Implementation of a continuum damage model for creep fractureand fatigue analyses to ANSYS

Petteri Kauppila1,Reijo Kouhia2, Juha Ojanpera1, Timo Saksala2, Timo Sorjonen1

1 Valmet Technologies Oy, P.O. Box 109, FI 33101 Tampere, Finland2 Tampere University, Civil Engineering, Structural Mechanics, P.O. Box 600, FI-33014

Tampere University

NAFEMS - Improving Simulation Predictions by Using Advanced Material ModeldsLund, Sweden, 2-3 April, 2019

Introduction

I Sustainable energy system is a combination of wide variety of energy resources.

I Result in flexible power generation.

I New requirements for boiler creep fatigue design due to intermittent power demand.

Thermodynamic formulation

Developed models are completely defined by two potential functions:the specific Helmholtz free energy ψ = ψ(T, εte, ω),(linear kinematics assumed ε = εe + εc + εth, εte = ε− εc, ω = 1−D)and the complementary dissipation potential ϕ(Y, q,σ;T, ω) defined as

γ =∂ϕ

∂q·q +

∂ϕ

∂σ:σ +

∂ϕ

∂YY.

Together with the Clausius-Duhem inequality

γ = −ρ(ψ + sT ) + σ: ε− T−1 gradT ·q ≥ 0

results the constitutive equations

− ρ(s+

∂ψ

∂T

)T +

(σ − ρ ∂ψ

∂εte

): εte +

(εc −

∂ϕ

∂σ

):σ

−(ω +

∂ϕ

∂Y

)Y −

(gradT

T+∂ϕ

∂q

)·q = 0.

Consistently coupled heat equation

From the energy balance equation (TD1):

ρcεT = −divq + ρr + σ:εc + ρ∂2ψ

∂εte∂Tεte +

(ρ∂2ψ

∂ω∂T− Y

)ω,

where the specific heat capacity cε is defined as

cε = −T ∂2ψ

∂T 2.

Specific modelsThe specific Helmholtz free energy

ρψ = ρcε

(T − T ln

T

Tr

)+

1

2(εte − εth): ωCe: (εte − εth),

εth = α(T − Tr), thermal strain, Ce elasticity tensor, α thermal expansion coefficients, Tr stress freereference temperature.

The complementary dissipation potential

ϕ(Y, q,σ;T, ω) = ϕth(q;T ) + ϕd(Y ;T, ω) + ϕc(σ;T, ω),

where the thermal part is

ϕth(q;T ) =1

2T−1q·λ−1q.

For creep the following Norton type potential function is adopted

ϕc(σ;T, ω) =hc(T )

p+ 1

ωσrc

tc

(σ

ωσrc

)p+1

,

σ =√

3J2, hc(T ) = exp(−Qc/RT ).

Damage potential

Kachanov-Rabotnov type

ϕd(Y ;T, ω) =hd(T )

r + 1

Yrtd ωk

(Y

Yr

)r+1

, model 1

ϕd(Y ;T, ω) =hc(T )

( 12p+ 1)(1 + k + p)

Yrtd ωk

(Y

Yr

) 12p+1

, model 2

td is a characteristic time for damage evolution,

hd(T ) = exp(−Qd/RT ),

Qd is the damage activation energy and R is the universal gas constant.The reference value Yr = σrd

2/(2E), where σrd is a reference stress for the damage process.

Monkman-Grant parameter

Experimental relationshipCMG = (εcmin)mtrup ≈ constant.

For the two models the Monkman-Grant parameter have the values (m = 1)

CMG = εcmintrup =1

1 + k + 2r

tdhctchd

(σ

σr

)p−2r

model 1

CMG =tdtc

model 2.

Model 2 can be obtained by imposing the following constrains to the model 1:

p = 2r,1

1 + k + 2r

tdhctchc

= constant.

