Effect of Coulomb friction and viscous forces on interfacial waves at steel-steel boundaries under external pressure

Thin Solid Films, 186 (1990) 53 58

53

METALLURGICAL AND PROTECTIVE LAYERS

E F F E C T OF C O U L O M B F R I C T I O N A N D VISCOUS F O R C E S O N I N T E R F A C I A L WAVES A T S T E E L - S T E E L B O U N D A R I E S U N D E R EXTERNAL PRESSURE* VIJAI KUMAR AND GURAJADA S. MURTY

Seismology Section, Bhabha Atomic Research Centre, Modular Laboratories, Trombay, Bombay 400085 (India) (Received June 2, 1989; accepted September 15, 1989)

Experimental results for the attenuation and velocity of propagation of interfacial waves along a steel-steel plane interface lubricated with a viscous fluid are presented. The bonding at the interface is varied by the application of an external pressure normal to the interface. These measurements are compared with values predicted by a theoretical model where the shear stress at the interface is assumed to be dependent linearly on the slip in displacement and velocity parallel to the interface. Further, the degree of bonding is assumed to be dependent non-linearly on the external pressure. Agreement between the theoretical model and the experimental data is found to be satisfactory. The implications of these results for testing the adhesion of surfaces are briefly discussed.

1. INTRODUCTION Interfacial waves are the elastic waves propagating along the boundary of contact between two half-spaces. These waves have a particle motion similar to that of Rayleigh surface waves. The study of propagation of interfacial waves along a solid-solid boundary is of considerable importance in the field of seismology and for their possible application in non-destructive methods of testing in the field of aeronautical, space and other engineering industries. The propagation of interfacial waves along a perfectly smooth and welded interface of elastic half-spaces has been studied quite extensively 1. Recently, a theoretical model of an imperfectly bonded interface between homogeneous and isotropic elastic half-spaces was proposed 2. In this model the boundary conditions are chosen in such a way that the well-known cases of a smooth interface and a welded interface are implicit as special cases. This model assumes continuity of the stress tensor and the component of velocity normal to the interface, but it allows a finite slip of the tangential component of velocity at the interface. These boundary conditions, and standard methods, can be used to compute the attenuation * Paper presented at the Seventh International Conference on Thin Films, New Delhi, India, December 7 I1, 1987. 0040-6090/90/$3.50

~) ElsevierSequoia/Printed in The Netherlands

54

V. KUMAR, G. S. MURTY

and velocity of propagation of the interface waves for a given set of elastic parameters of the half-spacesL It is seen that in general the velocity of propagation of interfacial waves increases monotonically with bonding whereas the attenuation increases with bonding initially and attains a peak before returning to zero 2' 3. A comparison of theoretical results with the experimental data on a steel-steel boundary with air as the viscous fluid at the interface revealed qualitative agreement, although the experimental values of the attenuation are considerably smaller than those predicted by the theory 4. In this paper we present some results for new boundary conditions incorporating the effects due to viscous forces as well as forces that give rise to Coulomb friction. Also, the experimental measurements of attenuation and velocity of propagation ofinterfacial waves along a steel steel plane interface lubricated with a viscous liquid are presented. These measurements, together with similar measurements for an interface with air as the viscous fluid, are compared with those obtained from the modified model. 2. EXPERIMENTAL DETAILS

The experimental set-up for the measurement of interfacial wave velocity and attenuation for various boundary conditions is shown in Fig. 1. The varying degrees of bonding are produced by means of a hydraulic ram pressing together the two steel blocks A and B. The interfacial waves are generated and detected by a suitable matched pair of surface acoustic wave piezoelectric transducers (T1 and T2) and the measurement of velocity of propagation and attenuation is performed with the help of a pulser-receiver electronic unit and oscilloscope. Further details are described elsewhere 4, 5. The accuracy of measurement of the velocity of propagation in this setup is approximately 0.2~o and that of the attenuation is 2~. Two types of interfaces, one with a thin film of the viscous liquid salol (silicone

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Fig. I. Schematic diagram of the experimental set-up. The shaded portion marked A and B represents the vertical cross-section of the two steel blocks, simulating half-spaces. The dimensions of block A are 3 cm x 3 c m x 3 cm and those of block B are 10 cm x 5 cm x 3 cm, with block B resting on the face of size 10cm x 5cm. The interface dimensions are 3cm x 3cm. T t and T2 are generating and receiving piezoelectric transducers respectively. A typical observed waveform is also shown.

VELOCITY AND ATTENUATION OF INTERFACIAL WAVES

55

oil) at the interface (steel-salol-steel interface), and the other without any liquid film (steel-air-steel interface) were studied. In each case, rectangular blocks of sizes 10cm x 5cm x 3cm for the lower block and 3cm x 3cm x 3cm for the upper block were prepared and mating surfaces were polished to an average surface finish of 0.1 pm. The type of steel chosen for the experiment has a longitudinal wave speed of 5.91 km s - 1, a shear wave speed of 3.21 km s - 1 and a density of 7.74 g c m - 3 in the unstressed state. During the experiment the interface (3 cm x 3 cm) was subjected to a steadily increasing external pressure up to 44.2x 10 7 N m -2 normal to the interface, the velocity of propagation as well as the attenuation of interfacial wave pulse were measured at various external pressures. The experiment was repeated four times for both types of interfaces and the mean values of the experimental data on velocity of propagation were plotted against corresponding attenuation values in Fig. 2. The values of the external pressure are also shown separately at the top. The average scatter in the attenuation values from experiment to experiment is 1 dB cm 1 and that in the velocity of propagation is

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Fig.2. V a r i a t i o n in interfacial wave speed with a t t e n u a t i o n . E x p e r i m e n t a l d a t a are s h o w n for a steel-salol steel interface ( x ) a n d for a steel a i r - s t e e l interface (O). The theoretical curves 1 3 c o r r e s p o n d to the following choice of parameters: curve 1, C 1 = 0.22(t0C2)11; curve 2, C 1 = 3.3coC2; curve 3, C1 = 0.

