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Materials Science and Engineering A 527 (2010) 1850–1855

Contents lists available at ScienceDirect

Materials Science and Engineering A

journa l homepage: www.e lsev ier .com/ locate /msea

omparison between impression creep and uni-axial tensile creep performed onickel-based single crystal superalloys

uzhu Yan, Shifeng Wen, Jun Liu, Zhufeng Yue ∗

epartment of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710129, PR China

r t i c l e i n f o

rticle history:eceived 8 July 2009eceived in revised form 2 November 2009ccepted 15 November 2009

a b s t r a c t

The purpose of the present work is to build the relationship between impression creep and conven-tional uni-axial tensile creep in determining crystallographic creep parameters for face centered cubic(FCC) nickel-based single crystal superalloys. To this aim, the impression creep performed on [0 0 1]-,[0 1 1]-, and [1 1 1]-oriented nickel-based single crystal superalloys were respectively investigated and

eywords:mpression creepni-axial tensile creeprystal plasticityinite element

the data was compared with those obtained with uni-axial tensile creep counterparts. It shows that thedetermination of crystallographic creep stress exponent is independent of crystallographic orientations,and the results agree reasonably well between impression creep and uni-axial tensile creep. However,some reveres dependence of crystallographic orientation was observed between impression creep anduni-axial tensile creep, such as the steady-state punch velocity for impression creep and steady-stateelongation rate for uni-axial tensile creep, and this induced different stress conversion factors among

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different crystallographic

. Introduction

The progressive use of nickel-based single crystal superalloys inero-engines service as turbine blades challenges the researcherso exploit new method to evaluate its in-service creep proper-ies [1–3]. In such applications, impression creep, an elegant testechnique wherein the conical or ball indenter is replaced by aylindrical flat bottomed punch owing to a constant punching stressuring creep, has been highlighted for exploring creep properties ofaterials for its simplicity, efficiency and non-destruction merits

4–7].A key problem for impression creep test is to bridge it and

ni-axial tensile creep in determining creep parameters of mate-ials. The first numerical simulation of impression creep wasarried out by Yu and Li [8,9]. Assuming a power law consti-utive equation between von Mises stress and creep rate, theyxploited:

˙ = Ci�niN (1)

= ni = � ln ε

� ln �eff= � ln h

� ln �N(2)

∗ Corresponding author. Tel.: +86 29 88431002; fax: +86 29 88431002.E-mail addresses: yanwuzhu24@mail.nwpu.edu.cn (W. Yan),

fyue@nwpu.edu.cn (Z. Yue).

921-5093/$ – see front matter. Crown Copyright © 2009 Published by Elsevier B.V. All rioi:10.1016/j.msea.2009.11.035

tations.Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.

where h is the steady-state velocity of the punch, �N is punch-ing stress, n is creep stress exponent, �eff is the von Misesequivalent stress, ε is the steady creep rate, ni and Ci are the expo-nent of the punching stress and the temperature and materialdependent pre-exponential factor for impression creep, respec-tively. Recently, this equivalence was further validated by Dornerand Liu [10,23] who performed the impression creep test anduni-axial tensile creep test on TiAl alloy and aluminum alloy2A12.

Recently, the impression creep technique has been appliedto extract crystallographic creep parameters of anisotropic sin-gle crystals. Xu et al. developed two numerical frameworks todetermine the crystallographic creep parameters of nickel-basedsingle crystal superalloys [11] and studied crystallographic orien-tation dependence of surface morphology of impression creep [12].However, the study of crystallographic orientation dependence ofdetermining crystallographic creep parameters using impressioncreep test and uni-axial tensile creep test has rarely been reported.Moreover, the correlation between impression creep test and uni-axial tensile creep test performed on anisotropic materials is stillunclear.

The present study started by carrying out numerical simula-tions of impression creep and tensile creep performed on FCC

nickel-based single crystal superalloys. Our emphasis was placedon investigating the crystallographic orientation dependence ofdetermining crystallographic creep parameters as well as relat-ing impression creep with uni-axial tensile creep for differentlyoriented single crystals.

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W. Yan et al. / Materials Science an

. Materials and finite element (FE) model

.1. Constitutive relationship of creep

Assuming the second stage of creep to obey the Norton’s creepaw, the evolution of resolved shear strain rate � is calculated from:

˙ (˛) = a(�(˛))n

(3)

here a and n are creep material parameters, the superscript ˛pecifies the slip system. When a stress � is applied, the resolvedhear stress �(˛) on slip system ˛ can be calculated by the followingquation:

(˛) = � : P(˛) (4)

ith

(˛) = 12

(m(˛)n(˛)T + n(˛)m(˛)T) (5)

here m(˛) denotes the slip direction of slip system ˛ and n(˛) theespective slip plane normal. The creep strain rate ε can be takeno evolve as:

˙ =N∑

˛=1

� (˛)P(˛) (6)

here N is the number of the active slip systems.Assuming that the creep deformation does not affect elastic

ehavior, we have:

� = D�ε (7)

here D is the anisotropic elastic stiffness tensor.The transformation between the local crystallographic coor-

inate system ([0 0 1]–[0 1 0]–[1 0 0]) and the global coordinate

ystem (x–y–z) can be written as:

′ = TDTT (8)

here D′ and T are the deviatoric elastic stiffness tensor and theransformation tensor, respectively.

