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

Available online at ww.scincirect.com ScienceDirect ELSEVER Nuclear Engineering and Design 237 (2007) 1969-1986 Nuclear Engineering and Design ww.elsevier.com/locate/nucengdes 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 associated failure mechanisms and to predict the lifetime in practice. Although meta creep has been studied for about 100 years, many problems are stil unsolved, in particular for those involving multi-axial stresses. In this work, a state-of-the-ar review of creep analysis and engineering design is carried out, with paricular emphasis on the effect of multi-axial stresses. The existing theories and creep design approaches are grouped into three categories, i.e., the classical plastic theory (CPT) based approach, the cavity growth mechanism (CGM) based approach and the continuum damage 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 and improvement of efficiency in energy conversion, higher operating temperature and design stresses have been adopted in chemical and petrochemical plants, power generation systems, etc. The main concern for the strength design of components has thus moved to the viscoplastic performance of materials in tic deformation under a fixed stress at temperatures of roughly 0.3-O.5T ro, where Trois the melting temperature of metals. The main cause of creep failure is the nucleation, growth and coalescence of cavities on the grain boundaries (Leckie and Hayhurst, 1984; Huddleston, 1985; Kassner and Hayes, 2003; Goodall and Skelton, 2004). The multi-axial creep design method based on the classical plastic theory is limited in practical application because it is derived from the criteria of yielding failure and does not account for the physical daage process. Staing from the innovative work of Hull and Rimmer order to prevent creep failure. On the other hand, the existence of multi-axial states of stress due to complexity of loadings and materials cannot be avoided. It is therefore essential to understand the multi-axial creep failure mechanisms and establish the multi-axial creep design criteria for the strength design and life prediction of high temperature components. Over the past several decades, considerable efforts have been made to gain a fundamental understanding of creep mechanisms and to develop an efficient engineering design criterion for high temperature components under multi-axial stress states. Because the multi-axial creep behavior is very similar to that of (1959), multi-axial creep design criteria using the models based on cavity growth mechanisms (CGM) were established. Then, the CGM-based models were improved by a lot of researchers (Rice and Tracey, 1969; Hayhurst, 1972; Gurson, 1977; Manjoine, 1975, 1982; Raj and Ashby, 1975; Ashby and Edward, 1978; Cocks and Ashby, 1980, 1982a,b; Edward and Ashby, 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), Margolin et a1. (1998), Spindler et al. (2001), Spindler (1994, 2004a,b) and Ragab (2002). Some of these models have been applied in high classical plasticity, the classical plastic theory (CPT) has been directly used in the multi-axial creep analysis during the first half of 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 temperature 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 Design Codes (Siemens PG) (Andreas, 2000), to include the effect of multi-axial states of stress on creep failure. (53) 5ij 51 t tc tn tr triaxial factor defined by Eg.. b. C'11. 2oo5a. ß material parameters in Eq. Hsiao and Gibbons. D. 7:2. 2005. Kc material parameters in multi- p". q". B". 2005a. Lin et aI.. 2004. (45) primary creep state variable (0 -c H -c H*) activation energy (1) multi state variables -t:H W damage state varable (0 -c w -c i) Wcr critical damage state varable W¡ cavitation damage state varable (0 -c Wi -c 1/3) W2 precipitate coarsening state variable (0 -c W2 -c 1) H* l¡ N saturation values of H at the end of primar creep first stress invarant (MPa) loading state parameter (N = 1 for a¡ tensile and N=O for a¡ compressive) material parameters in Eq.25 cavity area fraction am hydrostatic stress (MPa) ar creep rupture stress (MFa) ai. Mustata and Hayhurst. 2003.1970 H. B'. / Nuclear Engineering and Design 237 (2007) 1969-1986 Nomenclature y surface tension (N/m) a. continuum damage mechanics (CDM) based method. l! material parameters in multi-axial creep P radius of void (m) Po initial radius of void (m) equations with single state varable AI!. the CDMbased method is developed on the phenomenological way and presented from the viewpoints of mechanics. ê2. Yi. (37) grain diameter (m) ae efficient stress (MFa) a ij stress tensor d Dg Vf fc fl1 grain boundary diffusion coefficient (m2/s) chemical potential of vacancy (1) cavity area fraction at tc. C'. 2001a.q r R Rv W3 mobile dislocation multiplication state varable (O-cw3d) Q atomic volume (m3) i/ energy dissipation rate potential defined by Eq. (46) deviatoric stress tensor maximum deviatoric stress (MPa) time (s) time to coalesce void (s) time to nucleate void (s) time to rupture (s) temperature COC) subscripts c creep cr critical e effcient r rupture c coalescence n nucleation ss steady state T Tm melting temperature of metals (0C) Greek letters a. h'li. r H gO the value of g corresponding to simple tension material parameter in Eq. n material parameters A'. 2005). Kowalewski et aI..b. 1994a.b. 2006. i 999. 200a. 2004. l. 1996. The purpose of this paper is to present a state-of-the-art review of creep design methods or criteria under multi-axial . 7:3 shear stresses (MPa) v stress index in Eq. k axial creep equations with multi state variables material parameters in Eq. (31) material parameter in Eqs. material parameters in Eq. Hyde et aI.b. 1999. Contrasted with 8ij Kronecker Delta 8z grain boundar width (m) êc creep strain êe effective strain êf uniaxial failure strain êij creep strain tensor êr creep rupture strain êi.. Y3 shear strain rates (s -I) À creep damage tolerance parameter v Poisson ratio Kachanov-Robotnov equations A". ml. bl material parameters In classical YI.-T. £2. taken as 0. was developed from the initial work of Kachanov (1958). a3 principal stresses (MPa) fn g cavity area fraction at tn material parameter defined by Eg. Wang and Guo.. (38) A.p". (24) u von Mises equivalent stress (MPa) 7:¡. (36) vl/ stress index in multi-axial creep equations with gl.. a2. CDM-based method has been focused again during recent years (Othman et aI. (32) and (33) Boltzmann's constant (11K) p. Perrn and Hayhurst. B. Yao et at. Hayhurst et aI.b. m. (39) al material parameters in Eq. 1994. Hayhurst et aI.. a" material parameters in multi-axial creep equa- tions with single state variable a uniaxial stress (MPa) c. ê3 principal strains êf multi-axial failure strain £c creep strain rate (çI) £e effective creep strain rate (s. ling ei aI. BII. H*. Dill. Xu. nl. h principal strain rates (s-¡) the CGM-based multi-axial creep design method. 1993.I) £¡ initial creep strain rate (s-I) £ss steady state creep strain rate (Ç I) £1. 1996. 2001. (45) The third important method for multi-axial creep design. With the rapid development of modern computer technology and finite element analysis method. C. 1980: Boyle and Spence. (9) is called strain hardening because it contains stress and strain as variables.£3. so the rate of volume creep strain is zero: (2) £1 + £2 + £3 = 0. (7) which is known as Norton-Bailey law and commonly used in Eq. T). 1995). / Nuclear Engineering and Design 237 (2007) 1969-1986 1971 states of stress. Ti T2 T3 (11) where A and n are material constants. the effective stress criteria governing creep strain was proposed on the basis of the results of uniaxial creep test. For example.a3)2. a2-a3 h(t) = Dtm(1¡3:: m:: i¡2usually). i 980.6H is activation energy and R is Boltzman's constat. (8).tm all . (9) Eqs. Viswanathan.: (5) £e = g(t)f(ã). (-. law. many historical review papers are available on CPT-based method in books or monographs (Har. e. In the end. rated into: £c = f¡(a)h(i)f3(T). The function is popular for its simplicity in application to stress analysis. the creep strain is: published papers. 