Research Papers: Friction and Wear

On Constitutive Relations for Friction From Thermodynamics and Dynamics

[+] Author and Article Information
Michael D. Bryant

Fellow ASME
Mechanical Engineering,
University of Texas at Austin,
Austin, TX 78712–0292
e-mail: bryantmd@austin.utexas.edu

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received June 23, 2015; final manuscript received July 19, 2015; published online July 14, 2016. Assoc. Editor: Sinan Muftu.

J. Tribol 138(4), 041603 (Jul 14, 2016) (8 pages) Paper No: TRIB-15-1223; doi: 10.1115/1.4032821 History: Received June 23, 2015; Revised July 19, 2015

Constitutive and dynamic relations for friction coefficient are presented. A first thrust combines the laws of thermodynamics to relate heat, energy, matter, entropy, and work of forces. The equation sums multiple terms—each with a differential of a variable multiplied by a coefficient—to zero. Thermodynamic considerations suggest that two variables, internal energy and entropy production, must depend on the others. Linear independence of differentials renders equations that yield thermodynamic quantities, properties, and forces as functions of internal energy and entropy production. When applied to a tribocontrol volume, constitutive laws for normal and friction forces, and coefficient of friction are derived and specialized for static and kinetic coefficients of friction. A second thrust formulates dynamics of sliding, with friction coefficient and slip velocity as state variables. Differential equations derived via Newton's laws for velocity and the degradation entropy generation (DEG) theorem for friction coefficient model changes to the sliding interface induced by friction dissipation. The solution suggests that the transition from static to kinetic coefficient of friction with respect to slip velocity for lubricant starved sliding is a property of the motion dynamics of sliding interacting with the dynamics of change of the surface morphology. Finally, sliding with stick-slip was simulated to compare this model to others.

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Grahic Jump Location
Fig. 1

Friction coefficient μ versus dimensionless slip velocity V, transitioning from static friction μs to kinetic friction μk. Dotted curve via Eq. (1) with f=e−|V|, solid curve via Eq. (30) with V=v/vμ and Fa/N=1.4, medium dashed curve from Eqs. (1) and (2) with V=v/vo, θ(0)=1 and τθ  = 0.01 s, and long dashed curve from Eq. (33) with V=v/vμ and adhesion breaking and healing terms. Velocities vμ and vo will appear in Eq. (29) and Sec. 3.2.2.

Grahic Jump Location
Fig. 2

Slider of mass m sliding against a counter surface, with tribocontrol volume shown as the shaded region. Applied force Fa will later result from the spring pulled by velocity va.

Grahic Jump Location
Fig. 3

Friction coefficient μ versus dimensionless slip velocity V based on Eq. (30), transitioning from static friction μs to kinetic friction μk for various values of Fa/N

Grahic Jump Location
Fig. 4

(a) Displacement x and (b) friction coefficient μ versus time t. From the list of Sec.3.2.2, the black curves are model 1, Eq. (33) with adhesion breaking and healing terms; the light gray curves are model 2, Eq. (1) with f=e−|V|, and the dark gray curves are model 3, Eqs. (1) and (2) with θ(0)=1 and τθ  = 0.01 s.




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