A Review of Creep Analysis and Design Under Multi-Axial Stress States

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Engineeringand DesignELSEVER Nuclear Engineering and Design 237 (2007) 1969-1986

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A review of creep analysis and design under multi-axial stress states

Hua- Tang Yao, Fu-Zhen Xuan *, Zhengdong Wang, Shan-Tung Tu

School of Mechanical and Power Engineering, East China University of Science and Technology, 130. Meilong Street,

PO Box 402. Shanghai 200237, PR China

Received 18 August 2006; received in revised form lO Februar 2007; accepted 12 February 2007

Abstract

The existence of multi-axial states of stress cannot be avoided in elevated temperature components. It is essential to understand the associatedfailure mechanisms and to predict the lifetime in practice. Although meta creep has been studied for about 100 years, many problems are stilunsolved, in particular for those involving multi-axial stresses. In this work, a state-of-the-ar review of creep analysis and engineering designis carried out, with paricular emphasis on the effect of multi-axial stresses. The existing theories and creep design approaches are grouped intothree categories, i.e., the classical plastic theory (CPT) based approach, the cavity growth mechanism (CGM) based approach and the continuumdamage mechanics (CDM) based approach. Following above arangements, the constitutive equations and design criteria are addressed. In the end,challenges on the precise description of the multi-axial creep behavior and then improving the strength criteria in engineering design are presented.(i 2007 Elsevier B.Y. All rights reserved.

1. Introduction

With the increasing demand for reduced C02 emissions andimprovement of efficiency in energy conversion, higher oper-ating temperature and design stresses have been adopted inchemical and petrochemical plants, power generation systems,etc. The main concern for the strength design of componentshas thus moved to the viscoplastic performance of materials inorder to prevent creep failure. On the other hand, the existenceof multi-axial states of stress due to complexity of loadings andmaterials cannot be avoided. It is therefore essential to under-stand the multi-axial creep failure mechanisms and establish themulti-axial creep design criteria for the strength design and lifeprediction of high temperature components.

Over the past several decades, considerable efforts have beenmade to gain a fundamental understanding of creep mecha-nisms and to develop an efficient engineering design criterionfor high temperature components under multi-axial stress states.Because the multi-axial creep behavior is very similar to that ofclassical plasticity, the classical plastic theory (CPT) has beendirectly used in the multi-axial creep analysis during the first halfof the 20th century. However, creep is a time-dependent plas-

* Corresponding author. Tel. +86 2164253513; fax: +8621 64253425.E-mail address:fzxuaiiêecusLedu.cn (F.-Z. Xuan).

0029-5493/$ - see front matter iD 2007 Elsevier B.V. All rights reserved.doi: i O. I 0 16/j.iiucengdes.2007 .02.003

tic deformation under a fixed stress at temperatures of roughly0.3-O.5T ro, where Trois the melting temperature of metals. Themain cause of creep failure is the nucleation, growth and coales-cence of cavities on the grain boundaries (Leckie and Hayhurst,1984; Huddleston, 1985; Kassner and Hayes, 2003; Goodalland Skelton, 2004). The multi-axial creep design method basedon the classical plastic theory is limited in practical applicationbecause it is derived from the criteria of yielding failure and doesnot account for the physical daage process.

Staing from the innovative work of Hull and Rimmer

(1959), multi-axial creep design criteria using the modelsbased on cavity growth mechanisms (CGM) were established.Then, the CGM-based models were improved by a lot ofresearchers (Rice and Tracey, 1969; Hayhurst, 1972; Gurson,1977; Manjoine, 1975, 1982; Raj and Ashby, 1975; Ashby andEdward, 1978; Cocks and Ashby, 1980, 1982a,b; Edward andAshby, 1979; Cane, 1981a,b, 1982; Tvergaard and Needleman,i 984; Huddleston, 1985) from 1970s to 1980s. In recent years,further development has been made by Hales (1994), Margolinet a1. (1998), Spindler et al. (2001), Spindler (1994, 2004a,b) andRagab (2002). Some of these models have been applied in hightemperature strength design criteria or assessment procedures,e.g. R5 (Nuclear Electric pic, 1997), ASME BPYC-1I (1998),RCC-MR (1986) and Siemens AG Power Generation DesignCodes (Siemens PG) (Andreas, 2000), to include the effect ofmulti-axial states of stress on creep failure.

1970 H.-T. Yao et at. / Nuclear Engineering and Design 237 (2007) 1969-1986

Nomenclature

a, b, material parameters in Eq. (38)

A, B, C, D, m, n material parametersA', B', C', nl, ml, bl material parameters In classical

Kachanov-Robotnov equationsA", B",p", q", l! material parameters in multi-axial creep

equations with single state varableAI!, BII, C'11, Dill, h'li, H*, Kc material parameters in multi-

axial creep equations with multi state variablesmaterial parameters in Eq. (37)grain diameter (m)grain boundary diffusion coefficient (m2/s)chemical potential of vacancy (1)cavity area fraction at tc, taken as 0.25cavity area fraction

cavity area fraction at tnmaterial parameter defined by Eg. (24)the value of g corresponding to simple tensionmaterial parameter in Eq. (45)primary creep state variable (0 -c H -c H*)activation energy (1)saturation values of H at the end of primar creepfirst stress invarant (MPa)loading state parameter (N = 1 for a¡ tensile andN=O for a¡ compressive)material parameters in Eq. (31)material parameter in Eqs. (32) and (33)Boltzmann's constant (11K)triaxial factor defined by Eg. (46)deviatoric stress tensormaximum deviatoric stress (MPa)time (s)time to coalesce void (s)time to nucleate void (s)time to rupture (s)temperature COC)

melting temperature of metals (0C)

c, l, kdDgVf

fcfl1

fnggOgl, r

H-t:HH*

l¡N

p,qrRRv

5ij

51

ttc

tn

tr

TTm

Greek lettersa, ß material parameters in Eq. (39)

al material parameters in Eq. (45)

8ij Kronecker Delta

8z grain boundar width (m)

êc creep strainêe effective strainêf uniaxial failure strainêij creep strain tensor

êr creep rupture strain

êi, ê2, ê3 principal strainsêf multi-axial failure strain£c creep strain rate (çI)£e effective creep strain rate (s- I)£¡ initial creep strain rate (s-I)£ss steady state creep strain rate (Ç I)£1, £2, h principal strain rates (s-¡)

y surface tension (N/m)

YI, Yi, Y3 shear strain rates (s -I)À creep damage tolerance parameter

v Poisson ratio

P radius of void (m)Po initial radius of void (m)p", a" material parameters in multi-axial creep equa-

tions with single state variablea uniaxial stress (MPa)

ae efficient stress (MFa)a ij stress tensor

am hydrostatic stress (MPa)ar creep rupture stress (MFa)

ai, a2, a3 principal stresses (MPa)

u von Mises equivalent stress (MPa)

7:¡, 7:2, 7:3 shear stresses (MPa)

v stress index in Eq. (36)

vl/ stress index in multi-axial creep equations with

multi state variablesW damage state varable (0 -c w -c i)Wcr critical damage state varableW¡ cavitation damage state varable (0 -c Wi -c 1/3)W2 precipitate coarsening state variable (0 -c W2 -c 1)W3 mobile dislocation multiplication state varable

(O-cw3d)Q atomic volume (m3)i/ energy dissipation rate potential defined by Eq.

(53)

subscriptsc creepc coalescencecr criticale effcientn nucleationr rupturess steady state

The third important method for multi-axial creep design,continuum damage mechanics (CDM) based method, was devel-oped from the initial work of Kachanov (1958). Contrasted withthe CGM-based multi-axial creep design method, the CDM-based method is developed on the phenomenological way andpresented from the viewpoints of mechanics. With the rapiddevelopment of modern computer technology and finite elementanalysis method, CDM-based method has been focused againduring recent years (Othman et aI., 1993; Hayhurst et aI., 1994;Kowalewski et aI., 1994a,b; Perrn and Hayhurst, 1996, 1999;Hyde et aI., 1996, 2004, 2006; Hsiao and Gibbons, i 999; lingei aI., 2001a,b, 2003; Xu, 200a,b, 2001, 2004; Wang and Guo,2005; Hayhurst et aI., 2005a,b; Lin et aI., 2oo5a; Mustata andHayhurst, 2005).

The purpose of this paper is to present a state-of-the-artreview of creep design methods or criteria under multi-axial

H.-T. Yao et at. / Nuclear Engineering and Design 237 (2007) 1969-1986

states of stress. Since the subject is quite old and there are manypublished papers, it is impossible to conduct this review in acomprehensive manner. Also, many historical review papers areavailable on CPT-based method in books or monographs (Har,1980: Boyle and Spence, 1983; Penny and Marott, 1995).

