Relation between strain hardening exponent of metals and residual profiles of deep spherical indentation

List of symbols

Introduction

The measurement of contact area is the main factor to determine the hardness of materials ¹ and even to extract other mechanical properties from an indentation test. ² The contact area is strongly affected by the indentation morphologies developed during the test, piling-up or sinking-in. These morphologies were reported in 1928 by Norbury and Samuel ³ for a Brinell hardness test, and in 2006, ⁴ they were still cited as a reference for experimental points in order to calibrate some analytical models.

One of the mechanical properties that can be extracted from an indentation test is the strain hardening exponent n. The relation between n and hardness test is investigated since the establishment of the Meyer hardness in 1908. ⁵ More recently, many studies^6–9 presented a direct correlation between it and the indentation morphology, considering a spherical indentation as the experimental arrangement. An accurate revision of these models was presented by Hernot et al. ¹⁰ These researches identified limits to apply them and even their own proposed model. ¹¹ Within these considerations, the present work aims to provide a new series of experimental values in order to check the range of validity of available models and to update the Norbury and Samuels ones. Recently, Kim et al. ¹² used the residual profiles of indentation to determine the strain hardening exponent, but their study was limited to the nanoscale, using Berkovich indenter. Here, the experiments were performed using a spherical indenter in the macroscale of loads.

Experimental

Four materials were tested using spherical indentation arrangement: two aluminium alloys (AA 6063-T5 and AA 1350) and two steels (AISI 316L and AISI 1020). The indentation tests were performed using different diameters of spheres and different loads. The reason for this is to keep constant the relation between load L and the square of sphere diameter D ² , trying to make constant the plastic deformation level. Table 1 presents the values of L/D ² applied for each class of tested materials and the respective range of loads and sphere diameters. For each material, a minimum of 45 indentations were performed. The material of spheres was hardened steel, except in the conditions indicated in Table 1.

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Table 1

Values of loads L, sphere diameters D and L/D ² ratio used in indentation tests

Material	Loads/N	Sphere diameters/mm	L/D 2 /N mm−2
AA 1350	306·56, 494·42, 1226·25	2·5, 3·175, 5	49·05
AA 6063-T5	613·125, 2452·5, 9810	2·5, 5, 10	98·1
Steels	1834·5, 7357·5, 29 430	2·5*, 5*, 10*	294·3

*Tungsten carbide sphere was used in these conditions.

Aluminium AA 1350 was heat treated to take different levels of hardening. The as received condition is equivalent to the level H18 (37 HB). To get the level H24, the specimens were submitted to 330°C for 6 h in a furnace, obtaining a final hardness of 31 HB. Finally, the O condition (totally recovered) was reached after a treatment of 400°C for 6 h, resulting in 21 HB hardness. The other materials were tested in the as received condition, including the AA 6063 aluminium alloy, which can be found in commercial condition with T5 aging treatment, having 83 HB hardness. The hardness of steels was 145 and 237 HB for AISI 1020 and stainless steel respectively. For indentation experiments, the surfaces were prepared to have a finishing equivalent to a sanding performed with 1200 mesh.

After indentation tests, the residual profiles were determined using a bidimensional profilometer. The tip radius of stylus diamond is 5 μm. The set-up of measurements was performed by means of an optical system to localise the centre of indentation (Fig. 1). Figure 2 presents an example of the physical parameters extracted from a residual profile, showing the total evaluation length of 6 mm, which was used for all conditions.

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1

Illustration of residual profile measurement after spherical indentation test

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2

Physical parameters extracted from residual profile of spherical indentation, where a is indentation radius, a_c is indentation radius at contact, s is height associated to indentation morphology, h is indentation depth and h_c is contact depth

All materials were submitted to tensile tests. The values obtained were used as references, especially for the strain hardening exponent. The calculus of the strain hardening exponent obeyed the procedure recommended by ASTM E646 standard. ¹³ The obtained values are presented in Table 2. Moreover, the elastic modulus and the yield stress were determined to make use of the map provided by Taljat and Pharr. ¹⁴ The average values of these tensile properties were the result of a series of five measurements.

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Table 2

Values of strain hardening exponent obtained from tensile tests

Material	AISI 1020	AISI 316L	AA 6063-T5	AA 1350-H18	AA 1350-H24	AA 1350-O
n	0·08±0·01	0·48±0·01	0·13±0·01	0·08±0·01	0·17±0·01	0·28±0·04

It is worthwhile to remember that n is numerically equivalent to the uniform strain at the maximum load point in a tensile test, following the true stress–true strain relation.

