COMPARISON OF VARIOUS MODELS OF CREEP COUPLED WITH DAMAGE EVOLUTION

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1 ZESZYTY NAUKOWE POLITECHNIKI BIAŁOSTOCKIEJ 00 Nauki Techniczne Nr XXX Mechanika Z. XX Marcin Bielecki *, Janusz Badur * COMPARISON OF VARIOUS MODELS OF CREEP COUPLED WITH DAMAGE EVOLUTION Abstract: Four different eep models coupled with damage calibrated for P9 steel and implemented within the commercial ABAQUS code have been analysed for different element of live steam piping system within the superitical power plant... Introduction In the power plant elements the tertiary eep is assumed to occur mainly through acceleration of secondary eep by damage, when the eep strains exceed 3 %. Due to the combination of physical nonlinearity with 3D geometry of structural elements, the analysis becomes inappropriate for analytical solutions, and therefore a numerical methods have to be applied. As the Finite Element Method in now commonly used, the computer code ABAQUS, which is of high flexibility allowing for users modeling of materials, was used. Therefore we would like to verify in this paper, whether the various differences in eep models coupled with the damage Numerical example illustrates the main features of models and structures behavior in the desibed circumstances.. Creep models There are different eep models in the literature [,3,4,7,9]. Usually the differences between them are restricted to the steady-state eep strain rate, but they assume, the eep flow is the incompressible, isotropic process with its rate in function of the equivalent stress. This most basic assumption taken from theory of * Instytut Maszyn Przepływowych PAN, Zakład Przepływów z Reakcjami Chemicznymi, ul. Fiszera 4, Gdańsk.

2 Marcin Bielecki, Janusz Badur plasticity is not fulfilled in case of the heat resistant steel [3,4,8]. Five quantitatively different models are presented in this raport. Each of them does not obey one of these assumptions taken on eep flow (incompressible, isotropic, in function of Mises stress, with primary eep). The benchmark for the models under consideration is the most simple model the Norton-Bailey steady-state eep rule (eq. 0,) the standard eep procedure in the FEM codes. The solution taken from this models will be marked as NU model. Models: SKR, MKR, Gurson and ORNL have the steady-state stage desibed by the Norton-Bailey rule, therefore for all these models the material parameters of Norton law (A and m(t)) have the same values. As a result, the secondary (steady-state) eep has the same rate, and only difference between models take place in the form and rate of tertiary (accelerated, affected by damage) stage of eep... Model of Jones-Bagley eep with isotropic strain hardening coupled with the extended Kachanov-Rabotnov damage (JB model, [,5]) The evolution equation of eep for heat resistant material is expressed as a function of some equivalent time of thermo-mechanical service: ε& c = N γ t γ eq teq ( e ) e & t eq + N & t eq The term (-exp(-t eq )) γ is the probability distribution desibing the primary eep. N is a parameter which gives a measure of size of the dislocation population involved in primary eep. The parameter N is a measure of size of the dislocation population slipping to produce secondary eep. The Jones-Bagley model contracts the stresses, temperature and real time to one equivalent variable t eq, which may be called intrinsic time: c pi TA k σ ( ) & c eq + λ ε t eq ( σeq, T, ε, ω) = e T+ 73 B () A i ( ω) i It is observed that a power-law relationship for eep exists over a limited stress range, and that multiple power-law relationship can be a better approximation over wider stress ranges. Therefor there is the sum sign in the relationship for t eq. When the strain in a eep specimen ineases beyond about %, the true stresses visibly ineases by a factor equal to +λ ε in virtue of incompressible eep flow. Such consideration is common in plasticity, which is theoretical basis for incorporating dimensionless Jones-Bagley model to three dimensional structur- ()

