- Email: [email protected]
Analysis of frictional phenomena in friction welding of mild steel
Wear, 37 (1976) 265 - 278 @ Elsevier Sequoia S.A., Lausanne
ANALYSIS OF FRICTIONAL MILD STEEL
J. J. HEALY,
D. J. MCMULLAN
265 - Printed
in the Netherlands
PHENOMENA
IN FRICTION
WELDING
OF
and A. S. BAHRANI
The Queen’s University of Belfast (Gt. Britain) (Received
September
8, 1975)
Summary A dynamical analysis of the “equilibrium phase” of friction welding is presented. The fundamental idea is that the observed phenomena are controlled by the behaviour of a viscous layer of plasticised metal at the rubbing surfaces. This layer is postulated to obey a constitutive equation relating shear stress to rate of strain which is similar to the well-known “Bingham plastic” model. Formulae are thus obtained which predict the external driving torque, as well as the thickness and temperature distribution of the plasticised layer. Acomparison is made between the theoretical results and a number of experiments which have been carried out on mild steel tubular specimens, over a range of conditions. Good agreement is found in all cases. Preliminary results are presented for the apparent viscosity of plasticised mild steel.
1. Introduction In the friction welding process, one of the components to be welded is held stationary while the other is rotated at constant speed. Ideally, a constant axial load is applied, and this causes an axial shortening or “upset”, after the interface between the two workpieces has reached a sufficiently high temperature. The “equilibrium phase” of the process refers to the period of time during which the axial load, rubbing speed, torque, temperature and rate of axial movement are all constant (Fig. 1). Previous experimental work [l, 21 suggests that, during the equilibrium phase, a “boundary layer” exists on either side of the rubbing interface which contains hot plasticised metal undergoing a high rate of strain, due to the relative speed of the workpieces. This plasticised metal, together with some of the weakened parent material, extrudes in a direction normal to the rubbing velocity to produce a “collar” of extruded material surrounding the weld interface. In order to analyse the dynamical problem, consider the welding of two thin-walled tubes, of similar material, in which the wall thickness is much less than the mean radius. This restriction is made so that all portions of the contact surface can be considered to be subjected to the same rubbing
266
-----a$ To-L
frctioning
stage i forqlng stage -_-.._.---_-e
Fig. 1. Idealised traces of the variations with time of speed, shortening during the friction welding process. Fig. 2. Enlarged
section
torque,
axial force and axial
of the tube wall.
speed, and curvature effects can be neglected. Figure 2 shows an enlarged sketch of a portion of the tube wall in the region of the interface, and defines the coordinate system. The following physical mechanism during the equilibrium phase at the interface is postulated. (1) A layer of plasticised material of thickness Z* exists on either side of the interface. (2) Heat is generated by viscous dissipation within this layer, and removed as sensible heat by the extruded material. (3) The high temperature close to the interface causes a decrease in both the shear strength and the compressive strength of the solid material. Within the layer of thickness Z*, the shear strength of the material is less than that required to withstand the shear stress generated by the external torque. (4) Within the plastic&d layer the compressive strength of the solid material is insufficient to carry the axial load. Thus a hydrodynamic pressure must appear such that the sum of the pressure plus the compressive stress just balances the stress due to the applied axial load. This pressure then provides the driving force for the extrusion of plasticised material.
267
2. Mathematical
formulation
Referring to Fig. 2, consider the stresses acting on planes normal to the 2 axis. The external torque produces a shear stress 7 which is independent of 2. This stress is balanced by a viscous stress and the residual shear strength of the heated metal. Thus
where U(Z) is the velocity component in the direction of rubbing, G(0) is the shear strength of the metal at a temperature 0, and ~(5) is the apparent viscosity of the plasticised material, which may be a function of the local rate of strain 9. Assume that the plasticised layer is sufficiently thin that within it (e, - e) = Go + E(eo - e) In
Here Go is the shear strength of the material at the interface temperature BO,and I E I is the slope of the curve of shear strength versus temperature in the region of the melting point. It is assumed that such a curve is available for the given material, although in practice the precise value of E is difficult to determine. From eqns. (1) and (2) dU P(5) dZ+Go+E(eo-e)=7
O
(3) dU -= dZ
0
IzI>z*
Equation (3) is essentially a temperature-dependent modification of the well-known “Bingham plastic” constitutive equation, which has been widely used in the study of rheology [3]. Next consider the equation for conservation of energy. The dominant terms are those due to conduction, convection and dissipation of heat in the plasticised layer. Thus (yv2e = G. vfj -_-
1
@ (4) PGP where 0 (K) is the actual temperature minus the ambient temperature, CYis the thermal diffusivity, and @ is the rate of dissipation of mechanical energy per unit volume, which is given by the product of the stress and rate-ofstrain tensors. As an order of magnitude approximation 328 _ az2
-
eo, (z*y
a28 e. _%3y2 - T2
a28 52
z 0
Thus if the tube wall thickness 2T is much larger than 22*, it may be expected that a ‘0 /a y2 << a ‘8 /aZ2. Also, for I 2 I + 0, the symmetry
of the
268
geometry shows that V-V@-+ 0, provided atmosphere are small. Finally
C&g--
heat losses to the surrounding
(5)
121-0
since dU/dZ is by far the largest contribution The energy equation then simplifies to d2tI
dU
dZ2
dZ
k -~--___._
to the rate-of-strain
tensor.