Therefore, this review places parcular emphasis on the devel-opments of CGM-based and CDM-based methods. In the end,potential problems to be resolved in the near future are pointedout.

2. CPT-based multi-axial creep design method

In the strength design and remaining-life assessment of hightemperature components, the information and data obtainedfrom a tensile specimen under constant load are extensively useddue to be obtained easily in laboratory. For the case of multi-axialstress states, the effective stress criteria governing creep strainwas proposed on the basis of the results of uniaxial creep test. Todo this, some hypotheses and concepts developed in the theoryof instantaneous plastic deformation were introduced again inCPT-based method.

2.1. Uniaxial creep constitutive relationship

The deformation of a tensile specimen under constant loaddepends on stress a, time t and temperature T. Consequently, thecreep strain of materials can be written as (Harry, i 980; Boyleand Spence, 1983: Penny and Mariott, 1995):

£c = f(() t, T),

where £c is creep strain. Eq. (1) is usually assumed to be sepa-rated into:

£c = f¡(a)h(i)f3(T).

The stress and time dependence of creep under constat stresshas received considerable attention. As a result, there are anumber of altemative expressions (Kennedy, 1962: Penny andMarriott, i 995). The most commonly used function of stress isthe power law attributed to Norton (1929):

fi(a) = Aan,

where A and n are material constants. The function is popularfor its simplicity in application to stress analysis. An importnttime function is the so-called Bailey law (1935):

h(t) = Dtm(1¡3:: m:: i¡2usually),

where D and m are material constats. According to Arrhenius'slaw, the temperature dependence is given as (Dorn, 1955):

(-.6H)h(T) = C exp -- '

where -.6H is activation energy and R is Boltzman's constat.Then the creep strain may be written as:

(-.6H)£c = B exp -- tm all .

1971

For isothermal conditions, the creep strain is:

£c = Btm~, (7)

which is known as Norton-Bailey law and commonly used increep analysis (Harry, 1980).

Eq. (7) is developed for the constant stress and can describethe primar, secondar or tertar creep stages. For varying

stresses, differentiating the Eq. (7), we have

£c = mBtm-¡all. (8)

Removing the time variable t from Eq. (7) to Eg. (8), it canbe obtained:

£c = mB1lmaii/m £~m-l)/m. (9)

Eqs. (8) and (9) are the rate forms which can be used to modelthe primar creep of decreasing creep strain rate. This processis usually called hardening (Josef, 2003). For example, the Eq.(8) is called time hardening because it models hardening phaseusing time parameter. Eq. (9) is called strain hardening becauseit contains stress and strain as variables. Eqs. (7), (8) and (9) arethe accustomed creep constitutive relationships under uniaxialstress conditions.

2.2. Multi-axial creep constitutive relationship

(1)

Experimental results of multi-axial creep test indicate thatcreep is a shear-dominated process for isotropic and homoge-neous materials. Thus the following assumptions were made(Boyle and Spence, 1983; Viswanathan, 1989):

(2)

(1) Constant volume is maintaned during creep, so the rate ofvolume creep strain is zero:

£1 + £2 + £3 = 0, (10)

where £¡, £2 and £3 are principal strain rates.

(2) The principal shear strain rates are proportional to the prin-cipal shear stresses:

(3)

YI = Yi = h = 2 if,Ti T2 T3

where if is a constat; Yi, Yi and h are principal shear strainrates; Tl, T2 and T3 are principal shear stresses and:

(11)

(4)

a2-a3T¡ = --'

Yi = £2 - £3,

al-a2T3 =--'

Y2 = t¡ - £2,

0'3 -a¡T2=-,2

Y2=£3-£I,

(3) The effective strain rate £e is related to the effective stressin the same way as the uniaxial relation, e.g.:

(5)£e = g(t)f(ã), (12)

(6)

where ã is von Mises equivalent stress defined in the sameway as in plasticity theory:

ã = ~ V(ai - a2)2 + (a3 - a2)2 + (ai - a3)2. (13)


In terms of Eqs. (i 0)-( 12), the multi-axial creep constitutiverelationship is given by:

. 3 f(ã)81 = '2 g(t)--(a¡ - am),

3 f(ã)£3 = '2 g(t)--(a3 - am),

where am is the hydrostatic stress and defined by:

3 f(ã)£2 = '2 g(t)--(a2 - am),

0'1 + a2 + 0'3O'm =

3

Under steady stress conditions, i.e., g(t) = i, the constitu-tive relation (14) reduces to:

. 3 f(ã)81 = '2--(a¡ - am),

. 3 f(ã)83 = '2--(a3 - am). (16)A lot of examples for Eq. (16) in analyzing the steady statecreep of high temperature components can be found in thebooks by Boyle and Spence (1983), Harry (1980), and Pennyand Marriott (i 995), etc.

. 3 f(ã)82 = '2--(a2 - am),

3. CGM-based multi-axial creep design method

The studies on microstructural evolution of materials underexternal forces, e.g. stress and temperature, are a central sub-ject of materials science (Honeycombe, i 985; Loretto, 1985;Humpreys, i 996; Fabrizio et aI., 2002). From observations ata microscopic level, materials always contan some defects,e.g. dislocations (line-defect), grain boundaries and phaseboundaries (two-dimensional defects), inclusions and voids(three-dimensional defects). These microscopic discontinuitieswill develop and finally cause damage and failure of compo-nents. It is realized that failure of most components operated athigh temperature is caused by the nucleation, growth and coales-cence of cavities (Hales, 1994; K won et aI., 2000; Kassner andHayes, 2003; Michel, 2004). It is therefore necessar to under-stand the cavity growth mechanisms so as to establish a propermulti-axial creep design criterion.

3.1. Introduction to cavity growth theory

Cavity growth mechanisms have been used to model the dam-age resulting from the deformation of high temperature creep,cold metal forming, superplastic formng and hot metal forming(Lin ct aI., 2005b). Nucleation and growth process of cavitiesvares under different loadings and temperatures. In cold form-ing processes of metals, for example, the voids normally nucleatearound second-phase parcles and within the grains, as shown inFig. la. In creeping deformation processes, however, the voidsusually nucleate along the grain boundar, as shown in Fig. 1 b.

3.1.1. Process of creep cavity growthSmall voids are often observed at grain boundar, paricu-

larly transverse to the applied stress during creep tests. Thereare several methods whereby voids might be nucleated (Hull

(14)(a) (b)

(15)

Fig. 1. Scheme of voids nucleation (Lin, 2oo5b). (a) Voids nucleation in coldmeta formng process. (b) Voids nucleation in creep deformation process.

and Rimmer, i 959; Edward and Ashby, i 979; Nicolaou et aI.,2000; Kassner and Hayes, 2003). Impurity paricles which lack

cohesion with the matrix may act as voids. A dislocation pile-up breaking through the boundar might also be a suitablenucleus. Additionally, it has been shown by Edward and Ashby(1979) that grain boundar sliding is an important cause forvoids nucleation at grain boundaies. Grain boundar slidingcan be accommodated in varous ways: elastic accommodation,as shown in grain 1 in Fig. 2; diffusion flow, as shown in grain 2;or plastic flow, as shown in grain 3. Ifche incompatibility causedby grain bounda sliding cannot be accommodate in any ofthese ways, voids wil appear and grow at grain boundar, asshown in grain 4.

The nucleation and growth of voids reduce the load bear-ing section and accelerate the creep daage and this, in tur,

increases the void grow rate. When the voids grow from an ini-tial size to half the mean cavity spacing, the coalescence of voidsoccurs, as shown in Fig. 3c.

3.1.2. Creep cavity growth and creep failure

A stadard creep cure is shown in Fig. 4. According to Fig. 4,after the instantaneous elastic strain, there are thee creep stages:a decelerating strain rate stage I (primar creep), a steady min-imum strain rate stage II (secondar creep) and an acceleratingstrain rate stage II (tertiar creep). Studies have shown that thenucleation of cavities usually occurs during the creep stage I andII (Kassner and Hayes, 2003), and their growth and coalescence

. .i !

+i t tt t

¡

.,I, l

i

., ll ¡..

Fig. 2. Voids nucleation caused by grain bounda sliding (Edward and Ashby,1979).

