A Comparative Study on the Arrhenius-Type

A Comparative Study on the Arrhenius-Type Constitutive Model with Regression and Kriging for Flow Stress Prediction Mohanraj Murugesan1

Beom-Soo Kang1

Kyunghoon Lee∗

Department of Aerospace Engineering Pusan National University ∗

[email protected]

KSTP Conference, October 2015

Mohanraj Murugesan (PNU)

Constitutive Modeling for Flow Stress Prediction

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Overview

1

Introduction Objective Motivation

2

Constitutive Modeling Arrhenius-Type Constitutive Equation Determination of Material Constants

3

Surrogate Modeling Polynomial Regression Regression-Kriging Numerical and Graphical Verification Verification Results of WE91 Magnesium Alloy Verification Results of 70Cr3Mo Steel Material

4

Validation of Developed Constitutive Models

5

Conclusions



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Mini Abstract Comparative study on Arrhenius-type constitutive equation with regression and Kriging. Surrogate models employed to evaluate material constants. Materials: WE91 magnesium alloy obtained from previous papers)

1

and 70Cr3Mo steel 2 . (Experimental data

Material constants (Q,lnA,α,n) relationship modeled by regression and Kriging. Then, flow stress estimated by Arrhenius-type constitutive model. Prediction quality evaluated by both numerical and graphical validation. Mismatches b/w experimental and predicted flow stress values were observed. Requires modification of Zener-Hollomon parameter. 1 MA Ming-long et al., ”Establishment and application of flow stress models of Mg-Y-MM-Zr alloy,” Trans.Nonferrous Met.Soc. China 21(2011)857-862. 2 Fa-cai et al., ”Modeling Flow Stress of 70Cr3Mo Steel Used for Back-Up Roll During Hot Deformation Considering Strain Compensation,” Journal of Iron and Steel Research, International. 2013, 20(11):118-124. Mohanraj Murugesan (PNU)


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Motivation

Flow stress model importance in metal forming process. How micro-structural changes affect the mechanical properties. Failure stress influence in force and power requirements calculation. Surrogate model contribution in modeling material constants relationship. Figure 1: Flow-behavior at Elevated Temperature



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Examples of True Stress-Strain Curves Strain rate=0.1 s-1

180

240

160

120 100

1273 K

80 60

1373 K

40

1473 K 0

0.1

0.2

0.3

0.4

0.5

True Strain

0.6

0.7

0.8

REN Fa-ca et al. Journal of Iron and Steel Research, International (2013)

Figure 2: Flow curves with different temperatures

220

True Stress (MPa)

True Stress (MPa)

1173 K 140

20

Temperature=1173 K

260

10 s-1

200

1 s-1

180 160

0.1 s-1

140 120

0.01 s-1

100 80

0

0.1

0.2

0.3

0.4

0.5

True Strain

0.6

0.7

0.8


Figure 3: Flow curves with different strain rates

Effects of temperature & strain rate on flow stress is clear. Flow stress decreased with increasing deformation temperature. Flow stress decreased with decreasing strain rate. Flow curves showed a peak stress at low strain values. Mohanraj Murugesan (PNU)


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Arrhenius-Type Constitutive Equation In hot deformation, the relationships between the flow stress, temperature and strain rate can be expressed by Hollomon parameter Z as follows: Q Z = ε˙ exp = f (σ) (1) RT where  n1  A1 σ f (σ) = A2 exp(βσ)   A [sinh(ασ)]n

for low stress(ασ < 0.8) for high stress(ασ > 1.2) for all stress σ

(2)

σ is the material flow stress. R is the universal gas constant (8.31 J mol−1 K−1 ). ε˙ is the strain rate (s−1 ). T is the deformation temperature (K). Q is the activation energy during hot deformation (K J mol−1 ). A, α and n are the material constants, and α = β/n1 . Mohanraj Murugesan (PNU)


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Determination of Material Constants

Flow stress estimation: lnε˙ − lnA + Q/(RT ) 1 σ = arcsinh exp . α n

Taking the logarithm

Outputs

where σ = f (ε, ˙ T , R, lnA, Q, α, n)

(for ασ1.2)

lnε=lnA1+n1lnσ-[Q/(RT)] lnε=lnA2+βσ-[Q/(RT)]

n1, β and α=β/n1

*strain effects doesnot included. Including strain (ε) effects by expressing the material constants as a function of strain.

