Mechanics in frictional penetration with a blind rivet

Journal of Materials Processing Technology 222 (2015) 268–279

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Mechanics in frictional penetration with a blind rivet Junying Min a,∗ , Yongqiang Li b , Jingjing Li a , Blair E. Carlson b , Jianping Lin c,∗∗ a

Department of Mechanical Engineering, University of Hawaii at Manoa, 2540 Dole Street, Honolulu, HI 96822, USA Manufacturing Systems Research Lab, General Motors Global R&D, 30500 Mound Road, Warren, MI 48090, USA c School of Mechanical Engineering, Tongji University, Shanghai 201804, China b

a r t i c l e

i n f o

Article history: Received 15 November 2014 Received in revised form 5 February 2015 Accepted 7 February 2015 Available online 18 March 2015 Keywords: Friction stir Penetration Blind rivet Material removal rate

a b s t r a c t The mechanics of frictional penetration driven by a blind rivet to sheet metals is analyzed for a friction stir blind riveting process. Analytic models are deduced to calculate the material removal rate, penetration force and torque during the frictional penetration process. Frictional penetration tests with modified rivets and an Al alloy sheet were carried out at various rotation speed–feed rate combinations, where the penetration force and torque were recorded with a data acquisition system. An analysis of the contact condition between the rivet tip and the work material based upon the assumption of pure sliding contact in the initial penetration to partial sticking contact beyond a critical penetration depth of the rivet is completed, and the results are discussed based on the comparison of the analytically calculated and experimentally measured torque–force ratios. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Friction stir blind riveting (FSBR) as reported by Gao et al. (2009) and Lathabai et al. (2011) is a novel mechanical joining method and is being developed, which combines the advantages of FSW and the blind riveting. In FSBR (Fig. 1), a blind rivet is driven toward the work materials with a high-speed rotating tool (e.g. 3000 rpm or above). The rotating rivet generates frictional heat when engaging the work materials. The elevated temperature can significantly reduce the yield strength of the work materials. This allows the blind rivet to penetrate through the workpieces with reduced forces as compared to self-piercing riveting which occurs at room temperature. Herein lies the advantage of FSBR to achieve single sided joining. Once the blind rivet is fully seated, the internal mandrel is pulled upward to mechanically fasten the work materials, and the mandrel is broken to create a friction stir blind riveted joint. FSBR is capable of joining a variety of materials, e.g. dissimilar metals such as Mg alloy to Al alloy by Min et al. (2014a) and Al alloy to composites by Min et al. (2014b). Most existing studies on FSBR or FSR are experimental investigations aimed at demonstrating its feasibility for joining dissimilar sheet metals. For example, Gao et al. (2009) joined AA5052 sheets by FSBR and found that the FSBR joints carried higher tensile loads and exhibited greater fatigue resistance than joints produced by

∗ Corresponding author at: Lehrstuhl für Productions systeme, Ruhr-Universität Bochum, Bochum 44780, Germany. Tel.: +49 15738011709. ∗∗ Corresponding author. Tel.: +86 13901719457. E-mail addresses: [email protected] (J. Min), [email protected] (J. Lin). http://dx.doi.org/10.1016/j.jmatprotec.2015.02.011 0924-0136/© 2015 Elsevier B.V. All rights reserved.

resistance spot welding. Lathabai et al. (2011) investigated the effect of blind rivet design for FSBR of Al alloys to Mg AZ31. Min et al. (2014) joined cast Mg alloy AM60 to Al alloy sheets by FSBR and concluded that the FSBR joints carried greater tensile loads than joints fabricated using the conventional blind riveting method. Analytical and numerical modeling studies have focused on the FSW process, and in particular included thermal, thermo-mechanical, and friction models. Schmidt and Hattel (2008) reported basic thermal equations for friction stir welding and clarified several uncertainties regarding the different mechanisms of heat generation. Kuykendall et al. (2013) modeled the FSW process and found that the selection of constitutive law has a significant effect on the prediction of the temperature profile, the peak strain as well as the peak strain rate. Schmidt et al. (2004) developed an analytic model for the heat generation in FSW with several assumptions of contact conditions between the friction tool and the workpiece. Mishra and Ma (2005) showed that the frictional condition between the tool and the workpiece (2024Al-T3 alloy) changed from “stick” at lower rotation speeds (<400 rpm) to “stick/slip” at higher rotation speeds of the tool. Chen and Kovacevic (2003) established a 3-D finite element model incorporating the frictional heat source between work materials and the tool to study the thermal history and thermo-mechanical process in FSW of AA6061 alloys. To the best of the authors’ knowledge, there has been no published research on the mechanical modeling of the relatively new joining process, FSBR. The mechanical analysis will not only provide understanding of the FSBR process, but also other friction stir processes, such as friction stir drilling. The objective of this work is to analyze the mechanics of frictional penetration with a blind rivet. The upsetting step is not analyzed here and will be covered

