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Interpretation of the indentation size effect in vickers microhardness measurements-absolute hardness of materials
365
Thin Solid Films, 181 (1989) 365-374
I N T E R P R E T A T I O N OF T H E I N D E N T A T I O N SIZE E F F E C T IN VICKERS M I C R O H A R D N E S S M E A S U R E M E N T S - A B S O L U T E H A R D N E S S OF M A T E R I A L S G. FARGES A N D D.
DEGOUT
Etahlissement Technique Central de I'Armement, 16 bis Avenue Prieur de la C6te d'Or, 94114 Arcueil Cedex ( Fran~'e )
(Received March 17, 1989)
The apparent hardness increase with decreasing load observed with most microindentation measurements is in general referred to as the indentation size effect index (ISE). Empirical correction factors or sophisticated measurements suggested to date by several authors to correct this phenomenon are not entirely satisfactory. In this study a correction function for the measured diagonal length is introduced. A simple geometrical model of the deformation of the material around the impression is connected with the diagonal correction. The application of this method allows one to determine the absolute hardness of the materials and characterizes the reaction of the materials to the Vickers indentation test by a new coefficient. The model is accurately verified for different bulk materials: steels, copper, thick hard chromium but also titanium nitride films deposited on various substrates. This work demonstrates that the absolute hardness and this coefficient are also available via a very simple method based on the empirical relation
H = ' A + B d -1.
1. INTRODUCTION Microindentation is a widespread technique used to determine the hardness of bulk materials or coatings. Despite the extensive work performed in this field over the last thirty years, some practical problems encountered in indentation hardness measurements remain unresolved 1.2. One of the main problems associated with Vickers or K n o o p microhardness measurements is the apparent change in hardness number with change in indentation size. Therefore, comparisons between hardness values reported by different laboratories can be made only at similar load. Possible explanations of this well-known phenomenon, often termed the indentation size effect (ISE), have been discussed extensively in the literature. It is well established that a large number of mechanisms ~-4 and instrumental errors L6 contribute to the variation of hardness number with indentation size. It has been demonstrated that a better accuracy in diagonal length is obtained for scanning electron or transmission electron 0040-6090/89/$3.50
© ElsevierSequoia/Printedin The Netherlands
366
G. FARGES, D. DEGOUT
microscopes than for optical reading s'°. Another method proposed to compensate for this inaccuracy is to add a constant correction factor to the diagonal measurement 6. In most cases, since the sides of the imprint are not prefectly straight there is a difficulty in determining the load-bearing area of indentation. A point at issue is the amount by ~vhich the load is supported by the bulged area of the indentation surface during the loading of the indenter ~-9. However, recent work has demonstrated that the entire surface of an indentation bears the indenter load uniformly, including that created by any piled up 1°. Taking into account these results and on the basis of our experimental investigations, we propose an interpretation of the ISE in Vickers microhardness measurements based on a geometrical model. This simple geometric approach allows one to introduce a correction function to the diagonal length measured and then to determine the intrinsic hardness of various materials: steels, copper, thick hard chromium and also PVD titanium nitride films. Moreover, the application of this model gives a new coefficient characterizing the reaction of the material to the Vickers indentation. 2.
EXPERIMENTAL DETAILS
The surface of the samples were polished mechanically with SiC then with diamond pastes down to an average roughness of 0.05 ~tm. The designations and compositions of the bulk materials and coatings measured in this study are given in Table I. TABLE I DESIGNATIONAND CHEMICALCOMPOSITIONOF MATERIALS
Designation
Composition
H.S.P. steep 35 CD 4 steel 35 N C D 16 steel Tool steel, Z 85 WCDV Copper Chromium coating, 150 ~tm thick TiN coating, 7.4 ~tm thick
Unknown 0.35 C, 1 Cr, 0.25 Mo, 0.3 Si, 0.8 Mn 0.37 C, 3.84 Ni, 1.66 Cr, 0.34 Mn, 0.33 Si 0.85 C, 4Cr, 6W, 5 Mo, 2 V 99.9 Cu Hard electrolytic chromium PVD titanium nitride
a Hardness standard piece.
