A computational modeling of fully friction contact-interaction in linear friction welding of Ni-based superalloys

Materials & Design 185 (2020) 108244

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A computational modeling of fully friction contact-interaction in linear friction welding of Ni-based superalloys Peihao Geng a, b, Guoliang Qin a, b, *, Jun Zhou c a

Key Laboratory for Liquid-Solid Structure Evolution and Processing of Materials, Ministry of Education Shandong University, Jinan, 250061, China Institute of Materials Joining, Shandong University, Jinan, 250061, PR China c Harbin Welding Institute, Chinese Academy of Machinery Science and Technology, Harbin, 150028, PR China b

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


A computational FE modeling of fully coupled deformation of two deformable workpieces was applied to LFW of superalloys .
The 3D FE model successfully predicted thermo-mechanical response during LFW of same or dissimilar superalloys.
Qualitative validation of computational-analysis results showed excellent agreement with experimental results .
The model provided a platform on which further 3D modeling investigations of LFW for different geometries might be based.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 August 2019 Received in revised form 12 September 2019 Accepted 26 September 2019 Available online 18 October 2019

A three-dimensional coupled thermo-mechanical model was developed to simulate the linear friction welding (LFW) of same or dissimilar Ni-based superalloys. Full friction contact-interaction for two deformable workpieces rubbing against each other was considered using plastic/plastic friction pair model. Numerical simulated results have been well validated by experimental measured data. Owing to the inherent motion mechanism of LFW, interfacial heat flow varied with time periodically, resulting in the periodic evolution of interface temperature. The distribution of stress field at stationary side alternated dynamically with respect to that at oscillatory side. During LFW of GH4169 to FGH96 superalloys, more heat flowed into the GH4169 superalloy, which resulted in greater temperature gradient than that at side of FGH96 superalloy. More softened materials were extruded out from interface at GH4169 side owing to their better flow property. Peak stress near the interface of GH4169 superalloy side was higher than that at FGH96 side. Additionally, the characteristics of modeling of LFW based on plastic/plastic friction pair model were investigated by performing numerical simulations for LFW experimental phenomena. © 2019 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Computational modeling Linear friction welding Ni-based superalloys Thermo-mechanical behavior Arbitrary Eulerian-Lagrangian

1. Introduction * Corresponding author. Qianfoshan Campus of Shanetdong University, 17923 Jingshi Road, Jinan, 250061, China. E-mail address: [email protected] (G. Qin).

Linear friction welding (LFW) was a high-efficiency and lowenergy solid-state joining method, which was widely used for the

https://doi.org/10.1016/j.matdes.2019.108244 0264-1275/© 2019 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

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P. Geng et al. / Materials & Design 185 (2020) 108244

mass-production of excellent-quality weld joints [1]. In LFW, a reciprocated linear motion workpiece was rubbed against the other stationary one under a compressive pressure. Heat was generated by interfacial friction and consequently plastic deformation resulted in the flash formation. Microstructure of the parent material underwent the high temperature and flow deformation. As a result, the refined microstructure could occur and then improve the weld properties compared to the parent material. As reported previously [2], the physical nature of solid-state welding process made it ideal for joining difficult-to-weld Ni-based superalloys used in aircraft structural components. During the last few years, considerable research groups have studied LFW experimentally, with a focus on the measurements of process temperature and the influences of process parameters on the microstructures and properties. Chamanfar et al. [3] found that the weld strength was the same as the parent material above a critical shortening, which was related to the power input and welding time. Ma et al. [4] reported that the fine recrystallized grains caused by high temperature and deformation occurred in the weld zone, which improved weld strength of joints. As stated by Li et al. [5], the experimental obtained information to guide the process design was quite limited because several difficulties such as the determination of interfacial temperature and plastic deformation could not be addressed suitably. Therefore, a reliable and dedicated numerical model for LFW needed to be developed, which could provide a comprehensive interpretation of thermomechanical influence on microstructural evolution and joint properties. To date, numerous papers on numerical simulation of LFW have been published to study the main basic theoretical issues in LFW [6]. As the LFW process was undergoing rapid development, it was

reasonable that an updated literature review of process modeling could prove to be a useful tool to understand the burning questions in LFW. Alongside the review of finite element (FE) modeling of LFW, the relevant experimental work and numerical modeling in this paper were then described. 2. Literature review: modeling of LFW Significant effort has been expended to develop the modeling of LFW. Fig. 1 shows the representative papers found in the published literature from the Web of Science since the year of 2000. These papers have been divided by model dimension, i.e., twodimensional (2D) or three-dimensional (3D). To date, the 2D numerical models have been of preference because of the low computational cost relative to 3D modeling. Though the 2D models provided considerable information about the theoretical issues in LFW, the heat generation and material flow perpendicular to the oscillation direction were neglected in the 2D models. As a result, the investigation on spatial distribution of thermal field, stress state and flash formation were inevitably limited to such models. As was seen in Fig. 1, the 3D modeling to study the LFW has gained increasing attention in recent years. The literature showed that three 3D modeling approaches have been widely used to simulate the LFW process based on the selection of object modeling strategy, i.e., plastic-rigid contact [17,22,26,27], plastic-plastic contact [8,12,13], and singe block models [23,24,28,30]. The detailed review of the 3D models was mainly discussed in this section, because the same results obtained from 2D models could be readily achieved by the 3D models. Process models using the first approach were assumed as a symmetric model to reduce the modeling complexity and save computational

Fig. 1. Main papers [7e30] about LFW simulatin found in literature from Web of Science since the year of 2000.

