# Indentation Curves in Viscoplastic Alloys: Mathematical Model, Fitting Procedures, and Application to the Room-Temperature Creep of an Al-Sn Alloy

Authors: V. H. Jacobo; A. Ortiz; R. Schouwenaars
Journal: Advances in Materials Science and Engineering
 (2016)

---

## Abstract

Hardness is an important design parameter but, in rate-dependent materials, its value depends on the indentation speed and dwell time during measurement. Dimensional analysis for indentation testing provides rigorous descriptions for the load-displacement curves of elastoplastic materials; viscoplastic materials can be treated likewise by neglecting the plastic part of the deformation, which is not accurate for most engineering alloys. This work presents a methodology for constructing model indentation curves taking into account concurrent viscous and plastic strains, as well as corrections for tip roundness, load frame compliance, and the point of first contact. A procedure is presented to calculate the parameters of a single model curve by fitting to multiple experimental curves, incorporating the numerical solutions of the differential equation describing viscoplastic behaviour. The procedure is applied to Vickers indentation in brass and steel calibration blocks and to a SAE783 Al-Sn alloy for journal bearings, where creep at room temperature is observed. The soundness of the approach is demonstrated by the large reduction of statistical uncertainty on the parameters describing the indentation curves. A rate-independent hardness will be found and a brief comment is provided on the comparison between creep analysis by indentation and uniaxial tension.


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## Body

## 1. Introduction

Microindentation is a fairly easy and quick test which is widely used in quality control because it does not require the rigorous sample preparation needed for tensile testing and can be applied to small volumes. However, often the researcher needs detailed analysis of the tensile properties of a material, either for a better assessment of the material behaviour or as the input for further modelling. Therefore, efforts have been made over the last six decades to obtain additional information on material properties from this test [1], which typically consists of a controlled loading stage at a fixed loading range, a holding stage at fixed load, and unloading at a constant rate. Oliver and Pharr [2], using instrumented indentation, demonstrated that, besides hardness, the stiffness of the material can be obtained by careful analysis of the unloading curves. In a follow-up paper [3], they express caution with respect to the determination of further plastic properties from the curves, mainly due to the possible occurrence of material pile-up, which cannot be predicted without previous knowledge of the same properties that are to be determined.The latter problem can be partially solved by the observation of the geometry of the indentation [4]. Giannakopoulos and Suresh [5] addressed this question by a simplified analytical approach, while finite element models have been developed to determine the plastic properties by reverse modelling [6–9]. However, recent results seem to point out that the possibility for precise determination of the properties was somewhat overestimated in these studies. It has been shown that different sets of material properties can generate equal or at least highly similar indentation curves [10–12]. This means that small variations between measured curves may generate large variations in estimated mechanical properties, as has been addressed recently by Marteau et al. [13].Y.-T. Cheng and C.-M. Cheng [14] present a broad overview on the use of dimensional analysis to describe the indentation curves of elastoplastic and time-dependent materials. A review by Vandamme and Ulm [15] focuses mainly on linear viscoelasticity effects in indentation tests. By making use of Radok and Lee’s principle of equivalence [16, 17], interesting results can be derived for indentation in viscoelastic materials based on known elastic solutions, for example, Sneddon’s solution [18]. Examples of applications were given by Vandamme et al. for cementitious materials [19] and polystyrene [20] and Ngan and coworkers for polymers and metallic materials [21–23].The first two papers [19, 20] focused on the holding stage of the indentation test and analysed creep during this stage. Careful analysis of the fundamental equations governing the material behaviour (equilibrium, strain compatibility, constitutive equations, and boundary conditions) then allows for complementing the results of dimensional analysis. The latter three [21–23] study how creep affects the determination of the contact stiffness, which according to Oliver and Pharr’s method is determined from the slope of the curve at the start of unloading. The analysis of the jump conditions occurring at the transition between the holding and unloading stage then allows for providing important corrections to the contact stiffness when creep occurs in addition to plastic deformation. By assuming that creep strain can be superposed (summed) to plastic strain, the authors were able to include power-law creep in their analysis. This approach is also followed in the present work.The studies mentioned above go beyond dimensional analysis to include viscoplastic effects into the analysis of indentation curves but focus on the holding and the unloading stage. Most papers analysing creep during the loading stage use dimensional arguments [6–12] and focus on elastoplastic materials following simple power-law hardening:(1)
σ
=
k
0
ε
n
,where σ and ε are the true stress and true strain, respectively, and k
0 and n are constants. Dimensional analysis [14] then demonstrates that the theoretical curve always obeys the following relationship:(2)
P
=
C
1
h
2
,where P is the load, h is the indentation depth, and C
1 is a constant. For a Vickers indenter and without making a correction for material pile-up or sink-in, C
1
=
24.5
H, with H being the hardness of the material in MPa. For strain rate sensitive materials, obeying the following constitutive law,(3)
σ
=
k
1
ε
˙
m
′
,one finds(4)
P
=
C
2
h
α
,where ε
˙ represents the strain rate, k
1 and C
2 are constants which have to be determined by experiment, and the exponent α
=
2
/
(
m
′
-
1
), with m
′, is a materials constant [14, 24]. It shall be noted that (4) is only valid in deformation “dominated by creep” [14], that is, if the effect of time-independent plastic deformation can be neglected. This will rarely be the case in metallic materials at low homologous temperature.In general, dimensional analysis allows for the determination of the creep exponent only. Additional studies, combining dimensional arguments with finite element simulations, explore the possibility of determining the value ofk
1 in (3) [25, 26]. However, they are also limited to materials where the time-independent plastic behaviour can be neglected. In the present work, the details of the plastic behaviour and value of k
1 are not determined, but a rate-independent hardness is defined which follows from the analysis.The study is motivated by research in the Al-Sn system which is widely used for the journal bearings of compact combustion engines. The alloy is used in a bilayer with a steel sheet backing and is modified superficially when exposed to wear [27–30], which makes tensile testing impractical (although free-standing sheet can be easily produced for research purposes). Existing methods for the analysis of tensile properties through indentation testing [2–8] are not applicable to this material due to a viscous component in the total strain [31, 32]. For conventional Al-alloys at room temperature, this effect is considered unimportant but, apparently, the addition of 20 Wt% Sn and the resulting melting point reduction enhance the viscous behaviour of the alloy.The paper will follow the method presented by Sargent and Ashby [33]. However, in the latter paper, the hardness due to purely plastic (time-independent) strain is not taken into account. Corrections for the uncertainty of the starting point of the P
-
h curve and indenter tip roundness will be included in the analysis. Data fitting to the measured curves starts from the principle that a single material can only have a single set of fitting parameters and, therefore, all experimental curves must be fitted simultaneously to the theoretical one. An efficient method will be presented to effectuate this procedure.As a reference, tensile results on the same material will be briefly discussed. It will be seen that a critical assumption in all methods pretending to measure the strain rate sensitivity by means of indentation experiments is that the strain rate sensitivity exponent is a constant. In alloys showing the Portevin-Le Chatelier (PLC) effect [34], this is often not the case and results must be interpreted correspondingly.


