Application of Tukey’s Test for Statistical Measurement of Percentage Indexes Derived from the Difference Between Predicted Crystallographic Data in Five Nitihf Alloys Obtained by Two Processes

Download Full-Text PDF Cite this Publication

Text Only Version

Application of Tukeys Test for Statistical Measurement of Percentage Indexes Derived from the Difference Between Predicted Crystallographic Data in Five Nitihf Alloys Obtained by Two Processes

Roniere Leite Soares

Federal University of Campina Grande UFCG CCT UAEP – AEG

Campina Grande Paraíba – Brazil

Walman Benicio De Castro Federal University of Campina Grande UFCG CCT UAEM – LaMMEA

Campina Grande Paraíba – Brazil

Abstract The present paper uses the Tukey test to measure the minimal significative difference (MSD) between crystallographic data predicted by non-linear mathematical models generated from graphics published by Zarinejad (2008) and Potapov (1997) on 5 nominal compositions of Ni50Ti50-XHfX

.at% substitutional alloys fused by arc melting and melt spinning processes. Results show that the calculated MSD do not represent a statistically significant difference. Therefore, it is concluded that the independent variable (Hafnium content) does not change the analysed dependent variables (lattice parameters, crystalline structure volume and monoclinic angle ) for any fusing processes.

Keywords ANOVA; Monoclinic; NiTiHf; Statistical test; Tukey.

  1. PREDICTION OF THE B19 PHASE OF NI50TI50-XHFX .AT% ALLOYS

    Based on the graphs by Zarinejad (2008) [1], which measurements were obtained by XRD [3] refinement method, it was possible to calculate, through numerical proportions, the equivalence of the lattice parameters (a, b, c), volume of the monoclinic structure B19' and axial angle , as shown in table 1.

    1. INTRODUCTION

      The production of shape memory effect (SME) alloys depends sensibly on particular processes that, based on the variation of parameters involved in the equipment used (maximum melting temperature, vacuum magnitude, purge intensity, purity of the metallic charges, torch exposition time, etc), can or cannot generate different crystallographic features for samples which theoretically have the same chemical compositions. In this sense, it is important to compare the obtained results through predictive modelling functions to observe whether there are any significant statistical difference capable of indicating a great variability in crystallographic characteristics, such as: lattice parameters (a b c [Ã…]), cell volume (VOL [Ã…3]) and characteristic angle (beta [o]) of the monoclinic structure (space group P121/m1). In this work, they are studied as variables dependent on hafnium content, which is considered an independent variable in the nominal composition. The focus of the study is on room temperature B19 martensitic phase B19, which has a monoclinic structure (Pearson symbol mP8) and is considered by the literature as an unstable phase. In addition, the five substitutional compositions Ni50Ti50-XHfX .at% considered in this instance are X = 8, 11, 14, 17 and 20 .at%, which were obtained by two different processes: Arc Melting (Zarinejad, 2008) [1] and Melt Spinning (Potapov, 1997) [2].

      Hf

      a

      b

      c

      V

      ( at.% )

      ( Ã… )

      ( Ã… )

      ( Ã… )

      ( Ã…3 )

      ( o )

      5

      2.8228476

      4.1605263

      4.6628158

      54.195804

      98.068493

      10

      2.9049668

      4.1210526

      4.7220216

      55.758741

      98.49315

      15

      2.9980132

      4.1092105

      4.7707581

      57.972027

      99.189041

      20

      3.0814569

      4.0503947

      4.8754512

      59.807692

      100.04657

      Hf

      a

      b

      c

      V

      ( at.% )

      ( Ã… )

      ( Ã… )

      ( Ã… )

      ( Ã…3 )

      ( o )

      5

      2.8228476

      4.1605263

      4.6628158

      54.195804

      98.068493

      10

      2.9049668

      4.1210526

      4.7220216

      55.758741

      98.49315

      15

      2.9980132

      4.1092105

      4.7707581

      57.972027

      99.189041

      20

      3.0814569

      4.0503947

      4.8754512

      59.807692

      100.04657

      Table 1 – Values for the four compositions of Ni50Ti50-XHfX .at% alloys (Zarinejad, 2008) obtained by conventional solidification (X = 5, 10, 15 and

      20 at.%)

      From the obtained results, the same variables were estimated for the five alloys of study (X = 8, 11, 14, 17 e 20 at.%). From the third order nonlinear regression (shown in Figure 1) it was adopted the polynomial function = 0 + 1X + 2X2 + 3X3 ± for each parameter, which resulted in: a = 2,77219 + 0,00562x + 0,00104×2 2,73732E-5×3 (R2=1); b = 4,30224 0,04354x + 0,00354×2 9,94737E-5×3 (R2=1) and c = 4,52671 + 0,03934x 0,00287×2 + 8,85679E-5×3

      (R2=1). The determination coefficient R2 are equal to 1. This indicates that the proposed models are suitable to describe the phenomenon in which the parameters a and c increase, while the parameter b decreases. Equally, the models adopted for volume and angle for b19phase presented 100% fitting.

