data, which helps to eliminate spikes and outliers. An important step in data post-processing is determining the point of contact between the indenter and the surface, which defines the position of the curve and the strain at which plastic flow of the material begins.
Numerical modelling
Simulation of indentation using a spherical diamond tip was carried out using the finite element method in the COMSOL Multiphysics software package, in a two-dimensional axisymmetric configuration for a "nickel film/steel substrate" system. The indenter radius was 2.5 µm. An experimental validation of the computational model was carried out for a nickel film 9.2 µm thick at a maximum load of approximately 40 mN and a penetration depth of approximately 530 nm. Following the comparison of the computational and experimental "load-depth" curves (P–h), a series of calculations was carried out for film thicknesses ranging from 0.1 to 9.2 µm, and bulk nickel and steel were also investigated as limiting cases; the 9.2 µm film matches bulk nickel in terms of mechanical properties.
The nickel film and the steel substrate were modelled using an elastic-plastic model with isotropic hardening according to the Voise law. For nickel, the following parameters were used: E = 200 GPa, ν = 0.31, ρ = 8900 kg/m3, σ0 = 1.30 GPa, σsat = 2.0 GPa, β = 3.2; for 9XS steel: E = 210 GPa, ν = 0.30, ρ = 7750 kg/m3, σ0 = 0.34 GPa, σsat = 0.55 GPa, β = 5. The diamond indenter was modelled as a linearly elastic material with E = 1140 GPa and ν = 0.07. The contact was simulated without taking friction into account, and the lower boundary of the substrate was fixed. A local mesh refinement was applied in the contact zone, with a minimum element size of 35 nm on the sample surface and 70 nm on the indenter surface (Fig.1).
The accuracy of the model was assessed by comparing the calculated and experimental P–h curves. To analyse the mechanical response, the distributions of equivalent stress (Mises) and equivalent plastic strain were examined, enabling an assessment of the contact stress zone, the localisation of plastic flow and the extent of substrate involvement.
RESULTS
The influence of the substrate when measuring thin films is determined by the size of the contact area and the deformed region. Let us introduce a parameter, defined as the relative thickness of the coating (as a percentage of the film thickness), which determines the point at which the substrate begins to influence the measured mechanical properties of the coating:
η = 100 · hs / t,
where hs is critical depth at which the substrate begins to exert an influence; t – coating thickness.
Deviation of the load curves for films on a substrate from the curve of a bulk material makes it possible to assess the point at which the measured response ceases to correspond solely to the film and the substrate’s contribution begins to be detected, indicating the substrate’s involvement in the deformation process. Thicker coatings are most suitable for measurements, as even at a great indenter penetration depth, the substrate may not influence the measured properties of the coating, which will then correspond to the bulk material curve. Figure 2 shows the P–h curves (load-displacement curves) of nickel coatings, measured at a maximum indentation depth of hm ≈ 530 nm. Films of two thicknesses are shown: a thick film t1 = 9.2 µm and a thin film t2 = 3.7 µm relative to hm, at which the substrate is expected to influence the measured data. The curve for film t2 begins to diverge from that of film t1 at a depth of 130 nm, which corresponds to a parameter η = 3.5%. Although the standard recommendation for measuring thin coatings at hm/t ≈ 0.1 was followed, the value of η is significantly lower than expected.
A finite-element model was constructed to explain the interaction between the microspherical indenter and the ‘film-substrate’ system, as well as to explain the deviation of η from the expected values. The von Mises stress distribution for films t1 and t2 is shown in Figs.3a and 3b respectively; hm is similar to the experimental value. The region of highest stress in the film is concentrated beneath the indenter tip. As expected, interaction with a thick film does not lead to interaction with the substrate: the entire stress field is concentrated within the film. For a film with a thickness of t2 µm, the stress state spreads noticeably towards the interface and partially penetrates into the substrate. For this film, a noticeable interaction with the substrate is observed, rather than the onset of its influence; as can be seen in Fig.2, the influence of the substrate manifests itself before the depth hm.
