Asymmetric indentation: problem and solution

Abstract

Vickers and Knoop testers are the most common tools used to measure the hardness of materials. However, a very small tilt in the sample surface even in the order of a fraction of a degree produces a noticeable asymmetry, which affects the accuracy of the measured hardness. In this investigation, a mathematical approach has been used to correct asymmetry in the Vickers and Knoop indentations in both metallic and ceramic materials. Measurements were taken for metals such as aluminium 6061(Al-6061), 304 stainless steel as well as various zirconia toughened ceramic materials including tetragonal zirconia doped with: 2.5 mol% Y₂O₃ (TZ2.5Y), 3 mol% Y₂O₃ (TZ3Y) and the composite containing 20 wt% alumina (TZ3Y20A), all prepared from commercial powders. A hardness tester equipped with Vickers and Knoop indenters was used for hardness and elastic modulus determination. Optical and scanning electron microscopes have been used to get the indentation micrographs. The method enabled accurate determination of both hardness and elastic modulus from corrected Vickers and Knoop indentations. The hardness and elastic modulus values obtained in this study are in good agreement with reported data for similar materials. The results obtained in this study have been successfully validated using the Zwick reference block. The developed method is readily applicable for the most widely used Vickers hardness machines for the correction of asymmetric indentations if existing, consequently leading to accurate determination of the hardness.

Article highlights

A simple correction method for asymmetrical Vickers and Knoop indentations was developed and applied on metallic and ceramic materials.

This method enabled precise determination of hardness and elastic modulus values.

The obtained hardness and elastic modulus values were consistent with those obtained by other techniques for similar materials.

Introduction

Hardness measurement is very important in material characterization, especially mechanical properties, which include strength, toughness, elastic modulus and fatigue resistance [1, 2]. The hardness test, being a non-destructive method, easy to perform, makes it very popular and economic to be used in industry as an important quality control tool [2]. The principle to get the hardness value is very simple, just dividing the indentation load by the indentation projected surface area [3, 4].

Vickers and Knoop testers, are the most common tools, used to measure the hardness of materials in general and more specifically of metals, metal alloys and ceramic materials. The Vickers (HV) and Knoop (HK) hardness tests, are described in detail in the ASTM E-384 and E-92-16 standards [4, 5]. In order to get correct and accurate hardness values, the standards emphasis on taking the measurements of completely perfect symmetric indentations. In spite of the test simplicity, several sources of error can introduce asymmetry into Vickers and Knoop hardness indentations: including instrument limitations, operator error, sample preparation, and environmental conditions. These errors could be minimized by careful sample preparation; parallel sectioning and efficient grinding and polishing to assure scratch free mirror surface. The use of a stable appropriate non-vibrating mounting table, placed in a very low level vibration area, as well as a clean dust free optical path are of crucial importance. It is obvious that the indenter should be in the vertical direction and the sample support surface plane in the horizontal direction [4, 6]. In addition, it is imperative that the indentation maintains the original shape of the indenter even after the force has been released [4]. Furthermore, it is essential to perform a precise calibration using a standard test block, prior to conducting any test.

Neglecting these precautionary measures can led to errors in the hardness measurements. It should be noted that, a very small tilt in sample surface even in the order of a fraction of a degree produces a noticeable asymmetry. To obtain reliable and accurate hardness profile, the correction of the asymmetry would be required if a small tilt in the surface existed. Several works have studied the effect of sample tilt on indentation deformation of materials and have also provided valuable insights into potential solutions for the asymmetry in both micro and nano-indentations problems. Nano-indentation has advantages over traditional mechanical testing, providing both elastic modulus and hardness data from a single test. However, the size of the indenter tip and the sensitivity of the measuring system might be a disadvantage in case of heterogeneous or multi modal grain size materials as a result of the localized nature of its very small indentations.