J. Min et al. / Journal of Materials Processing Technology 222 (2015) 268–279

Nomenclature the slope of FZ1 –MRR1 curve A B the slope of MZ1 –MRR1 curve f the feed rate FZ , FZ1 and FZ2 the penetration forces on the rivet mandrel tip FZ1 max the peak penetration force Fd the driving force applied on the rivet mandrel by the spindle fixture Fs the force on the shank head applied by the spindle fixture h the height difference between the surfaces of the penetrated and non-penetrated areas of the workpiece H the heat generation rate Mz , Mz1 and Mz2 the torques on the rivet mandrel tip MZ1 max the peak torque Md the driving torque applied on the rivet mandrel by the spindle fixture Ms the torque on the shank head applied by the spindle fixture MRR1 and MRR2 the material removal rates P1 and P2 the pressures acting normal to the rivet shear plane Q the frictional heat R the ratio of the penetration force to torque S1 and S2 the shear stresses between the rivet shear planes and the work material St1 and St2 the shear stresses tangential to the rotational motion Sp1 and Sp2 the shear stresses along the shear planes t1 and t2 the wall thicknesses of the rivet tip twork the thickness of the workpiece the time T v1 and v2 the motions of the rivet shear planes relative to the work material vt1 and vt2 the motions tangential to the rotational motion vp1 and vp2 the motions along the shear planes vt work the speed of work material tangential to the rotational motion V1 and V2 the volumes of work material removed by the rivet tip the penetration depth of the rivet tip Z ZF=max the rivet travel distance corresponding to FZ1 max ZM=max the rivet travel distance corresponding to MZ1 max ZF=0 the rivet travel distance corresponding to FZ1 = 0 ZM=0 the rivet travel distance corresponding to MZ1 = 0 ZH=max the rivet travel distance corresponding to the peak heat generation rate the rivet travel distance corresponding to the critical ZR–c R ˛1 and ˛2 the rivet tip angles the ratio of  work to P1 ı  a state parameter  the friction coefficient between the rivet shear plane and the work material  the distance between the point on the rivet tip and the rotational axis  work the yield tensile stress of the work material the yield shear stress of the work material  work ω the rotational speed. Subscripts 1, 2 indicate the inner and outer shear plane with respect to the axis of rotation, respectively.

269

in a separate paper. Frictional penetration tests in single AA6022T4 sheets were conducted. Analytic models for material removal rate, penetration force and torque were established. Particularly, a torque–force ratio is proposed to evaluate the contact conditions between the rivet tip and the Al alloy sheet. 2. Mechanical analysis Through analyses of a large number of FSBR experiments incorporating several designs of blind rivets, Lathabai et al. (2011) concluded that blind rivets with hollow mandrel heads require significantly lower penetration force than those with solid mandrel heads. Illustrated in Fig. 2a is a blind rivet including a mandrel body, a hollow mandrel head, shank body, shank head, and break notch. In FSBR, the mandrel body is held with a spindle fixture rotating around the Z-axis at a rotational speed (ω) and fed along the Zaxis at a feed rate (f), as shown in Fig. 2b. The mandrel head is first brought into contact with the upper workpiece of a lap joint, and as it penetrates through the workpieces, the shank head eventually comes into contact with the top workpiece. Assuming that there is no slippage between the spindle fixture and the mandrel body, according to the force and torque equilibrium conditions, Eqs. (1) and (2) can be obtained FZ = Fd + Fs

(1)

MZ = Md + Ms

(2)

where FZ and MZ are the penetration force and torque acting on the rivet tip due to its interaction with the workpiece during frictional penetration; Fd and Md are the holding force and torque on the mandrel body applied by the spindle fixture; and Fs and Ms are the force and torque on the shank head applied by the spindle fixture, which are subtle and difficult to measure since they are due to static friction between the spindle fixture and the shank head and are also dependent on Fd and Md . The analysis of overall force and torque equilibrium is beneficial for avoiding quality issues of friction stir blind riveting joints, e.g. intrusion of the mandrel head into the shank during frictional penetration (refer to Min et al. (2015) for details). The following focuses on the interaction between the rivet tip and the workpiece. Fig. 3a illustrates the axial symmetrical force analysis between the mandrel tip and the workpiece when the penetration depth is Z. For simplicity, only one layer of workpiece is considered in the detailed mechanical analysis. Further, the mandrel tip has a simple sharp shape with two shear planes. With respect to the Z-axis as shown in Fig. 3a, the inner and outer shear planes have angles of ˛1 and ˛2 respectively. Hereinafter “1” and “2” indicate the inner and outer shear plane with respect to the axis of rotation and the radius of curvature where the two shear planes intersect is considered to be 0. During the feeding of the mandrel tip, there are pressures (P1 and P2 ) acting perpendicular to the shear planes. The rotation of the mandrel tip results in two tangential shear stresses (St1 and St2 ), and the feeding motion of the mandrel tip leads to two shear stresses (Sp1 and Sp2 ) along the shear planes. The mandrel tip in Fig. 3a can be separated into two simpler cases: a mandrel tip with only an inner shear plane as shown in Fig. 3b and a mandrel tip with only an outer shear plane in Fig. 3c. Based upon this breakdown, the effects of the inner and outer shear planes will be discussed and compared, while the experimental validation in this work will focus solely on the case described in Fig. 3b. 2.1. Calculations of material removal rates The rotating rivet during the frictional penetration portion of the FSBR process displaces the workpiece material which is very

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Fig. 1. Illustration of the FSBR process. (a) The rotating blind rivet is approaching the workpieces, (b) frictional penetration of the rivet, (c) pulling out the mandrel, and (d) completion (Min et al., 2015).

Fig. 2. (a) A typical blind rivet with a hollow mandrel head, and (b) schematic illustration of the frictional penetration process.

similar to the machining (e.g. milling and drilling) process. The material removal rate (MRR), which is defined as the material volume removed by the tool per unit time, has a significant effect on FZ , as reported by Bayoumi et al. (1994). As shown in Fig. 4, for a feed rate (f) and rotation speed (ω), the mandrel tip has a displacement of f · dT in the Z direction for an infinitesimal time, dT. The volume of work material (dV1 ) removed by the rivet tip, indicated in the dotted red region, is the difference between the removed volume V1 (T) at T and the removed volume V1 (T + dT) at T + dT. Appendix A details the calculation of the V1 (T), which is expressed by Eq. (3)



r1

2 · [ − (r1 − Z · tan ˛1 )] · cot ˛1 · d

V1 (T ) =

(3)

r1 −Z·tan ˛1

where  is the distance between the point on the rivet tip and the rotational axis, Z is the penetration depth of the rivet tip, r1 is the

outer radius of the rivet in the case of Fig. 3b. Similarly, V1 (T + dT) is deduced as



r1

V1 (T + dT ) =

2 · [ − (r1 − Z · tan ˛1 r1 −(Z+f ·dT )·tan ˛1

− f · dT · tan ˛1 )] · cot ˛1 · d

(4)