T24 material parametersThe calibrated model parameters for the 7CrMoVTiB10-10 steel (T24), qc = Qc/R and qd = Qd/R,

p(T ) = pr(1 + a(T − Tr)/Tr) and r(T ) = rr(1 + b(T − Tr)/Tr), σrc = σrd = σr = σy0(T ) = σ∗ − cT ,

with σ∗ = 1123 MPa, c = −1 MPa/K.

mod tc [s] pr td [s] a qc [K] rr qd [K] b

1 3039.9 14.77 37.768 −4.804 7137.6 7.545 9350.1 −5.2012 3414.1 14.59 41.26 −4.891 7137.6 - - -

εc,min [%/103h]

σ[M

Pa]

10110010−110−2

400

300

200

100

50

1

trup [h]

σ[M

Pa]

105104103102

400

300

200

100

50

1

Minimum creep strain-rate (lhs) and the creep strengths (rhs).Solid lines = model 1, dashed lines = model 2. Top 500◦C, middle 550◦C bottom 600◦C.

Monkman-Grant parameter

model 2300 MPa200 MPa100 MPa

T [◦C]

CM

G

600580560540520500

0.024

0.022

0.02

0.018

0.016

0.014

0.012

0.01

model 2model 1,T = 600◦Cmodel 1,T = 550◦Cmodel 1,T = 500◦C

εc,min [s−1]

t rup[s]

10−610−710−810−910−1010−1110−12

1012

1011

1010

109

108

107

106

105

1

model 2300 MPa200 MPa100 MPa

T [◦C]

CM

G

600580560540520500

0.024

0.022

0.02

0.018

0.016

0.014

0.012

0.01

model 2model 1,T = 600◦Cmodel 1,T = 550◦Cmodel 1,T = 500◦C

εc,min [s−1]

t rup[s]

10−610−710−810−910−1010−1110−12

1012

1011

1010

109

108

107

106

105

1

Constant stress creep test

t[h]

ε

107106105104103102101100

0.014

0.012

0.01

0.008

0.006

0.004

0.002

0

1

At temperatures 500◦C (blue curves) and 600◦C (red curves) with stress levels 150 MPa and200 MPa. Model 2 results with thicker line.

Implementation to ANSYS as USERMAT subroutine

Evolution equations for stress and integrity

σ = fσ(σ, ω),

ω = fω(σ, ω),

where

fσ(σ, ω) = ωCeεe + ωCeεe + ωCeεe

= ωCe(ε− εth − εc) +

(ω

ωCe+ωT

∂Ce

∂T

)C−1

e σ,

fω(σ, ω) = − Bd

td ωk

(Y

Yr

)r.

Time discretisation using backward Euler scheme and linearisation

[H11 h12

h21 H22

]{δσδω

}= ∆t

{fσfω

}−{

∆σ∆ω

},

where

H11 = I − ∆t∂fσ∂σ

,

h12 = −∆t∂fσ∂ω

, h21 = −∆t∂fω∂σ

,

H22 = 1 − ∆t∂fω∂ω

.

The Jacobian matrix, i.e. “the algorithmic tangent” matrix has the form

Calg = ωH−1

11 Ce, where H11 = H11 − h12H−122 h

T21. Unsymmetric!

Error in one step solution

FE analysis and results

The mesh consists of mainly 20 node hexahedral ANSYS SOLID186 elements & some 10 nodetetrahedal SOLID187 elements.Prescribed displacement history at the end of the tube nozzle.

The computed lifetime is roughly 150 cycles. Ramp time 1 hour and hold time 200 hours. Internal

pressure 14 MPa.

8 P. Kauppila et al. / Structural Integrity Procedia 00 (2016) 000–000 6. Implementation

The described model has been implemented in a structural FE-code ANSYS as a user-defined material subroutine. Integration of the rate dependent model is performed by using the implicit backward Euler method, which has proven to be accurate especially for large practically usable time step sizes in solving rate-dependent problems Kouhia et al. (2005), although it does not completely herit such a nice property when combined with damage Wallin and Ristinmaa (2001); Eirola et al. (2006).

