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Fig. 1. Hardness density distribution for a single phase material.
points was implemented. The resulting data sets were trans-
formed into continuous microhardness density distributions
using S-Plus software [1,2]. These distributions were the sumof individual microhardness density distributions for each phase
or microstructural component. Using PeakFit software [3] the
peaks were separated and used for further analysis.
Microhardness measurements of a perfectly homogeneous
single phase material would always yield one value (within
equipments measuring accuracy), represented by a vertical line
depicting a density value of one on the hardness density distri-
bution (see Fig. 1). In reality, a single phase metal alloy does
not comprise such homogeneity, there are variations in grain
size, grain orientation, chemical composition, elastic and plas-
tic deformation of various regions of the materials and so on.
All these factors influence the microhardness values at differ-ent locations resulting in a microhardness density distribution
(often Gaussian) as shown in Fig. 1. One would expect such a
distribution to vary depending on the processing history of the
material (as-cast, heat treated, hot rolled, etc.).
Two-phase materials, where each phase varies in micro-
hardness, should result in two distinct microhardness density
Fig. 2. Hardness distribution for a two-phase material.
distribution curves. These curves may or may not overlap
depending on the microhardness differences. Further, if these
twophases donotdevelopadditionalstructural forms(e.g. eutec-
tic or eutectoid) the area under the peak representing one of thephases is expected to be proportional to the phase content. For
example, if there was 70% of phase A and 30% of phase B, one
would expect that after normalisation the area under the distri-
bution curve for phase one would account for 70% of the entire
area and the area under the second curve would account for the
remaining 30% (see Fig. 2). The first peak on Fig. 2 is not part
of any of the two phases; it reflects the presence of oxides, pores
and other inclusions that may be present in the material. It may
also be a result of measuring hardness on the grain boundaries
directly.
However, most of the two or more phase metal alloys
form additional microstructural components. Essentially, theseare spatial arrangements of phases involved, which lead to
unique physical and chemical properties, including hardness.
The following microstructural components are usually present:
primary precipitates (e.g. carbides), eutectic, forms resulting
from eutectoid or other transformations (e.g. perlite) and sec-
ondary precipitates distributed in the primary phase (matrix).
Fig. 3. Example of hardness density distribution analysis.
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Fig. 4. Hardness distribution and numerical results for alloy #5 in the as-cast state.
Fitted parameters
r2 coef det 0.99979664
d.f. adj r2 0.99978198
Fit S.E. 3.6556E05
Fvalue 71393.0879
Peak Type a0 a1 a2
1 Gauss Amp 0.00059655 47.9486321 12.2691203
2 Gauss Amp 0.00036387 104.863587 23.4187213
3 Gauss Amp 0.00149296 164.141910 16.2284467
4 Gauss Amp 0.01314490 250.970324 23.4283062
5 Gauss Amp 0.00178980 315.050380 14.7067200
6 Gauss Amp 0.00077938 360.841214 21.0251870
7 Gauss Amp 0.00033430 487.999718 11.9617422
8 Gauss Amp 0.00033431 965.001055 11.9616167
Peak Type Amplitude Center FWHM Asym50 FW Base Asym10
Measured values
1 Gauss Amp 0.00059655 47.9486321 28.8915704 1 57.83249 1
2 Gauss Amp 0.00036387 104.863587 55.1468743 1 110.388 1
3 Gauss Amp 0.00149296 164.141910 38.2150715 1 76.49542 1
4 Gauss Amp 0.01314490 250.970324 55.1694451 1 110.4331 1
5 Gauss Amp 0.00178980 315.050380 34.6316791 1 69.32252 1
6 Gauss Amp 0.00077938 360.841214 49.5105318 1 99.10564 1
7 Gauss Amp 0.00033430 487.999718 28.1677504 1 56.38362 1
8 Gauss Amp 0.00033431 965.001055 28.1674548 1 56.38303 1
Peak Type Anlytc area % area Int area % Area Centroid Moment2Measured values
1 Gauss Amp 0.01834651 1.83559133 0.01832833 1.83381 47.98963 148.97
2 Gauss Amp 0.02135971 2.13706595 0.02135916 2.13705 104.8661 548.19
3 Gauss Amp 0.06073147 6.07626052 0.06073147 6.07637 164.1419 263.36
4 Gauss Amp 0.77194794 77.2343697 0.77194794 77.2358 250.9703 548.89
5 Gauss Amp 0.06597980 6.60136296 0.06597980 6.60149 315.0504 216.29
6 Gauss Amp 0.04107494 4.10959937 0.04107494 4.10968 360.8412 442.06
7 Gauss Amp 0.01002364 1.00287788 0.01002364 1.0029 487.9997 143.08
8 Gauss Amp 0.01002358 1.00287228 0.01002358 1.00289 965.0011 143.08
Total 0.99948758 100.000000 0.99946886 100
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Distribution of the secondary precipitates is not necessarily
homogeneous throughout the matrix. In fact, there are often dif-
ferent conglomerations of precipitates visible within the matrix,
as well as areas free of precipitates altogether (pure matrix).
Additionally, it is not unusual to detect shifting microhardness
properties of thesame grain in differentplanes, which is a reflec-
tion of the directional anisotropy of a given microstructure. It is
thus expected to observe a complex collection of microhardness
density distribution peaks within the matrix related to its struc-
tural and chemical composition, as well as individual peaks for
each remaining phase/structure (eutectic, carbides, etc.).
Fig. 3 provides an example of such analysis. It shows both
microhardness density distribution as determined by micro-
hardness measurements (dotted outline) and the separated
components (peaks) forming the distribution. The example
assumes Gaussian distribution and illustrates six visible peaks,
where a corresponding microhardness value with the highest
probability (in this case the mean value) is provided for each
peak. The r2 is an estimate of the correlation between calculated
and measured values of the resultant hardness density distribu-tion, while S.E. represents the estimated standard error of the fit
and Fis the F-statistic value estimating the effectiveness of the
fit.
Fig. 3 illustrates a series of peaks related to the com-
plex structure of the matrix and the individual peaks of other
microstructural components. The call-outs describing the var-
ious phases point to the microhardness density distribution
determined directly from hardness measurements and indicate
the presence of phases A, B, C and D. Phase A, comprising the
largest area, represents the matrix, while phases B, C and D may
represent other precipitates or microstructural components. For
example, phase B could be (and most likely is) the reflection ofthe eutectic presence within these alloys, while phases C and D
probably represent the carbide precipitates.
In the present case, the matrix is described by three of the
six peaks. Structure A1 is very likely to correspond to a pure
matrix, structure A2 to be the result of a presence of minor car-
bideprecipitates distributed throughout the matrix, and structure
A3 either results from texture within the matrix, the presence of
areas varying in precipitation density, or different precipitates
altogether.
It is beyond the scope of this article to consider the exact
nature of relations between peaks and matrix microstructure.
However,existence of a correlationbetween these results and the
chemical composition of alloys supporting the above assump-
tions will be demonstrated. Most importantly, this additional
information can be used in specific situations such as those
involving materials performance analysis.
Fig. 4 depicts another example of a numerical and graphi-
cal analysis output. From this, it is clear there is a rather large
quantity of information associated with all alloys tested. For
practical reasons, it is essential such information is transformed
into a small set of indicators/factors, which can be used to com-
parevariousalloysor tofindcorrelationsbetweenmicrohardness
density distribution and the selected materials properties.