56

V, KUMAR, G. S. MURTY

0.2~/o. As the material yields at an external pressure of about 33 x 107 N m 2, indicated by the arrow in Fig. 2, we shall pay attention only to the experimental data pertaining to external pressures varying from 0 to 33 x 107 N m 2. It is seen in Fig. 2 that the interfaciat wave speed increases steadily from 2.97 to about 3.2 km s 1. The attenuation exhibits a peak at a velocity of propagation of about 3.1 km s Although the behaviour of the attenuation vs. the velocity of propagation is similar for both types of interface, the value of the attenuation in the steel-salol-steel interface is higher than that for the case of the steel-air-steel interface. 3. THEORETICAL MODEL The model of a loosely bonded interface assumes a thin viscous layer which allows a finite discontinuity in velocity parallel to the interface 2"6. The theoretical variation in attenuation with velocity predicted by this model is plotted in Fig. 2 (full curve), the influence of external pressure on the elastic constants being neglected 4. It is seen that the experimental values of attenuation are significantly smaller than those predicted by this model. However, the curve for the steel-salol-steel interface is closer to the theoretical result than that for the case of the steel-air-steel interface. The viscous layer model has no further free parameter to be adjusted, and so it can be concluded that forces other than those of viscous origin are at play in the experiment. We note that, although the mating surfaces are polished to the required degree of smoothness, it has to be assumed that there are small valleys and humps interlocked with each other at the interface. In fact these interlocked valleys and humps give rise to Coulomb friction when one surface slides past the other. For oscillatory motion at the interface, these valleys and humps can give rise to elastic forces depending on the slip in the displacement at the interface. Therefore a linear relation between the shear stress at the interface and the slip in displacement [u] and also slip in velocity [/~] is assumed 7. The boundary conditions on velocity and stress normal to the interface are kept unchanged. Thus the shear stress parallel to the interface is, in this case, represented by Pxz = C1 [HI -[- C2[-//]

or

(1) Px: = ( C i q- iooC2)[u]

where C~ and C2 are "constants" which may depend non-linearly on the external pressure, i = ( - 1 ) 1/2 and o9 is the angular velocity of the elastic wave. It has already been shown that, in the case C~ = O, C2 represents the degree of bonding of two elastic half-spaces when they are separated by a vanishingly thin viscous layer of a fluid whose coefficient of viscosity is also vanishingly small 6. The inclusion in eqn. (1) of a term proportional to the slip in displacement at the interface amounts to the assumption that the separating layer can be considered as a viscoelastic medium 7. The justification for the particular form of the boundary

VELOCITY AND ATTENUATION OF INTERFACIAL WAVES

57

condition in this case can be given on the same lines as for C 2 but is omitted for brevity 6. It is confirmed that the well-known ideal boundary conditions of a welded interface are recovered when the constants Ca and C2 both tend to infinitely large values such that Px~ remains finite 2. Further, vanishingly small values of C a and C 2 lead to the boundary conditions of a smooth interface. For other values o f C a and C 2 the bonding at the interface is imperfect. Thus the values of Ca and C2 indicate the degree of bonding at the interface. If we change the degree of bonding at the interface by applying an external pressure normal to the interface the relation between Ca, C2 and the external pressure X is assumed to be of a non-linear form, such as 3 Ca = Da (X/Xo) r and C 2

=

D2(X/Xo) s

(2)

where Da, D2, r and s are constants and X o is a normalizing external pressure. Eliminating X / X o, we obtain C1 = F C 2 q

(3)

where F and q are new parameters to be determined from the experiment. 4.

RESULTS AND DISCUSSION

The theoretical relation between attenuation and velocity of propagation was computed with the boundary conditions of eqn. (1) according to the method described elsewhere a-4. A number of cases were studied for different values of the parameters F and q and only those that are "best fits" for the experimental data are plotted by broken lines in Fig. 2. It is seen that the experimental data are in quantitative agreement with the theoretical curve with the choice C 1 = 0.22(coC2) H

(4)

for the steel-air-steel interface and Cl = 3.300~C2

(5)

for the steel-salol-steel interface. It may be recalled that Ca and C2 in eqn. (1) are independent of each other. However, because we modified the interface conditions with the application of an external pressure, these constants have become related to each other and we saw that the appropriate relation between them is decided by the experimental observations as indicated by Fig. 2. Although the model described in this paper is for two halfspaces with imperfect welding at the interface, this can be generalized to the case of a number of layers with imperfect welding between them. This non-destructive method of measuring the attenuation of interracial waves promises to be of value in judging the degree of bonding of composite structures. REFERENCES W. M. Ewing, W. S. Jardetzky and F. Press, Elastic Waves in Layered Media, McGraw-Hill, New York, 1957.

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V. KUMAR, G. S. MURTY

G.S. Murty, Phys. Earth Planet. Inter., 1l (1975) 65. V. K u m a r and G. S. Murty, IEEE Trans. Sonics Ultrason., 24 (1982) 138. V. Kumar, J. Appl. Phys.,54(1983) I141. V. Kumar, J. Pure Appl. Ultrason., 4 (1982) 33. A . R . Banghar, G. S. Murty and I. V. V. Raghavacharyulu, J. Acoust. Soc. Am., 60 (1976) 1071. R . K . Miller, Bull. Seismol. Sac. Am., 69 (1979) 305.