1955): T2=-. Uniaxial creep constitutive relationship £c = mB1lmaii/m £~m-l)/m. differentiating the Eq. Then the creep strain may be written as: where ã is von Mises equivalent stress defined in the same way as in plasticity theory: £c = B exp -. stresses. (8) is called time hardening because it models hardening phase using time parameter. Boyle and Spence. To do this. (4) where D and m are material constats. the information and data obtained from a tensile specimen under constant load are extensively used due to be obtained easily in laboratory. Yao et at.6H) (12) where -. The stress and time dependence of creep under constat stress has received considerable attention. (7). 1983: Penny and Mariott. (7) is developed for the constant stress and can describe the primar. Eq.' (-. (2) The principal shear strain rates are proportional to the principal shear stresses: (3) YI = Yi = h = 2 if.a2)2 + (a3 . 1980). (8) and (9) are the rate forms which can be used to model the primar creep of decreasing creep strain rate.2. (1) is usually assumed to be sepa- (1) Constant volume is maintaned during creep. £2 and £3 are principal strain rates. it can be obtained: temperature components. i 995). Eqs. this review places parcular emphasis on the developments of CGM-based and CDM-based methods. the Eq. 1983.-T. Penny and Marott. Yi. 1989): (1) where £c is creep strain. CPT-based multi-axial creep design method In the strength design and remaining-life assessment of high (8) Removing the time variable t from Eq. it is impossible to conduct this review in a comprehensive manner. Experimental results of multi-axial creep test indicate that creep is a shear-dominated process for isotropic and homogeneous materials. time t and temperature T. £c = Btm~. (8) and (9) are the accustomed creep constitutive relationships under uniaxial stress conditions. we have £c = mBtm-¡all. According to Arrhenius's Y2=£3-£I.£2. For varying Therefore. The most commonly used function of stress is the power law attributed to Norton (1929): fi(a) = Aan. some hypotheses and concepts developed in the theory of instantaneous plastic deformation were introduced again in CPT-based method. h(T) = C exp -. An importnt time function is the so-called Bailey law (1935): where if is a constat.1. secondar or tertar creep stages. Yi and h are principal shear strain rates. potential problems to be resolved in the near future are pointed out. 2. 2. Eq. (7). Multi-axial creep constitutive relationship The deformation of a tensile specimen under constant load depends on stress a. 1962: Penny and Marriott. As a result. 1995): £c = f(() t. (13) . For the case of multi-axial stress states. (7) to Eg. Thus the following assumptions were made (Boyle and Spence.6H) (6) ã = ~ V(ai . (3) The effective strain rate £e is related to the effective stress in the same way as the uniaxial relation. 1983. creep analysis (Harry. This process is usually called hardening (Josef. where £¡. Consequently. the creep strain of materials can be written as (Harry.H. 2003).g. Tl. Since the subject is quite old and there are many For isothermal conditions. the temperature dependence is given as (Dorn.a2)2 + (ai . 0'3 -a¡ 2 T3 al-a2 =--' Y2 = t¡ . 2. T2 and T3 are principal shear stresses and: T¡ = --' Yi = £2 . Also. there are a (10) number of altemative expressions (Kennedy. i ! t + i t tt Cavity growth mechanisms have been used to model the dam- age resulting from the deformation of high temperature creep. Additionally. dislocations (line-defect). diffusion flow. Creep cavity growth and creep failure a microscopic level. 1994. (16) in analyzing the steady state creep of high temperature components can be found in the books by Boyle and Spence (1983). Process of creep cavity growth ¡ Small voids are often observed at grain boundar. 1985. 2003. 1. etc. In cold forming processes of metals. There are several methods whereby voids might be nucleated (Hull . cold metal forming. la.1972 H. 2003). In creeping deformation processes.. inclusions and voids (three-dimensional defects).. From observations at shown in grain 4. These microscopic discontinuities A stadard creep cure is shown in Fig.g. (i 0)-( 12).1.am). 3. g(t) = i. It is realized that failure of most components operated at high temperature is caused by the nucleation. (b) Voids nucleation in creep deformation process. 3c. Kassner and Hayes. increases the void grow rate. It is therefore necessar to under- a decelerating strain rate stage I (primar creep). however. 2oo5b). 2. 3 f(ã) 82 = '2--(a2 . materials always contan some defects. 2002). after the instantaneous elastic strain. the voids normally nucleate around second-phase parcles and within the grains. i. as shown in grain 3.1. 1979). 2004).. 3 f(ã) £3 = '2 g(t)--(a3 . Michel.am).am). 3. 2005b)..am). it has been shown by Edward and Ashby (1979) that grain boundar sliding is an important cause for voids nucleation at grain boundaies.. The nucleation and growth of voids reduce the load bearing section and accelerate the creep daage and this. When the voids grow from an initial size to half the mean cavity spacing. . . e. the coalescence of voids occurs. grain boundaries and phase boundaries (two-dimensional defects). / Nuclear Engineering and Design 237 (2007) 1969-1986 In terms of Eqs. 4. are a central subject of materials science (Honeycombe.2. i 959. 2. i 985. as shown in Fig. K won et aI. CGM-based multi-axial creep design method The studies on microstructural evolution of materials under external forces. stress and temperature.. superplastic formng and hot metal forming (Lin ct aI. Impurity paricles which lack . e. l ¡ . A dislocation pileup breaking through the boundar might also be a suitable nucleus. O'm = 3 (15) Under steady stress conditions. Introduction to cavity growth theory . and Penny and Marriott (i 995). 2000.. Voids nucleation caused by grain bounda sliding (Edward and Ashby. Loretto. there are thee creep stages: will develop and finally cause damage and failure of components... for example.am). i 979. Ifche incompatibility caused by grain bounda sliding cannot be accommodate in any of these ways. 4. . cohesion with the matrix may act as voids. 1 b. as shown in grain 2. 3 f(ã) £2 = '2 g(t)--(a2 . (a) Voids nucleation in cold meta formng process. 3 83f(ã) '2--(a3 .e. Nucleation and growth process of cavities vares under different loadings and temperatures. voids wil appear and grow at grain boundar. Grain boundar sliding can be accommodated in varous ways: elastic accommodation.-T. Scheme of voids nucleation (Lin. and their growth and coalescence stand the cavity growth mechanisms so as to establish a proper multi-axial creep design criterion. i 996. Edward and Ashby. as shown in grain 1 in Fig. the voids usually nucleate along the grain boundar. a steady minimum strain rate stage II (secondar creep) and an accelerating strain rate stage II (tertiar creep). growth and coalescence of cavities (Hales. Nicolaou et aI. as shown in Fig. 3 f(ã) (14) (a) (b) where am is the hydrostatic stress and defined by: 0'1 + a2 + 0'3 Fig. as 3. I l i . 3 f(ã) 81 = '2--(a¡ .1. Fabrizio et aI. Harry (1980). 3.am). the multi-axial creep constitutive relationship is given by: 81 = '2 g(t)--(a¡ . l Fig.1. Kassner and Hayes. paricularly transverse to the applied stress during creep tests. . Studies have shown that the nucleation of cavities usually occurs during the creep stage I and II (Kassner and Hayes. or plastic flow. Yao et at. . the constitutive relation (14) reduces to: and Rimmer. in tur. According to Fig. (16) = A lot of examples for Eq. as shown in Fig. Humpreys.g. 2000. 2003). Riedel. 5.-1 1 1 1 1 1 1 1 -ï---------------- . Rice and Tracey.-T. In order to calculate the chemical potential There have been. 2004). The square aray of cavities in the Hull-Rimmer analysis (Hull and Rimmer. 2002. This has been verified a mechanism by which diffusion leads to cavity growth.. 2002. Assuming the initial radius of voids po ~~ a . Vacancies Sklenieka et al. The roughly 0 -+ 0. Spindler.8 tr roughly 0. on the rupture time of a 3. a lot of publications in which cavity growth mechanisms were discussed (Evans. It was used to predict the effect of combined hydrostatic pressure. Cavity growth mechanisms and models copper wire in the temperature range 400500°C.-II 1 1 1 1 1 Fig. R the Boltzann's constant and S is the function of pIa.2yj p)S r . the time taken up by cavity growth is often much longer than that by cavity nucleation (Cane. 1998. a. the rate of change of void radius is given by (Hull and Rimmer. --I¡. C\ ~ -.2.A of vacancies 'Vf at the void surface. Michel. The first cavity growth model based on diffusion-controlled cavity growth mechanism was proposed by Hull and Rimmer (1959). Cane.g . assumptions were made by Hull and Rimmer (1959): the ring of void surface lies in the grain boundar. 