To illustrate the implementation of the model in ANSYS, a part of a typical steam boiler superheater header made of steel T24 and operating within a temperature range from 500 to 600 °C has been analyzed under creep-fatigue loading. A part of the superheater header Ø355.6x32 mm with two symmetry planes has been analyzed under a static internal pressure of 14.0 MPa(g), the static axial stress components caused by the internal pressure, time-varying creep temperature and a time-varying displacement at the end of the tube nozzle Ø44.5x6.3 mm. The displacement is considered to be a result of thermal expansion caused by variable operating temperature and local temperature differences in a large superheater structure. The analyzed lifetime of the header is 150 creep-fatigue cycles in which the duration of load changes is one hour and the duration of hold periods at constant load is 200 hours.

(a) (b) (c)

Fig. 2. The FEM-model mesh in ANSYS (a) and the time-varying displacement at the end of the tube nozzle (b). The displacement at the end of the tube nozzle and the creep temperature changes periodically and the periods consist of ramp and hold times (c).

The main results of the analyses, the values of the damage parameter and the equivalent creep strains at the most critical location are shown in Fig. 3. The most critical location in these analysis was on the surface of the weld between the header pipe and the tube nozzle in all situations, although it shall be noted that these models do not take the special characteristics of a welded joint into account. According to the results, the values of the damage parameter calculated using the model 1 are slightly higher than the values calculated using the model 2. In any case, the values of the damage parameter calculated using the model 1 are only 3–7 pp greater than the values calculated using the model 2, and thus the results of the both models are relatively precise for practical applications. In practical applications the analyzed material can be considered to be damaged, when the value of the damage parameter is over 0.3 and the onset of the tertiary creep phase has occurred. Despite the slight difference in the values of the damage parameter between the models, the equivalent creep strains at the most damaged location are almost equal between the models 1 and 2.

500

600

0 Tem

pera

ture

(°C)

Disp

lace

men

t

Time

DisplacementTemperature

Results

P. Kauppila et al. / Structural Integrity Procedia 00 (2016) 000–000 9

(a) (b) (c)

Fig. 3. The accumulated damage of the header at the most critical location (a). The accumulated damage at the most critical location (b) and the accumulated equivalent creep strain at the most damaged location (c) as functions of the length of the displacement in the analyses.

From a practical point of view, the both developed models yield relatively equal results within the temperature range 500–600 °C. However, the model 2 is accurate only in relatively high creep temperatures and it becomes slightly inaccurate as the temperature lowers, which is a result of the assumed Monkman-Grant relationship. The model 1 is more accurate also in low creep temperatures mainly because of its greater flexibility due to higher number of calibration parameters.

Acknowledgements. This work was carried out in the research program Flexible Energy Systems (FLEXe) and supported by Tekes and the Finnish Funding Agency for Innovation. The aim of FLEXe is to create novel technological and business concepts enhancing the radical transition from the current energy systems towards sustainable systems. FLEXe consortium consists of 17 industrial partners and 10 research organisations. The programme is coordinated by CLIC Innovation Ltd. www.clicinnovation.fi References

Altenbach, H., Gorash, Y., Naumenko, K., 2009. Steady-state creep of a pressurized thick cylinder in both the linear and the power law ranges. Acta Mechanica 195, 263–274.

Arndt, J., Haarmann, K., Kottmann, G., Vaillant, J., Bendick, W., Kubla, G., Arbab, A., Deshayes, F., 2000. The T23/T24 Book. 2nd ed., Vallourec & Mannesmann Tubes.

da Costa Andrade, E., 1910. On the viscous flow in metals, and allied phenomena. Proceedings of the Royal Society A 84, 1–12. Eirola, T., Hartikainen, J., Kouhia, R., Manninen, T., 2006. Some observations on the integration of inelastic constitutive models with damage, in:

Dahlblom, O., Fuchs, L., Persson, K., Ristinmaa, M., Sandberg, G., Svensson, I. (Eds.), Proceedings of the 19th Nordic Seminar on Computational Mechanics, Division of Structural Mechanics, LTH, Lund University. pp. 23–32.

