We are plainly aware that the New Testament helps us in choosing the right way in life, at any level of manifestation of the human being: from the individual to social level. In virtue of this observation we claim here that this maxim is valid for science also. Moreover, it is valid only for science. For, in conducting a rational life, if the man doesn’t have the New Testament, then he has not at his disposal but only the science.

            First and foremost we find in the New Testament the law that we like to call of the whole and the part: every part of the world around us is perceived by man as a whole. No other manifestation of Nature can be found but only by respecting this law. This is how the world around is presented to our senses, and certainly this is what Christ respects in the first place. In its moments of truth the modern science came out as we have it today, only respecting such a law. In the very first moments of natural philosophy, the whole was the human body, and it was taken as the expression of harmony, being by itself a whole part of another whole.
            Secondly, the universe is made obvious to our senses by what we call matter. The New Testament then shows that the matter is submitted to the law of accumulation, i.e. of growing, until it can reach our senses or is reached by them. This law, stated in one parable of the New Testament, is obvious by itself in the history of science. Indeed, if this law wouldn’t act, we couldn’t have at our disposal nowadays concepts like substances, elements, and such like; the chemistry as well as the physics, would not be possible.           
            Finally, there is a law that shows how the wholes behave with respect to each other, that we’d like to call the law of density. It shows that the relation between the wholes is actually decided internally. However, the man, being of “little faith” hasn’t even succeeded to bring this law among the basic facts of the modern science, to say the least: it is only a fact that needs to be explained.           
            It is interesting, in the first place, to bring the acts of New Testament which show these laws.

A Classical Description of the Surface Deformation

Let’s therefore describe the so-called 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

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

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

and therefore

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

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

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

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

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

The first two of these tell us that dp³ 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:

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

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

amounting to the system of three differential equations:

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):

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

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

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

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 2-form, 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:

Here  is the characteristic density of the magnetic flux lines – the so-called 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 Hooke-like 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 finite-element calculations:

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 ‘in-surface’, as they say. Then, according to one of the Cartan lemmas the friction force can be written in the form

where f¹ and f² are two conveniently chosen 1-forms. Indeed, only in this case the friction force is zero if the curvature is zero and reciprocally. If the ‘conveniently chosen’ 1-forms are the components of the first fundamental form s¹ and s², 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

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 2-form:

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

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

where Φ is the skew-symmetric 2×2 matrix having the entry Φ₁₂ ≡ 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:

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

The second fundamental form of the surface does not change:

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 1-forms f^1,2 as the components of the differential of position vector on the deformed surface:

then there is a friction force acting in-surface:

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

where, as shown before, p³ is a constant. Using equation (30) in equation (29), we have an explicit expression for the 2-form of the friction force acting in the surface of the tire:

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

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

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:

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

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

 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:

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

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:

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:

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

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

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

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

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:

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

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:

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

     First, while working, the tire is in a rolling motion, and this requires the representation of the force by differential forms, in order to be able to apply a transport theorem helping in dynamical calculations. Physically the force is represented either by a 1-form (the mechanical work) or by a 2-form (the flux), as in the case of the pressure of the tire cavity. The difference between the two representations is given by the fact that, in the first case the force is a vector upon which we construct the 1-form of elementary mechanical work, while in the second case the 2-form itself is the force, while the flux defining the 2-form is the pressure. Correctly speaking, the pressure is therefore a skew-symmetric second order tensor, which defines a 2-form representing the force, exactly the way in which the vector force defines the 1-form of mechanical work. Speaking of  the tire cavity air, there is a third approach of the pressure, that from a thermodynamical point of view, whereby the pressure is a tensor of the third order, generating a differential 3-form, which physically is an energy. It is this representation that allows us to understand the thermodynamics of the tire cavity and to connect it with the behavior of the materials from the construction of tire.

