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Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C
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International Journal of Solids and Structures 0 0 0 (2016) 1–7
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Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C F. Pöhl∗, S. Huth, W. Theisen Ruhr-Universität Bochum, Chair of Materials Technology, 44780 Bochum, Germany
a r t i c l e
i n f o
Article history: Received 18 November 2015 Revised 26 January 2016 Available online xxx Keywords: Indentation-size-effect Indentation Loading curvature Plastic properties
a b s t r a c t Numerous materials are affected by the Indentation-Size-Effect (ISE). Thus, the interpretation of ISEinfluenced indentation data plays an important role. This paper presents a method which allows for the characterization of the ISE using the loading curvature C of a single load-displacement curve (P–h curve) from sharp indentation. The method is based on the analysis of the change of the parameter C as a function of the indentation depth as described by Kick’s law (related to the universal hardness or also called Martens hardness HM). The intensity as well as the indentation depth where the ISE is saturated can be detected. Furthermore, it allows for the correction of ISE-affected loading curves with the use of the Vickers macro hardness. For example, this can enable the application of inverse methods for ISE-affected materials or phases. Mechanically polished samples show an additional increase in strength during nanoindentation due to hardened surface layers. The presented method also accounts for this influence and can correct affected loading curves. It was applied to the austenitic stainless steel X2CrNi18-9 (AISI 304L) which is heavily affected and the C45 carbon steel which is slightly affected by the ISE. The influence of hardened surface layers was investigated using electropolished and mechanically polished AISI 304L. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The ISE which is often observed with geometrically self-similar indenters is expressed through an increase in strength (or hardness) with decreasing indentation depth (Pharr et al., 2010). Several models have been developed leading to the explanations of the ISE and allow the explanation of test data. The most recognized models use the concept of a plastic strain gradient (Gao et al., 1999; Huang et al., 1999; 2004; Nix and Gao, 1998). The Nix-Gao model is the most adopted and employed model for the explanation of experimental results (Nix and Gao, 1998). In general, the occurrence of the ISE complicates analysis of test data especially when different materials are compared. The objective of this paper is not the analysis or development of models for the description of the ISE. It is the analysis of ISE-affected experimental data and its correction. In this context the loading curvature C is used for the detection of the ISE. This approach is based on Kick’s law which is also related to the Martens hardness HM (DIN EN ISO 14577, 2007; Kick, 1885). Thus, the ISE is analyzed by a change of total dissipated energy per deformed volume which is proportional to
∗
Corresponding author. Tel.: +49 2343225966. E-mail address: [email protected] (F. Pöhl).
C. Furthermore, the superposition of surface hardening which leads to an additional increase in strength is also addressed. The determination of the material parameters without any ISEinfluence often is of relevance. Thus, the characterization of the ISE is very important. Firstly, it is important to find out how strongly the data is influenced, and, secondly, to find the indentation depth where the ISE is saturated. An adequate tool is the ContinuousStiffness-Method (CSM), which offers both (Li and Bhushan, 2002). Though, the method cannot correct the load-displacement curve or at least the loading part and is limited to test equipments which can carry out the special test procedure (Pathak et al., 2009). This paper presents a method which can easily be applied to single P-h curves showing how heavily they are influenced and showing the transition depth up to which the ISE is significant. With the help of the Vickers macro hardness the method can be used to correct the loading curve and calculate the unaffected loading curvature C. This parameter is often used with inverse methods (Dao et al., 2001; Hyun et al., 2011; Le, 20 08; 20 09; Tho et al., 2005). Thus, the method can be applied in order to provide non ISE-affected input parameters for the inverse analysis of highly ISE-affected phases or materials. Mechanically polished samples show an additional increase in strength during nanoindentation due to hardened surface layers. The presented method also accounts for this influence and can
http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024 0020-7683/© 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024
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F. Pöhl et al. / International Journal of Solids and Structures 000 (2016) 1–7 Table 1 Chemical composition of examined materials in mass%.
Pmax
C
N
Cr
Ni
Si
Mn
Mo
Fe
AISI 304L C45
0.02 0.39
0.24 −
18.55 −
8.18 −
0.41 −
1.53 0.69
− −
bal. bal.
Loading
Load, P
Alloy
Unloading
Table 2 Parameters of the performed instrumented indentation tests. The indentation depth was varied in steps of 200 nm. Alloy
AISI 304L C45
Displacement [nm] 20 0–240 0 20 0–160 0
Loading-/unloading- rate [mN/min]
hp
S
hmax
Displacement, h
100 100
correct affected loading curves from the ISE and the influence of hardened surface layers. 2. Materials The investigated materials were the austenitic stainless steel X2CrNi18-9 (AISI 304L) and the carbon steel C45. The chemical composition of AISI 304L and C45 measured by sparc spectroscopy is listed in Table 1. AISI 304L shows a significant ISE, whereas, the C45 is slightly influenced. The AISI 304L was solution annealed and subsequently quenched in water in order to avoid carbide precipitation and to achieve a homogeneous microstructure. Afterwards, the specimens were mechanically grinded and polished to a grain size of 1 μm. Additionally, one sample of AISI 304L was electropolished in order to remove hardened surface layers. The C45 carbon steel was quenched in water from 860 °C (argon inert atmosphere). After the heat-treatment the C45 specimen was mechanically grinded and polished to a grain size of 1 μm.
Fig. 1. Schematic illustration of a P-h curve for a self-similar indenter showing important parameters (loading curvature C, maximum load Pmax , maximum indentation depth hmax , residual indentation depth hp , plastic indentation work Wpl , elastic indentation work Wel , and contact stiffness S).
The parameter C represents the specific (per volume) amount of work which is necessary for elasto–plastic deformation during indentation. The higher the parameter C the higher the amount of total work at specific indentation depths. The parameter C is linked to the mechanical properties of the material. For Ludwik powerlaw materials (Eq. 2, σ y is the yield stress) the parameter C is given in Eq. 3 (Pöhl et al., 2014). In Eq. 3, K is the strength coefficient, n the strain hardening exponent, and E the Young’s modulus. Vel is the elastically deformed volume with its mean elastic strain ε el and Vpl is the plastically deformed volume with its mean plastic strain ε pl (Pöhl et al., 2014).
σ= C=
E · ε, K · εn ,
3ε npl+1 Vpl n + 1 h3
σ ≤ σy σ ≥ σy ·K+
(2)
3ε 2pln Vpl K 2 3ε 2el Vel · + ·E n + 1 h3 E 2 h3
(3)
During self-similar indentation of a homogeneous material and 3. Experimental procedure
V
Instrumented indentation tests were performed on a CSM NHT indenter equipped with a Berkovich diamond tip. The indentation parameters are given in Table 2. At least five measurements were performed for each investigated indentation depth. The indentation hardness Hi was calculated with the Oliver and Pharr method and the parameter C with the use of Eq. 1 (tupels of P and h are substituted in Eq. 1) (Oliver and Pharr, 1992; Oliver and Pharr G.-M., 2004). The macro Vickers hardness number was measured with a KB 30 S FA test device of the company KB Prüftechnik GmbH. For the discussion of the correcting procedure introduced in Section 6.1 several atomic-force-microscope images of indentation imprints in AISI 304L were measured with an atomic-forcemicroscope (type nanos in contact mode) of the company Bruker. Since the indentation test device is not able to capture the saturation of the ISE for AISI 304L (force limit of the device is reached) the Finite-Element-Method (FEM) was used to calculate the macroscopic value of the parameter C. The material parameters used in the FEM model were derived by uniaxial tensile tests. The FEM model is described elsewhere (Pöhl et al., 2013). 4. Method for the characterization of the ISE Fig. 1 schematically shows the P-h curve obtained by sharp indentation (self-similar). For an homogenous, isotropic material and in absence of the ISE the loading part can be described by Kick’s law (Eq. 1) with the loading curvature C remaining constant (Cheng and Cheng, 2004; Kick, 1885).
