A physical-based constitutive model considering the motion of dislocation for Ni3Al-base superalloy

Materials Science & Engineering A xxx (xxxx) xxx

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A physical-based constitutive model considering the motion of dislocation for Ni3Al-base superalloy Jianfeng Xiao, Haitao Cui, Hongjian Zhang *, Weidong Wen, Jie Zhou Jiangsu Province Key Laboratory of Aerospace Power System, State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Energy & Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, PR China

A R T I C L E I N F O

A B S T R A C T

Keywords: Physical-based model Dislocation density Ni3Al-base superalloy Crystal plasticity finite element method

Ni3Al-base superalloys are widely used as high temperature structure materials in aerospace applications. In order to study the microphysical mechanism of Ni3Al-base superalloys, unidirectional tension tests are carried out on IC10 (a typical Ni3Al-base superalloy in China) at 300 K and 973 K with a strain rate 1 � 10 3 s 1 in the direction of [001]. Different strain levels (0.8% strain, 3.0% strain, 6.0% strain and 8.2% strain) are tested and the dislocation configurations under these different deformation conditions are observed by transmission elec­ tron microscope (TEM). Both edge dislocation and screw dislocation are found in the late stage of deformation. The effect of different types of dislocation on deformation has been investigated through measuring the dislo­ cation densities. Based on the experimental results, a new physical-based constitutive model has been established by considering the effects of dislocation movement. The evolution forms for edge-character dislocation and screw-character dislocation are studied respectively in each slip system. The laws of latent hardening between different slip systems are considered directly. Finally, the model has been implemented in a crystal plasticity finite element method numerical framework. And, the simulated results at 300 K and 973 K with a strain rate 1 � 10 3 s 1 show good qualitative agreement with experimental results.

1. Introduction Ni3Al-base superalloys are widely used as high temperature structure materials in aerospace applications, due to their advanced mechanical properties, such as high strength, good ductility from room temperature up to elevated temperature, high incipient melting temperature, excel­ lent oxidation resistance and high creep resistance over a wide tem­ perature range [1–3]. A lot of researches show that above mechanical properties of Ni3Al-base superalloys are closely related to the motion of dislocation during deformation [4–11]. In order to study the physical mechanism of Ni3Al-base superalloys, it is necessary to observe the dislocation configurations and investigate the motion of dislocation during different deformation conditions. Presently, there are two main methods to study dislocation: X-ray diffraction (X-RD) peak profile analysis and transmission electron microscope (TEM) analysis. T. Ung� ar [12] measured dislocation densities and some micro characteristic pa­ rameters of bulk materials by XRD, which provide data for macroscopic simulation calculation. W. Woo and his coworkers [13,14] measured dislocation density in a friction-stir welded aluminum alloy and studied the influence of the dislocation density on the strain hardening behavior

during tensile deformation. Dislocation in ferrous lath martensite was investigated by T. Berecz and his coworkers [15] using XRD. And their research revealed that the majority of dislocations formed by martens­ itic transformation were geometrically necessary dislocations (GNDs). Compared with XRD, TEM observation is a more direct method for dislocation observation. F. Long and his coworkers [16,17] investigated the evolution of dislocation and the effects of dislocation on tension-compression asymmetry response in a deformed Zr-2.5Nb alloy by TEM. To investigate the effect of irradiation hardening of structural materials due to cavity formation in BCC metals for nuclear applications, K. Tougou and his coworkers [18] observed the dynamic interaction between dislocation by TEM of pure molybdenum and pure iron. S. Jing and his coworkers [19] determined the active slip systems in situ TEM tensile deformation. Thus it can be seen that the dislocation observation has been an important way to study the micro physical mechanism of alloys. Besides, the different types of dislocation in Ni3Al-base superal­ loys played different roles on tensile deformation [20]. However, there are few literatures focused on determining the type of dislocation during deformation. To reflect the relationship between macro deformation and physical

* Corresponding author. College of Energy & Power Engineering, Nanjing University of Aeronautics & Astronautics, 210016, Nanjing, Jiangsu, PR China. E-mail addresses: [email protected], [email protected] (H. Zhang). https://doi.org/10.1016/j.msea.2019.138631 Received 28 March 2019; Received in revised form 8 October 2019; Accepted 4 November 2019 Available online 7 November 2019 0921-5093/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Jianfeng Xiao, Materials Science & Engineering A, https://doi.org/10.1016/j.msea.2019.138631

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Table 1 Nominal chemical composition of IC10 SC superalloy (wt %). Element Min Max

C

Cr

Co

W

Mo

Al

B

Ta

Hf

Ni

0.07 0.12

6.5 7.5

11.5 12.5

4.8 5.2

1.0 2.0

5.6 6.2

0.01 0.02

6.5 7.5

1.3 1.7

Bal. Bal.