Then dV1 is calculated as dV1 = V1 (T + dT ) − V1 (T ) = f

1 3

[r12 − (r1 − Z · tan ˛1 )2 + (r1 − Z · tan ˛1 )

× (r1 − Z · tan ˛1 − f · dT · tan ˛1 ) 2



+ (r1 − Z · tan ˛1 − f · dT · tan ˛1 ) ] · dT

(5)

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271

Fig. 3. (a) Stresses acting on the mandrel tip. (b) and (c) are two simplified cases of the mandrel tip based upon their geometry.

By ignoring the second and higher order terms of dT in Eq. (5), dV1 becomes dV1 = f [r12 − (r1 − Z · tan ˛1 )2 ] · dT

(6)

expressed by Eq. (7) and Z is limited to the range from 0 to twork , and MRR1 = 0 when Z > twork since the material beneath the rivet is cut off when the rivet tip penetrates through the workpiece. When twork ≥ t1 · cot ˛1 , MRR1 is a piecewise function

At last, the material removal rate MRR1 is expressed by MRR1 =

MRR1 =

dV1 = f [r12 − (r1 − Z · tan ˛1 )2 ] dT

(7)

It is worth to note that MRR1 is dependent on the comparison between the depth of the mandrel tip with shear plane, t1 · cot˛1 (here t1 is the wall thickness of the rivet in Fig. 3b), and the workpiece thickness (twork ), i.e., when twork < t1 · cot ˛1 , MRR1 is

MRR2 =

⎧ f [(r2 + Z · tan ˛2 )2 − r22 ] ⎪ ⎪ ⎪ ⎪ ⎨ f [(r2 + Z · tan ˛2 )2 − (r2 + Z · tan ˛2 − t

work

MRR2 =

⎧ f [(r2 + Z · tan ˛2 )2 − r22 ] ⎪ ⎪ ⎪ ⎪ ⎨ f [(r2 + t2 )2 − r 2 ] 2

⎪ ⎩

f [r12 − (r1 − t1 )2 ]

t1 · cot ˛1 ≤ Z ≤ twork

0

Z > twork

0 ≤ Z ≤ twork 2

· tan ˛2 ) ]

twork < Z ≤ t2 · cot ˛2

(9)

t2 · cot ˛2 < Z ≤ twork + t2 · cot ˛2 Z > twork + t2 · cot ˛2

when twork ≥ t2 · cot ˛2 0 ≤ Z ≤ t2 · cot ˛2 t2 · cot ˛2 < Z ≤ twork

⎪ ⎪ f [(r2 + t2 )2 − (r2 + Z · tan ˛2 − twork · tan ˛2 )2 ] twork < Z ≤ twork + t2 · cot ˛2 ⎪ ⎪ ⎩ 0

(8)

The second expression on the right of Eq. (8) indicates that MRR1 is a constant when the whole rivet tip with shear plane is embedded in the middle of the workpiece thickness. Similarly, the material removal rate (MRR2 ) for the case in Fig. 3c is deduced as (see Appendix B):when twork < t2 · cot ˛2 ,

⎪ ⎪ f [(r2 + t2 )2 − (r2 + Z · tan ˛2 − twork · tan ˛2 )2 ] ⎪ ⎪ ⎩ 0

⎧ f [r12 − (r1 − Z · tan ˛1 )2 ] 0 ≤ Z ≤ t1 · cot ˛1 ⎪ ⎨

Z > twork + t2 · cot ˛2

(10)

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Fig. 6. Illustration of the contact between an element in the work material and a rivet element.

equilibrium of the vertical forces on the shear planes by ignoring Sp1 and Sp2

Fig. 4. Illustration for the calculation of MRR in frictional penetration.

Before establishing force equilibrium conditions for the rivet tip, the shear stresses on the shear plane are analyzed in detail first. Fig. 5 is a view normal to the shear plane, which illustrates the motion of the shear plane relative (v1 ) to the work material and the shear stress (S1 ). S1 has two components, Sp1 and St1 , which are also shown in Fig. 3b and correspond to the components of the motion (v1 ), vp1 and vt1 , respectively. Sp1 , St1 , vp1 and vt1 are expressed by Sp1 = S1 sin 1

(11)

St1 = S1 cos 1

(12) f cos ˛1

vt1 = v1 cos 1 = ω

(13) (14)

where  is the distance from the point on the rivet shear plane to the rotation axis; 1 is the angle between v1 and vt1 , 1 = arctan

vp1 Sp1 = arctan vt1 St1

(16)

FZ2 = a2 · P2 · sin ˛2

(17)

where a1 and a2 are the areas of contacting shear planes “1” and “2”, respectively, and are calculated as

2.2. Force and torque calculations

vp1 = v1 sin 1 =

FZ1 = a1 · P1 · sin ˛1

(15)

vp1 is far less than vt1 in the actual friction penetration process, e.g. considering f = 780 mm/min, ˛1 = 30◦ , ω = 2␲ × 6000/min and  = 2 mm (Min et al., 2015), then vp1 = 9.01 × 102 mm/min and vt1 = 7.54 × 104 mm/min. As a result, 1 is a very small value close to 0 based on Eq. (15). Hence, St1 ∼ S1 , and Sp1 is close to 0 and is ignored in the following analysis. As shown in Fig. 3b and c, the penetration forces acting on the two shear planes of the mandrel tip can be calculated from the