P = C · h2
(1)
V
in absence of the ISE, all parameters (including hpl3 and hpl3 ) in Eq. 3 are constant and independent of the indentation depth (Pöhl et al., 2014). Thus, C is constant. At shallow indentation depths it is wellknown that the self-similarity of the indenter does not hold (tip blunting etc.) leading to a deviation from an ideal curve (elastic– plastic transition, dislocation nucleation and movement) (Bouzakis et al., 2002; Li et al., 2011). This regime cannot be described by the loading curvature C. At higher indentation depths, in the range of several hundred nanometers, those tip effects are more and more negligible and the parameter C becomes reasonably constant and can be used to describe the loading curve. The ISE leads to an increase in strength with decreasing indentation depth (influence of Geometric Necessary Dislocations, GNDs) (Nix and Gao, 1998). The strength of the material varies with the indentation depth. This can also be interpreted as the material parameters being a function of the indentation depth. The variables of Eq. 3 are not constant anymore. For example, the strength coefficient K is increasing (for n, E remaining constant) with decreasing indentation depth. This automatically leads to an increase of the parameter C. In general, any change of the material parameters as a function of the indentation depth will lead to a change of the parameter C as a function of the indentation depth. A hardened surface layer also leads to an increase of C. For this reason, the factor C can be used as a detector of changes of the material parameters. This might be caused by the ISE, a hardened surface layer, or a combination of both. At shallow indentation depths this is additionally superimposed by indenter tip form deviations. If the loading curve is affected by the ISE, the parameter C will increase with decreasing indentation depth due to the increase in strength. The parameter C as a function of indentation depth can
Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024
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3
Fig. 2. Schematic illustration of the determination of the loading curvature C as a function of the indentation depth h for a given loading curve.
Fig. 3. Schematic illustration of the loading curvature C as a function of the indentation depth h for a highly ISE-affected material. The saturation depth can not be reached with prevalent instrumented indentation testing devices.
easily be determined with Eq. 1 (substitution of depth h and load P into the equation). Fig. 2 shows schematically an ISE-affected loading curve and the parameter C as a function of the indentation depth h. The plot of C versus h reveals the intensity of the ISE and allows for the determination of the indentation depth where the ISE is negligible. In Fig. 2 this indentation depth is given by h∗4 . Some materials such as austenitic stainless steels (e.g. AISI 304L) show an intensive ISE (Manika and Maniks, 2006; Ye et al., 2007). The maximum force which can be applied by prevalent instrumented indentation testing devices is not able to reach the saturation depth of the ISE. This is schematically illustrated in Fig. 3. The determination of the not ISE-affected parameter C is relevant because it is an important input parameter for several inverse methods (Dao et al., 2001; Hyun et al., 2011; Le, 20 08; 20 09; Tho et al., 2005). 5. Correction procedure The following procedure allows for the correction of highly ISEaffected loading curves and is based on the Vickers macro hardness number which can easily be determined. The procedure is illustrated in Fig. 4. The indentation hardness is defined according to Eq. 4 and the Vickers hardness number is defined according to Eq. 5 with Pmax being the maximum load, Ap the projected contact area, and As the surface of the indenter as a function of the indentation depth (DIN EN ISO 14577, 2007; Oliver and Pharr, 1992; Tabor, 1951). In case of a perfect Vickers and Berkovich indenter with given projected contact area Ap and surface area As it is possible to derive a simple scaling function which enables the conversion of the Vickers hardness number to the indentation hardness mea-
Fig. 4. Flow chart of the correction procedure in order to correct ISE-affected loading curves. The ISE-affected P-h curves is used to determine hc , C and Hi (converted to HV with Eq. 6). The maximum indentation depth h in combination with hc can be used to calculated ϕ . Or C and the converted Hi to HV can be used to calculate the parameter ϕ . In combination with the Vickers macro hardness number the affected loading curve can be corrected.
sured by a Berkovich indenter (Eq. 6) (DIN EN ISO 14577, 2007). Eq. 6 is based on perfect indenter geometries. Therefore, it should be emphasized that at shallow indentation depths the indenter geometry can deviate significantly from a perfect Berkovich indenter and the conversion might become inaccurate. Eq. 6 can be transformed with Eq. 4 and Kick’s law to Eq. 7. Eq. 7 is a general relationship between the loading curvature C and the Vickers hardness number. Under the assumption that the ratio of the contact depth hc to the total indentation depth h is constant we obtain a linear relationship between the loading curvature C and the Vickers hardness number (Eq. 7). We define ( hh )2 as ϕ . After rearranging Eq. 7 c the parameter ϕ can be calculated in two different ways given by Eq. 8. For a given macroscopic Vickers hardness number it allows for the calculation of the corresponding macroscopic loading curvature C and, thus, allows for the correction of ISE-affected values of C (Eq. 9). In Section 6.1 the practical application of the presented correction procedure is shown.
Hi =
Pmax Ap
HV = 0.102 ·
(4)
Pmax As
(5)
Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024
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110000 4000
100000 90000
3600
80000 3200
70000 60000
2800
50000 40000 2500
2400 0
500
1000
1500
N mm²
120000
4400
Loading curvature, C in
Indentation hardness, Hi in MPa
Mean hardness Hi at different indentation depths Mean value of C at different indentation depths Values of C determined from single P-h curve (h = 2400 nm)
2000
Fig. 6. Determined values of ϕ 1 and ϕ 2 as a function of the indentation depth for electropolished AISI 304L.
Indentation depth, h in nm Fig. 5. Hardness Hi and loading curvature C as a function of the indentation depth h of electropolished AISI 304L.
