Analysis of contact deformation between a coated flat plate and a sphere and its practical application

Transcription

1 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 307 Analysis of contact deformation between a coated flat plate and a sphere and its practical application T. Motoda 1, M. Shima 2, T. Jibiki 2, S. Sasaki 3 & K. Miyake 3 1 Graduate School of Tokyo University of Mercantile Marine, Japan 2 Tokyo University of Marine Science and Technology, Japan 3 AIST, Japan Abstract Recently nano-indentation with a triangular pyramid indenter has been widely used as a method of measuring Young s modulus of materials as well as the hardness. However, this method becomes difficult to measure Young s modulus of thin films less than 1 µm in thickness accurately since an indenter penetration depth less than several tens nm is required avoiding the effect of the substrate. In this study we attempt to measure Young s modulus of coating film by the elastic indentation of a spherical indenter into a coated flat. Young s modulus of film can be accurately measured without being affected by the substrate material if the relation between the elastic normal approach and Young s modulus of the coating film can be obtained for a given thickness of the film and given properties of substrate material. In order to attain the purpose an analytical method, which takes the elastic deformation of the substrate into consideration, is developed to calculate such the contact deformation, and the possibility of applying the calculation to measurement of Young s modulus of films is examined. Keywords: coating film, spherical indenter, Young s modulus, Hertzian contact, elastic indentation, new analytical method, nano-indentation. 1 Introduction Young s modulus of films coated on the surface of materials is an essential physical parameter in analyses of the exfoliation of films or in analysis of the elastohydrodynamic lubrication of coated machine parts. Recently, nano-

2 308 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII indentation with an obtuse triangular pyramid is widely used for measuring Young s modulus and the hardness [1]. In this method, Young s modulus around the micro contact area is measured utilizing the subsequent behaviour of the elastic recovery during the unloading after the pressing the indenter into the surface of materials. However, it is becoming difficult to measure them with accuracy if a thickness of the film is less than one micron, since several tens nm indentation depth is required avoiding the effect of substrate. In addition, piling up or sinking in will occur in some materials. Therefore the contact area should be observed by SEM or AFM for measuring Young s modulus with high accuracy. We have already presented a handmade apparatus, which can measure Young s modulus quickly with the elastic indentation of a sphere (3/8inch in diameters) into a flat surface [2]. This apparatus measures the displacement δ between the sphere and the flat as the function of load P, and then calculates Young s modulus by using Hertzian contact theory. The advance of the elastic indentation with the spherical indenter is that it does not need to measure the contact area. On the other hand, this method cannot apply to the contact between a sphere and a coated flat since Hertzian contact theory assumes the contact of two homogeneous elastic bodies. Recently, Chudoba et al present an analytical approach to derive P-δ curve [3]. They fit the analytical P-δ curve into the experimental P-δ so that they estimate Young s modulus of film. However in that method they solve under assuming that the contact pressure distribution is semiellipse distribution i.e. Hertzian pressure distribution. In this paper we present a new numerical analysis method, which provide the real contact pressure distribution and δ. As its application, we attempt to measure Young s moduli of some coating film by the indentation of a spherical indenter into a coated flat. 2 Method of numerical analysis In this study, a numerical analysis that combines three-dimensional axialsymmetry FEM and three-dimensional elastic contact theory is developed with the purpose of analyzing the contact between a sphere and a coated flat. Namely, an area slightly larger than expected contact area is partitioned into n number of small regions; then, contact pressure in each region is taken to be constant and a compatibility condition expressed by the following equation is applied to the centroid position of each region within the contact area (Fig. 1). n = 1 ( i = 1, 2, n) b f δ p ( Di + Di ) = zi, (1) where i and are subscripts indicating small regions. D i b and D i f are respective influence factors in the surface displacement of the sphere and in that of the coated flat surface that occur at the centroid position in the region i with the action of contact pressure p uniformly distributed in the region. Also, z i is the