3. Results
Table 1 breaks down chemical compositions of the tested
alloys. These are 0.3% C, 18% Cr and 30% Ni cast alloys with
varying amount of Si, Nb, Ti and Al.
For reference purposes, the microstructure of tested alloys
consists of an austenitic matrix and a network of primary and
eutectic precipitates of carbides in the inter-dendritic regions.
The type and morphology of carbides varies with changing
chemical composition. Without the addition of stabilising ele-
ments, the carbides within alloys are of M23C6 nature, while the
simple NbC and TiC carbides are observed in the structure ofthe alloys with Nb and Ti content, respectively. When both Nb
and Ti elements are present, additional (Nb, Ti)C-based com-
plex carbides are also formed. When these alloys are annealed,
secondary M23C6 type carbides precipitate, while MC carbides
Table 1
Chemical composition of the tested alloys (wt.%)
Alloy Si Nb Ti Al C Mn Cr Ni P S Cu
1 1.685 1.750 0.830 0.280 0.270 0.92 17.50 29.30 0.02 0.0090 0.22
2 1.820 0.030 1.000 0.260 0.300 1.05 18.30 29.60 0.02 0.0120 0.21
3 1.815 1.840 0.050 0.021 0.300 0.96 18.20 29.30 0.02 0.0100 0.224 3.995 3.000 0.700 0.110 0.250 0.96 18.20 29.50 0.01 0.0100 0.23
5 1.720 0.100 0.700 0.160 0.350 0.94 18.30 29.20 0.01 0.0100 0.21
6 1.390 1.920 0.050 0.041 0.300 0.91 18.30 29.50 0.02 0.0090 0.19
7 3.110 2.480 1.420 0.240 0.280 1.04 17.80 29.50 0.02 0.0090 0.19
8 1.690 0.030 0.030 0.035 0.260 0.97 17.90 29.20 0.01 0.0090 0.21
9 4.255 1.590 1.070 0.130 0.300 1.02 17.80 29.30 0.02 0.0070 0.21
10 3.535 2.800 0.530 0.160 0.290 1.04 18.30 29.20 0.01 0.0090 0.22
11 1.570 0.550 0.300 0.100 0.330 0.97 18.30 29.40 0.02 0.0120 0.19
12 3.055 1.590 1.220 0.230 0.260 1.04 17.80 29.30 0.02 0.0100 0.19
13 3.530 1.470 0.400 0.120 0.290 0.98 17.90 29.20 0.02 0.0090 0.18
14 1.755 2.060 0.680 0.180 0.360 0.94 18.20 29.20 0.02 0.0100 0.21
15 3.150 1.540 0.370 0.068 0.300 0.89 17.80 29.20 0.02 0.0100 0.20
Average 0.30 0.98 18.04 29.33 0.02 0.01 0.21
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in the presence of Si, undergo partial transformation into phase
G [4].
Based on the above analyses, microhardness and respec-
tive volume fraction has been extracted and tabulated. For all
alloys, 16 subcomponents (F0F15) were found to be consis-
tently determined in terms of varying microhardness value (see
Table 2).
4. Discussion
The average microhardness of the microstructure subcom-
ponents increases from 60 HV for the F0 component, to
2500HV for the F15 component (see Table 2). Both micro-
hardness and volume fraction of each component vary with
an alloys chemical composition. It can be argued that these
(from left to right) describe matrix, eutectic and primary precip-
itates, respectively. Thefirst subcomponents canbe related to the
matrix, thenext ones to theeutectic andthe remainingto primary
precipitates. Without detailed optical and X-ray microstructural
analyses to determine the nature of corresponding subcompo-nents, there is no way to establish a clear boundary in terms of
microhardness between the three structural components of the
alloys. It is proposed that in general there would be four major
subcomponents (see Table 3).
A low alloyed matrix phase (C0). This matrix subcompo-
nent would have microhardness less than 200 HV and would
incorporate low hardness measurements resulting from vol-
ume defects in the form of pores, non-metallic inclusions and
grain boundaries. This would result in a significant variation
of hardness across tested alloys.
Highly alloyed matrix (C1), possibly with precipitates dis-tributed throughout the volume, including precipitates that
are part of the eutectic (transient volumes).
Subcomponents related to the eutectic (C2).
Subcomponents likely related to primary precipitates (C3).
These would have highly varying microhardness, as pre-
cipitate nature changes significantly with varying chemical
composition.
Following the above, the results comprising Table 2 were
transformed to conform to these assumptions and presented in
Table 3 (average weighed hardness and sum of volume frac-
tions, respectively). The results point to an apparent relationship
between chemical compositions, heat treatment, volume frac-
tion of the microstructural subcomponents, and their micro-
hardness.
Multivariate linear regression analysis was implemented to
determine quantitative relations between chemical composition
and the alloys microstructural components volume fraction
and hardness (arbitrary function without an underlying phys-
ical meaning). Since volume fraction and hardness is always
positive, the regression function should also ensure a positive
response. An exponential dependence between volume fraction
and chemical composition was selected in the following form:
Vf= ef(x)
or ln(Vf)= f(x),
HVi= ef(x) or ln(HVi)= f(x)
where Vf is the volume fraction (vol.%); HVi is hardness; f(x)
is the polynomial function of independent variables (chemical
composition in wt.%).
Following these analyses, relationships between selected
mechanical properties, volume and microhardness of micro-structural components (C0C3), and chemical composition of
the alloys were determined using a similar approach. The fol-
lowing mechanical properties were considered (see Table 4):
Yield stressat room temperature (R02) andat 900C (WR02).
Ultimate tensile strength at room temperature (Rm) and at
900 C (WRm).
Elongation at room temperature (A10) andat 900C (WA10).
Fracture toughnessat room temperature (KCV) andat 900C
(WKCV).
The regression analysis was limited to heat-treated alloysas there is an interest in determining relationships between
microstructural composition and mechanical properties of the
alloys following heat treatment. It is established that mechani-
calpropertiesof materialsdepend on theirchemicalcomposition
andprocessing history, which in turn determines themicrostruc-
ture of the material. Here we can study the influence of
microstructural composition (volume fractions) and its micro-
hardness collectively with chemical composition on materials
mechanical properties. Clearly, there remains an information
gap regarding spatial and size distribution of microstructural
components, nonetheless it is a significant leap forward when
compared with the ability to consider only chemical composi-tion and average microhardness of bulk material. Quantitative
metallography can be of some help and will be explored in the
future.
The regression analysis results and associated statistical
information are presented in the appendices. Very high corre-
lations (R20.9999, with one exception of 0.9987, and in case
of the mechanical properties the R20.99999 in all cases) and
low errors (less than 5%) were attained, indicating fundamental
interrelationships between the chemistry, materials history and
microstructuralcomposition of alloys. Only in thecase of micro-
hardnessof componentC3, the selected exponential dependence
could not be obtained at satisfactory correlation and error
levels. Instead, a direct dependency between microhardnessand chemical composition was established (HVw3=f(chemical
composition)).
As expected, the four alloying elements observed (Si, Nb, Ti,
Al) proved strongly influential. Silicon had an extremely pow-
erful influence on matrix properties, while Nb and Ti combined
with C and Cr showed strong influence on eutectic and pre-
cipitates properties (volume fraction and microhardness). An in
depth analysis of the effects of alloying elements will be done
elsewhere for now it suffices to say that volume fraction of the
four microstructural components and their microhardness can
be predicted with confidence based on chemical composition
(within the studied range of the alloying elements).