## 2. Experiments

Instrumented hardness tests were executed on the surface of annealed SAE783 sheet as well as polished surfaces of brass and steel calibration blocks. A Microphotonics Nanovea® platform with a Vickers pyramid was used. Loading and unloading rates were 10 N/min up to 10 N; the samples were held for 10 s at 10 N. The reference tensile tests were performed on the same sheets according to norm ASTM E9 on samples cut along the rolling direction.Although the operation of the instrument assumes that the user can identify the starting point of the curves by careful inspection of the signal during the start of the equipment, this procedure may be unreliable. The reason is that the value of the curve and its derivative are equal to 0 according to (2) and (4). Electronic noise and vibrations cause finite variations. Therefore, the estimation of where the signal starts to deviate from 0 is always fraught with uncertainty. For spherical nanoindentation, this problem has been analysed in depth by Kalidindi and Pathak [35]. The curves presented without data correction use this intuitive determination of the point of first contact; the data correction will take care of the precise determination of this point according to the procedure proposed by Marteau et al. [13]. This same procedure will be used to correct for tip roundness according to the proposal by Sun et al. [36].The effect of load frame compliance is minimised in the equipment by the use of a direct optical measurement of the indentation depth in which the optical pen is mounted as close as possible to the indenting tip. This reduces the effect of the compliance to the deformation of the tip and tip holder only. Nonetheless, even this small effect is detectable in the results. Instead of following the fairly complicated procedure established by Oliver and Pharr [2, 3], the method of Sun et al. [36] is incorporated into the data correction procedure, as shown as follows.


## 3. Data Analysis

### 3.1. Viscoplastic Analysis

The approach followed by Sargent and Ashby [33] for indentation creep will be expanded to incorporate plastic strain as well as variable loads. In their work, (3) is replaced by(5)
ε
˙
=
ε
˙
0
σ
σ
0
m
,where ε
˙
0 and σ
0 are reference values (constants) and the exponent m is the inverse of m
′ used in (3).From dimensional analysis, it is found that, for a given loadP,(6)
d
A
c
=
C
2
A
ε
˙
0
P
A
σ
0
m
d
t
,where d
A
c is the increment in contact area due to creep alone and A is the contact area. From this point on, the analysis of Sargent and Ashby is modified to include also an increment in plastic strain d
A
p. From the definition of hardness, it follows that(7)
d
A
p
=
d
P
H
=
P
˙
H
d
t
,where H is the rate-independent hardness of the material, that is, without any manifestation of creep. Notice that this definition is independent of any constitutive law and no assumptions on the distributions of stress and strain fields are made. For a constant loading rate, it is found that (8)
d
A
=
d
A
p
+
d
A
c
=
P
˙
H
+
C
2
ε
˙
0
σ
0
-
m
P
˙
m
A
1
-
m
t
m
d
t
.This summation is based on the following reasoning: if creep were absent, only (7) is needed. In presence of creep, some additional “growth” of the indentation will occur, just as the case in creep testing where an initial indentation is made at a given loading speed and subsequently the increase in contact area (decrease in apparent hardness) is measured. Being based on dimensional analysis, no physical information is used nor gained on how the plastic and viscoplastic deformation mechanisms interact inside the material volume tested.Generally, the indentation depth is measured andA
=
f
h
2, with f being a geometric factor which for the Vickers indenter is equal to 24.5. Then, (8) is rewritten as(9)
d
h
=
1
2
h
f
H
+
1
2
C
2
ε
˙
0
σ
0
-
m
f
-
m
P
˙
-
1
h
1
-
2
m
P
m
d
P
.This is a nonlinear ordinary differential equation describing the indentation curve for power-law hardening materials with power-law creep. It is interesting to regroup all the constants by defining the fitting parameters C
1 and k:(10)
d
h
d
P
=
1
2
C
1
h
+
k
h
1
-
2
m
P
m
.Attempts to solve this equation in analytical form are beyond the scope of this paper. Modern mathematical software can easily provide numerical solutions for any set of parameters m, C
1, and k.