(6)
lZI+O
On the other hand, for 121 > Z*, Cpz 0 and 3. VJe z - wm de/dZ, where wm is the rate of axial upset as seen from a reference frame located at the interface. Thus dB
d2fI Ly-~--&=dZ2 The boundary (7) are
IzI>z*
dZ conditions
(7)
to be used in conjunction
with eqns. (3), (6) and
Izl>z*
U(Z) = ‘/z U== Z=O
U(Z) = 0
de -= dZ
0
Z=O
dU -= dZ
0
IzI=z*
U” is the independently specified rubbing speed, in m s-r. Substituting for dU/dZ in eqn. (6) from eqn. (3) gives
d2e
---
dZ2
TE
=-J-(T-~o
+-e
kp
kp
-coo)
IZl"0
(9)
Equations (7) and (9) are asymptotic forms of the energy equation, each valid in a particular region. The solutions to eqns. (7) and (9) are found by elementary means, and four integration constants are involved. Two of these can be fixed from eqn. (8), and the remaining two by requiring that the temperature and temperature gradient given by the two equations should match at 2 = Z*. After some elementary algebra e(z)=
0
0
-TmGo y)
+Acos
(z-)“Z
O~IZI~IZ*l
(10)
269
O(Z)=Bexp
(-F)
lZ*lfiZI<
(11)
m
where
-
B=
(E)
cos
“z’[-l (12)
I(eo--!-.$22)+Acos(~)“Z*/
exp(q)
Next eliminate
(13)
B from eqn. (6) using eqn. (10). Then (14)
Since dU/dZ = 0 at 2 = Z*, from eqn. (14) Z’ = 0
(15)
Substituting eqn. (15) into eqns. (12) and (13), the expressions A and B simplify to yield eo----
E T--Go
1I
1+-
e(z)=
f
%9(Z)=
Go f 1exp ie*-=---
2 w*.Z* A -cos ff
I
- $(Z-Zz’)
for
77 2 i 2z*
1
iI
i.Zt>.Z*
(16)
(17)
(W From eqns. (14) and (15) (19) In principle eqn, (19) may be non-linear, since ~1may be expected to be a function of jr, which is approximately equal to dUldZ. As no information is available on P, assume that any dependence of P on i is small; and integrate eqn. (19) directly. Then.
(20)
From the boundary
=
condition
(Go + do
on U(Z) at 2 = Z*
w-k -
-7)
(21)
QET
Let 0, = B. + 6, where 8, corresponds to the melting temperature and 6 is the difference between the melting temperature and the actual interface temperature Bo. Then Go+~Oo=c& Substituting
+f(e,--6j=ee,
(22)
eqn. (22) in eqn. (21),
Solving for r, 7=_
2kB, a
wm --. U”
(23)
The experimentally
measured
torque
is given by
p = A,Rr =
(24)
where A, is the contact area over which rubbing occurs and R is the mean radius of the tube. Equation (24) enables a prediction of the torque to be made provided information is available on the upset velocity wm for each u”. This information has been determined for mild steel tubing [l] and will be used later to check the validity of eqn. (24), and hence the entire theoretical development. Before turning to this, a number of additional formulae are developed, which will be relevant to the subsequent discussion. From eqn. (16) at Z=O e.
=
fjo
i
-?_.I.?! E
and Go =
Hence
c(e,
-e,)
i
I+ i
2 wmz* ) 71
ff
271
e(J=
I
eo--
T
-
E(&, - 6,)
(
I+--
E
2 w-z” 7r (Y 1
1
(25)
The interface ~mpe~ture may be estimated from eqn. (25) and compared with experimental data which are available for mild steel. From eqn. (16) an expression for the viscosity of the plasticised metal is extracted: P=
22* 2 re ( 7T 1 -C-
(26)
This expression is used to calculate an approximate value of P from experimental data on 7 and Z* under various rubbing speeds and axial loadings. Finally, an upper bound on the plasticised layer thickness 22’ may be predicted from eqn. (25): l,!?&-L_)
(I+;?)
Therefore ‘>2 w-z* ,------_.-E0, n OL
2w”z*
7
71 a
Eem
IIence 2z*
T &,--T
(27)
3. Experimental techniques and measurements Experiments were carried out to study the frictional behaviour of mild steel in friction welding. Tubular specimens were used in order to reduce the variation of the rubbing speed with radius. The specimens had an inside diameter of 12.7 mm and an outside diameter of 19 mm. The friction welding was carried out using a 12 ton research friction welding machine of the sliding head type. The machine, which is described in detail elsewhere [ 41, is driven by a hydrostatic drive unit with electronic speed control. The drive unit is capable of providing infinitely variable speed from 250 to 4200 rev min-’ . In a series of tests the rubbing speed and the axial force were varied in turn and for each test the variations in speed of rotation, resisting torque, axial force and axial shortening during the welding cycle were recorded on a UV recorder. The rubbing speed at the mean radius of the tubular specimens was varied ~thin the range 0.42 - 3.36 m s- ’ and the axial force was varied within the range 2.4 - 19.0 kN.