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Table 2
Hardness (HVi) and volume fraction (Vi) of each microstructural component (upper table for as-cast alloys and lower table for heat treated alloys)
Alloy Low alloyed matrix C0 Matrix C1 Eutectic related C2 Precipitates C3
F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12
HVc0 Vc0 HVc1 Vc1 HVc2 Vc2 HVc3 Vc3 HVc4 Vc4 HVc5 Vc5 HVc6 Vc6 HVc7 Vc7 HVc8 Vc8 HVc9 Vc9 HVc10 Vc10 HVc11 Vc11 HVc12
Cast
1 127 1.1 265 47.4 325 37.9 438 5.8 515 1.9 715 1.0 921 3.0 1149 1.0
2 61 3.0 196 16.1 293 60.7 374 17.8 494 1 .3 731 1.0
3 97 10.2 160 5.7 242 72.3 337 3.6 380 5.4 464 0.9 557 1.0 783 1.0
4 160 3.5 275 55.7 374 31.0 695 4.5 944 4.2 1330
5 48 1.8 105 2.1 164 6.1 251 77.2 315 6.6 361 4.1 488 1.0 965 1.0
6 81 5.3 179 16.6 348 68.6 463 2.9 535 3.6 696 2.0 802 1.0
7 96 1.4 205 15.1 276 39.7 348 27.2 563 8.0 699 3.0 810 3.6 1371
8 128 3 .4 244 30.0 329 60.5 418 3 .1 543 2.0 677 1.0
9 84 8.5 323 69.5 452 3.6 549 11.4 791 4.1 1044 2.0 1371
10 129 5.4 183 4.0 253 73.9 391 7.3 469 3.3 603 1.0 677 2.0 748 1.0 1217
11 99 0.9 204 6.0 254 6.0 343 64.6 415 18.5 525 2.0 700 1.0 1330
12 55 2.7 247 71.1 347 10.3 437 6.6 560 1.4 675 5.7 810 0.3 881 1.0
13 88 2.8 167 2.8 291 17.2 317 61.9 470 6.3 591 2.0 765 2.0 883 1.0 1330
14 111 2.1 189 5.9 269 65.0 345 15.6 442 4.4 574 2.0 677 1.0 836 2.0 1126 2.0
15 105 0.8 168 7.1 260 72.1 325 5.9 402 10.2 471 0.9 579 2.0
55 102 175 263 333 380 456 558 694 787 905 1106 1325
Alloy Low alloyed matrix C0 Matrix C1 Eutectic related C2 Precipitates C3
F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12
HVw0 Vw0 HVw1 Vw1 HVw2 Vw2 HVw3 Vw3 HVw4 Vw4 HVw5 Vw5 HVw6 Vw6 HVw7 Vw7 HVw8 Vw8 HVw9 Vw9 HVw10 Vw10 HVw11 Vw11 HVw12
Heat treated
1 103 1.0 213 7.6 273 11.9 300 68.3 404 1.9 471 6.3 580 1.1 644 0.9 1102 1.0
2 93 3.4 178 22.4 272 57.2 358 10.3 434 2.7 574 2.0 765 1.0 1102 1.0
3 32 0.8 143 10.9 174 5.3 248 56.6 343 19.2 462 4.1 585 1.3 653 1.7
4 149 1.7 285 18.9 352 36.4 411 13.7 456 9.3 663 12.7 758 1.9 890 1.3 1182 1.0 1330
5 108 5.6 206 16.8 245 22.6 300 45.2 376 5.6 473 2.1 620 1.9
6 144 20.6 203 8.5 275 54.2 421 15.8 531 0.9
7 115 9.4 156 1.5 248 47.3 332 28.6 440 1.8 530 4.7 671 3.6 805 0.9 1235 2.0
8 60 13.9 124 2.5 259 72.2 296 8.7 419 1.8 579 1.0
9 86 2.6 221 7.8 378 61.0 551 20.2 784 2.2 911 2.2 1184 1.0 1331
10 190 8.6 304 50.3 437 29.4 654 2.4 789 4.3 1149 1.0 1371
11 144 10.4 308 60.4 341 6.7 417 16.7 626 3.1 763 1.9 1030 1.0
12 140 2.2 223 7.7 297 63.8 385 13.9 493 5.6 565 3.1 628 2.9 1003 1.0
13 90 10.5 190 12.3 290 64.5 544 6.2 616 1.5 761 0.9 1118 1.0
14 61 0.8 189 5.7 281 47.4 370 38.6 385 2.0 630 2.7 735 1.8
15 65 0.6 148 11.3 232 55.4 241 10.7 303 14.5 406 2.5 526 2.0 731 1.0 1371
54 122 198 265 326 394 454 556 640 766 901 1123 1351
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Table 3
Hardness and volume distribution of major microstructural components
Alloy Cast
C0 C1 C2 C3
HVc0 Vc0 HVc1 Vc1 HVc2 Vc2 HVc3 Vc3
1 127 1.1 292 85.3 457 7.6 1018 6.0
2 175 19.2 293 60.7 382 19.1 731 1.0
3 120 15.9 246 75.9 415 7.2 783 1.0
4 160 3.5 275 55.7 374 31.0 869 9.7
5 130 10.0 256 83.8 386 5.1 965 1.0
6 156 21.9 348 68.6 503 6.5 731 3.0
7 195 16.5 305 66.9 563 8.0 965 8.6
8 128 3.4 301 90.5 467 5.1 677 1.0
9 84 8.5 323 69.5 526 15.0 944 7.1
10 152 9.4 253 73.9 431 11.6 960 5.0
11 191 6.9 335 70.6 426 20.5 1015 2.0
12 55 2.7 260 81.4 459 8.0 982 7.9
13 128 5.6 311 79.1 499 8.3 1191 7.0
14 169 8.0 284 80.6 483 6.3 922 5.1
15 162 7.9 265 78.0 434 13.1 1609 1.0
AVG 142 290 454 958
Alloy Heat treated
C0 C1 C2 C3
HVw0 Vw0 HVw1 Vw1 HVw2 Vw2 HVw3 Vw3
1 200 8.6 296 80.2 470 9.3 883 1.9
2 167 25.8 285 67.5 493 4.7 934 2.0
3 148 17.0 272 75.9 493 5.4 653 1.7
4 149 1.7 329 55.3 429 23.1 859 19.8
5 181 22.4 282 67.9 403 7.8 620 1.9
6 161 29.1 275 54.2 427 16.7 NA 0.0
7 121 11.0 280 75.9 505 6.5 862 6.6
8 69 16.4 263 80.9 477 2.8 NA 0.0
9 187 10.4 378 61.0 551 20.2 1113 8.410 190 8.6 304 50.3 437 29.4 1151 11.7
11 144 10.4 308 60.4 395 23.3 738 5.9
12 204 9.9 297 63.8 436 22.5 726 3.9
13 144 22.8 290 64.5 544 6.2 1447 6.4
14 174 6.5 321 86.0 385 2.0 823 5.5
15 216 67.3 277 25.2 459 4.5 1237 3.0
AVG 164 297 460 926
Table 4
Selected mechanical properties of the heat treated alloys
Alloy R02 (MPa) Rm (MPa) A10 (%) KCV (J/cm2) WR02 (MPa) WRm (MPa) WA10 (%) WKCV (J/cm2)
1 186.3 418.2 6.7 31.9 57.9 98.7 22.9 39.82 236.4 389.0 6.4 39.0 68.7 108.9 30.4 42.5
3 231.9 429.2 6.3 31.7 74.8 119.1 23.4 37.7
4 263.9 455.5 4.4 25.0 56.2 103.5 29.2 25.5
5 242.6 428.3 6.3 37.0 72.6 111.0 24.3 42.6
6 239.1 459.0 8.8 34.3 80.5 120.3 20.7 53.4
7 251.4 373.0 1.7 18.8 66.4 127.4 17.9 23.2
8 253.5 467.8 6.7 40.9 78.9 118.7 24.5 45.8
9 241.3 356.2 3.4 23.4 50.8 96.8 24.2 27.0
10 233.8 353.8 3.9 23.6 49.9 98.6 20.9 28.6
11 245.5 442.7 6.6 41.4 82.3 134.7 16.3 47.3
12 208.8 330.5 3.9 26.2 52.5 101.2 17.9 29.3
13 224.7 456.1 9.9 46.7 58.4 94.3 31.0 48.9
14 225.1 439.0 9.0 42.5 62.6 104.8 33.4 52.6
15 226.9 447.7 9.7 44.0 62.6 103.9 23.9 43.0
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P. Christodoulou et al. / Materials Science and Engineering A 457 (2007) 350367 357
Inallcasesa significant influenceonmechanical properties of
volume fraction and microhardness of the microstructural com-
ponents was observed along with significant effects of chemical
composition. The extremely high correlation and exceptionally
low errors indicate strong relationships between mechanical
properties on the one hand, and volume fraction and micro-
hardness of microstructural components as well as chemical
composition of the alloys on the other. Therefore, the following
can be confidently stated.
