JSA
Original Article
Modelling of high-temperature inelastic behaviour of the austenitic steel AISI type 316 using a continuum damage mechanics approach
J Strain Analysis 47(4) 229–243 Ó IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0309324712440764 sdj.sagepub.com
Yevgen Gorash1, Holm Altenbach2 and Gennadiy Lvov1
Abstract A conventional material behaviour model can be extended to taking into account varying thermo-mechanical loading conditions in wide stress range. The motivation for developing this model is given by the well documented failure case study of high-temperature components at unit 1 of the Eddystone fossil power plant (Pennsylvania, USA), which have operated for 130,520 h in creep–fatigue interaction conditions. The developed model basis is a creep constitutive law in the form of hyperbolic sine stress response function originally proposed by Nadai (1938). The constitutive law is extended to assume the damage process by the introduction of scalar damage parameter and appropriate evolution equation according to Kachanov–Rabotnov concept. The research task is the introduction into the constitutive model of a few additional material state variables, able to reflect hardening and recovery effects under cyclic loading conditions. The first variable is represented by the relatively fast saturating back-stress K describing kinematic hardening. The second variable is represented by the relatively slow saturating parameter H describing isotropic hardening. Evolution equations for K and H are formulated in a modified form originally proposed by Chaboche and based on the Frederick– Armstrong concept. The uniaxial modelling results are compared with cyclic stress–strain diagrams and alternative experimental data in the form of creep curves, tensile stress–strain diagrams, relaxation curves, etc., for the austenitic steel AISI type 316 at 600 °C in a wide stress range.
Keywords 316 stainless steel, creep failure, internal stresses, low-cycle fatigue, plasticity, relaxation, remaining life assessment
Date received: 16 November 2011; accepted: 7 February 2012
Introduction The review of failures in fossil-fired steam power plants1 indicated that 81% of the failures in hightemperature components were mechanical in nature, and that the remainder occurred due to corrosion. Of the mechanical failures, 65% were classified as shorttime, elevated-temperature failures. Only 9% were due to creep, with the rest being due to such causes as fatigue, weld failures, erosion, etc. Thus, generally, a failure case occurs due to complex interaction of creep deformation mechanism and other internal material behaviour processes, which lead to acceleration of material properties degradation. Therefore, for the purpose of correct simulation of a creep failure case study (CFCS), appropriate lifetime assessment and precise prediction of failure location, it is necessary to apply a unified material behaviour model. It should be able to describe various creep deformation mechanisms and
the processes accompanying creep, like damage, strain hardening/softening, recovery, stress relaxation, the processes evolving independently from creep like plasticity, low cycle fatigue (LCF), oxidation, corrosion, embrittlement, etc. In order to simulate the CFCS of the Eddystone unit no. 1 components illustrated in Figure 1, it is necessary to develop a unified material model able to describe the inelastic behaviour of the austenitic steel AISI type 316 for temperatures up to 650 °C. Due to the details of this CFCS,2–4 the model has to include such 1
Faculty of Physical Engineering, National Technical University ‘‘KhPI’’, Ukraine 2 Faculty of Mechanical Engineering, Otto-von-Guericke-University Magdeburg, Germany Corresponding author: Yevgen Gorash, Faculty of Physical Engineering, National Technical University ‘‘KhPI’’, Frunze strasse 21, UKR-61002 Kharkiv, Ukraine. Email:
[email protected]
230
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Location of cracking
3 mm
0.1 mm
about 20 m
Main steam piping
Boiler Stop valve By-pass valve
Junction header
Turbine Stop valve Control valve
e bin tur P H
OD = 228.6 mm THK = 63.5 mm
Figure 1. Failure of the steel AISI type 316 components, namely main steam piping, at Eddystone no. 1 power station, after DeLong et al.,2 Masuyama et al.3 and Skelton.4
phenomena as creep, plasticity, LCF resulting in creep– fatigue interaction and evolution of corresponding damage parameters to assess the time of failure. Inelastic material behaviour of the austenitic steel AISI type 316 has been comprehensively studied experimentally in the 1990s by several material research laboratories, among them the Forschungszentrum Karlsruhe in Germany,5–7 the National Research Institute for Metals in Japan8–10 and the Institute of Physics of Materials in the Czech Republic.11,12 The uniaxial creep tests were force-controlled and conducted for a wide range of constant stress values from 50 MPa to 350 MPa and for the temperature range 500–750 °C. One of the material properties, which was also measured individually for each creep curve under defined stress and temperature, was instantaneous (or initial) strain eins . Figure 2 shows the comparison of instantaneous strain values eins obtained from creep experiments5,9,10 in the stress range 60–330 MPa at temperature 600 °C with elastic strain eel derived from Hooke’s law eel = s=E, where Young’s modulus is E = 150 GPa, see Karditsas and Baptiste.13 This illustrates the fact that certain initial plastic strain epl for 1
AISI 316 L(N), FZKA, Germany [5] SUS 316-HP, NRIM, Japan [9, 10] elastic strain ε el , E = 150 GPa [13]
Strain
0.1
0.01
the steel AISI type 316 at 600 °C is induced even for moderate stress values. This observation also coincides with a value of experimental yield limit sy = 114 MPa corresponding to temperature 600 °C from Karditsas and Baptiste.13 This assumption was also confirmed by Schirra,7 where the yield point Rp0:2 (defined as 0.2% plastic strain) occurred between 115 and 155 MPa for the range 400 °C \ T \ 750 °C. Thus, at high temperatures and moderate loadings corresponding to the power-law creep mechanism of material deformation, the material behaviour of the steel AISI type 316 is viscous and it is necessary to take into account the significant amount of plastic strain epl . Molybdenum-containing austenitic AISI type 316 (SUS316 HTB, 17Cr12Ni2Mo, X5CrNiMo17-12-2) stainless steel has been used within the powergenerating industry. A common industrial usage is as superheater pressurized tubing exposed to temperatures of 650 °C or higher.14 Compared to martensitic and ferritic steels, austenitic grades including ASTM 316 have lower yield strength sy , but excellent ductility, which is best observed on stress–strain curves. However, in some circumstances it can become embrittled after prolonged exposure at elevated temperatures as a result of the formation of carbide and intermetallic phases. Moreover, it is now known that large cast-to-cast variations exist in ASTM grade 316. Its long-term ductility can vary from below 10% to over 100% also depending on temperature. All basic thermal and mechanical properties of the steel AISI type 316 for the temperature range of 300–1000 °K are available from Karditsas and Baptiste13 and show significant temperature dependence. Plastic deformations are described by experimental ‘true stress–true plastic strain’ curves available from Moosbrugger15 also for different temperatures derived under 4 105 1=s strain rate corresponding to creep deformation mechanism.