Microhardness tests were conducted with a Shimadzu type M microhardness tester. Measurements of indentation size were performed using the optical system of this apparatus at a magnification of 400x or a Reichert-Jung optical microscope equipped with a Sony video imaging working at a magnification of 5000 ×. The indenter was a Vickers diamond pyramid. On each sample impressions were made with four or more loads ranging from 15 gf to 1 kgf with a dwell time of 15 s. A set of five indentations were measured for each load level by the same operator. Both diagonals were measured to take into account the eventual assymetry of the indentation.
ABSOLUTE HARDNESS OF MATERIALS
367
Impressions were also studied with a three-dimensional rugosity apparatus built at E.T.C.A. The spatial resolution of this equipment is as low as 0.1 ~tm 3.
GEOMETRICAL MODEL OF INDENTATION
The hardness number H is defined as the ratio between the applied load L and the area A of the resulting indentation L H = -A
(I)
If the hardness is independent of the indentation load, with the Vickers squarebased pyramid used in the present study, A is expressed by d2 A = -k
(2)
where d is the diagonal length and k = 2 cos 22 °. In practice, the hardness is in most of the cases dependent on the indentation load, particularly for small indentations less than about 100 ~tm. In this case we express eqn. (1) as L H' = - A'
(3)
with A' given by d'2
A' = - k
(4)
where d' is the measured diagonal length. This paper is concerned with only those cases where the apparent hardness H' increases when the applied load decreases, corresponding to the situation AA=A-A'>0
(5)
As an example, Fig. 1 shows the surface morphology of indentation made under a load of 1 kgf on the 35 CD 4 steel sample recorded using the three-dimensional rugosity tester. A better understanding of the geometry of the bulge area is given by the partial contour-line map recorded above the surface defined to a good approximation by the extremities of the diagonals (Fig. 2). The cross-sections recorded along a diagonal and along a direction perpendicular to the boundaries (Fig. 3) show that the piled-up material around this Vickers indentation is a maximum at the midpoints of the indentation sides and a minimum at the corners, as reported by Chaudri and Winter 1°. Assuming a significant amount of the indenter load is supported by the piled-up area, these results suggest the geometrical model described in Fig. 4. The area of the bulge A b is given by
4fd'21/2
Ab -- ~
(6)
368
G. FARGES, D. DEGOUT
Y(mm)
0.05
0 0
0.02
0.04
0.06
8.08
Y= 0 ~ O. lmm
X(mm) X= 0 ~ O. Imm
Fig. I. Surface morpholgy of the indentation recorded by the three-dimensional rugosity technique.
Y (mm~} & F-"
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
./
o.~.
/"
,.,4,1
!
i
•
i
/ I~'
i
0.0~.
i
/
I--. . . . . . . .
0
*
0,02
......
4 B,O4
~ i- 0.0S
t
0.O8
Xtmm)
Fig. 2. Partial contour-line m a p recorded above the surface with an isohigh step of 0.05 j~m.
369
ABSOLUTE HARDNESS OF MATERIALS
Z
Z
b-b"
7,
5u'TJL
a - CI"
?.Sum 5tam
Z.Su.
O. Im
8.13u
8.15~
B. IGu
8.El~
Bum 8.,m
6.O?=m 8.04~
O.O6m O.g6m
Fig. 3. Profilometer traces recorded along the two directions drawn in Fig. 2.