5. Kassncr and Hayes. So the creep failure of components operated at elevated temperature is usually controlled by creep cavity growth and studies on cavity growth mechanisms have received more attentions.H. 1 Ragab. tr. 3. Scheme of the intergranular damage development (Sklenieka et ai. the growth rate of cavities is influenced by the shape of voids and the diffusion process. Oz the grain boundary width.. (b) growth. Hayhurst. 2001). Michcl. According to this mechanism. 1987. as shown in Fig. McClintock. 3.5CrMo V steels. 2003). 6. lying on a grain boundar normal to the direction of applied stress a. 1972. 1984. (c) coalescence. 2003. 1975. migrate under the influence of the gradient and lead to the growth of voids. 1982.Manjoine. 1959): dp _ 2rrQ(Dgoz)(a am . later followed by Raj and Ashby (1975) and Speight and Beere (1975).A r dt . 2003). Dg the grain boundar diffusion coeffcient.8 -+ 1. represents the rupture ductility of materiaL.0 tr . Spindler et aI.RTap (17) Cavity nucleation -+ Cavity coalescence and growth and creep rupture where Q is atomic volume. 1969. Yao et at. diffsion-controlled cavity growth and constrained cavity growth (Hales. It is a square aray of spherical voids with a radius p. Delph. 'Vi. and uniaxial tension. / Nuclear Engineering and Design 237 (2007) 1969-1986 1973 a (a) (b) (el Fig. i 959. Scheme of the standard creep curve. and of a void is determined by the gradient of chemical potential of vacancies. x -----0---- ruptue II : Fig. the growth rate by Cane (1981 a) on 2. Difusion-controlled cavity growth mechanism and model Hull and Rimmer (1959) was one of the first. tr 4. surace tension of metals. Using the model and assumptions. p the void radius.1. in the past. .------------------------. 1994.. Sklenieka et aI. The process of creep cavity growth: (a) nucleation. lead to the creep stage II when creep ruptue occurs because of a significant increase of net section stress resulting from the decrease of the load bearing section.¡ 1 1 Margolinetal. 1959). to propose Nucleation and growth of cavities usually occur during most of the creep life. (2003) on 9-12% Cr steels. Cocks and Ashby.. 1981 a.5:---:--v: ro i I CGM-based models have been used by many researchers to predict the influence of multi-axial stress on the creep failure strain or creep rupture time (Hull and Rimmcr. 1968.25CrlMo and 0.2. 1981a. in the plane of the grain boundares. Br. It is believed that cavity growth rates is a complex coupling function of plastic strain of surrounding grains and vacancy diffusion along grain boundaries (Michel. 1980. The ruptue strain. 6. am. 2004). Cavity growth mechanisms are usually grouped into thee categories: plasticitycontrolled cavity growth.¡ . Furtermore. y the Fig. the atoms are deposited uniformly over the grain boundar and the voids retain spherical shape during the growth. 2004). The model is shown in Fig. at rupture time. 1994. (20): Xl £f = exp (~ _ 3~n) . 8 but with a long cylindrical void in a remote simple strain rate field leads to another equation similar to Eq. 8a.3. Cocks and Ashby (1980) proposed a model based on the constrained cavity growth mechanism. = 0. I'f 2 2a (21) 3. This model is an isolated when the cavity size is very small. ta2 8 (a . (2000). 1982a. 7. 8. Plasticity-controlled cavity growth mechanism and model Generally speaking. The cavity growth mechanism occurs when the local deformation rate exceeds the deformation rate of surrounding materials due to cavity growth. was presented by Rice and Tracey (1969) to determine the relation between void (b) growth and stress triaxiality.2y stress. McClintock (1968) has proposed a model based on plasticitycontrolled growth mechanism to investigate the relation between the cavity growth and imposed stress and strain.am) Dyson (1976). 2000). and then grow following the power law rule until they Hancock. diffusion-controlled growth dominates cavitated. as shown in Fig. diffusion-controlled growth mechanism decreases quickly. Three examples of constrained voids growth proposed by Edward and Ashby (1979). Then a more realistic model of an isolated spherical void in a remote-uniformed stress and strain rate field. as shown in Fig. 8b. 1969): (c) D = 0. The mechanism becomes more importnt under high strain-rate conditions where significant strain is observed. then developed by Cocks and Ashby (1980.2. 9.008 cosh 25' (19) (3am) (3am) Neglecting the second term on the right hand side of Eq. The model of spherical voids in a remote simple tension strain rate field proposed by Rice and Tracey (1969). the ratio of the average strain rate of sphere radii to the remote-imposed strain rate. and material ligament between cavities when the diffsion field of a growing cavity does not extends half way to the next cavity. 1976).-T. D.2. Fig. an equation describing the relationship between the multi-axial Fig.2Y)1 -+ + In . 7. and has been recently improved by Khaleel et al. The mechanism was first proposed by McClintock (1968) and Rice and Tracey (1969). Thus the cavity growth rate is constrained to produce the local strain at the same rate as the deformation caused by the remote x ya 4y2 (a(a .O'm)/2 . The model is a cylinder cavity in a non-hardening materiaL. as shown in Fig.. The following hypotheses were introduced by Cocks and Ashby: grain boundary cavities nucleate at inclusions at time t=tn. cylinder with a hole and an outer diameter of 2l. and Delph (2002).b). Yousefiani et al. (19). cavity growth is resulting from the plastic deformation of the surrounding materials (Rice and Tracey. as shown in Fig. as shown in Fig.2. and the plasticity-controlled growth becomes predominant (Nicolaou et aI.am) . 1969. (2002) to model the superplasticity. cavities on the boundares when less than half (18) the boundares are 3.521 I'f sinh(3am/25) (20) A model similar to Fig. the rupture time is therefore obtained: tr = kTa 2:nQ(Dgoz)(a . Tvergaard (1984). (2001) and Taylor et al. is given by (Rice and Tracey. Three conditions under which cavities grow by constrained mechanism have been analyzed by Edward and Ashby (1979) versus a decreasing size. / Nuclear Engineering and Design 237 (2007) 1969-1986 t é Xi failure strain £f and uniaxial failure strain I'f is obtained: 1'. . 8c. According to Rice-Tracey model. Yao et at. As the cavity size increases. and integrating Eq. They are: one par of the structure when its deformation is non-uniform.am)2 PO(a . According to plasticity-controlled growth model. (17).558 sinh 25 + 0. Constrained cavity growth mechanism and model Constrained cavity growth model was first proposed by x.am) (a .1974 H. Edward and Ashby (1979). fh)J £ss. It has been showed by Kwon et al. £f ai (32) .5)/(n + 0.5))) sinh(((2(n . (25) where fn is the cavity area fraction when cavities nucleate. and fc the value offh when cavities coalescence occurs and closes to 0. German TRD (Technical Rules for Steam Boilers) and SIEMENS PG (Siemens AG Power Generation Design Rules). (200 1) 3.fn)Il+ll (27) Under uniaxial stress condition. (22) at constant stress in terms of limits (24) fh = fn.3. French RCC-MR. (30) can be used to describe the relationship between the multi-axial failure strain a and uniaxial failure strain £f The right hand side of the Eq. respectively.. SIn . 200). The study by Yao et al. (21) whenp=O and q= 1. The first term and second term on the right side of Eq. ~= ~ . I (l-(1-fe)Il+I) £f = tc£ss = g(n + I) In (1 (1 .5))(amlt7e))' £f g (30) (22) d:tc t 1 + g 2: l(1_1fh)'1 .(1 (n l)g£ss (1 . t = te.-T.15 and q = 1. (27) and (29). (27). we have: £* . = g l(1 _lfh)n . (2001) and Spindler (2004a. It is clear that Eg. cavities keep spherical during their growth. t7" 1975 3.(1 .fh the cavity area fraction. American ASME II. t7" l n + 1/2 3 (28) Replacing g by go in Eg. Yao et at.. The time to fracture is then given by (Cocks and Ashby. (31) reduces to Eg.. (26) (through assuming tn =0). tc = tn++ In (1 .1. Presently. there are a number of assessment procedures and design codes for high temperature components.25. The model was proposed and developed by Spindler (1994). A model for the effect of multiaxial stress states on creep ductility. is included. Cavity growth theory and models are adopted in these procedures and codes and multiaxial creep deformation criteria (MCDC) and multi-axial creep rupture criteria (MCRC) are therefore established to analyze the creep behavior of high temperature materials under multi-axial stress states.