Inelasticity material model Low-cycle fatigue properties Since the steel AISI type 316 has low yield strength and high ductility characteristics, it undergoes plastic deformations under relatively low stress at high temperatures as explained in the Introduction. Therefore, the lowcycle fatigue properties of the steel under varying mechanical loading have to be taken into account, which are usually characterized by the Coffin–Manson relation (see Mansoon and Halford16 for details) Depl =2 = e0f (2N)c
0.001
0.0001 50
100
150
200
250
300
350
Stress (MPa)
Figure 2. Instantaneous strain of steel AISI type 316 at 600 °C from creep experiments.5,9,10
ð1Þ
where Depl =2 is the plastic strain amplitude; ef 9 is an empirical constant known as the fatigue ductility coefficient, the failure strain for a single reversal; 2N is the number of reversals to failure (N cycles); c is an empirical constant known as the fatigue ductility exponent, commonly ranging from 0:5 to 0:7 for metals in time-independent fatigue.
Gorash et al.
Power creep law [18–20]: Eq. (5) 500°C
5000 2500
0.02
0.015
0.01
0.005
0 10 2
500°C 550°C 600°C
Q f (T )
0
0
200
400
600
Temperature (°C)
Strain rate in experiments: 10− 3 (1/s) 10− 4 (1/s) 10− 5 (1/s) 10 3
10 4
10 5
Number of cycles to failure
Figure 3. Fitting of LCF experiments8 by S–N curves for the steel AISI type 316 at the temperature range of 20–700 °C.
Analogously to equation (1), the following dependence for number of cycles to failure N (De, T) is proposed to describe the available experimental S – N diagrams8 for the steel AISI type 316 G Qf (T) N (Deeff , T) = F exp ð2Þ exp Deeff RT where Deeff is an effective strain range, where the effective strain eeff is defined as the maximum absolute value of principal strains multiplied by the algebraic sign of the dominant component within the one loading cycle in the following form eeff = maxðjeI j, jeII j, jeIII jÞ sign(eI _ eII _ eIII )
ð3Þ
In equation (2) the fatigue activation energy Qf takes into account the temperature range under 400 °C in the following way Qf (T) = Qfht ð1 + exp½q(T Ttr )Þ1
ð4Þ
where the fatigue material constants in equations (2) and (4) are identified by fitting to uniaxial LCF experiments8 for the temperature range of 20–700 °C, as illustrated in Figure 3: F = 2, G = 0:04, Qfht = 10;000 J=mole , q = 0:1, Ttr = 46 8C and the universal gas constant R = 8:314 J mol 1 K 1 .
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
-9
10
600°C
650°C Ref. 700°C [5] 750°C
650°C
700°C
650°C Ref. 700°C [9] 750°C
500°C 550°C 600°C
750°C
650°C Ref. 700°C [11] 750°C
plasticity
“Sinh” creep model: Eqs (6) and (7) 500°C 550°C 600°C
own
Q fht = 10000 (J /mole)
7500
550°C
650°C 700°C 750°C
breakd
10000
pow er-l aw
0.025
20°C 400° C 500° C 600° C 700° C
Strain rate (1/h)
Total strain range
0.03
Activation energy (J/mole)
Temperature
0.035
231
Ref. [10] 600°C 650°C 700°C 750°C
1
diffusion
Stress (MPa)
n = 15
100
low
700
moderate
high
Figure 4. Fitting of creep constitutive equations (5) and (6) to creep experiments.5,9–11
validity for the industrial use of the steel. The temperature range for the steel AISI type 316 relevant to engineering practice lies in the range 500–700 °C. Within the phenomenological approach to creep modelling one usually starts with formulation of a constitutive equation for the minimum creep rate e_cr min . Figure 4 illustrates all collected experimental data from the three laboratories in Germany, Japan and the Czech Republic5–7,9–12 for the steel AISI type 316. It presents the dependence of e_cr min on the applied stress s at different temperatures generally in the ranges of ‘moderate’ and ‘high’ stress levels with designation of different deformation mechanisms according to Rieth.17 Referring to Sherby and Burke,18 the classical stress-response function proposed by Norton19 and Bailey20 is usually extended with an Arrhenius-type temperature-response function to describe creep behaviour in the ‘power-law’ area at different temperatures Qcr n e_cr = gðs, TÞ = c exp s RT
ð5Þ
Modelling of creep Creep constitutive equation The deformation mechanism map of steel AISI type 31612 illustrates the requirement to take into account both main creep mechanisms while formulating the material model: power-law creep, which includes generally high stress range, and linear or viscous creep, which includes generally moderate and low stress ranges. The industrial application conditions lay mostly in the ‘viscous creep’ region, close to the transition boundary, which provides relatively better creep resistance. On the other hand, the laboratory testing conditions lie mostly in the power-law region, so those results have limited
where the creep exponent n = 15, creep material parameter c = 1011 h 1 and creep activation energy Qcr = 330;000 J=mole were defined fitting the experimental data5–7,9–12 within only the high stress range, as illustrated in Figure 4. Gorash21 extended the ‘power-law’ equation (5) to the ‘double power-law’ model and applied the description of creep behaviour in a wide stress range of the steel type P91 (9Cr-1Mo-V-Nb). But this constitutive model does not provide the smooth switching between the ‘diffusion’ and ‘power-law’ creep deformation mechanisms, which is observed experimentally for the steel AISI type 316. Thus, similarly to the approach of
232
Journal of Strain Analysis 47(4) 600
n = 15
Stress (MPa)
transgra nular frac ture
Ref. [5] 500°C 550°C 600°C 650°C 700°C 750°C
1
wed
ge c rack ing
high
Ref. [9]
ca vit yn uc lea tio n
Eq. (9), M-G relation: 500°C 550°C 600°C 650°C 700°C 750°C
20 10
10
2
500°C 550°C 600°C 650°C 700°C 750°C
10
(C )
Ref. [6] 500°C 550°C 600°C
Ref. [10]
)
500°C 550°C 600°C 650°C 700°C 750°C
Eq. (10), Combination:
(σ
low
Eq. (8), Hoff’s model:
n io at cle nu ty vi ca
moderate
100
550°C 500°C 600°C 650°C 700°C 750°C
3
10
4
10
5
10
6
600°C 650°C 700°C 750°C
Time to fracture (h)
Figure 5. Fitting of time-to-rupture relations (8)–(10) to creep–rupture experiments.5,6,9,10
Naumenko et al.22 for X20CrMoV12-1 steel, the hyperbolic sine stress response function initially proposed by Nadai23 and developed by Hayhurst24 was selected as the basis for the constitutive model due to satisfactory fitting of the experimental data for the complete stress range at each temperature e_cr = f(s, T) = A(T) sinh ½B(T) s
ð6Þ
where the temperature-dependent creep material parameters A(T) and B(T) are Arrhenius-type temperatureresponse functions A(T) = a exp½Qln =R T B(T) = b exp½Qpw =R T
ð7Þ
In equations (6) and (7) all the necessary secondary creep material constants of the steel AISI type 316 are identified by graphical fitting of the available experimental points in a general diagram (see Figure 4) corresponding to several temperatures simultaneously: a = 75;776 h1 , Qln = 170;000 J=mole , b = 0:7 MPa 1 and Qpw = 12;000 J=mole .