,q
C
,,I
f
+
',
Fig. 4. Geometricalmodel for the Vickersindentation: horizontal (top) and vertical(bonorn)projections of the hardness indentation. where f is the projected length of the bulge as represented in Fig. 4. If at the initial time we assume that A b = AA the total indentation area is then
d,2 A = A'+AA =~--t
4fd,2X/2 k
(7)
Comparing relations (2), (4) and (7), and since geometrically the value 2f21/2 is equal to the length L, defined in Fig. 4, the following relation between d and d' is obtained
d 2 = d ' 2 + 2 L v d'
(8)
We name this expression the correction function for the measured diagonal length and L, the Vickers correction length. Theoretically, in the case of a homogeneous material this function allows one to calculate the true Vickers
370
G. FARGES, D. DEGOUT
hardness H knowing d' measured under several loads and Lv from the equation kL
H - d,2 + 2 L v d'
(9)
derived from eqns. (1), (2) and (8). The calculation is performed with an iterative method stopped for the minimum value of the slope of the linear regression carried out on the corrected hardness values. An alternative method of calculating H and L v could be derived from the simple empirical relationship between hardness/4' and indentation diagonal d', which is of the form i 1,12 B
H' = H + d--
(10)
where H is the intercept and B the slope of the linear H' (d- 1) plot. Using eqns. (1)-(4) we obtain
(11)
H d 2 = H ' d '2 = k L
Multiplying eqn. (10) by d '2 and then substituting it in eqn. (11), we can write Hd 2 = Hd'2+Bd
'
and dividing this equation by H Bd' d 2 ----- d ' 2 + - - ~ -
(12)
By identification with (8) L v is expressed as B
Lv - 2H
(13)
Therefore, eqn. (10) can be written in the form 2 LvH
H' = H +
d~
(14)
4. RESULTS AND APPLICATION OF THE MODEL
First, the model has been tested for the steel of the hardness standard piece. The measured data and the corrected hardness values obtained using the iterative method are given in Table II. As expected, the corrected hardness vs. L relation exhibits linear behaviour. The minimum slope, B ' = 0.002, is obtained for L v = 1.11645 ~tm. The negligible value for the residual slope implies that the corrected average hardness is independent of the indenter load. Therefore, according to the definition of hardness, the intercept value equal to 691.1 k g f m m 2 determines the absolute Vickers hardness number of the H S P steel. The values of H, B and Lv determind by plotting average values of H' against d '-~ (Fig. 5) and using eqn.(13), are as follows: intercept = 691.2 k g f m m - 2 ; slope = 1543.4 kgf m m - 2 lam - 1; correction length = 1.11646 ~tm.
371
ABSOLUTE HARDNESS OF MATERIALS
TABLE II MICROHARDNESS DATA
FOR THE HARDNESSSTANDARDPIECE
(gf)
d' (ktm)
H' (kgfmm -2 )
H (kgf m m -2 )
Ab (pro 2)
Ab/A' (%)
15 25 50 100 200
5.3 + 0.1 7.2+0.2 10.54+0.18 15.2+0.2 22.12+0.16
990 _+34 894+46 835+29 803+20 758+11
696.7 + 19 682.6+2 688.7+21 699.8+ 16 688.5+19
6.44 8.75 12.8 18.5 26.9
42.5 31.3 21.4 14.0 10.2
Load
The excellent agreement between the values obtained from the intercept of the
H(P) and H'(d'-1) plots must be pointed out. This result justifies the determination of the absolute hardness from extrapolation of the linear H(d -1) plot 1°, and therefore the form of empirical relation (10). Calculating the corrected hardness for Lv = 1.11646 ~tm, the following values were obtained by linear regression analysis: intercept, H = 691.1 kgfmm-2; residual slope, B' < 2 × 10 - 3 kgfmm -2 ~tm-1. Comparing H and Lv values resulting from the iterative method and the H'(d'-1) plot it appears clear that the two methods are equivalent. These results demonstrate that, despite the inaccuracy in diagonal length measurement, the geometrical model of Vickers microindentation proposed here is accurately verified
11001000Et
v ~ej
~/
900-
j~/~/t
/
800-
f..d Z 121 0 0
600500 • o
oi,
~2
o:3
INVERSE INDENTATION DIAGONAL (~ m- .i) Fig. 5. Vickers hardness t,.s. the inverse indentation diagonal for the Standard Hardness Piece steel. (Q) Measured values, (O) corrected values.
372
G. FARGES, D. DEGOUT
for the HSP steel. The rapid increase in the ratio between the calculated bulge area supporting the load and the measured indentation area shows clearly that this material has an important ISE (see Table II). The H (d'- 1) plot which directly allows the determination of H and L v in a very simple way is the most suitable for practical purposes. Therefore this method has been applied for the other materials reported in Table I. T A B L E lll C O R R E C T E D M I C R O H A R D N E S S D A T A FOR T H E V A R I O U S M A T E R I A L S
Substrate
Absolute hardness (kgfmm -2)
Correction length (I.tm)
Residual slope ( k g f m m -2 ktm t)
35 C D 4 steel 35 N C D 16 steel Tool steel no. 43 Tool steel no. 44 Copper Hard chromium Titanium nitride
245 400 780 852 103 1111 2251
2.353 1.053 1.578 0.989 1.844 0.492 1.376
- 6.8 7.9 9.3 - 11.4 4.4 4.2 5.0
The corrected microhardness data presented in Table III show clearly that the Lv value is very dependent on the nature and microstructure of the sample. This is particularly evident when comparing H and Lv for two tool steel samples tempered in the same run. The hard chromium exhibits the lowest Lv and the highest is for the annealed 35 CD 4 steel. The absolute microhardness for the titanium nitride determined from the measurements not taking into account the influence of the substrate agrees with the value reported in the literature for bulk TiN (Fig. 6). When the composite 5000 2500"
I
~" 4000
~=2000.