Jakes & Staufer [7] used a geometric model to create contour plots for Berkovich and cube corner, from which the geometric area correction factors can be directly determined using only the ratios of side lengths measured from an image of the triangular nano-indentation. The authors reported that, they successfully got corrected hardness and elastic modulus measurements with surface tilts as high as 6°. However, their corrected values of hardness and E- modulus using standard analysis at tilt angles greater than 2°, showed increasing divergence of 15–20% from the standard analysis without tilt correction. They also stated that the geometric factor may not work in case of pile-up which is affected by surface tilt. Borc [8], studied the effect of surface tilt also in the range of 0° − 6° on nano-indentation under Berkovich indenter at different loads made on more plastic (pile-up) and more elastic (sink-in) samples, to explain the large discrepancy between the theoretical and experimental data. He found a distinct increase in the measured hardness at surface tilt angle θ > 3°, due to the increase of projected contact area with the tilt angle. He reported that the hardness and elastic modulus values determined at relatively high tilt angles, showed large deviations from the standard values.

The mathematical treatments elucidated by Huang et al. [9], to draw the corrected projected area were somewhat tedious since they were dealing with the triangular axes of the indentation as well as the principal Cartesian coordinate system. Liu et al. [10], made correction of asymmetric Vickers indentation by introducing a rotational angle besides the tilt angle. After a sophisticated analysis they concluded that there was no dependence on the rotational angle. Finite-element analysis by Kashani and Madhavan [11], showed that a tilt angle 5° causes a 12% error in hardness measurements, which can be reduced to 2% by the geometric correction.

In spite of using different indenters or methods of analysis; the above-mentioned authors agreed on the importance of making these corrections. This implies that there is a problem that must be faced and a solution for it might be very useful. In the present work the asymmetric indentation problem will be tackled and a solution will be elucidated to get the corrected dimensions of the indentations for reliable hardness and E-modulus measurements. The present mathematical treatment has the advantage of using very simple trigonometric analysis in Cartesian coordinate system. The limitation of this method of correction could arise; when applied on materials containing multi-phase components or of very large grain sizes or else having large surface pores or scratches. In addition, it is preferable that the tilt angle does not exceed 2° in order to get reliable measurements, which is in accordance with most of the above-mentioned references.

The manuscript is divided into five sections: Sect. 2 consists of two parts: one for Vickers hardness indentation and the other for Knoop hardness indentation. In each part, a schematic representation of the ideal case for hardness indentation, the basic equation used to calculate hardness, the asymmetric indentations resulting from tilt in the sample surface, and the mathematical equations used to find the corrected indentation are clearly explained. Section 3 focuses on the experimental procedures including materials used, steps of samples preparation, and methods used for indentations. In Sect. 4, the results and its discussion are given. Section 5 includes the conclusions of this work.

Theory

Vickers indentation

Vickers symmetrical indentation

The correct geometrical setting of a Vickers indenter aligned in a perfect vertical position normal to a flat and polished surface of a sample, is the ideal configuration. The symmetrical indentation produced by such a setting is shown in Fig. 1. The Vickers hardness HV, in kg/mm², is obtained from Eq. (1)

1

$$HV=1.854\frac{P}{d^2}$$where: p is the load in kg and d is the mean diagonal [d=(d₁ + d₂)/2] in mm, h is the penetration depth of the indenter.

Fig. 1 [Images not available. See PDF.]

The geometry of the ideal case of Vickers indenter and its indentation

Vickers asymmetric indentation

Figure 2 shows a schematic representation of Vickers asymmetric indentations resulting from tilt around X or Y and both X, Y axes. Practically, most of the asymmetric indentations result from tilt around only one axis. This comes from the fact that any universal grinding and polishing machine cannot become tilted around more than one axis at the same time, otherwise it will be defected.

Fig. 2 [Images not available. See PDF.]

 A schematic representation of Vickers asymmetric indentations resulting from tilt around X or Y and both X, Y axes

Figure 3 shows the geometry of asymmetric Vickers indentation in the case of asymmetry around the Y-axis with a tilt angle γ. The measured asymmetric half diagonals a₁, a₂, longitudinal diagonal angle (2ψ) and the indenter penetration depth (h) in the figure will be used to calculate the corrected indentation half diagonal (a_c). The measured diagonal d which equals a₁ + a₂ will be replaced by the corrected diagonal, d_c, which equals 2a_c to be used in Eq. (1) for hardness calculation. To calculate a_c the following simple trigonometric analysis can be applied:

2

$$\frac{a_1}{sin(90+\psi )}=\frac{a_c}{\text{sin}[90-\left(\psi ,+,\gamma \right)]}$$

3

$$\frac{a_2}{sin(90-\psi )}=\frac{a_c}{\text{sin}[90+\left(\psi ,-,\gamma \right)]}$$

4

$$\frac{a_1}{\text{cos}\psi }=\frac{a_c}{\text{cos}\left(\psi ,+,\gamma \right)}$$

5

$$\frac{a_2}{\text{cos}\psi }=\frac{a_c}{\text{cos}\left(\psi ,-,\gamma \right)}$$

6

$$\frac{a_1}{a_2}=\frac{1+\text{tan}\psi \text{tan}\gamma }{1-\text{tan}\psi \text{tan}\gamma }$$

Fig. 3 [Images not available. See PDF.]

Geometry of asymmetric Vickers indentation- tilt around the Y-axis

Finally, we can calculate the angle of asymmetry (tilt angle, γ)

7

$$\gamma ={\text{tan}}^{-1}\left[\frac{(a_1-a_2)}{(a_1+a_2)}\right]\frac{1}{\text{tan}\psi }$$

Knowing the tilt angle, the corrected half diagonal (a_c) could be obtained either from Eq. (4) or Eq. (5) to be used in turn to calculate the corrected Vickers hardness, HV_C, taking in consideration that the Vickers indenter diagonal edge angle (2ψ) = 148° 06’.

Knoop indentation

Knoop symmetric indentation

Figure 4 shows schematic representation of the geometry of Knoop indenter. The Knoop indenter has longitudinal edge angles 2ϕ = 172° 27’ and 2θ = 130° 5’, while a is equal to half the long diagonal and b is equal to half the short one. The length ratio of the long diagonal to the short one equals 7.051. Further, Fig. 5 shows the geometry of symmetric Knoop indentation.

Fig. 4 [Images not available. See PDF.]

Geometry of the Knoop indenter

Fig. 5 [Images not available. See PDF.]

Geometry of symmetric Knoop indentation

Knoop asymmetric indentation

When the above ideal configuration is not fulfilled, the indentation will be asymmetric as shown in Fig. 6. In the case (a) the tilt is around the Y-axis and in (b) it is around the X-axis, or it is around both X and Y-axis as in (c).

Fig. 6 [Images not available. See PDF.]

Asymmetric Knoop indentations:  a tilt around Y axis, b tilt around X-axis c tilt around X and Y-axis

From Figs. 4, 5 and 7 and using simple trigonometric relations similar to those used in section-2.1.2, Eqs. (8) and (9) could be easily obtained. The equations give the values of the tilt angles α and β as functions of the measured values of $a{}^{‵}1$, $a{}^{‵}2$, b′₁, b′₂ and the known values of the longitudinal edge angles of the Knoop indenter, 2ϕ = 172 ° 27’ and 2θ = 130 ° 5ˋ.

8

$$tan\alpha =\left[\frac{(a_2^{\prime }-a_1^{\prime })}{\left(a_2^{\prime }+a_1^{\prime }\right)}\right]cot\varphi$$

9

$$tan\beta =\left[\frac{(b_2^{\prime }-b_1^{\prime })}{(b_2^{\prime }+b_1^{\prime })}\right]cot\theta$$

Fig. 7 [Images not available. See PDF.]

Geometry of asymmetric Knoop indentation

Then, knowing α and β, the corrected diagonal dimensions of the indentation: a’_c and b’_c can be calculated from the following equations:

10

$$a_c^{\prime }=a_2^{\prime }\left(\frac{cos\left(\varphi +\alpha \right)}{cos\varphi }\right)=a_1^{\prime }\left(\frac{cos\left(\varphi -\alpha \right)}{cos\varphi }\right)$$

11

$$b_c^{\prime }=b_2^{\prime }\left(\frac{cos\left(\theta +\beta \right)}{cos\theta }\right)=b_1^{\prime }\left(\frac{cos\left(\theta -\beta \right)}{cos\theta }\right)$$

