Adhesive Contact of Elastic Solids... with just a Trace of Viscoelasticity
Dr. J.A. Greenwood and Prof. K.L. Johnson
Cambridge University Engineering Department
The JKR theory describing the contact of an elastic sphere when surface energy is significant has recently become the heart of numerous methods of determining the surface energy of solids directly. AFM style methods use the JKR load/approach curve, or determine the contact area indirectly by normal or tangential oscillations: in SFA methods the contact area may be directly measured.
Frequently the loading and unloading curves differ, and not only because long range surface forces cause the bodies to jump into contact at a point different from the "pull-off" point. With truly elastic solids, this implies that the surface energy involved in separation differs from that in making contact: perhaps because the surface is roughened during separation, or because of entanglement/saturation of the molecular chains on the surface.
An alternative possibility is that when the technique is applied to elastomers, these are not in fact perfectly elastic, but, at very high rates of strain, behave viscoelastically. Direct measurements varying the loading/unloading rate suggest this is not the case. But classical fracture mechanics theory (of which the JKR theory is an example) relies on infinite strains at a crack-tip/contact-edge: and so has an infinite strain-rate when the tip moves, however slowly.
Modified fracture mechanics taking account of the actual surface forces rather than their integral, the surface energy, returns the strains to finite values, but the strain rates remain enormously higher than are implied by the bulk loading rate. The result is a severe limitation on the motion of the crack tip: unsuspected viscoelasticity can drastically modify the behaviour! Perhaps the most striking example is that a viscoelastic time constant of T=1E-7 sec will "freeze" the contact edges during a 50Hz load oscillation, exactly as found experimentally.
Using known results for the dependence of the crack velocity on the nominal stress intensity factor/apparent surface energy, "elastic hysteresis" in the loading/unloading curves can be explained as the effect of a "trace" of viscoelasticity.
Related effects occur in a two-dimensional contact as in the Chaudhuri rolling-contact pendulum experiment, where the distinctly non-sinusoidal damping curve found experimentally can be reproduced remarkably closely.