### 3.2. Corrections for Tip Roundness and Frame Compliance

It has been shown that if the indenter tip is rounded [14], the indentation curve will become equal to the one predicted by dimensional analysis once the indentation depth is large as compared to the radius of the tip. However, the value of h will then be shifted by an unknown amount [14, 36]. Such a shift will also occur if one considers that the starting point of the loading curve is not precisely known. Both effects will be characterised by a single offset h
0 and the symbol h must be substituted by (h
-
h
0) in all equations. Notice that h
0 includes the tip roundness which is constant from test to test as well as the uncertainty on the starting point of the indentation, which is variable. The combination of both effects means that no information on the tip roundness itself can be obtained in this way.Frame compliance is incorporated into the analysis by considering a linear elastic response from the loading frame [36]. Then, an additional Δ
h
=
C
f
P must be added to all curves, with C
f being the frame compliance which can be obtained by data fitting. Assuming briefly that h
0 is 0, the measured curve is described by the following equation:(11)
h
=
h
id
+
Δ
h
=
P
C
1
+
C
f
P
,where h
id is the ideal indentation depth obtained for an infinitely stiff measurement system (2). Inverting this relationship and substituting h by (h
-
h
0), the following relationship is found:(12)
P
=
1
+
2
C
1
C
f
h
-
h
0
-
1
+
4
C
1
C
f
h
-
h
0
2
C
1
C
f
2
.It is not straightforward to see that (12) tends to (2) if C
f tends to 0, but, by calculating the limit, this essential property can be verified.


### 3.3. Fitting Procedure

Formulas (10) and (12) provide models for the experimental curves. To simplify the analysis of the viscoplastic indentation curves, the correction for frame stiffness is assumed to be identical as the one obtained on calibration blocks. This correction is small in the present study. Then, the models can be described by the functions P
v
p
(
h
;
C
1
,
k
,
m
) and P
p
l
(
h
;
C
1
,
C
f
), where the values before the semicolon correspond to variables of the function and the ones behind it are parameters of the model, determined by fitting. The computational cost of evaluating each of the model curves for a given set of parameters is larger for P
v
p than for P
p
l, with the former requiring the numerical solution of a nonlinear ODE. As has been demonstrated by Marteau et al. [13], all the experimental curves have to be fitted to a single physical model representing the average properties of the material, determining a single set of the parameters of the model under consideration. On the other hand, each measurement curve is represented by a single value for h
0.Taking as an example the viscoplastic indentation curve, the following objective function must be minimised with respect to the parameters of the model:(13)
s
s
C
1
,
k
,
m
=
∑
k
=
1
K
∑
l
=
1
L
k
P
k
l
-
P
v
p
h
k
l
-
h
0
k
;
C
1
,
k
,
m
2
P
k
l
p
,where P
k
l is the load measured in point l of curve k, at a measured indentation depth h
k
l; h
0
k is the unknown starting point of curve k. L
k is the number of measurements in curve k and K is the number of measured curves. The exponent p is a characteristic parameter of the fitting procedure. Higher values of p assign more importance to the points close to the origin, where P
k
l are small. Setting p
=
0 gives a standard least squares procedure. This value was found to provide a poor fit close to the origin; other values tested were 1/2, 1, and 2. The value of 1 was used in the paper; this choice will be briefly addressed in Section 3.4.Minimisation is performed with respect to the free parametersC
1, k, and m and the individual h
0
i corresponding to each curve. An interesting simplification is obtained by taking into account that, for a given set of C
1, k, and m, the curve P
m
(
h
;
C
1
,
k
,
m
) provides a single model to which the individual measurements must be fitted. This means that, instead of performing an optimisation in a K
+
3-dimensional space, optimisation is done in two steps, involving K single-parameter fits for each of h
0
i, embedded in a three-parameter optimisation for C
1, k, and m. Taking into account that each evaluation of (10) involves the numerical solution of a nonlinear differential equation, this reduction in the dimensionality of the problem is important.