Fig. 1. (a–f) Impression creep and uni-axial tensile creep mode

ineering A 527 (2010) 1850–1855 1851

The above constitutive equations were implemented into thecommercial FE software ABAQUS via a user-defined subroutineUMAT [13], and its effectiveness has been validated through anindependent studies of crystallographic creep parameters andcreep damage evolution of nickel bases single crystals [11,14].

2.2. FE model

Two FE models were concerned in the present study. Theyare associated with impression creep and uni-axial tensile creep,respectively. For the impression creep, a three-dimensional (3D)semi-infinite solid pressed by a rigid flat-ended punch is con-structed as shown in Fig. 1(a)–(c). The radius of the punch is R,the size of the specimen was chosen 10R × 10R × 10R to eliminatethe boundary effect. The boundary conditions are as follows: thebottom surface is constrained in the z-direction, the left-side is con-strained in the x-direction and the back surface is constrained in they-direction. The mesh near the contact zone is refined. Frictionlesscontact condition was assumed between the punch and the speci-men. A simplified cylindrical specimen with a diameter d and lengthl0 = 10d was employed to simulate the uni-axial tensile creep test(shown in Fig. 1(d)–(f)). The boundary conditions are as follows:the gripped end is constrained in x-, y- and z-directions and theloading end is fixed in the x- and y-directions to eliminate the devi-ation from the loading axial which would be induced by anisotropyof single crystals.

2.3. Material

The mechanical parameters of nickel-based single crystal super-alloys listed in Table 1 are used throughout the present study [15].

simulate the deformation of the FCC-family metals under certainconditions. The octahedral slip systems can be written as {1 1 1}〈1 1 0〉. The same creep parameters are used for different crystal-lographic orientations to highlight the effect of crystallographicorientation on determining crystallographic creep parameters.

ls for [0 0 1]-, [0 1 1]- and [1 1 1]-oriented single crystals.

1 d Engineering A 527 (2010) 1850–1855

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852 W. Yan et al. / Materials Science an

. FE results

.1. Effect of crystallographic orientation on impression creepnd uni-axial tensile creep behaviors

In the past decades, the creep behavior of nickel-based sin-le crystal superalloys for different crystallographic orientationsas been investigated by experiment [16–18]. It is reported thathe crystallographic orientation has a dramatic influence on creepehavior of nickel-based single crystal superalloys. In the presenttudy, we assume the same crystallographic creep parameters (pre-ented in Table 1) for different crystallographic orientations so aso focus our study on the effect of crystallographic orientation.

Fig. 2 shows the impression creep (Fig. 2(a)) and tensile creep

Fig. 2(b)) behaviors for [0 0 1]-, [0 1 1]- and [1 1 1]-oriented singlerystals. From Fig. 2(a), it can be seen that both of the impressionepth and penetration velocity associated with crystallographicrientation [1 1 1] are the largest among the three different crys-

Table 1Mechanical parameters used throughout the present study.

Properties Value

Young’s modulus/MPa 86,300Possion’s ratio 0.33Creep pre-factor/(s−1 MPan) 1.0 × 10−15

Creep exponent 5.11

ig. 2. Effect of crystallographic orientation on creep behavior for (a) impressionreep (P = 600 MPa) and (b) uni-axial tensile creep (P = 200 MPa).

Fig. 3. Determination of the steady-state velocity of the punch using extrapolationmethod. The crystallographic orientation is [1 1 1], P = 600 MPa.

tallographic orientations for the same creep time and the samepunching stress. The main reason is that more slip systems are acti-vated for [1 1 1]-oriented single crystal. It can be summarized as:

h(h)-[1 1 1] > h(h)-[0 1 1] > h(h)-[0 0 1]

where h and h are the penetration depth and steady-steady velocityof the punch. For the uni-axial tensile creep, however, the oppositeresult was obtained (see Fig. 2(b)):

le(le)-[0 0 1] > le(le)-[0 1 1] > le(le)-[1 1 1]

where le and le are the displacement of the loading end and theelongation rate of the tensile specimen, respectively.