Assessment of time to fracture The next important engineering parameter, after e_cr min , is time-to-fracture t , which is derived from a set of creep rupture data for the steel AISI type 316 combined with its fracture mechanism map proposed by Maruyama.25 It shows that the stress exponent n and the activation energy Qfr for time-to-fracture decrease with increasing t . This fact is consistent with the assumptions about long-term strength of heat-resistant steels discussed by Gorash21 and the fracture mechanism maps discussed by Maruyama.25 The steel AISI type 316 is assumed to have the following fracture mechanisms: transgranular fracture (T), wedge cracking (W), cavity nucleation at the grain boundary carbides (C) and cavity nucleation
at the grain boundary s -phase (s). The changes in fracture modes coincide very well with the changes in the values of n and Qfr defined experimentally. Such a variety of creep fracture mechanisms in a wide stress range is characterized by the so-called phenomenological long-term strength curves describing mathematically the dependence of time-to-rupture t on applied stress s. These relations are intended for fitting the available set of creep–rupture data5,6,9,10 corresponding to the steel AISI type 316 at different temperatures, illustrated in Figure 5. The conventional approaches to formulation of the long-term strength functions are presented by the Hoff model26 and Monkman–Grant relation.27 The following equation based on the creep power-law was used by Hoff26 for the determination of time to ductile rupture for a rod under tension and creep conditions t (s, T) =
1 1 = cr n n g(s, T) n c exp Q RT s
ð8Þ
where the values of creep material parameters c, n and Qcr are taken from the Norton–Bailey model (5). However, in the case of advanced heat-resistant steels including the steel AISI type 316, equation (8) is applicable only for a quite narrow ‘high’ stress range usually not corresponding to the industrial application range, as shown in Figure 5. Referring to Gorash,21 these steels demonstrate the ductile rupture character at high stress levels resulting in necking of a uniaxial specimen caused by plastic instability at a micro-scale. Then, at moderate stress levels the transition to a brittle rupture character becomes observable, which is caused by the degradation of material microstructure. Finally, the rupture mode becomes brittle at low stress range accompanied by the nucleation and growth of cavities leading to initiation of cracks. One of the variants to describe the mixed and brittle rupture mechanisms,
Gorash et al.
233
which contribute to the creep-damage development, is given by a form28 of the Monkman–Grant relation
1
t (s, T) = D ½f(s, T)
m
ð10Þ
Employment of continuum damage mechanics (CDM) approach The next step within the phenomenological approach to creep modelling is the transition from time-to-rupture t relations (equations (8)–(10)) to the formulation of damage mechanism incorporated into a creep constitutive model. Previous studies in this field began with the concept of continuity introduced by Kachanov.29 Later the concept of continuity was modified and replaced with the concepts of damage and effective stress, introduced by Rabotnov.30 Based on previous concepts it was assumed by Gorash21 that the rupture modes of the long-term strength curve can be interpreted by the accumulation character of the damage parameter v, which is brittle for low values of tertiary creep constant l and ductile for its high values according to the expression for creep damage 1
vcr (t) = 1 (1 ½t=t )l + 1
ð11Þ
which is differentiated to obtain the following damage evolution equation expressing the transition from timeto-rupture t to damage parameter v v_ cr = ½t (l + 1) (1 vcr )l 1
ð12Þ
with time-to-rupture t in the form of equation (9) t
(sveq , T) = D1
½f(sveq , T)m
+ (sII sIII )2 =2 0:5
ð13Þ
sveq = a smax t + b svM + (1 a b)I1
dvf 1 = dN N (l + 1) (1 vf )l
smax t = max ½ (sI + jsI j)=2, (sII + jsII j)=2; (sIII + jsIII j)=2
ð18Þ
where vf is a fatigue damage parameter, N is a number of cycles to failure in the form of equation (2) and constant l = 1 in equation (18) is assumed to govern the evolution character of fatigue damage. The influence of creep and fatigue damage parameters (vcr and vf ) on the accumulation of creep strain ecr propagating according to the constitutive model (equation (6)) is proposed in the following form using the strain equivalence principle e_cr =
f(s, T) (1 vcr vf )l
ð19Þ
In relation to the Kachanov–Rabotnov concept,29,30 the value of damage parameter vjt = 0 = 0 for undamaged (virgin) material and vjt = t = v for fractured (cracked) material, where v = 1 is the ‘idealized’ critical value of the damage parameter. However, according to experimental observations the value of v is usually α = 1, β = 0
α = 0, β = 1
1
ð15Þ
α = 0, β = 0
0.5
Experiments [31] at 600°C and 593°C
0.25
σ II 0 σ0
I 1 , ref. to Eq. (14) σ max t , ref. to Eq. (15)
-0.25
σ vM , ref. to Eq. (16) σ ωeq , ref. to Eq. (17)
-0.5 -0.75 -1
or the maximum tensile stress
ð17Þ
where a = 0:15 and b = 0:7 are weighting factors considering the influence of different damage mechanisms (controlled either by smax t or svM or I1 ), which are identified by fitting the corresponding plane stress isochronous rupture locus (equation (17)) to the available multiaxial experimental locations31 obtained at 593 °C and 600 °C, as illustrated in Figure 6. From Penny and Mariott,33 the damage evolution equation caused by LCF can be introduced similarly to equation (12)