3 Comparison of various models of eep coupled with damage evolution al work. Specifically, λ = for tensile strain, λ = - for compressive strain. The distinction in eep flow for tension and compression effects in other deterioration rate in both states. In application tension and compression will be indicated by a sign of hydrostatic stress σ kk i.e. first stress invariant. The damage growth law is the Kachanov-Rabotnov type: σ ( + λ ε ) ( ω) σ u n(t) ω& = (3).. Model of Norton eep in function of Mises stress with isotropic strain hardening coupled with extended Kachanov-Rabotnov damage (MKR model, [,8]) Viscoplastic (here eep) strain rate tensor is a following function of the Mises equivalent stress and deviatoric stress tensor: vp 3 ε& ( σ,t, ω,p, ε ) ε& = s (4) σ Modified Norton eep law is given by formula: m( T) σ ( + λε ) ε& = A (5) ( ω) Damage evolution law has the form: n( T) σ ( + λε ) ω& = (6) σu ( ω).3. Model of isotropic eep with swelling void growth (Gurson model) [7] Damage in the Gurson model is desibed by a single parameter, the void volume fraction f, and the failure process is the result of a local loss of stiffness as the void volume fraction exceeds a certain itical value. At any stage of deformation, the miovoid volume fraction ineases partly because of the growth of existing miovoids and partly because of the nucleation of new miovoids: f& = f& + f& = Aε& + f ε&, f f min (7) n g ( ) ii 3

4 Marcin Bielecki, Janusz Badur f & n - nucleation void fraction rate f & - growth void fraction rate g f n ε ε n A = exp (8) Sn π Sn has been chosen so that void nucleation follows a normal distribution around a itical plastic strain ε n, S n being the related standard deviation and f n the volume of void nucleating particles. Viscoplastic strain rate tensor is defined as: vp sw ε & + ε& (9) 3 ε& ( σ,t) ε& = σ s (0) m( T) ε& = Aσ () sw ii ε & qf sinh σ δ () σ.4. Model of eep with Stobyriev yield iterion coupled with the homogeneous Kachanov-Rabotnov damage (SKR model, [3,4]) The eep flow is incompressible and isotropic but its rate is in function so called Stobyriev equivalent stress. This stress is defined as average of the Mises stress and the maximal principal stress (maximal tension): σ ST = β σ MAX + (-β)σ (3) β - material parameter, from range 0 to ; for heat resistant steels β = ~0.5 Running with Stobyriev stress allows within simulation to keep pace with the laboratory observations, that the eep strain rate for samples under the same Mises stress is the slowest for sheared or compressed (i.e. not tensioned) samples. The form of evolution equations is like in the MKR model: vp 3 ε& ( σst,t, ω) ε& = s (4) σ 4

5 Comparison of various models of eep coupled with damage evolution m( T) σst ε& = A (5) ω n ( T) σst ( ) ω& = (6) ω σ u.5. Model of eep with kinematic strain hardening without damage (ORNL theory) [6,5] By changing the definition of anisotropy evolution law, the ORNL model should boost the tertiary eep by inease of material anisotropy. The equations assume that eep flow is incompressible and in function of Mises stress. vp 3 ε& ε& = ( s α ) (7) σ m( T) ε& = Aσ (8) σ y σ u α & (9) σ u ε u E 3. Applications In order to compare all desibed in the paper models, after their implementation to the Abaqus program, the calculations of two engineering structures were carried out. The elbow of fresh steam pipeline made of P9 steel operates in high temperature (585 o C inside and 590 o C outside the 45mm thick wall), under the internal pressure 3 MPa and the shear force on the top end. The T-joint from other pipeline has constant temperature 550 o C, the internal pressure is 40 MPa and the ends of pipes are stretched out with averaged stress 3 MPa. The boundary conditions are set up this way, that both structures can relax stresses due to the presence of kinematics constraints - the ends of pipes must be flat during analysis. The initial conditions assumes shear force in the elbow, and constant temperature 0 o C in 5

6 Marcin Bielecki, Janusz Badur both structures. Heating is the reason of thermal stresses in elbow, but not in the T- joint on the virtue of combination of kinematics constraints. Fig.. Equivalent eep strain; elbow calculated by JB model state after h, black color = maximal strain 6 %, T-joint calculated by SKR model state after h, black color = % The analysis carried out showed that the ORNL and Gurson models practically do not exhibit any acceleration of eep rate for high strains (more than 3 %). Hence these two model desibes tertiary eep incorrectly, and the results obtained from the ORNL and the Gurson model, as covering the solution obtained from the Norton-Bailey rule (NU model), are not separately presented in the paper. 6