Z=A1σn1 Z=A2exp(βσ)

Parameter (β) @ Peak stress as reference

Parameter (n1) @ Peak stress as reference

α=α(ε) n=n(ε) Q=Q(ε) lnA=lnA(ε)



@ Peak stress as reference


@ Peak stress as reference

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Continuation... Q ) RT Differentiating above equation gives: ε˙ = A[sinh(ασ)]n exp(−

Q = nR

where

n=

∂ln[sinh(ασ) ∂(1/T )

∂lnε˙ ∂ln[sinh(ασ)

Parameter (n) @ Peak stress as reference

Z=ε exp(Q/RT)=A[sinh(ασ)]n Z = ε exp(Q/RT)

lnZ=lnA + n ln[sinh(ασ)] Plot of ln[sinh(ασ)] vs lnZ

ε˙

Taking the logarithm to both sides of above equation gives:

Here, lnA is measured from intercept. Repeat same procedure to calculate material constants at different strains.

Choose the valuable no of samples. T Parameter (Q) @ Peak stress as reference

REN Fa-ca et al. Journal of Iron and Steel Research, International (2013) Rong-xia Chaia et al.(Materials Science and Engineering, 2012)



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Polynomial Regression

We utilized nth order polynomial regression model for approximation function: yi = β0 +

k X

βi xi +

i=1

k X

βii xi2 + ε

(3)

i=1

where ε, β are the error term and regression coefficients. The response vector can be written in matrix notation: y = Xβ + ε where X is a matrix of the control variables. The least squares method, is used to estimate unknown vector β: βˆ = (X T X )−1 X T y .



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Regression-Kriging

Regression-Kriging predictor3 is yˆ(x) = µ ˆ + ψ T (Ψ + λI)−1 (y − 1ˆ µ).

(4)

Hyperparameters, such as µ, σ 2 , θ, and p are determined using maximum likelihood estimation. µ ˆ=

1T (Ψ + λI)−1 y 1T (Ψ + λI)−1 1

(y − 1µ)T (Ψ + λI)−1 (y − 1µ) . n where λ is the amount of noise and Ψ is the correlation matrix: σ ˆ2 =

(i)

(l)

ψ = cor[Y (x ), Y (x )] = exp −

k X

! (i) θj |xj

−

(l) xj |pj

.

j=1

3

ooDACE Documentation http://sumo.intec.ugent.be/?q=ooDACE Mohanraj Murugesan (PNU)


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Numerical and Graphical Verification

Verify that whether the predictions are good or not, When the analysis of experiment is complete. Coefficient of Determination (R 2 ): Ranges from 0 to 1 and a good model have a large R 2 . P (yi − yˆi )2 2 R = 1 − Pi ¯)2 i (yi − y Root Mean Squared Relative Error (RMSRE): v u 2 n u1 X yi − yˆi RMSRE = t n i=1 yi



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Validation with WE91 Magnesium Alloy Material 235 235 235 235

Kriging Kriging Polynomial Polynomial Kriging Kriging Polynomial Polynomial

lnA lnA lnA lnA

230 230 230 230 Q Q Q Q

225 225 225 225 220 220 220 220 215 215 00 215 215 00

0.2 0.2 0.2 0.2

0.4 0.4 0.6 0.6 ε 0.4 ε0.6 0.6 0.4 εε

0.8 0.8 0.8 0.8

11 11


3.8 3.8 3.8 3.8 3.6 3.6 3.6 3.6 3.4 3.4 3.4 3.4 3.2 3.2 3.2 3.2 33 33 2.8 2.8 00 2.8 2.8 00

0.2 0.2 0.2 0.2

Kriging Kriging Polynomial Polynomial Kriging Kriging Polynomial Polynomial 0.6 0.8 11 0.6 0.8