      Figure 1 – Scatter diagrams & Cubic regression curves: with the values of lattice parameters (a, b, c) determined for the Ni50Ti50-XHfX .at% alloys proposed by Zarinejad (2008)

      All the values estimated for the five dependent variables are organized in table 2.

      Ni50Ti50-XHfX .at% Ni-rich alloys of study are plotted in figure 2:

      Figure 2 – Scatter Plot with estimated values of the lattice parameters of the crystal structure B19' in Ni50Ti50-XHfX .at% alloys (X = 8, 11, 14, 17, 20 at.%)

      According to Figure 3(a), from the regressive

      Hf (at.%)

      a ( Ã… )

      b ( Ã… )

      c ( Ã… )

      VOLreg ( Ã…3

      )

      ( o )

      8

      2.897725

      0

      4.1295494

      6

      4.7030967

      6

      54.99855

      98.2847324

      6

      11

      2.996283

      7

      4.1192405

      0

      4.7300638

      7

      56.18361

      98.6143636

      3

      14

      3.129822

      0

      4.1135641

      6

      4.7579803

      1

      57.52923

      99.0337747

      7

      17

      3.302774

      5

      4.0964057

      1

      4.8011940

      9

      58.81347

      99.5192931

      4

      20

      3.519575

      4.0516500

      4.874053

      59.81439

      100.047246

      td>

      98.2847324

      6

      Hf (at.%)

      a ( Ã… )

      b ( Ã… )

      c ( Ã… )

      VOLreg ( Ã…3

      )

      ( o )

      8

      2.897725

      0

      4.1295494

      6

      4.7030967

      6

      54.99855

      11

      2.996283

      7

      4.1192405

      0

      4.7300638

      7

      56.18361

      98.6143636

      3

      14

      3.129822

      0

      4.1135641

      6

      4.7579803

      1

      57.52923

      99.0337747

      7

      17

      3.302774

      5

      4.0964057

      1

      4.8011940

      9

      58.81347

      99.5192931

      4

      20

      3.519575

      4.0516500

      4.874053

      59.81439

      100.047246

      function

      VOLreg

      = 54,31119 0,25944x + 0,05413×2

      Table 2 – Values of the parameters predicted by the 3rd order polynomials: lattice parameters (a, b, c), monoclinic volume and , for Ni50Ti50-XHfX .at% alloys (X = 8, 11, 14, 17 and 20 .at %)

      A comparison between the values of table 1 and table 2, confirms a coherence in the direct proportion between the percentage of hafnium and the volume of the monoclinic structure (phase B19'). The lattice parameters for the five

      0,00137×3 (R2 = 1), adopted for all four Zarinejad (2008) alloys, in which X = 5, 10, 15 e 20 at.%, it is possible to predict the volume (Ã…3) of the B19 monoclinic martensitic structure for the five compositions of the alloy Ni50Ti50-XHfX at.% (X=8, 11, 14 17 and 20 at.%), shown in Figure 3(b).

      Similarly, using the cubic polynomial function y = 98,02467 0,03662x + 0,00981×2 1,46128E-4×3 (R2 = 1)

      adopted for Zarinejad alloys (2008), as seen in Figure 4(a), the angles of the B19 phase for the five Ni50Ti50-XHfX at.% (Hf = 8, 11, 14, 17 e 20 .at%) alloys of study were predicted and plotted in figure 4(b).

      Figure 3 – Nonlinear regression (order 3) from which the volumes of the martensite phase (B19 ') of the five substitutive Ni50Ti50-XHfX .at% (X = 8, 11, 14, 17 and 20 %.at) alloys were calculated

      Figure 4 – Nonlinear cubic regression from which the angles of the B19martensitic phases were estimated for the five Ni50Ti50-XHfX .at% (X = 8, 11, 14, 17 and 20 at.%) alloys rich in nickel

      For the functions used as statistical models it was calculated the analysis of variance in each polynomial, according to table 3.