P. Geng et al. / Materials & Design 185 (2020) 108244

cost, in which one plastic body was considered as a rigid body or analytical surface. Li et al. [17] established a 3D numerical model using the first approach to obtain the temperature field in LFW. Buffa et al. [22] developed a similar 3D numerical model combined with a neural network to predict the weld quality of LFW joints. Recently, Bühr et al. [27] has proposed a computationally efficient thermal model considering single sided workpiece, which was used to predict the residual stresses in the Ti alloy LFW weld. To the knowledge of authors, Sorina-Müller et al. [8] developed the first 3D LFW model based on plastic/plastic friction pair to compare the weld temperatures of different geometrical workpieces for Ti alloys. A higher interface temperature occurred in the larger geometry. Grujicic et al. [12,16] also developed the LFW model using the second approach, but they only studied the spatial distribution of temperature in integral joint and no validated results were given. Compared with the first modeling approach, the LFW process for two deformable workpieces with different cross-section dimensions, and particularly with different materials could be modeled based on the plastic-plastic friction pair. The present authors namely Geng et al. [29] also developed a 3D model to simulate the LFW of GH4169 superalloy and have obtained a satisfactory agreement between simulated and experimental results. Recently, to obtain the flash formation of LFW for Ti alloys, Turner et al. [10] first noticed that there was negligible macroscopic plastic deformation prior to the merging of two workpieces for Ti alloys. Then, a 2D plastic flow model using a single block representing the two plastic workpieces as full contact achieved was developed. Based on this 2D model [18,19], McAndrew et al. [24] also developed a 3D LFW model of Ti alloys. Recently, similar model has been used to study influences of welding parameters in LFW by

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Baffari et al. [28]. Compared to the first two approaches, the third approach produced much better replications of the flash formation for Ti alloys workpiece owing to the merging of the material element. However, the continuity of thermo-mechanical behavior during process were ignored in the single block model. In addition to the selection of object modeling strategy, there were the four main issues those were should be considered for the thermo-mechanical modeling of LFW: (a) the interfacial friction state affecting the heat input; (b) the constitutive material models affecting the plastic deformation, (c) the realistic thermal and mechanical boundary conditions, (d) the validation of heat generation model and the resulting temperature. Table 1 summaries the corresponding solutions for these issues in the literature. The most concern modeling issue was the models of the contact condition in LFW. To date, there were two domain friction models, i.e. Coulomb friction model and Shear friction model with constant or various friction coefficient (factor). The friction model, either Coulomb friction model or Shear friction model, aimed to obtain the realistic friction force, which revealed the contact condition in LFW experiments of studied metals. Based on assumption that the friction condition in LFW could be represented by that in friction and wear tests, the friction coefficient obtained experimentally was calibrated in most cases to provide a suitable temperature history, which matched with measured one. Although the realistic friction condition in LFW was complex and even could not be fully reflected by analytical friction model, the approach of simple friction model to describe the contact condition was pragmatic in most cases. The second key point in modeling of LFW was the importance of material constitutive data. As studied in the recent work [26], it has proved that the selection

Table 1 Summary of the approaches to the core modeling issues in LFW. References

Materials

Object modeling strategy and contact conditions

Heat input

Material constitutive data

Boundary conditions

Validation

Sorina-Müller et al. [8]

Ti alloy

Friction (proportional to friction force and slip velocity)

No stated

Ti alloy; Steel

Friction (proportional to friction force and slip velocity) and plastic deformation

Johnson-Cook deformation model

Fratini et al. [13]

Al alloy

Li et al. [17]

Steel

Buffa et al. [22]

Al alloy

Two cases: One is only thermal and mechanical boundary conditions for plastic workpieces, rigid body only considering heat conduction or not. One is that considering same thermal (or mechanical) boundary conditions for two plastic workpices

Temperature

Grujicic et al. [12,16,21]

McAndrew et al. [24]

Ti alloy

Baffari et al. [28] Geng et al. [26]

Ti alloy

Bühr et al. [27]

Ti alloy

Plastic/plastic friction pair; Coulomb's model with temperature-dependent friction coefficient (calibrated via experiments.) Plastic/plastic friction pair; Coulomb's model with temperature-dependent friction coefficient and limited shear value Rigid/plastic friction pair; Shear friction model with empirical correlation between temperature and friction factor. Rigid/plastic friction pair; Coulomb's friction model with temperature-dependent friction coefficients Rigid/plastic friction pair; Shear friction model with calibrated friction factor Single block models; Not stated constant heat input with weld variables; Calibrated via experiment; Shear model with temperaturedependent friction factor Rigid/plastic friction pair; Modified Coulomb's model with constant friction coefficient Rigid/plastic friction pair; Constant heat fluxes, so assumed constant friction coefficient.

GH4169;

Fields-Backofen model considering softening

Johnson-Cook deformation model Friction and plastic deformation Two cases: friction only and friction plus plastic heating

Friction and plastic deformation

Stain, strain rare and temperaturedependent yield stress provided by DEFORM software.

Arrhenius model

Temperaturedependent yield stress from reference

No direct validation

Shortening length;

Temperature; Shortening length

Temperature

Temperature

Flash shape; Burn-off (rate) Temperature; Deformation

Temperature; Burn-off rate;

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P. Geng et al. / Materials & Design 185 (2020) 108244 Table 2 Chemical compositions of the two Ni-based superalloys (wt%). Materials

Elements

GH4169 superalloy

Ni 54.10 Cr 16.1

FGH96 superalloy

Cr 19.39 Co 13.0

Nb 5.21 W 4.05

Mo 3.05 Mo 3.94

Al 0.53 Al 2.3

Ti 1.02 Ti 3.7

Mg 0.01. Nb 0.8

Fe Bal Ni Bal.

Table 3 Experimental conditions used to validate the model. No.

J1 J2 J3 J4

Fig. 2. Schematic of LFW process: (a) self-designed LFW machine and (b) positions of preset thermocouple.

of constitutive data had an obvious impact on the flash formation. The constitutive response for high-temperature materials in LFW significantly not only influenced the flow behavior and deformation, but was strongly related to the heat of plastic deformation. However, since the plastic flow constitutive models for various alloys were relatively incomplete and even sparse within temperatures and strain rates regimes of friction welding processes,

Welding parameters Friction pressure/Pf (MPa)

Frequency/fo (Hz)

Amplitude/Ao (mm)

200 300 400 400

25 25 25 30

2.5 2.9 2.9 2.9

considerable researches have used the constitutive data of similar alloys from references or material library in simulation software. Additionally, although the different thermo-mechanical condition at two sides of interface during LFW has been reported experimentally, its effects usually were unconsidered in most models with plastic-plastic contact pair. This was discussed further in our model implementation below. As listed in Table 1, the literature showed that the prime objectives of all models were to predict the distribution of temperature and stress fields, the shape of deformation zone, and the critical condition of bonding. The corresponding experimental validation showed that these robust and reliable numerical models certainly could represent the valid design tools in the parameter optimization of the LFW process. Thus, numerous researchers have tried to establish an excellent numerical strategy that was able to provide a thorough-analysis on the LFW process. To date, there was very little work published on the 3D computational modeling of full friction contact-interaction and fully-coupled thermo-mechanical deformation of two deformable workpieces in LFW for Ni-based superalloy. Besides, few researches were focused on qualitatively revealing the periodic pulsed nature of thermo-mechanical coupling behavior in LFW owing to the limitation of numerical modeling.

