Application of anisotropic inclusion theory to the deformation of Ni based single crystal superalloys: Stress–strain curves determination

Mechanics of Materials 42 (2010) 237–247

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Application of anisotropic inclusion theory to the deformation of Ni based single crystal superalloys: Stress–strain curves determination N. Ratel a,*, M. Kawauchi b, T. Mori c, I. Saiki b, P.J. Withers c, T. Iwakuma b a

CIRIMAT-ENSIACET, 4 allées Emile Monso, BP 44632, 31432 Toulouse cedex 4, France Department of Civil and Environmental Engineering, Tohoku University, Sendai 980-8579, Japan c Manchester Materials Science Centre, Grosvenor Street, Manchester M17HS, UK b

a r t i c l e

i n f o

Article history: Received 29 January 2009 Received in revised form 10 September 2009

Keywords: Inclusion theory Mean field approach FEM Nickel superalloys Stress–strain curves Mean field

a b s t r a c t The development of plastic deformation after uniaxial plastic strain along [0 0 1] is analyzed in two types of domains in c–c0 nickel superalloys. These domains are the horizontal matrix channels normal to [0 0 1] and the vertical channels normal to [1 0 0] and [0 1 0]. By using a mean field method, an elastic energy increase due to the introduction of plastic strain (elongation or compression) in the two types of domains is calculated. The analysis of a mixed mode, where horizontal and vertical channel deformation occurs, is also conducted. Results show that plastic deformation is primarily initiated in a particular type of channel. The choice of the deformation configuration is related to the sign of misfit strain. After a critical amount of plastic strain is reached, the deformation expands through all the matrix channels. This conclusion is supported by the dependence of a flow stress on the deformation mode. FEM calculations are also conducted and compared with the analytical calculation. Except when the strain is extremely small, the two methods give almost the same result. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Deformation processes in single crystal nickel based superalloys have been investigated by several authors during the past decades. Most results show that a preferred channel deformation occurs when the magnitude of plastic strain is small. The choice of channel deformation is strongly related to the sign of precipitate misfit strain and plastic deformation (elongation or compression) (Pollock and Argon, 1994; Ratel et al., 2009). When the largest component in magnitude of plastic strain is along [0 0 1], horizontal channels are defined as matrix subdomains normal to [0 0 1], and vertical channels are defined as those normal to [1 0 0] and [0 1 0]. There are several crucial experimental observations, obtained by transmission electron microscopy (Feller-Kniepmeier and Link, 1989; Pollock and Argon, 1994), or X-ray diffraction (Khun * Corresponding author. Tel.: +33 534323459. E-mail address: [email protected] (N. Ratel). 0167-6636/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2009.10.001

et al., 1991), which have reported a localized plastic deformation in the horizontal matrix channels during the early stages of a tensile creep test in alloys having a negative lattice parameter mismatch (ac’ ac). There are important theoretical studies with respect to the channel deformation. FEM analysis was performed (Socrate and Parks, 1993; Pollock and Argon, 1994) in order to evaluate the total stress field in the matrix channels. The results of the calculations conducted by Pollock and Argon indicated that horizontal (vertical) matrix channels experienced a larger von Mises equivalent stress under an

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external tensile stress, when c’ particles had a negative (positive) lattice parameter misfit with respect to the matrix. Accordingly, they showed that the channels which experienced larger stress yielded before those experiencing smaller stress. Socrate and Parks used an energymomentum tensor, in the context of isotropic elasticity, in order to determine the force acting on a c–c0 interface and correctly predicted the occurrence of rafting when the amount of plastic strain in the matrix was larger than two times the initial misfit value. Under these conditions, the initial misfit and elastic properties cease to be important (Nabarro et al., 1996). In the previous investigation by (Ratel et al., 2009), the anisotropic inclusion theory combined with a mean field approach has been used to investigate the rafting phenomenon, through evaluation of the elastic energy. In this study, the elastic energy increase associated with the channel deformation was evaluated. It is found that the plastic deformation initiates in a particular type of matrix channels (horizontal or vertical), whose choice depends on the sign of precipitate misfit strain for a given macroscopic strain. The results indirectly show that the plastic deformation tends to become uniform through the matrix when macroscopic strain increases. However, this analysis does not propose any mechanism of the transition from a single type of matrix channel deformation to a quasi uniform matrix deformation. In the present paper, we aim at completing this analysis, by providing the evaluation of the elastic energy increase associated with the operation of both horizontal and vertical matrix channels during plastic deformation. The method employed here is essentially the same as that used in the previous analyses (Ratel et al., 2006; 2009). The calculations are first carried out analytically. In addition, an original approach leading to the determination of stress-strain curves is also presented, for all the mechanisms analyzed in the present investigation. The results obtained in this way are then compared with FEM calculations.

