Amontons s law

Combining the two previous relations for contact area and friction force gives Amonton s law ... 

More recently, arguments for the origin of Amontons s law have arisen that are based on experimental studies demonstrating that the shear stress as varies with the local pressure P according to Eq. [4]... 

Numerous simulations of boundary lubricants have shown that cto is indeed small, even at pressures close to the yield strengths of solids.14,21 Thus, it seems that Amontons s law should hold in a wide range of cases. However, exceptions are observed when adhesive interactions are strong, which leads to large values of cto. 

Microscopic Theory of Amontons s Laws for Static Friction. 

It would appear that no account of friction is complete without first stating Leonardo da Vinci s (or Amonton s) laws and Coulomb s law of friction and pointing out that, in general, polymers do not obey them. The laws are ... 

Assume that Amontons s law of sliding friction can be applied at each elementary area of the interface, such that... 

The microslip motion is in the circumferential direction and the surface distribution of traction in this annular region takes the limiting value from Amontons s law as... 

The friction parameter generally measured is the coefficient of friction. To measure the friction coefficient, a surface is brought into contact with another and moved relative to it. When the two surfaces are brought into contact, the perpendicular force is defined as the normal force (N). The friction force (F) is that force, which opposes relative movement between the two surfaces. From Amonton s law, the coefficient of friction (/u.) is defined as the ratio of the friction force to the normal force ... 

Friction measurements can offer quantitative insight into changes on the skin surface and the UMT offers technical advances over existing friction measurements. The control of the probe speed and the real-time monitoring of the normal load allow for real-time calculation of the friction coefficient. As seen in Figure 32.1, the control of the load is important because the friction coefficient does not adhere to Amonton s Law. Wolfram18 theoretically deduced that the friction coefficient would relate to the normal load as follows ... 

FIGURE 32.1 Friction coefficient versus normal load (50 g Sensor). The friction coefficient increased as the normal load was decreased suggesting that the skin does not follow Amonton s Law. The probe was moved at 5 mmrnin-1 (n = 4). Reproduced from Sivamani et al.24... 

Amonton s law, stipulating a constant value for the friction coefficient whatever the pressure applied to skin, is not verified for skin. 

Amonton s law relates pressure to absolute temperature. Consider the ideal gas law and then write a statement of Amonton s law in a form that is analogous to the statements of Charles s law and Boyle s law in the text. 

The earliest attempts to explain Amontons s laws were based on the idea that macroscopic peaks or asperities on one surface interlocked with valleys on the opposing surface [1]. As illustrated in Fig. la, the bottom surface then forms a ramp that the top surface must be lifted up over in order to slide. If the typical angle of the ramp is 9, and there is no microscopic friction between the surfaces. 

This provides a simple phenomenological explanation for Amontons s laws as well as some exceptions to them. Summing over Areal, one finds... 

Over a wide range of system parameters the dilatancy is smaller than the characteristic length A. Under this condition the generalized Prandtl Tomlinson model predicts a linear increase of the static friction with the normal load, which is in agreement with Amontons s law. It should be noted that, in contrast to the multi-asperity surfaces discussed in Section VII, here the contact area is independent of the load. The fulfillment of Amontons s law in the present model results from the enhancement of the potential corrugation, a2C7oexp(l — Z/K), experienced by the driven plate with an increase of the normal load. 

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