Welding wave on the contact spot of solids

Tribology International Vol. 31, No. 4, pp. 169–174, 1998  1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0301–679X/98/$19.00 + 0.00

PII: S0301–679X(98)00018–8

Welding wave on the contact spot of solids A. A. Shtertser*

In this paper the theoretical model of adhesion between clean solid surfaces is presented. Material behaviour is described by hydrodynamic equations for viscous liquid and by the Gruneisen equation of state. This approach is quite suitable for quantitative analysis of mechanical processes running under high pressures. The model states that growth of a bonded area can occur in a regime of self-propagating welding wave (WW). WW originates at any active point where initial bonding takes place and further moves at velocity Uww 苲 ␥/␩ (␥ is surface energy and ␩ is viscosity of material). According to the model presented surface energy plays the key role in adhesion.  1998 Elsevier Science Ltd. All rights reserved. Keywords: adhesion, surface tension, viscosity

For all symbols the mark ‘0’ refers to normal conditions (room temperature and atmosphere pressure) and marks ‘1’ and ‘2’ refer to parameters just before and behind the welding wave front, respectively.

Introduction The seizure phenomenon is caused by the welding of solid bodies (parts) over contacting area during friction or joint deformation. To avoid seizure in most cases lubricants are used in tribology systems. At the same time the seizure phenomenon is used in technologies of metal joining by roll, pressure and other kinds of cold welding. There presently exist different views on the mechanism of solid-state-welding. In Semenov1 and Krasulin2 the opinion was stated that the bonding process proceeds similarly to heat explosion. Seizure occurs as a result of a self-sustaining topochemical reaction which sometimes proceeds violently, and the diffusion process can play a substantial role on complete reaction only. It is emphasised by Semenov1 that interatomic bonding occurs between atoms which escape the certain energetic barrier (activated atoms). This activation is stimulated by heat transfer from bonded to unbonded area. Design and Technology Institute of High-Rate Hydrodynamics, Tereshkovoi 29, Novosibirsk 630090, Russia *Corresponding author. Tel: + 7 3832 35 52 32; Fax: + 7 3832 35 52 32; E-mail: [email protected] Received 13 May 1997; revised 27 November 1997; accepted 14 January 1998

Nomenclature WW d u Uww ␳ v p ␥ w q ␴ik ⑀ik ⑀′ T G cv ␣ K ␩, ␰ j ␹

welding wave; interatomic distance; mass velocity; velocity of WW front; density; specific volume; pressure; surface energy; internal energy (per mass unit); heat energy (per mass unit); stress tensor; deformation tensor; rate of deformation; absolute temperature; Gruneisen coefficient; heat capacity; volume thermal expansion coefficient; bulk modulus; viscosity coefficients (first and second respectively); flow density; specific volume relative jump in WW front;

The first acts of bonding appear at ‘active points’, which in many cases are the dislocation outlets on the solid surface2.

Tribology International Volume 31 Number 4 1998

169

Welding wave on the contact spot of solids: A. A. Shtertser

Tabor and his co-authors3 consider that at least for metals activation of surface atoms is not needed for bonding, and the moving force of the seizure process is connected with the existence of surface energy. There are the other thoroughly made experimental works which confirm this point of view4,5. To our opinion, the work most close to reality is the consideration given by Buche6, who stated that the mechanism of seizure depends on materials as well as on conditions under which these materials are treated. To provide bonding in some cases it is enough to destroy surface layers and to force clear surfaces into direct contact. In other cases activation of surface atoms is needed as an additional condition. In any case all researchers agree that surface contaminations are the main obstacles preventing bonding. There are a lot of publications on the behaviour of lubricants in tribology systems, and on the effect of oxides, hydroxides and other layers on the cold welding process (see for example references in Semenov1). Therefore we do not consider this question in the present paper. It should be borne in mind only that surface layers have an effect on seizure, even in such extremely intensive processes as explosive powder compacting and explosive welding7,8. Actually, in the model below we assume that surface contaminations are somehow removed from the contact zone and surface atoms are driven in direct contact. Thus we study the mechanism of clear surface bonding.