I
3000 o~ ~ s O 0 .
~3 z2000
~" 1000
to
;2 I o.1
~1 000 ~ I
t I I
500
i II 0.2
I 0.3
I ,, 0.4
INVERSE INDENTATION DIAGONAL(~ m -1)
0
I 0
I 0.1
I
II
I
0.2
I 0.3
INVERSE INDENTATION DIAGONAL (~ =- i)
(a) (b) Figl O. Vickers hardness vs. the inverse indentation diagonal for the (O) TiN coating, ( ~ ) tool steel substrate, (O) composite material: (a) measured values; (b) corrected values.
ABSOLUTE HARDNESS OF MATERIALS
373
microhardness is corrected for the TiN coating Lv value, the critical value of the indentation depth to film thickness ratio is equal to 0.1 (see Fig. 6). It is interesting to note that most of the materials exhibit a positive residual slope value but that two materials have a minus sign. This reflects the fact that generally materials are not perfectly homogeneous and microhardness measurements are incaccurate. If the residual slope is found to be connected with the microhardness gradient of the material, it would be very useful for the purposes of establishing quality control procedures. 5. CONCLUSION
In this study a new geometrical model has been proposed for relating the deformation of the material around t-he Vickers impression when it bulges upward. On the assumption that a significant amount of the indenter load is supported by the bulge area, a correction function for the measured diagonal length has been established. For homogeneous material this function allows one to compensate accurately for the indentation size effect and then to determine the absolute Vickers number. However, the most important result of this work is probably the introduction of the constant Lv, referred to as the Vickers correction length. This constant allows one to propose a very simple form for the empirical relationship H'(d' - ~) H'=H+--
2 LvH d'
From successful tests performed on various bulk materials and titanium nitride coating, it is expected that this model has general relevance. We hope this work will contribute to improving the quality control of materials and permitting better communication between researchers. ACKNOWLEDGMENTS
The authors are indebted to J. L. Bouvier and A. Michel of ETCA/CREA/PS respectively for the numerous microindentation measurements and threedimensional rugosity experiments. REFERENCES 1 L.E. Samuels, in P. J. Blau and B. R. Lawn (eds.), Microindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 5 25. 2 P.M. Sargent, in P. J. Blau and B. R. Lawn (eds.), Mieroindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 160-174. 3 D. Tabor, in P. J. Blau and B. R. Lawn (eds.), Microindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 129-159. 4 J.E. Sundgren, H. T. G. Hentzell, J. Vac. Sci. Technol. A, 4 (1986) 2259. 5 W . C . Oliver, R. Hutchings, J. B. Pethica, in P. J. Blau and B. R. Lawn (eds.), Mieroindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 90-108.
374 6 7 8 9 10 11 12
G. FARGES, D. DEGOUT
R . M . Westrich, in P. J. Blau and B. R. Lawn (eds.), Microindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 196 205. J.B. Pethica, R. Hutchings and W. C. Oliver, Philos. Mag. A, 48 (1983) 593. D. Newey, M.A. WilkingsandH. M.Pollock, J. Phys. E, 15(1982) l19. T.B. Crow and J. F. Hinsley, J. Inst. Met., 72 (1946) 461. M.M. Chaudhri and M. Winter, J. Phys. D, 21 (1988) 370. A. Thomas, Surf Eng.,3(2)(1987) 117. O. Vingsbo, S. Hogmark, B. J6nsson and A. Ingemarsson, in P. J. Blau and B. R. Lawn (eds.), Microindentation Techniques in Materidls Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 257-271.
