The rolling tire interacts dynamically with the road surface, under the constant constraint of the load of vehicle. There are a bunch of distinct characteristics of this interaction, reflected in the friction force, for the case of traction and braking properties of the tire, in the noise produced by the tire when rolling, in the hydroplaning properties of the tires, etc. All these depend upon some primary properties of the tire surface, of the rubber of tire, and of the road surface roughness and its physical condition (dry, wet, erroded and so on). In order to decide on the theoretical description of the way all these factors intermingle during the rolling of a tire, it is necessary to start from the basic process involved at any level in such an interaction: the local deformation of a surface.
A Classical Description of the Surface Deformation
Let’s therefore describe the socalled infinitesimal deformation of a surface, the way it is classically described (Guggenheimer, 1977). By definition this is the process of deformation in which the position vector on surface varies infinitesimally, something like


(1)

so that the change in the first fundamental form of the surface is always smaller than ε even if this quantity is arbitrarily small:


(2)

As from equation (1) the deformed first fundamental form can be written as


(3)

and therefore


(4)

the condition (2) is satisfied if, and only if, the two infinitesimal vectors are perpendicular:


(5)

The degree of arbitrariness of the deformation vector is thus somewhat reduced, in the sense that we can assume the existence of a vector, say, such that


(6)

About the vector we don’t know anything. However, about its differential we have a wealth of information. Indeed, as in the left hand side of equation (6) we have a vector with components exact differentials, applying the exterior differentiation to it should yield the null vector. This condition comes to


(7)

where the symbol ʻ×_{˄}’ shows that in the cross product of vectors, the regular multiplication is replaced by exterior multiplication of the differentials. Thus the equation (7) can be expanded over, just as usual, in the form of a 3×3 determinant


(8)

provided we replace the regular multiplication by exterior multiplication of the differential forms. The result is a set of three differential equations:


(9)

The first two of these tell us that dp^{3} is null, therefore the vector must be chosen such that its component along the normal to surface is constant. This circumstance is ideal in introducing, for instance the deformation of the tire due to the global loading force given by the dead weight of the vehicle, which is approximately constant. Then, on top of this deformation, the local details on the footprint come into play. Indeed, when using the Cartan’s lemma, the last of equations (9) tells us that there is a symmetric 2×2 matrix with entries A, B, C say, such that:


(10)

As, on the other hand, the vector is an exact differential vector, we can write


(11)

Expanding the last of these equations after the manner of the classical theory of surfaces, we get the vectorial equation


(12)

amounting to the system of three differential equations:


(13)

The first two of these equations show that is transported by parallelism along the (already deformed) surface, while the third provides a condition we need to impose on the symmetric matrix from equation (10). Indeed, as the equations of definition of the curvature vector of the surface are (Guggenheimer, 1977):


(14)

if we use the equation (10), the third of the conditions (13) comes to


(15)

For an algebraic – and physical – interpretation of this result, let’s notice that, because is an intrinsic vector with respect to surface (its component along the normal to surface is null), the cross product of this vector with the elementary displacement on surface is oriented along the normal to surface. This vector is


(16)

Consequently its magnitude is a quadratic form, algebraically conjugated to the second fundamental form of the surface (apolar to that form). One can therefore say that it adds to the second fundamental form, thus changing the local curvature of the surface. In order to assure this condition by default, one can define the coefficients A, B, C, up to a multiplicative factor, by equations


(17)

where λ, μ, ν are three external physical parameters, or even three differentials determined by the second fundamental form itself, when the situation is referred to a previous state of the surface.
These last parameters may describe the material of the deforming surface from a constitutive point of view. On the other hand, they may be even variations of the first fundamental form of the surface, therefore deformations properly speaking. This leads to the idea that the deformation described by displacements is not equivalent to that described by the variation of the first fundamental form of the surface. Usually there are also necessary some conditions relating the variation of the metric tensor with that of displacement vector. As long as one assumes that the metric tensor must depend exclusively only on the coordinates on surface, the conditions for infinitesimal deformation define, for instance, the Killing vectors of the first fundamental form. The equations (17) show nevertheless that, no matter of the physical nature of the deformation of surface, if this one is infinitesimal, the second fundamental form of the surface is involved directly and explicitly in its description, together with some external, perhaps physical, properties. This purely speculative conclusion mirrors the practical observation that there is no deformation of a surface in a point, that can proceed without being noticed by a variation of its normal direction in that point. Therefore this manner of describing the deformation is particularly useful in characterizing the friction process between two rough bodies, like the tire tread lugs and the road surface. Here the constant normal component of vector is simply the normal contact force – the loading force. The other two components don’t even matter but in the final end, and at a finer scale, after we have characterized their variations, possibly as stochastic processes due to roughness, and even after we have integrated these processes. As a matter of fact, the process of friction between two surfaces, offers an example where the force in surface should be specially defined by a differential 2form, as the following analogy shows.
A Physical Interpretation by Electrodynamical Analogy
The equation (6) reminds us of the definition of force in electrodynamics. This science was built upon the ideas of deformations and stresses, but the definition we are talking about does not rely on such concepts, but mainly on the kinematics extracted from experience. That experience shows us that a piece of wire through which an electric current passes, assumes a rotation when submitted to a magnetic field. This rotation is determined by a force proportional to the intensity of the current and the density of the magnetic field lines, and is perpendicular on both the lines of the magnetic field and the element of current:


(18)

Here is the characteristic density of the magnetic flux lines – the socalled magnetic induction – and I is the intensity of current passing through the the piece of wire characterized by the vector . As well known, the piece of wire is immaterial here, so that the formula (18) can also describe for instance the bending of the trajectories of a current of electrons or ions in vacuum.
Now, let’s associate to the deformation vector a force, say by a Hookelike formula, or by some other mathematical means, involving for instance a ‘matrix of stiffnesses’, regularly used by engineers in the estimation of deformation forces in finiteelement calculations:


(19)

Then the equation (6) shows that, just like in the case of magnetic field from electrodynamics, the vector is related to some density of ‘flux of lines’ entering the physical surface under deformation. This analogy has, in the case of tire, a nice connotation that can be properly speculated in the theoretical physics of tire. Let’s describe it.
The road roughness is ‘felt’ by the tire through the intermediary of its footprint, which is determined by the dead weight of the vehicle, by the material properties of the tread rubber, as well as by the tread design. In rolling conditions the road roughness influence on tire can be imagined as a flux of tiny ‘pokes’ in the portion of the surface of tire delimited by the footprint. This flux of tiny pokes, acting like the “rain on the roof”, as they sometimes say, is actually the analogous of the induction of the magnetic field as described above. More to the point, an appropriate expression would be ‘rain on the windshield’, because that ‘rain’ is strongly influenced by the motion of the vehicle. As the equation (16) shows, the details of local deformation on the very footprint or, even better, on the tread lugs, are then to be read in the variation of the second fundamental form of the external surface of the tire rubber.
Description of the Process of Wrinkling
Obviously, the rubber surface is prone to wrinkling: that much we notice even in the daily life. The wrinkling of a surface, is a process dependent on the state of previous wrinkling, and is physically assisted by the friction force. Therefore, one can assume that the friction force is null when the wrinkling is absent. Let’s show how this observation can be theoretically put into equation. First of all, the state of wrinkling is aptly described by the local curvature of the surface: the more wrinkled the surface the many more variations of curvature we notice in a certain portion of it. Secondly, the variation of curvature is described by the two differential forms of curvature (the components of the variation of normal unit vector of surface). On the other hand the wrinkling is physically determined by the friction (rubbing) force, which is a force acting tangent to surface, therefore a force ‘insurface’, as they say. Then, according to one of the Cartan lemmas the friction force can be written in the form


(20)

where f^{1} and f^{2} are two conveniently chosen 1forms. Indeed, only in this case the friction force is zero if the curvature is zero and reciprocally. If the ‘conveniently chosen’ 1forms are the components of the first fundamental form s^{1} and s^{2}, then equation (20) gives us the definition of the curvature matrix as a limiting case. Indeed, it can happen that there are no friction forces, even if the surface is curved. Therefore the friction forces are null, without the curvature being null, so we have zero in the left hand side of equation (20). In this case, another one of the Cartan lemmas, applied to equation (20), shows that we must have


(21)

which is the usual definition of the curvature matrix, used in the development above.
Let’s, however, assume that the friction force is nonvanishing, as surely is the case while the tire is rolling. In that case we can express the friction force as a differential 2form:


(22)

where the indices take the values 1 and 2. The equation (20) can thus be rewritten in the form


(23)

Now, according to the same one of the Cartan’s lemmas, we can write, using a compact notation:


(24)

where Φ is the skewsymmetric 2×2 matrix having the entry Φ_{12} ≡ Φ, and b is the usual curvature matrix from equation (21). We do recognize in the second one of these equations a curvature matrix which is no more symmetric. This means that it accounts also for the twist of surface, which is very important in the theory of deformations of a physical surface (Lowe, 1980). From among the usual measures of curvature, friction thus described does not touch the mean curvature, but has an important saying in the Gaussian curvature:


(25)

It is therefore proper to say that the associated wrinkling of the surface does not affect the mean curvature of the surface. In this respect, another important connection is to be noticed.
The road surface is rough, and one needs to model somehow this roughness. A good approximation of such a surface is a minimal surface, i.e. a surface of zero mean curvature, but nonzero absolute curvature. In this case it is proper to say that Φ is a stochastic process, characterizing the road surface in a time decided by the speed of the rolling tire; therefore the Gaussian curvature itself of the road surface is a stochastic process. This further means that the road surfaces can have different roughnesses to be modeled as stochastic processes, and this fact bestows upon them different friction properties. It is also proper to say that the road surface bestows upon a tire specific noise properties, inasmuch as it determines a specific deformation on the footprint rubber. The fact is well known: the tire prediction industry has a whole arsenal of techniques dedicated to it, mainly along the lines of characterizing the tire behavior in connection with standard road surfaces. We shall come again on this subject later on.
To conclude, in the cases when friction exists and acts continuously, like in the case of rolling tire, instead of equation (14) we must have


(26)

The second fundamental form of the surface does not change:


(27)

and thus the ‘rubbing force’ does not influence the geometrical nature of the local surface of the tire rubber. On the other hand, if in the equation (20) we choose the ‘convenient’ differential 1forms f^{1,2} as the components of the differential of position vector on the deformed surface:


(30)

then there is a friction force acting insurface:


(29)

In other words, a friction force is always present whenever the rubber of the tire is deformed. In such cases the deformation vector from equation (6) is known, therefore the vector can be known, as we have, by definition


(30)

where, as shown before, p^{3} is a constant. Using equation (30) in equation (29), we have an explicit expression for the 2form of the friction force acting in the surface of the tire:


(31)

One can draw from this formula some well known conclusions about friction. For instance, the force is directly proportional with the normal load but, more importantly, it is also proportional with the mean curvature of the surface, at a certain scale represented by ε. As the friction force is usually hard to quantify properly, this last property makes formula (31) particularly valuable, of course, when properly used.
Conclusions
The last phrase here alludes to a “proper use” of the theory, and we discuss this notion now, by the way of concluding the above developments. If not used correctly, and read ad literam, formula (31) would show that on a surface of zero mean curvature the friction force is zero, which is far from what we observe in practice. As a matter of fact the friction phenomenon involves two surfaces, one of which, at least, has to be in deformation. Indeed, by the way we obtained that formula, we actually appealed to deformation in every step. In order to clarify this point let’s think about the process of friction between the tire and the road.
When the road surface is perfectly smooth like, say, the face of ice, there is indeed no friction between road and tire. The friction starts to count significantly when the road surface becomes rough. This is because the road roughness determines a local deformation of the tire tread lugs, and it is this local deformation that contributes to friction. It can be diminished again by a lubricant, like the water or snow on the road, having the essential property to fill the little valleys of the road profile, and thus level it at a certain microscopic scale. This indicates the fact, well known to engineers for a long time, that the friction force is a scale property. For instance if the mean curvature is calculated on the whole area of footprint of the tire, it might end up being zero, thus indicating an improper zero friction force. However, we need to consider that formula on regions of such magnitude that the deformation becomes important: such a region is the tread lug surface area. On such regions the deformation imprinted by the road is far from being zero, so that the formula (31) gives actually nontrivial results depending on the tread rubber properties.
Now let’s take another example, in order to correct the particular choice of an ice surface, whose behavior is in significant measure due to the melting of ice. Assume that the road face is from glass, and the road and tire are perfectly dry. Now there is no question of melting and lubrication, and the glass surface is ideally smooth. On the other hand, there is also no question that the friction properties of the rubber on glass are way different form those on ice. Why? It is here the place where the ‘material’ properties come into play: forces at the molecular level first initiate a small deformation of rubber, which then triggers a subsequent macroscopic deformation enhancing the friction force. In the case of ice, the melting water precludes, or at least delays, the initial small deformation, so that the friction force is appropriately diminished. This is why, when driving on snow we have to act carefully: and the known way to take such care is not to touch the brakes!
References
Guggenheimer, H. W. (1977): Differential Geometry, Dover Publications, New York
Lowe, P. G. (1980): A Note on Surface Geometry with Special Reference to Twist, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 87, pp. 481 – 487