Fig. 1. Geometry of precipitates and different matrix channels configurations. (a) A horizontal channels P3 and a vertical channel P1 (the dashed line is the uniaxial loading axis) and (b) two types of vertical channels, P1 and P2. The loading direction is normal to the sheet plane.

The stress inside an inclusion is calculated by I

r ¼ CðS  IÞe ;

where S is the Eshelby tensor of disk shape inclusions and e* the eigenstrain tensor (here the plastic deformation in a matrix channel). The Eshelby tensor of S3333 = 1 and S3311 = S3322 = C12/C11 is used (Mura, 1987). The average stress is calculated using the mean field method (Mori and Tanaka, 1973; Brown, 1973).

hrij iV ¼ ð1  f ÞrIij in the particles; I ij

hrij iM ¼ f r in the matrix:

In order to conduct the analysis in a simple and intuitive way, we assume that the c matrix and c0 phases have the same elastic constants C. This is firstly because our previous paper has shown that this assumption gives nearly the same result as the analysis when c0 is 15% elastically harder than c. Secondly, because this assumption still predicts the occurrence of rafting in a successful manner without the complication of elastic mismatch (Ratel et al., 2006). In the beginning, the elastic state due to the presence of precipitate misfit strain e0 between the c and c0 phases is determined. Then, the elastic energy change associated with the introduction of plastic strain in the matrix, horizontal and/or vertical channels is calculated. The channel geometry used in the present investigation is described in Fig. 1. The total elastic energy increase related to a mode of plastic deformation includes a self energy term and an interaction term. Each term is evaluated using the following procedure

ð2Þ ð3Þ

The elastic self energy is calculated using

EP ¼ 

1 2

Z D

hrij ieij dV;

ð4Þ

where D is the whole body (matrix + particles). The additional energy of the interaction term between two internal stress sources (1) and (2) are given by

EI ¼  2. Elastic energy evaluation

ð1Þ

Z D

hrij ð1Þieij ð2ÞdV:

ð5Þ

In our case, the sources for internal stress are (1) plastic deformation in a matrix channel and (2) precipitate misfit strain. In the case of the deformation mode where both horizontal and vertical channels operate, an elastic energy of interaction between plastic deformation in horizontal matrix channels and vertical ones must be taken into account. It is similarly calculated, using (5). In this way, the total elastic energy change induced by the introduction of plastic strain in each material domain considered in the present analysis is evaluated. The deformation mode, leading to the lowest elastic energy increase, is then considered to be the selected one. 2.1. Horizontal channel deformation Here, we consider that plastic deformation occurs only in the horizontal channels. The plastic strain in the horizontal channels, called P3 (see Fig. 1), is

eP11 ¼ eP22 ¼ eP ðHÞ=2; eP33 ¼ eP ðHÞ:

ð6Þ

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Conducting the same calculations as that performed previously (Ratel et al., 2009), the two following expressions are obtained:

1  F 3 ðC 11 þ 2C 12 ÞðC 11  C 12 Þ 2 eP ; 4C 11 F3 ðC 11 þ 2C 12 ÞðC 11  C 12 Þ eP e0 : EI ðHÞ ¼ f C 11 EP ðHÞ ¼

ð7Þ ð8Þ

Here, f is the volume fraction of precipitates, and the volume fraction of horizontal matrix channels is expressed as