Axial
14 i t
a o
force 38kN
60kN
D
11 0
o
I6
kN
2 kN
0 Mean
Fig. 3. Variation and axial force.
rubbtncj
speed
It-n s-1)
of the rate of axial shortening
h
during the equilibrium
phase with speed
The welded specimens were sectioned and prepared for metallographic examination. The grain structures of the materials at the weld region and on either side of the rubbing interface were studied in detail. Some of the welded specimens were annealed at 880 “C for 45 min in a vacuum furnace. Metallographic examination of the annealed specimens showed that the material ;‘I the plasticised zone had recrystallised into a fine grain structure; thus it became possible to measure the thickness of the plasticised zone with a fair degree of accuracy. The results of the above experiments are given in detail elsewhere [l] and some are given in this paper for comparison with the theoretical predictions. 4. Comparison
of theory
and experiment
For mild steel, the thermophysical properties at temperatures close to melting are those of Touloukian [ 51. The data from various sources do not agree closely at elevated temperatures, and the numerical values quoted here are averages which may be in error by as much as f 50%: thermal
diffusivity
thermal
conductivity
melting temperature
(Y= 1.8 X lo- 6 m2 see1 k = 15
W m-r
minus ambient
Km-’ temperature,
0, z 1490 K
The value of E (the slope of the curve of shear strength uersus temperature in the vicinity of the melting point) is even more difficult to fix accurately, as relatively little information on mechanical properties at elevated temper-
273
force
Axial
_
B
3.6
o
6.0
kN
w 11.0
kN
o 16.2
kN
experimental
--------
theoreticat
kN
results results
L
0
1.0 Rubbmg
2.0 speed
U*
(m
3.0 s“l
___L
Fig. 4. Comparison between the experimental results and theoretically predicted values of the variation of equilibrium torque with rubbing speed and axial force.
atures is available. Hawkyard et al. [ 61 have published data on the static and dynamic tensile yield strength of mild steels at temperatures up to 700 “C. From these data, the value of E is estimated as ES
0.5 (avg. tensile strength
at 700 “C)
1510 - 700 s 1.6 X lo5 N mL2 K-l Substituting numerical predicted to be T
=
3.3 x lo4
values for steel into eqn. (24), the torque
1+ 98 5
Nm
is
(28)
The experimental information for w- as a function of U”, for various axial loadings, is shown in Fig. 3. When this is inserted in eqn. (28), the comparison of theoretical and experimental torque is as shown in Fig. 4. In view of the uncertainty in the numerical values of the physical properties and experimental error, the agreement is reasonably good. Experimental curves showing the thickness 22* of the plasticised zone as a function of rubbing speed and axial load are shown in Fig. 5. This, combined with the data on shear stress 7 and upset velocity woD,is inserted
Axial B
n II
0
‘-_A
1.0
.__
~_~.A~_..~_
force
60kN OkN
~~~
.~__I_
2.0 Mean
rubbmg
___
3.0 speed
(m s-‘1
___i
LO
-
Fig. 5. The variation of the thickness of the plasticised layer with rubbing speed and axial force. in eqn.
(25), and Fig. 6 shows the resulting prediction of interfacial temperature. A trend of increasing temperature with increasing U” is found, but the maximum temperature is always below the melting temperature over the range of conditions studied. The low values of torque measured at large rubbing speed are consistent with the predicted high temperature in this regime, since at higher temperatures the plastic&d metal viscosity, and thus the measured torque, would be expected to be lowered. A number of investigators [ 7, 81 have attempted to measure the interface temperature, and values of about 1300 “C have been reported, with no indications that the melting point is reached. Figure 5 also shows the upper bound on 22* at 6 kN axial load, as predicted by eqn. (27). The experimental curve approaches the upper bound at high rubbing speed. This is consistent with the behaviour of interface temperature at 6 kN in Fig. 6. The upper bound is obtained by setting Be = 19~) and from Fig. 6 B,-,is close to 6, for rubbing speeds above 1.5 m s-i. Thus the experimental value of 22* would be expected to approach the asymptotic value at rubbing speeds above 1.5 m s- ‘; this is the case. It is of interest to make a quantitative estimate of the magnitude of the plasticised metal viscosity. From the previous discussion, it is clear that the temperature can vary by several hundred degrees Kelvin over the range of rubbing speed encountered. Also the rate of strain (2 U-/22*) varies considerably from one test to the next. Since P is likely to depend on temperature, strain rate and pressure, the experimental information at hand is not really sufficient to extract P accurately as an explicit function of these parameters. Nevertheless some indication of the magnitude and behaviour of p can be gained by plotting eqn. (26) as a function of Urn and axial load, on the same curve as the interfacial temper-
275
___---
,_.W---”
_a--
-a--
Axialforce c, 6.0
kN
B16.2
kN
Fig. 6. The derived values of the variation of interfacial temperature, viscosity and strain rate with rubbing speed and axial force.
ature (Fig. 6). For completenesi the approximate strain rate U”/2Z* is also shown. One might tentatively conclude that P decreases with temperature and strain rate, which might be expected. However P appears to decrease with increase in axial loading, which is surprising. 5. Conclusion A theory covering the phenomena encountered in friction welding has been proposed and foliated* The resulting mathematics expressions have been tested against experimental measurements of torque, upset rate and plasticised layer thickness, on mild steel tubes. Although there is some doubt about the correct numerical values of the physical properties at the required conditions, the theory is clearly supported by the experimental evidence. The viscosity of the plasticised metal appears to decrease with increase in ~mpe~ture, strain rate and axial load, although these conclusions are tentative. Further work is in progress on aluminium and titanium tubular material, as well as on dissimilar materials, and will be communicated in a later paper.