### 3.4. Statistical Analysis

To get a first-order assessment of the relative importance of small perturbations on the parameters in the model, one can analyse (2) and write(14)
P
+
δ
P
=
C
+
δ
C
h
+
δ
h
2
,where δ
P corresponds to the variation of P due to the variation of C  (
δ
C
) and h  (
δ
h
). For example, δ
C, which indicates a variation of the material properties, may be caused by microstructural heterogeneities in the material or by surface alteration during sample preparation. If δ
P were only caused by the sensibility and noise in the load cell and δ
h by similar effects in the optical distance measurement system, they would be very small. However, the inaccuracy in the determination of h
0 may introduce much larger variations δ
h. Neglecting second-order terms in (14), one finds(15)
δ
P
≈
2
C
h
δ
h
+
h
2
δ
C
.The effect of the estimation of the initial point of contact can be probed as follows: one starts with the global fitting procedure of one set of parameters to all curves. This gives a table of K values of h
0
k. Using these values, the fitting procedure is repeated to each curve one by one. This gives K sets of parameters. Second, the fitting procedure is performed to each raw curve, with the value of h
0
k obtained from each single curve alone. Because the choice of this second value of h
0
k now does not take into account the data of the other curves, the second table of h
0
k is different from the first one and the second set of fitting parameters is different from the first one. Comparing standard deviations on the parameters for the first and second set provides a measure for the improvement achieved. More specifically, by using the same fitting procedure on curves for which h
0
k is determined by individual fits (one curve at the time) or by the collective fit to a single curve, the effect on the estimation of h
0
k is measured.To measure the total dispersion on the data, one can define the average curve:(16)
P
h
¯
=
1
K
∑
k
=
1
K
P
k
h
,with P
k
(
h
) being the fitted curve for measurement number k. An estimate for the statistical spread for each set of curves (corrected or uncorrected) can be obtained by the following integral:(17)
MSE
=
1
K
∑
k
=
1
K
1
b
-
a
∫
a
b
P
h
¯
-
P
k
h
P
p
2
d
h
.The use of an integral in (16) instead of the sum in (13) is because the individuals h
k
l are different for all curves and therefore the determination of the average curve (15) unavoidably involves interpolation; in this sense, (16) corresponds to a continuous interpolating function and not to individual points; because h
k
0 and h
k
L
k may differ from curve to curve, [a
,
b] is the common interval over which data are available for all curves.Ideally, all points along a curve should have the same weight in the calculation of the mean square error. If the first term in (15) predominates, then one should choose p
=
1
/
2; if the second term is more important, p
=
1. In case of the calibration blocks, the results show that both terms are of the same order of magnitude (Table 1), such that the choice of p is not self-evident. For the viscoplastic analysis, larger variations of C are observed and therefore p
=
1 seems to be more appropriate, so this is the value used here. Notice that the simple formula (2) was used in this section and not the more precise formulations (10) and (12), because this part of the analysis aims to establish order of magnitude estimates of the variations as opposed to the precise estimates of the model parameters which are obtained from the full model. Notice also that (13) can be used to estimate the precision of the results, but not theimprovement of the precision of the results, because no such value is available for the uncorrected data.Table 1
Mean value and standard deviation ofC
1 obtained from individual fitting of the curves before and after data correction on steel and brass calibration blocks. The standard deviation of the curves is reduced by a factor of 3.9 for steel and 4.7 for brass.
⁢Steel
⁢Brass
Raw data
Corrected data
Raw data
Corrected data
Mean (C
1, Pa)
1.51 × 1011
1.49 × 1011
2.97 × 1010
2.93 × 1010
St. Dev. (C
1, Pa)
1.8 × 109
4.4 × 108
2.0 × 109
4.2 × 108


## 3.1. Viscoplastic Analysis

The approach followed by Sargent and Ashby [33] for indentation creep will be expanded to incorporate plastic strain as well as variable loads. In their work, (3) is replaced by(5)
ε
˙
=
ε
˙
0
σ
σ
0
m
,where ε
˙
0 and σ
0 are reference values (constants) and the exponent m is the inverse of m
′ used in (3).From dimensional analysis, it is found that, for a given loadP,(6)
d
A
c
=
C
2
A
ε
˙
0
P
A
σ
0
m
d
t
,where d
A
c is the increment in contact area due to creep alone and A is the contact area. From this point on, the analysis of Sargent and Ashby is modified to include also an increment in plastic strain d
A
p. From the definition of hardness, it follows that(7)
d
A
p
=
d
P
H
=
P
˙
H
d
t
,where H is the rate-independent hardness of the material, that is, without any manifestation of creep. Notice that this definition is independent of any constitutive law and no assumptions on the distributions of stress and strain fields are made. For a constant loading rate, it is found that (8)
d
A
=
d
A
p
+
d
A
c
=
P
˙
H
+
C
2
ε
˙
0
σ
0
-
m
P
˙
m
A
1
-
m
t
m
d
t
.This summation is based on the following reasoning: if creep were absent, only (7) is needed. In presence of creep, some additional “growth” of the indentation will occur, just as the case in creep testing where an initial indentation is made at a given loading speed and subsequently the increase in contact area (decrease in apparent hardness) is measured. Being based on dimensional analysis, no physical information is used nor gained on how the plastic and viscoplastic deformation mechanisms interact inside the material volume tested.Generally, the indentation depth is measured andA
=
f
h
2, with f being a geometric factor which for the Vickers indenter is equal to 24.5. Then, (8) is rewritten as(9)
d
h
=
1
2
h
f
H
+
1
2
C
2
ε
˙
0
σ
0
-
m
f
-
m
P
˙
-
1
h
1
-
2
m
P
m
d
P
.This is a nonlinear ordinary differential equation describing the indentation curve for power-law hardening materials with power-law creep. It is interesting to regroup all the constants by defining the fitting parameters C
1 and k:(10)
d
h
d
P
=
1
2
C
1
h
+
k
h
1
-
2
m
P
m
.Attempts to solve this equation in analytical form are beyond the scope of this paper. Modern mathematical software can easily provide numerical solutions for any set of parameters m, C
1, and k.


## 3.2. Corrections for Tip Roundness and Frame Compliance

It has been shown that if the indenter tip is rounded [14], the indentation curve will become equal to the one predicted by dimensional analysis once the indentation depth is large as compared to the radius of the tip. However, the value of h will then be shifted by an unknown amount [14, 36]. Such a shift will also occur if one considers that the starting point of the loading curve is not precisely known. Both effects will be characterised by a single offset h
0 and the symbol h must be substituted by (h
-
h
0) in all equations. Notice that h
0 includes the tip roundness which is constant from test to test as well as the uncertainty on the starting point of the indentation, which is variable. The combination of both effects means that no information on the tip roundness itself can be obtained in this way.Frame compliance is incorporated into the analysis by considering a linear elastic response from the loading frame [36]. Then, an additional Δ
h
=
C
f
P must be added to all curves, with C
f being the frame compliance which can be obtained by data fitting. Assuming briefly that h
0 is 0, the measured curve is described by the following equation:(11)
h
=
h
id
+
Δ
h
=
P
C
1
+
C
f
P
,where h
id is the ideal indentation depth obtained for an infinitely stiff measurement system (2). Inverting this relationship and substituting h by (h
-
h
0), the following relationship is found:(12)
P
=
1
+
2
C
1
C
f
h
-
h
0
-
1
+
4
C
1
C
f
h
-
h
0
2
C
1
C
f
2
.It is not straightforward to see that (12) tends to (2) if C
f tends to 0, but, by calculating the limit, this essential property can be verified.