AISI 304 mechanically polished before correction procedure AISI 304 mechanically polished, corrected with AISI 304 mechanically polished, corrected with
ϕ=
h hc
Cmacro =
2
Ch2 0.0926 = ·ϕ ·C 2 24.5 24.5hc and
ϕ=
with
ϕ=
h hc
24.5 · HV 0.0926 · C
24.5 · HVmacro ϕ · 0.0926
2 (7)
(8)
(9)
Loading curvature, C in
HV = 0.0926 ·
(6)
N mm²
160000
Pmax HV = 0.0926 · Hi = 0.0926 · Ap
AISI 304 electropolished before correction procedure AISI 304 electropolished, corrected with AISI 304 electropolished, corrected with
140000
Macroscopic (FEM, tensile test data)
120000 100000 80000 60000 40000 20000 0
6. Results 6.1. Analysis of the ISE and correction of the loading curve The data shown in Fig. 5 reveals a significant ISE for the electropolished AISI 304L which is in agreement with the literature (Manika and Maniks, 2006; Ye et al., 2007). Since the CSM method could not be applied with the used test device the indentation hardness was measured with at least five P-h curves with different investigated indentation depths. The mean indentation hardness determined at the investigated indentation depths increases drastically with decreasing indentation depth (increase of ≈145% from 2400 nm to 200 nm). A saturation of the ISE cannot be detected in the investigated depth range. The hardness shows a strong sinking tendency over the whole investigated depth range which illustrates that even the P-h curves with the highest possible indentation depth of the test device are notably affected by the ISE. The ISE can also be characterized using the parameter C determined from the P-h curves measured at the different indentation depths. In Fig. 5 the mean value of C at each investigated indentation depth is shown as a function of the indentation depth. As introduced in Section 4, the parameter C can also be determined as a function of the indentation depth from a single P-h which is also shown in Fig. 5 for one of the P-h curves with the highest indentation depth. The progression is in good agreement with the mean values of C determined from each P-h curve measured at different indentation depths and is also in good agreement with the progression of the hardness plotted versus the indentation depth (Fig. 5). The analysis of the parameter C can practically be used to analyze single or mean P-h curves of given indentation depths in order to characterize the ISE. This procedure can easily show that
500
1000
1500
2000
2500
Indentation depth, h in nm Fig. 7. Loading curvature C as a function of the indentation depth of AISI 304L in electropolished and mechanically polished condition before and after correction procedure. As reference the macroscopic loading curvature derived by FEMsimulation with the material parameters from tensile test is included.
even curves of the highest, investigated indentation depth are still highly influenced by the ISE. The parameter C is an important parameter for many inverse methods which are based on sharp indentation (Dao et al., 2001; Hyun et al., 2011; Le, 20 08; 20 09; Tho et al., 20 05). Using ISEaffected values of C as input parameters for inverse methods leads to artificially influenced results. In such circumstances it is desirable to correct an ISE-affected loading curvature C. Following the procedure introduced in Section 4 (see Fig. 4) the affected values of the parameter C can be corrected. There are two possible options to calculate the correction parameter ϕ (see Eq. 8). From an ISE-affected curve there can be used h and hc as well as the loading curvature C and the Vickers hardness number HV to calculate ϕ . Fig. 6 shows the mean values of ϕ determined with h and hc as well as determined with the parameters C and HV as a function of the investigated indentation depths. There is a slight difference at low indentation depths (<500 nm), whereas, at higher indentation depths the values of ϕ determined with the different ways show a good agreement. With Eq. 3 the ISE-affected loading curvature C can be corrected with the parameter ϕ and the macroscopic Vickers hardness number which was determined to 145 HV30 for AISI 304L. Fig. 7 shows the parameter C before and after the correction procedure. It can be seen that both different ways of calculating ϕ lead to almost identical results at indentation
Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024
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Increase of loading curvature C in %
F. Pöhl et al. / International Journal of Solids and Structures 000 (2016) 1–7
Mean value of C at different indentation depths Corrected values of C ( ) Corrected values of C ( )
400
Increase due to surface hardening Increase due to the ISE
350
220000
N mm²
300
Loading curvature, C in
250 200 150 100 50 0 0
400
800
1200
1600
2000
2400
Indentation depth, h in nm
ISE 200000
180000
160000
140000
120000 0
Fig. 8. Separation of the increase of the parameter C due to the ISE and due to surface hardening for mechanically polished AISI 304L.
9200 180000
8400 160000 8000
250
500
750
1000
1250
1500
140000 1750
Indentation depth, h in nm Fig. 9. Hardness Hi and loading curvature C as a function of the indentation depth h of C45 (quenched).
depths higher than ≈500 nm. Since the indentation test device is not able to capture the saturation of the ISE the Finite-ElementMethod (FEM) was used to calculate the macroscopic value of the parameter C. The numerically calculated and the corrected values of the parameter C are in good agreement. Especially, using h and hc for calculating ϕ allows for an accurate correction of C even at low indentation depths. Mechanically grinded and polished samples exhibit a hardened surface layer leading to an additional increase in strength which is superimposed by the ISE. This is shown in Fig. 7. In case of the mechanically grinded and polished sample the parameter C is higher at all investigated indentation depths compared to the electropolished sample. Both, the ISE and surface hardening lead to an artificially increase of C. It can be seen that the additional increase in strength due to surface hardening can also be corrected leading to a good agreement to the corrected electropolished values and the FEM results (see Fig. 7). The comparison of the electropolished and mechanically polished samples allows the separation of both influences. Fig. 8 shows the increase of the loading curvature C compared to the macroscopic value due to the ISE and surface hardening. It can be seen that the influence of the ISE is dominant compared to surface hardening. The carbon Steel C45 does not exhibit an extensive ISE. The hardness and the loading curvature C as a function of the indentation depth are plotted in Fig. 9. The hardness as well as the param-
Factor of projected contact area
200000
7600
750
1000
1250
1500
1750
Indenter geometry used in this study Ideal indenter geometry
N mm²
ISE 9600
8800
500
Indentation depth, h in nm
220000
10000
Loading curvature, C in
Indentation hardness, Hi in MPa
250
Fig. 10. Loading curvature C as a function of the indentation depth of C45 (quenched) before and after correction procedure.
Mean hardness Hi at different indentation depths Mean value of C at different indentation depths Values of C determined from single P-h curve (h = 1600 nm)
0
5
30 devitation < 5 % 25
20
15
10
5
0 0
200
400
600
800 1000 1200 1400 1600 1800 2000
Indentation depth, h in nm Fig. 11. Deviation of the Berkovich indenter used in this study from an ideal Berkovich indenter with constant factor of projected contact area of 24.5.
eter C decrease with increasing indentation depth until they are reasonable constant at indentation depths larger than ≈800 nm. Analogous to the AISI 304L the analysis of C shows quantitatively the extend of the ISE, although, at shallow indentation depths it is increasingly overlayed with the violation of the self-similarity. Nevertheless, the influence of the ISE is faded away at indentation depths being larger than 800 nm. Although a correction procedure is not necessary Fig. 10 shows the corrected loading curvatures using the procedure shown in Fig. 4. The ‘macro’ Vickers hardness number was measured to 750 HV (conversion of Hi with Eq. 6) at the indentation depth of 1600 nm were the ISE is saturated. The corrected values of C are in agreement to those which are not affected by the ISE. At shallow indentation depths (<600 nm) the correction leads to higher values of C compared to the not ISEaffected values of C. Those deviations are the result of the violation of the self similarity which is discussed in Section 6.2. 6.2. Discussion of the correction procedure The correction procedure is based on the assumption that the ratio hhc is mostly independent of the indentation depth and not significantly affected by the ISE or hardened surface layers. As shown in Fig. 12 for AISI 304L and C45 this assumption is essentially correct. Solely, at indentation depths lower than 500 nm the ratio hhc slightly decreases. All in all, the ratio is nearly constant
Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024
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tion depths lower than ∼600 nm. Firstly, Eq. 9 is not valid in this regime due to the changed projected contact area. Secondly, the induced strain and deformed volumes are significantly dependent on the indenter geometry and, thus, leading to non self-similar deformation where Kick’s law is not valid. For higher indentation depths the procedure can effectively correct the influences of the ISE and surface hardening. Though, application and analysis of different materials are necessary in order to investigate the transferability. According to Fig. 10 the corrected loading curvature shows a certain scatter range. Perturbations of the loading curvature used as input parameter for inverse methods can highly affect the inverse solution. Due to the ill-posed nature of the inverse indentation problem small variations of the loading curvature can lead to larger errors of the inverse solutions (Liu et al., 2009).