Fig. 3. LFWed joints of Ni-based superalloys at Pf ¼ 400 MPa, fo ¼ 25 Hz and Ao ¼ 2.9 mm: (a) GH4169 to GH4169 joint and (b) GH4169 to FGH96 joint.

P. Geng et al. / Materials & Design 185 (2020) 108244

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Fig. 4. Characteristic times of LFW of GH4169 to GH4169 superalloys during one oscillation cycle: (a) the actual welding and (b) the schematic.(Lines in the dark was some line caps of asbestos cloth).

Therefore, in this present work, a 3D model based on the plastic/ plastic friction pair for LFW was developed and aimed to provide in-depth understandings on the thermo-mechanical coupling behavior during LFW. The issues in Table 1 have also been addressed in the following numerical modeling. In consideration of the limited information on modeling for LFW of Ni-based superalloys, the developed model was applied to the instrumented welds of same or dissimilar Ni-based superalloys. The thermo-mechanical coupling characteristics such as the evolution of the main field variables were discussed to reveal the physical nature of LFW. Several experimental phenomena those were existing, however overlooked in LFW experiments of Ni-based superalloys were first investigated using the developed model. On the basis, the characteristics of LFW modeling in this work were also discussed.

FGH96 at Pf ¼ 400 MPa, fo ¼ 25 Hz, Ao ¼ 2.9 mm. The curled flash was extruded out from the interface of either same superalloy joint or dissimilar superalloy joint. Fig. 4(a) shows the actual LFW process of GH4169 to GH4169 superalloys during one oscillation cycle. It was seen that the softened materials were alternately extruded out from two workpieces into flash. The friction contact area varied with time periodically as the oscillation workpiece reciprocated as schematically in Fig. 4(b). Compared to the LFWed joint of Ti alloys [10], the deformation behavior of the workpieces merging was not obvious in LFW of superalloys. This was related to the different

3. Experimental work A self-designed hydraulic drive LFW machine in Fig. 2(a) was used to join the rectangular blocks of Ni-based superalloys. All blocks had the dimensions of 33 mm in height (H), 14 mm in length (L) and 10 mm in width (W). In this study, the selected Ni-based superalloys were GH4169 and FGH96 superalloys, which were widely used in fabrication of turbine disc [1]. Table 2 lists the chemical compositions of the two Ni-based superalloys. The same joint of GH4169 to GH4169 and dissimilar joint of GH4169 to FGH96 were joined by LFW, respectively. In experiments, the oscillation motion driven by the hydraulic system was the standard sinusoidal mode with a given frequency (fo) and amplitude (Ao). The oscillatory direction was along the interfacial length of 14 mm. Table 3 shows the experimental welding parameters. The friction time was kept as 5 s for all experiments. Upsetting pressure was as same as friction pressure. Temperature history during the welding was recorded using K-type thermocouples with an outer diameter of 0.8 mm. As shown in Fig. 2(b), the thermocouples before welding were preset into the blind hole with a diameter of 1 mm, which was drilled at the mid-length along the oscillatory direction with a 5 mm distance from the interface. A high speed camera was used to acquire the transient welding process. To avoid the effect of clamping in LFW, the same clamping location of 10 mm was used. The asbestos cloth was used to avoid the loosening of clamping during process. Fig. 3 shows the LFW joints of GH4169/GH4169 and GH4169/

Fig. 5. Thermo-mechanical coupled FE model of LFW.

P. Geng et al. / Materials & Design 185 (2020) 108244

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thermo-physical properties between two different alloys. Based on experimental observation, the plastic-plastic contact pair should be considered as the object modeling strategy, which could reflect the full friction contact-interaction for heat generation. Two deformable workpieces formed the reciprocating friction pair, where the interface heated by friction work reached the plastic state in a short time. 4. Modeling approach of LFW The 3D thermo-mechanical coupling model of LFW was established using an Arbitrary Lagrangian-Eulerian FE method (ALE-FEM) based on software ABAQUS. ALE-FEM could simulate the oscillations of workpieces, and calculate the transient heat flow from friction movement. Additionally, adaptive re-meshing based on the ALE algorithm avoided unacceptable element distortion due to the plastic deformation near the friction interface. In this study, the studied workpieces of Ni-based superalloys were rectangular blocks exhibiting same dimensions as those in experiments. Fig. 5 shows the mesh condition of FE model. Brick type (8-nodes thermally coupled, trilinear displacement and temperature, reduced integration, hourglass control) elements with various sizes were applied for discretization of workpieces. Fine elements close to the friction interface were considered to ensure the accuracy of thermal and deformation fields calculated by the numerical model. According to the preliminary study of mesh sensitivity, the mesh element length at the region within 5 mm distance from contact surface and at fixed region of clamp from the bottom surface to 10 mm distance from interface were 0.75 mm and 2.5 mm, respectively. 4.1. Plastic/plastic friction pair model To consider the process features of LFW into numerical simulation, the contact model with plastic/plastic friction pair was developed to describe the thermo-mechanical interaction between two workpieces. Accurate calculation of contact forces such as normal force (Fn) and tangential force (Ft) in friction pair was important because the contact forces was known to have a considerable influence on the friction state between two workpieces. Penalty contact algorithm was used to define normal contact condition between the friction pair [31], and the normal and tangential forces were expressed as:

In consideration of inadequacy of the Classic Coulomb's model at higher normal pressure caused by the transition of friction condition [32], the calculation of frictional shear stress was modified using the following equation:



s tf ¼ min mpn ; psffiffiffi

 8 < Fn ¼ pn Sn ¼ En du2n  du1n  c   : F ¼ t S ¼ mF sign du2  du1 t n f n t t

(1)

where pn and Sn represented the normal pressure and contact area, respectively. m and tf represented the coefficient of friction and frictional shear stress, respectively.