Theoretical considerations Perhaps everyone has watched the coalescence of mercury balls. The moving force of this process is the surface tension of material. Coalescence occurs very quickly because mercury has a small shear strength and viscosity. For all solids which have positive surface energy the bonding over the contact zone is thermodynamically justified, but this process is usually suppressed by material strength (low mobility of atoms). Experience shows that this obstacle (strength) is overcome by the action of high compressive and shear stresses appearing at contact zones during plastic deformation or friction. Under such conditions, material can be approximately described by equations of viscous liquid. This approach was successfully used in the modelling of solid material flow9, and we use it in our description as well. Let us consider the two-dimensional situation and suppose that two pieces (parts 1 and 2 in Fig 1) of the same material are subjected to direct contact along their plane surfaces (plane XZ in Fig 1). Let us further suppose that surface atoms of these parts are bonded at X > 0 and they are not bonded at X ⬍ 0, but the distance between atomic layers is close to interatomic distance d. Thus the axis 0Z is the linear front between bonded and unbonded zones on the plane XZ. Actually at X ⬍ 0 we have a closed crack which should vanish with the release of surface energy. Surface atoms of crack are ready for bonding but are not activated and the welding process does not occur spontaneously. Nevertheless, the welding can occur by movement of point 0 to the left. We have named this phenomenon 170

Fig. 1 Contact zone of solid parts. 1. Unbonded surface atoms just before the WW front. 2. Bonded surface atoms just behind the WW front. AB0CD, Free surface curve presenting a closed crack; B0C, conventional presentation of crack vertex as a line with curvature radius d/2; d, interatomic distance; b, conventional width of WW front, i.e. the distance between unbonded and bonded atoms (it is assumed that b = d) the welding wave (WW) and the line 0Z is therefore the WW front. The movement of the WW front is caused by the material’s tendency to reduce surface energy. This tendency is taken into account below by introduction of the Laplas formula into the system of equations (see Equation (3) below). This formula gives the pressure jump in point 0 where the surface line radius of curvature is equal to d/2 (see Fig 1). Let us make the next assumptions: 1. The flow near the X-axis is one-dimensional, i.e. uy = 0; ux = u(x,y)⫽0, where ux, uy are the components of mass velocity → u. 2. The WW front 0Z can move in left direction with constant velocity Uww. In Figs 1 and 2 the flow picture is shown in a coordinate system of moving WW front. 3. In the WW front only the surface atoms receive a jump in velocity caused by Laplas pressure jump. Therefore below we consider only the flow line coinciding with the X-axis (X-line) and this flow line has a thickness of 2d (two atomic layers). 4. The mass velocity u(x) has linear dependence on x inside the WW front, it reaches its maximal value u2 just behind the WW front, and at last it comes down to the initial value u1. This drop is due to viscous transfer of momentum to interior atomic layers on both sides of the Y-axis (Fig 2). The thickness of the WW front is equal to the interatomic distance d of material. The physical sense of this assumption is that the distance between completely bonded and completely unbonded atoms can not be less than d (see Fig 1). Linear approximation of u(x) inside the WW front is quite symbolic because we consider the distances of interatomic scale. This kind of approximation is the simplest, and use of any other curve is quite possible. The only demand is that u(x) should be a continuous function to solve

Tribology International Volume 31 Number 4 1998

Welding wave on the contact spot of solids: A. A. Shtertser

describes the initial flow state of media. It is not used in the equations below directly, but it is needed for the assessment of viscosity. Now we have the next system of equations: Equation of movement (Navier–Stokes equation for viscous liquid)

␳u

冉

冊 冉 冊

∂2u ∂ 2u ␩ ∂2u ∂u ∂p + + ␰ + =− +␩ ∂x ∂x ∂x 2 ∂y 2 2 ∂x 2

(1)