Fig. 5. Shear stresses on the rivet shear plane (enlarged normal view from the axisymmetric axis of an infinitesimal section of the shear plane).

a1 = a2 =

r12 − (r1 − Z · tan ˛1 )2 sin ˛1 (r1 + Z · tan ˛2 )2 − r12 sin ˛2

(18)

(19)

When developing analytic models for the torque in the frictional penetration process of a rivet, the contact condition between the workpiece and the rotating rivet is the most critical (Schmidt et al., 2004). According to Schmidt et al. (2004), the contact condition between the tool and the workpiece could be: (1) pure sliding, (2) pure sticking or (3) mixed sliding and sticking in the friction stir welding process. As regards to Contact (1), the Coulomb’s law of friction can be applied to describe the shear stress between the rotating rivet and work material, St1 |Slide ≈ S1 |Slide =  · P1

(20)

St2 |Slide ≈ S2 |Slide =  · P2

(21)

where  is the friction coefficient between the rivet shear plane and the work material and is a function of temperature, velocity and choice of workpiece materials. When the pressure between the rivet shear plane and the work material is sufficiently high and St |Slide calculated from Eqs. (20) and (21) exceeds the yield shear stress ( work ) of the work material contacting with rivet shear plane, the sticking contact condition, i.e. Contact condition (2), is then satisfied. In this case, the work material closest to the shear plane sticks to the rivet shear plane and is accelerated by the rotating rivet until an equilibrium state is achieved between the tangential shear stress and  work ; finally, the speed of the work material closest to the rivet shear plane (vt work ) is equal to the speed on the rotating rivet shear plane (vt1 ) as shown in Fig. 6. As a result, a sticking affected zone (SAZ) exists, where the speed of work material decreases to zero as the distance (ϕ) from the rivet shear plane increases. In the Contact condition (3) (also the mixing or partial sticking condition), the material closest to the rivet shear plane still rotates with the rivet, however vt work is smaller than vt rivet . Therefore, relative sliding as well as sticking exists between the rivet shear plane and the work material. The following analyses on the torque are based on the above three contact conditions. In the pure sliding contact condition (Contact (1)), the torque is essentially a result of the tangential shear stress in the tangential direction (St1 |Slide and St2 |Slide ) on the contact surfaces of the rivet shear planes and the work material, and St1 |Slide and St2 |Slide are

J. Min et al. / Journal of Materials Processing Technology 222 (2015) 268–279

273

Table 1 Process parameters in the frictional penetration tests. Rotation speed [rpm] Feed rate [mm/min]

3000 120

9000 120

9000 780

Replacing da1 from Eq. (22) into Eq. (27) and integrating dz from 0 to Z, then the torque (Mz1 |Stick ) can be deduced as Mz1 |Stick =

2work 3 [r − (r1 − z · tan ˛1 )3 ] 3 sin ˛1 1

(28)

Similarly, the torque (Mz2 |Stick ) in the sticking contact condition in Fig. 3c is expressed as Mz2 |Stick =

Fig. 7. Illustration for calculation of torque in the frictional penetration process.

calculated based on Coulomb’s law as expressed in Eqs. (20) and (21), respectively. Assume the rivet penetration depth is Z and the infinitesimal displacement of the rivet tip is dz. The distance from the rivet shear plane to the rotation axis is calculated as r1 − Z · tan˛1 , refer to Fig. 7, and then the corresponding infinitesimal area on the rivet shear plane, da1 , is computed as Eq. (22) by ignoring the higher order terms of dz da1 = =

(r1 − z · tan ˛1 )2 [r1 − (z + dz) · tan ˛1 ]2 − sin ˛1 sin ˛1 2 (r1 − z · tan ˛1 ) dz cos ˛1

(22)

The infinitesimal change in torque (dMz1 |Slide ) based on the infinitesimal area (da1 ) and the tangential shear stress (St1 |Slide ) is expressed as Eq. (23). dMz1 |Slide = St1 |Slide · (r1 − z · tan ˛1 ) · da1

(23)

Replacing da1 from Eq. (22) into Eq. (23) and integrating dz from 0 to Z, the torque (Mz1 |Slide ) in the frictional penetration process is obtained. Mz1 |Slide =

2P1 3 [r − (r1 − z · tan ˛1 )3 ] 3 sin ˛1 1

(24)

Similarly, the torque in the case of Fig. 3c is deduced as Mz2 |Slide =

2P2 [(r2 + z · tan ˛2 )3 − r23 ] 3 sin ˛2

(25)

When the pure sticking condition is fulfilled, the tangential shear stress, St1 |stick (or St2 |stick ), is equivalent to the yield shear stress of the work material,  work , which is supposed to be Eq. (26) as reported by Schmidt et al. (2004). work work = √ 3

(26)

where  work is the yield tensile stress of the work material, and  work is independent of the pressure. Then the infinitesimal torque (dMz1 |Stick ) in the case of the sticking contact condition is expressed as dMz1 |Stick = St1 |Stick · (r1 − z · tan ˛1 ) · da1

(27)

2work [(r2 + z · tan ˛2 )3 − r23 ] 3 sin ˛2

(29)

A state parameter, , is introduced to describe the portion of sticking in the mixed sliding/sticking contact between the work material and the rotating rivet. The mixed contact is assumed to be a linear combination of the sliding contact and sticking contact, namely, the tangential shear stress (St |mix ) on the contact surface between the work material and the rivet shear plane is St |mix =  · St |stick + (1 − ) · St |Slide

(30)

Based on the previous deduction in Section 2.2, the following equation can also be obtained.  |mix =  ·  |stick + (1 − ) ·  |Slide