Based on chemical composition we can predict volume frac-
tion of the constituent microstructural components and their
microhardness. This, in conjunction with chemical composi-
tion, allows mechanical properties of the alloys studied to be
determined fairly accurately.
It is beyond the scope of this article to analyse in great depth
the developed relationships and cross check with other analyses
suchas X-raydiffraction, optical quantitative metallographyand
microhardness measurements of selected microstructural com-
ponents.However, for illustration purposestheequationforyield
stress at room temperature (R02, see Table A9) will be brieflydiscussed
ln(R02) = 0.050Vw0+ 0.048Vw1+ 0.048Vw2
+0.0709Vw3+ 0.000488HVw2 0.000137
HVw3+ 2.63 108 HV32 + 0.02801Nb2
0.1170Nb+ 0.2959 Ti2 0.4125 Ti
+1.7127 C
From the above equation it can be said that each of the
microstructure constituents contribute proportionally to their
volume fraction (the biggest being alloyed matrix and eutectic).This contribution is enhanced by microhardness of the eutec-
tic (HVw2), while increasing microhardness of the precipitates
(whichis an indication of thechangingnatureof theprecipitates)
has a deteriorating effect. Also, there appear to exist optimum
levels of Nb and Ti that maximise yield stress. However, this
needs to be considered in conjunction with volume fraction of
microstructural components as these also depend on chemical
composition. To achieve this, a series of equations should be
formulated (function to maximise and constraints) to determine
using optimisation procedures whether an optimal composition
for the set of required mechanical properties is attainable.
It is worth noticing that similar results could be achieved byusing quantitative X-ray analysis [5]. However, this approach
is much more difficult as it requires intricate (not always easily
available) information about phase components. It seems that
combining the two analysis tools with quantitative metallogra-
phywillyielda powerfuland comprehensivemeans formaterials
analysis.
5. Conclusions
A new microstructural analysis tool based on microhardness
measurements has been developed and presented. It appears
capable of uniquely describing materials in terms of microhard-
ness density distribution, which can then be used to determine
volume fraction and average microhardness of microstructural
components.
It has been shown that high quality (in the sense of correla-
tions and accuracy) quantitative relationships between chemical
composition and microstructural properties (as determined by
the analysis) can be achieved. Further, relationships between
microstructure, chemical composition and selected mechani-
cal properties can be developed. It follows that combining
the two sets of relationships (microstructurechemical com-
position and microstructuremechanical propertieschemical
composition) an in depth analysis of relationships between
chemical composition and mechanical properties can be carried
out.
It appears that the microhardness analysis presented offers
a relatively easy way to accomplish the complex task of pre-
dicting mechanical properties of materials based on chemical
composition for a given processing.
Appendix A. Regression analysis results
A.1. Linear multivariate modellingabbreviations
To increase clarity the results of the regression analysis are
presented in table form. The following abbreviations are used:
log natural logarithm function
d.f. degrees of freedom
standard deviation on the equations response scale
r.sq square of the correlation coefficientCALL expression for the equation, where denotes the
functional dependence between the left side (the
response) and right side (independent variables, alloy-
ing elements); the sign : denotes multiplication
(interaction between two variables). I() denotes iden-
tity function that is used internally by the software for
evaluating expressions. Thus I(Si2) is the same as Si2,
orI(((C)/(Nb))2) is the sameas (C/Nb)2. (1) indicates
that intercept is equal zero (no intercept)
value value of the coefficients for the respective independent
variables in the equation
S.E. standard error for a given coefficient
tvalue Student tvalue for a given coefficientPr(>|t|) probability that for a given t value the coefficient is
equal zero (insignificant effect)
sum of sq sum of squares
mean sq mean value of sum of squares
Fvalue value of the Fishers test
Pr(F) probability that for a given Fvalue the considered vari-
able has zero (insignificant) contribution.
A.2. Volume fraction
See Tables A1A4.