The experimental research work carried out at Queen’s University, Belfast, into the mechanism of friction welding was supported by a generous grant from the Science Research Council.
276
References 1 F. D. Duffin and A. S. Bahrani, Frictional behaviour of mild steel in friction welding, Wear, 26 (1973) 53 - 73. 2 F. D. Duffin and A. S. Bahrani, The mechanics of friction welding, Proc. 3rd Int. Conf. on Advances in Welding Processes, Harrogate, England, May 1974, Welding Institute, 1974, Paper 34, pp. 228 - 242. 3 R B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. 4 F. D. Duffin, G. R. Dickson and A. S. Bahrani, Research friction welding machine with hydrostatic drive, Met. Constr. Br. Weld. J., 4 (5) (1972) 161 - 166. 5 Y. S. Touloukian, Thermophysical Properties of Matter, Plenum Press, New York, 1970. 6 J. B. Hawkyard, D. Eaton and W. Johnson, The mean dynamic yield strength of copper and low carbon steel at elevated temperatures, Int. J. Mech. Sci., 10 (1968) 929 - 948. 7 A. Hasui, S. Fukushima and J. Kinugawa, Experimental studies on friction welding phenomena, Trans. Nat. Res. Inst. Met., 10 (4) (1968) 207 - 225. 8 F. Eichhorn and R. Schaefer, Fundamental investigations into friction welding, Schweissen + Schneiden, 21 (5) (1969) 189 - 198. 9 S. W. E. Earls and M. J. Kadhim, Friction and wear of unlubricated steel surfaces at speeds up to 655 ftls, Proc. Inst. Mech. Eng., London, 180 (1) (22) (1965-66) 531 - 548.
‘Appendix Criteria for formation of a friction weld Further support for the mechanism postulated is found by asking whether there exists a minimum rubbing speed, for a given axial load, below which a weld will not be formed. In the context of the present theory, a minimum value of u” does exist, and an estimate of its magnitude can be made as follows. Consider the initial stages of the welding process, before the interfacial temperature 0s has reached a sufficiently high value for yielding of the tube material to occur. Thus, when no plasticised layer has formed, the upset rate is zero and heat is being generated by “solid” rubbing friction at the interface (as opposed to viscous dissipation during the equilibrium period). The heat input at the interface can be conducted away axially, or it can be lost by convection at the tube surface. Now if the rubbing speed is sufficiently low, a stable condition can be reached where heat input at the interface is conducted away and eventually lost to the surrounding air by convection and the interface temperature is too low to cause yielding of the solid material under the applied loads. The two workpieces will then rub indefinitely without formation of a weld. Referring back to Fig. 2, and using the same notation as before, the equation governing the temperature distribution is now
(A.11 Separation
of variables gives
0 = c cos (hy) exp (-X2) where X is the separation
parai neter.
(A.2) At y = * T the boundary
condition
277
ae = he ay is applied, where h is an appropriate heat transfer coefficient piece to ambient air. From eqns. (A.2) and (A.3) I/% hZ & forXT<< 1 ( 1 -k
t-4.3) from work-
(A-4)
The heat input per unit area at 2 = 0 can be written as TJFU”/A,, where 1) is a coefficient of friction, F is the axial load and A, is the contact area. The average value of the temperature gradient at 2 = 0 must then be atZ=O
-k
(A.5)
Substituting eqn. (A.2) in eqn. (A.5) gives
(cos,,)
=&I
cosXydy=F-lforXT<
Therefore
(A.6) The interface temperature is thus G4.7) The shear strength of the interface material is given by Go = E(e, -. e,)
(A-8) The shear stress produced by rubbing is r = r/F/A, In the plasticised zone,
(A.91
r 2 G,, Therefore (A.lO)
278
From eqn. (A.lO) it is found that (A.ll) provided that (hT/h)” << 1. The magnitude of h will vary between the stationary and rotating workpieces, and will be partly due to natural convection and partly due to forced convection. For the specimens of interest here, the value of h is roughly 10 W mP2 K-l, baaed on a convective Reynolds number calculated at a rubbing speed of 1 m s- ‘. Assuming a value of 77of about 0.1 [ 9 J and an axial load F of 6 kN, the minimum value of U” from eqn. (A.ll) will be about 0.24 m s-l. Obviously a more exact calculation would require a more refined evaluation of h and r) , including their dependence on U _, but for present purposes it is clear that the lower limit of rubbing speed should be in the region of 0.2 m s-l. This is in good agreement with experimental observation; although no attempt has been made to establish the precise lower limits on U” for steel tubes, values of about 0.25 m s-l were typical and (as predicted by eqn. (A.ll)) a larger axial load F will permit a lower value of U” to be used before welding ceases to occur. Thus it may be concluded that the workpieces first heat up by “solid” frictional heating until the yield strength of the interface material drops below the stresses generated by applied load and torque. A boundary layer of material undergoing strain rate then forms, and the mechanism of energy input changes to viscous dissipation. The plasticised metal then extrudes, and the plasticised layer forms the joint between the two workpieces.