AISI 304 electropolished AISI 304 mechanically polished C45 (quenched)
0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.82 0.80 0
500
1000
1500
2000
2500
Indentation depth, h in nm Fig. 12. Ratio hhc as a function of the indentation depth h for C45 (quenched) and AISI 304L after mechanical polishing and electropolishing.
18 m 60 N m F N = 18 0 F m = 35 N 0 m N
This paper presents a method which allows for the characterization of the ISE using the loading curvature C of a single load-displacement curve. The method analyzes the change of the parameter C as a function of the indentation depth during sharp indentation. Furthermore, it allows for the correction of ISE-affected loading curves including the correction of hardened surface layers due to mechanical grinding and polishing. The following conclusions can be drawn:
=
F
F
=
0,25 0,00
Height in µm
-0,25 -0,50 -0,75 -1,00
top view x
-1,25 -1,50
path
-1,75 -2,00 0
5
10
15
20
25
30
35
Distance along path, x in µm Fig. 13. Hight-profiles of Berkovich imprints of different indentation depth in AISI 304L measured with atomic-force-microscopy. The sink-in behavior is largely unaffected by the ISE.
with ∼0.95 for mechanically and electropolished AISI 304L and constant with ∼0.88 for quenched C45. Fig. 13 shows several profiles of imprints made in AISI 304L with different indentation depths measured by atomic-force-microscopy. The profiles show that the ratio hhc , contrary to the hardness, is almost not affected
by the ISE. The pile-up and sink-in behavior and, thus, the ratio hhc is a function of the material parameters (Taljat and Pharr, 14) given by Eq. 10 for Ludwik power-law materials (Pöhl et al., 2014). The results indicate that the changes of the material parameters due to surface hardening and the ISE do not have a high impact on hhc .
hc K = f , n h E
7. Summary and conclusions
(10)
The correction procedure shows a good performance for indentation depths larger than ≈600 nm for AISI 304L and C45. At lower indentation depths the ISE-affected parameter C can be improved with the correction procedure, but still deviates from the not ISE-affected value. The explanation is the deviation of a real indenter from the perfect Berkovich indenter geometry. For an ideal Berkovich indenter the projected contact area is given by: Ap = 24.5 · h2c . Fig. 11 shows the deviation from the ideal value of 24.5 for the indenter used in this study. The projected contact area of the real indenter was determined with a calibration procedure in fused silica as it is described in DIN EN ISO 14577 (2007). It deviates less than 5% for indentation depths larger than ≈600 nm and explains the deviations of the correcting procedure at indenta-
• The analysis of the loading curvature C is a practical method for the characterization of the ISE. The intensity as well as the indentation depth where the ISE is saturated can be detected (when the saturation depth is within the measurement range of the test device). • The ISE was successfully characterized for the heavily affected steel AISI 304L and the quenched carbon steel C45 with low ISE. • ISE-affected loading curves (parameter C) can be corrected with the use of the Vickers macro hardness according to Eq. 9. The analysis of the loading curve and the correction procedure can be applied to single P-h curves. The correction procedure was successfully applied to heavily ISE-influenced P-h curves of AISI 304L and to slightly affected P-h curves of the quenched carbon steel C45. • The constant ratio hh is the basis for an accurate correction c procedure with Eq. 9. In contrast to the hardness, it is widely constant and not affected, although, the material parameters change in presence of the ISE. Those changes do not have a high impact on hh for the investigated materials. Though, the c transferability to other materials has to be examined. • Mechanically grinded and polished samples exhibit a hardened surface layer leading to an additional increase in strength which is superimposed by the ISE. It was shown that the influence of both the ISE and hardened surface layers can be corrected for the analyzed materials. References Bouzakis, K.-D., Michailidis, N., Hadjiyiannis, S., Skordaris, G., Erkens, G., 2002. Continuous fem simulation of the nanoindentation: actual indenter tip geometries, material elastoplastic deformation laws and universal hardness. Zeitschrift für Metallkunde 93 (9), 862–869. Cheng, Y.-T., Cheng, C.-M., 2004. Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng. R 44, 91–149. Dao, M., Chollacoop, N., van Fliet K.-J., Venkatesh, T.-A., Suresh, S., 2001. Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Materialia 49, 3899–3918. DIN EN ISO 14577, 2007. Instrumentierte eindringprüfung zur bestimmung der härte und anderer werkstoffparameter. Gao, H., Nix, W.-D., Hutchinson, J.-W., 1999. Machanism-based strain gradient plasticity. i. theory. J. Mechan. Phys. Solids 47, 1239–1263. Huang, Y., Gao, H., Nix, W.-D., Hutchinson, J.-W., 1999. Machanism-based strain gradient plasticity. ii. analysis. J. Mech. Phys. Solids 48, 99–128.
Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024
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F. Pöhl et al. / International Journal of Solids and Structures 000 (2016) 1–7 Huang, Y., Qu, S., Hwang, K.-C., Li, M., Gao, H., 2004. A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plasticity 20, 753–782. Hyun, H.-C., Kim, M., Lee, J.-H., Lee, H., 2011. A dual conical indentation technique based on fea solutions for property evaluation. Mech. Mater. 43, 313–331. Kick, F., 1885. Das Gesetz der proportionalen Widerstände und seine Anwendungen: Nebst Versuchen über das Verhalten verschiedener Materialien bei gleichen Formänderungen sowohl unter der Presse als dem Schlag-werk. A. Felix, Leipzig. Le, M.-Q., 2008. A computational study on the instrumented sharp indentations with dual indenters. Int. J. Solids Struct. 45, 2818–2835. Le, M.-Q., 2009. Materials characterisation by dual sharp indenters. Int. J. Solids Struct. 46, 2988–2998. Li, T.-L., Gao, Y.-F., Bei, H., George, E.-P., 2011. Indentation schmid factor and orientation dependence of nanoindentation pop-in behavior of nial single crystals. J. Mech. Phys. Solids 59, 1147–1162. Li, X., Bhushan, B., 2002. A review of nanoindentation continuous stiffness measurement technique and its applications. Mater. Characterization 48, 11–36. Liu, L., Ogasawara, N., Chiba, N., Chen, X., 2009. Can indentation technique measure unique elastoplastic properties? J. Mater. Res. 24, 784–800. Manika, I., Maniks, J., 2006. Size effects in micro- and nanoscale indentation. Acta Materialia 54, 2049–2056. Nix, W.-D., Gao, H., 1998. Indentation size effects in chrystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425. Oliver, W.-C., Pharr, G.-M., 1992. An improved technique for determination hardness and elastic-modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564–1583.