H.-T. Yao et at. / Nuclear Engineering and Design 237 (2007) 1969-1986 1973

(a) (b) (el

Fig. 3. The process of creep cavity growth: (a) nucleation, (b) growth, (c)coalescence.

ruptue1 1~ -;------------------------- --C\

II II :¡.----I11111111

.g .5:---:---v: ro i I.¡ .¡ 1 11tr

-ï----------------111

Fig. 4. Scheme of the standard creep curve.

lead to the creep stage II when creep ruptue occurs becauseof a significant increase of net section stress resulting from thedecrease of the load bearing section. The ruptue strain, Br, atrupture time, tr, represents the rupture ductility of materiaL.

Nucleation and growth of cavities usually occur during mostof the creep life, as shown in Fig. 5. This has been verified

by Cane (1981 a) on 2.25CrlMo and 0.5CrMo V steels, andSklenieka et al. (2003) on 9-12% Cr steels. Furtermore, thetime taken up by cavity growth is often much longer than thatby cavity nucleation (Cane, 1981 a; Sklenieka et aI., 2003). Sothe creep failure of components operated at elevated tempera-ture is usually controlled by creep cavity growth and studies oncavity growth mechanisms have received more attentions.

3.2. Cavity growth mechanisms and models

There have been, in the past, a lot of publications in whichcavity growth mechanisms were discussed (Evans, 1984; Riedel,1987; Delph, 2002; Kassncr and Hayes, 2003; Michel, 2004).It is believed that cavity growth rates is a complex couplingfunction of plastic strain of surrounding grains and vacancy dif-fusion along grain boundaries (Michel, 2004). Cavity growthmechanisms are usually grouped into thee categories: plasticity-controlled cavity growth, diffsion-controlled cavity growth andconstrained cavity growth (Hales, 1994; Michcl, 2004). The

rroughly 0 -+ 0.8 tr

.Aroughly 0.8 -+ 1.0 tr

.ArCavity nucleation -+ Cavity coalescence

and growth and creep rupture

Fig. 5. Scheme of the intergranular damage development (Sklenieka et ai.,2003).

a

-----0----x

Fig. 6. The square aray of cavities in the Hull-Rimmer analysis (Hull andRimmer, 1959).

CGM-based models have been used by many researchers to pre-dict the influence of multi-axial stress on the creep failure strainor creep rupture time (Hull and Rimmcr, i 959; McClintock,1968; Rice and Tracey, 1969; Hayhurst, 1972; Cocks and Ashby,1980; Cane, 1981a;Manjoine, 1975, 1982; Margolinetal., 1998;

Ragab, 2002; Spindler, 1994; Spindler et aI., 2001).

3.2.1. Difusion-controlled cavity growth mechanism andmodel

Hull and Rimmer (1959) was one of the first, later followed byRaj and Ashby (1975) and Speight and Beere (1975), to proposea mechanism by which diffusion leads to cavity growth. Accord-ing to this mechanism, the growth rate of cavities is influencedby the shape of voids and the diffusion process; the growth rateof a void is determined by the gradient of chemical potentialof vacancies, 'Vi, in the plane of the grain boundares. Vacanciesmigrate under the influence of the gradient and lead to the growthof voids.

The first cavity growth model based on diffusion-controlledcavity growth mechanism was proposed by Hull and Rimmer(1959). It was used to predict the effect of combined hydrostaticpressure, am, and uniaxial tension, a, on the rupture time of a

copper wire in the temperature range 400500°C. The modelis shown in Fig. 6. It is a square aray of spherical voids witha radius p, lying on a grain boundar normal to the directionof applied stress a. In order to calculate the chemical potentialof vacancies 'Vf at the void surface, assumptions were madeby Hull and Rimmer (1959): the ring of void surface lies inthe grain boundar; the atoms are deposited uniformly over thegrain boundar and the voids retain spherical shape during thegrowth.

Using the model and assumptions, the rate of change of voidradius is given by (Hull and Rimmer, 1959):

dp _ 2rrQ(Dgoz)(a am - 2yj p)Sdt - RTap (17)

where Q is atomic volume, Dg the grain boundar diffusioncoeffcient, p the void radius, Oz the grain boundary width, y thesurace tension of metals, R the Boltzann's constant and S isthe function of pIa. Assuming the initial radius of voids po ~~ a

t é


failure strain £f and uniaxial failure strain I'f is obtained:

1974

Xi

Xl

x.

Fig. 7. The model of spherical voids in a remote simple tension strain rate fieldproposed by Rice and Tracey (1969).

and integrating Eq. (17), the rupture time is therefore obtained:

kTatr =

2:nQ(Dgoz)(a - am)

ta2 ya 4y2 (a(a - O'm)/2 - 2Y)1x -+ + In .

8 (a - am) (a - am)2 PO(a - am) - 2y

(18)

3.2.2. Plasticity-controlled cavity growth mechanism andmodel

Generally speaking, diffusion-controlled growth dominateswhen the cavity size is very small. As the cavity sizeincreases, diffusion-controlled growth mechanism decreasesquickly, and the plasticity-controlled growth becomes predom-inant (Nicolaou et aI., 2000). According to plasticity-controlledgrowth model, cavity growth is resulting from the plastic defor-mation of the surrounding materials (Rice and Tracey, 1969;

Hancock, 1976). The mechanism becomes more importnt underhigh strain-rate conditions where significant strain is observed.The mechanism was first proposed by McClintock (1968) andRice and Tracey (1969), and has been recently improved byKhaleel et al. (2001) and Taylor et al. (2002) to model thesuperplasticity.

McClintock (1968) has proposed a model based on plasticity-controlled growth mechanism to investigate the relation betweenthe cavity growth and imposed stress and strain. The model is acylinder cavity in a non-hardening materiaL. Then a more real-istic model of an isolated spherical void in a remote-uniformedstress and strain rate field, as shown in Fig. 7, was presented byRice and Tracey (1969) to determine the relation between voidgrowth and stress triaxiality.

According to Rice-Tracey model, the ratio of the averagestrain rate of sphere radii to the remote-imposed strain rate, D,is given by (Rice and Tracey, 1969):

(3am) (3am)D = 0.558 sinh 25 + 0.008 cosh 25' (19)

Neglecting the second term on the right hand side of Eq. (19),an equation describing the relationship between the multi-axial

1'; = 0.521

I'f sinh(3am/25)

A model similar to Fig. 8 but with a long cylindrical void in aremote simple strain rate field leads to another equation similarto Eq. (20):

(20)

£f = exp (~ _ 3~n) .I'f 2 2a (21)

3.2.3. Constrained cavity growth mechanism and modelConstrained cavity growth model was first proposed by

Dyson (1976), then developed by Cocks and Ashby (1980,1982a,b), Edward and Ashby (1979), Tvergaard (1984),Yousefiani et al. (2000), and Delph (2002). The cavity growthmechanism occurs when the local deformation rate exceeds thedeformation rate of surrounding materials due to cavity growth.Thus the cavity growth rate is constrained to produce the localstrain at the same rate as the deformation caused by the remotestress. Three conditions under which cavities grow by con-strained mechanism have been analyzed by Edward and Ashby(1979) versus a decreasing size. They are: one par of the struc-ture when its deformation is non-uniform, as shown in Fig. 8a;cavities on the boundares when less than half the boundares arecavitated, as shown in Fig. 8b; and material ligament betweencavities when the diffsion field of a growing cavity does notextends half way to the next cavity, as shown in Fig. 8c.

Cocks and Ashby (1980) proposed a model based on theconstrained cavity growth mechanism. This model is an isolatedcylinder with a hole and an outer diameter of 2l, as shown inFig. 9. The following hypotheses were introduced by Cocks andAshby: grain boundary cavities nucleate at inclusions at timet=tn, and then grow following the power law rule until they

(b)

(c)

Fig. 8. Three examples of constrained voids growth proposed by Edward andAshby (1979).

t7, t7"

H.-T. Yao et at. / Nuclear Engineering and Design 237 (2007) 1969-1986 1975

0:, t7"

Fig. 9. A model of voids growing on grain boundar proposed by Cocks andAshby (1980).

coalescence at time t= tc; cavities keep spherical during theirgrowth.