Equation of continuity

Fig. 2 Mass velocity as a function of x. ABCD, Mass velocity curve plotted against x; BC, linear section of curve inside the WW front; b, width of WW front (equals to interatomic distance d); mass velocity first increases from u1 to u2 because of surface energy release. Then at x > x2 mass velocity comes down due to momentum viscous transfer. At points x1 and x2 a derivative ∂u/∂x = 0

j = ␳u = const Laplace equation

p2 − p1 = −

5.

the differential equations below. In any case the representation of u(x) by the curve ABCD in Fig 2 is rather conventional, but it gives us the possibility to find the solution of the equations below. Just before and behind the WW front ∂u/∂x = 0. That is because mass velocity u = const before the WW front (line AB in Fig 2), and because the first derivative of any function should be equal to zero at the point of its maximal value (point C in Fig 2).

These five assumptions give us a qualitative representation of the WW model of bond formation. Let us write p for pressure, u for flow velocity, ␳ for material density, v = 1/␳ for specific volume, ␥ for material surface energy, w for internal energy (per mass unit), q for heat energy (per mass unit), ␣ for volume thermal expansion coefficient, cv for heat capacity at constant volume, K for bulk modulus (compressibility factor), T for Kelvin temperature, G = ␣K/␳cv for Gruneisen coefficient, ⑀′ for rate of deformation, ⑀ik for deformation tensor, ␴ik for stress tensor, ␩ and ␰ for first and second viscosity, respectively, d for interatomic distance; pc for ‘cold’ pressure (depends on v only), wc for ‘cold’ energy (depends on v only), and j = ␳·u for flow density. For all symbols in this paper the mark 0 refers to normal conditions (room temperature and atmosphere pressure) and marks 1, 2 refer to parameters before and behind the WW front, respectively. It should be taken into account that ⑀′ is the rate of deformation generated by external forces producing compressive and shear stresses in material, whereas ⑀ik is brought about by WW propagation. Actually the local flow induced by WW propagation is superimposed on the initial flow caused by external forces. In our model, ⑀′ refers to the given parameters and

(2)

2␥ d

(3)

Equation of energy

dw = v␴ ik d⑀ ik + dq

(4)

Gruneisen equation of state for pressure10

p = p c (v) + Gc v T/v

(5)

Gruneisen equation of state for energy10

w = w c (v) + c v (T − T 0 ) + w 0

(6)

Furthermore, we have from Equation (1)

冋

册

∂ ∂2u ∂u ␳u 2 + p − (␰ + 4␩/3) =␩ 2 ∂x ∂x ∂y

(7)

Components of stress tensor in our case have the following forms11:

␴ xx = − p + (4␩/3 + ␰)∂u/∂x; ␴ yy = ␴ zz = − p + ( − 2␩/3 + ␰)∂u/∂x; ␴ xy = ␴ yx = ␩∂u/∂y; ␴ 31 = ␴ 13 = ␴ 32 = ␴ 23 = 0; According to assumption 1 and the symmetry of flow at Y = 0 we have d⑀xx = dv/v, whereas all the other components of deformation tensor are equal to zero. Thus, we have from Equation (4) for flow line along the X-axis (X-line) ∂w ∂u ∂v ∂v =−p + (4␩/3 + ␰) ∂x ∂x ∂x ∂x

(8)

Tribology International Volume 31 Number 4 1998

171

Welding wave on the contact spot of solids: A. A. Shtertser

After multiplying Equation (8) by ␳u and Equation (7) by u and adding them, we have

p − p0 = −

∂ ␳u 3 ∂2 u ∂u ␳uw + = u␩ 2 + pu − (␰ + 4␩/3)u ∂x 2 ∂x ∂y

w − w0 =

冋

册

(9) Integration of Equations (7) and (9) with respect to X from 0 to d gives (see assumptions 4 and 5):

冕 d

j(u 2 − u 1 ) + (p 2 − p 1 ) = ␩

∂2u dx ∂y 2

(10)

冉 冊

K T T0 (v − v 0 ) + Gc v − v0 v v0

K(v − v 0 ) 2 + ␣KT 0 (v − v 0 ) + c v (T − T 0 ) 2v 0 (16)