3 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 309 initial gap before the application of load at the centroid position in the region i. The displacement δ is determined by solving eqn (1), taking into account two below conditions, the condition of force equilibrium, n = 1 A p = P and condition, A : surface area of element, P: compressive load, (2) p 0 (3) i within the contact area. Simultaneously, contact surface area, contact pressure p i can be determined. The influence factor D i b of the surface displacement of the sphere is calculated by using a method described in a previous publication [4]. Namely, a surface area slightly larger than the expected one is partitioned into small rectangles or small squares (Fig. 2) and the influence factor D i b of the surface displacement of the sphere that occurs at the centroid position in the region i by a unit pressure acting in the region is expressed by the following equation: D b i 2 (1 ν b ) = π E b a b dx dy a b 2 2 ( X + i X x ) ( Yi Y y ), (4) where X i, Y i and X, Y are centroid positions in regions i and, respectively. Then, a and b are a half-length of the side going in parallel to the Y-axis and a half-length of the side going in parallel to the X-axis in the region, respectively. E b and ν b are Young s modulus and Poisson s ratio of the sphere. Figure 1: Calculating method of normal approach and coordinate system. Figure 2: Division of contact area and influence factor D i b.

4 310 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII f Since an analytical treatment of the influence factor D i (including the influence of substrate) in the displacement of the coated flat surface is difficult, calculations are performed using the three-dimensional axis-symmetrical FEM technique to simplify the calculations. In this study, the above-mentioned small regions are obtained by partitioning into squares of equal size, and calculations are made by modelling the action of a unit force in equal circles to area of square regions to perform the FEM analysis of the three-dimensional axis-symmetrical problem. 3 Verification of numerical analysis method In order to examine the validity of the established numerical analysis method, the contact of the sphere and the coated flat in which the film properties are the same as the substrate ones is calculated, and obtained results are compared with theoretical figures. The assumed analytical model is a model of penetration of a sphere made of silicon nitride mm (3/8 inch) in diameter into a flat plate formed by applying a steel film 100 µm in thickness on a steel substrate. Young s modulus and Poisson's ratio of a sphere made of silicon nitride are 300 GPa and 0.28, respectively, and those parameters for steel are 208 GPa and 0.3, respectively. Since the contact area formed by the contact between sphere and flat surface is circular and symmetrical to the X-axis and the Y-axis, only 1/4 of the area is calculated. An area larger than expected contact area is partitioned into equal regions (a = b ) and the analysis is performed using the total number of elements n = 900. Results obtained are shown in Fig. 3 (a) and (b). (a) Contact pressure p (b) Displacement δ Figure 3: Comparison between numerical and theoretical results. It follows that the present analytical method combining three-dimensional elastic contact theory and the three-dimensional axis-symmetrical FEM technique allows the contact condition to be obtained to a sufficient accuracy. And also these results suggest that the established numerical analysis can

5 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 311 calculate the contact condition between the sphere and the coated flat even if properties of coating film are different as the substrate ones. 4 Parametric analysis It has been presented that in the case of the contact between a sphere and a coated flat a shape of contact pressure distribution is different from a simple elliptical shape [5-7]. Here we perform a wide range of parametric analysis to examine the effects of film thickness and Young s modulus on the shape of contact pressure distribution in detail. The calculated conditions are shown in Table 1. Young s modulus of film E f is changed from times Young s modulus of the substrate E s to 16 times one. This covers a wide range from lower Young s modulus, such as high-polymers material, to higher one, such as DLC. The thickness of film is changed from times a H (contact radius by Hertzian contact theory in the case of no film), to 16times a H. The normal load is constant (100mN). The results on E f /E s = , 0.25, 4, and 16 are shown in Fig. 4. Table 1: Calculated conditions. In the case of E f > E s, if t/a H >2, the contact pressure distributions are almost equal to that of t/a H =. Thus we can regard the contacts as those between the sphere and the infinitely thick-coated film, so that those contact problems can be calculated by Hertzian contact theory. However, if t/a H < 1, the contact radius is gradually extended and the contact pressure rapidly decreases. In addition in the cases of E f /E s =16 and t/a H = and 0.25, it is clear that the contact pressure distributions are different from elliptical ones, especially at t/a H = The maximum contact pressure doesn t exist at the centre of the contact, but along the concentric circle. Those phenomena occur under the conditions of E f /E s 8 and at t/a H = In the case of E f < E s, the contact is almost the same as the contact between the sphere and the infinitely thick-coated film, if t/a H > 2. However, there is a certain deviation between them even if t/a H > 4. This suggests that the effect of the substrate is more remarkable in the case of E f < E s than E f > E s. The effect of the coating film on the displacement is examined using the coated flats shown in Table 2. The assumed indenter is a sphere made of silicon nitride 9.525mm in diameter. Young s modulus and Poisson s ratio of it are 297GPa and 0.28, respectively. The results, including the theoretical curves, are shown in Fig. 5(a). When the applied load is high, the effect of the coating films