0.75
ð14Þ
ð16Þ
But the most general variant of sveq , see Naumenko and Altenbach,32 is
1.25
where l is a tertiary creep constant identified below and the expression for the creep constitutive equation f(sveq , T) is taken in the form of equation (6). The damage equivalent stress sveq in equation (13) is usually taken in the form of either the first stress invariant I1 = sI + sII + sIII
svM = ½ (sI sII )2 =2 + (sI sIII )2 =2
ð9Þ
where the expression for f(s, T) can be taken in the form of the creep constitutive model (6), and m = 0:55 and D = 0:07 h m1 are tertiary creep material constants for the steel AISI type 316, which are identified by fitting equation (9) to the creep–rupture experimental data5,6,9,10 for the range of temperatures 500 °C to 750 °C. Combining both concepts and the corresponding equations (8) and (9) into one equation, the long-term strength curve is obtained for the complete stress range, as illustrated in Figure 5 t (s, T) = (n g(s, T) + D ½f(s, T)m )1
or the von Mises equivalent stress
α = 0.15, β = 0.7 -1
-0.75
-0.5
-0.25
0
σ I /σ 0
0.25
0.5
0.75
1
1.25
Figure 6. Comparison of the plane stress isochronous rupture loci with experimental values.31
234
Journal of Strain Analysis 47(4)
which mathematically is not significantly different from the value of t . Thus, the expression for creep rupture strain e (s, T) without assumption of fatigue damage vf is derived by integrating the constitutive model (equation (19)) in the time range from t = 0 to t = t0:7 according to equation (20) and taking into account the expression for damage parameter (equation (11)) l
ð21Þ
0
where the value of tertiary creep constant l = 4 is identified by fitting the result of numerical integration of equation (21) for the varying stress s and discrete temperature T values to the collected experimental creep rupture strain data,5,9,10 as illustrated in Figure 7. It should be also noticed that equation (21) takes into account creep strain ecr accumulated during the secondary and tertiary creep stages, neglecting the primary stage. This simplification is valid only for high values of stress, when the duration of primary stage is relatively short. In the case of low and moderate stress ranges, equation (21) should be modified with an assumption of some hardening mechanism, especially for such a ductile steel as AISI type 316. In the case of taking into account of the both damage parameters (vcr and vf ), their sum must not exceed the critical value of 0:9 during the simulation: 0:95vcr + vf . Besides the limitation by critical value of 0:9, according to Figure 7 the ‘real’ rupture strain e 500°C 550°C 600°C
650°C Ref. 700°C [5] 750°C
500°C 550°C 600°C
650°C Ref. 700°C [9] 750°C
Ref. [10]
1
600°C 650°C 700°C 750°C
The reported failure case study2–4 of high-temperature components made of the steel AISI type 316 at unit no. 1 of the Eddystone power plant, which has in total operated for 130,520 h with 326 cycles, specified that their failure had been caused by the damage accumulated under variable thermo-mechanical loading conditions inducing the complex interaction of different processes. A unified material model has to be employed for the thermo-mechanical creep–fatigue analysis of these components. The main steam piping (MSP) from this power plant, shown in Figure 1, is selected for the numerical FE analysis in CAE software ABAQUS in order to obtain a correct lifetime prediction, which complies with real fracture. So the principal task is to confirm the applicability of the proposed model presented by the constitutive equation (22) with equation (6), damage evolution equations (12) and (18) and the thermo-mechanical material properties.13,15 The axisymmetrical FE model of MSP, consisting of 52 type DCAX4/CAX4 four-node bilinear axisymmetric quadrilaterals, with corresponding geometrical parameters, boundary conditions (BCs) and transient z
100000
r
Geometrical parameters (mm): r i = 50.8 r o = 114.3 ri w thk = 63.5 Steam: ro h = 20 P stm Tstm , hstm
zmin
moving rigid block
ductile rupture
mix ed
Creep strain
Creep–fatigue analysis of main steam piping failure
0.1
0.02 10
zmax
w thk h
Stress (MPa)
brittle
100
low
500
moderate
hair
5 0
0
6
407 413 445 451
Time (h)
tsu
Rupture strain (21): 650°C 700°C 750°C
hstm
649
0.5
500°C 550°C 600°C
ð22Þ
Air: Tair , ε r = 0.5 Tair , hair
Temperature (°C)
t il + 1 f(s, T) 1 dt t
3 f(svM , T) s 2 (1 vcr vf )l svM
Tstm
tst tdt Tair
20 0
0
6
407 413 445 451
Time (h) 34.5
Pressure (MPa)
e (s, T) =
h
_ cr =
moving rigid block
Zt0:7
is also limited by the value of 0:5 for the uniaxial specimen instead of theoretical e ) ‘ derived by Hoff’s model (equation (8)) in ductile rupture mode. Finally, the multiaxial form of constitutive equation (19) is formulated as follows
Heat transfer coef. (W/m2 °C)
taken in range of 0.3–0.7 corresponding to the results of Lemaitre and Chaboche34 for heat-resistant steels including the steel AISI type 316. Thus, one can obtain the time to ‘real’ fracture t0:7 corresponding to v = 0:7 by converting equation (11) h i t0:7 = t 1 (1 v )l + 1 ð20Þ
P stm
0
tac
tht 0
6
407 413 445 451
Time (h)
high
Figure 7. Comparison of rupture strain equation (21) obtained by the creep model with experimental data.5,9,10
Figure 8. Formulation of structural analysis problem for MSP:2–4 geometry of the FE model, thermal and mechanical loadings and BCs.