Marshall et al. [12] developed a simple technique for the determination of elastic modulus (E) using Knoop indentation measurements. This method was based on the elastic recovery of the diagonal dimensions of the Knoop indentation (aˋ, bˋ). They showed that for many materials the ratio of the dimensions of the residual indentations, bˋ/aˋ is correlated linearly with the Vickers hardness to elastic modulus ratio, HV/E, according to the following equation:

12

$$\frac{b^{\prime }}{a^{\prime }}=\frac{b}{a}-\alpha \left(\frac{\mathit{HV}}{E}\right)$$where: b/a = 1/7.051 derived from the dimensions of the Knoop indenter geometry, α = 0.45 is the slope of the linear regression curve fitting of the data measured by those authors. The Vickers hardness to the elastic modulus ratio (HV/E) could be determined with an accuracy better than 10%, according to the authors. In case of asymmetry, replacing bˋ and aˋ in Eq. (8) by the corrected half diagonals bˋ_c and aˋ_c determined from Eqs. (10) and (11), the corrected HV/E could be calculated. Knowing the corrected Vickers hardness HV_C, determined from Eq. (1) in section-2.1.1, the elastic modulus (E) could then be obtained.

13

$$\frac{{b^{\prime }}_c}{{a^{\prime }}_c}=\frac{b}{a}-\alpha \left(\frac{\mathit{HVc}}{E}\right)$$

Experimental

The metallic samples of about 1.2 cm thick plates of aluminum 6061 and 304 stainless steel were cut into 3 × 2 cm rectangular samples. The samples were ground and polished to a mirror finish. The ceramic samples made from TZ2.5Y, TZ3Y and TZ3Y20A powders (TOSOH) were prepared by uniaxial cold pressing at a pressure of 100 MPa, followed by sintering at 1600 °C in air for 2 h. The sintered ceramic samples were then ground and polished to a mirror finish using diamond disc (50 μm), followed by micro-cloth finishing with 7 μm and 3 μm diamond paste and finally with γ-Al₂O₃ slurry. Since grinding induces tetragonal to monoclinic phase transformation on the sample surfaces; putting it under a compression stress, annealing after samples’ surface grinding and polishing was important to remove the surface stresses. So, the samples were annealed at 1450 °C for 1 h.

The indentations were made using Vickers indenter with longitudinal edge angle 136° 30’- diagonal edge angle (2ψ) = 148° 06’ and Knoop indenter with longitudinal edge angle 2ϕ = 172° 27’, transverse edge angle 2θ = 130° 5’. The indenters were each mounted in Zwick, hardness tester, keeping the maximum indenter load for 10 s. Optical and scanning electron microscopy SEM, JEOL-5400, were used to get the indentation micrographs. From the micrographs, the average of measurements taken from six indentations made on the polished sample surface of each material was used in the calculations.

Results and discussion

Vickers indentations

The Vickers indentations made on the surface of aluminum 6061 and 304 stainless steel samples, using 10 N load are shown in Figs. 8 and 9. The figures show a tilt around the Y-axis for both samples, being bigger for the aluminum 6061 sample than it is for the 304 stainless steel one. Figure 10 shows the indentation at a load of 20 N, made on the surface of TZ3Y sintered ceramic as an example for the ceramic materials used.

Fig. 8 [Images not available. See PDF.]

Vickers indentation made on aluminum 6061 using 10 N load

Fig. 9 [Images not available. See PDF.]

Vickers indentation made on 304 stainless steel using 10 N load

Fig. 10 [Images not available. See PDF.]

Vickers indentation made on TZ3Y ceramic using 20 N load

Taking the measurements and applying the analysis given above in section-2.1.1 led to the results shown in Table 1.

Table 1. Hardness values obtained from corrected Vickers indentation measurements

Sample	ψ	*am=dm/2 µm	*HV GPa	a1 µm	a2 µm	γ°	*ac=dc/2 µm	*HVc GPa
Aluminum 6061	74.05	70.30 ± 2.00	00.94 ± 0.05	79.80	69.30	1.15	74.18 ± 1.80	00.83 ± 0.04
Stainless Steel 304	74.05	49.50 ± 0.50	01.89 ± 0.40	52.00	49.00	0.49	50.46 ± 0.40	01.82 ± 0.30
TZ3Y	74.05	60.50 ± 0.30	12.63 ± 0.10	64.00	58.10	0.81	61.2 ± 0.20	12.35 ± 0.10

*a_mis the arithmetic mean of the measured half diagonals, d_m is the arithmetic mean of diagonals lengths

*HV is Vickers hardness calculated using asymmetric indentation dimensions a₁, a₂

*a_c is the corrected half diagonal length, d_c is the arithmetic mean of corrected diagonals length

*HV_C is the corrected Vickers hardness calculated using the arithmetic mean of corrected half diagonals a_c.