Based on the above model and hypotheses, a set of differentialequations describing the void growth and creep strain rate wereobtained:

d~, = g l(1 _lfh)n - (1 - fh)J £ss,

d:tc t 1 + g 2: l(1_1fh)'1 - 1 D £ss.

where £ss denotes the steady creep rate without cavities; d thegrain diameter;fh the cavity area fraction; n is material constantsand g is defined as:

. ht2n-1/2 (am)jg = SIn - .n + 112 Õ'

Integrating Eq. (22) at constant stress in terms of limits

fh = fn, t = tn, fh = fc, t = te,

where fn is the cavity area fraction when cavities nucleate, and

fc the value offh when cavities coalescence occurs and closes to0.25. The time to fracture is then given by (Cocks and Ashby,1980):

(1 - (1 fc1l+1)tc = tn + In(n + l)g£ss (1 - (1 -

3.3. Applications of cavity growth theory in design criteria

Presently, there are a number of assessment procedures anddesign codes for high temperature components, e.g. British R5,American ASME II, French RCC-MR, German TRD (Tech-nical Rules for Steam Boilers) and SIEMENS PG (SiemensAG Power Generation Design Rules). Cavity growth theory andmodels are adopted in these procedures and codes and multi-axial creep deformation criteria (MCDC) and multi-axial creeprupture criteria (MCRC) are therefore established to analyze thecreep behavior of high temperature materials under multi-axialstress states.

3.3.1. Applications of cavity growth theory in McDcAccording to Eq. (26) (through assuming tn =0), the rela-

tionship between the failure strain of multi-axial stress statesand that of uniaxial stress is given by:

*. I (l-(1-fe)Il+I)£f = tc£ss = g(n + I) In (1 (1 - fn)Il+ll (27)

Under uniaxial stress condition, replacing am I Õ' in Eq. (24) by1/3, we have

gO = sinh r2n -1/2 (~)J .l n + 1/2 3Replacing g by go in Eg. (27), failure strain under uniaxial loadcan be obtained:

(28)

1 (l-(1-fc)Il+¡J~= ~ .go(n + 1) (i - (1 - fn)n+¡J

By Eqs. (27) and (29), we have:

(29)

£* gO.. =£f g

sinh((2/3)((n - 0.5)/(n + 0.5)))sinh(((2(n - 0.5))/(n + 0.5))(amlt7e))'

(30)=

(22)The Eg. (30) can be used to describe the relationship betweenthe multi-axial failure strain a and uniaxial failure strain £f Theright hand side of the Eq. (30) is defined as a creep damage factorF CK and applied for depicting the influence of multi-axial states

of stress on creep ductility of materials (Andreas, 200). It hasbeen showed by Kwon et al. (2000) that the model can predctwell the creep ductility of Durehete 1055 steel under multi-axialstress states. The study by Yao et al. (2006) has also found thatthe model can predict well the multi-axial creep ductility of GX12CrMoWVNb 10-1-1 steel.

In British R5 procedure, a ductility exhaustion approach isused to assess creep damage. A model for the effect of multi-axial stress states on creep ductility, which is intended to beused when the ductility is a function of the creep strain rate, isincluded. The model was proposed and developed by Spindler(1994), Spindler et al. (2001) and Spindler (2004a,b) on Type316 and Type 304 stainless steels from biaxial creep data andtaes the following form:

£; L ( a¡ ) ( 1 3am ) J- = exp p 1 - - + q - - - ,£f ae 2 2ae

(23)

(24)

(25)

(31)

(26)where p and q are material parameters. Spindler et al. (200 1)and Spindler (2004a) suggests that p = 2.38 and q = 1.04 whenfailure strain is a function of creep strain rate and p = 0.15 andq = 1.25 when failure strain is independent of creep strain rate.The first term and second term on the right side of Eq. (31)reflect the influences of cavity nucleation and growth on thecreep failure behavior of materials, respectively. It is clear that

Eg. (31) reduces to Eg. (21) whenp=O and q= 1.It is wort noting that Egs. (20) and (21) are also applicable

for the prediction of multi-axial creep ductility. Similar workwas done by Hales (1994):

a = ( Õ' ) r+l,£f ai (32)


Sf = 2a¡ ( ã ) r+ ¡Sf 3S¡ a¡ ,where r is the material parameter. Eq. (32) is developed fordiffusion-controlled cavity growth and Eq. (33) is for con-strained cavity growth.

3.3.2. Applications of cavity growth theory in MeReThe creep rupture time of a high temperature component

depends on stress and temperature (Webster et aI., 2004). Underuniaxial stress condition, rupture time tr can be expressed as afunction of temperature T and uniaxial stress a:

tr = f(T, a).

Under multi-axial stress states, tr is given by:

tr = f(T, ae).

The effcient stress a e in this function depends on ã, a¡ and am'A common explicit form for the function (35) is the power-lawrelation:

tr = C(T)a;V,

where v is a stress index and CCD is a temperature dependentconstant.

The study of Cane (1981a) on 2.25CrlMo and 0.5CrMoVshown that cavity nucleation was determined by the maximumprincipal stress ai, while cavity growth depended on the com-bination of a¡ and von Mises stress ã. Accordingly the effcientstress is expressed as a function of ai and ã:

c-l Ilkae = (a¡ a) ,

where c, land k are material parameters.Huddleston (1985, 1993) further presented an improved effi-

cient stress model, which was adopted in ASME II:

ae=~S¡(::ir eXP(b(~: -1)),

where J¡ is the maximum deviatoric stress defined byJ¡ =ai +a2 +a3; Si is the first invariant of the stress ten-sor defined as Si = a¡ - J¡/3 and Ss is defined as Ss =

J a? + ai + a~; a and b are coefficients determned from curefitting. Pressurized tube tests showed that the model is moreaccurate than the classic criteria ofvon Mises, Tresca or Rankinein predicting the creep ruptue of 304 stanless under multi-axialstress states (Huddleston, 1985).

In RCC-MR, the efficient stress is calculated by:

(38)

ae = aai + 3ßam + (1 - a - ß)ã (0:: a + ß :: 1). (39)

where a and ß are material parameters. This model was pro-posed by Hayhurst (i 972) according to the plasticity-controlledcavity mechanism. Clearly, it is reduced to the maximum prin-cipal stress when a = 1 and ß = 0 and to von Mises stress whena = ß = O. When ß = 0, it is reduced to the model proposed bySdobyrev (1958):

ae = aa¡ + (1 - a)ã (0:: a :: 1). (40)

(33) böl.2

OJ"öl.2

Plasticity

Constrained

(b) loge(a) logi!

Fig. 10. The relationship between cavity growth mechanisms and rupture timeand rupture strain proposed by Hales (1994). (a) The effect of stress on time torupture. (b) The effect of strain rate on ductilty.

(34)Eq. (39) was adopted by Chellapandi et al. (2006) in the designof a prototype fast breeder reactor of 500 MW under 550°C.For the steam generator spigot of modified 9Cr-1Mo, efficientstress is given by:

(35)ae = O.4am + 0.867ã. (41 )

It is clearthat a = 0 and ß = 0.133 in Eg. (41).

(36)3.4. Summary

(37)

Cavity growth mechanisms and models and their applicationsin engineering have been discussed in this section.

For the three commonly accepted cavity growth mechanisms:plasticity-controlled growth, diffusion-controlled growth andconstrained cavity growth, the first and second are mainly dueto viscoplastic strain and diffusion respectively. However, con-strained cavity growth is the result of combination of viscoplasticstrain and diffusion.

Most multi-axial creep models were developed on one cavitygrowth mechanism (Hull and Rimmer, 1959; Rice and Tracey,1969; Hayhurst, 1972; Cocks and Ashby, 1980; Cane, 1981a).According to Beere (1981), the cavity growth rate in the wholecreep failure process was not always controlled by the fastestgrowth mechanism. A complete description of the effects of cav-ity growth mechanisms on rupture time and strain was developedby Hales (1994), as shown in Fig. 10. When the stress is low,the constrained cavity growth is predominant, and there is a lowstrain rate and a long time to failure. When the stress is high, theplasticity-controlled cavity growth mechanism is predominant,and there is a high strain rate and a short time to failure. Whenthe diffusion-controlled cavity mechanism is predominant, thereis an increasing rupture strain with the increasing creep strainrate.

Pessimisms in creep damage evaluation can be reducedsoundly when transitions between different mechanisms weretaken into account in modeling. Further improved models havebeen proposed by Margolin et al. (i 998), Spindler et al. (2001),Spindler (1994, 2004a,b), and Ragab (2002), etc., in which thecombined influence of cavity growth and cavity nucleation orcavity coalescence on multi-axial creep ductility is considered.

Cavity growth based models and their applications were gen-eralized in Fig. 11. In RCC-MR and ASME, models based onplasticity-controlled growth mechanism were used to calculatethe effcient stress. For the components in power plants, primarstresses are usually low and the failure is thus controlled by

H.-T. Yao et al. / Nuclear Engineering and Design 237 (2007) 1969-1986

Fig. ¡I. Models based on voids growth mechanism and their applications inengineering.

constrained cavity growth mechanism (Hales, 1994).

Cocks-Ashby model has been constrcted for SIEMENSPG. A less conservative model by Spindler including thecombined influence of cavity nucleation and growth wasadopted in R5 procedure.