It can also be seen that the impression depth h vs. creep time tcurves experience two creep stages: the transient creep stage witha sharply decreased penetration velocity of the punch followed bya steady-state creep stage. The accelerated third creep stage cannotbe observed for impression creep. For the uni-axial tensile creep,on the other hand, the accelerated creep stage was quite obvious.From Fig. 2(b) we can conclude that the third creep stage arrivesfor the [0 0 1]-oriented single crystals first, and latest for the [1 1 1]-oriented single crystals.

3.2. Determination of crystallographic creep stress exponent

Xu et al. [11] has proposed the following equations to describethe behavior of impression creep performed on nickel-based singlecrystal superalloys:

h = APn and � = �P (9)

where A and n is the impression constant and crystallographic creepstress exponent, P is the punching stress, � is the resolved shearstress. From Eq. (9), the following equation can be deduced:

h = Ci[�]n (10)

Thus, the crystallographic creep stress exponent can be deter-mined using a fitting procedure. For the impression creep test, aseries of simulations were carried out under different punchingloads. The steady-state punch velocities were extracted from FEMresults using extrapolation method, which was first exploited by Yu

and Li [8]. The velocity of the punch was plotted as a function of thereciprocal time as shown in Fig. 3. The data points were then fittedby multiple-order exponential decay function and extrapolated toinfinite time to find out the steady-state punch velocity. We thenplot the steady-state punch velocities in a double logarithmic grid

d Engineering A 527 (2010) 1850–1855 1853

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[�(˛)t ]

ni (17)

Fig. 5 shows the determination of stress conversion factorC1 of the [0 0 1]-, [0 1 1]-, and [1 1 1]-oriented single crystals

W. Yan et al. / Materials Science an

s a function of corresponding resolved shear stress. The crystallo-raphic creep stress exponent can be directly obtained as the slopef the log-log plots.

For the uni-axial tensile creep, the crystallographic creep stressxponent was obtained using an analogous fitting procedure asmpression creep test, such as:

t = �(lg ε)�(lg �)

(11)

here � is the resolved shear stress, ε is the steady-state strain rate.he strain ε was calculated as the percentage elongation:

= lel0

(12)

here l0 is the original length before loading and le is the dis-lacement of the loading end. Considering the presence of theccelerated third creep stage, the value of ε was taken as the small-st one throughout the tensile creep process.

In order to obtain creep parameters, a series of numerical sim-lations under different magnitude of loads were carried out. Forhe impression creep test, the applied load are P = 400, 500, 600, 700nd 800 MPa. The corresponding steady-state punch velocity wasxtracted using extrapolation method aforementioned. For the uni-xial tensile creep test, the tensile load are P = 100, 150, 200, 250 and00 MPa. The corresponding minimum strain rate was then foundut as the steady-state creep rate for tensile creep test. The crys-allographic creep stress exponent was determined using a fittingrocedure in a double logarithmic grid as shown in Fig. 4.

From Fig. 4(a), two important conclusions can be concluded:i) the determination of crystallographic creep stress exponent isndependent of crystallographic orientation for both of the two

ethods, (ii) the creep stress exponent determined using impres-ion creep technique is identical with that obtained by uni-axialensile creep.

.3. Effect of crystallographic orientation on stress conversionactor

It is a common practice to relate the impression creep data withhat of tensile creep using conversion factors [19–23]. The referenceni-axial tensile creep rate can be predicted from impression creepata by:

˙ = C0h

2R(13)

here h is the steady-state punch velocity, R is the radius of theunch, C0 is a conversion factor used with C0 = 1 [24–26], which

s an acceptable simplification for most materials. The resolvedhear stress on the primary slip plane can be converted betweenhe impressed specimen and the tensile creep counterpart by:

t = C1�i (14)

here C1 is the stress conversion factor that we concerned in thisection, �i and �t are the resolved shear stress of the impressed solidnd the tensile specimen, respectively.

From Eqs. (3)–(6) we know that the creep behavior of nickel-ased single crystal obeys Norton creep law with respect toesolved shear stress:

˙ =N∑

a(�(˛)t )

nP(˛) = C[�(˛)

t ]n

(15)

˛=1

˙ = Ci[�(˛)i

]ni (16)

here C and n are the creep pre-factor and creep stress exponentor uni-axial tensile creep.

Fig. 4. Determination of crystallographic creep parameters for different crystallo-graphic orientations using (a) impression creep method and (b) uni-axial tensilecreep method.

Submitting Eqs. (13), (14) and (16) to Eq. (15), we have:

Fig. 5. Determination of the stress conversion factor C1 for different crystallographicorientations.