235
Firstly, the transient thermal analysis was implemented in order to obtain the temperature distribution through the thickness (wthk ) of MSP for the total simulation duration tts . During the high-temperature (HT) operation periods tht the steady-state temperature distribution was obtained, as illustrated in Figure 9(a). In addition to inner pressure, the computed transient thermal field was also applied as loading for the creep–fatigue mechanical simulation of MSP. As a usual result of mechanical analysis, the transient stress distribution was obtained, as illustrated in Figure 9(b) for several cycles in terms of effective stress or absolute value of maximum principal stress, formulated similarly to equation (3) seff = max ( jsI j, jsII , jsIII j ) sign(sI _ sII _ sIII )
ð23Þ
This result shows that with the number of cycles the stresses reduce and relax due to creep during HT operation in contrast to cold downtime, when the stresses increase. However, on the last cycle the stress on the outer surface of MSP during HT operation is lower than the stress on the inner surface during cold downtime, but it has greater influence on the accumulation of creep damage vcr , as illustrated in Figure 9(c). It should be noticed that the fatigue damage vf is a localization of microcracks on the surface as opposed to creep damage vcr , which is uniformly distributed in form of microvoids through the volume. Thus, the values of vf accumulated on the surfaces really matters for the analysis. And in the case of MSP, the amount of vf accumulated on the inner surface is twice as high as on the outer surface. Creep and relaxation processes lead
610 590 570
Steady-state temperature during HT operation
ri
ro
a 50
55
60
65
70
75
80
85
90
95
100
105
110
115
HT operation (N = 1) cold downtime (N = 1) HT operation (N = 100) cold downtime (N = 100) HT operation (N = 326) cold downtime (N = 326)
150
Effective stress (MPa)
120 90
ri
60 30 0 -30 -60 -90
b
ro
-120 -150
50
55
60
65
70
75
80
85
90
95
100
105
110
115
Radius (mm) 0.6 0.5
ri
0.4
0.06
fatigue damage (ω f ) creep damage (ω cr )
0.05 0.04
0.3
0.03
0.2
c
0.1 0
ro
50
55
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Creep damage
start-up period tsu = 6 h ; HT operation period tht = 401 h ; shut-down period tsd = 6 h ; downtime period tdt = 32 h ; average cycle period tac = 445 h ; number of cycles N = 326; total HT operation time ttht = tht N = 130;520 h ; total simulation duration tts =ttc N=145;070 h.
630
180
Fatigue damage
650
Radius (mm)
0 115
Radius (mm)
Figure 9. Distribution through the thickness (wthk ) of MSP for (a) temperature, (b) effective stress and (c) damage parameters.
150
Δ ε eff (N = 1)
120
Δ ε eff (N = 326)
90
Effective stress (MPa)
thermo-mechanical loadings is illustrated in Figure 8. The thermal loading includes periodically changing temperature Tstm = 20 649 8C and heat transfer coefficient hstm = 5 100;000 W=m 2 8C for the forced convection on the inner surface of the pipe. The thermal parameters of the outer surface include ambient temperature Tair = 20 8C, surface emission coefficient er = 0:5 and coefficient of free convection hair = 5 W=m2 8C . The mechanical loading consists of periodically changing pressure Pstm = 0 34:5 MPa on the inner surface of the pipe. Mechanical BCs consist of free movement in the axial direction (z) and free expansion in radial direction (r) of the left and right pipe cross-sections without rotation, i.e. always remaining straight and perpendicular to the z -axis. Therefore, the periodical operation conditions are assumed to have the following parameters:
Temperature ( °C)
Gorash et al.
60 30
Δ σ eff (N = 1)
0 -30
1st cycle 2nd cycle 50th cycle 100th cycle 200th cycle 270th cycle 326th cycle
-60 -90 -120 -0.001
0
Δ σ eff (N = 326)
0001 . 0002 . 0003 . 0004 . 0005 . 0006 . 0007 . 0008 . 0009 . 001 .
001 . 1 0012 .
Effective strain
Figure 10. Hysteresis loops on the outer surface.
to an increase of the effective stress range Dseff and shift of stresses into compression with a number of cycles, while the accumulation of damage parameters (vcr and vf ) results in an increase of the effective strain range Deeff , as illustrated in Figure 10 with hysteresis loops on the outer surface of the MSP. The estimated values of damage parameters (vcr = 0:06 and vf = 0:6) after 326 cycles
236
Journal of Strain Analysis 47(4) 1
Isolated cavities Orientated cavities Microcracks Macrocracks Experimental data [35] at 593° C Damage linear summation (x = y = 1) ASME standard bi-linear locus Fitting of ASME locus (x = y = 0.58) Conservative locus (x = 0.6, y = 0.17) Creep-fatigue analysis result for MSP
0.7 0.6
Fracture Kinematic hardening parameter K
Strain ε
Fatigue damage
0.8
0.5 0.4
ε
III
cr
Isotropic hardening parameter H
ε*
0.3
II
0.2
I
0.1 0 0
0. 1
0 .2
0.3
0.4
0. 5
0.6
0.7
0.8
0.9
1
Creep damage
ε pl ε el
ε pr
ε˙cr min
Creep damage starts Damage accumulates
ε ins Damage parameter ω
Stress σ , Damage ω
0.9
Time t
t*
Figure 11. Creep–fatigue damage interaction diagrams compared with experiments.35
Figure 12. Idealized creep curve with illustration of internal state variables evolution.
corresponding to 130,520 h of HT operation on the outer surface of the MSP (see Figure 9(c)) comply with the real location of the component failure shown in Figure 1, and a scatter of experimental data35 on the creep–fatigue interaction diagram in Figure 11. According to this diagram the creep–fatigue damage interaction is more critical for the structural integrity, than the unilateral accumulation of one damage parameter. The result obtained for MSP damage interaction can be fitted with the equation vxcr + vyf = 1, where the corresponding damage parameter exponents have the values x = 0:6 and y = 0:17. They provide some conservative locus for creep–fatigue damage interaction, taking into account the scatter of experimental data in Figure 11.
temperature-dependent Young’s modulus E. And a portion of epl is defined by the hardening processes, which induce the evolution of both hardening parameters – relatively slow saturating isotropic H and relatively fast saturating kinematic K. The contribution of each parameter on epl is dependent on stress level, e.g. at high stresses contribution is mainly by H and at low stresses mainly by K. The time-dependent inelastic response is the slow increase of the creep strain ecr with a variable creep strain rate e_cr . Depending on the character of creep strain acceleration €ecr , three stages can be considered in a typical creep curve as illustrated in Figure 12: the first stage (primary or reduced creep), the second stage (secondary or stationary creep) and the third stage (tertiary or accelerated creep). Moreover, referring to Viswanathan,36 the shape of the creep curve is determined by following competing reactions or processes, as explained in Figure 12:
Discussion Phenomenological approach Referring to Naumenko and Altenbach,32 the phenomenological approach to the development of a unified material model is based on mathematical description of experimental creep curves obtained from uniaxial creep tests under constant loading s = const, since creep is a dominant material behaviour at high temperatures. The best way to illustrate the developed technique is to explain the processes affecting the form of an idealized creep curve in Figure 12 and to propose some phenomenological description for them. According to the available mechanical properties13 of the steel AISI type 316 explained in Figure 2, the values of the normal stress s in the specimen, corresponding to moderate axial loading, may exceed the yield limit of the material sy ’100 120MPa. The instantaneous inelastic material response is therefore elasto-plastic at high temperature range eins = eel + epl
ð24Þ
where eins is instantaneous, eel is elastic, and epl is plastic strain. Elastic strain eel is characterized by the
(a) creep strain hardening; (b) softening processes such as recovery, recrystallization, strain softening, and precipitate overaging; (c) damaging processes characterized by the damage parameter v resulting in cavities initiation and cracking, and specimen necking. Of these three factors, creep strain hardening (a) tends to decrease e_cr , whereas the other factors (b) and (c) tend to increase e_cr . The balance between these factors determines the shape of the creep curve. During the primary creep stage the decreasing slope of the creep curve is attributed to strain hardening, which decreases e_cr to a certain value (minimum creep rate e_cr min ). And a portion of creep strain epr accumulated during the primary creep stage is defined also by the evolution of both hardening parameters K and H, as in the case of epl . The contribution of each parameter on epr is also dependent on the stress level. A number of creep material properties can be deduced from the uniaxial creep curve. The most important of them are the duration of
Gorash et al.