ANALYSIS OF FRICTIONAL MILD STEEL
J. J. HEALY,
D. J. MCMULLAN
265 - Printed
in the Netherlands
PHENOMENA
IN FRICTION
WELDING
OF
and A. S. BAHRANI
The Queen’s University of Belfast (Gt. Britain) (Received
September
8, 1975)
Summary A dynamical analysis of the “equilibrium phase” of friction welding is presented. The fundamental idea is that the observed phenomena are controlled by the behaviour of a viscous layer of plasticised metal at the rubbing surfaces. This layer is postulated to obey a constitutive equation relating shear stress to rate of strain which is similar to the well-known “Bingham plastic” model. Formulae are thus obtained which predict the external driving torque, as well as the thickness and temperature distribution of the plasticised layer. Acomparison is made between the theoretical results and a number of experiments which have been carried out on mild steel tubular specimens, over a range of conditions. Good agreement is found in all cases. Preliminary results are presented for the apparent viscosity of plasticised mild steel.
1. Introduction In the friction welding process, one of the components to be welded is held stationary while the other is rotated at constant speed. Ideally, a constant axial load is applied, and this causes an axial shortening or “upset”, after the interface between the two workpieces has reached a sufficiently high temperature. The “equilibrium phase” of the process refers to the period of time during which the axial load, rubbing speed, torque, temperature and rate of axial movement are all constant (Fig. 1). Previous experimental work [l, 21 suggests that, during the equilibrium phase, a “boundary layer” exists on either side of the rubbing interface which contains hot plasticised metal undergoing a high rate of strain, due to the relative speed of the workpieces. This plasticised metal, together with some of the weakened parent material, extrudes in a direction normal to the rubbing velocity to produce a “collar” of extruded material surrounding the weld interface. In order to analyse the dynamical problem, consider the welding of two thin-walled tubes, of similar material, in which the wall thickness is much less than the mean radius. This restriction is made so that all portions of the contact surface can be considered to be subjected to the same rubbing
266
-----a$ To-L
frctioning
stage i forqlng stage -_-.._.---_-e
Fig. 1. Idealised traces of the variations with time of speed, shortening during the friction welding process. Fig. 2. Enlarged
section
torque,
axial force and axial
of the tube wall.
speed, and curvature effects can be neglected. Figure 2 shows an enlarged sketch of a portion of the tube wall in the region of the interface, and defines the coordinate system. The following physical mechanism during the equilibrium phase at the interface is postulated. (1) A layer of plasticised material of thickness Z* exists on either side of the interface. (2) Heat is generated by viscous dissipation within this layer, and removed as sensible heat by the extruded material. (3) The high temperature close to the interface causes a decrease in both the shear strength and the compressive strength of the solid material. Within the layer of thickness Z*, the shear strength of the material is less than that required to withstand the shear stress generated by the external torque. (4) Within the plastic&d layer the compressive strength of the solid material is insufficient to carry the axial load. Thus a hydrodynamic pressure must appear such that the sum of the pressure plus the compressive stress just balances the stress due to the applied axial load. This pressure then provides the driving force for the extrusion of plasticised material.
267
2. Mathematical
formulation
Referring to Fig. 2, consider the stresses acting on planes normal to the 2 axis. The external torque produces a shear stress 7 which is independent of 2. This stress is balanced by a viscous stress and the residual shear strength of the heated metal. Thus
where U(Z) is the velocity component in the direction of rubbing, G(0) is the shear strength of the metal at a temperature 0, and ~(5) is the apparent viscosity of the plasticised material, which may be a function of the local rate of strain 9. Assume that the plasticised layer is sufficiently thin that within it (e, - e) = Go + E(eo - e) In
Here Go is the shear strength of the material at the interface temperature BO,and I E I is the slope of the curve of shear strength versus temperature in the region of the melting point. It is assumed that such a curve is available for the given material, although in practice the precise value of E is difficult to determine. From eqns. (1) and (2) dU P(5) dZ+Go+E(eo-e)=7
O
(3) dU -= dZ
0
IzI>z*
Equation (3) is essentially a temperature-dependent modification of the well-known “Bingham plastic” constitutive equation, which has been widely used in the study of rheology [3]. Next consider the equation for conservation of energy. The dominant terms are those due to conduction, convection and dissipation of heat in the plasticised layer. Thus (yv2e = G. vfj -_-
1
@ (4) PGP where 0 (K) is the actual temperature minus the ambient temperature, CYis the thermal diffusivity, and @ is the rate of dissipation of mechanical energy per unit volume, which is given by the product of the stress and rate-ofstrain tensors. As an order of magnitude approximation 328 _ az2
-
eo, (z*y
a28 e. _%3y2 - T2
a28 52
z 0
Thus if the tube wall thickness 2T is much larger than 22*, it may be expected that a ‘0 /a y2 << a ‘8 /aZ2. Also, for I 2 I + 0, the symmetry
of the
268
geometry shows that V-V@-+ 0, provided atmosphere are small. Finally
C&g--
heat losses to the surrounding
(5)
121-0
since dU/dZ is by far the largest contribution The energy equation then simplifies to d2tI
dU
dZ2
dZ
k -~--___._
to the rate-of-strain
tensor.