(31)

 can be either of St1 , St2 , MZ1 and MZ2 . At last, for a general case where the rivet has a tip shown in Fig. 3a, the penetration force and torque are computed per Eqs. (32) and (33), respectively. FZ = FZ1 |mix + FZ2 |mix

(32)

MZ = MZ1 |mix + MZ2 |mix

(33)

It is easy to understand that the cases where  = 0 and  = 1 indicate pure sliding contact and pure sticking contact, respectively. Consequently, Eqs. (31)–(33) cover all three contact conditions. 3. Details of frictional penetration tests The work material used in the frictional penetration tests was AA6022-T4 sheet with a gage thickness of 1.2 mm, and the dimensions of the workpieces were 38 mm in width by 127 mm in length. Rivets with tips as shown in Fig. 3b and ˛1 = 30◦ were fabricated by machining commercial blind rivets (Advel SSPV-08-06), which had a shank diameter of 6.4 mm. The detailed dimensions of the machined rivets are presented in Fig. 8a. The frictional penetration tests were performed on a Makino A99 CNC machine. In the tests, a single workpiece was clamped on a backing plate with a Ф 10 mm hole to allow for penetration of the rivet. Furthermore, the CNC machine was equipped with a dynamometer, as shown in Fig. 8b and finally, the rivet mandrel was held by a spindle fixture. The rotation speeds and feed rates are listed in Table 1. The displacement of the rivet from the top surface of the workpiece was set to 5 mm in all tests, i.e. only part of the mandrel head penetrated the AA6022 workpiece. Use of a dynamometer and a data acquisition system allowed for measurements of both the penetration force and torque during the frictional penetration tests. 4. Results 4.1. Force and torque results The force and torque curves in the frictional penetration tests are presented in Fig. 9a and b, respectively. FZ1 increased with rivet

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Fig. 8. (a) Dimensions of the rivet modified from the blind rivet SSPV-08-06, and (b) the fixture used in the frictional penetration tests.

travel distance (Z) to the peak value (FZ1 max ) at ZF=max . The depth at which the peak value was achieved is a function of the frictional heat input thus, a slower feed rate provided greater heat input for a given depth and resulted in both a lower peak force and a shorter ZF=max . Once the peak force was achieved, FZ1 gradually decreased to 0 at ∼1.6 mm (ZF=0 ), where the rivet tip penetrated thorough the workpiece. Similar evolution of MZ1 with Z was also observed. It is noted that both ZF=0 and ZM=0 were larger than the thickness of the workpiece, which was attributed to the material crown at the bottom of the workpiece formed by the frictional penetration of the rivet as reported by Min et al. (2014c). At a fixed feed rate of f = 120 mm/min and as the rotation speed increased from 3000 to 9000 rpm, FZ1 max decreased from 0.38 to 0.18 kN, and MZ1 max decreased from 1.72 to 0.90 N m. Again, this reduction was attributed to a greater amount of frictional heat input. Likewise, at fixed ω = 9000 rpm, when the feed rate increased from 120 to 780 mm/min, FZ1 max increased from 0.18 to 0.45 kN, and MZ1 max increased from 0.90 to 1.47 N m. It is interesting to observe that ZF=max increased as rotation speed decreased and feed rate increased; however, ZM=max showed little dependence on the process parameters, which kept at ∼0.64 mm.

4.2. Heat generation The previous study by Min et al. (2015) on the FSBR of Al alloys showed that over 95% of the input energy was consumed by the torque, and that this proportion increased as rotation increases and/or feed rate decreased. Thus for this body of work, the heat generated by the penetration force was neglected with respect to analyzing the frictional penetration process. Furthermore, it is assumed that the work done by the rotation of the rivet was completely converted into heat. Hence, the heat generation rate (H) is expressed as Eq. (34). H = MZ · ω

(34)

Then the heat (Q) generated in the frictional penetration process is calculated as

 Q (T ) =

T

MZ (t) · ω dt 0

Fig. 9. (a) FZ1 vs. Z curves and (b) MZ1 vs. Z curves of the frictional penetration tests at various rotation speed-feed rate combinations.

(35)

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275

Fig. 10. Heat generation (Q) and heat generation rate (H) in the frictional penetration tests.

where T is time and a function of the penetration depth (Z) and the feed rate, namely, T=

Z f

(36)

Q is then rewritten as Eq. (37) by substituting Eq. (36) into Eq. (35).



Q (Z) = 0

Z/f

MZ (t) · ω dt f

Fig. 11. Calculated material removal rates (MRR). To calculate MRR2 , twork and t2 in Eq. (10) are assumed as 3 mm and 1 mm, respectively.

MRR1 (refer to Fig. 3b) is always larger than MRR2 (refer to Fig. 3c), and that MRR1 increases faster and then slower than MRR2 as Z increases until both reach the same maximum value, as shown by the long-dashed lines in Fig. 11. With a larger rivet tip angle, MRR1 increases much faster, and it requires a shorter penetration depth to reach the maximum value.