mechanism, various constitutive models have been proposed of Ni3Albase superalloy during the past few decades [21–27]. Although in the majority of these models, the flow rule and hardening rule are described by a simple power law empirically [15,16,18–20]. More and more in­ terests have been attracted in formulating constitutive relations based on the physical mechanism of deformation [4–8,21,24,28–30]. In these physical-based models, the flow rule is usually based on the Orowan equation and the hardening rule is normally based on the evolution of dislocation. Unfortunately, in most of these models, a conceptual dislocation density is usually used to formulate the flow rule without considering the types of dislocation [24,28–30]. In order to study the influences of different dislocations types, Choi [4,5] has taken both edge-character and screw-character dislocation into the flow rule for Ni3Al-base superalloy. However, in his model, the contributions of two kinds of dislocation to deformation are assumed to be identical. More­ over, in order to separate the individual slip systems and account for latent hardening directly for aluminum single crystal, dislocation structures have been divided into four parts by Ma [6,7]: the mobile dislocation, the immobile dislocation, the parallel dislocation and the forest dislocation. Because the plastic deformation of aluminum single crystal depended mainly on screw dislocation, edge dislocation was not taken into consideration in Ma’s model [6,7]. The motion of dislocation is not only related to the flow rule, but also affects the hardening law. According to former researches [1–3], there were many abnormal mechanical properties of Ni3Al-base superalloys and those properties were usually thought to be closely related to hardening behavior during deformation [4,5]. Thus, various dislocation motion mechanisms [6–8,27,28] were proposed to interpret the hard­ ening behavior of Ni3Al-base superalloys. Normally, these mechanisms can be categorized into four basic parts: dislocation storage, dislocation annihilation, formulation of geometrically necessary dislocations (GNDs) and formulation of KW-locks. Dislocation propagation and annihilation rate depend mainly on dislocation density during defor­ mation. GNDs are dislocations related to the size and spacing of par­ ticipates and grain boundaries. E.P. Busso and his coworkers [31,32] used a gradient crystallographic formulation of GNDs to simulate slip inhomogeneities of single crystal superalloys. Other authors [10,11,28] try to add various state variables into hardening laws based on the

concept of GNDs to reflect the evolution of GNDs. KW-locks mechanism was firstly proposed by Hirsch [9] to account for the anomalous yield stress behavior with respect to temperature, tension/compression asymmetry and orientation dependence for Ni3Al material. There were two important processes in this mechanism: the locking process of KW-locks and the unlocking process of KW-locks [9]. The mechanism of KW-locks plays a significant role in the hardening behavior for Ni3Al-­ base superalloys and has been widely adopted by many other re­ searchers [14,26,28,33]. In this paper, macro tensile properties of IC10 superalloy were tested at different strain levels at 300 K and 973 K in the direction of [001]. In order to study the physical mechanism of deformation, dislocation configurations were observed at different strain levels by TEM. The type of dislocation was determined and dislocation densities were measured. On this basis, a physical-base constitutive model will developed to simulate the mechanical behavior of Ni3Al-base superalloys. In the developed model, edge dislocation and screw dislocation are both taken into consideration in the flow rule. Unlike Choi’s model [4,5], the effects of two kinds of dislocation on plastic flow are not the same. Meanwhile, edge dislocation and screw dislocation are studied respectively in indi­ vidual slip systems. The laws of latent hardening between different slip systems are investigated directly. And, by the finite element simulation based on the homogenization method, the effectiveness and feasibility of the model will be verified and the influences of model parameters will be further discussed and analyzed. 2. Experiment procedure IC10 alloy is a typical directionally solidified Ni3Al-base superalloy in [001] orientation, which is provided by Beijing Institute of Aero­ nautical Materials (BIAM). The nominal composition of IC10 alloy is listed in Table 1. Unidirectional tension tests are carried out on IC10 at 300 K and 973 K with a strain rate 1 � 10 3 s 1. These tensile specimens were stretched to different strains (0.8% strain, 3.0% strain, 6.0% strain and 8.2% strain). Fig. 1 shows the dimensions and the macro photograph of the experimental specimen, as well as the unidirectional tension equipment (MTS809). After tensile tests, the dislocation configurations under different deformation conditions were observed by transmission

Fig. 1. (a) Schematic geometry of the tensile specimen (all dimensions in mm) and (b) macro photograph of the tensile specimen. 2

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that there are many hexagonal dislocation cells and three sets of parallel dislocation lines (marked as A, B and C). In order to determine the type of dislocation lines, samples with strain 0.8% (early stage of deformation) and strain 8.2% (late stage of deformation) were chosen to conduct further observations. Fig. 4 gives the dark field calibration diagrams of three band axis. In Fig. 4, “O” is the central spot of transmission, Ai, Bi and Ci (i¼1,2,3) are crystal plane indexes under different band axis diffraction. Diffraction patterns of different diffraction vectors and crystal band axes with stain 0.8% are given in Fig. 5 and Fig. 6 respectively. In Fig. 5, it can be easily found that parallel dislocation lines are invisible when the diffraction vectors is ½311�. However, if the diffraction vector is ½111�, those parallel disloca­ tion lines are clearly visible. Based on Table 2, the Boggs vector of parallel dislocation lines in Fig. 3(a) is determined to be ½110�. In order to determine the orientation of dislocation lines, vertical lines were made of parallel dislocation lines under different crystal band axis diffraction patterns to determine the corresponding crystal plane index. The intersection lines of different crystal planes are the orientation of dislocation lines. In figures, the solid line represents the dislocation line and the dotted line represents the vertical line. Based on the method of Two-beam Diffraction Method, it can be concluded from Fig. 6 that dislocation lines in Fig. 3(a) parallel to the traces of ð111Þ and ð331Þ crystal plane individually. Thus, the direction of dislocation lines is the intersection of ð111Þ plane and ð331Þ plane,

Fig. 2. The morphology of the microstructure of IC10.

electron microscope (TEM) and the dislocation densities are measured by Image-Pro Plus software.

