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
February 23, 2016
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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
Mohanraj Murugesan (PNU)
Constitutive Modeling for Flow Stress Prediction
February 23, 2016
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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)
Constitutive Modeling for Flow Stress Prediction
February 23, 2016
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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
Mohanraj Murugesan (PNU)
Constitutive Modeling for Flow Stress Prediction
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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
REN Fa-ca et al. Journal of Iron and Steel Research, International (2013)
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)
Constitutive Modeling for Flow Stress Prediction
February 23, 2016
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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)
Constitutive Modeling for Flow Stress Prediction
February 23, 2016
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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)
Mohanraj Murugesan (PNU)
REN Fa-ca et al. Journal of Iron and Steel Research, International (2013)
@ Peak stress as reference
Constitutive Modeling for Flow Stress Prediction
@ Peak stress as reference
February 23, 2016
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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)
Mohanraj Murugesan (PNU)
Constitutive Modeling for Flow Stress Prediction
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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 .
Mohanraj Murugesan (PNU)
Constitutive Modeling for Flow Stress Prediction
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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)
Constitutive Modeling for Flow Stress Prediction
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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
Mohanraj Murugesan (PNU)
Constitutive Modeling for Flow Stress Prediction
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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
Kriging Kriging Polynomial Polynomial Kriging Kriging Polynomial Polynomial
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)
Constitutive Modeling for Flow Stress Prediction
February 23, 2016
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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
Mohanraj Murugesan (PNU)
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
Constitutive Modeling for Flow Stress Prediction
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)
Constitutive Modeling for Flow Stress Prediction
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Validation with 70Cr3Mo Steel Material 440 440 440 440 420 420
Kriging Kriging Polynomial Polynomial Kriging Kriging Polynomial Polynomial
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
Kriging Kriging Polynomial Polynomial Kriging Kriging Polynomial Polynomial
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
Kriging Kriging Polynomial Polynomial Kriging Kriging Polynomial Polynomial
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
Kriging Kriging Polynomial Polynomial Kriging Kriging Polynomial Polynomial
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)
Constitutive Modeling for Flow Stress Prediction
February 23, 2016
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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
Mohanraj Murugesan (PNU)
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
Constitutive Modeling for Flow Stress Prediction
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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
Mohanraj Murugesan (PNU)
6.5
Constitutive Modeling for Flow Stress Prediction
February 23, 2016
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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
Kriging Polynomial 36
38
σ /MPa Observed σ\MPa a
40
Figure 10: T =733 K
42
50 60
p Predicted σ\MPa
True Stress/MPa
120
Kriging Polynomial 65
70
σ /MPa Observed σ\MPa a
Figure 11: T =693 K
Constitutive Modeling for Flow Stress Prediction
75
110 100 90 80 95
Kriging Polynomial 100
105
110
115
σ /MPa Observed σ\MPa a
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
σ /MPa Observed σ\MPa a
22
23
30 34
36
38
σ /MPa Observed σ\MPa a
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
Maximum average error is 8.28%.
σ /MPa
0
p Predicted σ\MPa
True Stress/MPa
120
Kriging Polynomial 68
70
σ /MPa Observed σ\MPa a
72
Figure 16: ε˙ = 0.1 s−1
Constitutive Modeling for Flow Stress Prediction
90 105
Kriging Polynomial 110
115
σ /MPa Observed σ\MPa a
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
Maximum average error is 3.93%.
Figure 18: Comparison plot at 0.01 s−1
105 100 95 90 90
Kriging Polynomial 95
100
105
110
σa /MPaσ\MPa Observed
115
Figure 19: T =1173 K
Mohanraj Murugesan (PNU)
32
65 60
30
40
Kriging Polynomial 55
60
65
σa /MPa Observed σ\MPa
70
75
Figure 20: T =1273 K
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
Kriging Polynomial 40
45
σa /MPa Observed σ\MPa
50
Figure 21: T =1373 K
Constitutive Modeling for Flow Stress Prediction
Kriging Polynomial
22 20 22
24
26
28
30
σ /MPa Observed σ\MPa
32
34
a
Figure 22: T =1473 K
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Continuation... Kriging Polynomial Experiment
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
Maximum average error is 9.1843%.
Figure 23: Comparison plot at 0.1 s−1
130 120 110 120
Kriging Polynomial 130
140
150
160
σa /MPaσ\MPa Observed
170
Figure 24: T =1173 K
Mohanraj Murugesan (PNU)
95 90 85 80 80
Kriging Polynomial 90
100
σa /MPa Observed σ\MPa
110
Figure 25: T =1273 K
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
105 p σ\MPa Predicted
110
160 p σ\MPa Predicted
170
Kriging Polynomial 50
55
60
σ /MPa Observed σ\MPa a
65
70
Figure 26: T =1373 K
Constitutive Modeling for Flow Stress Prediction
30 30
Kriging Polynomial 35
40
45
σa /MPa Observed σ\MPa
50
Figure 27: T =1473 K
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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
σa /MPaσ\MPa Observed
200
Figure 29: T =1173 K
Mohanraj Murugesan (PNU)
Kriging Polynomial
100 100
110
120
130
σa /MPa Observed σ\MPa
140
Figure 30: T =1273 K
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
Kriging Polynomial 80
85
90
σa /MPa Observed σ\MPa
95
100
Figure 31: T =1373 K
Constitutive Modeling for Flow Stress Prediction
Kriging Polynomial
56 54 55
60
65
σa /MPa Observed σ\MPa
70
Figure 32: T =1473 K
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Continuation... Kriging Polynomial Experiment
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
Kriging Polynomial 180
200
220
240
σa /MPaσ\MPa Observed
260
Figure 34: T =1173 K
Mohanraj Murugesan (PNU)
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
σa /MPa Observed σ\MPa
200
Figure 35: T =1273 K
100 90
p σ\MPa Predicted
True Stress/MPa
300
Kriging Polynomial 100
110
120
130
σa /MPa Observed σ\MPa
140
Figure 36: T =1373 K
Constitutive Modeling for Flow Stress Prediction
90
80
70 70
Kriging Polynomial 80
90
100
σ /MPa Observed a σ\MPa
110
Figure 37: T =1473 K
February 23, 2016
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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)
Constitutive Modeling for Flow Stress Prediction
February 23, 2016
24 / 26
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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Constitutive Modeling for Flow Stress Prediction
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Thank you for your attention!!! Questions?
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Constitutive Modeling for Flow Stress Prediction
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