      Variables

      ANOVA

      DF

      SQ

      MQ

      F

      Significance F

      a

      Regression

      3

      0.03776864

      0.012589547

      Residue

      0

      0

      Total

      3

      0.03776864

      b

      Regression

      3

      0.006228132

      0.002076044

      Residue

      0

      0

      Total

      3

      0.006228132

      c

      Regression

      3

      0.024311803

      0.008103934

      Residue

      0

      0

      Total

      3

      0.024311803

      Angle

      Regression

      3

      2.245370993

      0.748456998

      Residue

      0

      0

      Total

      3

      2.245370993

      VolReg

      Regression

      3

      18.21455606

      6.071518687

      Residue

      0

      0

      Total

      3

      18.21455606

      VolCalc

      Regression

      3

      18.46596165

      6.155320551

      Residue

      0

      0

      Total

      3

      18.46596165

      Table 3 – ANOVAS calculated for each model = 0 + 1x + 2×2 + 3×3 referring to the 5 variables of the B19martensitic phase of the 5 Ni50Ti50- XHfX at.% alloys

      To make explicit the coherence of the regressive model adopted for the volume, the calculated volume (Vol = a.b.c.sen) was added in Table 3, on top of which the model VOLcalc = 53,5806847 + 0,018979973x + 0,033586976×2 +

      8,39734E-04×3 (R2 = 1) was created.

      All of the R2 values were equal to 1. This makes unnecessary the calculation of Adjusted-R2, normally used to measure the real degree of modeling reliability. According to Table 3, since there was no standard error in the coefficients, the general error of the models is 0. For this reason, F value is not presented in table 3. Thus, it can be stated that the mathematical models adopted here are statistically significant as well as predictive for all five dependent variables observed.

      Taking the four Zarinejad alloys (2008) as a reference, the residues found for each regression, in each variable of the monoclinic structure B19, are presented in table 4 and figure 5.

      Hf (.at%)

      a ( Ã… )

      b ( Ã… )

      c ( Ã… )

      ( o )

      Volregressive

      Volcalculated

      5

      -1.78E-15

      8.88E-16

      1.78E-15

      -2.84E-14

      7.11E-15

      -2.84E-14

      10

      -8.88E-16

      1.78E-15

      2.66E-15

      1.42E-14

      1.42E-14

      0

      15

      -8.88E-16

      8.88E-16

      1.78E-15

      1.42E-14

      1.42E-14

      0

      20

      -8.88E-16

      0

      2.66E-15

      1.42E-14

      1.42E-14

      0

      Table 4 – Residues of each variable calculated for phase B19 'of the 4 substitutive Ni50Ti50-XHfX .at% (X = 5, 10, 15 and 20) alloys from Zarinejad

      (2008)

      Figure 5 – Dispersive plot of residues in each variable of the B19monoclinic structure (a, b, c, Volume and Beta [, o]) for the 4 alloys of Zarinejad (2008)

  2. PREDICTION OF THE PHASE B19ON NI50TI50-XHFX

    .AT% RIBBONS QUICKLY SOLIDIFIED BY MELT SPINNING

    Based on the Potapov plots (1997) for the six ribbons of substitutional compositions Ni49.8Ti50.2-XHfX .at% (X= 8, 9.5, 11, 15, 20 e 25 .at%) obtained by melt spinning, the calculation of lattice parameters (a, b, c), B19monoclinic crystal volume and characteristic angle are shown in table 5:

    Hf (.at%)

    a ( Ã… )

    b ( Ã… )

    c ( Ã… )

    VOL ( Ã…3 )

    ( o )

    8

    2.985357

    4.100357

    4.725

    56.961414

    99.992857

    9.5

    2.989285

    4.096428

    4.755642

    57.337329

    100.07142

    11

    3.001071

    4.094857

    4.779214

    57.769323

    100.38571

    15

    .027785

    4.0925

    4.827142

    58.499286

    102.03571

    20

    3.06000

    4.084642

    4.89000

    59.539886

    103.05714

    25

    3.099285

    4.072857

    4.937142

    60.25763

    104.78571

    Table 5 – Crystallographic characteristics of the martensitic phase obtained by XRD refinement for the six ribbons with Potapov (1997) nominal compositions Ni49.8Ti50.2-xHfx .at% obtained by fast melt spinning solidification

    Considering the similarity of the composition used by Potapov (1997) regarding the composition Ni50Ti50-XHfX

    .at%, adopted in this work (difference of 0,2 .at% in the contents of Ni e Ti), the values of table 5 were used to proportionally calculate the same variables having as reference an alloy rich in Ni of nominal composition Ni50Ti50- XHfX .at%, quickly solidified by melt-spinning. The results are shown in Table 6:

    Hf (.at%)

    a (Ã…)

    b (Ã…)

    c (Ã…)

    Vol (Ã…3)

    ( o )

    8

    2.99735

    4.08402

    4.74398

    57.19017

    100.39443

    9.5

    3.00129

    4.08011

    4.77474

    57.5676

    100.47331

    11

    3.01312

    4.07854

    4.79841

    58.00133

    100.78887

    15

    3.03994

    4.0762

    4.84653

    58.73422

    102.44549

    20

    3.07229

    4.06837

    4.90964

    59.779

    103.47102

    25

    3.11173

    4.05663

    4.95697

    60.49963

    105.20654

    Table 6 – Crystallographic characteristics of the martensitic phase calculated proportionally from Potapov's ribbons Ni49.8Ti50.2-xHfx .at% (1997) for the theoretical ribbons Ni50Ti50-XHfX .at% (X = 8, 9.5, 11, 15, 20 and 25 .at%) obtained by fast melt spinning solidification