F 3 ¼ 1  f 1=3 :

ð9Þ

The macroscopic plastic strain is expressed as

eP ¼ F 3 eP ðHÞ:

ð10Þ

The elastic energy of interaction for horizontal matrix channel deformation mode is negative when

eP e0 < 0:

ð11Þ

Thus, horizontal channel deformation is likely to occur when the condition (11) is satisfied. This is one of the main points underlined in our previous analysis (Ratel et al., 2009). 2.2. Vertical channel deformation

0 1   2  a eP ðVÞ 0   B P e ð1Þ ¼ @ 0  12 þ a eP ðVÞ 0

0 0

eP ðVÞ

and that in P2 as

0 1   2 þ a eP ðVÞ 0 1  B P e ð2Þ ¼ @ 0  2  a eP ðVÞ 0

0

0 0

eP ðVÞ

1

C A;

EI ðVÞ ¼ f

  ðC 11 þ 2C 12 ÞðC 11  C 12 Þ 1 a  eP e0 : C 11 2

ð15Þ

ð16Þ

Here, F1(= F2 = F3) is the volume fraction of the P1 channels. We also have the macroscopic plastic strain of

eP ¼ 2F 1 eP ðVÞ:

ð17Þ

2.3. Dependence of a on eP(V) In order to determine the a parameter as a function of plastic strain, the elastic energy increase DE = EP + EI due to the introduction of plastic strain, must be minimized with respect to a. The calculation of the root of @@DaE gives the following expression   1 1 je0 j : 2C 12 þ F 3 ðC 11  C 12 Þ  4fF 3 ðC 11 þ 2C 12 Þ þa ¼ eP 2 2ðC 11 þ C 12 Þ

ð12Þ

ð13Þ

tition the plastic strain along the [1 0 0] and [0 1 0] directions in a channel. Since the lateral plastic strain must have a negative sign to the imposed tensile (compressive) strain eP(V), the parameter a must obey the condition

ð0  1=2 þ a  1Þ:

eP e0

> l1 ;

e0 > 0:

ð19Þ

12 ; l1 ¼ 4fF 3 2C 12CþF113þ2C ðC 11 C 12 Þ

However, when eP =e0 is less than l1, the a value given by (18) does not satisfy the condition given by (14). Thus, it is natural to adopt

1 C A:

2.3.1. Case of positive precipitate misfit strain The condition (14) is satisfied when (Ratel et al., 2009)

if

a is a parameter introduced in the analysis in order to par-

1=2  a  1=2;

"  2 C 11  C 12 eP 2 1 ðC 11 þ C 12 Þ þa 2 C 11 4F 1   1 þ a þ ðC 11 þ C 12 Þ  2C 12 2    # 1  F 1 ðC 11  C 12 Þ þ a þ 2C 11 þ C 12 ; 2

ð18Þ

There are two types of vertical channels: one (hereafter called P1) normal to [1 0 0] and the other (P2) normal to [0 1 0]. From symmetry considerations and constancy of volume associated with plastic strain, we can assume the plastic strain in P1 as

0

EP ðVÞ ¼

ð14Þ

a is not always zero. This is because the vertical channels, say P1, have different dimensions along the [1 0 0] and [0 1 0] directions. This a parameter is determined so that the elastic energy change induced by plastic deformation is minimized. Accordingly, a depends on eP, as shown later. Conducting the same calculations as those performed in our previous analysis (Ratel et al., 2009), the elastic energies due to the plastic deformation in the vertical matrix channels (EP) and due to the interaction between the plastic deformation and precipitation misfit of the c0 particles (EI) are calculated using (4) and (5). These are respectively given as