6 312 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII on the displacement is negligible and the calculated P-δ curves approximately equal to the theoretical curves. On the other hand, when the applied load is quite low, the effect of the coating films on the displacement is marked and the calculated P-δ curves approach the theoretical curves. Figure 4: Contact pressure distribution. The relationship between a/t and the deviation of displacement (D = (δ - δ')/δ' 100) is shown in Fig. 5 (b), where a/t is ratio of the contact radius to the thickness of the film, δ and δ' the displacement between the sphere and the coated and non-coated flats, respectively. The large value of D means that the effect of the substrate on the displacement is low and that the effect of the film is high. Inversely, the small value of D means that the effect of the substrate is strong. In both cases of E f /E s = 4 and 0.25, the obtained results show that D rapidly drops with the increase in a/t, and then gradually decreases from around 3 of a/t. From these results, the value of a/t should be taken as small as possible to measure Young s modulus of films accurately.

7 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 313 Table 2: Calculated conditions. (a) P-δ curves Figure 5: (b) The deviation of displacement Analysis of normal approach. 5 Application to measurement of Young s modulus of film Here Young s modulus of coating film is measured as an application of established numerical analysis. The principle of the measuring system is as follows. Firstly, the indentation is performed to measure the displacement between the sphere and the coated flat. Secondly, the relationship between the displacement and Young s modulus of the film is calculated by the established numerical analysis. At this time Young s modulus and Poisson s ratio of the sphere and the substrate and also thickness of film must be known as a calculating condition. Then, Young s modulus is determined when that obtained empirical displacement is substituted for the calculated relationship. An experimental apparatus used for simultaneous measurements of the displacement and the applied load is shown in Fig. 6. The apparatus has been described in detail in an earlier publication [2] and it is only outlined in the present work. A spherical indenter is attached to a hardness tester (Vickers hardness testing machine) that is used as a loading device and brought in contact with a flat plate by applying very low load. Thereafter, the test load is applied. Two electrostatic capacity-type displacement sensors that are placed

8 314 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII symmetrically to the loading axis detect the displacement taken place at that time. Simultaneously, a load cell placed at the bottom surface of the guide axis measures applied load. Generated signals are converted by a 12 bit AD converter and sent to a PC, where the displacement δ- the load P curve is determined. Figure 6: Apparatus. Figure 7: Cross-section (WC +17Co). In the present study, Young s moduli of thermal sprayed coatings of two kinds: WC + 12% Co and WC + 17% Co, coating material being of the carbide ceramics type, were measured. In this case, the effect of the Co content and the coating thickness on Young s modulus of sprayed coatings is examined. A spherical indenter used in the experiments is a sphere made of silicon nitride (diameter mm, Young s modulus 297 GPa, Poisson's ratio 0.28, and hardness HV 1428). A disk made of bearing steel (SUJ 2: Young s modulus 208 GPa, Poisson's ratio 0.3 and hardness HV 542) is used as substrate. The substrate is provided with sprayed coating with target thicknesses of 50, 100, 200, and 300 µm. After the indentation tests, in order to measure the actual thickness of coating film the tested specimen is cut to obtain a cross section, and the coating film thickness is measured by using a microscope. A cross-section of specimen is shown in Fig. 7. The surface of thermal sprayed coating is finished by lapping. The centre line average height Ra in the surface profile is 0.03~0.05 µm. A test specimen is a disk 20 mm in diameter and 13 mm in thickness including sprayed coating. In addition, Young s modulus of sprayed coating is measured by the nano-indentation using the same test specimen to verify results obtained by using the present measuring system. Typical measurements of the load P-the displacement δ curve are shown in Fig. 8. The displacement is determined on the basis of the above curve for P s = 49 N, which is a comparatively low load, considering the possible breakage of sprayed coating, and then Young s modulus of coating film is measured. Young s moduli of sprayed coatings determined by the nano-indentation and by the present measuring system are shown in Fig. 9. Poisson's ratio of coating is Measurements using the present measuring system are performed five