Let us introduce the parameter ␹ = (v2 − v1)/v1 = (u2 − u1)/u1. Then using Equations (3) and (15) we get c v (T 2 − T 1 ) =

0

j(w 2 − w 1 ) +

j 2 (u − u 21 ) + p 2 u 2 − p 1 u 1 2 2

∂2u = ␩ u 2 dx ∂y According to assumptions 3 and 4 we can write the approximate expressions: ∂2u (u − u 1 ) =− (12) 2 ∂y d2 u = u 1 − (u 2 − u1 )x/d Using Equation (12) for integration of Equations (10) and (11) we finally have: ␩(u 2 − u 1 ) j(u 2 − u 1 ) + (p 2 − p 1 ) = − (13) 2d (p 2 + p 1 ) (v 2 − v 1 ) 2

(14)

j␩(v 2 − v 1 ) 2 +q− 12d

(17)

␩ + 4␳ 1 d

冪冉

冊

␩ 4␳ 1 d

2

+

2␥ ␳ 1 d␹

(18)

In deriving Equation (18) the negative sign before the square root was not taken as it has no physical sense. This equation can be simplified in cases of great and small viscosity values. Actually when 2␥ ␩ 2 À we get 4␳ 1 d ␳1 d␹ 4␥ U ww = (19) ␩␹

冉 冊

When ␩→0 we get the maximal possible value of WW front velocity U ww =

2␥

冪␳ d␹

(20)

1

These two equations are fundamental in the description of WW propagation. In analogy with adiabats of detonation and combustion let us name Equation (14) adiabat of adhesion or seizure adiabat. When heat transfer is absent then q = 0 and the process is adiabatic. In other cases the process should be specified and q should be found for this process. Fundamental Equations (13) and (14) can be used if the equation of the state of material is known. We have chosen that of Gruneisen, which looks in general as Equations (5) and (6), and which describes well the material behaviour at high pressures and elevated ∂v −1 temperatures10. Using bulk modulus K = − v0 and ∂p T volume thermal expansion coefficient ␣, we can write for cold pressure and cold energy: K(v − v 0 ) p c (v) = − − ␣KT 0 + p 0 v0

冉冊

Thus, if the adhesion process is thermodynamically specified then at first the relative change of specific volume ␹ should be found using Equations (14), (16) and (17), and secondly the welding wave velocity is found from Equations (18)–(20).

Isothermal process In this case ␹ is found from Equation (17) directly, as we have T2 = T1. An isothermal process is possible when thermal conductivity of material is of great value and all the released heat is quickly removed from the bonding zone. Equation (17) can be transformed into the form of

冋

␹2 + 1 +

册

␣v 20 T 1 2␥v 0 2␥v 0 − ␹− =0 v 21 dKv 1 Kdv 1

(21)

In Table 1 the physical parameters for several metals are given. As can be seen for all of them, approximate equality has place 2␥ ⬇ 0.1 Kd

K(v − v 0 ) 2 Kv 0 ␣ 2 T 20 + ␣KT 0 (v − v 0 ) + 2v 0 2

After substitution of these expressions to Equations (5) and (6) we get: 172

册

2␥v 1 dG

U ww = u 1 = −

0

w c (v) =

冋

Kv 21 2 2␥v 1 Kv 21 ␹ + cvT1 − ␹ + v0G dG v0G

The velocity of the WW front is derived from Equations (3) and (13) in the form of

d

w2 − w 1 = −

−

(11)

冕

(15)

(22)

Besides, we can take v1 ⬇ v0 because the bulk modulus K is very high for solids and applied external pressure

Tribology International Volume 31 Number 4 1998

Welding wave on the contact spot of solids: A. A. Shtertser Table 1 Physical parameters of metals

␳0, g/cm3 cv, J/kg grad K, 1010 N/m2 ␣, 10−5 grad−1 G c0, km/s ␥, J/m2 d, nm 2␥/Kd ␩, Pa s: ⑀′ = 1 s−1 ⑀′ = 10 s−1 ⑀′ = 104 s−1 ⑀′ = 105 s−1

