On determination of elastic modulus and indentation hardness by instrumented spherical indentation: influence of surface roughness and correction method

Although, surface roughness can influence the determination of elastic moduli and indentation hardness to some extent by instrumented spherical indentation test, limited work has been done to quantitatively reveal and minimize these influences. In the present work, through a large number of finite element (FE) simulations and analyses, we clarified the evolution trend of determined elastic moduli and indentation hardness corresponding to different normalized indentation depths (h/R) and normalized roughness (S q/R). On this basis, an area correction method was proposed to improve the measurement accuracy in the elastic moduli and indentation hardness. The FE results show that, with the newly proposed correction method, the maximum relative error in determined elastic moduli is reduced from about ±7% to ±2%, and that in the determined indentation hardness is reduced from about ±13% to ±5%, when S q/R ≤ 2.2 × 10−3 and h/R = 5%. Applications were then illustrated on four typical metallic materials (i.e., AA 7075, AA 2014, steel 316 L, and copper T2). The experimental results demonstrate that the proposed correction method is able to mitigate the effects of surface roughness on the determination of elastic moduli and indentation hardness to obtain more correct results.


Introduction
With the aid of portable instrumented micro-indentation devices, instrumented indentation testing (IIT) has been widely used to evaluate the mechanical behavior of in-service engineering structures, such as high-speed rail [1,2], pipelines and turbine rotors [3]. In laboratories, IIT were also applied on metals [4,5], ceramics [6,7], coatings [8,9], shape memory alloys [10], etc, to determine mechanical parameters such as indentation hardness [11,12], elastic-plastic parameters [13][14][15], fracture parameters [16] and indentation activation volume [17]. Theoretically, to increase the accuracy in the determination of elastic modulus and indentation hardness, the surface of the sample to be tested should be mirror-flat. In practice, however the surfaces of the engineering structures have different degrees of roughness [18][19][20]. Generally, the surface roughness produced by conventional machining ranges from 0.025 μm to 25 μm. This may scatter the load-depth curves determined by instrumented spherical indentation tests and affect the identification of elastic modulus and indentation hardness [21,22]. Previous studies have shown that the nanoindentation test results at lower indentation depths are sensitive to initial contact position and associated high stresses, even on surfaces with low roughness [23]. Moreover, ultra-fine polishing on the surfaces of engineering structures is often challenging, and the effects of surface roughness on the determination of elastic modulus and indentation hardness using instruction spherical indentation test are not clear yet.
Finite element analysis (FEA) offer a powerful approach to study the effects of surface roughness on IIT [24]. Various models with rough surfaces can be created in Commercial FEA software enables the creation of various models featuring rough surfaces, among which the axisymmetric model stands out as a cost-effective option for simulating indentation testing [25]. Within the axisymmetric method, rough surfaces can be generated using different methods, such as employing a sine function, a single spherical asperity, or a dent (cavity) [24,26]. The height and wavelength of the asperities of the rough surface, as well as the initial contact position, have been found to affect the shape of the load-depth curve. In addition to analytical models, semi-analytical models with stochastic rough surfaces can be generated using numerical methods [27][28][29]. These methods are commonly used to study the effect of each standard roughness parameters of rough surfaces on the IIT [27] and to determine the most significant influence parameters. However, the rough surfaces represented by these models commonly differ from of the surfaces of the real-world engineering structures. Using scanning microscopy to obtain the actual morphologies of rough surfaces and applying them to the models is a promising approach to studying the effect of surface roughness on the determined mechanical properties of engineering structures using IIT [30,31].
To mitigate the influence of surface roughness on IIT, the following three primary methods, to the best of our knowledge, have been proposed. When the surface roughness is significant compared to the indentation depth, the indentation depths at different test positions may vary under the same indentation load. To ensure the measurement accuracy of IIT, researchers correct the indentation depth for different test positions. However, this correction requires measurements of the mean height of the rough surface and the height of pile-up/sink-in around the indenter, which might be hard to obtain to obtain on the surface of engineering structures. The scatter of the indentation hardness can be reduced by shifting the load-depth curve [32]. The shifting distance is related to the material properties and the indenter tip geometry [33]. Flatting rough surfaces is another effective approach to reducing the influence of surface roughness on IIT. In this method, a flat indenter is used to locally flatten the rough surface, and then spherical indentation testing is performed at the same position [34]. However, using this method on metallic materials will introduce additional residual stress during the flattening process, and in addition, the required indentation loads often exceed the capacity of portable indentation devices [35], Therefore, it is essential to study the influence of surface roughness on the measurement of instrumented spherical indentations, and propose corresponding solutions to facilitate the determination of mechanical properties of in-service structures using portable indentation devices.
In this study, the effects of surface roughness on elastic modulus and indentation hardness, determined by spherical indentation, were clarified using FEA. On this basis, an area correction method was proposed to improve the measurement accuracy of the elastic modulus and indentation hardness. Optimal indentation parameters, including the radius of the spherical indenter and the indentation depth, are given according to the various root-mean-square deviation (S q ). Experiments were carried out on four commonly used metallic plates (i.e., AA 7075, AA 2014, copper T2 and steel 316 L) with different roughness to show the influence of surface roughness and validate the effectiveness of the area correction method. Guidelines were provided for instrumented spherical indentation tests on engineering structures with rough surfaces.