(1 . Spindler et al.. 1 (l-(1-fc)Il+¡J go(n + 1) (i . 1980): of stress on creep ductility of materials (Andreas.+ q . = gO = sinh((2/3)((n . e. (20) and (21) are also applicable for the prediction of multi-axial creep ductility.25 when failure strain is independent of creep strain rate. g =ht2n-1/2 (am)j n + 112 Õ' Integrating Eq.5))/(n + 0.. The Eg.fn)n+¡J (29) By Eqs. n is material constants and g is defined as: .3.38 and q = 1. 0:. (2000) that the model can predct well the creep ductility of Durehete 1055 steel under multi-axial stress states. A model of voids growing on grain boundar proposed by Cocks and Ashby (1980).(1 fc1l+1) £. failure strain under uniaxial load 9. (31) reflect the influences of cavity nucleation and growth on the creep failure behavior of materials. a ductility exhaustion approach is used to assess creep damage. replacing am I Õ' in Eq. a set of differential equations describing the void growth and creep strain rate were obtained: d~.= exp p 1 . / Nuclear Engineering and Design 237 (2007) 1969-1986 t7. (2006) has also found that the model can predict well the multi-axial creep ductility of G X12CrMoWVNb 10-1-1 steel.1 D £ss. Based on the above model and hypotheses. (24) by 1/3. fh = fc. In British R5 procedure. Fig. (30) is defined as a creep damage factor F CK and applied for depicting the influence of multi-axial states (23) where £ss denotes the steady creep rate without cavities. Applications of cavity growth theory in design criteria and Spindler (2004a) suggests that p = 2.0. d the grain diameter. which is intended to be used when the ductility is a function of the creep strain rate. can be obtained: coalescence at time t= tc.0.b) on Type 316 and Type 304 stainless steels from biaxial creep data and taes the following form: . Similar work was done by Hales (1994): a = ( Õ' ) r+l.04 when failure strain is a function of creep strain rate and p = 0. Applications of cavity growth theory in McDc According to Eq. Spindler et al. we have gO = sinh r2n -1/2 (~)J .H. t = tn.. It is wort noting that Egs. British R5.g. the relationship between the failure strain of multi-axial stress states and that of uniaxial stress is given by: *. L ( a¡ ) ( 1 3am ) J (26) £f ae 2 2ae (31) where p and q are material parameters. The study of Cane (1981a) on 2. Cavity growth based models and their applications were generalized in Fig.ß)ã (0:: a + ß :: 1).133 in Eg. 10. (33) is for constrained cavity growth. land k are material parameters. Huddleston (1985. 1969. (32) is developed for diffusion-controlled cavity growth and Eq. (i 998). 1981a). which was adopted in ASME II: ae=~S¡(::ir eXP(b(~: -1)). For the components in power plants.25CrlMo and 0. The relationship between cavity growth mechanisms and rupture time and rupture strain proposed by Hales (1994). / Nuclear Engineering and Design 237 (2007) 1969-1986 Sf = 2a¡ ( ã ) r+ ¡ Sf 3S¡ a¡ . Accordingly the effcient stress is expressed as a function of ai and ã: ae = (a¡ a) . Most multi-axial creep models were developed on one cavity growth mechanism (Hull and Rimmer. as shown in Fig. (41).V. ae). 1972. models based on plasticity-controlled growth mechanism were used to calculate the effcient stress. Yao et at. a). 1985). tr is given by: (34) Eq. Under multi-axial stress states. A complete description of the effects of cavity growth mechanisms on rupture time and strain was developed by Hales (1994). 10. and there is a low strain rate and a long time to failure.4. diffusion-controlled growth and constrained cavity growth. and Ragab (2002). c-l Ilk (37) where c. Applications of cavity growth theory in MeRe The creep rupture time of a high temperature component (a) Constrained logi! (b) loge depends on stress and temperature (Webster et aI. 1993) further presented an improved effi- cient stress model. a¡ and am' A common explicit form for the function (35) is the power-law relation: It is clearthat a = 0 and ß = 0. while cavity growth depended on the combination of a¡ and von Mises stress ã. (b) The effect of strain rate on ductilty. (33) b . Rice and Tracey. it is reduced to the model proposed by Sdobyrev (1958): ae = aa¡ + (1 .4am + 0. However.2 öl OJ" . 1959. (38) where J¡ is the maximum deviatoric stress defined by J¡ =ai +a2 +a3. (a) The effect of stress on time to rupture. (39) where a and ß are material parameters. Pessimisms in creep damage evaluation can be reduced soundly when transitions between different mechanisms were taken into account in modeling. Cocks and Ashby.2 öl Plasticity where r is the material parameter. rupture time tr can be expressed as a function of temperature T and uniaxial stress a: Fig.b). the first and second are mainly due to viscoplastic strain and diffusion respectively. When the stress is low. Si is the first invariant of the stress tensor defined as Si = a¡ .-T. the cavity growth rate in the whole creep failure process was not always controlled by the fastest growth mechanism. a and b are coefficients determned from cure Cavity growth mechanisms and models and their applications in engineering have been discussed in this section. primar stresses are usually low and the failure is thus controlled by In RCC-MR. In RCC-MR and ASME. tr = C(T)a. Spindler (1994. Further improved models have been proposed by Margolin et al. (39) was adopted by Chellapandi et al. Under uniaxial stress condition. J a? + ai + a~. 2004). the plasticity-controlled cavity growth mechanism is predominant. When ß = 0. For the three commonly accepted cavity growth mechanisms: plasticity-controlled growth. (2006) in the design of a prototype fast breeder reactor of 500 MW under 550°C. etc.1976 H.J¡/3 and Ss is defined as Ss = fitting. 11. (35) (41 ) The effcient stress a e in this function depends on ã. in which the combined influence of cavity growth and cavity nucleation or cavity coalescence on multi-axial creep ductility is considered. the efficient stress is calculated by: ae = aai + 3ßam + (1 . 3.. For the steam generator spigot of modified 9Cr-1Mo. (2001). constrained cavity growth is the result of combination of viscoplastic strain and diffusion. and there is a high strain rate and a short time to failure. According to Beere (1981). Hayhurst. it is reduced to the maximum principal stress when a = 1 and ß = 0 and to von Mises stress when a = ß = O. Cane.3. (40) .867ã.a)ã (0:: a :: 1). 3. When the stress is high.2. This model was pro- posed by Hayhurst (i 972) according to the plasticity-controlled cavity mechanism. Tresca or Rankine in predicting the creep ruptue of 304 stanless under multi-axial stress states (Huddleston. Pressurized tube tests showed that the model is more accurate than the classic criteria ofvon Mises. Summary (36) where v is a stress index and CCD is a temperature dependent constant.a . tr = f(T. Clearly. tr = f(T. there is an increasing rupture strain with the increasing creep strain rate. efficient stress is given by: ae = O.5CrMoV shown that cavity nucleation was determined by the maximum principal stress ai. Spindler et al. 1980. the constrained cavity growth is predominant.. When the diffusion-controlled cavity mechanism is predominant. 2004a. Eq. Models based on voids growth mechanism and their applications in engineering. 2003.1. Krajcinovic (1989). The existence of (Lemaitre. 1996). and ceramics.b). numerical analysis. constrained cavity growth mechanism (Hales. The explosive development in this field can be seen from a large number of textbooks. ling et aI. Betten. CDM-based multi-axial creep design method CDM-based approach was used in the prediction of failure time and rupture strain of and tensor representations of daage variables were introduced (Murakami and Ohno. creep-fatigue (ling et aI. During the past decades. CDM has become an essential complement to fracture mechanics. monographs and reviews. and petroleum and chemical industries. 2003. who presented a method to derive constitutive laws based on the framework of irreversible thermodynamics and the principle of strain equivalence. a brief introduction is given to continuum damage mechanics. and three-dimensional these micro defects is termed as damage defects.H.b. Zhang and Vallappan (1998a. On the microscale. 