     As the rolling tire is in motion, we can reckon that there should be a theorem of transport for every physical quantity, so much more for the force in all its aspects presented above, analogous to the classical one of Osborne Reynolds (Reynolds, 1903). This theorem is accounting for both the variation of the physical quantity itself, and for the variation of the space support of the force (the footprint, the matter volume of rubber, the surfaces – internal and external – of the tire). In the form we need it here, such a theorem was given by David Betounes (Betounes, 1983) who noticed that the general transport theorem is actually an explicit expression of recovering the Lie derivative of a certain differential form. Even though we presented this theorem elsewhere for the necessities of electrodynamics and solar physics, let’s repeat it here for the sake of completeness and ‘local use’ so to speak.

     The evolution of a certain domain D, like the internal cavity, the structural material volume or the surfaces of the tire, can be mathematically represented by a family of applications indexed by a real parameter ‘t’, playing the part of ‘time’: {f_t}_tÎR. The time derivative of a differential form, w say, is then replaced by the Lie derivative accounting for both the variation of the physical quantity represented by the differential form, and for the evolution of the domain itself. First, the evolution of the domain generates a velocity field:

(1)

 where  is the position vector in the initial domain D. Then, the Lie derivative of the differential form can be written as the time variation of the differential form when dragged by this vector field:

(2)

A star denotes here the so-called pullback, i.e. the result of replacing the new coordinates of the domain in the form ω. The actual computation of the Lie derivative can be performed by the “golden formula”

(3)

The symbol ‘Ù’ means here the exterior multiplication or exterior differentiation as the case may be, while the symbol  means the internal product between vector and any differential form (see Arnold 1976). The transport theorem describes the rate of variation of the quantity represented by ω – not by the quantity it defines, i.e. the quantity represented by its coefficients – in the domain D, while this very domain varies. Betounes gives the transport theorem in the form

(4)

where ∂f_t(D) denotes the border of f_t(D) and the Stokes theorem was used. Let’s illustrate this theorem for the case of tire.    We consider first the case of internal pressure of the tire cavity. The pressure is physically represented by the force exerted by the air from the cavity upon the internal wall of the tire cavity, which is usually given by a double integral written as

(5)

Here we neglected the global effect of pressure which, if considered, would ask for a closed surface of integration. But we shouldn’t be interested in this when it comes to the dynamical considerations on the rolling tire. For instance, a part of the surface bordering the tire cavity belongs to the rim, it is metallic. Consequently we expect no variations of this portion of surface that might qualify as deformation, by comparison with the internal surface of the tire carcass. Therefore, the integral from equation (5) is simply done over the internal surface of the tire per se, which is delimited by the circles of the bead reinforcements. This is an open surface.

     Perhaps the most important consequence of this line of thought is the fact that the pressure should be treated as a skew symmetric second order tensor, because the force given by pressure at the wall is a actually a differential 2-form. Indeed, the element of oriented surface of the interior wall of the tire is a skew symmetric tensor which, in view of the three-dimensionality of space, is equivalent to a vector that can be written in the form of a column matrix

(6)

where ε_ijk is the Levi-Civita totally antisymmetric symbol. Consequently the force exerted by pressure at the wall should rather be represented as a differential 2-form:

(7)

The matrix p is here a skew symmetric second order tensor. This makes out of ‘p’ from equation (5) the projection of the vector equivalent to p along the direction of the normal to surface, which is the natural way to consider the pressure. This is, for instance, the case in the kinetic theory of ideal gases. The bottom line is that the variation of force at the internal wall of the tire is given, according to the transport theorem, by the equation

(8)

Here we applied the formula (4) whereby we considered that the exterior differential of a 2-form oves a surface is zero. Consequently we consider this effect of pressure intrinsic to the surface. The vector ‘pressure’ in equation (8) is defined by equation

(9)

as the vector whose projection along the normal to surface is the internal pressure of tire. In the right hand side of equation (8) the cyclic line integral is performed over the circles of the bead reinforcements, and therefore the velocity field  is the internal tire surface velocity field at the bead reinforcements. 