The above assessment procedures or design rules generallylead to a conservative result. Nevertheless, the conservatism isusually acceptable in consideration of the safety requirements ofnuclear power stations, coal-fired power plants, and petroleumand chemical industries.

4. CDM-based multi-axial creep design method

CDM-based approach was used in the prediction of failuretime and rupture strain of high temperatue components by intro-ducing proper constitutive equations and damage variables. Theadvantage of the approach is that it is easily implemented withnumerical methods to simulate the process of damage evolutionand thus provide the information of the local stress and strainfield. The method has been highlighted again since 1990s withthe rapid development of computer technology and finite ele-ment method. A lot of multi-axial creep damage constitutiveequations have been presented to analyze the creep damage andfailure of different materials. In this section, a brief introductionis given to continuum damage mechanics. After this, a detaleddiscussion is carred out on the development of multi-axial creepconstitutive equations and their applications in engineering.

4.1. Creep damage theory and constitutive equations

4.1.1. 1ntroduction to continuum damage mechanicsAs discussed in Section 3, materials always contain some

micro defects from observation at a microscopic level, e.g.line-defects, two-dimensional defects, and three-dimensionaldefects. The existence of these micro defects is termed as damage

(Lemaitre, 1996). The process of these micro defects nucle-ation, growth and coalescence, which initiates the macrocracksand causes the progressive degradation of material propertes,is called damage evolution (Skrzypek and Ganczarski, 1999).On the microscale, material structure is piecewise discontinuousand heterogeneous because of the existence of micro-defects or

1977

damage. By introducing the damage variables based on equiva-lence principle and the concept of representative volume element

(RVE), the micro defects can be 'smeared out' and the stress andstrain state can be considered as homogeneous. This approachis known as the continuum damage mechanics (CDM).

The CDM was initiated by Kachanov (1958) for the caseof creep damage. A damage variable úJ varing from 0 for theundamaged matenal and i for the full broken material waspresented to depict the micro damage of materials. The inno-vative idea is a challenge to the traditional material mechanicsconcept 'perfect' and 'failure', and points out an intermediatestage existing between perfect and failure. Odqvist and Hult(1962) pointed out that Kachanov's concept implied the lifefraction rule of Robinson (1952). After that, a concept of effec-tive stress written as a e = al(1 - úJ) was proposed by Robotnov(1969), which makes it possible for the coupling of strainsand damages. Then, the strain equivalence principle was pre-sented by Lemaitre (1971), through the concept of effectivestress. It suggests that any constitutive equation for a dam-aged material may be derived in the same way as for a virginmaterial except that the usual stress is replaced by the effectivestress.

The further development was made by Hult (i 979), Chaboche

(1981) and Lemaitre (1985), who presented a method to deriveconstitutive laws based on the framework of irreversible thermo-dynamics and the principle of strain equivalence. In addition, totae damage-induced material anisotropy into account, vectorand tensor representations of daage variables were introduced(Murakami and Ohno, 1980; Krajcinovic and Fonseka, 1981 a,b;Betten, 1982; Krajcinovic, 1983; Murakami, 1983; Chaboche,i 982, 1984). The explosive development in this field can be seenfrom a large number of textbooks, monographs and reviews, forinstance, by Kachanov (1986), Chaboche (i 988a,b), Krajcinovic

(1989), Lemaitre (1996), Voyiadjis and Katta (1999), Skrzypekand Ganczarski (1999), Zhang and Vallappan (1998a,b), Kattan(2002), Allx et al. (2002), Lemaitre (2002), Lin et a1. (2005b)

and Hayhurst (2005), etc.CDM interlinks the experiences in micromechanics, con-

tinuum solid mechanics, physics, material science, numericalanalysis, etc. In many applications, CDM has become an essen-tial complement to fracture mechanics. The two branches ofsolid mechanics are complementar by the scale of the analysis:micro to meso for damage mechanics and meso to macro forfracture mechanics (Lemaitre, 2002). During the past decades,CDM has been used to descnbe different damage processes, e.g.creep (Murakami et aI., 2000; Becker et aI., 2002; Hayhurst etaI., 2005a,b; Mustata and Hayhurst, 2005), fatigue (Cheng andPlumtree, 1998; ling et aI., 2001a,b; Xiong and Shenoi, 2004;Dattoma et aI., 2006), creep-fatigue (ling et aI., 2003; Stolk etaI., 2004), ductile (Brunig, 2003; Gupta et aI., 2003; Pirondi etaI., 2006; Wu et al. 2006), and brittle (Vroonhoven and Borst,1999; Kaji et aI., 2001; Ciska and Skrzypek, 2004). The mate-rials include metals, concrete, rock, polymers, components, andceramics.

Here special focus is put on the creep damage analysis ofmetals and alloys under multi-axial stress conditions. Moreattentions are paid on the multi-axial creep damage constitutive

1978 H.-I: Yao et al. / Nuclear Engineering an Design 237 (2007) 1969-1986

equations using a scalar variable for damage measurement. Inthe cavity growth models discussed in Sections 3.2.1, 3.2.2and 3.2.3, the cavities are assumed as sphericaL. From the

geometrical viewpoint, the choice of a scalar variable forspherical cavities is a rational one (Krajcinovic, 1983). On theother hand, the assumption of isotropic damage is suffcientto give a good prediction of the carring capacity, or the time

to local failure in structure components. The calculations areeasily performed because of the scalar nature of the damagevariable (Lemaitre, i 984). Additionally, tensors offourth-orderor even eighth-order, eliminate any hope for the preservationof the physical clarity of the model (Krajcinovic, 1983). So theoriginal measurement on Kachanov damage is stil appealing,in paricular from a viewpoint of engineering application.

4.1.2. Creep damage constitutive equationsIn the design of high temperature components, consideration

must be given to the possibilities of failure due to excessive creepdeformation or creep rupture. Norton creep law can be used todescribe the stresses and deformation occurng in primary andsecond creep stages. However, for the life of components con-trolled by rupture, it is necessar to analyze their creep behaviorin the tertiar region of the creep curve.

To reflect the deterioration of materials and describe

the tertiar creep stage, Kachanov (1958) proposed a phe-

nomenological method. For uniaxial tension he suggested theconstitutive equations:

&c = f(rr T, w), w = g(a, T, w),

where f is the strain rate function and g denotes the function ofdamage rate. By the selection of functionsfand g, it is possibleto describe the tertiary region of creep curve and predict creeprupture life.

An explicit formulation for Eq. (42) is often given by:,

all& -AIc - (1 _ w)m"

p'w=B' a .(1 - w)a'

This is the so-called classical Kachanov-Robotnov Equations, inwhich A', B', nl, ml, p' and ql are time and temperature dependentmaterial constants and can be determined from uniaxial creeptension tests. Integrating the equation set under the conditionsof w = 0 when t = 0 and w = 1 when t = tr, the rupture time andstrain are obtained by:

1t -r - (1 + q')A'aP"

( (i t)I/ÀJt:r = t:R i - -i , (44)

where

p' + 1À = &0 = B'all, t:* = &ofr, t:R = Àt:*.p' + 1 - ql '

For the components subjected to complicated stress states,it is essential to generalize the creep damage constitutive equa-tions from uniaxial stress conditions to multi-axial stress. Theoriginal work was done by Leckie and Hayhurst (1974). Simpleformulae were developed to determine the time of crack initi-ation and a lower boundar on the rupture time of strctures.

Reference stress was introduced to take into account the effectof stress redistrbution and multi-axial stress states. However,this treatment is limited to kinematically determinate strctures.Also, it is applicable to other structures through the assumptionof stationar state pattem for deformation during creep rupture

process (Leckie and Hayhurst, 1974).

Then, a lot of multi-axial creep constitutive equations, withsingle or multi damage varables, were presented, for instance,by Hayhurst et al. (1975, 1984a, 1984b), Hayhurst (1975,

1983, 1984a, 1984b), Lemaitre (1979,1984,1985), Othman andHayhurst (1990), Kowalewski et al. (1994a,b), Hayhurst et al.(1994), Kowalewski et al. (i 994a,b), Perrin and Hayhurst (1996,1999), Hyde et al. (1996, 2004, 2006), Xu (20ooa,b, 2001,2004),etc. These constitutive equations were used to predict the creepfailure of different high temperature materials, e.g. nickel-basedsuperalloy, aluminum alloy, titanium alloy, austenitic stainlesssteel and ferritic steel, etc.