Analytical method
The Oliver-Pharr method [36], widely used in IIT, was employed to determine the indentation hardness and elastic modulus. The elastic modulus can be expressed as: where E P is the elastic modulus, n is the Poisson's ratio of the specimen, E r is the reduced modulus, E i and n i are the elastic modulus and Poisson's ratio of the indenter, respectively. For spherical indentation, the reduced modulus can be calculated by: where S is the contact stiffness of the upper portion of the unloading data (50% ∼ 95%), A P is the projected contact area calculated by the Oliver-Pharr method at the maximum indentation depth. In the case of the spherical indenter is modeled as a rigid body, the elastic modulus can be determined as: for an ideal spherical indenter, the projected contact area can be calculated by: where R is the radius of the ideal spherical indenter, h c is the contact depth. The indentation hardness, H , P can be expressed as: where F max is the maximum indentation load during the indentation process.

Finite element modeling
Four typical rough surfaces (roughness S q = 0.067 μm, 0.089 μm, 0.110 μm and 0.130 μm, respectively) were produced by polishing with SiC abrasive papers from mesh numbers of #1500 to #3000. The morphologies of these surfaces were extracted by a laser scanning confocal microscope (OLS4000, Olympus) with resolution of 10 nm in the Z direction. To meet the resolution requirements of finite element models at micro scale [30], an area of 128 × 128 μm 2 was scanned by the laser scanning confocal microscope with a resolution of 125 nm in the X/Y direction. A three-dimensional model, consisting of an analytically rigid indenter with a radius of 50 μm and a homogeneous deformable sample, was created in ABAQUS 6.14 (Dassault Systèmes Simulia Corp, Providence, Rhode Island, 2014). The size of the deformable sample was significantly larger than the indentation depth (h). The rough surfaces on the upper surface of the deformable samples were generated from the scanning data, and the altitude difference between the rough surfaces and the surrounding smooth surface was 0.5 μm [37]. Figure 1 shows the flow chart outlining the process of creating a representative three-dimensional finite element model with a normalized roughness (S q /R = 1.78 × 10 −3 ). The contact constraint between the indenter and the sample was defined as surface-to-surface contact, and Coulomb's friction law was applied between the contact surfaces with a friction coefficient of 0.15. To ensure reliable simulation results, the mesh created in the contact region was the densest with a minimum mesh size of 0.15 μm. The models were meshed with 8-node linear brick elements of type C3D8R, resulting in roughly 95 000 elements generated. The sample was three-dimensional-fixed at the bottom (figure 1). In each simulation, the spherical indenter linearly penetrated the sample to a prescribed depth. To investigate the effect of surface roughness on the identified elastic modulus and indentation hardness, a control simulation was performed using a model with a flat contact surface. Because surface roughness has a greater impact on the instrumented indentation test at shallow indentation depths [23], the normalized indentation depths (h/R) were set as: h/R = 1%, 3%, 5%, 7% and 10%. For each model and h/R, nine indentation simulations were performed by only shifting the initial contact positions (i.e., the test positions). Two commonly used metallic materials, namely AA 7075 and steel 316 L, were selected for simulation. The material properties determined by tensile testing and the indentation parameters used in these finite element models were given in table 1.