0.4 0.4 ε 0.4 ε0.6 0.6 0.4 εε

0.8 0.8

11

n n n n

α α α α

0.018 0.018 0.018 0.018 0.017 0.017 0.017 0.017 0.016 0.016 0.016 0.016 0.015 0.015 0.015 0.015 0.014 0.014 0.014 0.014 0.013 0.013 0.013 0.013 0.012 0.012 00 0.012 0.012 00

3636 3636 3535 3535 3434 3434 3333 3333 3232 3232 3131 00 3131 00

0.2 0.2 0.2 0.2

0.4 0.4 0.6 0.6 ε 0.4 ε0.6 0.6 0.4 εε

0.8 0.8 0.8 0.8

11 11

0.2 0.2 0.2 0.2

Kriging Kriging Polynomial Polynomial Kriging Kriging Polynomial Polynomial 0.6 0.8 11 0.6 0.8

0.4 0.4 ε 0.4 ε0.6 0.6 0.4 εε

0.8 0.8

11

Figure 4: Relationships between material constants and strain (ε) Mohanraj Murugesan (PNU)


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Numerical Validation Results Metrics details shows that how well the predictions are good. Table 1: Model validation results of material constants

Model Kriging Polynomial

Metric R2 RMSRE (%) R2 RMSRE (%)

Q 0.9785 0.2800 0.9363 0.4881

lnA 0.9892 0.2943 0.9743 0.4603

α 0.9995 0.1406 0.9989 0.2145

n 0.9910 0.5669 0.9989 0.2051

Table 2: Coefficients of the polynomial.

β0 β1 β2 β3 β4 β5

Q 211.6 97.57 101.5 -1263 2128 -1086

lnA 29.79 38.12 -59.6 -75.32 240.6 -144.3


α 0.0181 -0.0393 0.1025 -0.1159 0.0623 -0.0144

n 3.607 -1.899 20.18 -55.04 64.11 -27.19

Table 3: Estimated parameters of the regression-Kriging.

θ λ

Q 1.3028 1.033


lnA 0.6099 1.0048

α 0.4525 1.0001

n 0.0585 1.0046

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Graphical Validation Results 235

36 35 34

225

lnA

Q

p Predicted Q

Predictedp lnA

230

220

215 215

33 32

Kriging Polynomial 220

225

Qa Observed Q

230

31 31

235

(a) Q 0.016

34

35

36

3.6 p Predicted n

0.015

3.4

0.0145

n

α

33

lnA a lnA Observed

(b) lnA

0.0155 p Predicted α

32

3.8

0.0165

3.2

0.014

0.0135

Kriging Polynomial

0.013 0.0125

Kriging Polynomial

0.013

0.014

0.015

α Observed α a

(c) α

0.016

3 3

Kriging Polynomial 3.2

3.4

na n Observed

3.6

(d) n

Figure 5: Predicted vs observed plot for material constants Mohanraj Murugesan (PNU)


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Validation with 70Cr3Mo Steel Material 440 440 440 440 420 420


3434 34 34

Q Q Q Q

400 400 380 380 380 380 360 360 360 360 340 340 00 340 340 00

3232 32 32 0.2 0.2 0.2 0.2

0.4 0.4 εε 0.4 0.4 εε

0.014 0.014 0.014 0.014

0.6 0.6 0.6 0.6

0.8 0.8 0.8 0.8


0.013 0.013 0.013 0.013 0.012 0.012 0.012 0.012

3030 00 30 30 00

0.2 0.2 0.2 0.2

0.4 0.4 εε 0.4 0.4 εε

6.5 6.5 6.5 6.5 66 66 5.5 5.5

0.6 0.6 0.6 0.6

0.8 0.8 0.8 0.8


n n n n

α α αα

5.5 5.5 55 55 4.5 4.5

0.011 0.011 0.011 0.011 0.01 0.01 00 0.01 0.01 00


3636 36 36 lnA lnAlnA lnA

420 420 400 400

3838 38 38

0.2 0.2 0.2 0.2

0.4 0.4 εε 0.4 0.4 εε

0.6 0.6 0.6 0.6

0.8 0.8 0.8 0.8

4.5 4.5 44 00 44 00

0.2 0.2 0.2 0.2

0.4 0.4 εε 0.4 0.4 εε

0.6 0.6 0.6 0.6

0.8 0.8 0.8 0.8

Figure 6: Relationships between material constants and strain (ε) Mohanraj Murugesan (PNU)