ξ ¼ ½111� � ½331� ¼ ½110�

(1)

Here, ξ represent the direction of parallel dislocation lines in Fig. 3 (a). Above all, for parallel dislocation lines in Fig. 3(a), the direction and the Boggs vector are parallel to each other. Thus, they are determined to be screw dislocation. In other words, the dislocation is mainly screw dislocation in the initial stage of deformation. Similarly, to identify the type of dislocation in precipitates in Figs. 2 (d), Fig. 7 and Fig. 8 show the diffraction patterns of different diffraction vectors and crystal band axes through TEM with 8.2% strain. Following the same method mentioned above, the directions and the Boggs vectors of parallel dislocation lines (marked as A, B and C) are identified as follows,

3. Dislocation under different strain levels Fig. 2 gives the morphology of the microstructure of IC10. Obviously, The material consists of strengthening phases and matrix channels and the strengthening phases are ellipsoid with different sizes. Fig. 3 gives the typical TEM pictures of alloy IC10 with different strain levels at 973 K. In the early stage of deformation, most of dislocation lines are distributed in the matrix and a considerable number of them shear the precipitates in the form of parallel lines, as shown in Fig. 3(a). With the increase of deformation, new dislocation structures, such as stacking faults, K-W locks, dislocation dipoles and new parallel dislocation lines appear in IC10, as shown in Fig. 3(b) and (c). In Fig. 3(d), it is obvious

Fig. 3. Positive diffraction patterns for IC10 observed by TEM with diffraction vector [011] under different strain levels at 973K: (a) 0.8% strain, (b) 3.0% strain, (c) 6.0% strain and (d) 8.2% strain. 3

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Fig. 4. The dark field calibration diagrams of three crystal band axis.

Fig. 5. Diffraction patterns of different diffraction vectors with 0.8% strain observed by TEM: (a) g 1 ¼ ½111� and (b) g 2 ¼ ½311�. ⇀

bA ¼ ½011� bB ¼ ½101� bC ¼ ½110�

(2)

ξA ¼ ½200� � ½511� ¼ 2½011� ξB ¼ ½111� � ½121� ¼ 2½101� ξC ¼ ½111� � ½042� ¼ 2½112�

(3)

⇀

summarized as follows. Firstly, mechanical behavior of Ni3Al-base su­ peralloys is closely related to the motion of dislocation. Secondly, in the early stage, deformation is dominated by the screw dislocation and the screw dislocation contributes to the rapid multiplication of dislocation. With the increase of deformation, the edge dislocation begins to appear in the octahedral slip systems and the climbing mechanism of edge dislocation has beneficial effects on the deformation. Also, the type of dislocation and the form of dislocation movement tend to be diversified. Thirdly, during the deformation, there are many dislocation structures under different strain levels, such as stacking faults, K-W locks and dislocation dipoles. The mechanism of stacking faults and K-W locks is considered to be related to the abnormality of yield stress and the asymmetry of tension and compression. Dislocation dipoles may be related to macroscopic saturation stress. Lastly, dislocation density in­ creases with the increase of deformation.

Where, bA , bB and bC are Boggs vectors of parallel dislocation lines marked as A, B and C respectively; ξA , ξB , ξC are directions of parallel dislocation lines marked as A, B and C respectively. So, the dislocation lines marked as A and B are screw dislocation and the dislocation lines marked as C are edge dislocation. The types of dislocation lines are summarized in Table 3. Moreover, for subsequent model investigation, dislocation density is calculated by image analysis through Image-Pro Plus software. The principle of dislocation measurement is to calculate the length of all dislocation lines in a unit volume. Each sample has been measured several times under different field of view. Table 4 gives the statistical dislocation density at 973 K. Additionally, screw dislocation (A and B) density is calculated to be 6.243 � 1013 m 2 and edge dislocation (C) density is 1.868 � 1013 m 2 in Fig. 3(d). Above all, the experimental research in this paper can be

4. Development of the physical-based constitutive model From the experimental analysis, both edge and screw-character dislocation lines have effects on deformation for Ni3Al-base superal­ loys. And, the function of each type of dislocation is not identical. Thus, a new dislocation constitutive model will be developed in the following 4

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Fig. 6. Positive diffraction patterns of different crystal band axes with 0.8% strain observed by TEM: (a) ½011� crystal band axis and (b) ½123� crystal band axis.

chapters. In the new model, laws of flow and latent hardening are pro­ cessed separately for edge-character and screw-character dislocations. At last, the new model will be implemented in a crystal plasticity finite element framework.

Table 2 Diffraction vectors commonly used to distinguish Boggs vectors. g

g ⋅b ¼ N

⇀

⇀

110

110

101

101

011

011

200 202

1 1

1 1

1 0

1 2

0

0 1

002 111

0 0

0

1

111

1

0

1

0

1

0 1

1

1

1 0

1

0

1

4.1. The elastic constitutive law Based on the theory of crystal plasticity, the crystal elastic consti­ tutive relation can be expressed as

1

b σe ¼ b σ þ Wpσ

(4)

σW p ¼ L : De

Where, De ¼ D Dp , Wp ¼ W We . b σ e is the Jaumann rate of Kirchhoff σ is the Jaumann stress tensor based on the intermediate configuration. b rate of Cauchy stress tensor based on the initial configuration. L is the elastic moduli tensor. D, De and Dp are the rate of stretching tensor, elastic part of the rate of stretching tensor and plastic part of the rate of

“1” and “-1” represent that dislocations are visible, “0” represents that disloca­ tions are invisible, “2” represent that dislocations may produce double images.