From the table, it can be seen that even for small tilt angle (γ), of one degree or less, asymmetry was detected. Aluminum 6061 showed greater asymmetry than 304 stainless steel as a result of its larger tilt angle. Applying the corrected diagonal values in Eq. (1), the Vickers hardness values obtained for aluminum 6061and 304 stainless steel were found to correspond well to their known values [13, 14]. The value of Vickers hardness for TZ3Y is comparable to the published data [15, 16].

Liu et al. [10], studied the effects of sample tilt on Vickers indentation hardness using Vickers indenter with diagonal edge angle similar to that mentioned in section-2.1.2. They mentioned that the tilt angle should be lesser than 2° according to Laurent et al. [17], in order to limit the scatter of the hardness values within 10%. However, they expected more scatter than 10% in their results because of the large tilt angles. It seems from the abnormal indentation shapes that the indenter they used was partially damaged and misaligned around its edge. They obtained tilt angles 3.2 and 3.56 around the X-axis for two indentations. Substituting the asymmetric half diagonals of their two indentations in Eq. (7), section-2.1.2, tilt angles (around X-axis) of 2.9 and 2.96 were obtained, which shows a fair agreement between the two methods of analysis.

Knoop indentations

Figure 11a, b shows the Knoop indentations made on the polished surfaces of the 304 stainless steel and aluminum alloy (Al-6061) samples respectively, using 10 N load. A quasi-symmetric indentation with very small tilt around the Y-axis is noticed for the stainless-steel sample. In contrast, the Al-6061 sample showed large asymmetry due to a larger tilt angle resulting from imperfect sample preparation. Performing the analysis given before in section-2.2.2 and using Eq. (13), elastic modulus of 192 GPa was obtained for 304 stainless steel (Table 2) which is in good agreement with measured values for 304 stainless steel [14]. As for the Al-6061, elastic modulus of 67.9 GPa was obtained which corresponds well with the data sheets for this material [13].

Table 2. Elastic modulus values E obtained from corrected Knoop indentation measurements.

Material	a1 (µm)	a2 (µm)	α (deg.)	a′c (µm)	b′c (µm)	b′c/a′c	HVC (GPa)	E (GPa)
304 Stainless- steel	135.50	140.50	00.11	137.00	18.85	0.1376	1.82 ± 0.04	192.00 ± 1.00
Aluminum 6061	120.00	349.00	01.83	179.00	24.50	0.1363	0.83 ± 0.03	67.90 ± 0.60
TZ3Y20A (only polished surface)	51.20	112.50	1.41	70.60	8.50	0.1205	13.33 ± 0.12	279.00 ± 5.00
TZ3Y20A (polished & annealed)	58.67	90.67	0.81	71.23	8.60	0.1207	13.33 ± 0.07	284.00 ± 1.00
TZ2.5Y (polished & annealed)	77.33	84.00	0.16	80.35	9.40	0.1170	10.30 ± 0.08	191.00 ± 4.00
TZ3Y (polished & annealed)	72.00	84.50	0.30	77.73	9.00	0.1158	12.35 ± 0.10	218.00 ± 3.00

NB: b’= b, because of symmetry around X-Axis

Fig. 11 [Images not available. See PDF.]

Knoop indentations made on the surface of the samples a 304 stainless steel, and b aluminum 6061, using 10 N load

As for the sintered ceramic samples, they were almost of theoretical densities (above 99.5% TD), with smooth pore free surfaces, which is essential to avoid fluctuation in the measurements [1]. Figure 12a–c shows the SEM photomicrographs for the Knoop indentations at 20 N load, made on the polished and annealed sample surfaces of the single-phase materials TZ2.5Y, TZ3Y and the composite (two phase material) TZ3Y20A which contains 20 wt% Al₂O₃.