    0 1

    0 1

    Based on the data in table 6, these five variables were calculated with the nonlinear cubic model, which is composed by the third order function = + +

    2 + 3 + , ( = 1, 2, ). The values of R2 are,

    For each of the cubic functions, the values of Adjusted-R2 computed are, in the decrescent order, in table 8: 0.99643, 0.98445, 0.99682, 0.99658 e 0.97115.

    2 3

    for the dependent variables a, b, c, volume and , respectively, 0.9986, 0.9938, 0.9987, 0.9986 and 0.9885. This certifies a large percentage of correct answers. The estimated quantities for the five compositions are presented in table 7.

    Hf (.at%)

    a ( Ã… )

    b ( Ã… )

    c ( Ã… )

    VOL ( Ã…3 )

    ( o )

    8

    2.995688

    4.083342

    4.746384

    57.208961

    100.237795

    11

    3.012892

    4.07906

    4.795981

    57.93075

    100.997707

    14

    3.031679

    4.076048

    4.838146

    58.594993

    101.835911

    17

    3.051887

    4.07311

    4.87463

    59.199742

    102.726977

    20

    3.073354

    4.06905

    4.907188

    59.743049

    103.645474

    Table 7 – Estimative of the crystallographic characteristics of the B19phase in the five ribbons quickly solidified by melt spinning for the composition Ni50Ti50-XHfX .at% (8-20 .at%)

    Using the matrix adopted for monoclinic crystals [4], according to the Equation 01, the volumes were recalculated to show the coherence of the statistical model. The volumes found were, for the five compositions (Hf = 8, 11, 14, 17 and 20.at%), in this order: 57.13535181 Ã…3, 57.85901627 Ã…3,

    58.51514303 Ã…3, 59.10615326 Ã…3 e 59.63533041 Ã…3.

    2 0 . . 1/2

    = [| 0 2 0 |]

    . . 0 2

    (1)

    The model functions were determined based on the coefficients listed in table 8, accompanied by the respective errors.

    Coeffici ents

    0

    0

    1

    1

    2

    2

    3

    3

    Sum

    Value

    Erro

    ±

    Value

    Erro

    ±

    Value

    Erro ±

    Value

    Erro ±

    a

    2.958

    78

    0.034

    34

    0.003

    71

    0.007

    16

    1.21E-

    04

    4.61E

    -04

    – 1.00E-

    06

    9.25E

    -06

    b

    4.110

    06

    0.015

    79

    – 0.005

    38

    0.003

    29

    3.14E-

    04

    2.12E

    -04

    – 7.38E-

    06

    4.25E

    -06

    c

    4.564

    45

    0.058

    9

    0.028

    21

    0.012

    29

    – 7.70E-

    04

    7.90E

    -04

    1.08E-

    05

    1.59E

    -05

    Volume

    55.01

    768

    0.964

    61

    0.297

    08

    0.201

    2

    – 0.0028

    0.012

    94

    – 1.20E-

    05

    2.60E

    -04

    98.78

    753

    4.197

    37

    0.115

    09

    0.875

    51

    0.0095

    3

    0.056

    3

    – 1.57E-

    04

    0.001

    13

    Table 8 – Coefficient values calculated for third order statistical models for Ni50Ti50-XHfX .at%

    Figure 6 Left column presents the graphs with estimated values for the six Potapov ribbons (1997), and right column shows the calculated values for the five theoretical ribbons of nominal compositions Ni50Ti50-xHfx .at%

    For a graphic visualization, figure 6 presents the values calculated for the six Potapov ribbons (1997) plotted in the left column (figure 6(a), figure 6(c) and figure 6(e)) in a way that they are compared with the values predicted for the five ribbons studied here, plotted in the right column (figure 6(b), figure 6(d) and figure 6(f)).

    According to the obtained numbers , the values of angle, volume and "a" and "c" are directly proportional to the atomic percentage of hafnium from the alloy, except in the lattice parameter "b", which is inversely proportional to the content of Hf (.at%).