Specimen preparation
Four representative metallic plates (AA 7075, AA 2024, steel 316 L, and copper T2) were machined into blocks measuring approximately 50 mm × 40 mm × 5 mm. Different surface roughness (S q ≈ 0.35 μm-1.3 μm) was produced by SiC abrasive papers ranging from mesh numbers #150 to #400. Note that the surface roughness varied locally on each specimen. Prior to the indentation test, the surfaces were partitioned and the local roughness of each testing area was measured. The dimensions of each testing area were 600 μm × 600 μm. To reduce the effect of residual stress between testing positions during the test, a testing interval between testing positions was maintained as 3 mm. Furthermore, to minimize boundary influence, each testing position was located at least 5 mm away from the specimen boundary. Figure 2 shows a representative partitioned surface of a specimen.
The local roughness of each testing area was determined by the laser scanning confocal microscopy. To assess the influence of surface roughness on the identification of elastic modulus and indentation hardness, four control specimens were included in the experiments. The control specimens were sanded using SiC abrasive papers ranging from mesh numbers #400 to #7000, followed by polishing with diamond polishing pastes to achieve ultra-fine surfaces [38,39].

Instrumented spherical indentation tests
Instrumented spherical indentation tests were performed at room temperature using a universal indentation hardness testing machine (ZHU2.5/Z2.5, Zwick/Roell) equipped with a tungsten carbide spherical indenter (nominal radius of 500 μm). Each specimen underwent more than 150 tests for each h/R (1%, 3%, and 5%). The tests were carried out in depth control mode, with a loading time and unloading times set to 30 s, and a holding time of 10 s. To calculate the mean values and standard deviation of the elastic moduli and indentation hardness,

Results and discussion
5.1. Decoupling the effects of pile-up and roughness on the Oliver-Pharr method Pile-up is a typical form of surface deformation during indentation [40][41][42]. Both pile-up and surface roughness cause variations in the projected contact area [42], which significantly influences the elastic modulus and indentation hardness calculated by the Oliver-Pharr method. The influence of pile-up on the identification of elastic modulus and indentation hardness can be determined from the control model (with flat surface). This allows to separately study the influence of surface roughness on the identification of elastic modulus and indentation hardness by 'subtracting' the influence of pile-up. In the control model, at each normalized indentation depth, A PM , E P and H P were determined from the simulation results. A PM was then substituted into equations (1), (2), and (5), respectively, to calculate the nominal elastic modulus (E PM ) and nominal indentation hardness (H PM ). The absolute errors between E P and E PM were calculated for different h/R values, and the same was done for the indentation hardness. As shown in figure 3, E P and H P are significantly greater than E PM and H PM . The absolute errors between E P and E PM increase with increasing h/R. A similar variation trend can be found in the indentation hardness as well. Thus, for the case that pile-up will occur during the testing, performing shallow indentations is desired for the Oliver Pharr method.
The evolution trends of rough surface elastic modulus (E P ′ ) and rough surface indentation hardness (H P ′ ) corresponding to different S q /R and h/R are shown in figures 4(a), (b), (e), and (f). The dotted lines in figure 4 represent the reference values of elastic modulus (E re ) and indentation hardness (H re ). The elastic moduli in table 1 were used as E re , and H PM identified from the control models were used as H re in the simulation results. As evident from figures 4(a), (b), (e), and (f), the evolution trends of E P ′ and H P ′ are both affected by pile-up and surface roughness. The simulation results ( figure 3(a)) confirms that pile-up induces absolute errors between E P and E PM . To separately study the influences of surface roughness on the identification of elastic moduli and indentation hardness, it is essential to eliminate the influences of pile-up. We believe that pile-up has the same effect on the identification of elastic modulus and indentation hardness, regardless of whether performing instrumented spherical indentation tests on rough surfaces or smooth surfaces. For each h/R, the partially corrected elastic moduli (E pp ) only affected by surface roughness can be obtained by:

