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From thermodynamical point of view the pressure is a stress, or a density of energy. The differential form thus defined by pressure should be a 3-form. As a matter of fact, the stresses in a material, in general, can be defined likewise, by a formula which generalizes only slightly the 3-form of pressure:

(1)

The origin of the triadic tensor t, which here is no more totally skewsymmetric, can be explained in the engineering terms that follow. The forces transmitted through matter – in our case the tire rubber – can be written in the form

(2)

where p is a triadic tensor, skewsymmetric in the lower indices, and σ is the regular tensor of stresses, as defined classically. In view of the definition of the oriented elementary area as a skewsymmetric tensor of second order, we have the following definition for the tensor p:

(3)

where the summation convention over dummy indices is implicitly understood. In detail, the table defining p is

(4)

One can see that the covariant vector, defined by the contraction of p as

(5)

 is zero whenever the stress tensor is a symmetric matrix, and vice versa. If the classically defined stress tensor is not symmetric – for instance if the material has inhomogeneities leading to local internal moments – then the vector p_k is not zero. Now, in view of the definition (2) we can define the tensor t from equation (1) by lowering the upper index with the help of the metric tensor reflecting the state of deformation of the rubber:

(6)

Consequently, the thermodynamical transport theorem for the rubber can be written in the form

(7)

in view of the fact that the exterior differential of a 3-form in space is zero. Here t_kl is the second order tensor obtained from the triad t by contraction with the velocity field of the surface of rubber:

(8)

Just as in the case of the thermodynamics of the tire cavity air, there is a second derivative of the quantity from the left hand side of equation (7), coming from the transport theorem as applied to the quantity from the right hand side, which varies too, from different obvious reasons: the ‘little cube’ of the definition of stresses is deformed in a rolling tire, mostly in the region of the footprint, the whole tire vibrates, the heat deforms the tire, etc. Thus, we must have a time variation for the right hand side of equation (8), which normally induces a second time derivative for the left hand side:

(9)

This was the general idea, but we can simplify it by noticing that the fifferential 3-form from equation (1) can be written as:

(10)

Using now the definition (6) and the table (4) we get

(11)

This 3-form reduces to the usual mean stress when the metric is cartesian, i.e. when there is no deformation. Otherwise, if the metric is a deformed one but still of constant curvature (Coll, Llosa, Soler, 2002), the rubber stretches enter our considerations, and we have, for instance

(12)

where s_n is the classical mean stress (the trace of the stress tensor). The quantity from the curly brackets of this equation is the density of energy of the material due to stress in a state of deformation described by the stretches l_k. It explicitly depends on the stresses as well as on the very state of deformation.    Let’s denote by U the density of energy from the curly brackets of equation (12). The equation (7) thus takes the form

(13)

with   the velocity field at the surface of the tire material. As known, this surface has the internal part – the surface of the tire cavity – and an external part – the outer surface of sidewalls together with the tread. So, the velocity field  accounts here for the internal motions of the tire carcass as well as for the external vibrations. The equation (9) can be written as:

(14)

The right hand side here can again be treated by the transport theorem, and yields

(15)

where  is the velocity vector on the line boundaries of tire rubber, i.e. at the bead reinforcements, but not only there, because there is a subtle catch in this formula.
     Indeed, if we consider, just for the sake of understanding, the tire as a homogeneous structure, with no tread pattern and other accidents, the line boundary of the tire rubber is given not only by the bead reinforcements, where the two parts, internal and external, of the tire surface meet, but also by the limit closed line making the boundary of the footprint. So the line integral from the right hand side of the equation (15) is actually a sum of three cyclic integrals, two for the bead reinforcements and one for the tire footprint contour. This fact copes not only with the known fact that the rolling resistance of the tire is mainly concentrated in the footprint stress cycling, but also with the more subtle fact that, for instance, the flatspotting of the tire should be strongly influenced by the contact between the bead and wheel hub. Using equation (15) and the explicit form of the stress energy density from (12) into equation (14) we get the final form of the ‘acceleration’ of the energy in the tire structure:

(16)

This formula shows not only the intuitive fact that the second variation of the stress energy of the tire structure is given by the behavior of energy density at the rubber surface, but also that it should be decided, to a great extent, by the exchange of this energy through the limit of the footprint and the bead contact between the tire and the wheel.

Conclusions

 In order to get the gist of the issue, we reasoned, in the development above, on the tire as a kind of homogeneous structure. However, the tire engineers are well aware of the significant difference between different tire constructions. Where is that difference coming from? Well, as the equation (16) shows, it can only come from the particular properties of the bead reinforcements, of the footprint shape, and last, but by no means the least, from the material properties entering the very form of the density of energy.
     Just to illustrate the idea, let’s limit the present discussion to the footprint contour line, for it is by far the most significant in this respect. We’ll deal later on explicitly anyway, with all of the issues involved here, but for now let’s discuss just the footprint, because it is more obvious as an example. In a ideally homogeneous tire, and on an ideally smooth road, the footprint contour line is a continuous one, so that the line integral in equation (16) makes plainly sense, and can be performed as usual. In a real tire, however, the specific tread pattern interrupts the continuity of the contour of the footprint to only its segments on the tread lugs, while the road roughness interrupts the continuity of those very segments. This is an issue that can compel us to consider higher time variations of the right hand side of equation (16), inasmuch as the very interrupted contour varies. Therefore, the variation of energy from the left hand side of that formula may be decided by the third or even higher time derivatives. The bottom line is that the behavior of the tire depends strongly on its construction (tread pattern) as well as on the road roughness, and the fundamental formula (16) takes into consideration all of the possible cases. 
     But the things get even more complicate in the real tire. Indeed, by the very same token, one can figure out that, in a real tire, the ‘surface delimiting the material’ is to a large extent a matter of construction of the tire. In this construction we ought to decide what is the main material, i.e. the one producing the variable density of energy. One can say right away, by a century or so of practice, that this is the rubber. However, in the tire construction there are quite a few kinds of rubber having wildly varying material properties. Let’s therefore push the idea of homogeneity to still another level, again, just for the sake of argument. Assume, thus, that the tire construction involves only a kind of rubber and only the metal of wires of the belt and bead reinforcements. This way the argument that the rubber is the main material in this construction, having a variable energy density, remains indeed still in force. However, in this case it is obvious that the integral from the right hand side of the equation (13) involves not only the two parts of the outer surfaces of the tire rubber, but also the internal closed tiny surfaces between the rubber and metal, accounting for the loss of energy from rubber through the metal, which thus warms up. In case we treat the wires as structural inhomogeneities, the behavior at the limit interface metal-rubber is decided by the covariant vector from equation (5), for there the stress tensor is nonsymmetric. This is also the main microscopic fact that links the flatspotting of the tire at the footprint to the properties of contact between the bead and the hub. As we said before, we’ll have to deal with these issues separately later on. 

Reference

 Coll, B., Llosa, J., Soler, D. (2002): Three-Dimensional Metrics as Deformations of a Constant Curvature Metric, General Relativity and Gravitation, Vol. 34, pp. 269 – 282