  3. COMPARISON OF CRYSTALLOGRAPHIC CHARACTERISTICS OF THE MARTENSITIC PHASE IN

    ZARINEJADS ALLOYS AND POTAPOVS RIBBONS

    Due to the fact that these variables are different, i.e., one-dimensional (lattice parameters) two-dimensional ( angles) and three-dimensional (monoclinic structures volumes) entities. To enable a dimensionless comparison between the predicted data from Zarinejad (2008) and Potapov (1997), it was necessary to consider the percentage indexess of the differences between the estimated values for each parameter (a, b, c, Vol e ) of the fie compositions Ni50Ti50-XHfX .at%. This makes them arbitrary data. Therefore, it was necessary to organize the sample series from table 11 with the dimensionless format of table 9. Percentage indices have arbitrary units.

    Using descriptive statistics, the measures of central tendency and dispersion were summarized in table 10.

    The normal distribution (differences percentage indexes versus x indexes density) is represented by equation

    2 and plotted in figure 7. It is composed of: (standard deviation), (Constante de Euler-Mascheroni: 2.718), (proportion) and (population mean).

    Percentage indexes ranked in ascending order

    0.0012

    0.0068

    0.0137

    0.0236

    0.0327

    0.0043

    0.0091

    0.0151

    0.0275

    0.0347

    0.0055

    0.0091

    0.0166

    0.0302

    0.0386

    0.0057

    0.0098

    0.0182

    0.0312

    0.076

    0.0065

    0.0112

    0.0195

    0.0314

    0.1268

    Table 9 – List of 25 data organized sequentially (indexes of dimensional differences verified between both authors)

    Mean (average)

    0.0242

    Standard deviation

    0.0267

    Interval

    0.1256

    Standard error

    0.0053

    Sample variance

    0.0007

    Min and Max

    0.0012; 0.1268

    Median

    0.0166

    Kurtosis

    9.1143

    Sum

    0.605

    Mode (fashion)

    0.0091

    Asymmetry

    2.7719

    Confidence level (95 %)

    0.011025366

    Table 10 – Descriptive summary of the main statistical measurements

    1 1 )2

    () = ( , (, )

    2

    2

    (2)

    It is noticeable, in figure 7, that there is a concentration of minor differences close to bigger densities. This certifies that the values from both authors are close.

    Characteristics of phase B19´ (monoclinic)

    Zarinejad

    Potapov

    Zarinejad

    Potapov

    Zarinejad

    Potapov

    Zarinejad

    Potapov

    Zarinejad

    Potapov

    Hf (.a%)

    a

    b

    c

    VOL

    8

    Estimated values

    2.897.725

    2.995.688

    4.129.549

    4.083.342

    4.703.097

    4.746.384

    5.499.855

    5.720.896

    9.828.473

    1.002.378

    Absolute difference (Ã…)

    0.097963

    0.046207

    0.043287

    2.210.411

    1.953.065

    Proximity (%)

    96.73%

    98.88%

    99.09%

    96.14%

    98.05%

    Mean

    29.467.065

    41.064.455

    47.247.405

    561.037.555

    992.612.625

    Percentage difference

    3.27%

    1.12%

    0.91%

    3.86%

    1.95%

    11

    Estimated values

    2.996.284

    3.012.892

    4.119.241

    407.906

    4.730.064

    4.795.981

    5.618.361

    5.793.075

    9.861.436

    1.009.977

    Absolute difference (Ã…)

    0.016608

    0.040181

    0.065917

    174.714

    2.383.347

    Proximity (%)

    99.45%

    99.02%

    98.63%

    96.98%

    97.64%

    Mean

    3.004.588

    40.991.505

    47.630.225

    5.705.718

    998.060.335

    Percentage difference

    0.55%

    0.98%

    1.37%

    3.02%

    2.36%

    14

    Estimated values

    3.129.822

    3.031.679

    4.113.564

    4.076.048

    475.798

    4.838.146

    5.752.923

    5.859.499

    9.903.377

    1.018.359

    Absolute difference (Ã…)

    0.098143

    0.037516

    0.080166

    1.065.763

    2.802.141

    Proximity (%)

    96.86%

    99.09%

    98.34%

    98.18%

    97.25%

    Mean

    30.807.505

    4.094.806

    4.798.063

    580.621.115

    1.004.348.405

    Percentage difference

    3.14%

    0.91%

    1.66%

    1.82%

    2.75%

    17

    Estimated values

    3.302.775

    3.051.887

    4.096.406

    407.311

    4.801.194

    487.463

    5.881.347

    5.919.974

    9.951.929

    102.727

    Absolute difference (Ã…)

    3.302.775

    4.096.406

    4.801.194

    5.881.347

    9.951.929

    Proximity (%)

    92.40%

    99.43%

    98.49%

    99.35%

    96.88%

    Mean

    3.177.331

    4.084.758

    4.837.912

    59.006.606

    1.011.231.335

    Percentage difference

    7.60%

    0.57%

    1.51%

    0.65%

    3.12%

    20

    Estimated values

    3.519.576

    3.073.354

    405.165

    406.905

    4.874.053

    4.907.188

    5.981.439

    5.974.305

    1.000.472

    1.036.455

    Absolute difference (Ã…)