Effect of roughness on the measurement accuracy
The degree of deviation between the mean values and the reference values was used to assess the effect of surface roughness on the measurement accuracy. Figures 4(c), (d), (g), and (h) show that the mean values of E pp and H pp are both decreasing with increasing S q /R, and the absolute errors from the reference values gradually increase.  The slopes of the fitted linear-lines for the mean values increase with increasing h/R. When h/R = 5% and S q /R = 2.60 × 10 −3 , the maximum deviation of E pp with respect to E re is within ±6.68%, and the maximum deviation of H pp is within ±13.35% To reduce the influence of surface roughness on the measurement accuracy of elastic modulus and indentation hardness, a large h/R value and a small S q /R value are desired in instrumented spherical indentation tests. As shown in figure 5, the contact stiffness and the maximum indentation load are decreasing functions of S q /R (figures 5(a) through 5(d)), while the projected contact area remains constant (figures 5(e) and (f)). Figure 6 shows the difference in contact areas for different test positions when h/R = 1%. For AA 7075, the contact area in the control model is the region with deformation of the sample along the indentation direction greater than 0.115 μm. On the rough surface, we assumed that areas with the same amount of specimen deformation are also contact areas. The contact areas on the rough surface vary from testing position to testing position and are generally smaller than the contact area on the smooth surface. At shallow indentations, the contact area can be approximately equal to the projected contact area. Thus, the decrease in the measurement accuracy of E pp and H pp is attributed to the projected contact area calculated by the Oliver-Pharr method.

Effect of roughness on the measurement precision
Another side effect of surface roughness on the identification of elastic modulus and indentation hardness is decreasing the measurement precision. As shown in figures 4(c), (d), (g), and (h), the standard deviation of E pp and H pp generally increase with increasing S q /R. To quantitatively demonstrate these trends, the variation coefficients (i.e., the ratio of the standard deviation δ to the mean value μ) were adopted to estimate the measurement precision of elastic moduli and indentation hardness. Figure 7 shows that the δ E /μ E and δ H /μ H increase as S q /R increases, and the relationship between the variation coefficients and S q /R can be described by a quadratic polynomial relationship. Whereas δ E /μ E and δ H /μ H , which lower than 2% and 3%, respectively, decrease with increasing h/R. This implies that the effects of surface roughness on the measurement precision of elastic modulus and indentation hardness are gradually mitigated as h/R increases. In the case of shallow indentation (h/R = 1%), δ E /μ E and δ H /μ H dramatically increase with increasing S q /R. When S q /R = 2.6 × 10 −3 , δ E /μ E and δ H /μ H reach their maximum values of 16.99% and 33.64%, respectively. Therefore, increasing h/R is an effectively way to reduce the influence of surface roughness on the measurement precision, leading to reducing the number of tests to obtain accurate test results.