(2)

3

where ss was the material yield flow stress. The yield flow stress was calculated from the constitutive data in the following section. To obtain the frictional shear stress between two workpieces of dissimilar materials, the lower value of yield flow stress was considered, which was expressed as:

tmax ¼

  min s1s ; s2s pffiffiffi 3

(3)

where s1s and s1s represented the yield flow stress of two dissimilar materials. Besides, a mathematical equation obtained experimentally was used to calculate the friction coefficient [33], which was expressed as:

m ¼ apn b Tint c expðdvÞ

(4)

where m and v represented the friction coefficient and slip velocity, respectively. Tint was the interface temperature. The constants a, b, c, and d were determined as 0.12, 0.233, 0.471 and 0.739, respectively. 4.2. Heat generation model Interfacial heat generated by friction was transferred to the interior of two workpieces during LFW. The transient heat flow rate caused by friction work was expressed as:

h i qfr ¼ q1fr þ q2fr ¼ h ð1  dÞmpn þ dtf veq

(5)

where h represented the efficiency of heat conversion. d was a state variable describing the friction condition [32]. Additionally, the distribution of heat between two contact surfaces was calculated according to Ref. [34] for dissimilar metals as:

f1 ¼ f2





sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1 r1 cP1 l2 r2 cP2

(6)

where li ri, and cPi (i ¼ 1, 2) were thermal conductivity, density and specific heat, respectively. In this model, the total heat was distributed equally for LFW of GH4169 to GH4169 superalloys. Based the averages of thermal conductivity, density and specific heat in Table 3, the estimated heat flux into the workpiece of FGH96

Table 4 Thermo-physical properties of studied Ni-based superalloys. Material

Temp. ( C)

Thermal conductivity (W/m/K)

Specific heat (J/kg/K)

Density (g/cm3)

Young's modulus (GPa)

Poisson ratio

GH4169

25 200 400 600 800 1000 100 200 400 600 800 1000

13.4 15.9 18.3 21.2 23.6 30.4 6.9 9.3 14.8 21 27.3 34.0

468 471 493 539 615 707 480 480 490 540 695 e

8.24

204

0.31

8.32

211

0.30

FGH96

P. Geng et al. / Materials & Design 185 (2020) 108244

superalloy was approximately 48.5% for LFW of GH4169 to FGH96 superalloys. The volumetric plastic deformation heat also played an important role in heat transfer in friction welding processes. In this model, the volumetric heat generation by plastic deformation (qpl) was defined as: pl

qpl ¼ hsε_

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approximately 1100  C where its strength significantly reduced. Similarly, the flow stress data of FGH96 superalloy for temperature above 700  C were taken from Eq. (8). Below 700  C, flow stress data at a quasi-state strain rate from the Yan et al. [36] were used. In consideration of the achievement of steady-state flow stress with the increase in strain, unchanged flow stresses data at higher strain than 1.2 were used for various deformation conditions.

(7)

pl where s and ε_ represented the Von Mises effective stress and plastic strain rate, respectively.

4.3. Material properties Detailed thermo-mechanical properties of the studied Ni-based superalloys were used in this model. As listed in Table 4, the temperature-dependent thermo-physical properties including thermal conductivity, heat specific and Young modulus were taken from the publications [20,35,36]. Importantly, in consideration of the significant influence of the constitutive data on plastic deformation, the high-temperature flow stress-strain curves of the two studied Ni-based superalloys were obtained from hot compression experiments. Based on the experimental data, the constitutive modeling during high-temperature deformation has been performed as detailed in the recent work [26,37]. As expressed in Eq. (8), the strain-compensated Arrhenius model was used to calculate the yield flow stress of the two Ni-based superalloys.

4.4. Boundary conditions Thermal boundary condition involved the friction heating occurring at the friction interface between the friction pair, the heat convection and the heat radiation occurring simultaneously on the external surfaces of workpieces, which was defined by:

l

  vT ¼ qfr  hc ðTw  Ta Þ  qa T 4w  T 4a vn

(10)

where hc and a represented heat convection coefficient and the emissivity for heat radiation, respectively. q was the StefanBoltzmann constant. Tw was the workpiece temperature. Ta was the ambient temperature. To facilitate the numerical solution of thermal problem, heat convection coefficient needed to be determined. Because the oscillation workpiecc was reciprocating at high speed, it experienced force convection cooling for which the correlation for Nusselt number was given by Refs. [14,39]:

8 " #1=2 9  

 

< ε_ = 1 Q ðεÞ 1=nðεÞ ε_ Q ðεÞ 2=nðεÞ exp exp ln ss ¼ þ þ1 ; RT AðεÞ RT aðεÞ : AðεÞ

where Q(ε) was deformation activation energy, a(ε), n(ε), and A(ε) were material constants, respectively. R was universal gas constant. These material constants were expressed in Eq. (9) using the polynomial functions, whose coefficients were defined in Table 5.

aðεÞ ¼ B0 þ B1 ε þ B2 ε2 þ B3 ε3 þ B4 ε4 þ B5 ε5

nðεÞ ¼ C0 þ C1 ε þ C2 ε2 þ C3 ε3 þ C4 ε4 þ C5 ε5 Q ðεÞ ¼ D0 þ D1 ε þ D2 ε2 þ D3 ε3 þ D4 ε4 þ D5 ε5 ln AðεÞ ¼ F0 þ F1 ε þ F2 ε2 þ F3 ε3 þ F4 ε4 þ F5 ε5

Nu ¼

(8)

hc1 D

l

¼ 0:332Re1=2 Pr 1=3

(11)

where hc1 represented the force convection coefficients. The stationary workpiece experiences free convection, for which the convection efficient was significantly lower. The corresponding correlation for the Nusselt number was given by Ref. [40]:

(9)

Notably, flow stress data of GH4169 superalloy for temperature less than and equal to 650  C were taken from Brar et al. [38] at a quasi-static strain rate. From 650  C and beyond, flow stress data from Eq. (8) were used. Above 650  C, GH4169 superalloy has reduced strength due to the high temperature experiences and strengthening phases could be fully dissolved into the matrix at

Nu ¼

hc2 D

l

8 92 < = 0:387Ra1=6 ¼ 0:825 þ h i8=27 ; : 1 þ ð0:492=PrÞ9=16

(12)

where hc2 represented the free convection coefficients. The calculated heat convection coefficients could be expressed as function of temperature:

Table 5 Coefficients of polynomial functions of material constants. Material constants

a

n

Q

lnA

B0 ¼ 0.007 (0.004) B1 ¼ 0.021 (0.018) B2 ¼ 0.120 (0.130) B3 ¼ 0.253 (0.328) B4 ¼ 0.187 (0.379) B5 ¼ 0 (0.158)

C0 ¼ 4.486 (5.996) C1 ¼ 0.563 (29.454) C2 ¼ 15.871 (178.881) C3 ¼ 38.518 (503.897) C4 ¼ 23.371 (642.96) C5 ¼ 23.371 (305.701)

D0 ¼ 391.521 (884.039) D1 ¼ 389.697 (1362.808) D2 ¼ 1822.415 (14592.48) D3 ¼ 4829.101 (47941.27) D4 ¼ 4250.944 (59684.61) D5 ¼ 4250.944 (25574.006)

F0 ¼ 32.813 (76.454) F1 ¼ 34.324 (116.211) F2 ¼ 154.701 (1227.531) F3 ¼ 408.443 (4070.951) F4 ¼ 361.880 (5118.476) F5 ¼ 361.880 (2221.25)

Note: Data in brackets refer to coefficients of Eq. (10) for FGH96.