1 eP þ a ¼ 0; when 0   l1 : 2 e0

ð20Þ

That is, in this range, the constant value of a = 0.5 is used to calculate the elastic energy change associated to the introduction of plastic deformation. 2.3.2. Case of negative precipitate misfit strain In the case of a negative precipitate misfit strain, i.e. in the range of (11), tensile plastic deformation initiates in the horizontal matrix channels, as shown by our previous analysis (Ratel et al., 2009). As the case of a mixed mode where both channels deformation coexist is analyzed, plastic deformation in vertical matrix channels has to be considered when precipitate misfit strain is negative. The dependence of a on plastic strain is calculated by the same method as that use above. The condition (14) is satisfied when eP e0

> l2 ;

12 ; l2 ¼ 4fF 3 2C 12CF113þ2C ðC 11 C 12 Þ

if

e0 < 0:

ð21Þ

However, when eP =je0 j is less than l2 , the a value given by (18) does not satisfy the condition given by (14). Thus, it is natural to adopt

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1 þ a ¼ 1; when 0 < eP =je0 j < l2 : 2

ð22Þ

That is, in this range, the constant value of a = 0.5 is used to calculate the elastic energy increase associated to the introduction of plastic deformation. The dependence of a on eP is shown in Fig. 2. We used f = 0.7 and F3 = 0.11. As eP ðVÞ=e0 increases from l1 (l2), a increases (decreases) from 0.5 (0.5) to the asymptotic value,

a ¼ 

ð1  F 3 ÞðC 11  C 12 Þ : 2ðC 11 þ C 12 Þ

ð23Þ

2.4. Plastic deformation in horizontal and vertical matrix channels (Mixed Mode) As the transition from one type of channel to uniform deformations cannot occur in a sudden way, the two types of channel deformations develop at the same time, interacting with each other, to attain the overall deformation. Conducting similar calculations to obtain the interaction elastic energy between the channel deformation and precipitation misfit, we can obtain the interaction elastic energy between the plastic strain in the horizontal channels and that in the vertical channels. It is written as

ðC 11 þ 2C 12 ÞðC 11  C 12 Þ eP ðHÞeP ðVÞ; C 11

ð24Þ

where eP(H) and eP(V) are the plastic strains in the horizontal and vertical matrix channels respectively. In these conditions, the macroscopic strain is given by

ð25Þ

This interaction energy does not depend on a. This is due to the a dependent form of the plastic strains in the P1 and P2 channels. In other words, a can be solely determined as the value which makes the elastic energy change minimum for a given plastic deformation in vertical matrix channel. The total elastic energy change associated to the mixed mode deformation is written

DEðmixedÞ ¼ DEðHÞ þ DEðVÞ þ EI ¼ EP ðHÞ þ EI ðHÞ þ EP ðVÞ þ EI ðVÞ þ EI :

It should be noted that this value never reaches 0. This is due to the large difference in the dimensions of a vertical channel along the [1 0 0] and [0 1 0] directions, as mentioned before. The factor 1  F3 in the numerator indicates the interaction between the flat plastic domains. The dependence of a on eP(V) means that we cannot adopt the plastic strain having the form of (6) in the vertical channels, as pointed out in our previous analysis.

EI ¼ 2F 1 F 3

eP ¼ F 3 eP ðHÞ þ 2F 1 eP ðVÞ:

ð26Þ

2.5. Uniform matrix deformation The case of uniform matrix plastic deformation was treated in our previous analyses (Ratel et al., 2006; 2009). Although it is not expected, this case is treated for comparison. We remind here the elastic energy expressions obtained in our previous investigations.

3f ðC 11  C 12 Þð1 þ S1122  S1111 ÞeP 2 ; ð27Þ 4ð1  f Þ EI ðuniformÞ ¼ 0: ð28Þ EP ðuniformÞ ¼

Here, the Eshelby tensor components are obtained using numerical integration (Mura, 1987). Cuboidal particles are approximated as spheres. The numerical values used in the present analysis are S1111 = 0.445 and S1122 = 0.091. 2.6. Results Firstly, from the three modes examined above, we identify the plastic deformation mode, which leads to the lowest energy increase with no change in the shape of the c0 particles, when the macroscopic plastic elongation (compression) is introduced. This is shown in Fig. 3. In the regime of (11), the vertical channel deformation causes the largest energy increase (Ratel et al., 2009). Con-

Fig. 2. Dependence of a on eP(V)/|e0|.