(37)

The evolution of heat generation rate (dashed lines) and cumulative heat generation (solid lines) in the frictional penetration tests are shown in Fig. 10. Irrespective of the process parameters, the heat generation rate H increased as Z increased and reached a peak at ZH=max ∼0.64 mm, and then decreased as Z increased further. In the three frictional penetration tests, H was the largest when ω = 9000 rpm and f = 780 mm/min and it was the lowest when ω = 3000 rpm and f = 120 mm/min. As shown in Eq. (37), the heat generation (Q) or the energy input to the frictional penetration process, was affected positively by the rotation speed and negatively by the feed rate. As a result, the test run at ω = 9000 rpm and f = 120 mm/min generated the most frictional heat, and the test run at ω = 9000 rpm and f = 780 mm/min generated the least frictional heat, and the former is ∼3.7 times more than the latter. 4.3. Material removal rates It is easy to understand that the feed rate has a positive effect on the material removal rate as shown in Eqs. (7)–(10). To investigate the effect of the rivet tip angle (˛1 ) on MRR, the wall thickness of the rivets in Fig. 3b (t1 ) was assumed to be 1.0 mm, and the outer diameter of the rivets was assumed to be the same as that of SSPV-06-08 (refer to Fig. 8a), 6.1 mm, viz. r1 = 3.05 mm in Fig. 3b. In order to compare the material removal rate for the rivet geometries presented in Fig. 3b and c, namely, MRR1 and MRR2 , ˛1 and ˛2 were set to 30◦ , t2 was assumed to be equal to t1 , and the outer diameter of the rivet in Fig. 3c was also 3.05 mm, i.e. r2 + t2 = 3.05 mm. In the calculation of both MRR1 and MRR2 , the feed rate was fixed as 120 mm/min, and the thickness of the workpiece (twork ) was assumed to be larger than t1 · cot˛1 or t2 · cot˛2 , namely, Eqs. (8) and (10) were used. Fig. 11 compares the calculated material removal rates, MRR1 and MRR2 , for the cases of the rivet geometries presented in Fig. 3b and c, respectively, when ˛1 = ˛2 = 30◦ . Fig. 11 also includes plots of material removal rate, MRR1 , as a function of the rivet tip angle. Here no deformation on the bottom of the workpiece was assumed, which will be further discussed in Section 5. As shown in Eqs. (8) and (10), the maximum achievable MRR is dependent on the feed rate and rivet geometry, and the maximum MRR are equivalent in the case of the rivet geometries presented in Fig. 3b and c. However, before the MRR reaches the maximum value, it can be seen that

4.4. Contact conditions It is difficult to determine the exact contact condition between the rotating rivet and the work material in the frictional penetration process. However, it is possible to estimate the contact condition by comparing the theoretically calculated and experimentally measured ratios of torque to penetration force (R). For pure sliding contact, the ratio of the torque to the penetration force is obtained from Eqs. (16) and (24) R|Slide =

2 · [r13 − (r1 − Z · tan ˛1 )3 ] MZ1 |Slide = FZ1 3 sin ˛1 · [r12 − (r1 − Z · tan ˛1 )2 ]

(38)

As regard to the pure sticking contact, the ratio of the torque to the penetration force can be calculated from Eqs. (16) and (28) R|Stick = =

2ı · [r13 − (r1 − Z · tan ˛1 )3 ] MZ1 |Stick = FZ1 3 sin ˛1 · [r12 − (r1 − Z · tan ˛1 )2 ] ı · R|Slide 

(39)

where ı = work /P1 . Since  work is larger than P1 as mentioned above, ı >  and R|Stick > R|Slide . Similarly, the torque–force ratio in the mixing contact condition is deduced as Eq. (40) R|mix =

2[ · ı + (1 − )] · [r13 − (r1 − Z · tan ˛1 )3 ] MZ1 |mix = FZ1 3 sin ˛1 · [r12 − (r1 − Z · tan ˛1 )2 ]

=

·



ı + (1 − ) · R|Slide 

(40)

As shown in Eq. (38), for the case of pure sliding contact, R|Slide is dependent on the friction coefficient, the geometrical parameters of the rivet, as well as the penetration depth. The dimensions of the rivet used in the frictional penetration tests are shown in Fig. 7a. The friction coefficient  is set to 0.4 as suggested by Schmidt et al. (2004), which was used in a friction evaluation of the friction stir welding process between a steel tool and an aluminum workpiece. The theoretically calculated R|Slide and R-values in the FSBR tests are compared in Fig. 12. The theoretically calculated R|Slide decreases slightly as Z increases, i.e. at Z = 0.001 mm, R|Slide = 2.38 × 10−3 mm and at Z = 1.0 mm, R|Slide = 2.18 × 10−3 mm. However, as regards the

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Fig. 12. Comparison of experimental R-values and the theoretically calculated R|Slide . Table 2 Dependence of ZR–c on the process parameters. Process parameters

ZR–c [mm]

Rotation speed [rpm]

Feed rate [mm/min]

3000 9000 9000

120 120 780

0.3 0.15 0.4

experimental R-value, there was a threshold of the rivet penetration depth, ZR–c as listed in Table 2. When 0 < Z < ZR–c , R-values changed little with Z and were comparable to the calculated R|Slide , which indicates that the contact condition in this phase was probably pure sliding. Beyond ZR–c , the R-value increased first slowly and then quickly as Z increased, and saturated when the rivet was approaching the end of the frictional penetration process. However, it is still difficult to determine whether the contact condition was pure sticking or not when the rivet reached the bottom surface of the AA6022 specimen (Z = 1.0 mm), since  work was unknown in this work and was significantly dependent on the temperature of the work material closest to the rivet shear plane, which varied during the frictional penetration process. 5. Discussion In the actual frictional penetration process with a rivet, the material displaced by the rivet will flow out along the shear plane as indicated by the open arrow in Fig. 13a, which shows the cross section of a frictionally penetrated AA6022 sheet interrupted at Z = 0.6 mm where the displaced material produced flash. This material flash may lead to an increase of the contact area between the workpiece and the rivet shear plane, which has an effect on the actual material removal rate, penetration force, and torque. However, as the rivet tip penetrated further into the work material, the size of the material flash on the penetrated workpieces did not change significantly during the penetration process as seen by comparing Fig. 13a–c. Thus, it appears that the flash is generated during the first portion of the rivet insertion and that as the rivet continued to penetrate, the rivet pushed out material on the bottom side of the workpiece. Based upon this, flash generation is neglected in this work when calculating the material removal rate, penetration force and torque. Furthermore, as the rivet nears the bottom side of the workpiece, a plastic shear deformation (as indicated by the arrows in Fig. 13c) occurs on the remaining material. For example, at Z = 0.6 mm, there was no plastic deformation on the bottom of the penetrated workpiece, namely, there was no height difference (h) between the surfaces of the penetrated and non-penetrated areas