Fig. 7. Diffraction patterns of different diffraction vectors with 8.2% strain observed by TEM: (a) g 1 ¼ ½111�, (b) g 2 ¼ ½022�, (c) g 3 ¼ ½111� and (d) g 4 ¼ ½200�. ⇀

5

⇀

⇀

⇀

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Fig. 8. Positive diffraction patterns of different crystal band axes with 8.2% strain observed by TEM: (a) ½011� crystal band axis, (b) ½123� crystal band axis and (c) ½012� crystal band axis.

stretching tensor, respectively. W, We and Wp are the rate of spin tensor, elastic part of the rate of spin tensor and plastic part of the rate of spin tensor, respectively. The velocity gradient tensor, L, is usually used to describe the loading process and this tensor can be calculated as

Table 3 Summary of dislocations in precipitates of IC10. Sample 973K,

Type of dislocation

Boggs vector

Direction of dislocation lines

Screw

½110�

½110�

_ L ¼ FF

Screw (A)

½011�

½011�

Screw (B)

½101�

½101�

With

Edge (C)

½110�

½112�

ε½001� ¼0.8% 973K,

ε½001� ¼8.2%

1

e

¼ F_ ðF e Þ

e Le ¼ F_ ðFe Þ

1

p

þ F e F_ ðF p Þ 1 ðF e Þ

1

¼ Le þ Lp

(5)

(6)

1

p Lp ¼ Fe F_ ðFp Þ 1 ðFe Þ

(7)

1

Table 4 Statistical results of dislocation density at 973 K. Test conditions

ε¼0.8%

ε¼3%

ε¼4.5%

Total dislocation density/ 1014m 2 0.24 0.28 0.25 0.25 0.80 0.83 0.80 0.84 0.88 1.29 1.58 1.76 1.05

Average of total dislocation density/ 1014m 2

Test conditions

0.26

ε¼6%

0.83

ε¼8.2%

1.45

6

Total dislocation density/ 1014m 2 1.99 1.20 1.87 2.49 1.82 3.75 3.04 2.25 3.50 2.23 1.36 1.20 1.41

Average of total dislocation density/ 1014m 2 1.87

2.34

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� 1 vα ¼ λα ν0 exp 2

� � α � jτ j ταpass α Qslip exp V KB θ KB θ

(13)

With (14)

τα ¼ σ α : PðαÞ

Where, KB is the Boltzman constant, θ is the absolute temperature, Qslip is the effective activation energy for dislocation slip and ν0 is the attack frequency. τα and ταpass are the external driving stress and the athermal resistance or the passing stress for the mobile dislocations. The jump width λα and the activation volume V α can be calculated as the function of ραF as

Fig. 9. Schematic drawing of the slip mechanism for the FCC crystal structure.

c1 λα ¼ pffiffiffiαffiffi

p

N X

α α

N X

γ_ P ¼

L ¼ α

α

α

And

γ_ m � n

Where c1 and c2 are constants. From Eq. (12) and Eq. (13), it is concluded that the activation volume reflects the interactions between mobile dislocations and forest dislocations. According to the theory of dislocation motion [7], the passing stress for screw dislocation can be derived as

α

Here, Pα is the Schmid tensor for the slip system α, where mα rep­ resents the slip direction and nα is the slip plane normal with respect to the distorted configuration. N is the total number of slip systems, there are 12 slip systems for FCC crystals, ð111Þ½110�, i.e., N ¼ 12.

ταpass;s ¼ c3

4.2. The flow law

ταpass;e ¼ c3

N X

i

h

χ αβ ραedge jcosðnα ; nβ � mβ Þj þ ραscrew jcosðnα ; mβ Þj

N X

h

i

χ αβ ραedge jsinðnα ; nβ � mβ Þj þ ραscrew jsinðnα ; mβ Þj

∂ρM

(11)

ραM;s ¼

π

�1=2

ραp;e

(18)

τα ;ραF ;ραP ;θ

2KB θ qffiffiffiαffiffiffiffiffiffiffiαffi ρP;s ρF c1 c2 c3 μb3

for mobile edge dislocation, pffiffiffiαffiffi c 3 μb 1 KB θ ρF pffiffiffiαffiffiffiffiffiffiffiffiffiffiffiffiffiαffiffiffiffi ¼0 2 2ð1 νÞ ρP;e þ ρM;e c1 c2 b ραM;e

screw dislocation density in total. nα and nβ are normal unit vectors for slip plane α and β respectively. mα and mβ are slip directions. N is the total number of slip systems. χ αβ is the influence matrix between different slip systems and it can be calculated as [29]. ðmα ⋅mβ Þ2

νÞ

for mobile screw dislocation, pffiffiffiαffiffi c 3 μb 1 KB θ ρF pffiffiffiαffiffiffiffiffiffiffiffiffiffiffiffiαffiffiffiffiffi ¼0 2 2 ρP;s þ ρM;s c1 c2 b ραM;s

(10)

Where, ραF and ραP are forest dislocation density and parallel dislocation density for slip system α. ραedge and ραscrew are edge dislocation density and

2� 1

2πð1

Thus, a scaling relation for the mobile screw and edge dislocation can be derived as:

β¼1

χ αβ ¼

qffiffiffiffiffiffiffi

μb

4.2.2. A scaling relation for ραM Following the approach of Ma and Roters [6], mobile dislocation density is closely related to plastic deformation. It is assumed that the maximum plastic dissipation for the external resolved shear stress dur­ ing the plastic deformation is calculated by � α� ∂γ_ ¼0 (19) α

β¼1

ραP ¼

(17)

Where, subscript s and e represent screw dislocation and edge disloca­ tion respectively.