Thin Solid Films, 181 (1989) 365-374
I N T E R P R E T A T I O N OF T H E I N D E N T A T I O N SIZE E F F E C T IN VICKERS M I C R O H A R D N E S S M E A S U R E M E N T S - A B S O L U T E H A R D N E S S OF M A T E R I A L S G. FARGES A N D D.
DEGOUT
Etahlissement Technique Central de I'Armement, 16 bis Avenue Prieur de la C6te d'Or, 94114 Arcueil Cedex ( Fran~'e )
(Received March 17, 1989)
The apparent hardness increase with decreasing load observed with most microindentation measurements is in general referred to as the indentation size effect index (ISE). Empirical correction factors or sophisticated measurements suggested to date by several authors to correct this phenomenon are not entirely satisfactory. In this study a correction function for the measured diagonal length is introduced. A simple geometrical model of the deformation of the material around the impression is connected with the diagonal correction. The application of this method allows one to determine the absolute hardness of the materials and characterizes the reaction of the materials to the Vickers indentation test by a new coefficient. The model is accurately verified for different bulk materials: steels, copper, thick hard chromium but also titanium nitride films deposited on various substrates. This work demonstrates that the absolute hardness and this coefficient are also available via a very simple method based on the empirical relation
H = ' A + B d -1.
1. INTRODUCTION Microindentation is a widespread technique used to determine the hardness of bulk materials or coatings. Despite the extensive work performed in this field over the last thirty years, some practical problems encountered in indentation hardness measurements remain unresolved 1.2. One of the main problems associated with Vickers or K n o o p microhardness measurements is the apparent change in hardness number with change in indentation size. Therefore, comparisons between hardness values reported by different laboratories can be made only at similar load. Possible explanations of this well-known phenomenon, often termed the indentation size effect (ISE), have been discussed extensively in the literature. It is well established that a large number of mechanisms ~-4 and instrumental errors L6 contribute to the variation of hardness number with indentation size. It has been demonstrated that a better accuracy in diagonal length is obtained for scanning electron or transmission electron 0040-6090/89/$3.50
© ElsevierSequoia/Printedin The Netherlands
366
G. FARGES, D. DEGOUT
microscopes than for optical reading s'°. Another method proposed to compensate for this inaccuracy is to add a constant correction factor to the diagonal measurement 6. In most cases, since the sides of the imprint are not prefectly straight there is a difficulty in determining the load-bearing area of indentation. A point at issue is the amount by ~vhich the load is supported by the bulged area of the indentation surface during the loading of the indenter ~-9. However, recent work has demonstrated that the entire surface of an indentation bears the indenter load uniformly, including that created by any piled up 1°. Taking into account these results and on the basis of our experimental investigations, we propose an interpretation of the ISE in Vickers microhardness measurements based on a geometrical model. This simple geometric approach allows one to introduce a correction function to the diagonal length measured and then to determine the intrinsic hardness of various materials: steels, copper, thick hard chromium and also PVD titanium nitride films. Moreover, the application of this model gives a new coefficient characterizing the reaction of the material to the Vickers indentation. 2.
EXPERIMENTAL DETAILS
The surface of the samples were polished mechanically with SiC then with diamond pastes down to an average roughness of 0.05 ~tm. The designations and compositions of the bulk materials and coatings measured in this study are given in Table I. TABLE I DESIGNATIONAND CHEMICALCOMPOSITIONOF MATERIALS
Designation
Composition
H.S.P. steep 35 CD 4 steel 35 N C D 16 steel Tool steel, Z 85 WCDV Copper Chromium coating, 150 ~tm thick TiN coating, 7.4 ~tm thick
Unknown 0.35 C, 1 Cr, 0.25 Mo, 0.3 Si, 0.8 Mn 0.37 C, 3.84 Ni, 1.66 Cr, 0.34 Mn, 0.33 Si 0.85 C, 4Cr, 6W, 5 Mo, 2 V 99.9 Cu Hard electrolytic chromium PVD titanium nitride
a Hardness standard piece.