237
each of the creep stages, e_cr min , t and the value of strain before fracture e
e = eins + ecr = eel + epl + ecr
ð25Þ
which is related to the ductility of the material. The first iteration in the unified model development is formulated for the defined temperature value. Temperature value 600 °C is chosen as the basis for the definition of creep material constants for the steel AISI type 316, since this value is close to the mean service temperature of the power plant components from Eddystone unit no. 1. For the purpose of adjusting the basic creep law, two experimental sets of creep curves at 600 °C were employed for the creep constant identification. The first set includes six creep curves in the stress range 60–170 MPa provided by Rieth et al.,5 and the second set includes six creep curves in the stress range 200–300 MPa provided by Chaboche.37 Each curve from both sets was differentiated numerically with respect to time in order to identify the corresponding e_cr min values. The result of this identification is illustrated in Figure 13 with red and blue points. Both sets of e_cr min were fitted numerically by power law (equation (5)) and Sinh model (equation (6)) using weighted Levenberg–Marquardt algorithm as illustrated in Figure 13 with magenta and green lines, providing the following values of material constants: C = 5:647 1023 1=h and n = 7:515, A = 2:335 1010 1=h and B = 0:053 1=MPa . If a straincontrolled loading is applied to the Sinh creep model with a fixed value of total strain Det , then one can obtain a perfect plasticity stress response, i.e. the model provides irreversible deformation without any increase in stresses reaching a certain stress level, thus producing a stable closed hysteresis loop as shown in Figure 13. Following Khazhinskiy,38 depending on the form of stress response under cyclic strain-controlled loading all metals for high-temperature application are usually divided into cyclic (or isotropic) stable and instable categories. Typical examples of stable metals are medium-carbon steels, which have a width of
-3
Min. creep strain rate (1/h)
10
low stress tests from FZKA [5] high stress tests from ONERA [37]
-4
10
Sinh model, Eq. (6): ε˙ cr min = A sinh (B σ )
-5
10
n Power law, Eq. (5): ε˙cr min = C σ
-6
10
ε
-7
10
-8
Δ εt
Strain Control:
Δ εt
σ
ε
time
10
-9
10
10
20
30
40
50
100
200
elasto-plastic hysteresis loop almost not dependent on the number of loading cycles. Instable metals are subdivided into isotropic hardening and softening, while their instability is usually associated with microstructure transformations undergoing cyclic or significant inelastic deformations. Isotropic softening metals are characterized by the gradient narrowing of elasto-plastic hysteresis loop observable after first loading cycle. Such type of metals is usually represented by high-strength heat-resistant steels, e.g. martensite steel X20CrMoV12-1 described by Naumenko et al.22 with a combined model for hardening, softening and damage processes. Isotropic hardening metals are characterized by the gradient expansion of elasto-plastic hysteresis loop observable after the first loading cycle. Such type of metals is usually represented by ductile stainless steels, e.g. austenitic steel AISI type 316, the cyclic plasticity behaviour of which was described by Gong et al.39 for a wide temperature range with a unified viscoplasticity model after Chaboche.37 Thus, one can make an assumption that the plastic strain epl and the strain accumulated during the primary creep stage epr for cyclic stable metals are defined only by kinematic hardening, which provides the shift of initial yield surface. In the case of isotropic softening metals this assumption is also correct, but the softening phenomenon additionally influences epl , providing the reduction of initial yield surface, and the second creep stage. The steady-state segment almost vanishes due to creep strain acceleration caused by material transformation towards a softer microstructure and followed by damage accumulation at the tertiary stage, as illustrated in Figure 14 and explained by Naumenko et al.22 In the case of isotropic hardening metals the strains epl and epr are defined by both kinematic and isotropic hardening, which provides the additional growth of the initial yield surface. The influence of the hardening phenomenon reduces the creep strain deceleration of the second part of the primary stage, thus also causing the steady-state segment of the second creep stage to vanish. Such a behaviour is caused by material transformation towards a harder microstructure and followed by damage accumulation at the tertiary stage, as illustrated in Figure 14. Therefore, for an adequate description of the inelastic material behaviour of steel AISI type 316, it is necessary to formulate such a phenomenological model, which is able to reflect all the features of hardening metals specified above.
300
Stress (MPa)
Figure 13. Modelling of minimum creep strain rate using experimental creep curves for the steel AISI type 316 at 600 °C.5,37
400
Formulation of the unified model The creep constitutive model (equation (6)) based on the Sinh stress-response function is taken as the basis of the proposed unified viscoplasticity model for the steel AISI type 316 and extended below in order to take into account kinematic and isotropic hardening effects. Therefore, the model is formulated not for e_cr , but for inelastic strain rate e_in . It should be noticed that the
238
Journal of Strain Analysis 47(4) σ Cyclic (isotropic) hardening
kinematic hardening
3 2
isotropic hardening
1
III
ε
Cyclic (isotropic) softening
σ
1 2 3
Creep strain rate (log scale)
stability
I
II ε˙cr min isotropic stable metal hardening metal softening metal isotropic softening
III
stability ε
I
II ε˙cr min Creep strain (log scale)
Figure 14. Schematic representations of idealized creep rate versus creep strain dependence.