    On the other hand, the pressure should be conceived here thermodynamically, through the elementary thermodynamical work (pdV), as the tire is heating and its working and life are strongly dependent on the heat generated in rolling. In this case the pressure should be taken as a differential 3-form of energy, i.e. we should have

(10)

Here p_ijk is a totally antisymmetric third order tensor representing the pressure, while ε_ijk is the Levi Civita’s totally antisymmetric symbol. The transport theorem has now the form

(11)

where D is the space domain of the tire cavity, and we used the property of the differential 3-forms of yielding zero when exterior differentiated in space. Here  is the velocity at the limiting surface of the tire cavity, i.e. the internal surface of the tire carcass plus the surface of the rim. Under the cyclic integral from the right hand side of equation (11) we have the differential form

(12)

which, again, can be treated according to the transport theorem. Mention should be made that in the right hand side of equation (11) we have a skew-symmetric second order tensor representing a power:

(13)

This makes sense physically: the rate of variation of energy in a volume is the power dissipated or absorbed in that volume. However, as the tire rotates, even the integral of the 2-form from the right hand side of equation (11) varies, at the very least due to the vibration of the internal surface of the tire cavity, and this variation is described by the same transport theorem. Thus, it turns out that actually we must have even a rate of variation of the dissipated power, due to the fact that the border of the tire cavity varies during rolling:

(14)

Let’s concentrate on the integral from the right hand side here. First, we have reasons to believe that the metallic rim contribution to that integral has no variation. For instance the rim does not vibrate due to the road surface as much as the tire itself vibrates. Thus we are left with the time variation of an integral over the internal surface of the tire carcass which, by the Betounes’ transport theorem, can be expressed by a sum involving the variation of energy due to the variation of the internal surface of the tire and a cyclic line integral over the bead reinforcements:

(15)

Here  is the velocity of tire at the bead level. Thus the thermodynamics of the internal cavity of the tire is decided by an equation of the form

(16)

If the velocity vector of the surface of the tire at the level of bead reinforcements coincides with the velocity of the rim itself, which is a first thought about the two vectors, and can be true in certain situations, then the line integral from the right hand side of equation (16) is zero and the thermodynamical contribution of the air inside tire cavity is limited to the dissipation through the internal surface of the tire. However, in general, we can expect a certain jump of the velocity field of the tire internal surface close to the rim reinforcement, so that the two velocity fields can very well be different, and the line integral from equation (16) should be taken into consideration.

Conclusions

Let’s read the conclusions of the present part of the work, starting from the last equation and going towards the beginning of the developments. The air pressure inside the tire cavity – simply known to technicians as the tire pressure – is usually assumed to be constant. This may not be the case in view of the heat production of the rolling tire: the air evolves in a closed cavity, and the least we can assume is that it is an ideal gas. It is therefore to be expected that the elementary work involved in the thermodynamics of the air inside the tire cavity is a dynamic physical quantity, in the sense that it varies in time, due to the rolling conditions.

     The main result here is that, accounting for the paths of dissipation of the energy of the air inside the tire cavity, the basic quantity measuring the variation of the elementary thermodynamical work is the second time derivative of that quantity – an ‘acceleration of energy’ so to speak. Usually, in case they ever think of it, the engineers concentrate here on the first derivative, according to the classical Reynolds’ transport theorem. That theorem reflects a ‘genuine statical’ situation, whereby the details of the dissipation of energy do not include the evolution of the very channels of dissipation due to rolling. One can see that the Betounes’ formulation of the transport theorem has the advantage of forcing us to explicitly consider the time evolution of the borders of regions in which the variation of energy takes place. Thus, the ‘acceleration’ of the energetic content of the air inside cavity is dictated, even under ideally constant pressure, by the space variation of the internal surface velocity field, in the form of its divergence, and by the behavior of that surface at the bead reinforcements.