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Numerical Validation Results Metrics details shows that how well the predictions are good. Table 4: Model validation results for material constants

Model Kriging Polynomial

Metric R2 RMSRE (%) R2 RMSRE (%)

Q 0.9785 0.5864 0.9746 0.8669

lnA 0.9892 0.5875 0.9781 0.8431

α 0.9995 0.1575 0.9972 0.3419

n 0.9910 0.7270 0.9977 0.6066

Table 5: Coefficients of the polynomial.

β0 β1 β2 β3 β4 β5

Q 380.2 829.5 -5833 1.522e4 -1.853e4 8955

lnA 32.67 71.81 -501 1284 -1526 719.5


α 0.0149 -0.05411 0.239 -0.4936 0.4803 -0.1661

Table 6: Estimated parameters of the n 6.205 regression-Kriging. 1.829 Q lnA α n -58.98 θ 1.5653 1.4618 0.9154 0.0944 193.9 λ 1.07 1.0632 1.0004 1.0023 -263.4 134.6


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Graphical Validation Results 430

37

420

36

410

Qp Predicted Q

Predictedp lnA

35

400

34

lnA

390 380

32

370

Kriging Polynomial

360 350

33

360

380

400

Qa Q Observed

Kriging Polynomial

31 30 30

420

32

34

36

lnA a lnA Observed

(a) Q

(b) lnA 6.5 6 p Predicted n

0.012

5.5

0.0115

n

α

p Predicted α

0.0125

0.011

0.0105 0.01 0.01

Kriging Polynomial 0.0105 0.011 0.0115 0.012 0.0125 0.013

α Observed α a

(c) α

5

4.5 4 4

Kriging Polynomial 4.5

5

5.5

na Observed n

6

(d) n

Figure 7: Predicted vs observed plot for material constants


6.5


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Flow Stress Model for WE91 Magnesium Alloy 140

Kriging Polynomial Experiment

100

653 K

80

693 K

60 40

733 K

20

773 K 0.2

0.4

0.6

True Strain

0.8

1

R2 = 1 −

Figure 8: Comparison plot at 0.01 s−1

40

23

23.5

24

σ /MPa Observed σ\MPa a

24.5

25

Figure 9: T =773 K Mohanraj Murugesan (PNU)

30 34

65 60

35

20

120

70

σ /MPa

σ /MPa

22

130

75

p Predicted σ\MPa

σ /MPa

p Predicted σ\MPa

80

45

24

18 22.5

50

Kriging Polynomial

26

Maximum average error is 7.4868%.

p Predicted σ\MPa

28

573.699 = −21.8871. 25.067

σ /MPa

0

Why is negative R squared.

55


38


40

Figure 10: T =733 K

42

50 60

p Predicted σ\MPa

True Stress/MPa

120


70


Figure 11: T =693 K


75

110 100 90 80 95


105

110

115


120

Figure 12: T =653 K February 23, 2016

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Continuation... Kriging Polynomial Experiment

140

1/s

100 80

0.1/s

60 0.01/s

40

Proposed model doesnot work in all the cases.

0.001/s

20 0.2

0.4

0.6

True Strain

0.8

1

Negative R squared.

Figure 13: Comparison plot at 773 K

85

p Predicted σ\MPa

σ /MPa

22 20

40

35

19

20

21


22

23

30 34

36

38


40

42

55 66

110

100

60

Kriging Polynomial

Figure 14: ε˙ = 0.001s−1 Figure 15: ε˙ = 0.01 s−1 Mohanraj Murugesan (PNU)

70 65

18 16 18

120

75

σ /MPa

p Predicted σ\MPa

24

130

80

45

16

σ /MPa

50

Kriging Polynomial

18

p Predicted σ\MPa

20


σ /MPa

0

p Predicted σ\MPa

True Stress/MPa

120


70


72

Figure 16: ε˙ = 0.1 s−1


90 105


115


120

Figure 17: ε˙ = 1 s−1 February 23, 2016

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Flow Stress Model for 70Cr3Mo Steel Material 140

Kriging Polynomial Experiment

100

1173 K

80 60

1273 K

40

1373 K 1473 K

20 0.2

0.4

0.6

True Strain

Proposed model works at few cases.