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Oliver, W.-C., Pharr G.-M., 2004. Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology. J. Mater. Res. 19, 3–20. Pathak, S., Shaffer, J., Kalidindi, S.-R., 2009. Determination of an effective zero-point and extraction of indentation stress–strain curves without the continuous stiffness measurement signal. Scripta Materialia 60, 439–442. Pharr, G.-M., Herbert, E.-G., Gao, Y., 2010. The indentation soze effect: a critical examination of experimental observations and mechanistic interpretations. Annu. Rev. Mater. Res. 40, 271–292. Pöhl, F., Huth, S., Theisen, W., 2013. Finite element method-assisted acquisition of the matrix influence on the indentation results of an embedded hard phase. Mater. Sci. Eng. A 559, 822–828. Pöhl, F., Huth, S., Theisen, W., 2014. Indentation of self-similar indenters: an femassisted energy-based analysis. J. Mech. Phys. Solids 66, 32–41. Tabor, D., 1951. A simple theorie of static and dynamic hardness. Clarendon Press. Taljat, B., Pharr G.-M., 2004. Development of pile-up during spherical indentation of elastic-plastic solids. J. Solids Struct. 41, 3891–3904. Tho, K., Swaddiwudhipong, S., Liu, Z.-S., Zeng, K., 2005. Simulation of instrumented indentation and material characterization. Mater. Sci. Eng. A 390, 202– 209. Ye, D., Matsuoka, S., Nagashima, N., 2007. Determination of fatigue mesoscopic mechanical properties of an austenitic stainless steel using depth-sensing indentation (dsi) technique. Mater. Sci. Eng. A 456, 120–129.
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International Journal of Solids and Structures 0 0 0 (2016) 1–7
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Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C F. Pöhl∗, S. Huth, W. Theisen Ruhr-Universität Bochum, Chair of Materials Technology, 44780 Bochum, Germany
a r t i c l e
i n f o
Article history: Received 18 November 2015 Revised 26 January 2016 Available online xxx Keywords: Indentation-size-effect Indentation Loading curvature Plastic properties
a b s t r a c t Numerous materials are affected by the Indentation-Size-Effect (ISE). Thus, the interpretation of ISEinfluenced indentation data plays an important role. This paper presents a method which allows for the characterization of the ISE using the loading curvature C of a single load-displacement curve (P–h curve) from sharp indentation. The method is based on the analysis of the change of the parameter C as a function of the indentation depth as described by Kick’s law (related to the universal hardness or also called Martens hardness HM). The intensity as well as the indentation depth where the ISE is saturated can be detected. Furthermore, it allows for the correction of ISE-affected loading curves with the use of the Vickers macro hardness. For example, this can enable the application of inverse methods for ISE-affected materials or phases. Mechanically polished samples show an additional increase in strength during nanoindentation due to hardened surface layers. The presented method also accounts for this influence and can correct affected loading curves. It was applied to the austenitic stainless steel X2CrNi18-9 (AISI 304L) which is heavily affected and the C45 carbon steel which is slightly affected by the ISE. The influence of hardened surface layers was investigated using electropolished and mechanically polished AISI 304L. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The ISE which is often observed with geometrically self-similar indenters is expressed through an increase in strength (or hardness) with decreasing indentation depth (Pharr et al., 2010). Several models have been developed leading to the explanations of the ISE and allow the explanation of test data. The most recognized models use the concept of a plastic strain gradient (Gao et al., 1999; Huang et al., 1999; 2004; Nix and Gao, 1998). The Nix-Gao model is the most adopted and employed model for the explanation of experimental results (Nix and Gao, 1998). In general, the occurrence of the ISE complicates analysis of test data especially when different materials are compared. The objective of this paper is not the analysis or development of models for the description of the ISE. It is the analysis of ISE-affected experimental data and its correction. In this context the loading curvature C is used for the detection of the ISE. This approach is based on Kick’s law which is also related to the Martens hardness HM (DIN EN ISO 14577, 2007; Kick, 1885). Thus, the ISE is analyzed by a change of total dissipated energy per deformed volume which is proportional to
∗
Corresponding author. Tel.: +49 2343225966. E-mail address: [email protected] (F. Pöhl).
C. Furthermore, the superposition of surface hardening which leads to an additional increase in strength is also addressed. The determination of the material parameters without any ISEinfluence often is of relevance. Thus, the characterization of the ISE is very important. Firstly, it is important to find out how strongly the data is influenced, and, secondly, to find the indentation depth where the ISE is saturated. An adequate tool is the ContinuousStiffness-Method (CSM), which offers both (Li and Bhushan, 2002). Though, the method cannot correct the load-displacement curve or at least the loading part and is limited to test equipments which can carry out the special test procedure (Pathak et al., 2009). This paper presents a method which can easily be applied to single P-h curves showing how heavily they are influenced and showing the transition depth up to which the ISE is significant. With the help of the Vickers macro hardness the method can be used to correct the loading curve and calculate the unaffected loading curvature C. This parameter is often used with inverse methods (Dao et al., 2001; Hyun et al., 2011; Le, 20 08; 20 09; Tho et al., 2005). Thus, the method can be applied in order to provide non ISE-affected input parameters for the inverse analysis of highly ISE-affected phases or materials. Mechanically polished samples show an additional increase in strength during nanoindentation due to hardened surface layers. The presented method also accounts for this influence and can
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Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024
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F. Pöhl et al. / International Journal of Solids and Structures 000 (2016) 1–7 Table 1 Chemical composition of examined materials in mass%.
Pmax
C
N
Cr
Ni
Si
Mn
Mo
Fe
AISI 304L C45
0.02 0.39
0.24 −
18.55 −
8.18 −
0.41 −
1.53 0.69
− −
bal. bal.
Loading
Load, P
Alloy
Unloading
Table 2 Parameters of the performed instrumented indentation tests. The indentation depth was varied in steps of 200 nm. Alloy
AISI 304L C45
Displacement [nm] 20 0–240 0 20 0–160 0
Loading-/unloading- rate [mN/min]
hp
S
hmax
Displacement, h
100 100
correct affected loading curves from the ISE and the influence of hardened surface layers. 2. Materials The investigated materials were the austenitic stainless steel X2CrNi18-9 (AISI 304L) and the carbon steel C45. The chemical composition of AISI 304L and C45 measured by sparc spectroscopy is listed in Table 1. AISI 304L shows a significant ISE, whereas, the C45 is slightly influenced. The AISI 304L was solution annealed and subsequently quenched in water in order to avoid carbide precipitation and to achieve a homogeneous microstructure. Afterwards, the specimens were mechanically grinded and polished to a grain size of 1 μm. Additionally, one sample of AISI 304L was electropolished in order to remove hardened surface layers. The C45 carbon steel was quenched in water from 860 °C (argon inert atmosphere). After the heat-treatment the C45 specimen was mechanically grinded and polished to a grain size of 1 μm.
