AIP July 20-24,2009 www.wias-berlin.de togobyts@wias-berlin.de - - PowerPoint PPT Presentation

togobyts@wias-berlin.de www.wias-berlin.de (joint work with D. Hömberg, M. Yamomoto) Nataliya Togobytska

Parameter identification for the phase transformations in steel

AIP

July 20-24,2009 Vienna

Weierstrass Institute for Applied Analysis and Stochastics, Berlin

Outline

• Phase transformations in steel
• Dilatometer experiment
• Problem formulation
• Direct problem
• Inverse problem
• Stability result for the inverse problem
• Numerical implementation
• Future research

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Nataliya Togobytska Parameter identification from dilatometer investigations

Phase transformations in steel Phase transformations in steel

Austenite Ferrite, Pearlite, Bainite

Phase – chemical homogeneous region

• f ferrous material

microsection

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Nataliya Togobytska Parameter identification from dilatometer investigations

Heat treating of steel involves phase transformations

Martensite Time Temperature

Body-centered cubic iron atom face-centered cubic iron atom

Different physical properties: Pearlite,ferrite: soft and ductile Bainite,martensite: hard and brittle

Dilatometer experiment Dilatometer experiment

Measurements:

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Nataliya Togobytska Parameter identification from dilatometer investigations

Dilatometer is a device used for measuring thermal expansion and contraction of a solid during heating and subsequent cooling

Dilatometer measurements Dilatometer measurements

• Sometimes it is difficult and erroneous to fix transformation points
• the actual phase fractions of the different product phases cannot be drawn

directly from the dilatometer curve microscopic investigations have to be made Δl T

Ts Austenite

(γ)

Ferrite

(α)

Tf

Dilatometer curve

Tf Ts

Nataliya Togobytska Parameter identification from dilatometer investigations

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• start and end temperatures of the
• ccurring phase transitions

, = displacement over the temperature

Tf Ts ,

Problem formulation Problem formulation

We will consider 1D model, We assume that at most 2 phase fractions may occur during cooling

we define

• phase fractions

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Nataliya Togobytska Parameter identification from dilatometer investigations

Sample in dilatometer experiment

Length change

Idea: 1. step: to describe mathematically the inter-relation between displacement, temperature and phase transformation in the sample in dilatometer experiment. 2.step: to identify the evolution of product phases from the measurements Inverse Problem

Thermomechanical effects of phase transitions

Balance laws: Constitutive laws: Notations: Rate laws for phase fractions:

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Nataliya Togobytska Parameter identification from dilatometer investigations

Additive partitioning of strain:

• elastic strain
• mixture ansatz for the thermal strain

σ ε = K

el

(Hooke)

κ θ = − grad q

(Fourier)

el

ε ( )

el th

u ε ε ε = +

1 2

(1 )

th th th th

y z y z ε ε ε ε = + + − −

u - displacement

( )

th i i i ref

ε δ θ θ = −

i

δ >

• thermal expansion coefficient
• reference temperature of i-phase

i ref

θ

momentum balance energy balance

1D case 1D case

Thermoelasticity system

rate laws for phase fractions:

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Nataliya Togobytska Parameter identification from dilatometer investigations

u -displacement, θ - temperature, z, y – phase fractions

1 2

( ) , ( )

e t xx xt

c k u y z y L L z ρ θ θ δ ρ ρ γ θ θ − + Λ = + ′ + − ′ ( , ) ( ) 0 in (0, ) ,

xx x

y z y z u T δ θ η − + = Ω× , , ( ) (1 , ) ( ) 0 in (0, )

x

y z y z u t T δ θ η − + = ) , ( in ) , ( T t u = ) , ( in ) , 1 ( ) , ( T t t

x x

= =θ θ in ) , ( Ω = ⋅ θ θ

2

z f ( , , ) t z θ = &

1

y f ( , , ) t y θ = &

Direct problem

Assumptions:

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Nataliya Togobytska Parameter identification from dilatometer investigations