Fig. 12 [Images not available. See PDF.]

The SEM photomicrographs for the Knoop indentations at 20 N load for:  a TZ2.5Y, b TZ3Y, c TZ3Y20A

The Knoop indentations were in general asymmetric with different degrees of asymmetry depending on the sample type. Table II shows the elastic modulus values calculated from the corrected dimensions of the Knoop indentations (section-2.2.2) and the measured Vickers hardness values derived from the analysis given in section-2.1.2. From the table it can be concluded that the Knoop indentation made on the surface of aluminum 6061 was, like Vickers indentation, of greater asymmetry than the other materials tested. In the case of ceramic materials, the Knoop indentations made on the surface of the composite TZ3Y20A were more asymmetric than those made on the surfaces of TZ2.5Y and TZ3Y materials manifesting itself in the large difference between the measured dimensions a_l, a₂ and the tilt angle α as well. In other words, the degree of asymmetry was higher in the two-phase material TZ3Y20A than it was in the single-phase materials TZ2.5Y and TZ3Y which showed quasi symmetric indentations. The TZ3Y20A asymmetry might be a result of the large difference between elastic modulus of α-alumina (400 GPa) and that of the tetragonal zirconia (200 GPa), in agreement with recently published work on 3Y-TZP composites [18]. The elastic modulus calculated for the single phase TZ2.5Y and TZ3Y ceramic materials (Table II), being in the range 191–218 GPa are comparable to those obtained for stainless steel (192 GPa). From this, came the expression “ceramic steel” coined by Garvie et al. [19] in their famous work on zirconia ceramics.

The elastic modulus values obtained in this work are comparable to the values measured for similar materials, using different techniques like the resonance technique [15]. Table 3 shows a comparison between the Vickers Hardness, HV, and the elastic modulus values obtained in the present work and those in the literature. A reasonable agreement is noticed between the values obtained by applying the present correction approach, and those in the literature. However, the comparison given in Table III might not be a fair one due to the difference in the test conditions concerning the applied load and its dwell time in each case. So, to validate the method developed here for asymmetric indentation correction, a set of measurements were made on the surface of Zwick reference block (MPA NRW 34328.996) having a hardness 385 Kg/mm² (Fig. 13). The measurements were made on the reference block surface without tilt and on the same surface after introducing some tilt, using the same load (2 Kg) and dwell time 10 s. Table 4 shows the results obtained for the Vickers hardness values of the tilted surface of the reference block, HV_t, and the hardness calculated from the corrected asymmetric indentations HV_C, made on the reference block surface with different tilt angles. It can be seen that, the values of HV_C calculated from the corrected asymmetric indentation dimensions correspond well to the Zwick reference value.

Table 3. Comparison between the Vickers Hardness HV and elastic modulus E values obtained in the present work and those in the literature

Material	HV, GPa		E, GPa	Remarks
Stainless Steel 304	2.01		193.00	Ref [14]
Stainless Steel 304	1.82		192.00	present work
Aluminum 6061	0.94		68.90	Ref [13]
Aluminum 6061	0.83		67.90	present work
TZ3Y	12.80		210.00	Ref. [15, 16]
TZ3Y	12.35		218.00	present work
TZ3Y20A	13.30		279.00	present work
TZ3Y20A	13.60		265.00	Ref. [15]
TZ3Y20A short fibre	13.20		-	Ref. [20]

Table 4. Vickers hardness values of the reference block

Reference sample Zwick	*a1 µm	*a2 µm	*HVt GPa	*HVnt GPa	γ°	*ac µm	*HVc GPa
Tilt angle 1.07°	53.50 ± 1.00	46.00 ± 0.20	374.00 ± 16.00	–	1.07	49.01 ± 0.60	385.90 ± 9.00
Tilt angle 2.30°	57.00 ± 1.00	43.00 ± 0.50	370.00 ± 33.00	–	2.30	49.00 ± 0.70	386.10 ± 11.00
No tilt	49.10 ± 0.20	49.20 ± 0.30	–	384.70 ± 3.50	0	–	–

*a₁, a₂,and a_c are arithmetic meansof the asymmetric half diagonals and the corrected one, respectively.