Microhardness tests were conducted with a Shimadzu type M microhardness tester. Measurements of indentation size were performed using the optical system of this apparatus at a magnification of 400x or a Reichert-Jung optical microscope equipped with a Sony video imaging working at a magnification of 5000 ×. The indenter was a Vickers diamond pyramid. On each sample impressions were made with four or more loads ranging from 15 gf to 1 kgf with a dwell time of 15 s. A set of five indentations were measured for each load level by the same operator. Both diagonals were measured to take into account the eventual assymetry of the indentation.
ABSOLUTE HARDNESS OF MATERIALS
367
Impressions were also studied with a three-dimensional rugosity apparatus built at E.T.C.A. The spatial resolution of this equipment is as low as 0.1 ~tm 3.
GEOMETRICAL MODEL OF INDENTATION
The hardness number H is defined as the ratio between the applied load L and the area A of the resulting indentation L H = -A
(I)
If the hardness is independent of the indentation load, with the Vickers squarebased pyramid used in the present study, A is expressed by d2 A = -k
(2)
where d is the diagonal length and k = 2 cos 22 °. In practice, the hardness is in most of the cases dependent on the indentation load, particularly for small indentations less than about 100 ~tm. In this case we express eqn. (1) as L H' = - A'
(3)
with A' given by d'2
A' = - k
(4)
where d' is the measured diagonal length. This paper is concerned with only those cases where the apparent hardness H' increases when the applied load decreases, corresponding to the situation AA=A-A'>0
(5)
As an example, Fig. 1 shows the surface morphology of indentation made under a load of 1 kgf on the 35 CD 4 steel sample recorded using the three-dimensional rugosity tester. A better understanding of the geometry of the bulge area is given by the partial contour-line map recorded above the surface defined to a good approximation by the extremities of the diagonals (Fig. 2). The cross-sections recorded along a diagonal and along a direction perpendicular to the boundaries (Fig. 3) show that the piled-up material around this Vickers indentation is a maximum at the midpoints of the indentation sides and a minimum at the corners, as reported by Chaudri and Winter 1°. Assuming a significant amount of the indenter load is supported by the piled-up area, these results suggest the geometrical model described in Fig. 4. The area of the bulge A b is given by
4fd'21/2
Ab -- ~
(6)
368
G. FARGES, D. DEGOUT
Y(mm)
0.05
0 0
0.02
0.04
0.06
8.08
Y= 0 ~ O. lmm
X(mm) X= 0 ~ O. Imm
Fig. I. Surface morpholgy of the indentation recorded by the three-dimensional rugosity technique.
Y (mm~} & F-"
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
./
o.~.
/"
,.,4,1
!
i
•
i
/ I~'
i
0.0~.
i
/
I--. . . . . . . .
0
*
0,02
......
4 B,O4
~ i- 0.0S
t
0.O8
Xtmm)
Fig. 2. Partial contour-line m a p recorded above the surface with an isohigh step of 0.05 j~m.
369
ABSOLUTE HARDNESS OF MATERIALS
Z
Z
b-b"
7,
5u'TJL
a - CI"
?.Sum 5tam
Z.Su.
O. Im
8.13u
8.15~
B. IGu
8.El~
Bum 8.,m
6.O?=m 8.04~
O.O6m O.g6m
Fig. 3. Profilometer traces recorded along the two directions drawn in Fig. 2.