4. material structure is piecewise discontinuous and heterogeneous because of the existence of micro-defects or CDM interlinks the experiences in micromechanics.b. Creep damage theory and constitutive equations 4. A lot of multi-axial creep damage constitutive equations have been presented to analyze the creep damage and failure of different materials. rock. 1982. Krajcinovic. and points out an intermediate stage existing between perfect and failure. It suggests that any constitutive equation for a damaged material may be derived in the same way as for a virgin material except that the usual stress is replaced by the effective stress. By introducing the damage variables based on equivalence principle and the concept of representative volume element (RVE). two-dimensional defects. e. 2001a. The two branches of solid mechanics are complementar by the scale of the analysis: micro to meso for damage mechanics and meso to macro for fracture mechanics (Lemaitre. Cocks-Ashby model has been constrcted for SIEMENS PG. In addition.1. The above assessment procedures or design rules generally lead to a conservative result. A less conservative model by Spindler including the combined influence of cavity nucleation and growth was adopted in R5 procedure. 1983. Odqvist and Hult (1962) pointed out that Kachanov's concept implied the life fraction rule of Robinson (1952). Becker et aI. Stolk et aI. 2004). Yao et al.b. concrete. The method has been highlighted again since 1990s with the rapid development of computer technology and finite ele- and Ganczarski (1999). by Kachanov (1986). 2003. etc.. Voyiadjis and Katta (1999).b). polymers.. The further development was made by Hult (i 979). is called damage evolution (Skrzypek and Ganczarski. Allx et al. the micro defects can be 'smeared out' and the stress and strain state can be considered as homogeneous. which initiates the macrocracks and causes the progressive degradation of material propertes. Skrzypek high temperatue components by intro- ducing proper constitutive equations and damage variables. 1999). Mustata and Hayhurst. Kattan (2002). through the concept of effective Fig. which makes it possible for the coupling of strains and damages.g. Lin et a1. 1980. the conservatism is usually acceptable in consideration of the safety requirements of nuclear power stations. Wu et al. a concept of effective stress written as a e = al(1 . 2006). 1ntroduction to continuum damage mechanics As discussed in Section 3. Then. The materials include metals. growth and coalescence. e. physics. the strain equivalence principle was presented by Lemaitre (1971). material science. Chaboche (i 988a. Chaboche (1981) and Lemaitre (1985). Nevertheless. to tae damage-induced material anisotropy into account. etc. 1999. In this section. 2004). constitutive equations and their applications in engineering. 2005). Pirondi et aI. (2005b) ment method. The advantage of the approach is that it is easily implemented with numerical methods to simulate the process of damage evolution and thus provide the information of the local stress and strain field. 2006). Chaboche. fatigue (Cheng and Plumtree. line-defects. After that. 2002). / Nuclear Engineering and Design 237 (2007) 1969-1986 1977 damage. for instance. 1983.. i 982. (2002). Kaji et aI. 2002. Lemaitre (2002). 1984).. materials always contain some micro defects from observation at a microscopic level. Gupta et aI. 2004..1. 1994).. CDM has been used to descnbe different damage processes. coal-fired power plants.g. and brittle (Vroonhoven and Borst. stress. Ciska and Skrzypek. This approach is known as the continuum damage mechanics (CDM). A damage variable úJ varing from 0 for the undamaged matenal and i for the full broken material was presented to depict the micro damage of materials. Hayhurst et aI. The process of these micro defects nucleation. Murakami. In many applications. After this. 1981 a.. Dattoma et aI. a detaled discussion is carred out on the development of multi-axial creep and Hayhurst (2005). 2000. Xiong and Shenoi. More attentions are paid on the multi-axial creep damage constitutive . continuum solid mechanics.. Krajcinovic and Fonseka. The CDM was initiated by Kachanov (1958) for the case of creep damage. 2001. creep (Murakami et aI.. Here special focus is put on the creep damage analysis of metals and alloys under multi-axial stress conditions. Lemaitre (1996).úJ) was proposed by Robotnov (1969). 2005a. vector 4. components.-T. 1998. The innovative idea is a challenge to the traditional material mechanics concept 'perfect' and 'failure'. 2006. ¡I. ductile (Brunig.. On the other hand. (1975. g' and r' are material constants. 1984b).) t . were presented. Assuming that w equals to unity when p' + 1 À = &0.1985). no attempt is made to identify the physical nature of the damage parameter and to distinguish between different damage mechanisms. in paricular from a viewpoint of engineering application. Lemaitre constitutive equation & -AI _ w)m" c . 1984a. 2004.g. the cavities are assumed as sphericaL. (45) ( . (1994a. 1984a. p' + 1 ql For the components subjected to complicated stress states. it is possible to describe the tertiary region of creep curve and predict creep rupture life. For uniaxial tension he suggested the constitutive equations: &c = f(rr T.2. Hayhurst et al. creep failure occurs (t = tr). (42) is often given by: are applied where one damage mechanism dominates in the creep rupture process. 1983).2. etc. or the time Also. ml. (1996. Rv is the triaxial factor which reflects the effect of stress states and defined as: Rv = 3(1 + v) + i-(1 . tensors offourth-order or even eighth-order. Norton creep law can be used to describe the stresses and deformation occurng in primary and second creep stages. Kowalewski et al. However. there are two kinds of multiaxial creep damage constitutive equations with single varable: Kachanov-Robotnov constitutive equation and Lemaitre constitutive equation. By the selection of functionsfand g. However. p' (1 . we have: tr = . Hyde et al.1. 1983). a dominant damage parameter is defined to depict the state change of materials and the performance degeneration of strctures. Additionally. (i 994a. An explicit formulation for Eq.2v) Õ' ' t- 1 r . to local failure in structure components. However. Othman and Hayhurst (1990). e.2. (48) . Hayhurst (1975. i 984). T. In the cavity growth models discussed in Sections 3. The constitutive equations with single damage varable To reflect the deterioration of materials and describe the tertiar creep stage. nickel-based superalloy.(1 + q')A'aP" t:r = t:R i . t:R = Àt:*. (44) 3 (am)2 ( (i t)I/ÀJ where (46) where v is Poisson ratio. by Hayhurst et al. 1985): material constants and can be determined from uniaxial creep tension tests. t:* = &ofr.= ' B'all. Lemaitre (1979. / Nuclear Engineering an Design 237 (2007) 1969-1986 equations using a scalar variable for damage measurement. 4. Integrating the equation set under the conditions of w = 0 when t = 0 and w = 1 when t = tr. 1984. the rupture time and strain are obtained by: w=l. it is necessar to analyze their creep behavior in the tertiar region of the creep curve.2 and 3. These constitutive equations were used to predict the creep failure of different high temperature materials. (1994). The original work was done by Leckie and Hayhurst (1974).1978 H.2004). titanium alloy. Simple formulae were developed to determine the time of crack initiation and a lower boundar on the rupture time of strctures. w).-i . Kachanov (1958) proposed a phe- nomenological method. 1974).b. 2006). the assumption of isotropic damage is suffcient to give a good prediction of the carring capacity. with single or multi damage varables.2. Xu (20ooa.r' J i/(l+a) where a'. for instance.-I: Yao et al. p' and ql are time and temperature dependent Lemaitre constitutive equation under multi-axial stress states usually takes the following form (Lemaitre. In general. Perrin and Hayhurst (1996. Creep damage constitutive equations In the design of high temperature components. w).3. for the life of components controlled by rupture. a lot of multi-axial creep constitutive equations. 2001. it is essential to generalize the creep damage constitutive equations from uniaxial stress conditions to multi-axial stress.1-Rv(1+a/)(f. (Õ'jÀ)-r Rv(a + 1) (47) Then Eg. eliminate any hope for the preservation of the physical clarity of the model (Krajcinovic.b). B'. w=B' a .2.