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Twelfth Pan American Congress of Applied Mechanics,
Port of Spain, Trinidad, September 2-6, 2011
Non-equilibrium thermodynamic approach to Coulomb’s friction
Michael Nosonovsky+,
Department of Mechanical Engineering
College of Engineering & Applied Science
University of Wisconsin-Milwaukee, Milwaukee, WI 53211, U.S.A.
Abstract
The Coulomb-Amontons friction law is usually treated as an approximate empirical rule,
rather than a fundamental law of nature. However, friction is quite universal and same laws
apply at different scales (from nN to GN of force), for diverse classes of materials, and different
mechanisms of dissipation (e.g., adhesion, deformation, and fracture). Friction reflects the
tendency of energy to irreversibly dissipate, so it is related to the 2nd law of thermodynamics.
The phenomenological Onsager relationships between the generalized thermodynamic
forces and flows in a non-equilibrium system suggest that the friction force is proportional to the
sliding velocity. This is the case for bulk viscous materials and fluids. The asymptotic transition
from a 3D bulk viscous material to a 2D interface between two such materials results in an
indefinite (“zero over zero”) value of sliding velocity and in shear force proportional to the
normal load, i.e. in the Coulomb friction. This suggests that the Coulomb-Amontons law has a
fundamental relationship with the Onsager relationships. The thermodynamic methods have a
significant potential for both understanding the fundamentals of friction, as well as for the
development of new smart materials.
1. Introduction
Mechanics, as we know it, has its roots in Ancient Greece. Aristotle (384-322 BC) in his
“Physics” introduced the concept of force as a cause of motion. According to his teaching,
bodies tend to move with the velocity V, which is proportional to the applied force
F=V
+
(1)
Corresponding author. E-mail: [email protected], Tel.: +1-414-229-4987, Fax: +1-414-229-6958
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where  is a coefficient of proportionality (the viscosity). Aristotle’s understanding of force
lacked the concepts of inertia and friction. The concept of inertia was suggested by several
medieval scholars (Philoponus, Jean Buridan, Nicole Oresme, etc.), and finally Galileo Galilei
(1564-1642) postulated his Principle of Inertia. Galileo also introduced the idea of the friction
force, which in many practical circumstances prevents bodies from their inertial motion with a
constant velocity [1].
The concept of the friction force and its proportionality to the applied normal load was
suggested by Leonardo da Vinci (1452–1519), who believed that the friction force is equal to the
quarter of the normal load. In a clearer way it was proposed by Guillaume Amontons (1663–
1705) and later by Charles-Augustin de Coulomb (1736-1806) who formulated the laws of
friction stating that the friction force is proportional to the normal load and does not depend on
the area of contact, and it is almost independent of the sliding velocity [1].
The introduction of irreversibility (the 2nd Law of thermodynamics) into the classical
mechanics is of importance for the study of friction, since the latter is an irreversible dissipative
process. Lord Rayleigh (1842-1919) introduced the concept of the dissipative function as the
quadratic form of velocities; however, it was defined only for the linear viscous friction. A. Lurie
(1901-1980) generalized the dissipative function for an arbitrary dependence of friction forces on
the velocity, including the Amontons-Coulomb friction law [2]; however, such an introduction
did not occur through any fundamental law of mechanics.
Friction force is a fundamental mechanical force which plays a role comparable with that
of the inertia forces. However, the study of friction did not go much further beyond the
observation by Leonardo, Amontons and Coulomb that the friction force is linearly proportional
to the normal load and almost independent of the sliding velocity and the apparent contact area.
In the literature, friction is usually not considered a general and universal phenomenon. Instead,
the theories of friction are viewed as empirical and ad hoc models, despite the prominent role
that friction plays in the mechanical processes. Although friction is often introduced into
differential equations of mechanics, it is done in a somewhat arbitrary ad hoc manner.
Friction and wear reflect the tendency of matter for irreversible deterioration and energy
for dissipation, so it is natural to relate the origin of friction and wear to thermodynamics and in
particular, to the thermodynamics of irreversible processes. Given that there are four
fundamental laws of mechanics: (1) the law of momentum (2) the law of angular momentum (3)
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the first law of thermodynamics and (4) the second law of thermodynamics, it is necessary to
seek the justification of the friction law from these laws and in particular, the 2nd law of
thermodymamics [1-2].
2. The Coulomb-Amontons law and the thermodynamic equations of motion
The Coulomb-Amontons law states the linear proportionality of the dry friction force F
with respect to the normal load force W is the best known empirical observation about friction
F = W
(2)
where  is the coefficient of friction. Despite the fact that the Coulomb law is valid only as an
approximation, it has a striking universality. The Coulomb law is valid for a very diverse range
of material combinations and for normal loads ranging from nano-newtons to thousands of tons.
Furthermore, friction is a complicated phenomenon that involves various apparently unrelated
physical mechanisms, such as the van der Waals and covalent adhesion, elastic and plastic
deformation, brittle fracture, the so-called ratchet, cobblestone and “third-body”. However, all
these diverse mechanisms of friction are described by the simple Coulomb law [3].
Most linear empirical laws of physics, such as the Ohm’s law of electrical resistance, the
Fourier law of heat conduction, or Fick’s law of diffusion, that relate thermodynamic flows Jk
with thermodynamic forces Yk are consequences of the assumptions of the linear
thermodynamics that lead to the thermodynamic equations of motion
J k   LkiYi
(3)
i
where Lki are Onsager coefficients [4]. However, the Coulomb law cannot be deduced from the
linear non-equilibrium thermodynamics. Indeed, in the case of dry or lubricated friction, the
sliding velocity is the thermodynamic flow, V=J, which, in accordance with Eq. 3 should be
proportional to the friction force F=Y (as it is the case for the viscous friction), so that the energy
dissipation rate is given by the product of the thermodynamic flow and force
  JY  VF
(4)
However, the Coulomb friction force is independent of sliding velocity.
Consider now sliding friction of a material point upon a surface of a deformable body.
We will take into account the normal degree of freedom of sliding surfaces, since it is known to
play a significant role in dynamic effects. The normal load and friction forces are given by N and
F respectively, whereas corresponding displacements are the normal separation  and the
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coordinate x. Introducing the normal degree of freedom is a standard procedure in the study of
dynamic friction, where normal vibrations are often coupled with in-plane vibrations [6]. We
further define the generalized flows as J1  x , J 2  y and forces as Y1=F, Y2=W [5].
The thermodynamic equations of motion Eq. 3 immediately yield the law of viscous
friction in the form of
 x   L11 L12  F 
   