A series of load curves was calculated for various film thicknesses: from 0.1 to 9.2 µm. The reference load curve is the curve for the 9.2 µm film, for which no substrate contribution is observed, as shown in Fig.3a. The calculations are shown in Fig.4. It is shown that for films of 5 µm thick and thicker, with a penetration depth of up to 530 nm, no significant deviation from the reference curve is observed. For thinner films of less than 5 µm, the curves begin to diverge at depths of hs; as expected, this depth decreases as the coating thickness decreases, and the curve approaches that of steel, i.e. the measured mechanical properties approach those of the substrate.
Table 2 shows the values of the parameter η for films of various thicknesses up to 4.5 µm, where the influence of the substrate is observed at a maximum depth of hm ≈ 530 nm. This parameter is difficult to determine for films thinner than 0.5 µm, as the load branch of such a film is barely distinguishable from that of the bare substrate. A value of η close to the experimental result corresponds to a 0.4 µm film. The significant discrepancies can be explained by the fact that the computational model is idealised: it does not take into account surface roughness, local thickness inhomogeneities, film defects, residual stresses, possible adhesion characteristics, and actual deviations in the indenter’s shape from an ideal sphere. These factors require further theoretical investigation. Thus, the model reveals the fundamental cause of the variation in the parameter η, which is linked to the spherical shape of the microindenter.
DISCUSSION
The characteristic contact size of a spherical indenter, unlike that of Berkovich’s self-similar ideal indenter, increases non-linearly with increasing contact depth:
as ≈ √(2Rhc),
where R is the radius of indenter.
When indenting thin films on a substrate with a spherical indenter, the value of η decreases as the film thickness decreases, in contrast to ideal self-similar indenters, where this value remains constant. This is due to the geometry of the spherical indenter, for which the characteristic contact size increases non-linearly. Consequently, for the same relative penetration depth, a larger relative contact area (as/t) is formed on thinner films. As a result, the stress-strain state reaches the interface with the substrate more rapidly, leading to a systematic decrease in the parameter η for thinner coatings. In particular, work [19] shows that even for self-similar Berkovich tips, different tip sharpness and tip-tip rounding alter the localisation of deformation and, consequently, the degree of substrate influence. This is consistent with the assumption that η changes with changes in film thickness for both spherical and rounded tips.
CONCLUSIONS
When indenting thin films of varying thicknesses with a microspherical indenter, it is to be expected that the influence of the substrate will manifest itself at different depths; under certain indentation conditions, this causes discrepancies in the P–h diagrams and, consequently, leads to discrepancies in the calculated mechanical properties of coatings made from the same materials. These circumstances explain the discrepancies in the measured mechanical properties of nickel coatings of different thicknesses. It has been found that, for microspherical indenters, the value of η during the indentation of coatings of varying thicknesses is not a constant, unlike in the case of self-similar indenters, where η is approximately constant and averages 10% of the coating thickness. This imposes new constraints on the process of measuring thin coatings using microspherical indenters.
Finite element modelling confirms this trend; moreover, as the model of the indenter and specimens is idealised, this effect can be attributed solely to the geometry of the spherical tip, for which the characteristic contact size increases non-linearly. Consequently, for the same relative penetration depth, a larger relative contact area is formed on thinner films. As a result, the stress-strain state reaches the interface with the substrate more rapidly.
The influence of the substrate in the experiment at depths shallower than those in the idealised model has important implications for the measurement of thin coatings using spherical microindenters. For a spherical microindenter with a radius of 2.5 µm, it is important to know the actual shape of the indenter and the surface quality of the coatings (roughness), particularly for films thinner than 4.5 µm, in which, even in the idealised case, the substrate will have an influence when η is less than 10%. Thus, a microspherical indenter with a radius of 2.5 µm is an effective tool for analysing the mechanical response of thin films; however, for coatings that are too thin, the contribution of the substrate becomes apparent earlier. Therefore, for measuring films whose roughness value is comparable to η (hs), it is recommended to use an indenter with a smaller radius.
ACKNOWLEDGMENTS
This work was carried out with financial support from the Russian Science Foundation (Grant No. 25-29-00291).
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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