Results and discussion

Figure 3 presents the variation of h_c/h ratio as a function of tensile strain hardening exponent. Furthermore, four curves are presented, corresponding to the models proposed by Matthews, ⁶ Hill et al., ⁷ Taljat et al. ⁸ and Alcalá et al. ⁹

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3

Relation between h_c/h ration determined from residual indentation profiles and tensile strain hardening exponent: analytical curves of Matthews, ⁶ Hill et al., ⁷ Alcalá et al. ⁹ and Taljat et al. ⁸ models are presented

The experimental points followed the trend predicted by the models: the higher the strain hardening exponent, the smaller the h_c/h values. The difference among the models can be perceived when one compares the results of AA 6063-T5 and AA 1350-H24 aluminium alloys once these materials have a small difference in n values. Considering the h_c/h value, the Taljat et al. ⁸ model predicted better strain hardening exponent for the AA 6063-T5 alloy. If the contact depth determined for the AA 1350-H24 alloy was put on the Taljat et al.'s curve, the strain hardening exponent should be smaller than that observed for AA 6063-T5, but a contrary behaviour was observed. Thus, it can indicate that a specific model may be more adequate for a particular group of materials.

Another interesting result was obtained for AA 1350-O alloy. This material almost did not present neither piling up nor sinking in, and its value of strain hardening exponent (0·28) met the prevision made by three models for the transition among the indentation morphologies, characterising a positive aspect of them.

For two metallurgical conditions, AISI 316L steel and AA 1350-H24 aluminium alloy, there are two h_c/h values indicated in the figure because the variation in the residual indentation profiles for these cases was very significant. Both points resulted from the experiments performed with a sphere of 2·5 mm diameter and the smallest applied load. These findings imply that all the models failed to predict this variation. At this moment, there is a need to search the possible reasons for this unexpected behaviour when fcc metals were tested with small spheres.

Mishra et al. ¹⁵ determined the n values of AISI 316L and 304L steels and for AA 1050 aluminium alloy, considering three directions in a tensile test. They demonstrated a large variation in the strain hardening exponent for 316L steel as the direction was varied, which is much higher than those observed for the other tested materials. As a result, in the same test, AISI 316L showed a variation from 0·2 up to 0·45 in n values. This variation is a consequence of a large positive stress deviation from the Hollomon relation at low strains, and Mishra et al. attributed it to the grain average misorientation (GAM), which was quantified by these researches using electron backscatter diffraction technique. For each strain hardening exponent value, a particular GAM value was identified, and a linear relationship was found for the mechanical property and this microstructural aspect. This result is corroborated by Yoda et al. ¹⁶ They proposed for the 316NG steel a linear relation between the true plastic strain and the GAM: the higher the former, the higher the latter. Moreover, as the true plastic strain increased, the scattering of GAM was higher.

The aspect of grains of the stainless steel can be seen in Fig. 4. For this material, the indentation radius was ∼0·5 mm after the test performed with a sphere of 2·5 mm diameter, a large dimension in comparison with its microstructural features.

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4

Microstructure of tested AISI 316L steel revealed in optical microscope

If the GAM can explain the variation observed for AISI 316L steel and AA 1350 aluminium in the spherical indentation tests, the strain hardening exponent should increase for larger plastic strains or even for conditions closer to the fully plastic behaviour. We can check this hypothesis by observing Fig. 5, where the experimental points were put on the mechanical behaviour map proposed by Taljat and Pharr. ¹⁴ These researches used the parameter (E/σ_y)(2h_c/a) to express the indentation regimes. When this parameter is >1000, it means that the material experiences the fully plastic regime.

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5

Experimental values on Taljat and Pharr's ¹⁴ map, which correlates parameter s/h with (E/σ_y)(2h_c/a), considering friction coefficient of 0·2 between indenter and specimen

One can observe in Fig. 5 an increase in the strain hardening exponent as the value of (E/σ_y)(2h_c/a) parameter increased for the cases of 316L steel and 1350-H24 aluminium alloy. Therefore, the hypothesis of the GAM effect on the indentation morphology seems plausible. If the grains of AA 1350-H24 aluminium alloy are considered, it is possible to affirm that there is a bimodal distribution of them because the heat treatment was enough to start the recrystallisation process, but it was not completed. In this way, the modelling of mechanical behaviour should consider this aspect and a probable deviation from the Hollomon relation.

In fact, more parameters are necessary to describe the mechanical behaviour of fcc metals, as presented by Choudhary et al., ¹⁷ for instance. Moreover, Han et al. ¹⁸ verified an increase in the strain hardening exponent for 5083 aluminium alloy as the grain size increased, showing a clear grain size dependence on this property. Since the investigation related by Hall, ¹⁹ a well known relation between the grain size and the hardness of metals was widely established by many investigations. However, for some cases, in particular for nanocrystalline materials, a so called inverse Hall–Petch relation is verified, ²⁰ and our current results show that these relations could be considered even in macroscale of loads.

Conclusions

New experimental values of residual profiles of indentations were provided using macroscale of loads for different levels of L/D ² ratio. Indentation morphologies were determined from those residual profiles, allowing the main conclusions.

The strain hardening exponent of 0·28 corresponded to the transition between piling-up and sinking-in, confirming the prediction made by the literature for this value.