## 3.3. Fitting Procedure

Formulas (10) and (12) provide models for the experimental curves. To simplify the analysis of the viscoplastic indentation curves, the correction for frame stiffness is assumed to be identical as the one obtained on calibration blocks. This correction is small in the present study. Then, the models can be described by the functions P
v
p
(
h
;
C
1
,
k
,
m
) and P
p
l
(
h
;
C
1
,
C
f
), where the values before the semicolon correspond to variables of the function and the ones behind it are parameters of the model, determined by fitting. The computational cost of evaluating each of the model curves for a given set of parameters is larger for P
v
p than for P
p
l, with the former requiring the numerical solution of a nonlinear ODE. As has been demonstrated by Marteau et al. [13], all the experimental curves have to be fitted to a single physical model representing the average properties of the material, determining a single set of the parameters of the model under consideration. On the other hand, each measurement curve is represented by a single value for h
0.Taking as an example the viscoplastic indentation curve, the following objective function must be minimised with respect to the parameters of the model:(13)
s
s
C
1
,
k
,
m
=
∑
k
=
1
K
∑
l
=
1
L
k
P
k
l
-
P
v
p
h
k
l
-
h
0
k
;
C
1
,
k
,
m
2
P
k
l
p
,where P
k
l is the load measured in point l of curve k, at a measured indentation depth h
k
l; h
0
k is the unknown starting point of curve k. L
k is the number of measurements in curve k and K is the number of measured curves. The exponent p is a characteristic parameter of the fitting procedure. Higher values of p assign more importance to the points close to the origin, where P
k
l are small. Setting p
=
0 gives a standard least squares procedure. This value was found to provide a poor fit close to the origin; other values tested were 1/2, 1, and 2. The value of 1 was used in the paper; this choice will be briefly addressed in Section 3.4.Minimisation is performed with respect to the free parametersC
1, k, and m and the individual h
0
i corresponding to each curve. An interesting simplification is obtained by taking into account that, for a given set of C
1, k, and m, the curve P
m
(
h
;
C
1
,
k
,
m
) provides a single model to which the individual measurements must be fitted. This means that, instead of performing an optimisation in a K
+
3-dimensional space, optimisation is done in two steps, involving K single-parameter fits for each of h
0
i, embedded in a three-parameter optimisation for C
1, k, and m. Taking into account that each evaluation of (10) involves the numerical solution of a nonlinear differential equation, this reduction in the dimensionality of the problem is important.


## 3.4. Statistical Analysis

To get a first-order assessment of the relative importance of small perturbations on the parameters in the model, one can analyse (2) and write(14)
P
+
δ
P
=
C
+
δ
C
h
+
δ
h
2
,where δ
P corresponds to the variation of P due to the variation of C  (
δ
C
) and h  (
δ
h
). For example, δ
C, which indicates a variation of the material properties, may be caused by microstructural heterogeneities in the material or by surface alteration during sample preparation. If δ
P were only caused by the sensibility and noise in the load cell and δ
h by similar effects in the optical distance measurement system, they would be very small. However, the inaccuracy in the determination of h
0 may introduce much larger variations δ
h. Neglecting second-order terms in (14), one finds(15)
δ
P
≈
2
C
h
δ
h
+
h
2
δ
C
.The effect of the estimation of the initial point of contact can be probed as follows: one starts with the global fitting procedure of one set of parameters to all curves. This gives a table of K values of h
0
k. Using these values, the fitting procedure is repeated to each curve one by one. This gives K sets of parameters. Second, the fitting procedure is performed to each raw curve, with the value of h
0
k obtained from each single curve alone. Because the choice of this second value of h
0
k now does not take into account the data of the other curves, the second table of h
0
k is different from the first one and the second set of fitting parameters is different from the first one. Comparing standard deviations on the parameters for the first and second set provides a measure for the improvement achieved. More specifically, by using the same fitting procedure on curves for which h
0
k is determined by individual fits (one curve at the time) or by the collective fit to a single curve, the effect on the estimation of h
0
k is measured.To measure the total dispersion on the data, one can define the average curve:(16)
P
h
¯
=
1
K
∑
k
=
1
K
P
k
h
,with P
k
(
h
) being the fitted curve for measurement number k. An estimate for the statistical spread for each set of curves (corrected or uncorrected) can be obtained by the following integral:(17)
MSE
=
1
K
∑
k
=
1
K
1
b
-
a
∫
a
b
P
h
¯
-
P
k
h
P
p
2
d
h
.The use of an integral in (16) instead of the sum in (13) is because the individuals h
k
l are different for all curves and therefore the determination of the average curve (15) unavoidably involves interpolation; in this sense, (16) corresponds to a continuous interpolating function and not to individual points; because h
k
0 and h
k
L
k may differ from curve to curve, [a
,
b] is the common interval over which data are available for all curves.Ideally, all points along a curve should have the same weight in the calculation of the mean square error. If the first term in (15) predominates, then one should choose p
=
1
/
2; if the second term is more important, p
=
1. In case of the calibration blocks, the results show that both terms are of the same order of magnitude (Table 1), such that the choice of p is not self-evident. For the viscoplastic analysis, larger variations of C are observed and therefore p
=
1 seems to be more appropriate, so this is the value used here. Notice that the simple formula (2) was used in this section and not the more precise formulations (10) and (12), because this part of the analysis aims to establish order of magnitude estimates of the variations as opposed to the precise estimates of the model parameters which are obtained from the full model. Notice also that (13) can be used to estimate the precision of the results, but not theimprovement of the precision of the results, because no such value is available for the uncorrected data.Table 1
Mean value and standard deviation ofC
1 obtained from individual fitting of the curves before and after data correction on steel and brass calibration blocks. The standard deviation of the curves is reduced by a factor of 3.9 for steel and 4.7 for brass.
⁢Steel
⁢Brass
Raw data
Corrected data
Raw data
Corrected data
Mean (C
1, Pa)
1.51 × 1011
1.49 × 1011
2.97 × 1010
2.93 × 1010
St. Dev. (C
1, Pa)
1.8 × 109
4.4 × 108
2.0 × 109
4.2 × 108