(6)
lZI+O
On the other hand, for 121 > Z*, Cpz 0 and 3. VJe z - wm de/dZ, where wm is the rate of axial upset as seen from a reference frame located at the interface. Thus dB
d2fI Ly-~--&=dZ2 The boundary (7) are
IzI>z*
dZ conditions
(7)
to be used in conjunction
with eqns. (3), (6) and
Izl>z*
U(Z) = ‘/z U== Z=O
U(Z) = 0
de -= dZ
0
Z=O
dU -= dZ
0
IzI=z*
U” is the independently specified rubbing speed, in m s-r. Substituting for dU/dZ in eqn. (6) from eqn. (3) gives
d2e
---
dZ2
TE
=-J-(T-~o
+-e
kp
kp
-coo)
IZl"0
(9)
Equations (7) and (9) are asymptotic forms of the energy equation, each valid in a particular region. The solutions to eqns. (7) and (9) are found by elementary means, and four integration constants are involved. Two of these can be fixed from eqn. (8), and the remaining two by requiring that the temperature and temperature gradient given by the two equations should match at 2 = Z*. After some elementary algebra e(z)=
0
0
-TmGo y)
+Acos
(z-)“Z
O~IZI~IZ*l
(10)
269
O(Z)=Bexp
(-F)
lZ*lfiZI<
(11)
m
where
-
B=
(E)
cos
“z’[-l (12)
I(eo--!-.$22)+Acos(~)“Z*/
exp(q)
Next eliminate
(13)
B from eqn. (6) using eqn. (10). Then (14)
Since dU/dZ = 0 at 2 = Z*, from eqn. (14) Z’ = 0
(15)
Substituting eqn. (15) into eqns. (12) and (13), the expressions A and B simplify to yield eo----
E T--Go
1I
1+-
e(z)=
f
%9(Z)=
Go f 1exp ie*-=---
2 w*.Z* A -cos ff
I
- $(Z-Zz’)
for
77 2 i 2z*
1
iI
i.Zt>.Z*
(16)
(17)
(W From eqns. (14) and (15) (19) In principle eqn, (19) may be non-linear, since ~1may be expected to be a function of jr, which is approximately equal to dUldZ. As no information is available on P, assume that any dependence of P on i is small; and integrate eqn. (19) directly. Then.
(20)
From the boundary
=
condition
(Go + do
on U(Z) at 2 = Z*
w-k -
-7)
(21)
QET
Let 0, = B. + 6, where 8, corresponds to the melting temperature and 6 is the difference between the melting temperature and the actual interface temperature Bo. Then Go+~Oo=c& Substituting
+f(e,--6j=ee,
(22)
eqn. (22) in eqn. (21),
Solving for r, 7=_
2kB, a
wm --. U”
(23)
The experimentally
measured
torque
is given by
p = A,Rr =
(24)
where A, is the contact area over which rubbing occurs and R is the mean radius of the tube. Equation (24) enables a prediction of the torque to be made provided information is available on the upset velocity wm for each u”. This information has been determined for mild steel tubing [l] and will be used later to check the validity of eqn. (24), and hence the entire theoretical development. Before turning to this, a number of additional formulae are developed, which will be relevant to the subsequent discussion. From eqn. (16) at Z=O e.
=
fjo
i
-?_.I.?! E
and Go =
Hence
c(e,
-e,)
i
I+ i
2 wmz* ) 71
ff
271
e(J=
I
eo--
T
-
E(&, - 6,)
(
I+--
E
2 w-z” 7r (Y 1
1
(25)
The interface ~mpe~ture may be estimated from eqn. (25) and compared with experimental data which are available for mild steel. From eqn. (16) an expression for the viscosity of the plasticised metal is extracted: P=
22* 2 re ( 7T 1 -C-
(26)
This expression is used to calculate an approximate value of P from experimental data on 7 and Z* under various rubbing speeds and axial loadings. Finally, an upper bound on the plasticised layer thickness 22’ may be predicted from eqn. (25): l,!?&-L_)
(I+;?)
Therefore ‘>2 w-z* ,------_.-E0, n OL
2w”z*
7
71 a
Eem
IIence 2z*
T &,--T
(27)
3. Experimental techniques and measurements Experiments were carried out to study the frictional behaviour of mild steel in friction welding. Tubular specimens were used in order to reduce the variation of the rubbing speed with radius. The specimens had an inside diameter of 12.7 mm and an outside diameter of 19 mm. The friction welding was carried out using a 12 ton research friction welding machine of the sliding head type. The machine, which is described in detail elsewhere [ 41, is driven by a hydrostatic drive unit with electronic speed control. The drive unit is capable of providing infinitely variable speed from 250 to 4200 rev min-’ . In a series of tests the rubbing speed and the axial force were varied in turn and for each test the variations in speed of rotation, resisting torque, axial force and axial shortening during the welding cycle were recorded on a UV recorder. The rubbing speed at the mean radius of the tubular specimens was varied ~thin the range 0.42 - 3.36 m s- ’ and the axial force was varied within the range 2.4 - 19.0 kN.