Fig. 1. Schematic illustration of a P-h curve for a self-similar indenter showing important parameters (loading curvature C, maximum load Pmax , maximum indentation depth hmax , residual indentation depth hp , plastic indentation work Wpl , elastic indentation work Wel , and contact stiffness S).
The parameter C represents the specific (per volume) amount of work which is necessary for elasto–plastic deformation during indentation. The higher the parameter C the higher the amount of total work at specific indentation depths. The parameter C is linked to the mechanical properties of the material. For Ludwik powerlaw materials (Eq. 2, σ y is the yield stress) the parameter C is given in Eq. 3 (Pöhl et al., 2014). In Eq. 3, K is the strength coefficient, n the strain hardening exponent, and E the Young’s modulus. Vel is the elastically deformed volume with its mean elastic strain ε el and Vpl is the plastically deformed volume with its mean plastic strain ε pl (Pöhl et al., 2014).
σ= C=
E · ε, K · εn ,
3ε npl+1 Vpl n + 1 h3
σ ≤ σy σ ≥ σy ·K+
(2)
3ε 2pln Vpl K 2 3ε 2el Vel · + ·E n + 1 h3 E 2 h3
(3)
During self-similar indentation of a homogeneous material and 3. Experimental procedure
V
Instrumented indentation tests were performed on a CSM NHT indenter equipped with a Berkovich diamond tip. The indentation parameters are given in Table 2. At least five measurements were performed for each investigated indentation depth. The indentation hardness Hi was calculated with the Oliver and Pharr method and the parameter C with the use of Eq. 1 (tupels of P and h are substituted in Eq. 1) (Oliver and Pharr, 1992; Oliver and Pharr G.-M., 2004). The macro Vickers hardness number was measured with a KB 30 S FA test device of the company KB Prüftechnik GmbH. For the discussion of the correcting procedure introduced in Section 6.1 several atomic-force-microscope images of indentation imprints in AISI 304L were measured with an atomic-forcemicroscope (type nanos in contact mode) of the company Bruker. Since the indentation test device is not able to capture the saturation of the ISE for AISI 304L (force limit of the device is reached) the Finite-Element-Method (FEM) was used to calculate the macroscopic value of the parameter C. The material parameters used in the FEM model were derived by uniaxial tensile tests. The FEM model is described elsewhere (Pöhl et al., 2013). 4. Method for the characterization of the ISE Fig. 1 schematically shows the P-h curve obtained by sharp indentation (self-similar). For an homogenous, isotropic material and in absence of the ISE the loading part can be described by Kick’s law (Eq. 1) with the loading curvature C remaining constant (Cheng and Cheng, 2004; Kick, 1885).
P = C · h2
(1)
V
in absence of the ISE, all parameters (including hpl3 and hpl3 ) in Eq. 3 are constant and independent of the indentation depth (Pöhl et al., 2014). Thus, C is constant. At shallow indentation depths it is wellknown that the self-similarity of the indenter does not hold (tip blunting etc.) leading to a deviation from an ideal curve (elastic– plastic transition, dislocation nucleation and movement) (Bouzakis et al., 2002; Li et al., 2011). This regime cannot be described by the loading curvature C. At higher indentation depths, in the range of several hundred nanometers, those tip effects are more and more negligible and the parameter C becomes reasonably constant and can be used to describe the loading curve. The ISE leads to an increase in strength with decreasing indentation depth (influence of Geometric Necessary Dislocations, GNDs) (Nix and Gao, 1998). The strength of the material varies with the indentation depth. This can also be interpreted as the material parameters being a function of the indentation depth. The variables of Eq. 3 are not constant anymore. For example, the strength coefficient K is increasing (for n, E remaining constant) with decreasing indentation depth. This automatically leads to an increase of the parameter C. In general, any change of the material parameters as a function of the indentation depth will lead to a change of the parameter C as a function of the indentation depth. A hardened surface layer also leads to an increase of C. For this reason, the factor C can be used as a detector of changes of the material parameters. This might be caused by the ISE, a hardened surface layer, or a combination of both. At shallow indentation depths this is additionally superimposed by indenter tip form deviations. If the loading curve is affected by the ISE, the parameter C will increase with decreasing indentation depth due to the increase in strength. The parameter C as a function of indentation depth can
Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024
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3
Fig. 2. Schematic illustration of the determination of the loading curvature C as a function of the indentation depth h for a given loading curve.
Fig. 3. Schematic illustration of the loading curvature C as a function of the indentation depth h for a highly ISE-affected material. The saturation depth can not be reached with prevalent instrumented indentation testing devices.
easily be determined with Eq. 1 (substitution of depth h and load P into the equation). Fig. 2 shows schematically an ISE-affected loading curve and the parameter C as a function of the indentation depth h. The plot of C versus h reveals the intensity of the ISE and allows for the determination of the indentation depth where the ISE is negligible. In Fig. 2 this indentation depth is given by h∗4 . Some materials such as austenitic stainless steels (e.g. AISI 304L) show an intensive ISE (Manika and Maniks, 2006; Ye et al., 2007). The maximum force which can be applied by prevalent instrumented indentation testing devices is not able to reach the saturation depth of the ISE. This is schematically illustrated in Fig. 3. The determination of the not ISE-affected parameter C is relevant because it is an important input parameter for several inverse methods (Dao et al., 2001; Hyun et al., 2011; Le, 20 08; 20 09; Tho et al., 2005). 5. Correction procedure The following procedure allows for the correction of highly ISEaffected loading curves and is based on the Vickers macro hardness number which can easily be determined. The procedure is illustrated in Fig. 4. The indentation hardness is defined according to Eq. 4 and the Vickers hardness number is defined according to Eq. 5 with Pmax being the maximum load, Ap the projected contact area, and As the surface of the indenter as a function of the indentation depth (DIN EN ISO 14577, 2007; Oliver and Pharr, 1992; Tabor, 1951). In case of a perfect Vickers and Berkovich indenter with given projected contact area Ap and surface area As it is possible to derive a simple scaling function which enables the conversion of the Vickers hardness number to the indentation hardness mea-
Fig. 4. Flow chart of the correction procedure in order to correct ISE-affected loading curves. The ISE-affected P-h curves is used to determine hc , C and Hi (converted to HV with Eq. 6). The maximum indentation depth h in combination with hc can be used to calculated ϕ . Or C and the converted Hi to HV can be used to calculate the parameter ϕ . In combination with the Vickers macro hardness number the affected loading curve can be corrected.
sured by a Berkovich indenter (Eq. 6) (DIN EN ISO 14577, 2007). Eq. 6 is based on perfect indenter geometries. Therefore, it should be emphasized that at shallow indentation depths the indenter geometry can deviate significantly from a perfect Berkovich indenter and the conversion might become inaccurate. Eq. 6 can be transformed with Eq. 4 and Kick’s law to Eq. 7. Eq. 7 is a general relationship between the loading curvature C and the Vickers hardness number. Under the assumption that the ratio of the contact depth hc to the total indentation depth h is constant we obtain a linear relationship between the loading curvature C and the Vickers hardness number (Eq. 7). We define ( hh )2 as ϕ . After rearranging Eq. 7 c the parameter ϕ can be calculated in two different ways given by Eq. 8. For a given macroscopic Vickers hardness number it allows for the calculation of the corresponding macroscopic loading curvature C and, thus, allows for the correction of ISE-affected values of C (Eq. 9). In Section 6.1 the practical application of the presented correction procedure is shown.