N. Ratel et al. / Mechanics of Materials 42 (2010) 237–247

sequently, only the energy increases for the uniform, horizontal channel deformation and mixed modes are examined (Fig. 3a). When 0:28  eP =e0  0, the horizontal channel deformation mode increases the elastic energy the least. In the above range, plastic deformation initiates in the horizontal matrix channels. However, in the range 1:3  eP =e0  0:28, the mixed mode show the lowest elastic energy increase. This means that after a critical amount of plastic strain in the horizontal matrix channels, the vertical matrix channels start to deform. Moreover, when eP =e0  1:3, the uniform plastic deformation would be energetically favoured from an elastic energy viewpoint. However, even when the macroscopic plastic strain is large, the deformation is not uniform in the matrix, as shown by the dependence of a on eP(V). When the precip-

241

itate misfit strain is negative, it is shown that the plastic deformation resulting from a tensile stress initiates in the horizontal matrix channels. After a critical amount of plastic deformation, vertical channels start to yield, until a reaches its asymptotic value. It has been found that when eP e0 > 0, the horizontal channel deformation mode leads to the largest energy increase and is thus excluded as a possible deformation mode in reality. The elastic energy change associated with the vertical channel, mixed and uniform deformation modes are plotted against normalized plastic strain in Fig. 3b. When eP =e0 is smaller than 0.5, the vertical channel deformation mode results in a smaller increase in the energy than the mixed and uniform matrix deformation modes. That is, we have to take into account the role of ver-

Fig. 3. Normalized elastic energy change DE=½ðC 11  C 12 Þe20  versus normalized plastic strain particles are approximated by spheres: (a) e0 < 0, (b) e0 > 0.

eP =e0 for different deformation modes, when the cuboidal

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tical channel deformation when small plastic strain is imposed. However, when 0:5  eP =e0  1:25, the mixed mode is selected from an energetic point of view. In this range, the horizontal matrix channels start to operate, in addition to the vertical channels.

where k is the flow stress in the c matrix and is assumed constant. The latter,

dE ¼ dEP ðHÞ þ dEI ðHÞ þ dEP ðVÞ þ dEI ðVÞ þ dEI ; is written as

dE ¼ F 3 EH eP ðHÞdeP ðHÞ þ F 3 EHI e0 deP ðHÞ

3. Examination from a macroscopic stress-strain relationship

þ 2F 1 EV eP ðVÞdeP ðVÞ þ 2F 1 EI e0 deP ðVÞ þ 2F 1 F 3 IHV eP ðVÞdeP ðHÞ þ 2F 1 F 3 IHV eP ðHÞdeP ðVÞ;

Since the elastic energy change DE = EP + EI induced by plastic deformation has been evaluated, we can determine the stress–strain relation, or flow curve. From these, we can further examine a possible deformation mode discussed above. Suppose that the uniaxial plastic strain increases by deP under the uniaxial applied stress rA. The work per unit volume supplied by the external load is

dW A ¼ rA deP :

ð29Þ

This work is converted to the change in the elastic energy dE = dEP + dEI and the energy dissipation dED,

dW A ¼ dEP þ dEI þ dED :

ð30Þ

The first two terms are written as in the form proportional to deP using EP and EI calculated before. The latter is assumed to have the form of

dED ¼ kdeP ;

ð31Þ

where k corresponds to the flow stress of a material consisting of the c matrix only. Since the uniaxial deformation is assumed as in (6), we will use the same value of k for all the deformation modes examined. For convenience, k is assumed to be 2(C11–C12)|e0|, even though the magnitude of k does not materially affect the discussion. For simplicity, we will focus on the case of a positive applied stress rA > 0, along [0 0 1].