Axial
14 i t
a o
force 38kN
60kN
D
11 0
o
I6
kN
2 kN
0 Mean
Fig. 3. Variation and axial force.
rubbtncj
speed
It-n s-1)
of the rate of axial shortening
h
during the equilibrium
phase with speed
The welded specimens were sectioned and prepared for metallographic examination. The grain structures of the materials at the weld region and on either side of the rubbing interface were studied in detail. Some of the welded specimens were annealed at 880 “C for 45 min in a vacuum furnace. Metallographic examination of the annealed specimens showed that the material ;‘I the plasticised zone had recrystallised into a fine grain structure; thus it became possible to measure the thickness of the plasticised zone with a fair degree of accuracy. The results of the above experiments are given in detail elsewhere [l] and some are given in this paper for comparison with the theoretical predictions. 4. Comparison
of theory
and experiment
For mild steel, the thermophysical properties at temperatures close to melting are those of Touloukian [ 51. The data from various sources do not agree closely at elevated temperatures, and the numerical values quoted here are averages which may be in error by as much as f 50%: thermal
diffusivity
thermal
conductivity
melting temperature
(Y= 1.8 X lo- 6 m2 see1 k = 15
W m-r
minus ambient
Km-’ temperature,
0, z 1490 K
The value of E (the slope of the curve of shear strength uersus temperature in the vicinity of the melting point) is even more difficult to fix accurately, as relatively little information on mechanical properties at elevated temper-
273
force
Axial
_
B
3.6
o
6.0
kN
w 11.0
kN
o 16.2
kN
experimental
--------
theoreticat
kN
results results
L
0
1.0 Rubbmg
2.0 speed
U*
(m
3.0 s“l
___L
Fig. 4. Comparison between the experimental results and theoretically predicted values of the variation of equilibrium torque with rubbing speed and axial force.
atures is available. Hawkyard et al. [ 61 have published data on the static and dynamic tensile yield strength of mild steels at temperatures up to 700 “C. From these data, the value of E is estimated as ES
0.5 (avg. tensile strength
at 700 “C)
1510 - 700 s 1.6 X lo5 N mL2 K-l Substituting numerical predicted to be T
=
3.3 x lo4
values for steel into eqn. (24), the torque
1+ 98 5
Nm
is
(28)
The experimental information for w- as a function of U”, for various axial loadings, is shown in Fig. 3. When this is inserted in eqn. (28), the comparison of theoretical and experimental torque is as shown in Fig. 4. In view of the uncertainty in the numerical values of the physical properties and experimental error, the agreement is reasonably good. Experimental curves showing the thickness 22* of the plasticised zone as a function of rubbing speed and axial load are shown in Fig. 5. This, combined with the data on shear stress 7 and upset velocity woD,is inserted
Axial B
n II
0
‘-_A
1.0
.__
~_~.A~_..~_
force
60kN OkN
~~~
.~__I_
2.0 Mean
rubbmg
___
3.0 speed
(m s-‘1
___i
LO
-
Fig. 5. The variation of the thickness of the plasticised layer with rubbing speed and axial force. in eqn.
(25), and Fig. 6 shows the resulting prediction of interfacial temperature. A trend of increasing temperature with increasing U” is found, but the maximum temperature is always below the melting temperature over the range of conditions studied. The low values of torque measured at large rubbing speed are consistent with the predicted high temperature in this regime, since at higher temperatures the plastic&d metal viscosity, and thus the measured torque, would be expected to be lowered. A number of investigators [ 7, 81 have attempted to measure the interface temperature, and values of about 1300 “C have been reported, with no indications that the melting point is reached. Figure 5 also shows the upper bound on 22* at 6 kN axial load, as predicted by eqn. (27). The experimental curve approaches the upper bound at high rubbing speed. This is consistent with the behaviour of interface temperature at 6 kN in Fig. 6. The upper bound is obtained by setting Be = 19~) and from Fig. 6 B,-,is close to 6, for rubbing speeds above 1.5 m s-i. Thus the experimental value of 22* would be expected to approach the asymptotic value at rubbing speeds above 1.5 m s- ‘; this is the case. It is of interest to make a quantitative estimate of the magnitude of the plasticised metal viscosity. From the previous discussion, it is clear that the temperature can vary by several hundred degrees Kelvin over the range of rubbing speed encountered. Also the rate of strain (2 U-/22*) varies considerably from one test to the next. Since P is likely to depend on temperature, strain rate and pressure, the experimental information at hand is not really sufficient to extract P accurately as an explicit function of these parameters. Nevertheless some indication of the magnitude and behaviour of p can be gained by plotting eqn. (26) as a function of Urn and axial load, on the same curve as the interfacial temper-
275
___---
,_.W---”
_a--
-a--
Axialforce c, 6.0
kN
B16.2
kN
Fig. 6. The derived values of the variation of interfacial temperature, viscosity and strain rate with rubbing speed and axial force.
ature (Fig. 6). For completenesi the approximate strain rate U”/2Z* is also shown. One might tentatively conclude that P decreases with temperature and strain rate, which might be expected. However P appears to decrease with increase in axial loading, which is surprising. 5. Conclusion A theory covering the phenomena encountered in friction welding has been proposed and foliated* The resulting mathematics expressions have been tested against experimental measurements of torque, upset rate and plasticised layer thickness, on mild steel tubes. Although there is some doubt about the correct numerical values of the physical properties at the required conditions, the theory is clearly supported by the experimental evidence. The viscosity of the plasticised metal appears to decrease with increase in ~mpe~ture, strain rate and axial load, although these conclusions are tentative. Further work is in progress on aluminium and titanium tubular material, as well as on dissimilar materials, and will be communicated in a later paper.