Fig. 13. Optical micrographs of frictionally penetrated AA6022 sheets by the modified rivets with penetration depth of (a) Z = 0.6 mm, (b) Z = 0.8 mm and (c) Z = 1.0 mm at ω = 9000 rpm and f = 120 mm/min.

i.e. h ∼ 0. When Z = 0.8 mm, plastic deformation was observed and h ∼ 0.08 mm, which increased to 0.25 mm at Z = 1.0 mm as illustrated in Fig. 13. After the rivet penetrates through the workpiece, a material crown was then formed (Min et al., 2014c). In the theoretical calculations of the material removal rates in Section 4.3, it was assumed that there was no plastic deformation on the bottom of the workpiece. Obviously, this plastic deformation resulted in a decrease of the actual material removal rate during the frictional penetration process, especially, when Z > 0.6 mm. As mentioned previously, the material removal rate has a significant effect on the penetration force (Bayoumi et al., 1994). Fig. 14 presents the FZ1 vs. MRR1 curve when ω = 9000 rpm and f = 120 mm/min. It is seen that the penetration force reached its peak at ZF−max = 0.35 mm, although MRR1 increased continuously during the frictional penetration process. We noted the negative effect of plastic deformation upon the workpiece bottom on MRR1 ; however, this negative effect occurred when Z > 0.6 mm. Thus, the decrease of FZ1 beyond Z = 0.35 mm could not be attributed to the plastic deformation upon the workpiece bottom. As shown in Fig. 10, the input energy, which was converted to heat and transferred to the rivet tip and workpiece, continuously increased throughout the frictional penetration process. As a result, the temperatures of both the rivet and remaining work material increased. Apparently, softening due to a temperature increase of the work material played an important role in reducing FZ1 during the frictional penetration process. Therefore, the trend of FZ1 during friction penetration is affected by both the increasing MRR1 and the

Fig. 14. FZ1 vs. MRR1 and Z vs. MRR1 curves when ω = 9000 rpm and f = 120 mm/min.

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Fig. 15. (a) FZ1 vs. MRR1 curves and (b) MZ1 vs. MRR1 curves with various rotation speed–feed fate combinations. Table 3 Dependences of A and B values on the process parameters. Process parameters

A

B

Rotation speed [rpm]

Feed rate [mm/min]

(FZ1 = A·MRR1 )

(MZ1 = B·MRR1 )

3000 9000 9000

120 120 780

3.65 × 10−2 3.58 × 10−2 6.90 × 10−3

8.32 × 10−2 7.95 × 10−2 1.62 × 10−2

The fitted A and B values by the least squares method were listed in Table 3. According to Fig. 15 and Table 3, the rotation speed exhibits little influence on the FZ1 vs. MRR1 and MZ1 vs. MRR1 curves, while the feed rate shows a significant effect. At a higher feed rate, FZ1 and MZ1 increased with a lower rate as MRR1 increased. 6. Conclusions

increasing temperature of the remaining work material. The two aspects have opposing effects on FZ1 and the competition of these two opposing effects led to the peak of the penetration force. Material softening also affects the contact condition, since the yield shear stress,  work , decreases as a function of increasing temperature of the work material. At the beginning of the frictional penetration process, the input energy was not enough to result in sufficient softening of the work material, and  work was much higher than the tangential shear stress computed by Eq. (20). Hence, it was pure sliding between the rivet shear plane and the work material as shown in Section 4.4 when Z < ZR–c . According to the analysis in Section 4.4, the R-value is the lowest for the pure sliding contact compared to the other two contact conditions. When Z > ZR–c , the experimental R-values are larger than the calculated R|Slide , which indicates that the contact condition developed from pure sliding to mixing contact or pure sticking contact. Once  work or ı is known, the state parameter indicating the portion of sticking in the mixing contact, , can be computed from Eq. (40). As Z increased two things happened; (1) more energy was input generating additional heat which transferred to the work material and (2)  work continued to decrease. However, as long as  work was larger than St1 by Eq. (20), the contact condition was mixing contact and  > 0; only when  work decreased to be comparable with St1 , would it lead to a pure sticking contact. The effects of process parameters on the FZ1 –MRR1 and MZ1 –MRR1 relationships are presented in Fig. 15a and b, respectively. Here, note that only the data corresponding to the penetration depth smaller than ZR–c (namely, in pure sliding contact condition) were considered so as to exclude the sticking effect and the effect of a considerable temperature increase of the work material. It is observed that MRR1 has a linear effect on both FZ1 and MZ1 irrespective of rotation speed–feed rate combination, i.e. FZ1 = A · MRR1

(41)

MZ1 = B · MRR1

(42)