9 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 315 times under the same conditions and the reproducibility of measured results is determined. In the figure, the range of scatter and the average value are shown. Nano-indentation tests are performed 11 times under the same conditions by using the continuous stiffness measurement (CSM) [8] and measurement data are obtained at the indentation depth of about 2 µm, at which Young s modulus is practically constant. In the figure, the range of scatter and the average value are shown. Results obtained by the present measuring system and the average values obtained by the nano-indentation conform comparatively well. Since the value of both results over thicknesses of films is approximately 130 GPa and a H calculated by Hertzian contact theory is 109 µm, these experiments are performed in E f /E s = and 0.7 < t/a H < 2.9. WC + 12% Co coatings show a constant Young s modulus for coatings of different thickness, but WC + 17% Co coatings show a lower Young s modulus at a coating thickness being 300 µm than Young s modulus of coating with other thickness. This tendency is observed in results obtained by both of the methods. In addition, results obtained using the present measuring system show a narrower scatter than results obtained by the nano-indentation. This fact is explained as follows. The present measuring system generates a large contact area by using comparatively larger loads and sphere of larger diameter. Therefore, the acquisition of information about the average deformation of a wider region is much less affected by defects in sprayed coating, and, as a result Young s elastic modulus shows less scatter. On the other hand, in the nano-indentation, the region of indentation is very small and the site where the indenter penetrates is the structural heterogeneity or defects affect results. In this case, the scatter of data can be wide. Figure 8: Example of P-δ curves measured by the apparatus. Figure 9: Young s modulus. It follows from the above results that the measuring system presented in this work can be used for measuring Young s modulus of coatings and that results obtained practically conform to results obtained by nano-indentation tests. It can

10 316 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII also be stated that the present measuring system is useful for measurements of Young s modulus of coatings with many defects, for example, thermal sprayed coatings. Sprayed WC + Co coatings show no practical influence of the coating thickness and the Co content on Young s modulus, which amounts to 120~140 GPa. On the other hand, Young s modulus of sintered WC + Co hard alloy changes with the Co content and is in a range of 560~620 GPa [9]. This is much higher than Young s modulus of sprayed coatings obtained in the present experiments. These results indicate that Young s modulus changes significantly even in materials of similar structure in dependence on the method of their production. 6 Conclusions In the present study, in order to analyze the contact between the elastic sphere and the coated flat, a numerical analysis combining the three-dimensional elastic contact theory and the three-dimensional axis-symmetrical FEM is constructed. By using this established analysis, the contact pressure distribution and the displacement are analyzed in a wide range condition. The contact pressure distribution and the displacement are greatly dependent on the values of E f /E s and t/a H, and can be quite different from those estimated by Hertzian theory. Then measuring Young s moduli of the coating film of thermal sprayed coatings of two kinds: WC + 12% Co and WC + 17% Co is attempted as an application of the established analysis. The results show that those Young s moduli are almost constant without dependence of Co contents and film thickness. And also the obtained results are compared with nano-indentation tests, and Average of both results showed a good agreement. It suggests that if the applied load and the diameter of the sphere are reduced, Young s modulus of very thin film can be measured as well. Acknowledgement This study was supported by Takahashi Industrial Economic Research Foundation. The authors are deeply grateful. References [1] W. C. Oliver, G. M. Pharr, An improved technique for determining hardness and elastic modulus using load displacement sensing indentation experiments. J. Mater. Res., 6, 1564, [2] M. Shima, T. Motoda, T. Jibiki, T. Sugawara, New apparatus for measuring elastic modulus based on elastic contact deformation of sphere and flat. Journal of the Japanese Society of Tribologist, 47, 8, 663, 2002.

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