 
 y   L21 L22 W 
(5)
Note that Eq. 5 describes viscosity and it is valid, in a general case, for the bulk of 3D
deformable medium, and not necessarily for the interface between two solids.
The interface between sliding bodies has highly anisotropic properties, because a small
force in the direction of the interface causes large displacements, whereas a small force in the
normal direction causes only small displacements. To compensate for this anisotropy, we
substitute coordinates using a small parameter  as x, y  x, y  . The force-displacement
relationships are now given by
 x   L11 / 
   
 y   L21
L12 /   F 
 
L22 W 
(6)
In the limit of →0, Eq. 6 yields
F 
L12
W
L11
(7)
x  ( L11F  L12W ) /   0 / 0

L L 
y   L22  12 21 W
L11 

According to Eq. 7, any velocity x satisfies Eq. 6, provided L11F+L12W=0, which is exactly the
case of Coulomb friction if =- L12/ L11.
3. Discussion
Thus we showed that the Coulomb-Amontons friction law (Eq. 2) can be deduced from
the thermodynamic equations of motion (Eq. 3) provided two assumptions are made: (I) motion
in the normal degree of freedom (y) is coupled with the tangential degree of freedom (x) and (ii)
the change of coordinates x, y  x, y  is introduced and the limiting case of →0 is investigated.
The normal degree of freedom was introduced into the analysis of dynamic friction since the
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pioneering works of Tolstoi [6] who discovered the existence of natural normal micro-vibrations
coupled with the tangential vibrations, which strongly affect the magnitude of the friction force
as well as the stability of sliding. The introduction of a small parameter  and an asymptotic
decomposition is the standard way of the transition from 3D to a 2D problem (for example, from
the bulk elastic 3D body to a thin elastic plate). The Coulomb friction is an interface (2D)
phenomenon and it is natural to obtain its properties using an asymptotic decomposition from a
3D case. The methods of non-equilibrium thermodynamics can also be used to investigate selforganization during friction, and, in particular, to design new materials with embedded selforganization capacity, such as self-healing, self-lubricating, and self-cleaning materials [5. 7-9].
These are either metamaterials or biomimetic materials.
4. Conclusions
We conclude that thermodynamic methods have a significant potential for both
understanding the fundamentals of friction, giving new answers to the questions which have been
at the core of mechanics since Aristotle, as well as for the development of new smart materials.
References
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of Sciences. 2006)
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Hierarchical Surfaces: Friction, Superhydrophobicity, and Biomimetics (Springer-Verlag,
Heidelberg, Germany. 2008).
4. S. R. De Groot, P. Mazur, Non-Equilibrium Thermodynamics (Interscience, NY,
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978-1-4614-0925-0).
6. D. M. Tolstoi, “Significance of the normal degree of freedom and natural normal
vibrations in contact friction,” Wear, Vol. 10 (1967) 199–213
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7. M. Nosonovsky, “Self-organization at the frictional interface for green tribology” Phil.
Trans Royal. Soc. A., Vol. 368 (2010) 4755-4774
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(2009) 1607-1627
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