     In order to understand the interrelation between these phenomena in the rolling tire, let’s assume that we are in a situation where we can neglect the rim effects, which are transmitted through the bead reinforcements. Then the second rate of dissipation of the cavity content of energy is calculated only through the internal surface of the tire. In rolling conditions, that surface is in vibration, and this vibration can be modeled by the field of velocities of each and every one of its points. Due to a certain degree of randomness of the local vibrations, one can guarantee that the divergence of the velocity field is always nonnull on the internal surface of the tire. Therefore the energy in and out of the tire cavity is guaranteed to be at least a quadratic function of time, depending on the vibrational properties of the tire itself. In other words, the tire vibrations strongly influence the heat transfer in and out of the tire cavity.

     One way to correlate the variations of pressure with the local deformations of the internal tire surface, is through the force developed by pressure upon a small portion of surface. In this respect, the pressure is to be considered as a skew-symmetric tensor, described strictly in connection with the intimate geometry of the surface. As we will show here, this tensor contributes to the local deformation of the surface. This representation of pressure is different from the ‘thermodynamical’ one, but it is surely connected to it, as we will also show here.

References

Arnold, V. (1976): Les Méthodes Mathématiques de la Mécanique Classique, Editions MIR, Moscou

Betounes D. E. (1983): The Kinematical Aspect of the Fundamental Theorem of Calculus, American Journal of Physics, Vol. 51,  pp. 554–560

Reynolds, O. (1903): The Sub-Mechanics of the Universe, Cambridge University Press, UK

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(1)

and assume that it is defined in any point of the surface. This ‘definition’ means that we know the coefficients X, Y, Z as functions of the coordinates on the surface, and also the differentials s¹ and s² of the position vector of the points on surface, which play the part of coordinates in the tangent plane. These differentials are the main tool in problems of the local theory of surfaces. They allow for instance the definition of the curvature matrix (Guggenheimer, 1977): this is the transition matrix between the vector (s¹, s²) from the tangent plane and the components of the variation of the normal vector to surface.
     When there is a variation of this quadratic form, it is described by both the variation of the coefficients: X, Y, Z → X+dX, Y+dY, Z+dZ, and by the second variation of the position vector itself. The total variation of the quadratic form can then be obtained according to the known rules, by differentiating it to obtain

(2)

One can then decide if the variation of the quadratic form is strictly due to the variation of its coefficients, by a simple test: the second part of the equation (2) should be zero. Thus the variation of the quadratic form is strictly due to the variation of the coefficients along the curves from the tangent plane in a point, given by the differential equation

(3)

Along these curves the equation (2) becomes:

(4)

One can say that along these curves the variation of the quadratic form is ‘perceived’ only through the variation of its coefficients. Therefore, along these curves, the quadratic form is constant only if its coefficients are constant. The equation (3) represents a Hamiltonian motion in the tangent plane of a point of surface. If the quadratic form is defined, and its coefficients are constant, the equation (3) can represent two harmonic oscillators on the surface. Indeed, assuming that we discovered a time parameter with respect to which we can characterize the motion continuously, the differential equation (3) can be written in the form of a Hamiltonian system in the ‘phase plane’ of coordinates x ≡ s¹ and y ≡ s²:

(5)

Now, either by direct exponentiation or by finding the second order differential equation for each component, one can see that indeed, we have to deal with two harmonic oscillators having the same frequency. We choose the exponentiation, because it makes more obvious the fact that the quadratic form (1) is both the generator of motion and is conserved along motion. Indeed, denoting ω² ≡ XZ – Y², the solution by exponentiation is

(6)

where E is 2×2 identity matrix, and the index ʻ0’ marks the ‘initial conditions’ at t = 0. One can directly verify the equality

(7)

which is the clear expresion of the conservation of the generator along the motion it generates.
     If the coefficients are not constants, but vary with the point of surface, in order to integrate equation (3) they must satisfy some integrability conditions, amounting to the fact that they can be expressed as functions of the local coordinates in the tangent plane, or that these coordinates can be expressed with respect to them or their variation. In these integrability conditions it is implicitly understood that the differentials ds¹ and ds², i.e. the second order differentials with respect to position on surface, are taken as fundamental, while the first order differentials s¹ and s², which are fundamental in the regular theory of surfaces, should be expressed with respect the differentials of the second order. Practically the integrability condition amounts to the vanishing of the exterior differential of the left hand side of equation (3):