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hc1 ¼ 0:11T þ 54:6 hc1 ¼ 0:0044T þ 11:7

(13)

Besides, the x-direction of the oscillation workpiece in Fig. 5 was considered as the oscillation direction, along which a sinusoidal mode with a given frequency and amplitude was applied. Meanwhile, the y-direction and z-direction of clamped region of oscillation workpiece were constrained. The x-direction and z-direction of clamped region of stationary workpiece were constrained and the y-direction was free. A time-dependent pressure along the ydirection was applied on the bottom surface of the stationary workpiece. Notably, the distance of clamping location from interface was same as that in experiments to avoid the clamping effects. Additionally, the conductive heat transfer between two contacting surfaces was expressed as,

q ¼ hg ðTa ; Tb ; pÞðTa  Tb Þ

(14)

where Ta, and Tb were the temperatures of the contact points. hg was the temperature-dependent gap thermal coefficient and its value was taken from Li et al. [17]. 4.5. Numerical parameters Owing to a large number of natural time increments in the numerical solution of rate-dependent quasi-static problems in ABAQUS/Explicit, mass scaling was attractive to reduce the

computational cost. As stated by Schmidt and Hattel [41], the ratio of kinematic energy to internal energy less than 5% was commonly used as a criterion to confirm the suitable mass scaling factor. Herein a fixed mass scaling factor of 800 was used for the developed model to keep the computational cost reasonable while ensuring the computational accuracy and procedure stability. 5. Results and discussion 5.1. LFW of same superalloy 5.1.1. Heat flux field Based on the developed model, the main thermo-mechanical fields during LFW of GH4169 to GH4169 have been obtained. Fig. 6(a) shows the distribution of heat flux field during the LFW process at Pf ¼ 400 MPa, fo ¼ 25 Hz, Ao ¼ 2.9 mm. As well known, the heat flux around the welding interface was because of the friction work and plastic deformation heat during LFW. At initial friction time, the heat flow distributed evenly at the contact area. As the welding proceeds, the heat flowed into two workpieces. Softened material began to be extruded out from interface, which indicated that the internal workpieces also underwent the influence of heat caused by the increasing deformation strain. Compared to the initial friction phase, the maximum value of heat flux gradually decreased with time owing to the transition of friction condition. As the welding time reached 2.03 s, the interfacial heat flux became non-uniform, which was attributed to the non-

Fig. 6. Heat flux during LFW of GH4169 to GH4169: (a) heat flux field around the friction interface and (b) evolution of interfacial heat flux at equilibrium phase.

P. Geng et al. / Materials & Design 185 (2020) 108244

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Fig. 7. Temperature distribution during LFW of GH4169 to GH4169: (a) temperature field around the interface, and evolution of interfacial temperature at the overall process (b) and at equilibrium phase (c).

uniform contact pressure affected by thermo-plastic deformation. Subsequently, it was observed from the contour of heat flux field that the welding process achieved the equilibrium phase, at which a dynamic balance between heat generation and losses was reached. At the friction interface, the variation of interfacial heat generation and frictional shear stress also reached a steady state. As clearly shown in Fig. 6(b), frictional heat flux and shear stress all varies with time periodically under certain scopes. This indicated that the interfacial heat transfer behavior in LFW owned the periodic pulsed characteristic, which was different from that in the conventional rotary friction welding. 5.1.2. Temperature and stress fields Temperature field during the LFW process under the same welding parameters mentioned above was shown in Fig. 7(a). Interfacial temperature rose quickly affected by the high interfacial heat flux at the initial phase. Under the effects of thermo-elastic deformation, the contact area was lower than the geometric area of workpicce friction surface. As a result, the high temperature region concentrated on the center of interface. With the heat transfer into workpiece as shown in Fig. 7, the materials near the friction interface were softened under high temperature. Consequently, the full contact was achieved at the friction interface. There was a large thermal gradient near the friction interface, owing to the cold materials being extruded to the interface and the hot materials being expelled out from interface. Fig. 7(b) shows the evolution of interfacial temperature with time during the LFW process. A large increase rate of temperature could be clearly observed at the interface as shown in Fig. 7(b). Compared to the

beginning time of flow deformation of materials, the interface temperature first reached a quasi-steady state. Under the effect of friction heat flux, the interfacial temperature varied with time periodically. Additionally, the interfacial temperature distribution at the equilibrium phase in Fig. 7(c) was non-uniform, which indicated the localization of friction heat generation. Fig. 8(a) shows the distribution of Von Mises stress field during the LFW process. The dry friction condition and the applied pressure along the feed direction resulted in a high stress level near the interface when the two workpeices were contacted initially. Due to the effect of high temperature, the decrease of flow stress of materials resulted in the decrease of Von Mises stress close to the friction interface. As the plastic deformation began, it was seen that the stress level at the clamping region of oscillatory workpiece increased gradually with friction time. This could be related to the enhancement of clamping effect under the applied friction pressure. Fig. 8(b) shows the stress evolution at three different locations corresponding to the three points in Fig. 7(b) during process. It was also found that the stress evolved with time periodically. The changing scope of stress was related to the variation of temperature and plastic strain rate. According to the fluctuated amplitude of stress values in Fig. 8(b), the more intense deformation appeared at the region closer to the edge along the oscillation direction. Additionally, the distribution of stress at two contact surfaces along the oscillation direction at equilibrium stage was presented in Fig. 9. At the welding time of 4 s, oscillatory workpiece returned back to the alignment location and then moved along x-axis positive direction. When oscillatory workpiece moved to the farthest location at the time of 4.01 s, maximum stress occurred at the corresponding

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Fig. 8. Stress distribution during LFW of GH4169 to GH4169: (a) stress field around the interface and (b) evolution of stress at different interface positions.