## 4. Results

### 4.1. Calibration Blocks

The results will be presented as a comparison between the raw measured data with the starting point of the indentation curve as determined by the equipment on the one hand and the corrected data with the fitting curve superimposed (Figure1). Only the loading part was studied in the data correction and therefore the unloading part is not presented. Table 1 gives a quantitative comparison between the dispersion of C
1-values obtained on individual curves without data correction and with data correction.Figure 1
Original data for steel (a) and brass (b) as compared to corrected data ((c) steel, (d) brass), with the fitting curve plotted in red. The measured data for the steel calibration block (4 curves) show a remarkably low dispersion and so the effect of the data correction appears to be small. For the brass calibration block (6 curves), the benefits of the procedure are easily observable (quantitative details are given in Table1).
(a)
(b)
(c)
(d)For the indentations in steel, it was found thatC
f is effectively 0 (3.6 × 10−13 m/N), for the data fitting on brass C
f = 7.9 × 10−8m/N. Rather than looking for an explanation for this apparent contradiction, one can conclude that the effect of frame compliance is too small to be determined precisely by the present data for the measurement system used. The effect of compliance of the system used in this work is low as compared to what was reported in early publications [2–4]. The possibility of obtaining better estimates of C
f, either by using a larger number of indentation curves or by including materials with a broader range of hardness, must be considered in future work.


### 4.2. Viscoplastic Analysis

Results for the raw and corrected loading curves measured on the Al-Sn alloy are presented in Figure2. Table 2 provides the fitting parameters and standard deviations for fitting to the individual curves before and after data correction.Table 2
Average data and standard deviations for the fitting parameters in the viscoplastic case before and after data correction.C
1
=
f
H, with H being the time-independent hardness.
Raw data
Corrected data
C
1 (Pa)
k
m
C
1 (Pa)
k
m
Mean
1.22 × 1010
3.47 × 10−6
0.45
1.28 × 1010
2.65 × 10−6
0.47
St. Dev.
1.4 × 109
1.1 × 10−6
0.03
1.3 × 109
9.4 × 10−7
0.03Figure 2
Raw curves for the Al-Sn alloy (a) and corrected data (b), with the fitting curve plotted in red.
(a)
(b)As can be seen, the comparison between the standard deviation on the fitting parameters before and after data correction shows an insignificant improvement of the results. Nonetheless, visual inspection of the corrected curves shows a clear reduction of the statistical spread. This reduction is quantified by the MSE as defined by (17). The corresponding values, for the three sets of indentation curves, are provided in Table 3.Table 3
Mean square error according to (17) for the brass and steel calibration blocks and the SAE 783 alloy.
Brass
Steel
SAE783
MSE before correction
0.008
0.006
0.014
MSE after correction
0.002
0.004
0.005
MSE before/MSE after
3.9
1.5
2.9


### 4.3. Tensile Tests on SAE783

Tensile tests were performed on the same material according to the norm ASTM E9 at crosshead speeds of 1.25, 2.5, 5, 10, and 20 mm/min, corresponding to engineering strain rates of 0.4 × 10−3, 0.8 × 10−3, 1.6 × 10−3, 3.2 × 10−3, and 6.4 × 10−3s−1 with four samples tested at each speed. The true stress-true strain curves were assumed to obey the following equation:(18)
σ
=
k
3
ε
˙
ν
ε
n
.All four values curves at a given speed were fitted to a single curve using (1). By plotting k
0 versus strain rate on a ln-ln plot, the values of k
3 and ν are obtained. The value of the strain rate sensitivity ν was estimated at −0.05 and serrations were seen in the tensile curves, revealing presence of a PLC effect (Figure 3).Figure 3
Selected tensile curves at different strain rates showing the inverse strain rate sensitivity and PLC effect (serrated flow). The inset presents a ln-ln plot ofε
˙ versus k
0 in formula (1); the slope of the plot corresponds to the exponent ν in formula (18); error bars represent the standard deviation on ln
⁡
k
0.


## 4.1. Calibration Blocks

The results will be presented as a comparison between the raw measured data with the starting point of the indentation curve as determined by the equipment on the one hand and the corrected data with the fitting curve superimposed (Figure1). Only the loading part was studied in the data correction and therefore the unloading part is not presented. Table 1 gives a quantitative comparison between the dispersion of C
1-values obtained on individual curves without data correction and with data correction.Figure 1
Original data for steel (a) and brass (b) as compared to corrected data ((c) steel, (d) brass), with the fitting curve plotted in red. The measured data for the steel calibration block (4 curves) show a remarkably low dispersion and so the effect of the data correction appears to be small. For the brass calibration block (6 curves), the benefits of the procedure are easily observable (quantitative details are given in Table1).
(a)
(b)
(c)
(d)For the indentations in steel, it was found thatC
f is effectively 0 (3.6 × 10−13 m/N), for the data fitting on brass C
f = 7.9 × 10−8m/N. Rather than looking for an explanation for this apparent contradiction, one can conclude that the effect of frame compliance is too small to be determined precisely by the present data for the measurement system used. The effect of compliance of the system used in this work is low as compared to what was reported in early publications [2–4]. The possibility of obtaining better estimates of C
f, either by using a larger number of indentation curves or by including materials with a broader range of hardness, must be considered in future work.