(8)

Then, by one of the Cartan lemmas one has

(9)

where λ, μ, ν are three external parameters, the entries of a matrix ‘assisting’ in integrability. Therefore, assuming that the matrix here is nonsingular, s¹ and s² must satisfy the system of differential equations

(10)

Performing the matrix multiplication, we get

(11)

The elementary ‘second order’ area – the so-called symplectic form – is given by equation

(12)

This quadratic form is algebraically conjugated to the variation due to coefficients of the original quadratic form (1)

(13)

and to the quadratic form introduced by the matrix ‘assisting’ in integrability:

(14)

It is this quadratic form that accounts for the physical conditions determining the integrability.
     Now, in order to solve (11) the matrix of evolution is essential. It can be written as

(15)

where by ω_1,2,3 we denoted the differential forms

(16)

In cases where the quadratic forms (13) and (14) are algebraically apolar, the matrix of evolution (15) reduces to

(17)

The parameters λ, μ, ν may represent new conditions of a geometrical or physical nature, affecting the deformation process of the surface. These conditions may be external (for instance the road surface) or internal to the surface (air pressure variation), reflecting its geometry (the first and second fundamental forms) or physics (for instance the deformation matrix, or the material condition of the tire).

Conclusions

Two main points are worth fixing in mind, as a conclusion of the previous exercise.
     First, is the fact that the frequency of vibration of the tire is a surface phenomenon, and needs to be treated as such. The usual wisdom puts first a simple model (Kelvin-Voigt or Maxwell) describing it together with the material underneath, in a time that has nothing to do with the real phenomenon, and then trying to fit experimental data to results. These are then improved by complicating the things with series of simple models. This philosophy is mainly entertained, in the prediction ‘industry’, by the preexisting commercial calculational instruments. It has, however an essential shortcumming: the frequency model is one-dimensional, and there is no way to take into consideration the complicate structure of the spectrum which is mainly a statistical process in a real time. In other words, the phase space associated to the classical constitutive models is actually four-dimensional, not two-dimensional, as the usual wisdom holds true. 
     Secondly, the variations of a quadratic form, be it the first or the second fundamental form, needs an external quadratic form, ‘assisting’ in integrability, in order to clarify physically the mechanics of deformation of the surface. The integrability is not just a mathematical process, but mainly a physical one. We think that, in order to understand this point, an example will do best. Consider the tread of the rolling tire on the rough surface of the road. First we have a stationary deformation due to the load of vehicle. This is a first-order process of deformation, according to which we need to ‘update’ the first and second fundamental forms of the tire surfaces snd thus describe the tire footprint shape. The road surface can be modeled by such a quadratic form which, naturally, is external to tire tread surface. Then the coefficients of this quadratic form go, in the manner shown above, into the variation of the first and second fundamental forms of the tire tread from the footprint region, thus imprinting a statistical variation of the frequency of vibration of the tire. This statistical variation is characteristic to the road surface.

References

Guggenheimer, H. W. (1977): Differential Geometry, Dover Publications, New York
Struik, D. J. (1988): Lectures on Classical Differential Geometry, Dover Publications, New York

     Let’s say that the first fundamental form of the surface is given by

(1)

 while the second fundamental form, which is the second variation in position along the normal to surface, is given by

(2)

where (,) means the usual dot product. Let’s take first the case where the second fundamental form varies: the pure bending of surface. This means that the normal to surface varies, perhaps due to some external perturbations, like local roughness of the road or some wave on the tire determining the local vibration in the case of tire. Applying the general considerations, the second fundamental form variation due exclusively to the variation of the curvature parameters can only be noticed along the curves from the tangent plane given by the differential equation

(3)