0.8



105 100 95 90 90


100

105

110

σa /MPaσ\MPa Observed

115

Figure 19: T =1173 K


32

65 60

30

40


60

65

σa /MPa Observed σ\MPa

70

75


30 35

28 26 24

35

55 50 50

34

45

σ /MPa

σ /MPa

σ /MPa

110

50

p Predicted σ\MPa

70

p Predicted σ\MPa

75

115 p σ\MPa Predicted

120

σ /MPa

0

p Predicted σ\MPa

True Stress/MPa

120


45


50



Kriging Polynomial

22 20 22

24

26

28

30

σ /MPa Observed σ\MPa

32

34

a


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True Stress/MPa

200

150

1173 K

100

1273 K

1373 K

50

1473 K

0

0.2

0.4

True Strain

0.6

Proposed model works at 1273K.

0.8



130 120 110 120


140

150

160


170



95 90 85 80 80


100


110


45

60

σ /MPa

140

σ /MPa

100

σ /MPa

σ /MPa

150

50

55

40

35

50 45 45

p σ\MPa Predicted

65

p Predicted σ\MPa

70


110


170


55

60


65

70



30 30


40

45


50


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Continuation... 250

Kriging Polynomial Experiment 1173 K

150

1273 K

100

1373 K 1473 K

50 0

0.2

0.4

True Strain

0.6

Proposed model works well except at 1373K.

0.8

Maximum relative error is 5.0566%.

Figure 28: Comparison plot at 1 s−1

95

170 160

120 110

Kriging Polynomial

150 160

180


200



Kriging Polynomial

100 100

110

120

130


140


68 66 64

90

σ /MPa

180

130

p

σ /MPa

σ /MPa

100

140

σ /MPa

p σ\MPa Predicted

190

140 140

150 p Predicted σ\MPa

200

Predicted σ\MPa

210

85

62 60 58

80 75 75

p σ\MPa Predicted

True Stress/MPa

200


85

90


95

100



Kriging Polynomial

56 54 55

60

65


70


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250

1173 K

200 1273 K

150

1373 K

100

1473 K

50 0

0.2

0.4

True Strain

0.6

Proposed model works at most of the test conditions.

0.8

Negative R 2 at 1473 K.

Figure 33: Comparison plot at 10 s−1 150

190

140


200

220

240


260



σ /MPa

σ /MPa

160

120 110

Kriging Polynomial

140 130

100

130

170

150

180

110

p σ\MPa Predicted

200

160 160

200

180

220

σ /MPa

σ /MPa

p σ\MPa Predicted

240

p Predicted σ\MPa

260

Maximum relative error is 8.7713%.

140

160

180


200


100 90

p σ\MPa Predicted

True Stress/MPa

300


110

120

130


140



90

80

70 70


90

100

σ /MPa Observed a σ\MPa

110


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Summary and Conclusions

Comparative study was performed with regression and Kriging. Material constants were evaluated by including strain compensation. Inaccurate prediction was found for every test conditions of WE91 magnesium alloy. 70cr3Mo steel material was found to have quite accurate prediction in most cases. Prediction quality was checked with the help of numerical and graphical validation.

Conclusions: Prediction accuracy should be verified for each test conditions separately instead of lumped method. Flow stress prediction was not only satisfactory with material constants evaluation. Constitutive model requires modification of Zener-Hollomon parameter for better agreement with measurements. Mohanraj Murugesan (PNU)


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Future Work In future, we would like to prolong the present approach to evaluate the material constants with modification on Zener-Hollomon parameter.

Z'=ε(6/5) exp(Q/RT)

ε(1/5)

lnZ'=lnZ + (1/5)*lnε Examples:

ε(1/2) , ε(1/3) , ε(2/3)

For ε˙(1/5) , the constitutive model becomes: (6/5)lnε˙ − lnA + Q/(RT ) 1 σ = arcsinh exp . α n



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Thank you for your attention!!! Questions?



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