EH ¼ ð1  F 3 Þ

ðC 11 þ 2C 12 ÞðC 11  C 12 Þ ; 2C 11

EHI ¼ f

ðC 11 þ 2C 12 ÞðC 11  C 12 Þ ; 2C 11

EV ¼

 2   C 11  C 12 1 1 ðC 11 þ C 12 Þ þ a  2C 12 þa 2 2 C 11     1 þðC 11 þ C 12 Þ  F 1 ðC 11  C 12 Þ þ a þ 2C 11 þ C 12 ; 2 ð40Þ

as from (15).

EVI ¼ f

  ðC 11 þ 2C 12 ÞðC 11  C 12 Þ 1 a : C 11 2

In the case of a tensile and uniaxial stress applied along [0 0 1], we have

ð33Þ

Suppose that eP(H) increases by deP(H) and eP(V) by deP(V), the work supplied by the applied stress is

dW A ¼ rA ðF 3 deP ðHÞ þ 2F 1 deP ðVÞÞ:

ð34Þ

ð41Þ

As from (16) and

IHV ¼ 

ðC 11 þ 2C 12 ÞðC 11  C 12 Þ ; 2C 11

ð42Þ

as from (24). Equating dWA to dWD plus dE, we obtain

from the variation of eP(H), and

3.2. Case of the mixed deformation mode

ð39Þ

as from (8).

Using (27) and noting EI = 0 in the case of uniform deformation and cuboidal particles, (30) results in

ð32Þ

ð38Þ

as from (7)

rA ¼ k þ EHI e0 þ EH eP ðHÞ þ 2F 1 IHV eP ðVÞ;

3f rA ðuniformÞ ¼ k þ ðC 11  C 12 Þð1 þ S1122  S1111 ÞeP : 2ð1  f Þ

ð37Þ

where

3.1. Case of the uniform deformation mode

eP ðHÞ; eP ðVÞ  0:

ð36Þ

rA ¼ k þ EVI e0 þ EV eP ðVÞ þ F 3 IHV eP ðHÞ;

ð43Þ

ð44Þ

from the variation of eP(V). (43) and (44) are the set of simultaneous equations to solve eP(H) and eP(V) under a given rA. EHI, (39), is positive, while EVI, (41), is negative for all possible values of a. This indicates that, when e0 > 0, the vertical channels operate in the early stages of plastic deformation while the horizontal channels do so when e0 < 0. Case of negative precipitate misfit strain In this case, when

eP ðHÞ <

EVI  EHI je0 j; EH  F 3 IHV

ð45Þ

This work is converted to the energy dissipation dWD and the elastic energy increase dE. The former is given by

the two Eqs. (43) and (44) give a negative value of eP(V). This physically unsatisfactory result means that, in the range of (45),

dW D ¼ kðF 3 deP ðHÞ þ 2F 1 deP ðVÞÞ;

eP ðVÞ ¼ 0;

ð35Þ

ð46Þ

N. Ratel et al. / Mechanics of Materials 42 (2010) 237–247

and

rA ¼ k þ EHI e0 þ EH eP ðHÞ:

ð47Þ

When

eP ðHÞ >

EVI  EHI je0 j; EH  F 3 IHV

ð48Þ

eP(H) and eP(V) coexist and are given as the roots of (43) and (44). In this case, the overall plastic strain is defined by (25), as the two types of channel operate. In Fig. 4, the stress-strain curve is plotted in the case of uniform deformation, horizontal channel deformation, and mixed deformation modes. The case of horizontal channel deformation is obtained using (47) in the full range of macroscopic deformation. In this plot, the applied stress is normalized by (C11–C12)|e0|. We can see one knee in the flow curve of the mixed channel mode, corresponding to the initiation of the vertical channels when the strain in the horizontal channels reaches its critical value. Case of positive precipitate misfit strain Similar to the above case, when

eP ðVÞ <

EHI  EVI e0 ; EV  F 3 IHV

ð49Þ

ð50Þ

The range of (49) corresponds to the vertical channel deformation mode. However, when