Fig. 9. Stress distribution at two contact surfaces at the equilibrium phase along the oscillation direction: (a) stationary side and (b) oscillatory side.

peripheral location of friction surface of stationary workpiece. Subsequently, as the moving direction was reversed, the oscillatory workpiece moved to the other farthest location. As a result, the location of maximum stress was changed into the other periphery of surface at stationary workpiece. Owing to the physical mechanism of periodic reciprocating motion in LFW, it could be clearly seen that the stress field at the side of stationary workpiece alternated dynamically with respect to that at the side of oscillatory workpiece. Based on the simulated results above, the stress field in

LFW could be characterized by the following two aspects. One was that the evolution of stress with time owned the periodical pulsed features; the other was that the stress field in LFW displayed periodically alternating distribution around the interface. To validate the robustness and reliability of the developed numerical model, simulated results were compared with the experimental ones. Fig. 10 shows the comparisons of temperature and axial shortening histories between experimental and simulated results. As shown in Fig. 10(a), the heating stage in each curve was

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Fig. 10. Comparison of thermal histories and axial shortening lengths obtained from experiments and FEM simulation for five different welding conditions.

Table 6 Calculated errors of peak temperature at the present location of thermocouple and shortening length. No. Peak temperature during process ( C) Total axial shortening length (mm)

J1 J2 J3 J4

Measured

Simulated

Error (%)

Measured

Simulated

Error (%)

620 663 786 867

639 679 771 895

3.06 2.41 1.78 3.70

1.65 2.52 4.75 5.2

1.8 2.61 4.9 5.07

9.10 3.57 3.15 2.50

in good agreement with the measured result, which indicated that the heat generation calculated by the numerical model was reasonable. As for the curves of axial shortening length, the simulated results showed the same changing trend as experimental results. This suggested the developed model could provide a good prediction for calculating the burn-off rate and plastic deformation in the LFW of GH4169 superalloy. Table 6 also lists the simulated quantitative error compared to the experimental results. It could be seen clearly the simulated result errors predicted by the developed model were within a margin of just a few percent. Hence, the comparisons demonstrated that the developed 3D numerical model based on the plastic/plastic friction pair could reliably simulate the LFW process of GH4169 superalloy. 5.2. LFW of dissimilar alloys Solid state joining of dissimilar Ni-based superalloys was important to improve the integral performance of components in aero-engine gas turbines. Although experimental study has been

undertaken on LFW of dissimilar Ni-based superalloys, the related numerical simulation has not been reported yet. In this study, the LFW joint of GH4169 to FGH96 superalloys was successfully obtained as shown in Fig. 3. To study the thermo-mechanical behavior during LFW of dissimilar superalloys, the developed numerical modeling based on the plastic/plastic friction pair was a valid tool. 5.2.1. Heat flux field Fig. 11 shows the heat flux field around the friction interface during LFW at Pf ¼ 400 MPa, fo ¼ 25 Hz, Ao ¼ 2.9 mm. Heat generated by friction work dominated at the initial phase, and was transferred into two workpieces. As two alloys near the interface being softened, the heat transfer reached a quasi-steady state. This phenomenon was similar to that in LFW of same superalloys. However, the heat flux into two workpieces was different owing to the difference of thermo-physical properties of two superalloys. As shown in Fig. 12(a), the evolution of heat flux into each workpiece with time was obtained from simulation. It was indicated that the heat flux at the side of GH4169 superalloy was a little larger than that at the side of FGH96 superalloy. The obtained variation of heat partition ratio between two workpieces (Fig. 12(b)) showed that the heat partition into two workpieces was essentially uniform while more heat flowed into GH4169 workpiece during LFW of GH4169 to FGH96. 5.2.2. Temperature and stress field Fig. 13 shows the temperature and stress fields during LFW of GH4169 to FGH96. Owing to the difference of thermal and mechanical properties between two superalloys, the temperature and stress fields at both sides of interface showed the asymmetrical

Fig. 11. Heat flux field distribution of LFWed joint of GH4169/FGH96 superalloys.

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Fig. 12. Evolution of heat flux into two workpieces (a) and heat partition ratio between two workpieces (b).

Fig. 13. Temperature and stress fields during LFW of GH4169 to FGH96: (a) Temperature field and (b) Stress field.

distribution. Although the interface temperature reached a value exceeding 1200  C close to the melting point of base metals, a small flash size occurred at the side of FGH96 superalloy because of its higher yield strength at high temperatures. Importantly, more heat flowed into the workpiece of GH4169 superalloy, which also promoted the softening of materials. Under the effect of high temperature, the stress near the interface decreased to the value below 200 MPa. The different magnitudes of stress induced during

process at two sides of interface were mainly due to the different constitutive behaviors at high temperatures for two superalloys. The result of deformation at same temperature showed that the GH4169 superalloy owned a better plastic flow property. To study the thermo-mechanical characteristic in dissimilar superalloys LFW, the distributions of temperature and stress across the friction interface were obtained. As shown in Fig. 14(a), it could be observed that a great temperature gradient formed at two sides of interface,

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Fig. 14. Temperature and stress distributions across the interface in LFW of GH4169 to FGH96: (a) temperature and (b) stress.

and the temperature gradient at GH4169 superalloy side was greater than that at FGH96 superalloy side. This could be mainly related to the more high-temperature materials being extruded out at GH4169 superalloy sides. Additionally, from the variation of temperature gradient with time, the effects of heat conduction into the workpieces decreased owing to the achievement of equilibrium phase. Base on the temperature distribution, the heat affected zone could be identified to discuss the microstructural evolution in the future work. As shown in Fig. 14(b), a higher stress appeared at FGH96 superalloy side at the initial phase. As the process proceeded, the stress distribution in the weld zone tended to the Mtype distribution. In other words, owing to the great temperature gradient close to the interface, the plastic deformation was resisted by the surrounding colder materials. As a result, the stress first increased to a maximum value and then gradually decreased from the interface to the clamping location. The simulated results showed that peak stress near the interface of GH4169 superalloy side was higher than that at FGH96 side. The LFW process of GH4169 to FGH96 superalloys has been

performed to validate the simulated results. Fig. 15 shows the comparison between simulated and experimental results. In Fig. 15(a), the measured temperature at the location of 5 mm distance from the original interface was compared with the prediction from simulation. A good agreement could be observed, which indicated that the heat generation predicted by model was reliable. Similar result also could be found at the comparison of shortening length history in Fig. 15(b). Additionally, the experimentally observed deformation zone (Fig. 15(c)) matched well with the plastic strain field after welding obtained from simulation. The comparisons between experimental and numerical results were found to be good, which proved the reliability of developed model to simulate LFW of dissimilar superalloys. 5.3. Discussion on the characteristics of modeling with plastic/ plastic friction pair Recently, more attention has been taken into the experimental studies of LFW of various Ni-based superalloys. Several

Fig. 15. Comparison between experimental (Exp.) and simulated (Sim.) (a) temperature, (b) shortening length histories and (c) deformation zone.