(1 all . in which A'. The former can be regarded as the generalization of classical Kachanov-Robotnov equation from uniaxial stress to multi-axial condition. The latter is based on the framework of irreversible thermodynamics. Then. From the Reference stress was introduced to take into account the effect of stress redistrbution and multi-axial stress states. the choice of a scalar variable for spherical cavities is a rational one (Krajcinovic. w = g(a.1. In creep equations with single variable. geometrical viewpoint. 1999).1. nl. austenitic stainless steel and ferritic steel. 4.1984. this treatment is limited to kinematically determinate strctures. 4. So the original measurement on Kachanov damage is stil appealing. it is applicable to other structures through the assumption of stationar state pattem for deformation during creep rupture process (Leckie and Hayhurst. Kowalewski et al. 1979. Multi-axial creep equations with single variable deformation or creep rupture.b). The calculations are easily performed because of the scalar nature of the damage variable (Lemaitre. (42) where f is the strain rate function and g denotes the function of damage rate. 1984b).w)a' (43) This is the so-called classical Kachanov-Robotnov Equations. aluminum alloy. 1983. consideration must be given to the possibilities of failure due to excessive creep etc. 3. (45) can be rewritten as: W=l-(l tr ~) 1/(l+it).2. 16.25V steel at 530 °C._ g In Egs.H. Hyde et al.b) obtained: V) w=l. However. p" and q" are material constants. of first category.j = ~AII(ä)ll-i Sij((1 _ pI!) + (1 _ w)-n). 4.. i 980.1423 x o M -q -q e. (49) when pI! = g" = 1. (2002) based on Eq. (200la.2.p" dw II II (a ai +" . To tae into consideration of the effects of these different damage mechanisms on creep failure. (49) should be rewritten in the following form: d. Dimensions of the T-joint. 1960) shown that the rupture of materials can be classified into two categories. as shown in Fig. with where £ ij is the creep strain tensor.3. Hayhurst.5MoO. (50) reduces to Eg. For the creep design of T-joint.a -w . is governed by the maximum principal stress criterion. In use of Eg. traxial and multi-materials creep and damage situations were caried out. is directly related to shear stress. lower than the local strain prescribed in ASME Code N-47 (5%). which can be described by the following general form (Hayhurst. . 1996): The benchmarks of numerical modeling for creep continuum damage mechanics was studied by Becker et al. consideration is given to the physical nature of damage parameter.15 for nickel-based alloy and a" = 0 for titanium alloy.25V steel at 640°C. Multi-axial creep equations with multi variables " (1 . The stress triaxial factor Rv in an axisymmetrc semicircular notched specimen was computed by using the FE software. 1986. Hyde et al. Consequently. Based on the uniaxial creep test data of ZbNCT25 alloy at 650°C.5CrO. (1996) proposed a less expen- the creep rupture life of alloy steel can be predicted accurately by Lemaitre constitutive equation with the modification of the triaxial factor Rv. (1996) suggested that a" = 0. dú) II (ar)P" di = B (1 + q")(1 . and tr = (ä /787.6% when 1 % allowable strain is used in the straight pipe.w)a" .w)a''' wcr = 1 (49) representing uniaxial. 1983. (50). (2003) analyzed the strain allowances of pipes of 0. A" . which are time consuming and costly. Hayhurst and Felce.. It is found that the dangerous location is the throat of the notched specimen where Rv is 1. Good agreement was achieved between the test results and those from two independent damage codes.i (1 1i)1/(J+q") Wcr . etc.-T. Johnson et al. (2) rupture stress is the function of effective stress and maximum principal stress. tr ( t)0. The value of a" can be determned by the tests of uniaxial tension. 12. the rupture criterion of second category. (1956. In the study of Wang and Guo (2005).5CrO. 1974. biaxial. The value is damage value is often less than unity. Moreover.569 7. pure shear and equal biaxial tension (Hayhurst. Wcr denotes the critical value of damage.2 'j )-Il dt .2. B". Hyde et aI. (50) and the finite element frame. grain boundar slide. ductile void growth. It is obvious that Eg. However.1-.. gll and a/I are material constats. and ar is the creep rupture stress.. studies on metal physics and void growth theory high temperature material results shown that the deterioration of from different mechanisms. the Lemaitre constitutive equation was used to analyze the creep performance of an aero-engine material (lMI834).g. sive procedure to determine the value of a" by comparson of FE results with experimental results obtained from Bridgman notch specimen. (45) and (50). an in-house code (FE-DAMAGES) and a commercial code (ABAQUS. including aluminium alloys. (200 I a. 1972). Value of a" is very close to zero for the first category and to unity for the second. 4. Four different types of test d£ij 3AiiS (-)"-i(l --. e. / Nuclear Engineering and Design 237 (2007) 1969-1986 1979 The constitutive equation has been used by ling et aL. Yao et al. diffusion of vacancies along the boundary and car- The parameter a" is introduced to reflect the influence of multi-axial stress states on creep rupture of materials. it was suggested that the allow- is usually heterogeneous. the multi-axial creep Kachanov-Robotnov constitutive equations can be expressed as (Leckie and Hayhurst. Tests by bide precipitate coarsening. multi-varable constitutive equations were thus developed.a )a) di = g B (1 + q")(1 . 12.03Rv Fig... Furthermore.2)-8. Hariy. and (3) critical able strain in knuckle region should be less than 4. Jing ct aL. there is only one damage varable and no (50) where pl/. Chen et al. Kachanov-Robatnov constitutive equation Typically. The matcrials used for creep damage calculation are titanium alloy at 650°C and 0.UMAT).b) in the multi-axial creep prediction of an aero-engine turbine disc. Considering that: (1) damage of materials the help of commercial code ABAQUS. the analysis of Rv with the help of finite element software ADINA indicates that complicated stress states wil accelerate the damage process and thus significantly reduce the creep life of disc. The rupture time . including copper.5MoO. the Eq. wi(i = 1.)aKK (58) (54) where W3 is the mobile dislocation accumulation state variable where Sij is the so-called deviatoric stress tensor and defined by: S. T) Wi_ C"IN' (ai) . 2. Perrin ancl Hayhurst. a Sinh-function is adopted to model the creep strain rate over a wide stress range. n) is the ith damage variable.wn.. the majority ofIifetime consumed in the tertiar stage. Perrin and Hayhurst. A length discussion on the above physical mechanisms can also be found in the publications of Dyson (1988). where H* is the saturation values of H at the end of primar creep and subsequently maintans such a value until the occurrence of creep failure. C'" is a material parameter. Il = hÊe (1 _ ~) For the construction of damage rate function. By introducing different damage state variables in the governing equation and in conjunction with the corresponding damage rate function. W2."'.' a'j .coth(BII/Ö'). B'll (53) Based on the assumption of normality and the associated flow rule. It is believed that the mobile dislocation accumulation should be responsible for the tertiar creep stage (Dyson and McLean.b) presented that the nucleation and growth of cavity are dependent on the maximum principal varable and vares from zero at the beginning of creep process to H*. Hayhurst. i 996).. . I) 3Sij 2 a. and gi(i = 1. For most nickel-based superalloys undergoing creep. Wi.WI. the multi-axial governing equation is proposed by Othman et al. oij the Kronecker delta and aKK obeys the rule of summation convection. . So precipitate coarsening is also an importt cause of creep damage and a damage varable is needed to reflect the effect of this process. 2005)... T) stress and von Mises equivalent stress. DJ/ is a material parameter. a . instead of the sequential one. Wn. Wn. a function for damage variable evolution is introduced by Kowalewski et al.. / Nuclear Engineering and Design 237 (2007) 1969-1986 2005): Êij = f(aij.. 1988): vff Wi =gl(aij.o. n) is the ith damage rate function.Se (51 ) Ö' (56) Wn = gn(aij. (1993): W3 = D'II(1 (3)2èe. Engineering alloys are often strengthened by a dispersion of precipitated particles which are unstable with respect to time and temperature. . By selecting the appropriate strain rate functionfand damage rate function gi. wi(1 . wi.) = 3 (55) and vares from 0 to 1.wi.-T. 1994a.W2)4' = Kc 3 (57) where W2 is the precipitate coarsening state variable and varies where Alii and B'll are material parameters.1. 