eP ðVÞ <

EHI  EVI e0 ; EV  F 3 IHV

the horizontal channels start to operate. eP(H) and eP(V) are then given by solving (43) and (44) for a given rA. The overall elongation is given by (25). Fig. 5 is similar to Fig. 4, but in the case of positive precipitate misfit strain. The case of horizontal channel deformation mode is omitted as it was found to be energetically unfavourable in the case of positive precipitate misfit strain and tensile applied stress. 3.3. Uniformity of deformation Figs. 3–5 indicate that, as the deformation proceeds, the strain in one type of channels, which yield after that the strain in the other type of channels has reached its critical value, increases. This means that, as the total deformation increases, the deformation mode tends to become uniform. Here, we show the degree of uniformity of plastic deformation after tensile deformation, defined by

bI ¼

eP ðVÞ ; when e0 < 0; eP ðHÞ

ð52Þ

eP ðHÞ ; when e0 > 0: eP ðVÞ

ð53Þ

and

the resolution of the two Eqs. (43) and (44) give the unsatisfactory result of eP(H) < 0. Thus, in this range, we use (44) by setting eP(H) = 0. This is because (44) is obtained by the variation of eP(V). We thus obtain

rA ¼ k þ EVI e0 þ EV eP ðVÞ:

243

ð51Þ

bII ¼

It can be seen in Fig. 6 that the ratio approaches unity as deformation proceeds, suggesting that a state of uniform deformation is approached when the overall plastic strain is large. This phenomenon is due to the parameter a. This means that, as long as the coexistence of two types of vertical channels is assumed, no uniform plastic deformation in the c matrix is achieved in a strict sense. This is due to the characteristic structure of the c–c0 alloy investigated in the present study. The c0 cuboids are almost periodically arranged, so that P1 and P2 channels should have different plastic strain components along the directions normal to the tensile or compressive loading direction.

Fig. 4. Stress–strain curve for uniform, horizontal channel and mixed deformation mode. Precipitate misfit strain is negative.

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Fig. 5. Stress–strain curve for uniform, horizontal channel and mixed deformation mode. Precipitate misfit strain is positive.

Fig. 6. Degree of uniformity of plastic deformation. b = 0 corresponds to a single type of matrix channel deformation and b = 1 to uniform plastic deformation.

4. Finite element analysis Numerical analyses by a finite element method were performed. A representative volume was taken as the volume of a c0 cuboid, together with contacting c channels. Fig. 7 illustrates such a finite element of the representative volume, which contains 13,824 linear isoparametric rectangular parallelepiped elements and 15,626 nodes. The periodic boundary condition is employed so that three pairs of surfaces have the same displacements except those caused by the macroscopic deformation (Saiki et al., 2006). In the present analysis, the effect of geometrical nonlinearity is neglected, since the magnitude of the strain is not so large. The classical associated J2 plasticity with isotropic

linear hardening is employed for the material model of the c phase. The initial yield stress is 103C44, and the hardening parameter is set 102C44. The c0 phase is assumed to be elastic. For the elastic parts of the material models of the c phase and the c’ phase, the same elastic properties, C11/C44 = 2.0197 and C12/C44 = 1.25, are assumed. In order to prepare the initial state of the superalloy for the numerical uniaxial stress test, the precipitate misfit is introduced to the c0 cuboid without stress prior to the application of the uniaxial stress. Both cases of negative and positive precipitate misfit were examined. The magnitude of the precipitate misfit e0 is assumed 0.002, which is commonly observed in most engineering Ni superalloys. In Fig. 8a, the plastic strains in the horizontal and vertical

N. Ratel et al. / Mechanics of Materials 42 (2010) 237–247

Fig. 7. Geometry used in the finite element analysis.

channels and the overall specimen strain are plotted against the applied stress. The plastic deformation starts in the horizontal channel near the precipitate and propagates into the inside of channel. However, the plastic strain in Fig. 8 is the value averaged inside the channel. It is seen