(9)

Where ρM;e and ρM;s are mobile dislocation densities for edge and screw dislocation, respectively; b is the Boggs vector; ve and vs are dislocation velocity for edge and screw dislocation, respectively. Immobile dislocation for slip system α can be divided into two parts: parallel dislocation and forest dislocation. For parallel dislocation, the dislocation lines are parallel to slip plane, and the dislocation lines for forest dislocation are parallel to slip plane normal, shown in Fig. 9 [6,7]. As the mobile dislocation density is much lower than the immobile, the interactions of dislocations in different slip systems can be expressed as

ραF ¼

μb qffiffiffiαffiffiffiffiffiffiffiffiffiffiffiffiαffiffiffiffiffi μb qffiffiffiαffiffiffi ρp;s þ ρM;s � c3 ρp;s 2π 2π

Likewise, for edge dislocation, the passing stress is

4.2.1. The framework of the flow rule As is well known, the plastic deformation of crystals is induced by the slip of screw and edge dislocation. Thus, the plastic shear strain, γ_ , can be rewritten as γ_ ¼ bρM;e ve þ bρM;s vs

(16)

V α ¼ c2 b2 λα

(8)

α

(15)

ρF

Based on the theory of crystal plasticity, the phenomenological variable and the physical phenomena are usually connected by the following constitutive assumption:

ραM;e ¼

(12)

2ð1 νÞKB θ qffiffiffiαffiffiffiffiffiffiffiαffiffi ρP;e ρF c1 c2 c3 μb3

(20) (21)

(22) (23)

From Eq. (21) and Eq. (23), if the number of edge dislocation is equal to that of screw dislocation, ραM;s is larger than ραM;e . This means the mobility of screw dislocation is higher that of edge dislocation.

Based on the framework of thermally activated dislocation motion, the average velocity vα then reads

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4.2.3. Quantitative relationship between edge and screw dislocation density The relationship between total dislocation density, edge dislocation density and screw dislocation density in one slip system can be expressed as

ρα ¼ ραedge þ ραscrew

b σe ¼ b σ þ Wpσ Lp ¼

ραedge fe ¼ α ρ

γ_ α Pα ¼

N X

α

(24)

γ_ α mα � nα

α

Flow rules

Simultaneously, we define

α

γ_ ¼ bρM;e vαe þ bρM;s vαs (25)

α

N X

σW p ¼ L : De

And, it is assumed that f αe starts from the initial values (f αeo ) and evolves towards finial saturation values (f αes ). Further, as f αe eventually evolves from strain hardening, it is assumed that f αe decays asymptotically be­ tween two limits during plastic deformation. Accordingly, the formalism for the evolution between the two bounds can be derived as � α � df αe f f αes ¼ θf αe (26) α α dγ f eo f es Where, θf is the initial decay rate. It is believed that the asymptote to­ wards a saturation limit is a common phenomenological nature of major physical (or mechanical) quantities involved in dislocation-based crystal plasticity [4].

� 1 vαe ¼ λα ν0 exp 2

� � � α jτ j ταpass;e α Qslip exp V KB θ KB θ

� 1 vαs ¼ λα ν0 exp 2

� � � α jτ j ταpass;s α Qslip exp V KB θ KB θ

ραM;e ¼

2ð1 νÞKB θ qffiffiffiαffiffiffiffiffiffiffiαffiffi ρP;e ρF c1 c2 c3 μb3

ραM;s ¼

2KB θ qffiffiffiαffiffiffiffiffiffiffiαffi ρP;s ρF c1 c2 c3 μb3

Hardening laws α

τ ¼ σ α : PðαÞ ταpass;e ¼ c3

4.3. The evolution of the dislocation density The hardening behavior of the flow stress depends on nucleation, multiplication and interactions of dislocation. For Ni-based superalloys, there are four major mechanisms for the motion of dislocation [28]. Storage of dislocation: dislocation become immobilized by forest dislo­ cation interactions upon traveling a distance (mean free path) propor­

ταpass;s ¼ c3 ραF ¼

qffiffiffiffiffiffiffi

μb 2πð1

ραp;e

νÞ

μb qffiffiffiαffiffiffi ρp;s 2π i

h

N X

χ αβ ραedge jcosðnα ; nβ � mβ Þj þ ραscrew jcosðnα ; mβ Þj

β¼1

1=2

tional to ρF . Dislocation annihilation: it leads to dynamic recovery, and is directly proportional to dislocation density ρ. GNDs: during deformation, the γ γ’ interfaces also restrict the motion of dislocation. GNDs density relates to the size and spacing of γ’ precipitates [31,32] and GNDs density contributes to micro size effects. K-W locks: this particular mechanism associates with the cross-slip, pinning and unpinning process of dislocation and stems from the L12 structure of the γ ’ precipitate phase [9]. To sum up, the complete dislocation evolution rate equation considering the four processes can be rewritten as pffiffiffiffiffi ρ_ α ¼ c4 ραF γ_ α

c6 α c5 ρα γ_ α þ γ_ þ X α γ_ α bd1

ραP ¼

i

h

N X

χ αβ ραedge jsinðnα ; nβ � mβ Þj þ ραscrew jsinðnα ; mβ Þj

β¼1

χ αβ ¼

2� 1

ðmα ⋅mβ Þ2

π

�1=2

ρα ¼ ραedge þ ραscrew df αe ¼ dγ α

(27)