1988. to describe the primary creep due to the initial strain hardening and the formation of dislocation microstrcture.5-3. This parallel process requires the creep strain having a hyperbolic sine dependence on the applied stress (Perrin and Hayhurst. The function for cavitation damage can be derived from the study of Cane (Dyson. . . particular consideration should be given to the following mechanisms: cav- a. wi"". Kowalcwski et al. Canc (i 98 i a. (1994a) and Perrin and Hayhurst (1996): where aij is the stress tensor. A rate function for dislocation accumulation damage was proposed by Dyson (1988) and Othman et al. So the effect of cavitation damage should be explicitly represented in the constitutive equations. H is the primar creep state itation damage from cavity nucleation and growth. . N = 1. (52) . The evolution of the proposed damage varable may be derived from the coarsening theory (Dyson.. Wi is the cavitation damagc variable and varies from 0 to 0.. for ai being compressive. . (1994a): Ê" = 3. Dyson and Osgerby (1993) proposed that dislocation climb and restrict the deformation within the grain interior.. Kowalewski et al. N = 0).. (I 993) and Kowalewski et al. The strain rate equation without consideration of damage variable can be written in the follow uniaxial form: Ê = AI/I sinh(B"I a).3. 1983).1980 H.T) lÓ = g2(aij. 1993. Strain rate and damage rate functions In the multi-variables damage equation. The equation can be from 0 to 1. and 2. N is a parameter to indicate the state of loading (for ai being tensile. Yao et at.. T) where Ê ij is the creep strain tensor. d" is the so-called sensitivity index of multi-axial stress and takes a value in the range of 0. Wn. The precipitates on grain boundares provide a site for nucleation of cavities and the precipitate coarsening may 4.. The nucleation and growth of cavities reduce the load bearing section and accelerate the creep damage.H*' (59) where h is the material parameter. 1996. Kc is a constant related to the initial paricle spacing extended to multi-axial conditions through assuming an energy dissipation rate: A'II tj = . the creep behavior of materials can be accurately described. 1996): glide in creeping materials occur as a parallel process. .. dislocation accumulation and strain hardening during the primar creep (Othman et aI. (1994a) and Perrin and Hayhurst (1996).3. Wi. precipitate coarsening.tj = ~A'I/ (Sij) sinh(B!!a-) and temperatue. In addition. Wi. aij the stress tensor. where Êe is the effective creep strain rate defined as Êe = (2Êijèij/3)1/2. (61). h(B"1-). and combined tension and torsion. 1999. the material constants in Eq..b ): Replacing the te in Eqs. (1 . crosswelded tension plates and T-branched welded pressure vesseL..b) from the experimental results and an automated numerical optimization technique.. Constitutive equations with double variables Table i Comparson of estimated failure time for notched tube with actual time by Hsiao and Gibbons (1999) (565 "C.1 . and the stress states of this point. I! .1-wi WI = C A ..1) 2(1-wi)" . Eq. (1994) made use of Eq. (63) has been used in the prediction of the creep rupture life of internally and externally notched tubes of 2.H)J effective stress levels and for three stress states of tension. a trivarables type of constitutive equation was developed by Pcrrin a. as shown in Table 1. (l994a. a constitutive equation including variables WI. . The constitutive equations were adopted again by Lin et al. / Nuclear Engineering and Design 237 (2007) 1969-1986 1981 lot of explicit forms for multi-axial creep deformation with the similar style of equation set (51) can be obtained.H)J ill (ai)ul! .(1 (64) Eg.H)coth H)J =-W2iBIIa-(1 . Ci A 3 AI! Wi(1= Wi)nNa. Subsequently.sinh . the material parameters were determined following the approach of Kowalewski et al. The estimated results were compared with the experimental data of notched tubes. The comparison showed that and Hayhurst (1996). (62).wi)" . to study the creep behavior of axisymmetric ally n BIIIa-(1 .li) sinh r BI!a-(1 . (64) have been determined by Kowalewski et al. (2005a) to depict the creep rupture of pure copper at 250 °C and aluminum alloy at 150 °C under combined loading. Mustata and Hayhurst. Iii I!(ai )U'" N sinh(BI!a-) a. In addition.5MoO. the CDM-based approach provided a very close result to the experimental data.25CrlMo steel under the support of a finite element program. Hayhurst et aI.H)J -l sinh . Numerical results indicate that creep behavior can be represented in terms of a 'skeletal stress' located at a point within the notch throat.(Sii) sinh(BI!a-) t.Wi) . The comparison of experimental data and computed effective creep strain was performed and indicated that Eg.WI)(1 .b). together with a continuum damage mechanics finite element based solver.2.Kc . the tr-varab1es tye of governing equation was widely applìed in the study of multi-axial creep analysis of 0.g.W2 . Hayhurst et al.wi ' (63) Eq. .= -A .3. pure torsion. L BIIa-(1 . Tri-variable constitutive equations To interpret the creep behavior of aluminium alloy under multi-axial stress states. = 2 a.5CrO. (56) and (58) by Eg. butt-welded pipes.3 -. Hayhurst (2005) verified the existing CDM-based tri-variables models in use of the welded ferritic steel components. Therefore 1/2 = AI! sinh(BI!a-) . H* 1 ..H)J II = ~AI!! (1 .25V and 2. (64) can predict the isochronous surface of materials precisely.coth(BI!a-). 2005.wi ' w?=Kc . (62) could not be used when the uniaxial ductility of the material was less than 1 % (Hayhurst et aI.-T. ABB-MARC..H. a. e.Wi)n' 1) 2 a. Based on the physical mechanism analysis of deformation and rupture of ferritic steel.3. . Hsiao and Gibbons (1999) developed a constitutive equation For the aluminum alloy at 150°C. the life predicted by ASME N-47 and that from Larson-Miller approach.. wi)4 W2(1= .1 .wi)4 d£ij 3 iii (Sii)' iBIIa-(1 .W3)" 4. (1993) to improve the prediction of multi-axial creep behavior of nickel-based super- where n is given by: alloy. 2005a. an explicit form for constitutive equation set (5 i ) is thus obtaned: t. 4. To predict the multi-axial creep behavior of ferritic steel.sinh . (1994a. iB1IIa-(1. (1994a).. 3 (62) ( Si). W2 and H was developed by Kowalewski et al. = ~AI! (1 . (1993) obtained a constitutive equation by introducing a cavitation damage state variables wi and a dislocation multiplication state variable W3: ME Code N 47 Lason miler CDM Expenmental results 33 128 2535 2907 325 3520 3844 t.(1 . DAMAGE XX. However.25CrlMo ferrtic steels and their weldments over a wide range of stress and temperature (Perrin and Hayhurst. h H=AI! -( wi)" iBIIa-(l-H)J' 1--H). te = (2tI3Jtii) (61) (1 . Creep tests were cared out for both materials at thee including the precipitate coarsening state variable W2 and the primar creep state variable H: dt 2 a. Othman et al.W3)(1 . 1994).W3)(1 .sin a W3 = (1n W3) . Yao et at. 1 I-wi (65) notched tension bars of a nickel-based superalloy. Dil A (1 .(Sii) sinh(BI!a-) 1) ~AI! (1-WI)(1-W3)n' (60) where n = B"!a. (62) has been used by Othman et al.W2 sinh a.3.H* L 1 . 100 MPa) Internally notched tube (h) AS Externally notched tube (h) 33 Following the above scheme. H) J f Table 2 age and creep rupture. (1996) on Waspaloy alloy (700 DC) and IM1834 alloy (650 "c) Hyde et al.5Mo steel (550 'C) Hyde et al. (2oola. 2004). (2002) on CrMo V steel (640°C) Chen et al. K c wi)4 wi=(1 . (2005) for 0.sinh w=wifi. 2001). To eliminate such a discrepancy. 2001. (I 994b) Kowalewski et al.25 V steel (640 GC) and titanium alloy (650 'C) Hyde et al.J 1. the model originally proposed by Cane (198 i b). exp 3am J 2& and (67) erning equation by Xu (2oo0a. Yao et al.5CrO.V steel (575-640 "C) Hyde et al. ê"3=iiiA . (2005a) for CrMoV (565-640 DC) Mustata et al.5CrO. it was found that Eq. (Bill &(1 . r-I 2& (68) Thefunctionfi derived from Huddleston's work (1985. was used to reflect the elTect of stress states on cavitation evolution.-T.5MoO. New development oftri-variable constitutive equations From the nature of cavity nucleation and growth. (2005b) for Cr-Mo.1982 H.25V (565-675 GC) Hayhurst et al.5CrO.(Sij). (2002) on aluminium alloy (150 DC) Un et al. (5 i) could not lead to a sound prediction for the creep strain at failure and significant discrepancy was reported between the estimated results and experimental data (Xu.(1 --wi)(1 . . (62) and (64). (I 994a) Hayhurst et al.