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Fig. 16. Comparison of temperature and stress fields between two FE models: (a) based on rigid/plastic friction pair and (b) based on plastic/plastic friction pair. (Pf ¼ 400 MPa, fo ¼ 25 Hz, and Ao ¼ 2.9 mm).

experimental phenomena those were existing, however overlooked in LFW of Ni-based superalloys were investigated to better provide guidelines for process. The characteristics of modeling with plastic-plastic contact pair to predict the thermo-mechanical behavior in LFW of Ni-based superalloys were discussed based on numerical simulations for experimental phenomena. 5.3.1. Fully-coupled thermo-mechanical deformation As reported previously, the FE model with a rigid/plastic contact pair has been widely used to simulate the LFW due to its simplification. In this study, the thermo-mechanical field was also predicted by the FE model with a rigid-plastic friction pair, in which a rigid body was meshed for thermal analysis under unchanged boundary conditions. Fig. 16 shows the comparisons of temperature and stress fields between these two FE models. It could be seen that temperature and stress fields of joint simulated by the 3D model with a plastic/plastic friction pair showed asymmetric distribution at both sides of interface. The magnified region of temperature field as shown in Fig. 16(b) showed that the peripheral temperature of interface at oscillatory side was lower than that at stationary side. The temperature difference could be attributed to that more heat within flash at oscillatory side was dissipated into surroundings caused by forced heat convection. Moreover, the stress state of joint also showed significant difference at both sides of interface, which

was due to the different forced constraint in LFW. However, these results could not be obtained by the conventional 3D model with the rigid/plastic friction pair. Additionally, Fig. 17 shows the plastic strain field distribution at both sides at the end of friction stage. There were some differences in the maximum of plastic strain between two workpieces. The maximum plastic strain of workpiece at stationary side was higher than that of workpieces at oscillatory side. In fact, owing to the larger high temperature region at stationary side, the plastic flow of material was more apparent at the friction surface of stationary side. Consequently, the larger plastic strain appeared at stationary side. This finding also showed the asymmetric material flow deformation both sides of interface, though two workpieces with same superalloy were joined by LFW. According to the analysis mentioned above, the developed FE model with the plastic/plastic friction pair exhibited the asymmetric distribution of thermo-mechanical behavior during the process. Compared with the conventional 3D model based on the rigid/plastic friction pair, it was indicated that the 3D model based on the plastic/plastic friction pair could describe the physical nature of friction behavior and the constraint state of boundaries in LFW more suitably. 5.3.2. Interface corner phenomena According to the experimental findings of LFW, a small

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Fig. 17. Plastic strain distribution at both sides of interface at the end of friction stage. (Pf ¼ 400 MPa, fo ¼ 25 Hz, and Ao ¼ 2.9 mm).

Fig. 18. Welding interface corner bonding: (a) An unbonded region for low burn-off and (b) Sound bonding for high burn-off.

unbonded region might occur at the corners of the welding interface as shown in Fig. 18(a). In this region, the remained oxides or micro-cracks due to poor plastic flow resulted in the welded interface zone with poorer mechanical properties. The weak interface corner of linear friction welded joint was very noticeable at low burn-off values. As such, the developed 3D model could be used to study the material flow and deformation at the corners of the welding interface during LFW of a Ni-based superalloy, something that has not been investigated before. As shown in Fig. 18(b), according to the model, as the burn-off was increased, the material at the corner of interface could be softened by the heat from interface and the flash and then extruded out plastically, which resulted in a sound bonding at the corner of interface, eliminating a source of interior mechanical properties. This also suggested that the 3D model with plastic-plastic contact pair could capture the multi-directional flow behavior of the LFW and provided the optimized method to remove the unbonded region, which was very important for a successful LFW.

5.3.3. Flash dimensional phenomena According to the analysis above, it was accepted that there was difference in distribution of temperature field and deformation amount at both sides of interface, but this phenomenon has been always ignored. Herein the current authors have noted that the flash dimensional size for each half showed more obvious difference when two workpieces were fixed at different positions using clamps. Two elastic-plastic clamps were used in the model and their geometry were presented in Fig. 19(a). The actual clamp was simplified as a block structure, through which the workpiece was fixed by the tie constraints. The material properties and constitutive data of clamp material were taken from the publication [17]. Displacement boundary conditions as described in Section 4.4 were applied at the external surfaces of the two clamps. Friction pressure was also applied at the bottom of stationary workpiece. Friction model remained unchanged as described in Section 4. The studied welding parameters were Pf ¼ 400 MPa, fo ¼ 25 Hz, Ao ¼ 2.9 mm. As shown in Fig. 19(b), the clamp fixed position of oscillatory workpiece exhibited a distance of 10 mm from the friction

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Fig. 19. Flash dimensional phenomena in LFW of GH4169 superalloy: (a) FE model with clamp structures, (b) comparison of experimental and simulated results, (c) temperature/ stress field distribution and (d) total shortening length under different fixed positions of clamp.

interface, while the clamp fixed position of stationary workpiece exhibited a distance of 6 mm. After welding, it was seen in Fig. 19(b) that the flash dimension of oscillatory side was smaller than that of stationary side. In other words, the total deformation amount of stationary workpiece was larger. As also shown in Fig. 19(b), the simulated result showed good agreement with experimental result by observing the strain field distribution of joint. Fig. 19(c) shows the temperature and stress fields under different fixed positions of clamp. In the simulation, only the distance of fixed position of clamp from contact interface at the side of oscillation workpiece was changed. As shown in Fig. 19(d), based on the variation of the total shortening length, it was demonstrated that the fixed position of clamp in LFW of GH4169 superalloy had a significant influence on the plastic deformation of joints. At Pf ¼ 400 MPa, fo ¼ 25 Hz, Ao ¼ 2.9 mm, the shortening length decreased with the increase in the distance of clamp fixed position from interface. The possible reasons for the phenomena might be the variation of thermal condition and stress state caused by the clamp constraint on the plastic deformation. Therefore, the relationship between flash

dimensional size for each half and welding condition could be studied using the developed numerical model, which suggested the possibility of controlling the deformation amount for each workpiece by changing fixed position of clamp or designing workpiece dimensions in LFW experiments.