The experimental research work carried out at Queen’s University, Belfast, into the mechanism of friction welding was supported by a generous grant from the Science Research Council.
276
References 1 F. D. Duffin and A. S. Bahrani, Frictional behaviour of mild steel in friction welding, Wear, 26 (1973) 53 - 73. 2 F. D. Duffin and A. S. Bahrani, The mechanics of friction welding, Proc. 3rd Int. Conf. on Advances in Welding Processes, Harrogate, England, May 1974, Welding Institute, 1974, Paper 34, pp. 228 - 242. 3 R B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960. 4 F. D. Duffin, G. R. Dickson and A. S. Bahrani, Research friction welding machine with hydrostatic drive, Met. Constr. Br. Weld. J., 4 (5) (1972) 161 - 166. 5 Y. S. Touloukian, Thermophysical Properties of Matter, Plenum Press, New York, 1970. 6 J. B. Hawkyard, D. Eaton and W. Johnson, The mean dynamic yield strength of copper and low carbon steel at elevated temperatures, Int. J. Mech. Sci., 10 (1968) 929 - 948. 7 A. Hasui, S. Fukushima and J. Kinugawa, Experimental studies on friction welding phenomena, Trans. Nat. Res. Inst. Met., 10 (4) (1968) 207 - 225. 8 F. Eichhorn and R. Schaefer, Fundamental investigations into friction welding, Schweissen + Schneiden, 21 (5) (1969) 189 - 198. 9 S. W. E. Earls and M. J. Kadhim, Friction and wear of unlubricated steel surfaces at speeds up to 655 ftls, Proc. Inst. Mech. Eng., London, 180 (1) (22) (1965-66) 531 - 548.
‘Appendix Criteria for formation of a friction weld Further support for the mechanism postulated is found by asking whether there exists a minimum rubbing speed, for a given axial load, below which a weld will not be formed. In the context of the present theory, a minimum value of u” does exist, and an estimate of its magnitude can be made as follows. Consider the initial stages of the welding process, before the interfacial temperature 0s has reached a sufficiently high value for yielding of the tube material to occur. Thus, when no plasticised layer has formed, the upset rate is zero and heat is being generated by “solid” rubbing friction at the interface (as opposed to viscous dissipation during the equilibrium period). The heat input at the interface can be conducted away axially, or it can be lost by convection at the tube surface. Now if the rubbing speed is sufficiently low, a stable condition can be reached where heat input at the interface is conducted away and eventually lost to the surrounding air by convection and the interface temperature is too low to cause yielding of the solid material under the applied loads. The two workpieces will then rub indefinitely without formation of a weld. Referring back to Fig. 2, and using the same notation as before, the equation governing the temperature distribution is now
(A.11 Separation
of variables gives
0 = c cos (hy) exp (-X2) where X is the separation
parai neter.
(A.2) At y = * T the boundary
condition
277
ae = he ay is applied, where h is an appropriate heat transfer coefficient piece to ambient air. From eqns. (A.2) and (A.3) I/% hZ & forXT<< 1 ( 1 -k
t-4.3) from work-
(A-4)
The heat input per unit area at 2 = 0 can be written as TJFU”/A,, where 1) is a coefficient of friction, F is the axial load and A, is the contact area. The average value of the temperature gradient at 2 = 0 must then be atZ=O
-k
(A.5)
Substituting eqn. (A.2) in eqn. (A.5) gives
(cos,,)
=&I
cosXydy=F-lforXT<
Therefore
(A.6) The interface temperature is thus G4.7) The shear strength of the interface material is given by Go = E(e, -. e,)
(A-8) The shear stress produced by rubbing is r = r/F/A, In the plasticised zone,
(A.91
r 2 G,, Therefore (A.lO)
278
From eqn. (A.lO) it is found that (A.ll) provided that (hT/h)” << 1. The magnitude of h will vary between the stationary and rotating workpieces, and will be partly due to natural convection and partly due to forced convection. For the specimens of interest here, the value of h is roughly 10 W mP2 K-l, baaed on a convective Reynolds number calculated at a rubbing speed of 1 m s- ‘. Assuming a value of 77of about 0.1 [ 9 J and an axial load F of 6 kN, the minimum value of U” from eqn. (A.ll) will be about 0.24 m s-l. Obviously a more exact calculation would require a more refined evaluation of h and r) , including their dependence on U _, but for present purposes it is clear that the lower limit of rubbing speed should be in the region of 0.2 m s-l. This is in good agreement with experimental observation; although no attempt has been made to establish the precise lower limits on U” for steel tubes, values of about 0.25 m s-l were typical and (as predicted by eqn. (A.ll)) a larger axial load F will permit a lower value of U” to be used before welding ceases to occur. Thus it may be concluded that the workpieces first heat up by “solid” frictional heating until the yield strength of the interface material drops below the stresses generated by applied load and torque. A boundary layer of material undergoing strain rate then forms, and the mechanism of energy input changes to viscous dissipation. The plasticised metal then extrudes, and the plasticised layer forms the joint between the two workpieces.