## 4.2. Viscoplastic Analysis

Results for the raw and corrected loading curves measured on the Al-Sn alloy are presented in Figure2. Table 2 provides the fitting parameters and standard deviations for fitting to the individual curves before and after data correction.Table 2
Average data and standard deviations for the fitting parameters in the viscoplastic case before and after data correction.C
1
=
f
H, with H being the time-independent hardness.
Raw data
Corrected data
C
1 (Pa)
k
m
C
1 (Pa)
k
m
Mean
1.22 × 1010
3.47 × 10−6
0.45
1.28 × 1010
2.65 × 10−6
0.47
St. Dev.
1.4 × 109
1.1 × 10−6
0.03
1.3 × 109
9.4 × 10−7
0.03Figure 2
Raw curves for the Al-Sn alloy (a) and corrected data (b), with the fitting curve plotted in red.
(a)
(b)As can be seen, the comparison between the standard deviation on the fitting parameters before and after data correction shows an insignificant improvement of the results. Nonetheless, visual inspection of the corrected curves shows a clear reduction of the statistical spread. This reduction is quantified by the MSE as defined by (17). The corresponding values, for the three sets of indentation curves, are provided in Table 3.Table 3
Mean square error according to (17) for the brass and steel calibration blocks and the SAE 783 alloy.
Brass
Steel
SAE783
MSE before correction
0.008
0.006
0.014
MSE after correction
0.002
0.004
0.005
MSE before/MSE after
3.9
1.5
2.9


## 4.3. Tensile Tests on SAE783

Tensile tests were performed on the same material according to the norm ASTM E9 at crosshead speeds of 1.25, 2.5, 5, 10, and 20 mm/min, corresponding to engineering strain rates of 0.4 × 10−3, 0.8 × 10−3, 1.6 × 10−3, 3.2 × 10−3, and 6.4 × 10−3s−1 with four samples tested at each speed. The true stress-true strain curves were assumed to obey the following equation:(18)
σ
=
k
3
ε
˙
ν
ε
n
.All four values curves at a given speed were fitted to a single curve using (1). By plotting k
0 versus strain rate on a ln-ln plot, the values of k
3 and ν are obtained. The value of the strain rate sensitivity ν was estimated at −0.05 and serrations were seen in the tensile curves, revealing presence of a PLC effect (Figure 3).Figure 3
Selected tensile curves at different strain rates showing the inverse strain rate sensitivity and PLC effect (serrated flow). The inset presents a ln-ln plot ofε
˙ versus k
0 in formula (1); the slope of the plot corresponds to the exponent ν in formula (18); error bars represent the standard deviation on ln
⁡
k
0.