1854 W. Yan et al. / Materials Science and Engineering A 527 (2010) 1850–1855

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Fig. 7. The surface von Mises stress contour plots under impression for (a)[0 0 1]-, (b) [0 1 1]-, (c) [1 1 1]-oriented single crystals and the (d) isotropic coun-terpart. P = 800 MPa.

ig. 6. Comparison of the steady-state creep stage between impression creep andni-axial tensile creep with respect to Eq. (13).

sing Eq. (17). It can be seen that the stress conversion factorsssociated with the three different crystallographic orientationsxhibits the following crystallographic orientation dependence:1-[1 1 1] > C1-[0 1 1] > C1-[0 0 1]. This was induced by the reveres depen-ence on crystallographic orientation between impression creepnd uni-axial tensile creep behaviors, which has been learned inection 3.1. The consistency check was done between the steady-tate punch velocity and the steady-state elongation rate withespect of Eq. (13) while the stress conversion factor was takennto account (see Fig. 6). It can be seen that the steady-state punchelocity agree reasonably well with that predicted from tensilereep.

.4. Characteristics of von Mises stress and deformation

In order to get some implications for the creep behavior char-cteristics of nickel-based single crystal super alloy, we now moven to study the stress distribution and deformation for impressionreep and uni-axial tensile creep performed on differently orientedingle crystals.

Fig. 7 gives the von Mises stress contour plots of the three dif-erent crystallographic orientations for impression creep. It can beeen that the surface von Mises stress distribution exhibits squaren shape for [0 0 1], butterfly-like in shape for [0 1 1], and trianglen shape for [1 1 1]-oriented single crystals. This is in consistence

ith Xu’s results, who revealed that the indentation surface mor-hology for the [0 0 1]-, [0 1 1]- and [1 1 1]-oriented single crystalsre fourfold symmetry, twofold symmetry and threefold symmetry,espectively [12]. For the isotropic counterpart, the von Mises con-our was constituted by concentric circles surrounding the punch.nother important character is the magnitude of von Mises stress.

t is seen that the [0 0 1]-oriented single crystals brings the small-st von Mises stress, while the largest magnitude was observedor the [1 1 1]-oriented single crystals. This explained the results ofection 3.1, which revealed that the largest penetration depth andhe largest velocity of the punch were obtained from the [1 1 1]-riented single crystals, while the minimum ones were associatedith the [0 0 1]-oriented single crystals. Moreover, the stress for the

nisotropic material is always larger than that of isotropic coun-erpart. This can be attributed to different slip systems as well as

tress concentration induced by anisotropy for differently orientedrystals.

Fig. 8 shows the deformation of the middle section of the tensilepecimen for the three different crystallographic orientations. It iseen that the section is square in shape for [0 0 1], diamond-like

W. Yan et al. / Materials Science and Eng

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ig. 8. von Mises stress contour plots and deformation of the cross-section of theensile specimen for (a) [0 0 1]-, (b) [0 1 1]-, and (c) [1 1 1]-oriented single crystals.= 200 MPa. Deformation scale factor: 10.

n shape (elongated along the axial [1 0 0] orientation) for [0 1 1],nd elliptical in shape (elongated along the axial [110] orientation)or [1 1 1]-oriented single crystals. Referring to Fig. 2(b) and Fig. 8,e found that the lateral deformation plays an important role in

ensile creep behavior. Comparing with [0 0 1]-oriented crystal, theore severe lateral deformation for [0 1 1] and [1 1 1]-oriented ten-

ile specimen stores a larger magnitude of transverse deformationnergy, yielding a smaller elongation and a lower elongation rateor the same creep time.

. Summaries and conclusions

The crystal plasticity slip creep theory was implemented intoE simulations to investigate the effect of crystallographic orien-ation on impression creep and uni-axial tensile creep behaviors.he stress conversion factor, which relates the impression creepata with those of tensile creep, was also investigated on its crys-

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ineering A 527 (2010) 1850–1855 1855

tallographic orientation dependence. The main conclusions can bedrawn as follows.

For the impression creep, the steady-state punch velocitiesh shows the following crystallographic orientations dependence:h-[1 1 1] > h-[0 1 1] > h-[0 0 1]. For the uni-axial tensile creep, on theother hand, the opposite result was obtained. The impression creeptechnique is capable of yielding crystallographic creep stress expo-nent which is in consistent with that obtained from uni-axial creeptest. Furthermore, the crystallographic orientation has no influenceon determination of crystallographic creep stress exponent. Thestress conversion factors that relate the impression creep data tothose of uni-axial tensile creep obey: C1-[1 1 1] > C1-[0 1 1] > C1-[0 0 1].

Acknowledgements

The authors appreciate the financial supports from NationalNature Science Fund of China (50775183, 50805118) and ResearchFund for Doctoral Program of Bigger Education (N6CJ0001).

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