    0.446222

    0.0174

    0.033135

    0.071341

    3.598.274

    Proximity (%)

    87.32%

    99.57%

    99.32%

    99.88%

    96.53%

    Mean

    3.296.465

    406.035

    48.906.205

    597.787.195

    101.846.337

    Percentage difference

    12.68%

    0.43%

    0.68%

    0.12%

    3.47%

    Table 11 – Comparison between Ni50Ti50-XHfX .at% alloys and ribbons based on the models by Zarinejad (2008) and Potapov (1997)

    However, there is a clear comprehension that the crystallographic characteristics depend not only on the hafnium content, but they are also sensitive to the peculiarities of the two production processes as well as the typical variables of each one of them: electric arc melting (Arc Melting) and quick solidification Melt Spinning.

    Figure 7 – Plot of the probability curve (100 points) of the difference indexes estimated by the statistical models adopted for Zarinejad alloys (2008) and Potapov ribbons (1997)

    Depending on how the peculiarities involving each one of the processes are treated, one can directly influence the main properties of the samples such as: homogeneity, amount of residual stress and minimization of oxidation, among others. These physical and mechanic properties, among others are derived from atomic arrangementknown as crystallographic structures (or crystalline structures).

    It can be mentioned as variables of these processes: the number of times that the bulk was melted, exposure time to the torch, type of material of the mold , efficiency of the applied vacuum, rotating speed of the copper flywheel in quickly solidified ribbons etc. Hence, all the possibilities of variation and instrumental errors generate conditions of solidification which interfere in the micro structure of the alloys and obtained ribbons.

  4. CONCLUSIONS

The analysis of variance (ANOVA) is an unilateral test based on the F-Snedecor table, which after calculating the F-value, it measures whether it is inside or outside of the acceptance area, according to the following hypothesis test:

{ H0: Potapov = Zarinejad

H1: there is at least one difference between the means

As it is an OneWay ANOVA, the only factor (independent variable) is the atomic percentage of Hf (.at%). As a rule, in terms of null hypothesis (H0), the calculated F- value is inside the area of acceptance, i.e., minor than critical

  1. Otherwise, if H0 is rejected, the alternative hypothesis (H1) will be accepted having necessarily, the F-value calculated outside the area of acceptance, i.e., bigger than the critical F.

    In this case, we considered that F(5%); DF1 (DFW); DF2 (DFR) = 3,12 (tabulated), according to the standard table for Tukeys Test (=0.05) [5]. According to Table 12, as the calculated F = 3.13 is bigger than the critical tabulated value, we concluded that there is a difference between the averages of both treatment groups: Potapov and Zarinejad. This conclusion is also confirmed by the P-value, which is smaller than 0.05. The lower the P-value, lower is the possibility for H0 be true. It is important to note that DF1 [horizontal =

    Variation source

    Sum of sQrs

    DF

    Mean square

    Fcalc

    valor-P (same)

    critical Ftabulated

    Between groups:

    6739.09

    2

    3369.55

    3.131

    0.04968

    3.123907449

    Within groups:

    77488

    72

    1076.22

    Permutation p (n=99999)

    Total:

    84227.1

    74

    0.04909

    Table F-Snedecor – F(5%): 2.74 = 3.12

    Table 12 – OneWay ANOVA calculated for two series of samples of the crystalline dimensions of the B19 'monoclinic phase in Ni50Ti50-

    XHfX .at% alloys and ribbons obtained by conventional and rapid solidifications (Zarinejad and Potapov, respectively)

    1. Between groups (W) = Treatment = Between treatments = Between

    2. Within groups (R) = Residue = Error = Within

    3. DF = degree of freedom

Therefore, the null hypothesis is rejected (H0). However, to ascertain whether this difference is statistically significant, it is necessary to perform a parametric test to certify this significance. This decision was taken based on the normality test of Anderson-Darling [6], which predicts as normal the set of 25 data organized in table 9, as shown in figure 8:

Figure 8 – Anderson-Darling Normality Test for the percentage indexes

A comparative analysis was made between two different solidification methods (arc melting e melt spin), when applying the TUKEYs Test (through table 13) [7] to, in a complementary form to ANOVA, calculate the minimum significant difference (m. s. d.) there is between the averages in the martensitic crystallographic characteristics, according to the original dimensional data predicted by the adjustment models, shown in table 2 and table 7. It is concluded that the method to obtain the alloys doesnt interfere in the dimensions here evaluated, that is, there is no significant statistical difference. This is due to the fact that the

numerator] is the degree of freedom between groups (DFW)

and that DF2 [vertical = denominator] is the degree of

calculated difference (|

|) was of 0.75.

freedom of the residues (DFR).