� θf pffiffiffiffiffi

f αe f αeo

ρ_ α ¼ c4 ραF γ_ α

Where, the first term represents the storage of dislocation and the second term represents the annihilation and arrangement of dislocation. The third term is the evolution of GNDs density. This expression of GNDs is also adopted and discussed by Wang [28]. The last term is the evolution of K-W locks. Specifically, Xα can be derived as � � � � 1 Hl Ls Hu exp X α ¼ klock kunlock ρα exp (28) bL0 KB θ b KB θ

X α ¼ klock

f αes f αes

�

c5 ρα γ_ α þ

� � 1 Hl exp bL0 KB θ

c6 α γ_ þ X α γ_ α bd1 � � Ls Hu kunlock ρα exp b KB θ

There are mainly total eleven fitting parameters (c1 c6 , klock , kunlock θf f αeo , f αes ) in the present model. For a physically based model, all of these parameters have a physical meaning. Parameter c1 connects the average forest dislocation spacing with the jump width λα of a mobile dislocation and parameter c2 represents the effective obstacle width measured in Burges vectors. Parameter c1 and c2 are closely related to the activation volume in Eq. (16). Parameter c3 is a geometric param­ eter. Values of c3 determines the passing stress in Eq. (17) and Eq. (18) and it can be calculated from the initial passing stress of Ni3Al-base superalloys. Parameters c4 c6 , klock and kunlock are used to determine the evolution of different dislocation densities during deformation for Ni3Albase material. All of them are determined based on the measured dislocation densities under different strains. Parameters θf , f αeo and f αes are related to the quantitative relation between edge dislocation and screw dislocation.

Here, c4 c6 , klock and kunlock are material constants, d1 is the size of precipitates, Hl is the activation enthalpy for locking KW locks and Hu is the activation enthalpy for the unzipping process, and they are defined by Hirsch [9]. L0 is the activation length of sessile cross-slipped seg­ ments, which is of atomic dimension at 0 K. Ls is the critical unpinning spacing of a double super-kink, that is, the spacing between two pinning points. 4.4. Summary of the model All the equations in the model proposed in this paper are listed as follows. Elastic laws 8

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Table 5 Parameters of physical-based constitutive model at 973 K. Parameter

value

Parameter

value

C11

248.403GPa

υ

0.33

153.404GPa

1.38 � 10

C44

111.864GPa

KB

C12

5.32

Parameter

23

J/K

973K

θ

μ

66.47GPa

feo

0

b

2.5 � 10

fes

0.3

303J/mol

c1

1.8

Qslip

c2

0.025

c4

9.95 � 109/m

θf

c3

m

1 � 1010/s

v0

2.7

10

1.09 � 104/m

c5

value 3

c6

5.4 � 10

dδ

1.4 � 10

klock

4.8 � 10

Lo

0.4 � 10

Hl

30 � 103 J/mol

7 4

m

kunlock

0

Ls

0.5 � 10

Hu

28 � 103 J/mol

9 9

m m

Table 6 Parameters of physical-based constitutive model at 300 K. Parameter

value

Parameter

value

C11

262.357GPa

υ

0.33

C12

153.528GPa

KB

1.38 � 10

C44

112.126GPa 5.32

23

μ

66.47GPa

0

b

2.5 � 10

fes

0.3

Qslip

303J/mol

c1

1.8

J/K

973K

θ

feo

θf

Parameter

10

ν0

1 � 1010/s

4

m

c2

0.025

c4

9.95 � 109/m

c3

2.7

c5

1.09 � 10 /m

value

c6

5.4 � 10

dδ

1.4 � 10

klock

0

3 7

m

9

m

kunlock

0

L0

0.4 � 10

Ls

0.5 � 10

Hu

28 � 103 J/mol

Hl

9

m

30 � 103 J/mol

5. Application, comparisons and discussion Simulations of the alloy IC10, a typical Ni3Al-base superalloy, are conducted to verify the effectiveness of the new model. The simulations are conducted by finite element software ABAQUS (UMAT). During simulation, element C3D10 is adopted and the periodic boundary con­ ditions are considered. Extraction of stress and strain from simulated results is based on energy method and more details can be found in Zhang’s research [26]. 5.1. Determination of model parameters for simulation According to Zhang’s researches [34], the activation volume of alloy IC10 was 80b3 in 973 K with strain rate 1 � 10 3 s 1 and the density of forest dislocation was thought to be around 1 � 1014 m 2 [35]. Thus, the value of c1 c2 is calculated to be between 1.0 � 10 2 and 10 � 10 2. As the initial passing stress was reported to be around 30 MPa for Ni3Al-­ base superalloy [4], values ranging from 1.0 to 5.0 are reasonable for c3 . Parameters c4 c6 , klock and kunlock are used to determine the evolution of different dislocation densities during deformation for Ni3Al-base material. All of them are determined based on the measured dislocation densities in Table 4. Parameters θf , f αeo and f αes are related to the quan­ titative relation between edge dislocation and screw dislocation. Here, percentage of edge dislocation and screw dislocation in each slip plane is assumed to be identical. Thus, fe ¼ f αe , feo ¼ f αeo and fes ¼ f αes ðα ¼ 1; 2:::12Þ. The measured value of fe was 0.23 for IC10 at 973 K with axial strain 8.2% and the value of fe was zero with axial strain 0%. Based on Eq. (26), θf can be rewritten as, θf ¼

f αes

f αeo γα

⋅ln

f αes f αes

f αe f αeo

Fig. 10. Numerical and experimental results for IC10 at 300K and 973 K with strain rate 1 � 10 3 s 1.

