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

Parameter identification for the phase transformations in steel Nataliya Togobytska Weierstrass Institute for Applied Analysis and Stochastics, Berlin (joint work with D. Hömberg, M. Yamomoto) AIP July 20-24,2009 www.wias-berlin.de togobyts@wias-berlin.de Vienna

Outline • Phase transformations in steel • Dilatometer experiment • Problem formulation • Direct problem • Inverse problem • Stability result for the inverse problem • Numerical implementation • Future research 1 Nataliya Togobytska Parameter identification from dilatometer investigations

Phase transformations in steel Phase transformations in steel Heat treating of steel involves phase transformations Phase – chemical homogeneous region Body-centered cubic iron atom of ferrous material Austenite face-centered cubic iron atom Temperature Ferrite, Pearlite, Bainite microsection Martensite Time Different physical properties: Pearlite,ferrite: soft and ductile Bainite,martensite: hard and brittle Nataliya Togobytska Parameter identification from dilatometer investigations 2

Dilatometer experiment Dilatometer experiment Dilatometer is a device used for measuring thermal expansion and contraction of a solid during heating and subsequent cooling Measurements: Nataliya Togobytska Parameter identification from dilatometer investigations 5

Dilatometer measurements Dilatometer measurements Dilatometer curve = displacement over the temperature Δ l Ferrite ( α ) Austenite ( γ ) T s T f , - start and end temperatures of the occurring phase transitions T T f T s • Sometimes it is difficult and erroneous to fix transformation points T s , T f • the actual phase fractions of the different product phases cannot be drawn directly from the dilatometer curve microscopic investigations have to be made Nataliya Togobytska Parameter identification from dilatometer investigations 6

Problem formulation Problem formulation 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 Sample in dilatometer experiment We will consider 1D model, we define We assume that at most 2 phase fractions may occur during cooling Length - phase fractions change Nataliya Togobytska Parameter identification from dilatometer investigations 8

Thermomechanical effects of phase transitions Balance laws: Rate laws for phase fractions: Constitutive laws: σ = K ε el Additive partitioning of strain : (Hooke) = − grad κ θ (Fourier) q ε = ε + ε el th ( ) u Notations: ε el • elastic strain • mixture ansatz for the thermal strain ε = ε + ε + − − ε th th th th y z (1 y z ) 1 2 0 ε = δ θ − θ th i ( ) i i ref δ > 0 -thermal expansion coefficient i θ i u - displacement -reference temperature of i-phase ref Nataliya Togobytska Parameter identification from dilatometer investigations 7

Thermoelasticity system 1D case ′ ′ ρ θ − θ + Λ δ = ρ + ρ + γ θ − θ e c k ( y z , ) u L y L z ( ) energy balance t xx xt 1 2 θ ⋅ = θ Ω ( , 0 ) in 0 θ = θ = ( 0 , t ) ( 1 , t ) 0 in ( 0 , T ) x x 1D case − δ θ + η = Ω× u ( , ) y z ( y z , ) 0 in (0, ) T momentum xx x balance = u ( 0 , t ) 0 in ( 0 , T ) − δ θ + η = u ( y z , ) (1 , ) t ( y z , ) 0 in (0, ) T x = t y θ & rate laws for y f ( , , ) 1 phase fractions: = t z θ & z f ( , , ) 2 u -displacement, θ - temperature, z, y – phase fractions Nataliya Togobytska Parameter identification from dilatometer investigations 9

Direct problem Assumptions: Nataliya Togobytska Parameter identification from dilatometer investigations 10

Inverse problem l Phase fractions Theorem (Global stability estimate) Reference: D. Hoemberg, N. Togobytska, M. Yamamoto, On the evaluation of dilatometer experiments, to appear in Applicable Analysis Nataliya Togobytska Parameter identification from dilatometer investigations 11

Inverse problem ll The problem of identification of phase fractions from dilatometer curves can be represented as an optimal control problem : ⎧ ⎫ T T ∫ ∫ ω − λ + ω θ − τ ˆ ⎨ 2 2 ⎬ ˆ min ( u ( 1 , t ) ( t )) dt ( ( x , t ) ( t )) dt 1 2 0 ⎩ ⎭ 0 0 subject to the state system ′ ′ ρ θ − θ + Λ δ = ρ + ρ + γ θ − θ e c k ( y , z ) u L y L z ( ) t xx xt 1 2 − δ θ + η = Ω × u ( y , z ) ( y , z ) 0 in ( 0 , T ) xx x θ ⋅ = θ Ω ( , 0 ) in 0 θ = θ = ( 0 , t ) ( 1 , t ) 0 in ( 0 , T ) x x = u ( 0 , t ) 0 in ( 0 , T ) − δ θ + η = u x ( y , z ) ( 1 , t ) ( y , z ) 0 in ( 0 , T ) ∈ y , z U and the control constraints ad Nataliya Togobytska Parameter identification from dilatometer investigations 12

Numerical implementation Strategy: „First discretize then optimize“ Nataliya Togobytska Parameter identification from dilatometer investigations 13

Numerical implementation = p ( ( ), ( ),..., ( ), ( ), ( ),..., ( )) y t y t y t z t z t z t defining the parameter vector 1 2 n 1 2 n we consider finite-dimensional optimization problem ∈ 2 n p R subject to a discretized version of the state system ∈ % and the constraints p U ad Nataliya Togobytska Parameter identification from dilatometer investigations 14

Parameter identification for the steel C1080 with test data model data: λ ˆ( ) = t u (1 , ) t τ = θ ˆ( ) t ( x t , ) 0 Numerical results: Nataliya Togobytska Parameter identification from dilatometer investigations 14

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%) optimal solution dilatometer curve Nataliya Togobytska Parameter identification from dilatometer investigations 15

Input data for parameter identification with experimental data Thermomechanical system includes some parameters, that have to be known: physical parameters ρ κ c - density , heat capacity , thermal conductivity etc. (measurements) δ θ i - thermal expansion coefficients and reference temperatures for each i-phase i ref (These data should be obtained from the dilatometer curves) heat source in the heat equation ′ ′ θ − κ θ + Λ δ ω − ρ − ρ = γ θ − θ e ( ) u L y L z ( ) t xx x t 1 2 θ = ° e 20 C γ = θ + b ( ) Nu d η 1 2 1 1 = + ∞ Nu (0.4Re 2 0.06Re )Pr ( 3 4 ) 4 η 17

Parameter identification with experimental data Parameter identification for a steel 16MnCr5 with real dilatometer data 2 2 n n n ∑ ∑ ∑ & ω − λ ˆ + ω θ − τ + α 2 min{ ( (1 u , , ) t p ( )) t ( ( x , , ) t p ˆ( t )) p t ( ) } 1 i i 2 0 i i i = = = i 1 i 1 i 1 Dilatometer curve for 16MnCr5 (IEHK Aachen) Iterations: 58 Residual:0.0914 Nataliya Togobytska Parameter identification from dilatometer investigations 16

Conclusions We have assumed the existence of the phase fractions satistying ( , ) y z the thermoelastic system and realizing given dilatometer data and exploited only stability, which is an important theoretical issue for numerical computations Future research aims: investigation of the existence of solution for inverse problem parameter identification for 2D model Thank you very much for your attention Nataliya Togobytska Parameter identification from dilatometer investigations 17