values of creep material constants A and B for the Sinh function remain the same as defined in the previous section by fitting the available experimental creep curves at 600 °C. Thus, the extension consists in replacement of the value of applied stress s by the value of viscous stress sv , which is initially significantly higher than the value of applied stress s, but saturates towards it under a constant loading after a certain time depending on the value of applied stress and loading rate sv = s=(1 h + H) + s0 sign (s K) K
ð26Þ
where s0 and h are material constants for the steel AISI type 316 at 600 °C, which present the saturation values for kinematic backstress K and isotropic hardening variable H respectively. The influence of additional material state variables (K and H) and corresponding saturation values s0 and h results in so-called overstress X = sv s, which saturates towards zero under a constant loading after a certain time. The value of overstress at the initial moment of time before the elastic response of material considering the finite loading rate is X = sh=(1 h) + s0
ð27Þ
In the case of the following form for viscous stress sv = s=(1 H) + s0 sign (s K) K
K_ = C2 ½s0 sign (s K) Kje_in j
ð28Þ
ð29Þ
where the value of saturation constant h varies from 0 to 1, specifying the conventional value of initial yield stress sy on the first 0.25 hysteresis loop or tensile
ð30Þ
which is slightly different from the conventional form, but provides the same effect of kinematic shift after application of reverse loading and positive value of s, because the sign of e_in is defined by the difference between s and K. In equation (30) the constant s0 represents some kinematic shift of the initial yield surface, and the constant C2 defines the rate of this shift, thus providing the gradual transition from elastic slope to plastic in the tensile stress–strain diagram and the initial strain rate at the first stage of creep curve. In addition, the relation between elastic eel and inelastic strain ein for a strain-controlled test is formulated in following form s_ = E(e_t e_in )
the proposed model provides kinematic hardening combined with isotropic softening. The evolution equations for hardening state variables (K and H) are taken in the same form as implemented in research22,39 using the Frederick–Armstrong concept.40 The evolution equation for isotropic hardening parameter H is H_ = C1 (h H)je_in j
stress–strain diagram. Constant C1 defines the rate of saturation for parameter H towards the value of h, thus providing the stabilization of hysteresis loops and reaching the conventional value of ultimate stress su on the tensile stress–strain diagram. The evolution equation for kinematic hardening backstress K is proposed in the following form
ð31Þ
where E is an elasticity modulus and et is an applied total strain. The unified viscoplasticity model proposed in equations (6), (26), (29)–(31) provides all the required features of material behaviour under constant and monotonic loading, as illustrated in Figures 12 and 15, and hardening behaviour under cyclic loading, as illustrated in Figure 15. For convenience, the resulting hysteresis loop illustrating anisotropic hardening can be decomposed into two idealized hysteresis loops presenting the separate influence of kinematic backstress K and isotropic parameter H on the basic constitutive model (equation (6)).
Gorash et al. Kinematic backstress K
239
σ
Isotropic parameter H
σ
Combination σ of H and K ε
ε Non-linear Kinematic Hardening
ε
Bilinear Isotropic Hardening
Anisotropic Hardening
Figure 15. Schematic representations of hardening behaviour with idealized hysteresis loops.
Finally, the constitutive equation (6) of the proposed model can be coupled with creep damage (12) and fatigue damage (18) evolution equations using the strain equivalence principle (19). In the case of creep damage, the scalar parameter vcr influences the accumulation of inelastic strain ein resulting in the tertiary creep stage of the creep curve. In the case of fatigue damage, the influence of the scalar parameter vf on ein results in the fast reduction of the last hysteresis loops before the fracture.
Verification of the model by uniaxial tests Referring to Lemaitre and Chaboche34 for a qualitative analysis, several characteristic tests have to be performed to describe the phenomenology of viscoplastic materials at high temperature. The classical characteristic tests are essentially conducted in simple tension or tension–compression at constant temperature. The specimen is subjected to an axial load (force or displacement) which produces a uniform state of stress or strain within the whole useful volume of the specimen. The most general types of uniaxial tests required to characterize a material and to verify a corresponding phenomenological model are creep, relaxation, hardening and cyclic tests. These types of tests can be conducted in different combinations with different duration for the complex investigation of material behaviour. If the tests, with the exception of relaxation, are conducted
300
over an undefined period of time, then the fracture of the specimen is achieved and the corresponding characteristics of material can be obtained, including stress and strain at fracture, time or number of cycles to fracture and energy dissipated in fracture. Therefore, simulation results obtained by the unified viscoplasticity model proposed in equations (6), (26), (29)–(31) with appropriate creep material constants have to be compared with all the above-mentioned types of uniaxial experimental data for the steel AISI type 316 at 600 °C. The values of creep material constants A = 2:335 1010 1=h and B = 0:053 1=MPa for constitutive equation (6) were defined in the section ‘Phenomenological approach’ by fitting the available creep curves.5,37 The value of elasticity modulus E = 150;000 MPa is taken from guidance13 on the basic mechanical properties of the steel. The values of hardening material constants C1 = 20, h = 0:35 in equation (29) and C2 = 1100, s0 = 170 MPa in equation (30) were defined in the first iteration by the optical fitting of the first 1.25 loop from the experiment,39 conducted under 0:3% of et amplitude with strain rate e_t = 2:66 104 1=s. The experimental data of Gong et al.39 were optimally fitted by the Chaboche unified viscoplasticity model with the set of 10 corresponding material constants also provided by Gong et al.,39 as illustrated in Figure 16. According to comparison of the cyclic simulation results for the two unified models in Figure 16, the Chaboche model provides much better results for cyclic stress response and the form of the saturated loop after 50 cycles. However, for the available experimental data of relaxation41 and hardening tests42 the proposed model provides much more preferable simulation results than the Chaboche model. The relaxation curve obtained by the proposed model, corresponding to the value of total strain et = 0:003 achieved with the stain rate e_t = 8 106 1=s, provides the necessary rate of stress decrease, which coincides with experimental relaxation curve,41 as shown in Figure 17. The stress–strain
400
350
(a)
(c)
(b)
200
0 -100
175
Stress (MPa)
Stress (MPa)
Stress (MPa)
300 100
Total strain: ε t = 0.006 Total strain rate: ε˙t = 2.66 · 10− 4 1/s
200
100
-175
Experiment [39]
-200
0
Chaboche model Proposed model
-300 -3 -3 -4 10 -2 10
0
-3
2 10
Total strain
4 10
-3
0
0
10
20
30
Number of cycles
40
50
-350 -3 -3 -4 10 -2 10
0
2 10
-3
4 10
-3
Total strain
Figure 16. Fitting of the model equations (6), (26), (29)–(31) to the uniaxial strain-controlled cyclic experiment:39 (a) first 1.25 loop, (b) cyclic stress response and (c) saturated loop.