As is reported by former researches, the locking process was acti­ vated only in high temperatures for Ni3Al-base material, thus, klock is set to be zero when temperature is 300 K in Table 6. As the unlocking process was activated only when temperature was higher than 1023 K, kunlock is set to be zero in both 300 K and 973 K. For IC10, initial pass stress [4], total dislocation density [35] and the activation volume [34] were almost the same in 300 K and 973 K. Thus, for simplify, the pa­ rameters (c1-c6) are set to be identical at 300 K and 973 K.

(29)

5.2. Simulated results and discussion

Above all, it is set that feo ¼ 0, fes ¼ 0:3 and θf ¼ 5:32. Here, all types of dislocation with axial strain 8.2% are considered to be almost saturated. All the other parameters in the model are determined by ex­ periments or referring to former papers [4,28,34]. The parameters used in simulation are shown in Table 5 and Table 6.

Fig. 10 gives a comparison of the simulated stress-strain curves and the experimental ones. As is shown in Fig. 10, the simulation results capture general trends of the stress strain relation for IC10 in 300 K and 973 K. Also, it is easily found that there are two important points, 9

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Materials Science & Engineering A xxx (xxxx) xxx

marked as point A and point C, in simulated stress-strain curves. Plastic flow begins if external stress is larger than stress at point A and strain hardening process keeps almost stable when stress is larger than that at point C. At point A, resolved shear stress just reaches to the value of initial pass stress, screw dislocation begins to glide and propagate rapidly in different slip systems. After point C, the occurrence of dislo­ cation dipoles increases the annihilation rate of dislocation, leading to the gradually dynamic balance of the propagation and annihilation of dislocation. Thus, the strain hardening rate keeps almost stable. In order to present the stress-strain data in a more sensitive form, Fig. 11 gives the Kocks-Mecking-plot, where the hardening rate is plotted against stress. Obviously, the simulated results show a very good agreement with experimental ones. It is suggested that there are four hardening processes successfully reproduced by this model during deformation. Before point A, hardening rate keeps unchanged, this is elastic process. Between point A and point B, hardening rate increases slowly with the stress increasing, and this is the hardening stage I. However, between point B and point C, it increases sharply, and this is the hardening stage II. Again, the hardening rate almost keeps a constant when stress is larger than that at point C, this is the hardening stage III. Meanwhile, between point A and point B, the simulated hardening rate is lower than experimental ones. While between point B and point C, the simulated values is higher than tested ones. Thus, more complex hard­ ening mechanisms should be investigated for Ni3Al-base superalloy in the following researches. Hardening rate in the simulation is turned out to be closely related with the motion of dislocation. And shear rate in each slip system de­ pends directly on the motion of dislocation. It is found that only eight slip systems are active during simulation and the changes of shear rate in these eight slip systems are almost the same. Fig. 12 shows the simulated different shear rates, γ_ , γ_ edge and γ_ screw , of slip system ð111Þ½101� against the average strain ε½001� at 973 K. It is convinced from the simulated results that different kinds of dislocation (edge dislocation and screw dislocation) have different contributions to the shear rate. Thus, it is necessary to consider specific motion of edge dislocation and screw dislocation respectively. Furthermore, it is obvious that screw disloca­ tion has more effects on shear rate and this is probably due to the reason that the mobility of screw dislocations is higher than edge dislocation. Although the values of different shear rates are different, the variation trends of them are performed to be similar. All of them move toward a saturation value after average strain 1.5%. There are two kinds of classified method of dislocation, one is ρtotal , ρGND , ρSSD and ρKW , another is edge dislocation, screw dislocation and total dislocation. Fig. 13(a) gives the evolution of simulated dislocation density at 973 K follows the first classified method. Fig. 13(b) follows the

Fig. 11. Comparison of hardening rate between numerical and experimental results for the strain rate 1 � 10 3 s 1.

Fig. 12. Different slip shear rates (_γ, γ_ edge and γ_ screw ) of slip system ð111Þ½101� for IC10 at 973 K with the stain rate 1 � 10 3s 1.

Fig. 13. Comparison between (a) various simulated dislocation densities and experimental ones, (b) total dislocation density, edge dislocation density and screw dislocation density. 10

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Materials Science & Engineering A xxx (xxxx) xxx

Fig. 14. The comparison results between the new model and two single dislocation models.

second method. It is suggested by Fig. 13(a) that the simulated total dislocation density is in good agreement with the experimental values in Table 4. Furthermore, the simulated total dislocation density increases rapidly at the beginning. And, the increasing process will be slow when the strain is larger than 1%. For simulated SSDs, the similar increasing process is found at the beginning. However, when the strain is larger than 1%, dislocation density of SSDs is almost unchanged. It is noted that with the increase of strain, the deformation inhomogeneity of phase boundaries and grain boundaries increases, thus, the dislocation density of GNDs keeps increasing. Also, at 973 K, the unlocking process of KWlocks and dislocation arrangement process are not active, leading to the increasing of ρKW . The simulated results in Fig. 13 (b) suggest that edge dislocation density is still increasing while screw dislocation density reaches a saturation value during deformation at 973 K. Thus, the strain hardening after yielding is closely related to the storage of edge dislocation. Meanwhile, during deformation for IC10 at 973 K, the locking process of KW-locks is active, while the unlocking process is inactive. The density of long straight edge dislocation, which is called super-kinks [9], in­ creases with the increase of deformation. The increase of super-kinks (edge dislocation or near-edge dislocation) has significant influence in strain hardening process after yielding. One feature of the new model is that the laws of latent hardening of edge dislocation and screw dislocation between different slip systems are considered directly in each slip system. Fig. 14 gives the comparison results between different models. If only one type of dislocation is considered in deformation, the micro shear deformation is limited and the internal mobility of the material is poorly considered. In this case, the model is over-hardening.

