# Thermodynamical Description of Running Discontinuities: Application to Friction and Wear

## Abstract

**:**

## 1. Introduction

## 2. General Features on Moving Surfaces and Moving Layers

**The local constitutive law**The state of each body is characterized by the displacement $\underline{u}$, from which the strain field ε is derived. The other parameters are the temperature θ and a set of internal parameters α. The behaviour of ${V}_{i}$ is defined by the free energy density ψ as a function of strain ε, the temperature θ and the set of internal parameters α. The mass density of each phase (undamaged and damaged) is the same ρ. The state equations of each phase are

**σ**is decomposed in the reversible part ${\mathit{\sigma}}^{r}$ and an irreversible part ${\mathit{\sigma}}^{ir}$. The irreversible stresses ${\mathit{\sigma}}^{ir}$ are essentially due to viscosity. In non linear mechanics, the internal state is generally associated with irreversibility. The evolution of internal state must satisfy the second law of thermodynamics. Such requirement is fulfilled by the existence of a potential of dissipation.

**Analysis of the dissipation and potential of dissipation**The fundamental inequality of thermodynamics implies that the internal production of entropy must be non negative. The equations of state do not provide all the constitutive equations; some complementary laws are necessary to describe the irreversibility. In the total dissipation ${D}_{T}$ the contribution of the conduction and those of internal forces must be distinguished.

#### 2.1. The conservation laws

**Mass conservation**

**Momentum conservation**

**Energy balance**

**Continuity of displacement**

**Continuity of temperature**

#### 2.2. On moving interface

**Entropy production**Using the definition of the entropy of the system $S={\int}_{{V}_{i}}\rho s\phantom{\rule{0.277778em}{0ex}}\mathrm{d}\Omega $ the second law of thermodynamic is written as

**Interface propagation law**To control the matter loss, a criterion based on ${G}_{i}$ can be formulated. For example, we can considered a Griffith’s type law for the propagation of the moving interface

**Convected differentiation**To study the evolution of the mechanical state around the moving surface ${\Gamma}_{i3}$ a convected derivative of any mechanical quantity f is needed. A point ${\underline{X}}_{\Gamma}$ is on ${\Gamma}_{i3}$ if its coordinates satisfy the scalar equation:

**Consistency condition**The law of propagation shows that the condition ${\varphi}_{i}\ne 0$ implies ${G}_{i}={G}_{c}$, that relation is conserved during the propagation. The consistency condition associated with the Griffith’s propagation law is then written at point $s\in {\Gamma}_{i3}$ where ${G}_{i}={G}_{c}$ in the form

#### 2.3. On moving layer

**Analysis of dissipation**The dissipation has the classical expression

**Others expressions of dissipation**Introducing the local Eshelby momentum tensor

**P**

**Comments on local stationarity**If Y is the driving force associated to d, then the contribution of damage to dissipation is

## 3. The Macroscopic Interface Study

- At the microscopic scale, the contact between asperities govern the wear mechanisms. Some studies with plastic strains and with micro cracks propagation have been tempted [5,14]. These are the fundamental ideas of the decomposition of ${\Omega}_{3}$ in ${\Omega}_{13}\cup {\Omega}_{33}\cup {\Omega}_{23}$.
- At the mesoscopic scale, this is the description of the third body. This model was proposed in [1] developed by [15] and [16]. The local physics is that ${\Omega}_{i3}$ is a porous medium. In ${\Omega}_{33}$ the solid particles are in suspension forming a shear layer as inside a viscous fluid flow. This point is developed in [3]. In other situations, a very large plastic shear deformation is present, so the profile of local deformation in this thin layer depends strongly on the loading conditions.
- The macroscopic modelling is based on models of friction law which are depending on parameters to account of the evolution of the interface. These models can be inferred from the smaller scale by some averaging technics as those proposed under local stationarity hypothesis.

**Sharp transition**At the mesoscopic level, the dissipation is given by

**Comments**If matter loss occurs, ${m}_{1}$ or ${m}_{2}$ is positive, then the corresponding discontinuity ${\left[\underline{v}\right]}_{{}_{{\Gamma}_{i}}}$ exists too and the local quantity ${G}_{i}={m}_{i}{(\left[\psi \right]}_{{}_{{\Gamma}_{i}}}-\mathit{\sigma}:{\left[\epsilon \right]}_{{}_{{\Gamma}_{i}}}$) is positive. There is a dissipation due to the loss of material: that is a characterization of wear.

**Diffuse damage**Consider now that ${\Omega}_{i3}$ is a zone of diffuse damage. The description of damage can be made with a smooth transition governed by a damage parameter d in ${\Omega}_{i3}$, a set of internal parameters α, which contains the plastic strain ${\epsilon}_{p}$. For example a free energy of the form

## 4. A Macrolevel Approach

**The local problem**The layer is in equilibrium with external loading and contact conditions. The displacement is continuous on ${\Gamma}_{i}$ , the surface energy ${\psi}_{s}$ depends upon the given displacements along ${\Gamma}_{i}$. The other parameters are a set of internal parameters $\alpha ({\underline{X}}_{\Gamma},z)$, the thickness $H\left({\underline{X}}_{\Gamma}\right)$ and the temperature. The local strain ε derives from the displacement ${\underline{u}}_{3}$. The displacement is continuous along the interfaces ${\Gamma}_{i}$ then

**Analysis of dissipation**The global dissipation is derived from the macroscopic description. It is useful to introduce the global free energy $\mathcal{P}$ of the tribologic system

**Expansion of displacement with respect to z**This macroscopic point of view suggests to develop the internal state over ${\Omega}_{3}$ as an asymptotic expansion of the coordinate z

## 5. Examples and Applications

#### 5.1. Sliding contact in steady relative motion

**The constitutive law**The free energy of the mixture is given by

- At zero order, the Hertz’s solution is recovered
- At first order, a dependance with f is obtained. Wear occurs, and the profile of the pressure ${\mathit{\sigma}}_{yy}\left(x\right)$ evolves. The presence of viscous fluid induces a displacement of the maximum of pressure like under the dry contact with friction [25].

#### 5.2. Cyclic loading

## 6. Conclusions

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Stolz, C.
Thermodynamical Description of Running Discontinuities: Application to Friction and Wear. *Entropy* **2010**, *12*, 1418-1439.
https://doi.org/10.3390/e12061418

**AMA Style**

Stolz C.
Thermodynamical Description of Running Discontinuities: Application to Friction and Wear. *Entropy*. 2010; 12(6):1418-1439.
https://doi.org/10.3390/e12061418

**Chicago/Turabian Style**

Stolz, Claude.
2010. "Thermodynamical Description of Running Discontinuities: Application to Friction and Wear" *Entropy* 12, no. 6: 1418-1439.
https://doi.org/10.3390/e12061418