If the coefficients α, β, γ are constants, the equation can be integrated giving the quadratic form from the right hand side of equation (2). This means that along this conic, the surface shows the same distance from the tangent plane. The conic is known as the Dupin indicatrix of the surface in a point - a classical geometrical result. Assume now that the coefficients α, β, γ vary, but that the quadratic form ‘assisting’ in the integrability is given by the first fundamental form of the surface so that

(4)

This simply means that the bending of the surface is only compatible with certain states of deformation represented by the first fundamental form of the surface. Therefore the condition of integrability of the equation (3) becomes

(5)

The matrix of evolution in (5) is thus given by

(6)

with the differential forms given by

(7)

Now, the trace of matrix (6) gives the known geometrical result regarding the mean curvature of the surface:

(8)

On the other hand, the determinant of the matrix (6) gives us the absolute or Gaussian curvature:

(9)

These formulas contain, again, the known classical results in case where the metric of the surface is not euclidean, provided we consider the euclidean case as a reference and identify the metric tensor and the curvature matrix with their variation.

Conclusion

It is to be noticed that considering the coefficients of the fundamental quadratic forms as variations rather than finite quantities, is as general as it can be, comprising all the classical results in matter of geometry of surfaces. Indeed, if we consider the reference plane in a point of a surface, other than the tangent plane, for instance the mean plane of a rough road surface in the case of a rolling tire, then the variations of the curvature parameters are indeed these very parameters, and the formulas (8) and (9) are the usual ones from the classical theory of surfaces. Consequently, for physically realistic situations, we are entitled to take any quantity referring to the surface as being a variation rather than an absolute quantity. More than this, the theory provides the opportunity of describing the ‘wrinkling’ of a surface as a local deformation assisted by bending (remember that, in this jargon, the case above is bending assisted by deformation, inasmuch as the assisting matrix is that of the first fundamental form of the surface, which usually contains the deformation). However, the pure ‘geometrical’ description of the deformation of surfaces is obviously not sufficient from a physical point of view. This leads us, as will be shown elswhere on this page, to an interesting way of describing the friction forces between two surfaces in relation to their roughness and the physical properties of the materials they are delimiting. Such is plainly the case of the tire-road interaction, so important in braking and skidding properties of the tire.

We will limit here our considerations to the space of three dimensions. The vectors are conceived as entities written either by components, or in the Dirac form as matrices. For instance the position vector can be written in one of the two forms:

The first of these is the regular geometrical writing, in terms of the base unit vectors  . Here, and everywhere else, we adopt the summation convention over repeated indices, and x^k are the (contravariant) components of our position vector. The second writing – the matrix notation – disregards the reference frame of the three base unit vectors. It is therefore worth considering for calculations in the same reference frame, or in situations when the reference frame is fixed once and for all. This is, for instance the case of the Euclidean space, where the reference unit vectors do not vary. In general, however, the reference frame is local: it can vary from a point to another, and may not be always orthogonal. Using for the reference frame a matrix notation of the kind above, one can thus write in general

Here the matrix g is the metric tensor – a matrix having 1 as diagonal elements. In case this matrix is the identity matrix, we have to do with the Euclidean case in Cartesian coordinates.

      The grounds of Cartan’s considerations is the observation that an elementary displacement means both a variation in position per se, and a variation in the reference frame:

According to general geometrical rules, the variations of the unit vectors can be expressed with respect to the unit vectors themselves by linear relations:

Obviously, the matrix ω has zeros along the main diagonal, but it is by no means symmetrical. In view of the definition of the metric tensor, we have

So, only if the reference frame is orthonormal, the matrix ω is skew symmetric, otherwise it has no symmetry. It is thus sometimes very convenient to discuss the geometry in an orthonormal reference frame.