## 5. Discussion

The main goal of this work is to present a model for the indentation curves in materials where neither the viscous nor the plastic component of strain can be neglected. To apply this model to the material of interest, the problem of fitting a single set of parameters to multiple measured curves had to be addressed. This aspect will be treated first in the discussion of this paper.It was found that if different indentation curves are obtained from the same material and if this material is homogeneous, these curves must be fitted to a single curve, instead of fitting each curve to an individual model and then averaging the parameters. The fitting is performed in a two-step procedure, with the inner loop consisting of a series of single-parameter adjustments (yielding the value ofh
0
k) and an outer loop to find the value of the fitting parameters of the single model curve. This reduces the complexity of the optimisation problem, making it comparable to the complexity of fitting K single curves. Only the incorporation of the solution of a differential equation as part of this fitting procedure is somewhat unconventional, but using mathematical software, this aspect is also relatively easy to implement.A simple statistical analysis of this method was presented. In classical nonlinear regression analysis, which is fairly well understood, a single curve is fitted to a set of measurement data. Here, the nonlinear fitting procedure fits a single curve to many experimental curves; the difference between the curves is much larger than the statistical spread of the individual measuring points along a single curve. This topic is not covered in standard nonlinear regression; a more profound analysis is beyond the scope of this paper.When looking at the reduction of the statistical spread on the parameters for calibration blocks, the differences between standard deviations before and after data correction are clear even without using quantitative statistics. In terms of the standard deviation obtained for the parameters for the viscoplastic material, this reduction is insignificant. The reason for this observation is explained accurately by the error propagation formula presented in Section3.4. (15). For the SAE783 alloy, which is a composite material with a dispersed soft Sn-phase in an Al-matrix [37], a certain degree of heterogeneity is naturally present. This means that the different values for C
1 correspond to real differences in local mechanical properties (if they were measurement errors, a similar dispersion would occur in the calibration blocks). This kind of dispersion cannot and should not be removed by the presented approach, because heterogeneity is as much a characteristic of an alloy as are its average properties. Nonetheless, the MSE shows a significant reduction for both the calibration blocks and the Al-Sn-alloy, which demonstrates the soundness of fitting multiple experimental curves to a single model curve.The main contribution of this work is the proposal of the viscoplastic model. It is clear that, for most metals, even at fairly high homologous temperatures, plastic deformation and viscous flow will occur simultaneously if an increasing load is applied during indentation. Therefore, pure creep tests (at constant load) or pure relaxation tests (at constant displacement) must be preferred to determine the creep exponent. Such tests do not provide information on the rate-independent plastic behaviour, that is, while one may determine the creep exponent with good accuracy, no reliable data are obtained about the rate-independent hardness of the material. According to the definition of hardness (7), only this rate-independent component is correct; additional growth of the indentation by creep would lead to an underestimation. If one performs hardness tests on materials which show creep in addition to plastic behaviour, the result will depend on the indentation speed and the dwell time at maximum load. In Al-Sn alloys, it was found by the authors that variation of loading speed and dwell time caused serious variations in classical Vickers hardness measurements.The value of the creep exponent obtained here must be interpreted with some caution. Tensile tests were performed in the strain rate range of 0.4 × 10−3 to 6.4 × 10−3. Finite element models show that the maximum strain under an indentation is of the order of 50%, which is reached in 60 s. At the limit of the plastic zone, this strain decreases to 0. This corresponds to values between 0 and 8.3 × 10−3 s−1 for the strain rate and, therefore, the range of strain rates coincides fairly well. However, the PLC effect is typically limited to low strain rates [38] and the strain rate sensitivity is known to vary with strain rate. Therefore, the value obtained for m and the one for ν cannot be compared directly, with the former one being an average value for the entire indented zone. Also, ν can only be identified directly with m
′ if n
=
0, which is not the case under the assumptions of the present investigation but may be the case in high-temperature creep tests. At room temperature and under conditions where the time-dependent deformation is smaller than or equal to the plastic component of deformation, direct comparison between tensile tests and indentation tests may be difficult.There have been some attempts at estimating the viscous and elastic properties of metals by inverse modelling using finite element models [12, 27, 28]. It has been clearly shown that this technique confronts a problem of uniqueness in the inverse analysis; that is, different sets of parameters in the finite element model may produce equal indentation curves [12]. This problem is avoided here by providing a theoretical equation to describe the viscoplastic indentation curve. Inverse analysis is not required and the number of parameters is limited to 3. The model allows obtaining an average creep exponent m and the rate-independent hardness (H
=
C
1
/
f) with good accuracy. The parameter k is a combination of geometrical and physical parameters as well as loading speed (9). With all other factors being constant, variation of the loading speed will produce different values of k and therefore may allow determining the material constants in this equation by appropriate data fitting procedures. Alternative methods, using stress jumps during creep experiments, have been proposed by Alkorta et al. [39].Comparing the present approach to other studies on indentation of elastoviscoplastic materials [19–23], it is seen that considerable differences are present. This is at least partially due to a difference in application. Here, the development of the analysis was guided by the study of strain hardening during wear of metallic materials [27–30]. For bulk metallic materials, the elastic properties are generally well known and show almost insignificant influence of thermal and mechanical processes. Hardness is then the parameter to be determined and corrections for the estimation of the contact stiffness due to creep are not essential to the study. Also, for the case of a two-phase material where the composite properties are of interest, microindentation may be preferred over nanoindentation. On the other hand, to determine viscous or elastic properties of the microconstituents of heterogeneous materials such as concrete or biomaterials [40], nanoindentation is the method of choice.In the work of Vandamme and coworkers [19, 20], the creep compliance can be related to the micromechanics of the materials studied and was therefore analysed at conditions of constant load. Plastic effects have been shown to occur even under such conditions, but they are small and the viscoelastic approach can be used. Ngan and coworkers focus on contact compliance and the viscosity of the material, both of which can be obtained from the indentation curve just before and just after the transition point form holding to unloading; the jump condition studied by these authors is imposed by the change in loading rate but the load, and hence the plastic strain, is constant and disappears from the equations describing the jump. In the present study, the plastic strain and creep increase simultaneously and their relative importance will depend on the material, loading rate, and testing temperature. Plasticity can therefore not be removed from the equations, but it was incorporated in a fairly straightforward way through its relationship with the classical definition of hardness, thereby avoiding the use of the full field equations for elastoviscoplastic materials, which is mathematically prohibitive. Combination of all three elements discussed here (creep during loading, holding, and effect of loading rate jumps) is a topic for future investigation which may provide interesting additional understanding on the particular nature of creep under highly localised loads.


## 6. Conclusions

A nonlinear ordinary differential equation describing the indentation curve for alloys where viscous and plastic deformations occur simultaneously has been proposed based on the dimensional analysis of both phenomena. The lack of a closed-form solution of this equation is no impediment to use the results in the data fitting of the time-independent hardness of the alloy and its creep exponent. The fitting procedures developed to analyse the experimental data in relationship to the theoretical model also allow correcting for the uncertainty on the point of first contact, indentation tip roundness, and, to a lesser extent, loading frame compliance. A general rule to be derived from this work is that the simultaneous fitting of all results to a single model curve is to be preferred to the separate fitting of each experimental curve to an individual model curve and obtaining the mean values of the fitting parameters afterward. This allows a reduction of the statistical uncertainty on the results by a factor which ranges from 1.5 to 4.7, with the higher value corresponding to experimental datasets with higher initial dispersion. The method proposed here allows for the definition of a rate-independent hardness, instead of the classical hardness which depends on the loading rate during testing if the material shows viscoplastic behaviour. The creep exponent obtained in indentation tests must be treated with caution, as it is a value obtained at temperatures and strain rates where the plastic part of the deformation is still significant. If the creep exponent is not constant with temperature and strain rate, then a direct comparison with classical creep tests is not possible.



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*Source: 6909854-2016-11-22.xml*