Al

Cu

2.71 896 7.30 6.93 2.09 5.2 1.15 0.286 0.11

8.93 382 13.7 4.95 1.98 3.95 1.79 0.256 0.10

8.9·106 7.8·105 苲 104

Pb 11.34 129 4.13 8.7 2.46 1.91 0.58 0.350 0.08

Fe 7.86 450 13.9 3.57 1.40 4.2 2.49 0.248 0.14

5·106 4.3·106

9.3·106 苲 104

Note: Bulk sonic velocity was calculated as c0 = √K/␳0. The physical parameters for metals were taken from References 10,12,13. Viscosities for different deformation rates ⑀′ were taken from Reference15 and calculated using data from Reference 14.

does not change material density substantially. Finally, because of the small thermal expansion coefficient the term ␣T1 is negligible and we can write ␹2 + 0.9␹ − 0.1 = 0. This equation gives the same value ␹ = 0.1 for all metals when the adhesion process is isothermal, and then for great viscosity we have from Equation (19) the simple formula U ww =

40␥ ␩

the calculated isothermal Uww values for different metals are given.

Adiabatic process In this case q = 0 and Equation (17) should be solved together with Equations (14) and (16). Besides we should take into account that the flow line consists of two atomic layers (see assumption 3 above) and that released surface energy should be included in consideration. It means that the zero level of energy w0 in Equation (16) is different for material before (state 1) and behind (state 2) the WW front. Denoting these energies as w01 and w02 we have ␥ w 01 − w 02 = (25) d␳ 0

(23)

Maximal velocity of the WW front is found from Equations (20) and (22) and looks like U ww =

冪␳

K

(24)

0

Thus we can see that the maximal value of Uww is close to bulk sonic velocity of material. In Table 2

Making a number of operations we get the quadratic equation for specific volume jump ␹.

Table 2 Calculated Uww values of metals

␹ Uww, ␮m/s: ⑀′ = 1 s−1 ⑀′ = 10 s−1 ⑀′ = 104 s−1 ⑀′ = 105 s−1

Al

Cu

Pb

Fe

0.1

Isothermal adhesion 0.1

0.1

0.1

5.2 59.0 4600.0

4.6 16.7

10.7 9960.0

Adiabatic adhesion

␹ Uww, ␮m/s: ⑀′ = 1 s−1 ⑀′ = 10 s−1 ⑀′ = 104 s−1 ⑀′ = 105 s−1

0.12

0.17

4.3 49.1 3833.3

9.8

0.14

0.21

3.3 5.1 4742.9 Tribology International Volume 31 Number 4 1998

173

Welding wave on the contact spot of solids: A. A. Shtertser

␹2 + − −

再

冉

2G c vT 1 1 v0 + ␣T 0 + (G + 2) G v 1 Kv 1

冎

冊

2␥v 0 (G + 3) (v 1 − v 0 ) p 1 v 0 + ␹ + 3dKv 1 G v1 Kv 1

冉

冊

2G␥v 20 2v 1 1+ =0 2 dKv 1 (G + 2) Gv0

(26)

In deriving this equation we used Equation (19) because the viscosity of metals is very high (see Table 2). In the general case Equation (26) can be solved for the given values of applied pressure p1 and temperature T1. Specific volume v1 under these conditions is found with the help of Equation (15). In many practical cases we have p1¿K and T1 = T0. This yields v1 ⬇ v0 and Equation (26) can be written in the simplified form:

␹2 +

再 冉 冎

1 2G cv + ␣+ (G + 2) G Kv 0

冊

(27)

2␥(G + 3) 2␥ T0 − ␹− =0 3dKG dK The calculated adiabatic values of ␹ and Uww are presented in Table 2, together with data for isothermal adhesion.