5.3.4. Workpiece geometry effects Additionally, the realistic workpieces needing to be welded had different cross-section geometries. For example, LFW technology was widely used to the joining of blisks structure, in which the geometries of blade and disk were different [2]. Based on the modeling with plastic-plastic contact pair, the LFW of two workpieces with different cross-sectional geometries was modeled. In the model, the original geometry of oscillation workpiece in Fig. 5 was replaced by other geometry while the geometry of stationary workpiece remained unchanged. Two groups of workpieces with different cross-section geometries, viz. rectangle versus quadrate in Fig. 20(a), and rectangle versus simplified blade like in Fig. 20(b) were used. Friction and material models, and loading and boundary

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Fig. 20. Temperature and stress fields of LFWed joint of two GH4169 workpieces with different cross-sectional geometries: (a) rectangle versus quadrate, and (b) rectangle versus simplified blade like.

conditions remained unchanged as described in Section 4. The used welding parameters were Pf ¼ 400 MPa, fo ¼ 25 Hz, Ao ¼ 2.9 mm. Owing to the different distributions of temperature and stress fields at both sides of interface, a greater extruded amount of the softened materials occurred at the side of workpiece with smaller crosssection area. According to the results, the developed FEM model not only could be used as a more effective way to predict the thermal or mechanical behavior of integral joint during LFW, but hopefully provided a feasible platform on which further 3D modeling researches of LFW for dissimilar materials or different geometries might be based.

6. Conclusions A 3D thermo-mechanical coupling model based on the plastic/ plastic friction pair was successfully developed to simulate the LFW process of Ni-based superalloys: GH4169 to GH4169, and GH4169 to FGH96. This model demonstrated a number of innovative characteristics in simulating the LFW of Ni-based superalloys:

(1) Owing to the physical nature of LFW, interface heat flux and frictional shear stress all changed with time in a periodical fluctuated manner, which resulted in a periodic evolution of interface temperature. The stress field at the side of stationary workpiece alternated dynamically with respect to that at the side of oscillatory workpiece. Numerical simulated results showed excellent agreement with experimental data. (2) During LFW of GH4169 to FGH96, more heat flowed into the GH4169 side and the resulting temperature gradient at GH4169 side was greater than that at FGH96 side. More materials were extruded out at GH4169 side owing to its better flow property. The stress first increased to a maximum value and then gradually decreased from the interface to the clamping location. Peak stress near the interface of GH4169 superalloy side was higher than that at FGH96 side. (3) The established model with plastic/plastic friction pair demonstrated the asymmetric distribution of temperature and stress fields during LFW of same materials. Plastic strain of stationary workpiece was always higher than that of

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oscillatory workpiece owing to the different thermomechanical condition at two sides of interface. (4) Based on the model with plastic-plastic contact pair, the relationship between flash dimensional size for each half and welding condition could be determined, which suggested the possibility of controlling the deformation amount for each workpiece by changing fixed position of clamp or designing workpiece dimensions in LFW experiments. (5) On the basis of simulation, unbonded regions occurring at the corners of the welding interface could be eliminated by increasing the burn-off. This suggested that the model could capture the multi-directional flow behavior of the LFW. Additionally, the model provided a feasible way to simulate the LFW for different geometries, which was of interest to end users of the process.

CRediT authorship contribution statement Peihao Geng: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Validation, Visualization, Writing original draft, Software. Guoliang Qin: Writing - review & editing, Supervision, Funding acquisition, Project administration, Resources. Jun Zhou: Funding acquisition, Project administration, Resources. Acknowledgements The authors would like to thank financial support from the National Natural Science Foundation of China (Grant No. 51475196) and the National Science and Technology Major Project on Highend Numerically Controlled Machine Tools and Basic Manufacturing Technology (No.2017ZX04004001). References [1] I. Bhamji, M. Preuss, P.L. Threadgill, et al., Solid state joining of metals by linear friction welding: a literature review, Mater. Sci. Technol. 27 (2019) 2e12. [2] A. Chamanfar, M. Jahazi, J. Cormier, A review on inertia and linear friction welding of Ni-based superalloys, Metall. Mater. Trans. A 46 (2015) 1639e1669. [3] A. Chamanfar, M. Jahazi, J. Gholipour, et al., Maximizing the integrity of linear friction welded Waspaloy, Mater. Sci. Eng. A 555 (2012) 117e130. [4] T.J. Ma, X. Chen, W.Y. Li, et al., Microstructure and mechanical property of linear friction welded Ni-based superalloy joint, Mater. Des. 89 (2016) 85e93. [5] W.Y. Li, A. Vairis, M. Preuss, et al., Linear and rotary friction welding review, Int. Mater. Rev. 61 (2016) 71e100. [6] W.Y. Li, F.F. Wang, S.X. Shi, et al., Numerical simulation of linear friction welding based on ABAQUS environment: challenges and perspectives, J. Mater. Eng. Perform. 23 (2) (2014) 384e390. [7] A. Vairis, M. Frost, Modelling the linear friction welding of titanium blocks, Mater. Sci. Eng. A 292 (2000) 8e17. [8] J. Sorina-Müller, M. Rettenmayr, D. Schneefeld, et al., FEM simulation of the linear friction welding of titanium alloys, Comput. Mater. Sci. 48 (2010) 749e758. [9] W.Y. Li, T.J. Ma, J.L. Li, Numerical simulation of linear friction welding of titanium alloy: effects of processing parameters, Mater. Des. 31 (2010) 1497e1507. [10] R. Turner, J.C. Gebelin, R.M. Ward, et al., Linear friction welding of Tie6Ale4V: modelling and validation, Acta Mater. 59 (2011) 3792e3803. [11] F. Schroeder, R.M. Ward, R.P. Turner, et al., Linear Friction Welding of Titanium Alloys for Aeroengine Applications: Modelling and Validation//9th ICTWR, 2012, pp. 886e892. [12] M. Grujicic, G. Arakere, B. Pandurangan, et al., Process modelling of Ti-6Al-4V linear friction welding (LFW), J. Mater. Eng. Perform. 21 (2012) 2011e2023.

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