The mechanics of frictional penetration by a blind rivet in friction stir blind riveting process is analyzed, and analytic models are proposed to calculate the material removal rate, penetration force and torque. Frictional penetration tests were performed on AA6022 sheets with modified blind rivets, and conclusions are drawn as follows: (1) Analytic models for the material removal rate are proposed for the frictional penetration process by a blind rivet having a sharp tip with inner and outer shear planes. (2) Analytic expressions for the torque in the frictional penetration process are deduced based on pure sliding, pure sticking and mixed contact conditions between the rivet tip and the work material. (3) The ratio of torque to penetration force is proposed to determine the contact condition in the frictional penetration tests by rivets. The initial frictional penetration of the rivet was pure sliding contact which developed to mixed contact (i.e. partial sticking) beyond a critical penetration depth, which is dependent on the process parameters as listed in Table 2. (4) In the pure sliding condition, the material removal rate exhibits a linear effect on the penetration force and torque, and this effect is independent of the rotation speed but significantly depends upon the feed rate. (5) During the frictional penetration of an AA6022 sheet, the penetration force and torque increased to a peak and then decreased to zero as the penetration depth increased. Acknowledgements The authors would like to thank Mark Hull from GM R&D for machining the rivet tip, and Anthony J. Blaszyk and John S. Agapiou from GM R&D for their help in the FSBR tests. Financial support for this research was provided by the U.S. National Science Foundation Civil, Mechanical and Manufacturing Innovation grant No. 1363468 and the China National Natural Science Foundation under grant No. 51375346.

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Fig. B2. Case (2): the rivet tip penetrates through the workpiece but the end of shear plane has not penetrated into the workpiece yet, i.e. twork < Z ≤ t2 · cot ˛2 .

Fig. A1. Illustration for calculation of V1 (T).

Appendix A. Deduction of V1 (T) The volume of removed material at time T, e.g. V1 (T), is limited to the dashed area as illustrated in Fig. A1. Suppose there is an infinitesimal zone with a thickness of d shown by the shadow region, and its distance to the rotation axis is . Then the volume of the infinitesimal zone dV1 (T) is, dV1 (T ) = 2 · [ − (r1 − Z · tan ˛1 )] · cot ˛1 · d

(A1)

Integrate the Eq. (A1), V1 (T) can be obtained as



r1

V1 (T ) =

2 · [ − (r1 − Z · tan ˛1 )] · cot ˛1 · d

(A2)

r1 −Z·tan ˛1

Appendix B. Deduction of MRR2 For the case in Fig. 3c, the deduction of the material removal rate (MRR2 ) is more complicated and described as follows.When twork < t2 · cot ˛2 , MRR2 is calculated based on the three cases illustrated

Fig. B3. Case (3): the end of shear plane penetrates into the workpiece, i.e. t2 · cot ˛2 < Z ≤ twork + t2 · cot ˛2 .

in Figs. B1–B3. For the Case (1) in Fig. B1, i.e., 0 ≤ Z ≤ twork , similar as the deduction of dV1 , dV2 |(1) is expressed as dV2 |(1) = f [(r2 + Z · tan ˛2 )2 − r22 ] · dT

(B1)

For the Case (2) in Fig. B2, i.e., twork < Z ≤ t2 · cot ˛2 , dV2 |(2) + dV2 is expressed as dV2 |(2) + dV2 = f [(r2 + Z · tan ˛2 )2 − r22 ] · dT

(B2)

where dV2 = f [(r2 + Z · tan ˛2 − twork · tan ˛2 )2 − r22 ] · dT

(B3)

Then dV2 |(2) = f [(r2 + Z · tan ˛2 )2 − (r2 + Z · tan ˛2 − twork · tan ˛2 )2 ] · dT

(B4)

For the Case (3) in Fig. B3, i.e., t2 · cot ˛2 < Z ≤ twork + t2 · cot ˛2 , dV2 |(3) + dV2 + dV2 is expressed as dV2 |(3) + dV2 + dV2 = f [(r2 + Z · tan ˛2 )2 − r22 ] · dT

(B5)

where Fig. B1. Case (1): only the rivet tip penetrates into the workpiece, i.e. 0 ≤ Z ≤ twork .

dV2 = f [(r2 + Z · tan ˛2 )2 − (r2 + t2 )2 ] · dT

(B6)

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279

Then dV2 |(3) = f [(r2 + t2 )2 − (r2 + Z · tan ˛2 − twork · tan ˛2 )2 ] · dT (B7) Following the definition of the material removal rate, namely, MRR2 =

dV2 dT

(B8)

MRR2 is expressed as a piecewise function by Eq. (B9)

MRR2 =

⎧ f [(r2 + Z · tan ˛2 )2 − r22 ] ⎪ ⎪ ⎪ ⎪ ⎨ f [(r2 + Z · tan ˛2 )2 − (r2 + Z · tan ˛2 − t

work

0 ≤ Z ≤ twork 2

· tan ˛2 ) ]

⎪ ⎪ f [(r2 + t2 )2 − (r2 + Z · tan ˛2 − twork · tan ˛2 )2 ] ⎪ ⎪ ⎩

twork < Z ≤ t2 · cot ˛2

(B9)

t2 · cot ˛2 < Z ≤ twork + t2 · cot ˛2 Z > twork + t2 · cot ˛2

0

When twork ≥ t2 · cot ˛2 , besides the cases in Figs. B1 and B3, it is easy to understand that when t2 · cot ˛2 ≤ Z ≤ twork , MRR2 is a constant similar as the second expression on the right hand side of Eq. (8) and expressed as MRR2 = f [(r2 + t2 )2 − r22 ]

(B10)

Consequently, MRR2 is also a piecewise function as expressed by Eq. (B11)

MRR2 =

⎧ f [(r2 + Z · tan ˛2 )2 − r22 ] ⎪ ⎪ ⎪ ⎪ ⎨ f [(r2 + t2 )2 − r 2 ]

0 ≤ Z ≤ t2 · cot ˛2 t2 · cot ˛2 < Z ≤ twork

2

⎪ ⎪ f [(r2 + t2 ) − (r2 + Z · tan ˛2 − twork · tan ˛2 ) ] twork < Z ≤ twork + t2 · cot ˛2 ⎪ ⎪ ⎩ 2

0

2

(B11)

Z > twork + t2 · cot ˛2

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