Correction of projected contact area
When the sample to be tested has a relatively smooth surface, the measurement accuracy of the instrumented spherical indentation test can meet the requirements (within ± 5%, figure 4). However, the Oliver-Pharr method does not consider the influence of surface roughness in the calculation of A P . This is the main reason for the decrease in measurement accuracy of elastic modulus and indentation hardness on rough surface. To improve the measurement accuracy of elastic modulus and indentation hardness, an area correction method was proposed for the cases whose S q /R 1.34 × 10 −3 and h/R 5%. The principle of the partial area correction factor (λ) is: by correcting A PP , the partially corrected projected contact area that eliminates the effect of surface roughness, the E PP approaches E re . The modified elastic modulus can be expressed as: where λ, the square of the ratio of E pp to E re (equation (8)), is used to eliminate the influences of surface roughness on A PP , A PP-C and E PP-C are the corrected projected contact area and the corrected elastic modulus, respectively, which are unaffected by surface roughness and pile-up. As the decrease in measurement accuracy of the indentation hardness is induced by the calculation error of the projected contact area, this area correction method can be applied to the indentation hardness correction as well. Figure 8 shows the relationship between λ and S q /R can be described by a linear expression when 1.34 × 10 −3 S q /R 2.60 × 10 −3 . Thus, λ can be expressed as:  where j 1 is the slopes of the fitted line, and j 2 is the y-axis intercept of the fitted line. j 1 and j 2 can be fitted with h/R, and the best fitting result is achieved when h/R varies in the range of 1% to 5%. The λ of the two metallic materials exhibit approximate j 1 and j 2 values. By calculating the average value of j 1 for the two metallic materials, j 1 can be determined by: Both pile-up and surface roughness have significant effects on the determination of elastic moduli and indentation hardness through spherical indentation tests. When pile occurs, the elastic moduli and indentation hardness exhibit linear increasing trend with h/R (figure 3). To simultaneously eliminate the effects of pile-up and surface roughness on the identification of elastic modulus and indentation hardness, an expression related to h/R should be incorporated into λ. The evolution trends of the relative error (e E ) between E P and E re corresponding to different h/R values are shown in figure 9. The e E shows a near-linear relationship with h/R when h/R 5%. The fitting lines can be expressed as: for AA 7075, j 3 = 99.55 and j 4 = −0.13; for steel 316 L, j 3 = 105.21 and j 4 = 0.11. By calculating the mean of the fitted coefficients, the total area correction factor λ P can be expressed as:  Figure 11. Typical indentation morphology in the experiments.  Using λ P , the E PP-C and H PP-C corresponding to different S q /R and h/R values were obtained (figure 10). All the mean values of E PP-C and H PP-C approach the reference values as S q /R increases. The maximum deviations of the mean values of E PP-C and H PP-C with respect to the reference values are within ±1.38% and ±4.76%, respectively, when h/R = 5% and S q /R 2.60 × 10 −3 . Compared to E PP and H PP , the measurement accuracy of elastic moduli and indentation hardness is improved by over 4.09% and 6.23%, respectively, with the use of λ P . This indicates that λ P significantly increases the measurement accuracy of E P ′ and H P ′ . However, the use of λ P does not improve the measurement precision of E P ′ and H P ′ , thus a large number of tests are still necessary to obtain reliable test results at shallow indentation depths.