Hi =
Pmax Ap
HV = 0.102 ·
(4)
Pmax As
(5)
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2400 0
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N mm²
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Loading curvature, C in
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Mean hardness Hi at different indentation depths Mean value of C at different indentation depths Values of C determined from single P-h curve (h = 2400 nm)
2000
Fig. 6. Determined values of ϕ 1 and ϕ 2 as a function of the indentation depth for electropolished AISI 304L.
Indentation depth, h in nm Fig. 5. Hardness Hi and loading curvature C as a function of the indentation depth h of electropolished AISI 304L.
AISI 304 mechanically polished before correction procedure AISI 304 mechanically polished, corrected with AISI 304 mechanically polished, corrected with
ϕ=
h hc
Cmacro =
2
Ch2 0.0926 = ·ϕ ·C 2 24.5 24.5hc and
ϕ=
with
ϕ=
h hc
24.5 · HV 0.0926 · C
24.5 · HVmacro ϕ · 0.0926
2 (7)
(8)
(9)
Loading curvature, C in
HV = 0.0926 ·
(6)
N mm²
160000
Pmax HV = 0.0926 · Hi = 0.0926 · Ap
AISI 304 electropolished before correction procedure AISI 304 electropolished, corrected with AISI 304 electropolished, corrected with
140000
Macroscopic (FEM, tensile test data)
120000 100000 80000 60000 40000 20000 0
6. Results 6.1. Analysis of the ISE and correction of the loading curve The data shown in Fig. 5 reveals a significant ISE for the electropolished AISI 304L which is in agreement with the literature (Manika and Maniks, 2006; Ye et al., 2007). Since the CSM method could not be applied with the used test device the indentation hardness was measured with at least five P-h curves with different investigated indentation depths. The mean indentation hardness determined at the investigated indentation depths increases drastically with decreasing indentation depth (increase of ≈145% from 2400 nm to 200 nm). A saturation of the ISE cannot be detected in the investigated depth range. The hardness shows a strong sinking tendency over the whole investigated depth range which illustrates that even the P-h curves with the highest possible indentation depth of the test device are notably affected by the ISE. The ISE can also be characterized using the parameter C determined from the P-h curves measured at the different indentation depths. In Fig. 5 the mean value of C at each investigated indentation depth is shown as a function of the indentation depth. As introduced in Section 4, the parameter C can also be determined as a function of the indentation depth from a single P-h which is also shown in Fig. 5 for one of the P-h curves with the highest indentation depth. The progression is in good agreement with the mean values of C determined from each P-h curve measured at different indentation depths and is also in good agreement with the progression of the hardness plotted versus the indentation depth (Fig. 5). The analysis of the parameter C can practically be used to analyze single or mean P-h curves of given indentation depths in order to characterize the ISE. This procedure can easily show that
500
1000
1500
2000
2500
Indentation depth, h in nm Fig. 7. Loading curvature C as a function of the indentation depth of AISI 304L in electropolished and mechanically polished condition before and after correction procedure. As reference the macroscopic loading curvature derived by FEMsimulation with the material parameters from tensile test is included.
even curves of the highest, investigated indentation depth are still highly influenced by the ISE. The parameter C is an important parameter for many inverse methods which are based on sharp indentation (Dao et al., 2001; Hyun et al., 2011; Le, 20 08; 20 09; Tho et al., 20 05). Using ISEaffected values of C as input parameters for inverse methods leads to artificially influenced results. In such circumstances it is desirable to correct an ISE-affected loading curvature C. Following the procedure introduced in Section 4 (see Fig. 4) the affected values of the parameter C can be corrected. There are two possible options to calculate the correction parameter ϕ (see Eq. 8). From an ISE-affected curve there can be used h and hc as well as the loading curvature C and the Vickers hardness number HV to calculate ϕ . Fig. 6 shows the mean values of ϕ determined with h and hc as well as determined with the parameters C and HV as a function of the investigated indentation depths. There is a slight difference at low indentation depths (<500 nm), whereas, at higher indentation depths the values of ϕ determined with the different ways show a good agreement. With Eq. 3 the ISE-affected loading curvature C can be corrected with the parameter ϕ and the macroscopic Vickers hardness number which was determined to 145 HV30 for AISI 304L. Fig. 7 shows the parameter C before and after the correction procedure. It can be seen that both different ways of calculating ϕ lead to almost identical results at indentation
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Mean value of C at different indentation depths Corrected values of C ( ) Corrected values of C ( )
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Increase due to surface hardening Increase due to the ISE
350
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N mm²
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Loading curvature, C in
250 200 150 100 50 0 0
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1200
1600
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Indentation depth, h in nm
ISE 200000
180000
160000
140000
120000 0
Fig. 8. Separation of the increase of the parameter C due to the ISE and due to surface hardening for mechanically polished AISI 304L.
9200 180000
8400 160000 8000
250
500
750
1000
1250
1500
140000 1750
Indentation depth, h in nm Fig. 9. Hardness Hi and loading curvature C as a function of the indentation depth h of C45 (quenched).
depths higher than ≈500 nm. Since the indentation test device is not able to capture the saturation of the ISE the Finite-ElementMethod (FEM) was used to calculate the macroscopic value of the parameter C. The numerically calculated and the corrected values of the parameter C are in good agreement. Especially, using h and hc for calculating ϕ allows for an accurate correction of C even at low indentation depths. Mechanically grinded and polished samples exhibit a hardened surface layer leading to an additional increase in strength which is superimposed by the ISE. This is shown in Fig. 7. In case of the mechanically grinded and polished sample the parameter C is higher at all investigated indentation depths compared to the electropolished sample. Both, the ISE and surface hardening lead to an artificially increase of C. It can be seen that the additional increase in strength due to surface hardening can also be corrected leading to a good agreement to the corrected electropolished values and the FEM results (see Fig. 7). The comparison of the electropolished and mechanically polished samples allows the separation of both influences. Fig. 8 shows the increase of the loading curvature C compared to the macroscopic value due to the ISE and surface hardening. It can be seen that the influence of the ISE is dominant compared to surface hardening. The carbon Steel C45 does not exhibit an extensive ISE. The hardness and the loading curvature C as a function of the indentation depth are plotted in Fig. 9. The hardness as well as the param-
Factor of projected contact area
200000
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1750
Indenter geometry used in this study Ideal indenter geometry
N mm²
ISE 9600
8800
500
Indentation depth, h in nm
220000
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Loading curvature, C in
Indentation hardness, Hi in MPa
250
Fig. 10. Loading curvature C as a function of the indentation depth of C45 (quenched) before and after correction procedure.