Fremond, M., 2002. Non-Smooth Thermomechanics. Springer, Berlin.´ Garofalo, F., 1965. Fundamentals of Creep and Creep-Rupture in Metals. Macmillan series in Materials Science, Macmillan, New York. Gorash, Y., 2008. Development of a creep-damage model for non-isothermal long-term strength analysis of high-temperature components

operating in a wide stress range. Ph.D. thesis. Martin-Luther-Universitat. Halle-Wittenberg, Germany.¨ Kachanov, L., 1958. On the creep fracture time. Iz. An SSSR Ofd. Techn. Nauk. , 26–31(in Russian). Kachanov, L., 1986. Introduction to continuum damage mechanics. volume 10 of Mechanics of Elastic Stability. Martinus Nijhoff Publishers. Kouhia, R., Marjamaki, P., Kivilahti, J., 2005. On the implicit integration of inelastic constitutive equations. International Journal for Numerical¨

Methods in Engineering 62, 1832–1856. Lemaitre, J., 1992. A Course on Damage Mechanics. Springer-Verlag, Berlin. Lemaitre, J., Chaboche, J.L., 1990. Mechanics of Solid Materials. Cambridge University Press. Murakami, S., 2012. Continuum Damage Mechanics. volume 185 of Solid Mechanics and Its Applications. Springer Netherlands. Nabarro, F., Villers, H., 1995. The Physics of Creep and Creep Resistant Alloys. Taylor & Francis Ltd. Naumenko, K., 2006. Modeling of high temperature creep for structural analysis applications. Ph.D. thesis. Martin-Luther-Universitat. Halle-¨

Wittenberg, Germany. Ottosen, N., Ristinmaa, M., 2005. The Mechanics of Constitutive Modeling. Elsevier. Riedel, H., 1987. Fracture at High Temperatures. MRE Materials Reserch and Engineering, Springer-Verlag, Berlin, Heidelberg. Wallin, M., Ristinmaa, M., 2001. Accurate stress updating algorithm based on constant strain rate assumption. Computer Methods in Applied

Mechanics and Engineering 190, 5583–5601.

0,100,150,200,250,300,350,40

0 0,2 0,4 0,6 0,8

Dam

age

(-)

Displacement (mm)

Model 1Model 2

0,0030,0040,0050,0060,0070,0080,0090,010

0 0,2 0,4 0,6 0,8

Eq. c

reep

stra

in (-

)

Displacement (mm)

Model 1Model 2

Damage distribution near the most critical location of the header. The accumulated damage and the

equivalent creep strain at the most critical location as functions of the prescribed displacement.

More close to the real structure

108

Kuva 7.2. Tulistinkammion Usermat-materiaalimallin versiolla 1 mallinnettujen osienteholliset von Mises -jännitykset analyysin ensimmäisen käyttöjakson alussa alkuperäi-sen tulistingeometrian tapauksessa.