Discussion It can be seen from Table 2 that Uww depends much on the rate of deformation ⑀′ because material viscosity is very sensitive to this flow parameter. As mentioned above ⑀′ is the given parameter characterising the state of material under certain loading conditions. Viscosity decreases with the growth of ⑀′ and this provides less time for complete welding. This time can be estimated if the surface density of active points is known. According to Krasulin2 dislocation outlets play the role of active points. Usually we have arrays and networks of dislocations in metals12. Thus the characteristic dimension of material on the microlevel is the size of network cell l. This size varies from 苲 1 ␮m in nondeformed to 苲 0.01 ␮m in heavily deformed metal. The time needed for complete welding under applied compressive stresses is estimated as tw ⬇ l/2Uww. Using this formula and data from Table 2 we find for A1 tw 苲 1 ms (at ⑀′ ⬇ 1 s−1) and tw 苲 1 ␮s (at ⑀′ ⬇ 104 s−1). These two characteristic times correspond to processes of rolling and explosive welding, respectively15. In calculations we used l = 0.01 ␮m because material usually undergoes high deformation in such workings.

174

There exists another rate of deformation ⑀′ww connected with deformation tensor ⑀ik induced by velocity jump in the WW front. This quantity can be estimated as ⑀w′ w 苲 (u 2 − u 1 )/d = U ww ␹/d. Taking the values from Table 2 we can see that for AL at ⑀′ = 1 s−1 we have ⑀′ww 苲 1.8·10 3 s − 1 and for ⑀′ = 104 s−1 6 −1 ⑀′ww = 1.6·10 s . Obviously ⑀′ww is much greater than ⑀′, which is why the deformation induced by external forces was not included in deformation tensor ⑀ik. In a word, inside the WW front and in close distance behind it the media flow is controlled much more by surface tension than by external loading. To our opinion the model above gives quite a good explanation of the mechanism of adhesion in the solid state. The simple Equations (19), (20), (23) and (24) have clear physical sense and point evidently to the important role of surface energy in bonding processes.

References 1. Semenov, A. P., Skhvatyvanije metallov i metody ego predotvrashenija pri trenii. Trenije i Iznos, 1980, 1(2), 236–246. 2. Krasulin, Yu. L., Vzaimodeistvije Metalla s Poluprovodnikom v Tverdoi Faze. Nauka, Moscow, 1971. 3. Gane, N., Pfaelzer, P. F. and Tabor, D., Adhesion between clear surfaces at light loads. Proc. Roy. Soc., London, 1974, A340, 495–517. 4. Johnson, K. I. and Keller, D. V., Effect of contamination on the adhesion of metallic couples in ultra-high vacuum. J. Appl. Phys., 1967, 38(4), 1896–1904. 5. Nishikawa, O. and Rendulic, K. D., Some FIM observations on cold-welding of metals. Surface Science, 1971, 26(2), 677–682. 6. Buche, N. A., Podshipnikovyje Splavy Dlja Podvizhnogo Sostava. Transport, Moscow, 1967. 7. Shtertser, A. A., Combustion, explosion, and shock waves, Vol. 29. Plenum Publ. Corp., New York, 1993, pp. 734–739. 8. Zakharenko, I. D. and Kiselev, V. V., Svarochnoje Proizvodstvo, 1985, 9, 4–5. 9. Lavrentjev, M. A. and Shabat, B. V., Problemy Gidrodinamiki i ih Matematicheskije Modeli. Nauka, Moscow, 1973. 10. Zeldovich, J. B. and Reiser, Y. P., Fizika Udarnykh Voln i Vysokotemperaturnykh Gidrodinamicheskikh Javlenij. Nauka, Moscow, 1966. 11. Landau, L. D., and Lifshiz, E. M., Gidrodinamika. Nauka, Moscow, 1988. 12. Schulze, G., Metallphysik. Akademie-Verlag, Berlin, 1967, Mir, Moscow, 1971. 13. Miedema, A. R., Surface energies of solid metals. Zeitschrift fur Metallkunde, 1978, 69(5), 287–292. 14. Frost, H. J. and Ashby, M. F., Deformation-mechanism Maps. Pergamon Press, Cheljabinsk, Metallurgija, 1989. 15. Deribas, A. A., Fizika Uprochnenija i Svarki Vzryvom, Russia. Nauka, Novosibirsk, 1980.

Tribology International Volume 31 Number 4 1998