According to equation 3, the m.s.d [] necessary for both groups (data series) to have a significant difference is 19.027.

()

=

(03)

In this calculation, q [q(5%); k; DF of residues = 2.899943] is a tabulated value (F-Snedecor table, =0.05), the QMR is 1076.22 (highlighted in table 12) and the number of repetitions is equal to 25. This count of 25 is contained in table 13, before the summary (resume).

Hf ( at.% )

Dimensions

Zarinejad (2008)

Potapov (1997)

8

a (Ã…)

2.897725078

2.995688

b (Ã…)

4.129549466

4.083342

c (Ã…)

4.703096765

4.746384

Volume (Ã…3)

54.99855

57.208961

( º )

98.28473246

100.237795

11

a (Ã…)

2.996283729

3.012892

b (Ã…)

4.119240505

4.07906

c (Ã…)

4.730063875

4.795981

Volume (Ã…3)

56.18361

57.93075

( º )

98.61436363

100.997707

14

a (Ã…)

3.129822061

3.031679

b (Ã…)

4.113564167

4.076048

c (Ã…)

4.757980318

4.838146

Volume (Ã…3)

57.52923

58.59493

( º )

99.03377477

101.835911

17

a (Ã…)

3.302774532

3.051887

b (Ã…)

4.096405712

4.07311

c (Ã…)

4.801194093

4.87463

Volume (Ã…3)

58.81347

59.199742

( º )

99.51929314

102.726977

20

a (Ã…)

3.5195756

3.073354

b (Ã…)

4.0516504

4.06905

c (Ã…)

4.8740532

4.907188

Volume (Ã…3)

59.81439

59.743049

( º )

100.047246

103.645474

Resume

Sum

843.0616395

861.829798

Mean (average)

33.72246558

34.47319192

Standard deviation

39.50325635

40.6129109

Variance

1560.507263

1649.408532

Difference between means

|39.05 40.61| = 0.75072634

Table 13 – Paired values of the crystallographic characteristics of the B19 martensitic phase in Ni50Ti50-XHfX .at% alloys predicted by the fitting

REFERENCES

  1. Zarinejad, M., Y. Liu, and T.J. White, The crystal chemistry of martensite in NiTiHf shape memory alloys. Intermetallics, 2008. 16(7): p. 876-883.

  2. Potapov, P., et al., Effect of Hf on the structure of Ni-Ti martensitic alloys. Materials Letters, 1997. 32(4): p. 247-250.

  3. 3. Rietveld, H., A profile refinement method for nuclear and magnetic structures. Journal of applied Crystallography, 1969. 2(2): p. 65-71.

  4. Koch, E., Twinning, in International Tables for Crystallography Volume C: Mathematical, physical and chemical tables. 2006, Springer. p. 10-14.

  5. Tukey, J.W., The future of data analysis. The annals of mathematical statistics, 1962. 33(1): p. 1-67.

  6. Pettitt, A., Testing the normality of several independent samples using the AndersonDarling statistic. Journal of the Royal Statistical Society: Series C (Applied Statistics), 1977. 26(2): p. 156-161.

  7. Rojas, I., et al., Analysis of the functional block involved in the design of radial basis function networks. Neural Processing Letters, 2000. 12(1): p. 1-17.

ATTACHMENTS

DF

k levels

(n – k)