4.2. Multi-axial creep equations with single variable

(42)

In creep equations with single variable, a dominant dam-age parameter is defined to depict the state change of materialsand the performance degeneration of strctures. However, noattempt is made to identify the physical nature of the damageparameter and to distinguish between different damage mecha-nisms. The constitutive equations with single damage varableare applied where one damage mechanism dominates in thecreep rupture process. In general, there are two kinds of multi-axial creep damage constitutive equations with single varable:Kachanov-Robotnov constitutive equation and Lemaitre consti-tutive equation. The former can be regarded as the generalizationof classical Kachanov-Robotnov equation from uniaxial stressto multi-axial condition. The latter is based on the framework ofirreversible thermodynamics.

(43) 4.2.1. Lemaitre constitutive equationLemaitre constitutive equation under multi-axial stress states

usually takes the following form (Lemaitre, 1979, 1984, 1985):

( - r' J i/(l+a)w=l- 1-Rv(1+a/)(f,) t , (45)

where a', g' and r' are material constants; Rv is the triaxial factorwhich reflects the effect of stress states and defined as:

3 (am)2Rv = 3(1 + v) + i-(1 - 2v) Õ' ' (46)

where v is Poisson ratio. Assuming that w equals to unity whencreep failure occurs (t = tr), we have:

(Õ'jÀ)-rtr = .Rv(a + 1)

Then Eg. (45) can be rewritten as:

(47)

W=l-(l~) 1/(l+it).tr

(48)

H.-T. Yao et al. / Nuclear Engineering and Design 237 (2007) 1969-1986 1979

The constitutive equation has been used by ling et aL.

(200 I a,b) in the multi-axial creep prediction of an aero-engineturbine disc. Based on the uniaxial creep test data of ZbNCT25alloy at 650°C, Jing ct aL. (200la,b) obtained:

( t)0.1423w=l- 1-- ,

tr

and

(ä /787.2)-8.569tr =7.03Rv

Furthermore, the analysis of Rv with the help of finite elementsoftware ADINA indicates that complicated stress states wilaccelerate the damage process and thus significantly reduce thecreep life of disc.

In the study of Wang and Guo (2005), the Lemaitre constitu-tive equation was used to analyze the creep performance of anaero-engine material (lMI834). The stress triaxial factor Rv in anaxisymmetrc semicircular notched specimen was computed byusing the FE software. It is found that the dangerous location isthe throat of the notched specimen where Rv is 1.16. Moreover,the creep rupture life of alloy steel can be predicted accuratelyby Lemaitre constitutive equation with the modification of thetriaxial factor Rv.

4.2.2. Kachanov-Robatnov constitutive equationTypically, the multi-axial creep Kachanov-Robotnov consti-

tutive equations can be expressed as (Leckie and Hayhurst, 1974;Hariy, i 980; Hayhurst, 1983; Hayhurst and Felce, 1986; Hydeet aI., 1996):

d£ij 3AiiS (-)"-i(l )-Il--- ..a -wdt - 2 'j ,dú) II (ar)P"di = B (1 + q")(1 - w)a'''

wcr = 1 (49)

where £ ij is the creep strain tensor; A" , B", p" and q" are materialconstants; Wcr denotes the critical value of damage; and ar is thecreep rupture stress. Considering that: (1) damage of materialsis usually heterogeneous; (2) rupture stress is the function ofeffective stress and maximum principal stress; and (3) criticaldamage value is often less than unity, the Eq. (49) should berewritten in the following form:

d;;j = ~AII(ä)ll-i Sij((1 _ pI!) + (1 _ w)-n),

" " - p"dw II II (a ai + (1 - a )a)di = g B (1 + q")(1 - w)a" ,

- i - (1 _ 1i)1/(J+q")Wcr - g (50)

where pl/, gll and a/I are material constats. It is obvious that Eg.

(50) reduces to Eg. (49) when pI! = g" = 1.The parameter a" is introduced to reflect the influence of

multi-axial stress states on creep rupture of materials. Tests byJohnson et al. (1956, 1960) shown that the rupture of mate-rials can be classified into two categories. The rupture time

V)-qxoM-qe.

Fig. 12. Dimensions of the T-joint.

of first category, including copper, is governed by the maxi-mum principal stress criterion. However, the rupture criterionof second category, including aluminium alloys, is directlyrelated to shear stress. Value of a" is very close to zero forthe first category and to unity for the second. The value ofa" can be determned by the tests of uniaxial tension, pureshear and equal biaxial tension (Hayhurst, 1972), which are timeconsuming and costly. Hyde et al. (1996) proposed a less expen-sive procedure to determine the value of a" by comparson ofFE results with experimental results obtained from Bridgmannotch specimen. Consequently, Hyde et al. (1996) suggestedthat a" = 0.15 for nickel-based alloy and a" = 0 for titaniumalloy.

The benchmarks of numerical modeling for creep continuumdamage mechanics was studied by Becker et al. (2002) basedon Eq. (50) and the finite element frame. The matcrials usedfor creep damage calculation are titanium alloy at 650°C and0.5CrO.5MoO.25V steel at 640°C. Four different types of testrepresenting uniaxial, biaxial, traxial and multi-materials creepand damage situations were caried out. Good agreement wasachieved between the test results and those from two indepen-dent damage codes, an in-house code (FE-DAMAGES) and acommercial code (ABAQUS- UMAT).

In use of Eg. (50), Chen et al. (2003) analyzed the strainallowances of pipes of 0.5CrO.5MoO.25V steel at 530 °C, withthe help of commercial code ABAQUS. For the creep designof T-joint, as shown in Fig. 12, it was suggested that the allow-able strain in knuckle region should be less than 4.6% when1 % allowable strain is used in the straight pipe. The value islower than the local strain prescribed in ASME Code N-47(5%).

4.3. Multi-axial creep equations with multi variables

In Egs. (45) and (50), there is only one damage varable and noconsideration is given to the physical nature of damage param-eter. However, studies on metal physics and void growth theoryshown that the deterioration of high temperature material resultsfrom different mechanisms, e.g. grain boundar slide, ductilevoid growth, diffusion of vacancies along the boundary and car-bide precipitate coarsening, etc. To tae into consideration ofthe effects of these different damage mechanisms on creep fail-ure, multi-varable constitutive equations were thus developed,which can be described by the following general form (Hayhurst,


2005):

Êij = f(aij, Wi, wi"", Wn, T)

Wi =gl(aij,WI,wi,"',wn,T)

lÓ = g2(aij, Wi, wi, ..., Wn, T)(51 )

Wn = gn(aij, Wi, W2, ..., Wn, T)

where Ê ij is the creep strain tensor, aij the stress tensor; wi(i = 1,2, . . ., n) is the ith damage variable; and gi(i = 1, 2, .. ., n) isthe ith damage rate function. By selecting the appropriate strainrate functionfand damage rate function gi, the creep behaviorof materials can be accurately described.

4.3.1. Strain rate and damage rate functionsIn the multi-variables damage equation, a Sinh-function is

adopted to model the creep strain rate over a wide stress range.Dyson and Osgerby (1993) proposed that dislocation climb andglide in creeping materials occur as a parallel process, insteadof the sequential one. This parallel process requires the creep

strain having a hyperbolic sine dependence on the applied stress(Perrin and Hayhurst, i 996). The strain rate equation withoutconsideration of damage variable can be written in the followuniaxial form:

Ê = AI/I sinh(B"I a), (52)

where Alii and B'll are material parameters. The equation can beextended to multi-axial conditions through assuming an energydissipation rate:

A'IItj = - coth(BII/Ö').

B'll(53)

Based on the assumption of normality and the associated flowrule, the multi-axial governing equation is proposed by Othmanet al. (I 993) and Kowalewski et al. (1994a):

Ê" = 3...tj = ~A'I/ (Sij) sinh(B!!a-)I) 3Sij 2 a- 'where Sij is the so-called deviatoric stress tensor and defined by:

(54)

a'j - o,)aKKS,) =

3(55)

where aij is the stress tensor; oij the Kronecker delta and aKKobeys the rule of summation convection.

For the construction of damage rate function, particularconsideration should be given to the following mechanisms: cav-itation damage from cavity nucleation and growth, precipitatecoarsening, dislocation accumulation and strain hardening dur-ing the primar creep (Othman et aI., 1993; Kowalcwski et al.,1994a; Perrin ancl Hayhurst, 1996; Hayhurst, 2005).