Tulistinkammion aisaputken ja kammioputken välisen hitsin, jossa suurimmatjännitykset esiintyvät, tehollisen von Mises -jännityksen arvot tulistimen käyttöajanfunktiona on esitetty kuvassa 7.3. Kuvan mukaisesti rasitetuimmassa hitsisaumassaesiintyvät suurimmat 263 MPa:n suuruiset jännitykset relaksoituvat hyvin nopeasti joensimmäisen tulistimen 2000 tunnin käyttöjakson aikana vain noin 170 MPa:n suurui-siksi. Kuvasta havaitaan myös tulistimen suurimpien jännitysten relaksoituvan vain noin151 MPa:n suuruisiksi jo tulistimen toisen 2000 tunnin käyttöjakson jälkeen, jonka jäl-keen tulistimen rasitetuimman hitsin reunaviivalle esitettyyn maksimijännitysten esiin-tymiskohtaan muodostuu käyttötilannetta suuremmat jännitykset kattilan ollessa alas-ajettuna. Kattilan käyttöjaksojen edelleen lisääntyessä tulistimen rasitetuimmassa hitsis-sä esiintyvät käyttötilanteen jännitykset jatkavat relaksoitumistaan saavuttaen 200 000tunnin käyttöiän jälkeen tehollisen von Mises -jännityksen arvon 95 MPa. Käyttöjakso-jen määrän lisääntyessä samassa hitsin kohdassa esiintyvä jännitys kattilan ollessa alas-ajettuna kasvaa kuitenkin jatkuvasti ollen 200 000 tunnin käyttöiän lopussa jopa 225MPa. Aisaputken hitsissä esiintyvä suurin jännitys kattilan ollessa alasajettuna käyttöiänlopussa on kuitenkin selkeästi matalampi kuin teräksen SA-213 T24 20 °C:ssa esiintyvämyötöjännitys 450 MPa (Arndt et al. 2000, s. 27). Näin ollen tulistinkammion rasitetuinaisaputki tai sen liitoshitsi ei kuitenkaan myödä kattilaa alas ajettaessa ja kattilan alas-ajot eivät täten kehitetyn materiaalimallin tuottamien tulosten mukaan vaurioita raken-

111

tin plastisen venymän maksimiarvot, mikä on luonnollista vaurion ja ekvivalentin plas-tisen venymän arvojen ollessa riippuvaisia tehollisen jännityksen arvosta. Vaurion mak-simiarvo esiintyy kuitenkin kuvan 7.7 mukaisesti putken seinämän sisällä eikä aivan senpinnassa kuten ekvivalentin plastisen venymän ja tehollisen von Mises -jännityksenmaksimiarvot. Kuvien mukaisesti vaurion suurimmat arvot esiintyvät myös hyvin pie-nellä alueella aisaputken hitsin rajaviivan lähistöllä ja vaurion arvo pienenee voimak-kaasti vaurioituneimmasta kohdasta etäämmälle siirryttäessä kuvan 7.6 mukaisesti.Vaurion arvo laskee myös voimakkaasti vaurioituneimmasta kohdasta aisaputken sei-nämän sisäpintaa kohti siirryttäessä siten, että vaurio putken sisäpinnalla on vaurionmaksimiarvon läheisyydessä vain noin 0,05. Kuvan 7.6 tuloksista myös havaitaan mate-riaalin vaurioitumisen kammioputkessa ja taivutettujen aisaputkien alaosissa olevanhyvin vähäistä.

Kuva 7.6. Tulistinkammion Usermat-materiaalimallin versiolla 1 mallinnettujen osienvaurion arvot käyttöiän lopussa alkuperäisen tulistingeometrian tapauksessa.

Concluding remarks

I Thermodynamically consistent model for high-temperature creep fatigue analyses has beendeveloped.

I A specific model with Norton-Bailey type creep and Kachanov-Rabotnov type damage models areused.

I Two version of the damage evolution equations. One-version satisfies the Monkman-Granthypothesis exactly. Howewer, it restricts the model such that the behaviour is not realistic atlower temperatures.

I Materials parameters for the 7CrMoVTiB10-10 steel (T24) have been estimated in thetemperature range 500-600 0C.

I Developed models have been implemented in the ANSYS FE-software by using the USERMATsubroutine.

Acknowledgements. This work was carried out in the research program Flexible Energy Systems

(FLEXe) and supported by Tekes and the Finnish Funding Agency for Innovation. The programme is

coordinated by CLIC Innovation Ltd. www.clicinnovation.fi

Thank You for Your Attention!

Watercolor by Pia Erlandsson