240
Journal of Strain Analysis 47(4) 250
-5
1 x 10
Strain rate (1/h)
Stress (MPa)
-6
Experiment [41]
200
Chaboche model Proposed model
150
100
1 x 10
-7
1 x 10
Tests [5]:
-8
1 x 10
170 MPa 120 MPa 100 MPa 80 MPa 60 MPa
-9
1 x 10
Total strain: ε t = 0.003 Total strain rate: ε˙t = 8 · 10− 6 1/s
50
0 0.01
0.1
1
10
-4
4 x 10
1 x 10
-3
Total strain
Proposed model:
170 MPa 120 MPa 100 MPa 80 MPa 60 MPa
0.01
0.1
Figure 19. Comparison of the model simulation results for the strain rate from creep tests.5
100
1000
10
4
10
5 0.01
Time (h)
diagram obtained by the proposed model, corresponding to the value of stain rate e_t = 0:02 1=min = 3:33 104 1=s selected regarding the EN 10002-5 standard, provides the necessary level of yield stress sy and hardening slope of the experimental stress–strain diagram,42 but slightly overestimates it, as shown in Figure 18. Finally, the creep test simulation results obtained by both models are compared with experimental strain rate versus strain dependencies extracted from two sets of experimental creep curves. The first set includes creep curves in the low stress range (60–170 MPa) provided by Rieth et al.5 and shown in Figure 19, and the second set includes creep curves in the high stress range
Stress (MPa)
400
300 Experiment [42] Chaboche model Proposed model
200
100
0
Total strain rate: ε˙t = 0.02 1/min = 3.33 · 10− 4 1/s
0
0.02
0.04
0.06
0.08
0.1
Total strain
Figure 18. Comparison of the model simulation results for the hardening test.42
1 x 10
Strain rate (1/h)
Figure 17. Comparison of the model simulation results for the relaxation test.41
-3
Tests [37]:
1 x 10
-4
1 x 10
-5
1 x 10
-6
300 MPa 270 MPa 255 MPa 230 MPa 200 MPa
K
Proposed model:
0.01
300 MPa 270 MPa 255 MPa 230 MPa 200 MPa 0.02
H
Total strain
ε˙cr min 0.07
0.1
0.2
0.3
Figure 20. Comparison of the model simulation results for the strain rate from creep tests.37
(200–300 MPa) provided by Chaboche37 and shown in Figure 20. The Chaboche model with corresponding material constants does not provide any reasonable creep simulation results for either set of creep curves. The creep simulation results presented in Figures 19 and 20 provided by the proposed model are far from optimal matching of experiments. The simulated inelastic strains ein overestimate the experimental strains before reaching e_cr min for low stresses and underestimate the experimental strains for high stresses. The positive features of the obtained simulation results include two facts observed in Figures 19 and 20. First, the required values of e_cr min are reached for the complete stress range with the exception of some moderate stresses. Second, the simulation curves for low stress range demonstrate the separate influence of kinematic and isotropic hardening resulting in smooth changing of the curves’ slope before reaching e_cr min , as was assumed in Figure 14. This assumption is confirmed by the form of experimental curve corresponding to 200 MPa in Figure 20, which demonstrates the influence of different hardening types. Therefore, the creep simulation results may be improved by setting the saturation constants s0 and h corresponding to kinematic backstress K and isotropic parameter H as functions depending on the applied stress s. Then, these functions can be separately fitted to
Gorash et al. corresponding segments of the experimental creep curves. The fitting to experimental data from one laboratory would also improve simulation results.
241 09452, which resulted in the PhD thesis21 that became the basis of this work.
References
Conclusions The CFCS of components used in the Eddystone unit no. 1 power station have been selected for the application of creep lifetime assessment using a CDM approach. An accurate modelling of the selected components failure requires accounting of not only the creep process, but also processes caused by transient operation mode, such as hardening and relaxation leading to LCF. Thermal, elastic, plastic, low-cycle fatigue and creep properties of the steel AISI type 316 have been investigated and described for a wide temperature range. The CDM-based creep–fatigue model has been formulated to describe the high-temperature inelastic material behaviour using a conventional approach. Model material constants for the steel AISI type 316 have been identified from creep– and fatigue–rupture experiments. The comprehensive FEM simulation of MSP in service conditions has been implemented in ABAQUS applying a conventional model. The results of the MSP FEM simulation comply with a conservative locus on the creep–fatigue damage interaction diagram. In order to take into account the varying mechanical loading conditions in a wide stress range a Sinh-based creep model has been extended by the introduction of the relatively fast saturating backstress K describing kinematic hardening and relatively slow saturating parameter H describing isotropic hardening. The combination of both variables in the constitutive model produces anisotropic hardening effects as responses on inelastic strains. The model simulation results satisfactorily comply with cyclic stress response, creep curves, stress–strain diagram and relaxation curve at 600 °C. In comparison to the unified Chaboche model, the proposed unified model has fewer material constants (7 versus 10) and provides better results in modelling of creep and relaxation. However, the proposed model requires some significant improvements, such as the introduction of the stress dependence for material constants. The hardening evolution equations have to be extended with the temperature dependence. Then the proposed model equations have to be formulated in multiaxial form in order to develop the UMAT subroutine for ABAQUS FE code. Finally, the proposed comprehensive approach has to be verified using FE simulation of the components from the Eddystone unit no. 1 failure case study. Funding This work was supported by the German Academic Exchange Service (DAAD) under research grant no. A/10/05422. Moreover, deep appreciation to the DAAD is expressed for the study scholarship no. A/06/
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Appendix Notation A, B, a, b c C, c, n C1 , C2 D, m, l E F, G, q H I K k N Q R s T t X x, y z, r
secondary creep material constants specific heat power-law creep material constants hardening saturation values damage material constants Young’s (elasticity) modulus fatigue material constants isotropic hardening parameter stress invariant kinematic hardening backstress thermal conductivity number of cycles activation energy universal gas constant stress deviator temperature time overstress damage parameter exponents axial and radial directions
a, b s s0 , h sv e v
weighting factors stress hardening saturation values viscous stress strain strain tensor damage parameter
Gorash et al. m r u
243 Poisson’s ratio density coef. of thermal expansion
Subscripts, superscripts cr el eff eq ins in
corresponding to fracture creep elastic effective equivalent instantaneous inelastic
I, II, III ln f fr max t pl pr pw t vM y
principal values linear fatigue fracture maximum tensile plastic corresponding to primary creep stage power law total von Mises yield
v
corresponding to damage