7 Comparison of various models of eep coupled with damage evolution % 0% 8% 6% 4% ε [ ] MKR model SKR model JB model NU model % 0% t [h] Fig.. Maximal eep strains in elbow vs. time of service. Models JB, NU, MKR and SKR deliver results more or less similar each other. In case of the T-joint the MKR model has terminated analysis after h of eep from unknown numerical reason, however the NU model was able to run more than h and the results obtained from both models were nearly the same. The SKR model suggests twice higher life-time of the joint than the MKR and NU models. The elbow calculations of maximal equivalent eep strain made with aid of NU, MKR, JB and SKR models forecast the , , 5.000, hour life-time respectively. If the non-destructive testing iterion *) were employed for the life-time prediction, the elbow could work , 9.000, , hours respectively. Actually, this is rather small scatter of results, in comparison to the scatter of time to rupture in laboratory test (about ±50 % of average values). Tab.. Strains in elbow on the analysis end Creep model NU MKR SKR JB Simulation of eep terminated after [0 3 h] Maximal equivalent strain measured after eep 8,0%,5% 7,64% 5,77% Averaged circumferential strain after eep in cold state,69%,75%,4%,3% Averaged circumferential strain rate [/h] 0,35 0,67 0,3 0,3 *) For pipelines - the averaged circumeferential strain may not exceed % in cold state. 7

8 Marcin Bielecki, Janusz Badur Tab.. Strains in T-joint on the analysis end Creep model MKR and NU model SKR Simulation of eep terminated after [0 3 h] Maximal equivalent strain measured in weld between pipes,30% 6,4% The stress relaxation plays significant role in the work of structures. As both have been designed for h service, they relax stresses from elastic to the longterm values due to the eep (table 3), within very short time circa 5000 h, that is less than 5 % of their livetime. Tab. 3. Stress relaxation σ [MPa] in T-joint Localization After elastic After eep analysis NU and MKR model SKR model Inner surface on the end of small pipe 89,7 73,60 77,38 Inner surface of the hole in the big pipe for the small pipe outlet 0,90 84,30 9,96 Outer surface of the weld between pipes 5,85 79,95 7,9 Inner surface on the end of the big pipe 89,88 74,5 78,06 REFERENCES. Jones D.IG., Bagley R.L. (996); A renewal theory of high temperature eep and inelasticity [in:] Sixth International Conference on Creep and Fatigue, IMechE, Swansea, -4. Brear J.M., Aplin P.F. (99); Incorporating primary eep into the Kachanov-Rabotnov model - effects on rupture file and ductility prediction for /CrMoV and CrMo steels in Strang A. (ed.) Rupture ductility of eep resistant steels The Institute of Metals, London 3. Jakowluk A. (993); Processes of eep and fatigue in materials (in Polish), WNT, Warszawa 4. Hayhurst D.R. (997) High-temperature design and life assessment of structures using continuum damage mechanics (private information) 5. Bielecki M. (000); Modelowanie numeryczne zniszczenia materiału przy obciążeniu termomechanicznym pod kątem oceny żywotności urządzeń energetycznych; praca doktorska, IMP PAN, Gdańsk 6. Lamy C., Hrycaj P., Oudin J., Gelin J.C., Ravalard Y. (99); Finite element computation of void growth damage in thermo-viscoplasticity, Int. J. Mech. Sci. 33, Gurson A.L. (!977); Continuum theory of ductile rupture by void nucleation and growth: Part I Yield iteria and flow rules for porous ductile media, Transactions of the ASME, 43, -5, 8. Danilov V.L., Dobrov M.V, Zarubin S.V. (996); The processes of high-temperature eep and break-down of nuclear reactor vessel under an anticipated accident, [in:] Jakowluk A. (ed.) Creep and coupled processes Bialystok. 9. Tvergaard V. (98); On localization in ductile materials containing spherical voids, Int.J. Fract., 8, 37-5, 8