Inverse problem l

Theorem (Global stability estimate)

Phase fractions

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Nataliya Togobytska Parameter identification from dilatometer investigations

Reference: D. Hoemberg, N. Togobytska, M. Yamamoto, On the evaluation of dilatometer experiments, to appear in Applicable Analysis

The problem of identification of phase fractions from dilatometer curves can be represented as an optimal control problem:

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Nataliya Togobytska Parameter identification from dilatometer investigations

Inverse problem ll

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − + −

∫ ∫

T T

dt t t x dt t t u

2 2 2 1

)) ( ˆ ) , ( ( )) ( ˆ ) , 1 ( ( min τ θ ω λ ω

subject to the state system and the control constraints

ad

U z y ∈ ,

) ( ) , (

2 1

θ θ γ ρ ρ δ θ θ ρ − + ′ + ′ = Λ + −

e xt xx t

z L y L u z y k c

) , ( in ) , ( ) , ( T z y z y u

x xx

× Ω = + − η θ δ

in ) , ( Ω = ⋅ θ θ

) , ( in ) , 1 ( ) , ( T t t

x x

= =θ θ ) , ( in ) , ( T t u = ) , ( in ) , ( ) , 1 ( ) , ( T z y t z y ux = + − η θ δ

Numerical implementation

„First discretize then optimize“ Strategy:

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Nataliya Togobytska Parameter identification from dilatometer investigations

Numerical implementation

=

1 2 1 2

( ( ), ( ),..., ( ), ( ), ( ),..., ( ))

n n

p y t y t y t z t z t z t defining the parameter vector

∈

2n

p R

we consider finite-dimensional optimization problem

subject to a discretized version of the state system and the constraints

∈ %

ad

p U

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Nataliya Togobytska Parameter identification from dilatometer investigations

Parameter identification for the steel C1080 with test data

Numerical results:

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Nataliya Togobytska Parameter identification from dilatometer investigations

model data:

ˆ( ) (1 , ) t u t λ = ˆ( ) ( , ) t x t τ θ =

Numerical results of the identification process

Final result of parameter identification process with the model data perturbed with white noise (measurement errors for temperature and displacement are about 5%)

• ptimal solution

dilatometer curve

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Nataliya Togobytska Parameter identification from dilatometer investigations

Input data for parameter identification with experimental data

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physical parameters Thermomechanical system includes some parameters, that have to be known:

• density , heat capacity , thermal conductivity etc.

(measurements)

• thermal expansion coefficients and reference temperatures

for each i-phase

ρ

c

κ

i

δ

i ref

θ

(These data should be obtained from the dilatometer curves)

heat source in the heat equation

1 2

( ) ) (

t xx t e x

u L y L z θ κ γ θ θ θ δ ω ρ ρ ′ ′ − + − = − Λ −

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e

C θ = ° ( ) b Nu d γ θ = +

2 1 1 1 3 2 4 4

(0.4Re 0.06Re )Pr ( ) Nu η η

∞

= +

Parameter identification with experimental data

Parameter identification for a steel 16MnCr5 with real dilatometer data

Dilatometer curve for 16MnCr5 (IEHK Aachen)

2 1 2 2 1 2 1 1

ˆ min{ ( (1 , , ) ( )) ( ( , , ) ˆ( ) )) } (

n i i n n i i i i i i

u t p t x p t t p t ω λ ω θ τ α

= = =

− + − +

∑ ∑ ∑ &

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Nataliya Togobytska Parameter identification from dilatometer investigations

Iterations: 58 Residual:0.0914

Conclusions

Future research aims: investigation of the existence of solution for inverse problem parameter identification for 2D model We have assumed the existence of the phase fractions satistying the thermoelastic system and realizing given dilatometer data and exploited

• nly stability, which is an important theoretical issue for numerical computations

( , ) y z

Thank you very much for your attention

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Nataliya Togobytska Parameter identification from dilatometer investigations