      Now, in view of (4), the equations (3) can be written as

Obviously, both the components of the vector  and those of the vectors  are exact differentials. In the language of the differential forms this fact can be stated by the equations:

It is the simple consequences of these two equations that are the ground of the whole Cartan geometrical construction. Just following the rules of working with the exterior differentiation and exterior multiplication, one can find from (7) the following equations relating the components of the vector  to the matrix ω:

Here we maintain the rule of summation on dummy indices, only the monomials are not defined by the usual multiplication but by exterior multiplication. For obvious reasons the first equation is called the compatibility equation: it gives the compatibility conditions between the variation of the reference frame and the displacements in space.

      It can be easily proved, just following the same rules of exterior multiplication and exterior differentiation, that the first equation (8) is a direct consequence of the second one and the definition of s^k from equation (6). However, the equations (8) are very general: they are valid regardless of definition (6). So, the two equations from (8) may not be always equivalent in an obvious way. This means that there are situations, mostly corresponding to physical reasons, when one has to define the variations of coordinates directly in terms of the reference frame, in which case s^k don’t have that neat structure given by equation (6). In such cases we don’t know precisely how much from the displacement vector is pure displacement and how much is contribution of the variation of reference frame. All we can assume is that the components of displacement vector are differential 1-forms, and the equations (8) still stand.

      The physical reasons we were mentioning above have always geometrical expressions. In order to catch the idea, let’s take for instance an arbitrary motion. It is done along a certain path, whose image is a curve in space. This curve can be represented, at least in limited local extensions, as intersection of two surfaces. Even more precisely, if we consider the local situation in a certain point, the curve is a straight line represented by the intersection of the tangent planes of those surfaces in that point. And the tangent planes in a point can be represented always by 1-forms. Thus we can come up with the idea that, in spite of the fact that the space is three-dimensional, in order to describe the motion locally we only need two 1-forms, not three. True, these 1-forms are linear combinations of the three basic components of the elementary differential of position vector, but this is an entirely different story. The main point of the exercise is that the local description of the motion only needs two differential forms, and those are linearly independent. In a way this means that they are sufficient for the purpose of description of motion.

      Thus the immediate point is to define the linear independence of two differential 1-forms. As the above example suggests in terms of the tangent planes in a point, these have to be necessarily different, otherwise they cannot properly determine a line. This can be expressed simply by the statement: two 1-forms ω and f are linearly independent if, and only if, their exterior product is non-null. The same way three 1-forms, ω, f, y are linearly independent if their exterior product is non-null. This simply expresses the fact that the volume of the parallelepiped defined by the three 1-forms as edges is nonzero: the parallelepiped is nontrivial. Such three 1-forms define a basis of 1-forms in space, which can be characterized as usual: any linear combination of them, with numerical coefficients, uniquely represents a 1-form.

      Here an interesting fact occurs. We can say that a 1-form is null if all the coefficients of its linear expansion in a base of 1-form are zero. This fact can however be ascertained otherwise. Namely, if {y^k; k = 1, 2, 3} is such a base, then the 1-form W is zero if, and only if, the following relations are satisfied:

In a way, this relation is equivalent to the known fact that in a vectorial space a vector is characterized by its projections on the base vectors. If all these projections are zero, then the vector is zero. Only, here the dot multiplication of vector is replaced by the exterior multiplication of the differential forms.

      The exterior product of two 1-forms is a 2-form. The 2-forms in space are usually associated with fluxes. Well known examples are: the flux of magnetic lines through a surface, whose magnitude is the magnetic induction, the flux of electric lines whose magnitude is the electric induction, the flux of particles, whose magnitude is known as current, an so on. The magnitude is usually a skew symmetric tensor, because the base of two forms in space is given by the three 2-forms representing the projections of a surface element on the three coordinate planes. Thus a 2-form, ‘b’ say, can be written in this basis as

where the summation convention is respected. The magnitude here is given by the skew symmetric tensor b, formed with the coefficients of the 2-form.

      By the same reasoning, a differential 3-form represents densities. These have magnitudes represented by the coefficients of the 3-form in terms of the elementary oriented volumes:

where the summation is done over the six permutations of the indices.