Mean hardness Hi at different indentation depths Mean value of C at different indentation depths Values of C determined from single P-h curve (h = 1600 nm)
0
5
30 devitation < 5 % 25
20
15
10
5
0 0
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400
600
800 1000 1200 1400 1600 1800 2000
Indentation depth, h in nm Fig. 11. Deviation of the Berkovich indenter used in this study from an ideal Berkovich indenter with constant factor of projected contact area of 24.5.
eter C decrease with increasing indentation depth until they are reasonable constant at indentation depths larger than ≈800 nm. Analogous to the AISI 304L the analysis of C shows quantitatively the extend of the ISE, although, at shallow indentation depths it is increasingly overlayed with the violation of the self-similarity. Nevertheless, the influence of the ISE is faded away at indentation depths being larger than 800 nm. Although a correction procedure is not necessary Fig. 10 shows the corrected loading curvatures using the procedure shown in Fig. 4. The ‘macro’ Vickers hardness number was measured to 750 HV (conversion of Hi with Eq. 6) at the indentation depth of 1600 nm were the ISE is saturated. The corrected values of C are in agreement to those which are not affected by the ISE. At shallow indentation depths (<600 nm) the correction leads to higher values of C compared to the not ISEaffected values of C. Those deviations are the result of the violation of the self similarity which is discussed in Section 6.2. 6.2. Discussion of the correction procedure The correction procedure is based on the assumption that the ratio hhc is mostly independent of the indentation depth and not significantly affected by the ISE or hardened surface layers. As shown in Fig. 12 for AISI 304L and C45 this assumption is essentially correct. Solely, at indentation depths lower than 500 nm the ratio hhc slightly decreases. All in all, the ratio is nearly constant
Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024
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tion depths lower than ∼600 nm. Firstly, Eq. 9 is not valid in this regime due to the changed projected contact area. Secondly, the induced strain and deformed volumes are significantly dependent on the indenter geometry and, thus, leading to non self-similar deformation where Kick’s law is not valid. For higher indentation depths the procedure can effectively correct the influences of the ISE and surface hardening. Though, application and analysis of different materials are necessary in order to investigate the transferability. According to Fig. 10 the corrected loading curvature shows a certain scatter range. Perturbations of the loading curvature used as input parameter for inverse methods can highly affect the inverse solution. Due to the ill-posed nature of the inverse indentation problem small variations of the loading curvature can lead to larger errors of the inverse solutions (Liu et al., 2009).
AISI 304 electropolished AISI 304 mechanically polished C45 (quenched)
0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.82 0.80 0
500
1000
1500
2000
2500
Indentation depth, h in nm Fig. 12. Ratio hhc as a function of the indentation depth h for C45 (quenched) and AISI 304L after mechanical polishing and electropolishing.
18 m 60 N m F N = 18 0 F m = 35 N 0 m N
This paper presents a method which allows for the characterization of the ISE using the loading curvature C of a single load-displacement curve. The method analyzes the change of the parameter C as a function of the indentation depth during sharp indentation. Furthermore, it allows for the correction of ISE-affected loading curves including the correction of hardened surface layers due to mechanical grinding and polishing. The following conclusions can be drawn:
=
F
F
=
0,25 0,00
Height in µm
-0,25 -0,50 -0,75 -1,00
top view x
-1,25 -1,50
path
-1,75 -2,00 0
5
10
15
20
25
30
35
Distance along path, x in µm Fig. 13. Hight-profiles of Berkovich imprints of different indentation depth in AISI 304L measured with atomic-force-microscopy. The sink-in behavior is largely unaffected by the ISE.
with ∼0.95 for mechanically and electropolished AISI 304L and constant with ∼0.88 for quenched C45. Fig. 13 shows several profiles of imprints made in AISI 304L with different indentation depths measured by atomic-force-microscopy. The profiles show that the ratio hhc , contrary to the hardness, is almost not affected
by the ISE. The pile-up and sink-in behavior and, thus, the ratio hhc is a function of the material parameters (Taljat and Pharr, 14) given by Eq. 10 for Ludwik power-law materials (Pöhl et al., 2014). The results indicate that the changes of the material parameters due to surface hardening and the ISE do not have a high impact on hhc .
hc K = f , n h E
7. Summary and conclusions
(10)
The correction procedure shows a good performance for indentation depths larger than ≈600 nm for AISI 304L and C45. At lower indentation depths the ISE-affected parameter C can be improved with the correction procedure, but still deviates from the not ISE-affected value. The explanation is the deviation of a real indenter from the perfect Berkovich indenter geometry. For an ideal Berkovich indenter the projected contact area is given by: Ap = 24.5 · h2c . Fig. 11 shows the deviation from the ideal value of 24.5 for the indenter used in this study. The projected contact area of the real indenter was determined with a calibration procedure in fused silica as it is described in DIN EN ISO 14577 (2007). It deviates less than 5% for indentation depths larger than ≈600 nm and explains the deviations of the correcting procedure at indenta-
• The analysis of the loading curvature C is a practical method for the characterization of the ISE. The intensity as well as the indentation depth where the ISE is saturated can be detected (when the saturation depth is within the measurement range of the test device). • The ISE was successfully characterized for the heavily affected steel AISI 304L and the quenched carbon steel C45 with low ISE. • ISE-affected loading curves (parameter C) can be corrected with the use of the Vickers macro hardness according to Eq. 9. The analysis of the loading curve and the correction procedure can be applied to single P-h curves. The correction procedure was successfully applied to heavily ISE-influenced P-h curves of AISI 304L and to slightly affected P-h curves of the quenched carbon steel C45. • The constant ratio hh is the basis for an accurate correction c procedure with Eq. 9. In contrast to the hardness, it is widely constant and not affected, although, the material parameters change in presence of the ISE. Those changes do not have a high impact on hh for the investigated materials. Though, the c transferability to other materials has to be examined. • Mechanically grinded and polished samples exhibit a hardened surface layer leading to an additional increase in strength which is superimposed by the ISE. It was shown that the influence of both the ISE and hardened surface layers can be corrected for the analyzed materials. References Bouzakis, K.-D., Michailidis, N., Hadjiyiannis, S., Skordaris, G., Erkens, G., 2002. Continuous fem simulation of the nanoindentation: actual indenter tip geometries, material elastoplastic deformation laws and universal hardness. Zeitschrift für Metallkunde 93 (9), 862–869. Cheng, Y.-T., Cheng, C.-M., 2004. Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng. R 44, 91–149. Dao, M., Chollacoop, N., van Fliet K.-J., Venkatesh, T.-A., Suresh, S., 2001. Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Materialia 49, 3899–3918. DIN EN ISO 14577, 2007. Instrumentierte eindringprüfung zur bestimmung der härte und anderer werkstoffparameter. Gao, H., Nix, W.-D., Hutchinson, J.-W., 1999. Machanism-based strain gradient plasticity. i. theory. J. Mechan. Phys. Solids 47, 1239–1263. Huang, Y., Gao, H., Nix, W.-D., Hutchinson, J.-W., 1999. Machanism-based strain gradient plasticity. ii. analysis. J. Mech. Phys. Solids 48, 99–128.
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Please cite this article as: F. Pöhl et al., Detection of the indentation-size-effect (ISE) and surface hardening by analysis of the loading curvature C, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.01.024