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Table A1
Volume fraction for C0 microstructural component
CALL
lm log(Vw0)Nb+Ti+Cr+Fe+log(I((C)/(Cr)))+I(Ni2) +I((C)/(Cr))+I(I((C)/(Si))2)+ Ti:Cr+ Si:Fe+ Nb:log(I((C)/(Cr)))+ Nb:Ti + Nb:Cr 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
Nb 30.591323 2.467706 12.396664 0.006444
Ti 99.259423 1.815902 54.661213 0.000335
Cr 10.367372 0.165921 62.483847 0.000256
Fe 10.190755 0.081474 125.079141 6.40E05
log(I((C)/(Cr))) 155.279701 1.144282 135.700545 5.40E05
I(Ni2) 0.104586 0.002115 49.453508 0.000409
I((C)/(Cr)) 9818.588833 78.042079 125.811472 6.30E05
I(I((C)/(Si))2) 57.436742 2.578585 22.27452 0.002009
Ti:Cr 6.301932 0.103762 60.734257 0.000271
Si:Fe 0.23436 0.001979 118.417655 7.10E05
Nb:log(I((C)/(Cr))) 1.347527 0.102639 13.128799 0.005752
Nb:Ti 0.206821 0.013207 15.660157 0.004053
Nb:Cr 0.792291 0.114958 6.891989 0.020411
0.012978 NA NA NA
r.sq 0.999997 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Nb 1 59.641305 59.641305 354125.415 3.00E06
Ti 1 8.570229 8.570229 50886.47694 2.00E05
Cr 1 34.81712 34.81712 206729.6634 5.00E06
Fe 1 0.323476 0.323476 1920.669004 0.00052
log(I((C)/(Cr))) 1 0.069186 0.069186 410.799122 0.002425
I(Ni2) 1 0.484405 0.484405 2876.197907 0.000348
I((C)/(Cr)) 1 0.443368 0.443368 2632.537006 0.00038
I(I((C)/(Si))2) 1 2.637793 2.637793 15662.12225 6.40E05
Ti:Cr 1 0.205554 0.205554 1220.495013 0.000818Si:Fe 1 2.665877 2.665877 15828.87459 6.30E05
Nb:log(I((C)/(Cr))) 1 0.056282 0.056282 334.181324 0.002979
Nb:Ti 1 0.034777 0.034777 206.493492 0.004808
Nb:Cr 1 0.008 0.008 47.499517 0.020411
Residuals 2 0.000337 0.000168 NA NA
Table A2
Volume fraction for C1 microstructural component
CALL
lm log(Vw1)Si+Nb+Cr+Ni+Fe+Nb:Ni+Si:Nb+Si:Ni+Nb:Cr+Ti:Cr
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
(Intercept) 999.777498 31.000082 32.250802 6.00E06
Si 61.478523 3.414741 18.00386 5.60E05
Nb 78.275569 3.637342 21.51999 2.80E05
Cr 9.246534 0.298245 31.003136 6.00E06
Ni 13.873398 0.432768 32.057349 6.00E06
Fe 8.282341 0.277617 29.833688 8.00E06
Nb:Ni 1.942818 0.104818 18.535184 5.00E05
Si:Nb 0.40945 0.018761 21.824024 2.60E05
Si:Ni 1.843364 0.115083 16.01768 8.90E05
Nb:Cr 0.761204 0.067167 11.33297 0.000346
Ti:Cr 0.564654 0.019782 28.543259 9.00E06
0.027015 NA NA NA
r.sq 0.997626 NA NA NA
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Table A2 (Continued)
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Si 1 0.179544 0.179544 246.007857 9.70E05
Nb 1 0.003404 0.003404 4.664183 0.096925Cr 1 0.018011 0.018011 24.678414 0.007664
Ni 1 0.007624 0.007624 10.446652 0.031911
Fe 1 0.178697 0.178697 244.848286 9.70E05
Nb:Ni 1 0.061934 0.061934 84.861075 0.000772
Si:Nb 1 0.000788 0.000788 1.080229 0.357341
Si:Ni 1 0.078538 0.078538 107.61204 0.000488
Nb:Cr 1 0.103469 0.103469 141.771474 0.000285
Ti:Cr 1 0.594605 0.594605 814.717631 9.00E06
Residuals 4 0.002919 0.00073 NA NA
Table A3
Volume fraction for C2 microstructural component
CALL
lm log(Vw2)Si+Cr+I(Cr2)+Nb+Ti+log(I((C)/(Cr)))+I(Ti2)+ Ti:Cr 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
Si 0.531748 0.11229 4.735483 0.002119
Cr 11.38901 1.182861 9.62836 2.70E05
I(Cr2) 0.568066 0.058186 9.762952 2.50E05Nb 0.550525 0.120745 4.559399 0.002607
Ti 212.806018 23.809414 8.937894 4.50E05
log(I((C)/(Cr))) 5.143122 0.97753 5.261347 0.001172
I(Ti2) 3.337254 0.558581 5.974522 0.000556
Ti:Cr 11.574035 1.300106 8.902379 4.60E05
0.280226 NA NA NA
r.sq 0.993371 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Si 1 71.204338 71.204338 906.752631 0
Cr 1 3.406404 3.406404 43.378904 0.000308
I(Cr2) 1 0.541149 0.541149 6.891273 0.034154
Nb 1 0.190737 0.190737 2.428946 0.163074
Ti 1 0.058788 0.058788 0.748638 0.415573
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
log(I((C)/(Cr))) 1 0.662052 0.662052 8.430914 0.022868
I(Ti2) 1 0.087199 0.087199 1.110435 0.326996
Ti:Cr 1 6.22343 6.22343 79.252358 4.60E05
Residuals 7 0.549687 0.078527 NA NA
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Table A4
Volume fraction for C3 microstructural component
CALL
lm log(Vw3) log(Al)+I((C)/(Ti))+ Al+ log(Ti)+ log(Fe) + Ti+I((C)/(Ti)):log(Al)+I((C)/(Ti)):log(Fe)+ Al:log(Fe)+I((C)/(Ti)):log(Ti)
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
(Intercept) 256.088745 16.157542 15.849487 9.30E05
log(Al) 4.625657 0.428653 10.791141 0.000418
I((C)/(Ti)) 136.173345 6.993815 19.470538 4.10E05
Al 651.497393 79.179895 8.228066 0.001189
log(Ti) 2.344309 0.597771 3.921749 0.017222
log(Fe) 64.493814 4.305391 14.97978 0.000116
Ti 2.361372 0.631308 3.740446 0.020113
I((C)/(Ti)):log(Al) 2.056596 0.065923 31.196727 6.00E06
I((C)/(Ti)):log(Fe) 35.825226 1.986633 18.033136 5.60E05
Al:log(Fe) 164.73605 20.47168 8.047022 0.001295
I((C)/(Ti)):log(Ti) 3.073436 0.207698 14.797631 0.000121
0.06913 NA NA NA
r.sq 0.999735 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
log(Al) 1 23.241029 23.241029 4863.25095 0
I((C)/(Ti)) 1 35.283766 35.283766 7383.227566 0
Al 1 1.089047 1.089047 227.886053 0.000112
log(Ti) 1 0.650128 0.650128 136.041085 0.000309
log(Fe) 1 1.350978 1.350978 282.696015 7.30E05
Ti 1 1.320794 1.320794 276.379928 7.70E05
I((C)/(Ti)):log(Al) 1 6.82561 6.82561 1428.278098 3.00E06
I((C)/(Ti)):log(Fe) 1 1.360613 1.360613 284.712092 7.20E05
Al:log(Fe) 1 0.004774 0.004774 0.998878 0.374142
I((C)/(Ti)):log(Ti) 1 1.046437 1.046437 218.969878 0.000121
Residuals 4 0.019116 0.004779 NA NA
Table A5
Hardness of C0 microstructural component
CALL