      Examples of differential forms of all degrees will be provided here all along the development of the physical theory of tire. But in order to properly develop that theory we need a few results of Cartan, helping in draw particular conclusions from the equations (8). First there is the theorem that Finikov terms as Cartan’s lemma (Finikov, 1948). We give it in a little modernized formulation of Guggenheimer (Guggenheimer, 1977): if ω^r, r = 1, 2, 3 are r linearly independent 1-forms, and there are r 1-forms π^r such that

then there is a r´r symmetric matrix c such that:

The proof is as follows: if r = 3, then ω^r can be taken as a basis of the 1-forms and thus ω^r can be expressed linearly with respect to them. Therefore equation (13) is valid, without any qualification on the matrix c. However, the equation (12) shows that this matrix is symmetrical. Indeed, that condition becomes

whence the symmetry of the matrix, in view of the skew symmetry of the exterior product. If r < 3, then the system (ω) can be completed to a basis, and the proof follows the same lines.

      Another important theorem which, in our opinion, is instrumental in physical applications, is the following: the 2-form

is zero as a consequence of vanishing of the r (£ 3) linearly independent 1-forms f¹, f²,…,f^r, if, and only if, it can be written in the form:

where f₁, f₂,…, f_r are r appropriately chosen 1-forms, and the summation convention is respected. The proof goes as before: in case r = 3 the system of 1-forms {f^k} can be taken as a basis, in which case F can be written as

with b the skew symmetric matrix derived from a by changing the basis of 1-forms. For r < 3, the system {f^a} can be completed up to a basis, and the proof goes exactly like in the case r = 3.

References

Arnold, V. (1976): Les Méthodes Mathématiques de la Mécanique Classique, Editions MIR, Moscou

Cartan, E. (1983): Geometry of Riemannian Spaces, Mathematical Science Press, Brookline, Massachusetts, USA

Cartan, E. (2001): Riemannian Geometry in an Orthogonal Frame, World Scientific Publishing, Singapore

Finikov, S. P. (1948): Cartan’s Method of Exterior Forms in Differential Geometry, OGIZ, Moscow (Метод Внешних Форм Картана в Дифференциальной Геометрии, ОГИЗ, Москва)

However, perhaps one of the most important consequences might have been that of our image of the microscopic world. Such is, for instance, the classical explanation of scattering of alpha particles, which led the scientific community to the planetary image of the atom. Within the framework of Newtonian theory of central forces it does not need a nucleus in order to be explained, but only the possibility of migration of the electric charges in the target metal. Possibly not even that.

Come to think of it: Dewey Larson was right after all! The alpha scattering experiments are indication, not of the nucleus, but of an atomic structure in its entirety. He saw this in the contradictory data on the crystalline lattices and the subsequent dimensions of different atoms deduced from these data. However, this fact can well be a fundamental theoretical one, if we take the central forces as what they are supposed to be, with no strings attached.

Therefore both these, by now classical, images of light are in contradiction with the Mach’s principle. According to this the classical free motion is an expression of the overall interaction of a material body with the whole matter in the Universe, rather than being an expression of no interaction, as the first principle of classical dynamics seems to claim. Indeed there is no way to isolate a body in order to check how it moves in those conditions: the first principle of classical dynamics is not a falsifiable proposition. This contradiction will remain as such as long as we cannot judge the uniform motion in terms of forces, exactly as we judge, say the Kepler motion. If the light, for instance, can be judged this way, we have a model, if not of free motion, of a motion with constant speed anyway.

It is a pleasant surprise to see that Newton’s theory of the forces “invented” by him fits this bill, pointing towards an image of a light ray in full accordance with the electromagnetic theory of light as in the attached figure. At any moment the electric vector is acted on by central forces directed towards the source of light – they can be repelling as well as attractive forces – while rotated by the magnetic vector. This image of a light ray and, implicitly, of the constant speed motion is based upon dismissal of two common misconceptions. The first one of these is that the central forces have magnitude depending exclusively on the distance, and the second one is that the magnetic vector is a… vector, while in this instance is a skew-symmetric tensor.