The nucleation and growth of cavities reduce the load bear-ing section and accelerate the creep damage. So the effect ofcavitation damage should be explicitly represented in the con-stitutive equations. Canc (i 98 i a,b) presented that the nucleationand growth of cavity are dependent on the maximum principal

stress and von Mises equivalent stress. The function for cavi-tation damage can be derived from the study of Cane (Dyson,1988):

vff_ C"IN' (ai)Wi - Se - ,

Ö'(56)

where Êe is the effective creep strain rate defined as Êe =(2Êijèij/3)1/2; Wi is the cavitation damagc variable and variesfrom 0 to 0.3; C'" is a material parameter; N is a parameter toindicate the state of loading (for ai being tensile, N = 1; and

for ai being compressive, N = 0); d" is the so-called sensitiv-ity index of multi-axial stress and takes a value in the range of0.5-3.

Engineering alloys are often strengthened by a dispersion ofprecipitated particles which are unstable with respect to timeand temperature. The precipitates on grain boundares provide asite for nucleation of cavities and the precipitate coarsening mayrestrict the deformation within the grain interior. So precipitatecoarsening is also an importt cause of creep damage and adamage varable is needed to reflect the effect of this process.The evolution of the proposed damage varable may be derivedfrom the coarsening theory (Dyson, 1988; Perrin and Hayhurst,1996):

. (1 - W2)4wi = Kc 3 ' (57)

where W2 is the precipitate coarsening state variable and variesfrom 0 to 1; Kc is a constant related to the initial paricle spacingand temperatue.

For most nickel-based superalloys undergoing creep, themajority ofIifetime consumed in the tertiar stage. It is believedthat the mobile dislocation accumulation should be responsiblefor the tertiar creep stage (Dyson and McLean, 1983). A ratefunction for dislocation accumulation damage was proposed byDyson (1988) and Othman et al. (1993):

W3 = D'II(1 (3)2èe, (58)

where W3 is the mobile dislocation accumulation state variableand vares from 0 to 1; DJ/ is a material parameter.

In addition, to describe the primary creep due to the initialstrain hardening and the formation of dislocation microstrc-ture, a function for damage variable evolution is introduced byKowalewski et al. (1994a) and Perrin and Hayhurst (1996):

Il = hÊe (1 _ ~)a- H*' (59)

where h is the material parameter; H is the primar creep statevarable and vares from zero at the beginning of creep processto H*, where H* is the saturation values of H at the end ofprimar creep and subsequently maintans such a value until theoccurrence of creep failure.

A length discussion on the above physical mechanisms canalso be found in the publications of Dyson (1988), Kowalewskiet al. (1994a) and Perrin and Hayhurst (1996). By introducingdifferent damage state variables in the governing equation andin conjunction with the corresponding damage rate function, a


lot of explicit forms for multi-axial creep deformation with thesimilar style of equation set (51) can be obtained.

4.3.2. Constitutive equations with double variables

Following the above scheme, Othman et al. (1993) obtaineda constitutive equation by introducing a cavitation damage statevariables wi and a dislocation multiplication state variable W3:

t.. = ~AI! (Sii) sinh(BI!a-)1) 2 a- (1-WI)(1-W3)n'

where n = B"!a- coth(BI!a-). Therefore

te = (2tI3Jtii)

1/2 = AI! sinh(BI!a-) .

(1 - WI)(1 - W3)"

Replacing the te in Eqs. (56) and (58) by Eg. (61), an explicitform for constitutive equation set (5 i ) is thus obtaned:

t.. = ~AI! (Sii) sinh(BI!a-)1) 2 a- (1 - W3)(1 - Wi)n'

. Iii I!(ai )U'" N sinh(BI!a-)WI = C A - ,a- (1 - W3)(1 - wi)"

. Dil A I! (1 - W3) . h(B"1-)W3 = n sin a ,(1 - Wi)

Eg. (62) has been used by Othman et al. (1993) to improve theprediction of multi-axial creep behavior of nickel-based super-alloy. Hayhurst et al. (1994) made use of Eq. (62), togetherwith a continuum damage mechanics finite element based solver,DAMAGE XX, to study the creep behavior of axisymmetric ally

notched tension bars of a nickel-based superalloy. Numericalresults indicate that creep behavior can be represented in termsof a 'skeletal stress' located at a point within the notch throat,and the stress states of this point. However, Eq. (62) could notbe used when the uniaxial ductility of the material was less than1 % (Hayhurst et aI., 1994).

Hsiao and Gibbons (1999) developed a constitutive equationincluding the precipitate coarsening state variable W2 and theprimar creep state variable H:

d£ij 3 iii (Sii)' iBIIa-(1 - H)J-- = -A - sinhdt 2 a- 1 - wi '. (1 - wi)4w?=Kc- 3II = ~AI!! (1 - li) sinh r BI!a-(1 - H)Ja- H* L 1 - wi ' (63)

Eq. (63) has been used in the prediction of the creep rupture lifeof internally and externally notched tubes of 2.25CrlMo steelunder the support of a finite element program, ABB-MARC.The estimated results were compared with the experimentaldata of notched tubes, the life predicted by ASME N-47 andthat from Larson-Miller approach. The comparison showed thatthe CDM-based approach provided a very close result to theexperimental data, as shown in Table 1.

1981

Table iComparson of estimated failure time for notched tube with actual time by Hsiaoand Gibbons (1999) (565 "C, 100 MPa)

AS ME Code N 47Lason milerCDMExpenmental results

Internally notchedtube (h)

33

128

25352907

Externally notchedtube (h)

33

32535203844

(60)

(61)

4.3.3. Tri-variable constitutive equationsTo interpret the creep behavior of aluminium alloy under

multi-axial stress states, a constitutive equation including vari-ables WI, W2 and H was developed by Kowalewski et al.(1994a,b ):

( Si). L BIIa-(1 - H)J

-l sinh ,a- 1-wi3 AI!

t.- -1) - 2(1-wi)"

. C��i A ill (ai)ul! . iB1IIa-(1- H)JWi = - N sinh ,(1 - Wi)n a- 1 - W2

(62)

. (1 - wi)4W2 = Kc ,3

. h AI! ( H). iBIIa-(l-H)JH= 1-- sinha- (1 - wi)" H* 1 - W2 ' (64)

where n is given by:

BIIIa-(1 - H) iBIIa-(1 - H)Jn = 1 coth .-W2 I-wiFor the aluminum alloy at 150°C, the material constants in

Eq. (64) have been determined by Kowalewski et al. (l994a,b)from the experimental results and an automated numerical opti-mization technique. The constitutive equations were adoptedagain by Lin et al. (2005a) to depict the creep rupture of purecopper at 250 °C and aluminum alloy at 150 °C under combinedloading. Creep tests were cared out for both materials at thee

effective stress levels and for three stress states of tension, puretorsion, and combined tension and torsion. The comparison ofexperimental data and computed effective creep strain was per-formed and indicated that Eg. (64) can predict the isochronoussurface of materials precisely.

To predict the multi-axial creep behavior of ferritic steel, a tri-varables type of constitutive equation was developed by Pcrrinand Hayhurst (1996). Based on the physical mechanism analysisof deformation and rupture of ferritic steel, the material param-eters were determined following the approach of Kowalewskiet al. (1994a). Subsequently, the tr-varab1es tye of governingequation was widely applìed in the study of multi-axial creepanalysis of 0.5CrO.5MoO.25V and 2.25CrlMo ferrtic steelsand their weldments over a wide range of stress and temper-ature (Perrin and Hayhurst, 1999; Mustata and Hayhurst, 2005;Hayhurst et aI., 2005a,b). In addition, Hayhurst (2005) veri-fied the existing CDM-based tri-variables models in use of thewelded ferritic steel components, e.g. butt-welded pipes, cross-welded tension plates and T-branched welded pressure vesseL.