lm log(HVw0)Si+Nb+Ti+Fe+I(I((C)/(Ti))2) +I(Si2)+log(Nb)+I(Cr2)+ Ti:Fe+ Si:Ti+ Nb:log(Nb) 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
Si 0.848168 0.082276 10.30877 0.0005
Nb 1.432943 0.163379 8.770649 0.000932
Ti 33.306471 1.538345 21.650844 2.70E05
Fe 0.119999 0.010358 11.584767 0.000317
I(I((C)/(Ti))2) 0.024937 0.002287 10.901503 0.000402
I(Si2) 0.289269 0.0231 12.522723 0.000234
log(Nb) 0.506211 0.064708 7.823002 0.001441
I(Cr2) 0.007238 0.001548 4.676608 0.009472
Ti:Fe 0.622629 0.028252 22.038114 2.50E05
Si:Ti 1.776184 0.116218 15.283152 0.000107
Nb:log(Nb) 0.212479 0.074257 2.861399 0.045865
0.025175 NA NA NA
r.sq 0.999993 NA NA NA
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Table A5 (Continued)
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Si 1 339.180836 339.180836 535177.9262 0
Nb 1 0.73282 0.73282 1156.283125 4.00E06
Ti 1 1.899333 1.899333 2996.870692 1.00E06
Fe 1 43.424614 43.424614 68517.71243 0
I(I((C)/(Ti))2) 1 0.537226 0.537226 847.663851 8.00E06
I(Si2) 1 0.007348 0.007348 11.593792 0.027153
log(Nb) 1 0.009674 0.009674 15.264645 0.0174438
I(Cr2) 1 0.00352 0.00352 5.554652 0.077932
Ti:Fe 1 0.223796 0.223796 353.118196 4.70E05
Si:Ti 1 0.143302 0.143302 226.108987 0.000114
Nb:log(Nb) 1 0.005189 0.005189 8.187604 0.045865
Residuals 4 0.002535 0.000634 NA NA
Table A6
Hardness of C1 microstructural component
CALL
lm log(HVw1) Si+Nb+Cr+Fe+I(Si2)+ log(Ti)+ log(Si) + Si:log(Ti) 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
Si 2.134921 0.961995 2.219264 0.061944
Nb 0.121283 0.022847 5.308422 0.001113
Cr 0.126808 0.033867 3.744248 0.007222
Fe 0.104124 0.014581 7.141188 0.000187
I(Si2) 0.24281 0.085695 2.833417 0.025282
log(Ti) 0.008922 0.038832 0.22975 0.824855
log(Si) 2.443218 1.309833 1.86529 0.104397
Si:log(Ti) 0.046 0.019852 2.317136 0.05362
0.0394 NA NA NA
r.sq 0.999978 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Si 1 426.820222 426.820222 274954.5826 0
Nb 1 0.559062 0.559062 360.143874 0
Cr 1 58.172673 58.172673 37474.42628 0
Fe 1 0.028576 0.028576 18.408228 0.003609
I(Si2) 1 0.01827 0.01827 11.76917 0.010976
log(Ti) 1 0.074899 0.074899 48.249338 0.000222
log(Si) 1 0.001952 0.001952 1.257706 0.299078
Si:log(Ti) 1 0.008335 0.008335 5.369119 0.05362
Residuals 7 0.010866 0.001552 NA NA
Table A7
Hardness of C2 microstructural component
CALLlm log(HVw2) Si+Nb+C+Cr+Ni+log(I((C)/(Ti))) +I(Al2)+Nb:C 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
Si 0.278574 0.033576 8.296903 7.20E05
Nb 0.248116 0.136916 1.812175 0.112855
C 5.040324 1.183285 4.259603 0.003748
Cr 0.245647 0.058191 4.221371 0.00393
Ni 0.282206 0.033096 8.526988 6.10E05
log(I((C)/(Ti))) 0.255808 0.037596 6.804094 0.000252
I(Al2) 8.933878 1.462583 6.108286 0.000487
Nb:C 1.038008 0.464585 2.234272 0.060588
0.036376 NA NA NA
r.sq 0.999984 NA NA NA
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Table A7 (Continued)
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Si 1 494.418133 494.418133 373656.6002 0
Nb 1 0.461261 0.461261 348.598228 0
C 1 64.022906 64.022906 48385.32356 0
Cr 1 3.953503 3.953503 2987.86053 0
Ni 1 0.184913 0.184913 139.747963 7.00E06
log(I((C)/(Ti))) 1 0.029087 0.029087 21.982205 0.002238
I(Al2) 1 0.047075 0.047075 35.576756 0.000562
Nb:C 1 0.006605 0.006605 4.991972 0.060588
Residuals 7 0.009262 0.001323 NA NA
Table A8
Hardness of C3 microstructural component
CALL
HVw3Si+Nb+Ti+Cr+Fe+I(Cr2) +I(I((C)/(Si))2) +I(Ti2) +I(I((C)/(Cr))2)+C+Si:Ti+Ti:Cr+Si:Nb 1
row.names Value S.E. tvalue Pr(>|t|)
CoefficientsSi 1542.184313 47.959154 32.156204 0.000966
Nb 2002.956319 43.825787 45.702689 0.000478
Ti 50630.758565 1387.765570 36.483654 0.000750
Cr 11392.691153 251.858609 45.234472 0.000488
Fe 1906.160803 39.461400 48.304440 0.000428
I(Cr2) 315.798658 7.347174 42.982330 0.000541
I(I((C)/(Si))2) 43881.687615 1123.381906 39.062128 0.000655
I(Ti2) 678.250362 31.264292 21.694090 0.002118
I(I((C)/(Cr))2) 30559747.607236 1681834.032040 18.170489 0.003015
C 41518.142630 2968.722628 13.985188 0.005074
Si:Ti 552.456689 17.917693 30.833026 0.001050
Ti:Cr 2846.510871 77.999344 36.494036 0.000750
Si:Nb 16.427385 4.941069 3.324662 0.079792
7.949860 NA NA NA
r.sq 0.999989 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Si 1 10553256.72 10553256.72 166981.1755 6.00E06
Nb 1 5851.497967 5851.497967 92.586586 0.010629
Ti 1 56245.43967 56245.43967 889.955573 0.001122
Cr 1 32270.50981 32270.50981 510.607086 0.001953
Fe 1 971.342249 971.342249 15.369272 0.059333
I(Cr2) 1 4391.239953 4391.239953 69.481339 0.014089
I(I((C)/(Si))2) 1 448.329988 448.329988 7.093798 0.116784
I(Ti2) 1 483258.2763 483258.2763 7646.458069 0.000131
I(I((C)/(Cr))2) 1 589212.3263 589212.3263 9322.938825 0.000107
C 1 43589.33184 43589.33184 689.701583 0.001447
Si:Ti 1 22522.12799 22522.12799 356.361217 0.002794Ti:Cr 1 83855.87844 83855.87844 1326.827682 0.000753
Si:Nb 1 698.576509 698.576509 11.053377 0.079792
Residuals 2 126.400556 63.200278 NA NA
Table A9
Yield stress at room temperature (R02) as function of microstructural and chemical composition
CALL
lm log(R02)Vw0+ Vw1+HVw2+ Vw2+HVw3+ Vw3+Nb+ Ti+ C+I(Ti2) +I(Nb2) +I(HVw32) 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
Vw0 0.050447 0.000269 187.204163 0
Vw1 0.048363 0.000294 164.371812 0
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Table A9 (Continued)
row.names Value S.E. tvalue Pr(>|t|)
HVw2 0.000488 4.10E05 11.798301 0.001309
Vw2 0.048137 0.000273 176.530078 0
HVw3 0.000137 1.10E05 12.193866 0.001187
Vw3 0.07096 0.000602 117.794653 1.00E06
Nb 0.117014 0.004543 25.757382 0.000128
Ti 0.412528 0.014804 27.866835 0.000101