*HVt is the Vickers hardness measured on tilt sample surface and HVc is the corrected value

*HVnt is the Vickers hardness measured on the no tilted reference block surface

Among the different methods of analysis, the one given here proved its validity in application on both metals and ceramic materials. Since the method is distinguished by applying a simple trigonometric analysis based on measurements taken from indentations made by pyramid indenter of classical Vickers or Knoop hardness tester, it could be also applied for other material, that were not tested in this work, such as; metal matrix ceramic composite (MMCC), ceramic matrix/ceramic fibre composite (CMCF) as well as polymers and polymer/metal composites. However, the method has its limits in application: It cannot be applied in the case of testing a thin layer coating or thin film, where very low test loads should be used and consequently very small indentations will be obtained. This will result in errors in the indentations dimensions measurements, besides the influence of the indentation size effect (ISE). Other limitations concern the application on materials containing large proportion of a glassy phase, having large grain size or large surface pores and scratches as well as multiphase materials.

Fig. 13 [Images not available. See PDF.]

Indentation made on the surface of Zwick reference block; a Symmetric Indentation (no tilt)  b Asymmetric Indentation (Tilt 1.07°)

Conclusions

In this study, a correction method for the measurements of asymmetric Vickers and Knoop indentations has been developed, using a simple trigonometric analysis. The method enabled accurate determination of both hardness and elastic modulus from corrected Vickers and Knoop indentations. It has proved its validity for both metallic alloys of aluminum and stainless steel as well as advanced ceramic materials (Zirconia ceramics). The degree of asymmetry was higher in the two-phase material (TZ3Y20A) than it was in the single-phase materials (TZ2.5Y and TZ3Y) which showed quasi symmetric indentations. The TZ3Y20A asymmetry might be a result of the large difference between Elastic modulus of α-alumina (400 GPa) and that of the tetragonal zirconia (200 GPa). The asymmetric indentation correction method developed in this study has been successfully validated using Zwick reference block. However, the limitation of this method of correction could arise, when applied on a material of very large grains, a multi-phase material, and the one containing large surface pores or scratches.

A detailed study to explore the effect of sample surface microstructure on the degree of the asymmetry will be a subject of future research.

Author contributions

MA: Main idea, supervision SH: Mathematical treatment-results discussion. OI: Knoop and hardness measurements KIO: Wrote the main manuscript, prepared all figures, prepared sample for SEM &Optical examination,

Funding

Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). Open access funding provided by The Science, Technology and Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). No funds, grants, or other support were received.

Data availability

All data generated or analysed during this study are included in this article.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Kharchenko, VV; Katok, OA; Kravchuk, RV; Sereda, AV; Shvets, VP. Analysis of the methods for determination of strength characteristics of NPP main equipment metal from the results of hardness and indentation measurements. Procedia Struct Integr; 2022; 36, pp. 59-65. [DOI: https://dx.doi.org/10.1016/j.prostr.2022.01.003]

2. Broitman, E. Indentation hardness measurements at macro-, micro-, and nanoscale: a critical overview. Tribol Lett; 2017; 65, 1 23. [DOI: https://dx.doi.org/10.1007/s11249-016-0805-5]

3. Chicot D, Tricoteaux A (2010) Mechanical properties of ceramics by indentation: Principle and applications. Ceramic Mater 115–153. https://doi.org/10.1007/978-3-319-04492-7

4. A Standard, E384-17 (2017) Standard test method for microindentation hardness of materials. ASTM Int US 1–40. https://doi.org/10.1520/E0384-17

5. Standard, A ASTM E92-16. Standard test methods for Vickers hardness and Knoop hardness of metallic. In Am Soc Test Mater; 2016; [DOI: https://dx.doi.org/10.1520/E0092-16]

6. Ferrero C, Herrmann K (2002) EA Guideline to the determination of the uncertainty of the Brinell and the Vickers measuring method. Presentation at the European co-operation for Accreditation EA. 2–16. https://doi.org/10.13140/RG.2.2.30087.09126. -Mechanical Measurements