Experimental validation
Instrumented spherical indentation testing were performed on four metal surfaces with surface roughness. Figure 11 shows a typical indentation morphology obtained in the experiments. The evolution trends of E P ′ and E PP-C are shown in figures 12(a) through 12(d), and the evolution trends of H P ′ and H PP-C are shown in figures 12(e) though 12(h). The dotted lines in figure 12 represent the reference values of elastic moduli (E re ) and indentation hardness (H re ).
Consistent with the simulation results, the mean values of E P ′ and H P ′ decrease with the increase of S q /R. The relative errors of the mean values of E P ′ and H P ′ with respect to the reference values decrease with the increase of h/R. When h/R » 5%, the maximum relative errors of the mean values of E P ′ and H P ′ with respect to the reference values are within ±7.76% and ±7.26%, respectively. By applying our method to correct the projected contact area, the maximum relative errors decreased to ±2.11% and ±3.92%, respectively. In figure 12, S q /R, E PP-C , and H PP-C gently increase as the surface roughness increases (particularly at h/R = 1%). These trends may be attributed to the surface work hardening that occurs during the machining process of the surface roughness. Figure 13 shows the evolution trends of δ E /μ E and δ H /μ H corresponding to different S q /R and h/R. Akin to the findings in the simulation, the variation coefficients of δ E /μ E and δ H /μ H increase as the surface roughness increases. δ E /μ E and δ H /μ H show a dramatic increase with increasing S q /R, especially at h/R ≈ 1%, with maximum values reaching 17.04% and 16.59%, respectively. When h/R 3%, the measurement precision of δ E /μ E and δ H /μ H becomes less sensitive to the variation in surface roughness. When h/R ≈ 5%, the maximum values of δ E /μ E and δ H /μ H are both roughly within 4%. Therefore, within the applicable range of the area correction method, a large h/R should be selected to improve the precision of the test results.
Engineering structures always have rough surfaces. Performing in situ ultra-fine sanding and polishing on these surfaces were considerably challenging. Under-polishing may lead to an overestimation of the projected contact area by the Oliver-Pharr method, witch further affects the identification of the elastic modulus and indentation hardness. Correcting the projected contact area is considered a promising approach to improve the measurement accuracy of elastic modulus and indentation hardness. The experiment results demonstrate that a significant improvement in the measurement accuracy of elastic modulus and indentation hardness by applying the area correction method, when h/R is < 5% and S q /R 2.2 × 10 3 . For instrumented spherical indentation tests performed on the rough surfaces of engineering structures, minimizing the surface roughness is recommended. Additionally, setting h/R ∼5%, using a spherical indenter with a larger radius, and applying a greater load are beneficial for the accurate identification of elastic modulus and indentation hardness.

Future work
When conducting instrumented spherical indentation test on engineering structures, the test conditions are complex than those in the laboratory. To improve the reliability of the measurement results, it is crucial to comprehensively consider the influences of various factors, such as residual stress and the indentation slope. Moreover, during inspections of in-service engineering structures, it is essential to consider various mechanical properties beyond indentation hardness and elastic modulus. In near future, the influences of surface roughness on the identification of other mechanical parameters, such as strain hardening index and yield strain, will be studied. By addressing these aspects, the measurement accuracy and reliability of instrumented spherical indentation tests in engineering structures can be further improved.

Conclusions
The effects of surface roughness on the identification of elastic modulus and indentation hardness were separately investigated by eliminating the effect of pile-up. Subsequently, an area correction method was proposed to improve the measurement accuracy of elastic modulus and indentation hardness through the understanding of these effects. According to results of this investigation, the following conclusions are drawn: (i) During instrumented spherical indentation testing, the measurement accuracy of elastic modulus and indentation hardness decreases as the surface roughness increases and increases as the indentation depth increases. To mitigate these influences, it is advisable to use larger h/R value and a smaller S q /R value in the testing.
(ii) The measurement precision of elastic modulus and indentation hardness decreases as the surface roughness increases, particularly in shallow indentation. To reduce this influence, increasing the indentation depth is recommended.
(iii) The application of the proposed area correction method has significantly improved the measurement accuracy of the mean values of elastic moduli and indentation hardness by 4% to 22% and 6% to 38%, respectively. However, achieving reliable elastic modulus and indentation hardness at shallow indentation depths still requires a great number of tests.
(iv) The instrumented spherical indentation tests performing on four metals (AA 7075, AA 2014, copper T2 and steel 316 L), with over 150 tests performed on each sample for each h/R value (1%, 3%, and 5%), confirmed the influences of surface roughness on the determination of elastic moduli and indentation hardness, as well as the reliability of the area correction method.