C 1.712759 0.046811 36.588992 4.50E05
I(Ti2) 0.295947 0.010228 28.935577 9.10E05
I(Nb2) 0.028074 0.001876 14.963134 0.000648
I(HVw32) 2.63E08 7.90E09 3.3282 0.044779
0.003838 NA NA NA
r.sq 1 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Vw0 1 258.316364 258.316364 17533134.24 0
Vw1 1 178.054351 178.054351 12085377.75 0
HVw2 1 5.625469 5.625469 381826.798 0
Vw2 1 3.195897 3.195897 216920.4021 0
HVw3 1 0.056517 0.056517 3836.041955 9.00E06
Vw3 1 0.739913 0.739913 50221.3401 0
Nb 1 0.013174 0.013174 894.212549 8.20E05
Ti 1 3.00E05 3.00E05 2.061339 0.246589
C 1 0.018072 0.018072 1226.632142 5.10E05
I(Ti2) 1 0.011938 0.011938 810.320512 9.50E05
I(Nb2) 1 0.00327 0.00327 221.925824 0.000656
I(HVw32) 1 0.000163 0.000163 11.076913 0.044779
Residuals 3 4.40E05 1.50E05 NA NA
Table A10
Yield stress at 900 C (WR02) as function of microstructural and chemical composition
CALL
lm log(WR02)HVw0+Vw1+Si+I(Vw22)+Vw2+I(1/(HVw3)) + HVw3 +I(1/(C)) + HVw0:C+ Vw1:Si
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
(Intercept) 6.650373 0.040639 163.644154 0
HVw0 0.001625 0.000232 6.999824 0.002192
Vw1 0.004036 0.000286 14.134486 0.000145
Si 0.047351 0.007289 6.496529 0.002896
I(Vw22) 0.000557 1.40E05 38.951135 3.00E06
Vw2 0.013553 0.000468 28.942802 8.00E06
I(1/(HVw3)) 0.001669 3.30E05 50.35335 1.00E06
HVw3 0.000185 4.00E06 41.956049 2.00E06
I(1/(C)) 0.373903 0.013526 27.643056 1.00E05
HVw0:C 0.017394 0.000834 20.844406 3.10E05
Vw1:Si 0.000752 0.000107 7.032001 0.002155
0.001932 NA NA NA
r.sq 0.999962 NA NA NArow.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
HVw0 1 0.118711 0.118711 31811.56228 0
Vw1 1 0.001137 0.001137 304.699706 6.30E05
Si 1 0.23786 0.23786 63740.71091 0
I(Vw22) 1 0.004551 0.004551 1219.685028 4.00E06
Vw2 1 0.010365 0.010365 2777.477269 1.00E06
I(1/(HVw3)) 1 0.008871 0.008871 2377.163874 1.00E06
HVw3 1 0.003184 0.003184 853.109608 8.00E06
I(1/(C)) 1 0.010478 0.010478 2807.842281 1.00E06
HVw0:C 1 0.001618 0.001618 433.5125 3.10E05
Vw1:Si 1 0.000185 0.000185 49.449043 0.002155
Residuals 4 1.50E05 4.00E06 NA NA
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Table A11
Ultimate tensile strength at room temperature (Rm) as function of microstructural and chemical composition
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
HVw0 1 517.881769 517.881769 2687690.165 0
Vw0 1 0.01304 0.01304 67.677119 0.00119
Vw1 1 21.597211 21.597211 112084.6801 0
Vw2 1 4.191071 4.191071 21750.71575 0Vw3 1 1.077479 1.077479 5591.874973 0
Ti 1 0.063943 0.063943 331.847515 5.30E05
I(HVw02) 1 0.00249 0.00249 12.92438 0.022859
I(1/(HVw2)) 1 0.008223 0.008223 42.677887 0.002836
HVw0:Vw1 1 0.024211 0.024211 125.650814 0.000361
HVw0:Vw2 1 0.006495 0.006495 33.706644 0.004379
Vw0:Vw2 1 0.001857 0.001857 9.639857 0.036053
Residuals 4 0.000771 0.000193 NA NA
CALL
lm log(WRm)HVw0 + Si+ log(HVw0)+I(HVw02)+Vw1+I(HVw32) +I(Vw32) + 1 cbind(WRm= MP.df[, WRm], HT.ccn.df)
F statistic value= 77730.61295, numdf 8, dendf 7
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
HVw0 0.027795 0.001988 13.980777 2.00E06
Si 0.079952 0.015692 5.095078 0.001407
log(HVw0) 1.708268 0.043802 38.999374 0
I(HVw02) 4.30E05 5.00E06 7.978474 9.30E05
Vw1 0.005054 0.000591 8.552277 5.90E05
I(HVw32) 0 0 8.39169 6.70E05
I(Vw32) 0.000756 0.000115 6.586949 0.000308
r.sq 0.999999 NA NA NA
Table A12
Ultimate tensile strength at 900
C (WRm) as function of microstructural and chemical compositionCALL
lm log(WRm)HVw0 + Si+ log(HVw0)+I(HVw02)+Vw1+I(HVw32) +I(Vw32) + Si:log(Vw3) 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
HVw0 0.027795 0.001988 13.980777 2.00E06
Si 0.079952 0.015692 5.095078 0.001407
log(HVw0) 1.708268 0.043802 38.999374 0
I(HVw02) 4.30E05 5.00E06 7.978474 9.30E05
Vw1 0.005054 0.000591 8.552277 5.90E05
I(HVw32) 0 0 8.39169 6.70E05
I(Vw32) 0.000756 0.000115 6.586949 0.000308
Si:log(Vw3) 0.024783 0.003822 6.484306 0.000339
0.02304 NA NA NAr.sq 0.999989 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
HVw0 1 313.010038 313.010038 589629.9937 0
Si 1 1.542081 1.542081 2904.881165 0
log(HVw0) 1 15.463757 15.463757 29129.721 0
I(HVw02) 1 0.038717 0.038717 72.93225 6.00E05
Vw1 1 0.014045 0.014045 26.45701 0.001333
I(HVw32) 1 0.014209 0.014209 26.766707 0.00129
I(Vw32) 1 0.006426 0.006426 12.10554 0.010276
Si:log(Vw3) 1 0.022321 0.022321 42.046218 0.000339
Residuals 7 0.003716 0.000531 NA NA
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Table A13
Elongation at room temperature (A10) as function of microstructural and chemical composition
CALL
lm log(A10)Vw1+HVw2+Vw2+HVw3+Vw3+Ti+C+I(Vw02) +I(1/(HVw1)) + Vw0+I(1/(Ti))+ Vw1:Vw3 + Vw2:Ti 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
Vw1 0.171633 0.000762 225.213079 2.00E05HVw2 0.011216 8.90E05 126.28491 6.30E05
Vw2 0.134002 0.00089 150.526647 4.40E05
HVw3 0.000865 1.40E05 61.101118 0.000268
Vw3 0.065038 0.002073 31.368707 0.001015
Ti 1.048054 0.012172 86.100139 0.000135
C 15.148203 0.08961 169.046618 3.50E05
I(Vw02) 0.001102 8.00E06 142.035043 5.00E05
I(1/(HVw1)) 1802.012882 10.868802 165.796824 3.60E05
Vw0 0.261548 0.001318 198.468982 2.50E05
I(1/(Ti)) 0.00524 0.000715 7.326765 0.018124
Vw1:Vw3 0.001164 3.80E05 30.597925 0.001066
Vw2:Ti 0.002387 0.000498 4.796904 0.040817
0.003713 NA NA NA
r.sq 0.999999 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Vw1 1 42.598259 42.598259 3090156.152 0
HVw2 1 1.834693 1.834693 133091.9884 8.00E06
Vw2 1 0.077135 0.077135 5595.504557 0.000179
HVw3 1 0.005646 0.005646 409.558876 0.002433
Vw3 1 0.225936 0.225936 16389.80395 6.10E05