7. Jakes, JE; Stauffer, D. Contact area correction for surface tilt in pyramidal nanoindentation. J Mater Res; 2021; 36, 11 pp. 2189-2197. [DOI: https://dx.doi.org/10.1557/s43578-021-00119-3]

8. Borc, J. Effect of Surface Tilt on Nanoindentation Measurement on Copper and Glass. Adv Sci Technol Res J; 2022; 16, 3 pp. 78-88. [DOI: https://dx.doi.org/10.12913/22998624/148167]

9. Huang, H; Zhao, H; Shi, C; Zhang, L. Using residual indent morphology to measure the tilt between the triangular pyramid indenter and the sample surface. Meas Sci Technol; 2013; 24, 10 pp. 05602-05607. [DOI: https://dx.doi.org/10.1088/0957-0233/24/10/105602]

10. Liu M, Zhu G, Dong X, Liao J, Gao C (2018) Effects of Sample tilt on Vickers indentation hardness. In Advanced Mechanical Science and Technology for the Industrial Revolution 4.0 Part IV: 271. https://doi.org/10.1007/978-981-10-4109-9

11. Kashani, MS; Madhavan, V. Analysis and correction of the effect of sample tilt on results of nanoindentation. Acta Mater; 2011; 59, 3 pp. 884-895. [DOI: https://dx.doi.org/10.1016/j.actamat.2010.09.051]

12. Marshall, DB; Noma, T; Evans, AG. A simple method for determining elastic-modulus–to‐hardness ratios using Knoop indentation measurements. J Am Ceram Soc; 1982; 65, 10 pp. c175-c176. [DOI: https://dx.doi.org/10.1111/j.1151-2916.1982.tb10357.x]

13. Leon, JS; Jayakumar, V. Investigation of mechanical properties of aluminium 6061 alloy friction stir welding. Int J Students’ Res Technol Manage; 2014; 2, 04 pp. 140-144.

14. Singh, R; Goel, S; Verma, R; Jayaganthan, R; Kumar, A. Mechanical behaviour of 304 austenitic stainless steel processed by room temperature rolling. IOP Conf Series: Materials Science and Engineering; 2018; 330, 012017. [DOI: https://dx.doi.org/10.1088/1757-899X/330/1/012017]

15. Tsukuma, K; Ueda, K; Matsushita, K; Shimada, M. High-temperature strength and fracture toughness of Y₂O₃‐partially‐stabilized ZrO₂/Al₂O₃ composites. J Am Ceram Soc; 1985; 68, 2 pp. C-56. [DOI: https://dx.doi.org/10.1111/j.1151-2916.1985.tb15284.x]

16. Abdallah, SA; Motawea, IT; Abdel Ghany, MM; Amin, RA. Phase transformation, surface topography and mechanical properties of hydrothermal aged laser treated cubic versus tetragonal zirconia ceramics. ADJ for Girls; 2020; 7, 2–C pp. 213-227. [DOI: https://dx.doi.org/10.21608/adjg.2020.14226.1189]

17. Brocq, M; Béjanin, E; Champion, Y. Influence of roughness and tilt on nanoindentation measurements: a quantitative model. Natl Inst Health; 2015; 37, 5 pp. 350-360. [DOI: https://dx.doi.org/10.1002/sca.21220]

18. Gafur, MA; Sarker, SR; Alam, Z; Qadir, MR. Effect of 3 mol% yttria stabilized zirconia addition on structural and mechanical properties of alumina-zirconia composites. Mater Sci Appl; 2017; 8, pp. 584-602. [DOI: https://dx.doi.org/10.4236/msa.2017.87041]

19. Garvie, RC; Hannink, R; Pascoe, RJN. Ceramic steel. Nature; 1975; 258, 5537 pp. 703-704. [DOI: https://dx.doi.org/10.1038/258703a0]

20. Ibrahim, OH; Othman, KI; Hassan, AA; El-Houte, S; El Sayed Ali, M. Synthesis and mechanical properties of zirconia–yttria matrices/alumina short fibre composites. Arab J Sci Eng; 2020; 45, 6 pp. 4959-4965. [DOI: https://dx.doi.org/10.1007/s13369-020-04542-2]