(65)

1982 H.-T. Yao et al. / Nuclear Engineering and Design 237 (2007) 1969-1986

4.3.4. New development oftri-variable constitutiveequations

From the nature of cavity nucleation and growth, a lot ofmaterial damage models have been developed to predict thecreep failure. In the constitutive Egs. (62) and (64), the modeloriginally proposed by Cane (198 i b), namely Eq. (56), wasused to reflect the elTect of stress states on cavitation evolution.However, it was found that Eq. (5 i) could not lead to a sound pre-diction for the creep strain at failure and significant discrepancywas reported between the estimated results and experimentaldata (Xu, 2000a,b, 2001). To eliminate such a discrepancy, twofunctions of stress statesfi andh were introduced into the gov-erning equation by Xu (2oo0a,b, 2001, 2004), which can bewritten as:

. 3 iii (Sij). (BII &(1 - H) Jê" = - A - sinhI) 2 & (1-W2)(1-W) ,

. CIIAIIN' h ( BII&(1 - H) J fWi = sin 1,(1-W2)(1-w) -

. K (1 - wi)4wi= c 3

. h iii ( H). (Bill &(1 - H) JH = - A I - - sinh ,& H* (1 - wi)(1 - w)w=wifi,

where

(66)

( 2& ) a (( 3am ) Jfi = 3Si exp b s: - 1 ,and

(67)

r (( ai ) ( I 3tTm ) J 1. -Ih = 1. exp p 1 - & + q 2' - 2& r (68)

Thefunctionfi derived from Huddleston's work (1985, 1993)

is used to couple the effects of tertiary deformation, creep dam-age and creep rupture. The function h proposed by Spindler(1994) and Spindler et al. (2001) is employed to describe

the effect of multi-stress states on the damage evolution. The

Table 2

Summar of the multi-axial creep damage constitutive equations and applications in engineering

Constitutive equations and damage variables Contrbutors and applications

Constitutive equations with single varableKachanov-Nobotnov constitutive equations

A dominant damage variable W is used to describethe state changes occurred in materials

Lemaitre constitutive equationsA dominant damage variable w is used to describe

the state changes occurred in materials

Constitutive equations with double varableOthman constitutive equations

Variable wi for cavitation damage and (Ù3 formobile dislocation multiplication

Hsiao constitutive equationsVarable W2 for precipitate coarsening and H for strain

hardening from primary creepXu constitutive equations

Vanable W¡ for cavitation damage and Hfor strainhardening during the primar creep stage

Constitutive equations with tr-varable

Varable (ù¡ for cavitation damage; W2 for precipitatecoarsening and H for strain hardening during theprimar creep stage

Hyde et al. (1996) on Waspaloy alloy (700 DC) and IM1834 alloy (650 "c)Hyde et al. (1999) on 2.25Cr1Mo (640 'OC and I CrO.5Mo steel (550 'C)Hyde et al. (2002) on CrMo V steel (640°C)Chen et al. (2003) on O.5CrO.5Mo 0.25 V steel (530 DC)Becker et al. (2002) on 0.5CrO.5Mo 0.25 V steel (640 GC) and titanium alloy (650 'C)Hyde et al. (2004) on CrMoV steel weldment (640°C)Orlando and Goncalves (2005) on Ti-6AI-2Cr-2Mo alloy (400 "C)Hyde and Sun (2006a) on CrMoV steel weldment (640 "C)Hyde et al. (2006) on P91 steel (650 'C and 625 'C)

ling et al. (2oola,b) on ZbNCT25 alloy (650°C)ling et al. (2003) on 30CrlMo i V steel (525 "C)Wang and Guo (2005) on IMI 834 alloy (650°C)

Othman et al. (1993) on nickel-based superalloyHayhurst el al. (1994) on nickel-based superalloy

Hsiao and Gibbons (1999) on 2.25Cr1Mo steel (565 "C)

Xu and Hayhurst (2003) on 316 stainless steel (550 'C)

Kowalewski et al. (I 994a) on aluminium alloy (150 'C)Kowalewski et al. (I 994b) on aluminium alloy (l50"C)Li et al. (2002) on aluminium alloy (150 DC)Un et al. (2005) on copper (250°C) and aluminium alloy (150°C)PelTin and Hayhurst (1996) on 0.5CrO.5MoO.25V steel (600-75 °C)PelTin and Hayhurst (1999) on 0.5CrO.5MoO.25V steel (600-75 "C)Hayhurst et al. (2005a) for CrMoV (565-640 DC)Mustata et al. (2005) for 0.5CrO.5MoO.25V (565-675 GC)

Hayhurst et al. (2005b) for Cr-Mo- V steel (575-640 "C)Hyde et al. (2006) for P91 steel (650°C and 625°C)


improved equation (66) was applied to the prediction of creepbehavior of 0.5CrO.5MoO.25V ferritic steel at 590°C. As wasexpected, good agreement between the experimental observa-tions and estimation from the developed Eq. (66) was achieved(Xu, 2001. 2004).

4.4. Summary

The CDM-based method and constitutive equations for creepdesign under multi-axial stress states were reviewed in this Sec-tion. The existing multi-axial creep constitutive equations andtheir applications are summarized in Table 2. It can be seen thatthe single varable Kachanov-Robotnov constitutive equations,namely equation Set (46), and tri-variable equations, namelyequation Set (64), are used frequently, in paricular in the pre-diction of multi-axial creep behavior of low-alloy ferritic steel.Generally speaking, such a kind of steel is widely used in com-ponents in fossil and nuclear power stations operated under

creep regime. Compared with the single variable equation, twoadvantages are revealed in multi-variable equations:

(i) Multi-variables are introduced to distinguish differentmechanisms and effects on the damage evolution.

(ii) A Sinh-function is adopted to replace the traditional powerlaw and to describe the stress sensitivity of creep rates overa wide stress range.

Hyde et a1. (2006) compared the single variable Eq. (49) andtr - variable Eq. (64) in the prediction of creep fail ure of P91 steel.The results shown that for the single variable model, the creepevolution is dominated by the damage variable w. While for thetri-variable model, the value of cavitation damage variable wiis much larger than that of the precipitate coarsening variableW2. Therefore creep failure is mainly controlled by the accumu-lation of varable wi. However, the predicted results of failuretime by Egs. (49) and (64) very close to each other at the higherstress levels (e.g. 70-100MPa). On the other hand, the singlevariable Eg. (49) will overestimate the lifetime of componentsat the lower stress level (Hyde et aI., 2006). Nevertheless, whenthe creep failure is controlled by one dominant damage mech-anism, and the creep tests to determine material parameters arecarred out at a stress close to the operating level, both singlevariable equation and tr-variab1e equation are appropriate forthe description of multi-axial creep behavior.

It has been shown (Goodall et aI., 1975; Dyson, 1988) thatcreep damage tolerance parameter (À = er!S¡tr, where s¡ is ini-tial strain rate) should to be greater than 5-10 to extend locallifetimes sufficiently and to justify the safe usage of upperbound estimates of component lifetimes. A 'nodal release tech-nique' is usually needed in implementing CDM-based methodin creep analysis (Hsiao and Gibbons, 1999; Yatomi et aI., 2003).According to the technique, once damage occurred in the mate-rial, the corresponding elements are considered to be 'death' andwill be removed in the FE modeL. In addition, finer finite elementsize wil lead to an improved prediction, but the size of finiteelement should not be less than the minimum of the RVE (rep-resentative volume element) (Skrzypek and Ganczarski, 1999).

1983

5. Conclusions

Since the primar, secondar and tertar uniaxial creepprocesses were identified by Andrade (1910, i 914), many exper-imental investigations have been cared out and focused on thedata collection. These data can be used for the creep designof different engineering components operated at high temper-ature through extrapolating in four ways, i.e., to multi-axialstate of stresses, to life extrapolation, to various loadings andto aggressively external environment. In this work, the first waywas discussed, with paricular emphasis on the CGM-based andCDM-based multi-axial creep design methods.

The CGM-based method is developed on the basis ofthe physical modeling for microstrcture evolution of mate-rials under external loading. The main failure causes of hightemperature components, namely, the nucleation, growth andcoalescence of cavities under stress and temperatue wereaccommodated in the CGM-based models. Therefore it hasbeen adopted in many design codes or assessment procedures topredict the creep deformation and rupture of high temperaturecomponents under multi-axial stress conditions.

The CDM-based method is derived from the innovative workof Kachanov. The great advantage of CDM-based approach isthat it can be used in conjunction with finite element methodto provide information on the local stress and strain fields. Itreflects the effects of damage evolution on stress redistributionand strain accumulation in components so as to reduce con-servatism in creep design. For multi-varable equations beingnon-linear and strongly coupled, an automated numerical opti-mization technique proposed by Kowalewski et al. (1994) isusually needed to determine the material parameters required increep analysis.

For both CGM-based and CDM-based methods, the exist-ing multi-axial creep analysis models or constitutive equationsare generally sensitive to material and temperature. Since creepdamage mechanisms are dependent on materials and tempera-ture, a general model or constitutive equation is yet unavailable.For a given material under the given temperature and stressrange, it is importnt to know the nature of creep damage andthus to develop a proper analysis modeL. In addition, it is bene-ficial to propose a CGM-based method for initial design whilstto develop a CDM-based method for life extension or integrityassessment of the serviced strctures.

Acknowledgements

The authors are grateful for the supports provided by ChinaNatural Science Foundation (50225517, 50505012). FZ wouldalso wish to thank the supports provided by Shanghai Rising-Sta Program (05QMXI416) and Fok Ying Tung EducationFoundation (101054).

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A Review of Creep Analysis and Design Under Multi-Axial Stress States

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