,q
C
,,I
f
+
',
Fig. 4. Geometricalmodel for the Vickersindentation: horizontal (top) and vertical(bonorn)projections of the hardness indentation. where f is the projected length of the bulge as represented in Fig. 4. If at the initial time we assume that A b = AA the total indentation area is then
d,2 A = A'+AA =~--t
4fd,2X/2 k
(7)
Comparing relations (2), (4) and (7), and since geometrically the value 2f21/2 is equal to the length L, defined in Fig. 4, the following relation between d and d' is obtained
d 2 = d ' 2 + 2 L v d'
(8)
We name this expression the correction function for the measured diagonal length and L, the Vickers correction length. Theoretically, in the case of a homogeneous material this function allows one to calculate the true Vickers
370
G. FARGES, D. DEGOUT
hardness H knowing d' measured under several loads and Lv from the equation kL
H - d,2 + 2 L v d'
(9)
derived from eqns. (1), (2) and (8). The calculation is performed with an iterative method stopped for the minimum value of the slope of the linear regression carried out on the corrected hardness values. An alternative method of calculating H and L v could be derived from the simple empirical relationship between hardness/4' and indentation diagonal d', which is of the form i 1,12 B
H' = H + d--
(10)
where H is the intercept and B the slope of the linear H' (d- 1) plot. Using eqns. (1)-(4) we obtain
(11)
H d 2 = H ' d '2 = k L
Multiplying eqn. (10) by d '2 and then substituting it in eqn. (11), we can write Hd 2 = Hd'2+Bd
'
and dividing this equation by H Bd' d 2 ----- d ' 2 + - - ~ -
(12)
By identification with (8) L v is expressed as B
Lv - 2H
(13)
Therefore, eqn. (10) can be written in the form 2 LvH
H' = H +
d~
(14)
4. RESULTS AND APPLICATION OF THE MODEL
First, the model has been tested for the steel of the hardness standard piece. The measured data and the corrected hardness values obtained using the iterative method are given in Table II. As expected, the corrected hardness vs. L relation exhibits linear behaviour. The minimum slope, B ' = 0.002, is obtained for L v = 1.11645 ~tm. The negligible value for the residual slope implies that the corrected average hardness is independent of the indenter load. Therefore, according to the definition of hardness, the intercept value equal to 691.1 k g f m m 2 determines the absolute Vickers hardness number of the H S P steel. The values of H, B and Lv determind by plotting average values of H' against d '-~ (Fig. 5) and using eqn.(13), are as follows: intercept = 691.2 k g f m m - 2 ; slope = 1543.4 kgf m m - 2 lam - 1; correction length = 1.11646 ~tm.
371
ABSOLUTE HARDNESS OF MATERIALS
TABLE II MICROHARDNESS DATA
FOR THE HARDNESSSTANDARDPIECE
(gf)
d' (ktm)
H' (kgfmm -2 )
H (kgf m m -2 )
Ab (pro 2)
Ab/A' (%)
15 25 50 100 200
5.3 + 0.1 7.2+0.2 10.54+0.18 15.2+0.2 22.12+0.16
990 _+34 894+46 835+29 803+20 758+11
696.7 + 19 682.6+2 688.7+21 699.8+ 16 688.5+19
6.44 8.75 12.8 18.5 26.9
42.5 31.3 21.4 14.0 10.2
Load
The excellent agreement between the values obtained from the intercept of the
H(P) and H'(d'-1) plots must be pointed out. This result justifies the determination of the absolute hardness from extrapolation of the linear H(d -1) plot 1°, and therefore the form of empirical relation (10). Calculating the corrected hardness for Lv = 1.11646 ~tm, the following values were obtained by linear regression analysis: intercept, H = 691.1 kgfmm-2; residual slope, B' < 2 × 10 - 3 kgfmm -2 ~tm-1. Comparing H and Lv values resulting from the iterative method and the H'(d'-1) plot it appears clear that the two methods are equivalent. These results demonstrate that, despite the inaccuracy in diagonal length measurement, the geometrical model of Vickers microindentation proposed here is accurately verified
11001000Et
v ~ej
~/
900-
j~/~/t
/
800-
f..d Z 121 0 0
600500 • o
oi,
~2
o:3
INVERSE INDENTATION DIAGONAL (~ m- .i) Fig. 5. Vickers hardness t,.s. the inverse indentation diagonal for the Standard Hardness Piece steel. (Q) Measured values, (O) corrected values.
372
G. FARGES, D. DEGOUT
for the HSP steel. The rapid increase in the ratio between the calculated bulge area supporting the load and the measured indentation area shows clearly that this material has an important ISE (see Table II). The H (d'- 1) plot which directly allows the determination of H and L v in a very simple way is the most suitable for practical purposes. Therefore this method has been applied for the other materials reported in Table I. T A B L E lll C O R R E C T E D M I C R O H A R D N E S S D A T A FOR T H E V A R I O U S M A T E R I A L S
Substrate
Absolute hardness (kgfmm -2)
Correction length (I.tm)
Residual slope ( k g f m m -2 ktm t)
35 C D 4 steel 35 N C D 16 steel Tool steel no. 43 Tool steel no. 44 Copper Hard chromium Titanium nitride
245 400 780 852 103 1111 2251
2.353 1.053 1.578 0.989 1.844 0.492 1.376
- 6.8 7.9 9.3 - 11.4 4.4 4.2 5.0
The corrected microhardness data presented in Table III show clearly that the Lv value is very dependent on the nature and microstructure of the sample. This is particularly evident when comparing H and Lv for two tool steel samples tempered in the same run. The hard chromium exhibits the lowest Lv and the highest is for the annealed 35 CD 4 steel. The absolute microhardness for the titanium nitride determined from the measurements not taking into account the influence of the substrate agrees with the value reported in the literature for bulk TiN (Fig. 6). When the composite 5000 2500"
I
~" 4000
~=2000.