245

that the horizontal channel operates at the lowest stress. When the stress increases, the vertical channel starts to yield, in agreement with the mean field analysis presented in the previous section. In Fig. 8b, the stress–overall plastic strain relationship is shown. Unlike the result by the mean field method, the transition from the horizontal channel to the mixed mode deformation is not so sharp. This transition seems to occur when the strain is approximately 0.07 e0. When only the horizontal channel becomes plastic, the stress strain curve is not linear. This result is also different from that by the mean field analysis. The FE analysis can take into account the internal stress due to the precipitate misfit, the stress that should depend on the location within the c0 channel. Fig. 9 shows results in the case of the positive precipitate misfit. Fig. 9a is similar to Fig. 8a. Similar to the result by the mean field analysis, only the vertical channels yield at the beginning until the stress increase up to approximately 1.1. This characteristic is seen more clearly in Fig. 9b, where the start of the horizontal channel activity is seen as a knee of the stress–overall plastic strain curve at the strain of approximately 0.07 e0. The parameter a is also calculated as shown in Fig. 10. A general trend is the same as Fig. 2 except when the magnitude of the strain is extremely small, say, less than 0.3. In

Fig. 8. The relationship between applied stress and plastic strain for the case of the negative precipitate misfit. (a) Horizontal channel and vertical channel strain and overall strain are plotted against the applied stress. (b) The stress–overall strain curve.

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Fig. 9. The relationship between applied stress and plastic strain for the case of the positive precipitate misfit. (a) Vertical channel and horizontal channel strain and overall strain are plotted against the applied stress. (b) The stress–overall strain curve.

this case, a is extremely large for the negative precipitate misfit and extremely small for the positive precipitate misfit. In the mean field analysis, the parameter a is taken as 0.5 (negative misfit) or 0.5 (positive misfit), when the strain is small. It is believed that this difference originates from the assumption of the mean field method, which assumes that plastic deformation occurs uniformly in a channel. On the contrary, a FEM analysis takes into account the heterogeneous deformation in the c channels. The heterogeneous deformation is due to the built-in internal stress caused by the precipitate misfit. When the precipitate misfit is negative, a decreases from 0.5 asymptotically to 0.25, as the strain increases. When

Fig. 10. Dependence of a on eP(V)/|e0| as calculated by FEM.

the precipitate misfit is positive, a increase from 0.5 to 0.3 when the strain is very large. 5. Conclusions In order to discuss primarily the deformation mechanisms in c–c0 Ni alloys, the development of the plastic deformation in horizontal channels and vertical channels in the matrix is examined quantitatively. The coexistence of deformation in both horizontal and vertical matrix channels is also examined. The deformation of these channels is discriminated by the precipitate misfit and the reason for the discrimination is discussed by examining the interaction energy between the precipitate misfit and plastic strain. The plastic strain tends to be uniform in the matrix as the deformation proceeds. A finite element analysis is also conducted in the same problem with the same geometry. Except in an extremely small strain regime, the mean field method and the finite element analysis give almost the same results. The deformation behaviour predicted by these analyses is qualitatively in agreement with experimental observations. In the small strain regime, plastic deformation initiates in a single type of channels, depending on the sign of both lattice parameter mismatch and applied stress. As strain increases, plastic deformation expands to the whole matrix.

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The difference in the prediction between the mean field method and FEM becomes more remarkable at the very beginning of straining. In this regime, the FEM can simulate a more detailed deformation, while the mean field method ignores the position dependent strain. However, as the amount of strain increases, this difference increasingly diminishes. Even in the FEM, the relative change in the strain inside a channel decreases. Thus, the result of the finite element analysis approaches the uniform deformation in a channel adopted in the mean field method from the beginning. Acknowledgement T. Mori acknowledges visiting fellowship funding from an EPSRC grant. References Brown, L.M., 1973. Back stresses, image stresses, and work hardening. Acta Metal. 21, 879–887. Feller-Kniepmeier, M., Link, T., 1989. Dislocation structures in c–c0 interfaces of the single crystal superalloy SRR99 after annealing and high temperature creep. Mater. Sci. Eng. A 113, 191–195.

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