2

3

4

5

6

7

8

9

10

5

3.64

4.6

5.22

5.67

6.03

6.33

6.58

6.8

6.99

6

3.46

4.34

4.9

5.3

5.63

5.9

6.12

6.32

6.49

7

3.34

4.16

4.68

5.06

5.36

5.61

5.82

6

6.16

8

3.26

4.04

4.53

4.89

5.17

5.4

5.6

5.77

5.92

9

3.2

3.95

4.41

4.76

5.02

5.24

5.43

5.59

5.74

10

3.15

3.88

4.33

4.65

4.91

5.12

5.3

5.46

5.6

11

3.11

3.82

4.26

4.57

4.82

5.03

5.2

5.35

5.49

12

3.08

3.77

4.2

4.51

4.75

4.95

5.12

5.27

5.39

13

3.06

3.73

4.15

4.45

4.69

4.88

5.05

5.19

5.32

14

3.03

3.7

4.11

4.41

4.64

4.83

4.99

5.13

5.25

15

3.01

3.67

4.08

4.37

4.59

4.78

4.94

5.08

5.2

16

3

3.65

4.05

4.33

4.56

4.74

4.9

5.03

5.15

17

2.98

3.63

4.02

4.3

4.52

4.7

4.86

4.99

5.11

18

2.97

3.61

4

4.28

4.49

4.67

4.82

4.96

5.07

19

2.96

3.59

3.98

4.25

4.47

4.65

4.79

4.92

5.04

20

2.95

3.58

3.96

4.23

4.45

4.62

4.77

4.9

5.01

24

2.92

3.53

3.9

4.17

4.37

4.54

4.68

4.81

4.92

30

2.89

3.49

3.85

4.1

4.3

4.46

4.6

4.72

4.82

40

2.86

3.44

3.79

4.04

4.23

4.39

4.52

4.63

4.73

60

2.83

3.4

3.74

3.98

4.16

4.31

4.44

4.55

4.65

120

2.8

3.36

3.68

3.92

4.1

4.24

4.36

4.47

4.56

2.77

3.31

3.63

3.86

4.03

4.17

4.29

4.39

4.47

Table 14 – Tukey test table ( = 0.05): degrees of freedom and levels [studentized]

199.5

Table VI

Degrees of Freedom (DF) in the numerator

DF in denominator

1

2

3

4

5

6

7

8

9

10

1

161.45

215.71

224.58

230.16

233.99

236.77

238.88

240.54

241.88

2

18.51

19

19.16

19.25

19.3

19.33

19.35

19.37

19.38

19.4

3

10.13

9.55

9.28

9.12

9.01

8.94

8.89

8.85

8.81

8.79

4

7.71

6.94

6.59

6.39

6.26

6.16

6.09

6.04

6

5.96

5

6.61

5.79

5.41

5.19

5.05

4.95

4.88

4.82

4.77

4.74

6

5.99

5.14

4.76

4.53

4.39

4.28

4.21

4.15

4.1

4.06

7

5.59

4.74

4.35

4.12

3.97

3.87

3.79

3.73

3.68

3.64

8

5.32

4.46

4.07

3.84

3.69

3.58

3.5

3.44

3.39

3.35

9

5.12

4.26

3.86

3.63

3.48

3.37

3.29

3.23

3.18

3.14

10

4.96

4.1

3.71

3.48

3.33

3.22

3.14

3.07

3.02

2.98

11

4.84

3.98

3.59

3.36

3.2

3.09

3.01

2.95

2.9

2.85

12

4.75

3.89

3.49

3.26

3.11

3

2.91

2.85

2.8

2.75

13

4.67

3.81

3.41

3.18

3.03

2.92

2.83

2.77

2.71

2.67

14

4.6

3.74

3.34

3.11

2.96

2.85

2.76

2.7

2.65

2.6

15

4.54

3.68

3.29

3.06

2.9

2.79

2.71

2.64

2.59

2.54

16

4.49

3.63

3.24

3.01

2.85

2.74

2.66

2.59

2.54

2.49

17

4.45

3.59

3.2

2.96

2.81

2.7

2.61

2.55

2.49

2.45

18

4.41

3.55

3.16

2.93

2.77

2.66

2.58

2.51

2.46

2.41

19

4.38

3.52

3.13

2.9

2.74

2.63

2.54

2.48

2.42

2.38

20

4.35

3.49

3.1

2.87

2.71

2.6

2.51

2.45

2.39

2.35

21

4.32

3.47

3.07

2.84

2.68

2.57

2.49

2.42

2.37

2.32

22

4.3

3.44

3.05

2.82

2.66

2.55

2.46

2.4

2.34

2.3

23

4.28

3.42

3.03

2.8

2.64

2.53

2.44

2.37

2.32

2.27

24

4.26

3.4

3.01

2.78

2.62

2.51

2.42

2.36

2.3

2.25

25

4.24

3.39

2.99

2.76

2.6

2.49

2.4

2.34

2.28

2.24

26

4.23

3.37

2.98

2.74

2.59

2.47

2.39

2.32

2.27

2.22

27

4.21

3.35

2.96

2.73

2.57

2.46

2.37

2.31

2.25

2.2

28

4.2

3.34

2.95

2.71

2.56

2.45

2.36

2.29

2.24

2.19

29

4.18

3.33

2.93

2.7

2.55

2.43

2.35

2.28

2.22

2.18

30

4.17

3.32

2.92

2.69

2.53

2.42

2.33

2.27

2.21

2.16

35

4.12

3.27

2.87

2.64

2.49

2.37

2.29

2.22

2.16

2.11

40

4.08

3.23

2.84

2.61

2.45

2.34

2.25

2.18

2.12

2.08

45

4.06

3.2

2.81

2.58

2.42

2.31

2.22

2.15

2.1

2.05

50

4.03

3.18

2.79

2.56

2.4

2.29

2.2

2.13

2.07

2.03

100

3.94

3.09

2.7

2.46

2.31

2.19

2.1

2.03

1.97

1.93

Table 15 – F distribution of Snedecor ( = 0.05)

Leave a Reply

Your email address will not be published. Required fields are marked *