Ti 1 1.754432 1.754432 127269.7524 8.00E06
C 1 0.915662 0.915662 66423.77561 1.50E05
I(Vw02) 1 0.243122 0.243122 17636.5291 5.70E05
I(1/(HVw1)) 1 0.001081 0.001081 78.452807 0.01251
Vw0 1 0.996135 0.996135 72261.47837 1.40E05
I(1/(Ti)) 1 0.006926 0.006926 502.397808 0.001985
Vw1:Vw3 1 0.013246 0.013246 960.885924 0.001039Vw2:Ti 1 0.000317 0.000317 23.010287 0.040817
Residuals 2 2.80E05 1.40E05 NA NA
Table A14
Elongation at 900 C (WA10) as function of microstructural and chemical composition
CALL
lm log(WA10)Vw2+I(Vw32) +I(Ti2)+ log(Ti) +I(HVw32)+C+log(Si)+I(Nb2)+ log(HVw3)+ Vw2:C+ Ti:log(HVw3) 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
Vw2 0.128963 0.004478 28.798743 9.00E06
I(Vw32) 0.001881 7.50E05 25.018903 1.50E05
I(Ti2)
0.718015 0.042406
16.93183 7.10E
05log(Ti) 0.436711 0.010965 39.828023 2.00E06
I(HVw32) 0 0 27.613905 1.00E05
C 6.978825 0.120024 58.145196 1.00E06
log(Si) 0.434188 0.024941 17.408292 6.40E05
I(Nb2) 0.029256 0.002068 14.148109 0.000145
log(HVw3) 0.021797 0.001457 14.956858 0.000116
Vw2:C 0.460194 0.014765 31.167742 6.00E06
Ti:log(HVw3) 0.305277 0.011308 26.996544 1.10E05
0.011656 NA NA NA
r.sq 0.999996 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Vw2 1 94.166469 94.166469 693087.7129 0
I(Vw32) 1 0.239087 0.239087 1759.739776 2.00E06
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Table A14 (Continued)
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
I(Ti2) 1 10.424839 10.424839 76729.30272 0
log(Ti) 1 27.355742 27.355742 201344.7985 0
I(HVw32) 1 10.645921 10.645921 78356.5246 0
C 1 6.887299 6.887299 50692.16769 0
log(Si) 1 0.009083 0.009083 66.855692 0.001218I(Nb2) 1 0.046779 0.046779 344.306775 5.00E05
log(HVw3) 1 0.02406 0.02406 177.090995 0.000184
Vw2:C 1 0.484119 0.484119 3563.227958 0
Ti:log(HVw3) 1 0.09902 0.09902 728.813401 1.10E05
Residuals 4 0.000543 0.000136 NA NA
Table A15
Fracture toughness at room temperature (KVC) as function of microstructural and chemical composition
CALL
lm log(KCV)HVw2+ Vw2 + HVw3+ Nb+ Ti+ log(Nb) + log(C)+ Vw3 + HVw0+ HVw2:HVw3 + HVw2:Nb + Vw2:log(C)
row.names Value S.E. tvalue Pr(>|t|)
Coefficients(Intercept) 4.487551 0.008311 539.959368 3.00E06
HVw2 0.002945 1.80E05 161.102782 3.90E05
Vw2 0.030717 0.000431 71.313991 0.000197
HVw3 0.001874 6.00E06 297.400569 1.10E05
Nb 0.981808 0.002679 366.445905 7.00E06
Ti 0.248427 0.00051 487.103051 4.00E06
log(Nb) 0.114402 0.000418 273.874131 1.30E05
log(C) 0.835761 0.004967 168.274781 3.50E05
Vw3 0.003563 9.70E05 36.894088 0.000734
HVw0 0.000295 1.00E05 30.115229 0.001101
HVw2:HVw3 5.00E06 0 342.646505 9.00E06
HVw2:Nb 0.002878 6.00E06 444.846854 5.00E06
Vw2:log(C) 0.037465 0.000373 100.571071 9.90E05
0.000575 NA NA NA
r.sq 0.999999 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
HVw2 1 0.043343 0.043343 131233.5362 8.00E06
Vw2 1 0.357375 0.357375 1082062.274 1.00E06
HVw3 1 0.012597 0.012597 38142.3278 2.60E05
Nb 1 0.225693 0.225693 683354.0243 1.00E06
Ti 1 0.295872 0.295872 895841.9697 1.00E06
log(Nb) 1 0.011756 0.011756 35594.51727 2.80E05
log(C) 1 0.01056 0.01056 31973.83961 3.10E05
Vw3 1 0.000877 0.000877 2654.168743 0.000377
HVw0 1 0.002022 0.002022 6123.152477 0.000163
HVw2:HVw3 1 0.058261 0.058261 176403.11 6.00E06
HVw2:Nb 1 0.06383 0.06383 193265.8097 5.00E06
Vw2:log(C) 1 0.003341 0.003341 10114.54032 9.90E05Residuals 2 1.00E06 0 NA NA
Table A16
Fracture toughness at 900 C (WKVC) as function of microstructural and chemical composition
CALL
lm log(WKCV)Vw0+Vw1+Vw2+I(HVw02)+ log(Vw2)+I(Vw12) +I(C2)+Ti+log(Vw0)+I(Ti2)+ Vw1:log(Ti) + Vw2:log(Vw2) 1
row.names Value S.E. tvalue Pr(>|t|)
Coefficients
Vw0 0.04839 0.000644 75.10713 5.00E06
Vw1 0.087421 0.002141 40.835982 3.20E05
Vw2 0.562229 0.020774 27.063947 0.000111
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Table A16 (Continued)
row.names Value S.E. tvalue Pr(>|t|)
I(HVw02) 5.00E06 1.00E06 8.61852 0.003285
log(Vw2) 1.37143 0.045409 30.201424 8.00E05
I(Vw12) 0.000333 2.00E05 16.97169 0.000446
I(C2) 2.716264 0.212326 12.792871 0.001031
Ti 0.887369 0.072294 12.274456 0.001165
log(Vw0) 0.059745 0.009517 6.277516 0.008162
I(Ti2) 0.116978 0.036211 3.230439 0.0482
Vw1:log(Ti) 0.004552 0.000145 31.471761 7.00E05
Vw2:log(Vw2) 0.113904 0.004751 23.973393 0.000159
0.009306 NA NA NA
r.sq 0.999999 NA NA NA
row.names d.f. Sum of sq Mean sq Fvalue Pr(F)
Analysis of variance
Vw0 1 120.481613 120.481613 1391184.731 0
Vw1 1 75.706776 75.706776 874175.8 0
Vw2 1 2.211015 2.211015 25530.28787 1.00E06
I(HVw02) 1 0.012479 0.012479 144.098424 0.001244
log(Vw2) 1 0.071561 0.071561 826.299631 9.20E05
I(Vw12) 1 0.060211 0.060211 695.244335 0.00012I(C2) 1 0.093624 0.093624 1081.062152 6.20E05
Ti 1 0.165061 0.165061 1905.933441 2.60E05
log(Vw0) 1 0.072814 0.072814 840.77275 9.00E05
I(Ti2) 1 0.062089 0.062089 716.930674 0.000114
Vw1:log(Ti) 1 0.084843 0.084843 979.666227 7.20E05
Vw2:log(Vw2) 1 0.049773 0.049773 574.723558 0.000159
Residuals 3 0.00026 8.70E05 NA NA
A.3. Hardness
See Tables A5A8.
A.4. Mechanical properties
See Tables A9A16.
References
[1] M. John, Chambers, Trevor J. Hastie, Statistical Models in S, Chapman &
Hall Inc., 1993, ISBN 0-412-04261-4.
[2] John M. Chambers, Programming With Data. A Guide to the S Lan-
guage, Hamilton Printing Co., Rensselaer, NY, 1998, ISBN 0-387-
91577-X.
[3] Users Manual, PeakFitPeakSeparation andAnalysisSoftware, SPSSInc.,
1997, ISBN 1-56827-183-2.[4] B. Piekarski, M. Garbiak, Metalurgija 41 (2) (2002) 7782.
[5] P. Christodoulou, N. Calos, Mater. Sci. Eng. A 301 (2001) 103117.