(1-W2)(1-W) J.I + q h = 1. two functions of stress statesfi andh were introduced into the gov- .3 .25V steel (600-75 "C) Kowalewski et al. (2002) on 0. (56). which can be written as: . where (66) fi = (3Si ) a ((b s: -)1 . 2000a. (BII &(1 .H) I) 2 & sinh r exp p 1 . (( ai ) ( & 3tTm2'). The Summar of the multi-axial creep damage constitutive equations and applications in engineering Constitutive equations and damage variables Constitutive equations with single varable Contrbutors and applications Kachanov-Nobotnov constitutive equations A dominant damage variable W is used to describe the state changes occurred in materials Hyde et al. (2005) on copper (250°C) and aluminium alloy (150°C) PelTin and Hayhurst (1996) on 0.5Mo 0.5CrO.b. However. H h=iii (AH).4.5CrO. namely Eq.5MoO. & H* I .25Cr1Mo steel (565 "C) Hsiao constitutive equations Varable W2 for precipitate coarsening and H for strain hardening from primary creep Xu constitutive equations Vanable W¡ for cavitation damage and Hfor strain Xu and Hayhurst (2003) on 316 stainless steel (550 'C) hardening during the primar creep stage Constitutive equations with tr-varable Varable (ù¡ for cavitation damage. (1994) on nickel-based superalloy Hsiao and Gibbons (1999) on 2.5MoO. / Nuclear Engineering and Design 237 (2007) 1969-1986 4.H) . (2003) on O. (2006) on P91 steel (650 'C and 625 'C) Lemaitre constitutive equations A dominant damage variable w is used to describe the state changes occurred in materials ling et al. CIIAIIN' h ( BII&(1 .w)J . 1993) is used to couple the effects of tertiary deformation.3. The function h proposed by Spindler (1994) and Spindler et al. (1-W2)(1-w) . In the constitutive Egs.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) Constitutive equations with double varable Othman constitutive equations Variable wi for cavitation damage and (Ù3 for mobile dislocation multiplication Othman et al. W2 for precipitate coarsening and H for strain hardening during the primar creep stage on aluminium alloy (150 'C) on aluminium alloy (l50"C) Li et al.25V steel (600-75 °C) PelTin and Hayhurst (1999) on 0. (2001) is employed to describe the effect of multi-stress states on the damage evolution.b. (2006) for P91 steel (650°C and 625°C) . (1999) on 2.5Mo 0.25Cr1Mo (640 'OC and I CrO. creep dam- Wi = sin 1. (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. a lot of material damage models have been developed to predict the creep failure. (1993) on nickel-based superalloy Hayhurst el al.25 V steel (530 DC) Becker et al. once damage occurred in the material. The existing multi-axial creep constitutive equations and their applications are summarized in Table 2. an automated numerical optimization technique proposed by Kowalewski et al.25V ferritic steel at 590°C. The great advantage of CDM-based approach is creep regime. the creep evolution is dominated by the damage variable w. two advantages are revealed in multi-variable equations: (i) Multi-variables are introduced to distinguish different mechanisms and effects on the damage evolution. are used frequently. 1988) that creep damage tolerance parameter (À = er!S¡tr.-T. Conclusions Since the primar. Generally speaking. 70-100MPa).. 1999. It reflects the effects of damage evolution on stress redistribution and strain accumulation in components so as to reduce conservatism in creep design. (ii) A Sinh-function is adopted to replace the traditional power law and to describe the stress sensitivity of creep rates over a wide stress range. It has been shown (Goodall et aI.e. 2004). (49) and tr . 0. Therefore creep failure is mainly controlled by the accumu- that it can be used in conjunction with finite element method to provide information on the local stress and strain fields. the existing multi-axial creep analysis models or constitutive equations are generally sensitive to material and temperature. .. 4. but the size of finite element should not be less than the minimum of the RVE (representative volume element) (Skrzypek and Ganczarski. These data can be used for the creep design of different engineering components operated at high temperature through extrapolating in four ways.5MoO. finer finite element References AlIx. Continuum Damage Mechanics of Materials and Structures. According to the technique. i 914). However. namely data collection. As was expected. 1999). In addition. the single variable Eg. Summary The CDM-based method and constitutive equations for creep design under multi-axial stress states were reviewed in this Section. both single For both CGM-based and CDM-based methods. namely. 2002. The results shown that for the single variable model. In: Allix. While for the tri-variable model. (49) and (64) very close to each other at the higher stress levels (e. it is importnt to know the nature of creep damage and thus to develop a proper analysis modeL. when the creep failure is controlled by one dominant damage mechanism. Dyson. Dragon.variable Eq. 50505012). A 'nodal release technique' is usually needed in implementing CDM-based method in creep analysis (Hsiao and Gibbons. with paricular emphasis on the CGM-based and CDM-based multi-axial creep design methods. it is beneficial to propose a CGM-based method for initial design whilst to develop a CDM-based method for life extension or integrity assessment of the serviced strctures. lation of varable wi. to life extrapolation.g. F.4. Hild. 1975. and tri-variable equations. variable equation and tr-variab1e equation are appropriate for the description of multi-axial creep behavior. namely equation Set (46). the value of cavitation damage variable wi is much larger than that of the precipitate coarsening variable W2. the nucleation. Continuum damage mechanics of materials and structures: present and future. size wil lead to an improved prediction. where s¡ is ini- Acknowledgements The authors are grateful for the supports provided by China Natural Science Foundation (50225517. secondar and tertar uniaxial creep processes were identified by Andrade (1910. For a given material under the given temperature and stress range. In this work. The CDM-based method is derived from the innovative work of Kachanov. the predicted results of failure time by Egs. good agreement between the experimental observa- 5. (64) in the prediction of creep fail ure of P91 steel. On the other hand. Therefore it has been adopted in many design codes or assessment procedures to predict the creep deformation and rupture of high temperature components under multi-axial stress conditions. 2006). Compared with the single variable equation. Hild. The CGM-based method is developed on the basis of equation Set (64). (2006) compared the single variable Eq. (1994) is usually needed to determine the material parameters required in creep analysis. Since creep damage mechanisms are dependent on materials and temperature. growth and coalescence of cavities under stress and temperatue were accommodated in the CGM-based models. (Eds. (49) will overestimate the lifetime of components at the lower stress level (Hyde et aI.. A. Yao et at. the corresponding elements are considered to be 'death' and will be removed in the FE modeL. 2003).H.. the first way was discussed. Nevertheless. Sta Program (05QMXI416) and Fok Ying Tung Education Foundation (101054).. FZ would also wish to thank the supports provided by Shanghai Rising- tial strain rate) should to be greater than 5-10 to extend local lifetimes sufficiently and to justify the safe usage of upper bound estimates of component lifetimes. (66) was achieved (Xu. For multi-varable equations being non-linear and strongly coupled. It can be seen that the single varable Kachanov-Robotnov constitutive equations. 1-15. to multi-axial state of stresses. / Nuclear Engineering and Design 237 (2007) 1969-1986 1983 improved equation (66) was applied to the prediction of creep behavior of 0. many experimental investigations have been cared out and focused on the tions and estimation from the developed Eq. F. i. in paricular in the prediction of multi-axial creep behavior of low-alloy ferritic steel.5CrO. 2001. 0. such a kind of steel is widely used in components in fossil and nuclear power stations operated under the physical modeling for microstrcture evolution of materials under external loading. to various loadings and to aggressively external environment. The main failure causes of high temperature components. pp. Yatomi et aI. Hyde et a1.. 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