I
3000 o~ ~ s O 0 .
~3 z2000
~" 1000
to
;2 I o.1
~1 000 ~ I
t I I
500
i II 0.2
I 0.3
I ,, 0.4
INVERSE INDENTATION DIAGONAL(~ m -1)
0
I 0
I 0.1
I
II
I
0.2
I 0.3
INVERSE INDENTATION DIAGONAL (~ =- i)
(a) (b) Figl O. Vickers hardness vs. the inverse indentation diagonal for the (O) TiN coating, ( ~ ) tool steel substrate, (O) composite material: (a) measured values; (b) corrected values.
ABSOLUTE HARDNESS OF MATERIALS
373
microhardness is corrected for the TiN coating Lv value, the critical value of the indentation depth to film thickness ratio is equal to 0.1 (see Fig. 6). It is interesting to note that most of the materials exhibit a positive residual slope value but that two materials have a minus sign. This reflects the fact that generally materials are not perfectly homogeneous and microhardness measurements are incaccurate. If the residual slope is found to be connected with the microhardness gradient of the material, it would be very useful for the purposes of establishing quality control procedures. 5. CONCLUSION
In this study a new geometrical model has been proposed for relating the deformation of the material around t-he Vickers impression when it bulges upward. On the assumption that a significant amount of the indenter load is supported by the bulge area, a correction function for the measured diagonal length has been established. For homogeneous material this function allows one to compensate accurately for the indentation size effect and then to determine the absolute Vickers number. However, the most important result of this work is probably the introduction of the constant Lv, referred to as the Vickers correction length. This constant allows one to propose a very simple form for the empirical relationship H'(d' - ~) H'=H+--
2 LvH d'
From successful tests performed on various bulk materials and titanium nitride coating, it is expected that this model has general relevance. We hope this work will contribute to improving the quality control of materials and permitting better communication between researchers. ACKNOWLEDGMENTS
The authors are indebted to J. L. Bouvier and A. Michel of ETCA/CREA/PS respectively for the numerous microindentation measurements and threedimensional rugosity experiments. REFERENCES 1 L.E. Samuels, in P. J. Blau and B. R. Lawn (eds.), Microindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 5 25. 2 P.M. Sargent, in P. J. Blau and B. R. Lawn (eds.), Mieroindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 160-174. 3 D. Tabor, in P. J. Blau and B. R. Lawn (eds.), Microindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 129-159. 4 J.E. Sundgren, H. T. G. Hentzell, J. Vac. Sci. Technol. A, 4 (1986) 2259. 5 W . C . Oliver, R. Hutchings, J. B. Pethica, in P. J. Blau and B. R. Lawn (eds.), Mieroindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 90-108.
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R . M . Westrich, in P. J. Blau and B. R. Lawn (eds.), Microindentation Techniques in Materials Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 196 205. J.B. Pethica, R. Hutchings and W. C. Oliver, Philos. Mag. A, 48 (1983) 593. D. Newey, M.A. WilkingsandH. M.Pollock, J. Phys. E, 15(1982) l19. T.B. Crow and J. F. Hinsley, J. Inst. Met., 72 (1946) 461. M.M. Chaudhri and M. Winter, J. Phys. D, 21 (1988) 370. A. Thomas, Surf Eng.,3(2)(1987) 117. O. Vingsbo, S. Hogmark, B. J6nsson and A. Ingemarsson, in P. J. Blau and B. R. Lawn (eds.), Microindentation Techniques in Materidls Science and Engineering, ASTM STP 889, Philadelphia, PA 1986, pp. 257-271.