Fig. 15. Sensitivity analysis of parameters: (a) feo , (b) fes and (c) θf .

smaller the value of θf , the greater the rate of edge dislocation multi­ plication. The occurrence of edge dislocation evolves from the genera­ tion of KW-locks during deformation and the mechanism of KW-locks increase the initial yield stress [9]. Above all, the determination of θf , feo and fes depends highly on the measurement of dislocation density during deformation. Based on the measured results of dislocation density in Table 4, it is suggested that feo ¼ 0, fes ¼ 0:3, θf ¼ 5:32.

5.3. Sensitivity analysis of related parameters, θf , feo and fes In the new model, parameters θf , feo and fes are introduced to explain the influences on deformation of screw and edge dislocation respec­ tively. As can be seen from Fig. 15, some parameters, such as feo and fes , have little influence on the mechanical behavior of Ni3Al-base superal­ loy, but the parameter θf has obvious effect. Here, for the principle of controlling variables, the value of fe is keep unchanged when axial strain is 8.2% in Fig. 15(a) and (b). Thus, the changing of the initial and saturation value of fe during simulation has little impact on the distri­ bution ratio of edge dislocation and screw dislocation under different strain levels. θf is the initial decay rate and it is closely related to the propagation rate of edge dislocation during deformation. Obviously, with the decrease of the value of θf , the initial yield stress increases. The

6. Conclusion (1) Unidirectional tension tests of IC10, a typical Ni3Al-base super­ alloy in China, were conducted at 300 K and 973 K with a strain rate 1 � 10 3 s 1. Meanwhile, dislocation configurations were observed and dislocation densities were measured under different strain levels (0.8% strain, 3.0% strain, 6.0% strain and 8.2% strain) by TEM. In the initial stage of deformation, only screw dislocation is found in IC10. With the increase of strain, edge 11

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Materials Science & Engineering A xxx (xxxx) xxx

dislocation begins to emerge. In the late stage of deformation, the ratio of edge dislocation density to screw dislocation density is about 0.3. Meanwhile, total dislocation density increases with the increase of deformation. (2) A new physical-based constitutive model has been established. The model has been implemented in a crystal plasticity finite element method numerical framework at 300 K and 973 K with a strain rate 1 � 10 3 s 1. Simulated macro mechanical behavior and total dislocation density agree with tested results. It is revealed from simulation that the shear rates of edge dislocation and screw dislocation are different during deformation. With the increase of deformation, total dislocation density and screw dislocation density increase sharply at first and then tend to a saturation value. Unlike screw dislocation density, the edge dislocation density increases slowly and monotonically with the increase of deformation.

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Declaration of competing interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “A physical-based constitutive model considering the motion of dislocation for Ni3Al-base superalloy”. Acknowledgment Thanks are given to the National Natural Science Foundation of China (91860111), the Fundamental Research Funds for the Central Universities (NO. NS2016026), and the Innovation Fund of Jiangsu Province, China (KYLX-0304). References [1] S.J. Davies, S.P. Jeffs, M.P. Coleman, R.J. Lancaster, Effects of heat treatment on microstructure and creep properties of a laser powder bed fused nickel superalloy, Mater. Des. 159 (2018) 39–46. [2] Y.K. D Kimb, H.K. Kimb, E.Y. Yoonc, Y. Leec, C.S. Ohd, B.J. Lee, A numerical model to predict mechanical properties of Ni-base disk superalloys, Int. J. Plast. 110 (2018) 123–144. [3] S. Chandra, M.K. Samal, R. Kapoor, N.N. Kumar, V.M. Chavan, S. Raghunathan, Deformation behavior of nickel-based superalloy Su-263: experimental characterization and crystal plasticity finite element modeling, Mater. Sci. Eng. A 735 (2018) 19–30. [4] Y.S. Choi, D.M. Dimiduk, M.D. Uchic, Modeling plasticity of Ni3Al-based L12 intermetallic single crystals. I. Anomalous temperature dependence of the flow behavior, Philos. Mag. 87 (2007) 1939–1965. [5] Y.S. Choi, D.M. Dimiduk, M.D. Uchic, Modeling plasticity of Ni3Al-based L12 intermetallic single crystals. II. Two-step (T1 and T2) deformation behavior, Philos. Mag. 87 (2007) 4759–4775. [6] A. Ma, F. Roters, A constitutive model for fcc single crystal based on dislocation densities and its application to uniaxial compression of aluminum single crystals, Acta Mater. 52 (2004) 3603–3612. [7] A. Ma, F. Roters, D. Rabbe, On the consideration of interactions between dislocations and grain boundaries in crystal plasticity finite element modelingtheory, experiments and simulations, Acta Mater. 54 (2006) 2181–2194.

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